Introduction
Isotropic diffusion (thermal conduction, viscosity, Ohmic resistivity, and ambipolar diffusion) is present in many MHD code bases, including Athena++ [1 ] , which we will be using henceforth. The GPU version of Athena++, AthenaK, has support for parallel anisotropic thermal conduction [2 ] . Anisotropic thermal conduction and viscosity split into parallel and perpendicular components are also present in some codes such as the finite volume astrophysical codes PLUTO [3 ] and AREPO [4 , 5 , 6 ] ; and the multi-physics code Gizmo [7 ] .
The FLASH code [8 ] supports the Hall effect, anisotropic resistivity, the Righi–Leduc effect, and thermoelectric effects (including the Biermann battery). In its explicit extended-MHD timestep, the non-ideal magnetic terms are recast as advection by characteristic drift velocities. These are summed with the fluid velocity into an effective electron velocity, and the CFL condition uses the larger of the fluid and effective electron velocities plus the fast-magnetosonic speed. For stability, the thermoelectric drifts are each passed through a flux limiter that restricts them relative to a user-set fraction of the electron thermal speed (some of these terms are disabled at shocks). The Hall and battery terms enter through a wave speed: the fast-magnetosonic speed is replaced by the maximum of itself, the whistler phase speed, a Hall drift-wave speed, and a thermal–magnetic (battery) wave speed.2
The Hall effect on its own is well known to be unstable to first-order (and second-order) naive time integration [9 ] ; however, it can be stabilised by the presence of ambipolar diffusion [10 ] . It has also been shown that the Hall effect is conditionally stable at RK3 using spectral methods [11 ] . With finite-volume methods the Hall effect still suffers from non-linear instabilities using RK3, despite being shown to be linearly stable via von Neumann stability analysis [12 ] . Several modifications to the MHD solver have been proposed to stabilise the Hall effect at the grid scale including artificial hyper-resistivity [13 ] [14 ] and modifying the Riemann solver to use the whistler wave phase speed (Hall-HLL) [15 , 16 , 17 ] .
Explicit integration of dissipative terms is restricted by the timestep Δ t p a r ∝ Δ x 2 / D \Delta t_{\rm par} \propto\Delta x^2/D Δ t par ∝ Δ x 2 / D D D D s s s ∝ s 2 \propto s^2 ∝ s 2 s s s s s s [18 ] is first-order accurate and requires a tunable damping parameter, whereas the Runge–Kutta–Legendre (RKL) schemes [19 , 20 ] need no such parameter. RKL1 and RKL2 are first- and second-order accurate in time respectively. Athena++ implements both as an operator-split update of the dissipative terms: one super-step equal to the MHD timestep for RKL1, or two of half that length for RKL2 - one before, and one after the MHD update, giving a speed-up roughly proportional to s s s [1 ] . Because the RKL stability regions are built along the negative real axis, they contain no part of the imaginary axis: like RKL1 and RKL2, RKL1 and RKL2 amplify every purely imaginary mode. The non-dissipative terms, whose eigenvalues lie on the imaginary axis, therefore gain nothing from STS and must be integrated with the main MHD update.
In this work we present a new implementation of anisotropic dissipative and non-dissipative effects in Athena++ (“Athena-Aniso”). Effects acting on the momentum and energy have their fluxes calculated directly at cell faces and are integrated with either the main MHD solver or the STS solver. Effects acting on the magnetic field have their EMFs calculated directly at cell edges and passed to the upwind constrained transport (UCT) solver [21 , 22 ] (also with STS support). In Section 2 we present the non-ideal MHD equations that are solved. In Section 3 we perform von Neumann stability analysis on the non-ideal terms to derive their CFL conditions, and determine which time integrator is required for stability. The dissipative and non-dissipative parts are first separated to provide the CFL conditions required for super-time-stepping the dissipative parts; then the unsplit system is analysed for non-STS use. In Section 4 we derive the modified HLLE signal speeds. In Section 5 we present our implementation in Athena++. In Section 6 we test the new physics against analytical results. In Section 7 we conclude and summarise our results. We provide the expanded form of the viscosity terms in Appendix § 11 . Intermediate steps of the derivations of § 3 and § 4 are given in Appendices § 12 and § 13 , and the set-up of the noise tests of Section 6 in Appendix § 14 . We provide numerical validation of our derived CFL conditions with physical diffusion coefficients in Appendix § 15 .
Governing Equations
Neglecting all operator-split source terms (such as radiative heating/cooling, self-gravity, and orbital advection) and relativistic effects, Athena-Aniso solves the following equations of non-ideal MHD [1 , 23 , 24 , 25 ] [26 pp. 405–406]
∂ ρ ∂ t + ∇ ⃗ ⋅ [ ρ v ⃗ ] = 0 , (1) \frac{\partial\rho}{\partial t}+\vec{\nabla}\cdot[\rho\vec{v}]=0, \tag*{(1)} ∂ t ∂ ρ + ∇ ⋅ [ ρ v ] = 0 , (1) ∂ ρ v ⃗ ∂ t + ∇ ⃗ ⋅ [ ρ v ⃗ ⊗ v ⃗ − B ⃗ ⊗ B ⃗ + ( P + ∣ B ⃗ ∣ 2 2 ) I − Π a n i s o ] = 0 ⃗ , (2) \frac{\partial\rho\vec{v}}{\partial t}+\vec{\nabla}\cdot\left[\rho\vec{v}\otimes\vec{v}-\vec{B}\otimes\vec{B}+\left(P+\frac{|\vec{B}|^2}{2}\right)\mathbf{I}-\mathbf{\Pi}_{\mathrm{aniso}}\right]=\vec{0}, \tag*{(2)} ∂ t ∂ ρ v + ∇ ⋅ [ ρ v ⊗ v − B ⊗ B + ( P + 2 ∣ B ∣ 2 ) I − Π aniso ] = 0 , (2) ∂ ε ∂ t + ∇ ⃗ ⋅ [ ( ε + P + ∣ B ⃗ ∣ 2 2 ) v ⃗ − B ⃗ ( B ⃗ ⋅ v ⃗ ) + q ⃗ a n i s o + q ⃗ J − Π a n i s o ⋅ v ⃗ + ( η J ⃗ + η A D b ^ × ( J ⃗ × b ^ ) + η H J ⃗ × b ^ + F ⃗ T ) × B ⃗ ] = 0 , (3) \begin{aligned}
\frac{\partial\varepsilon}{\partial t}+\vec{\nabla}\cdot\Bigg[ \left(\varepsilon+P+\frac{|\vec{B}|^2}{2}\right)\vec{v}-\vec{B}(\vec{B}\cdot\vec{v})+\vec{q}_{\mathrm{aniso}}+\vec{q}_{J}-\mathbf{\Pi}_{\mathrm{aniso}}\cdot\vec{v} \\
+\left(\eta\vec{J}+\eta_{\mathrm{AD}}\hat{b}\times(\vec{J}\times\hat{b})+\eta_{\mathrm{H}}\vec{J}\times\hat{b}+\vec{F}_{T}\right)\times\vec{B}\Bigg]=0, \tag*{(3)}
\end{aligned} ∂ t ∂ ε + ∇ ⋅ [ ( ε + P + 2 ∣ B ∣ 2 ) v − B ( B ⋅ v ) + q aniso + q J − Π aniso ⋅ v + ( η J + η AD b ^ × ( J × b ^ ) + η H J × b ^ + F T ) × B ] = 0 , (3) ∂ B ⃗ ∂ t − ∇ ⃗ × [ v ⃗ × B ⃗ − η J ⃗ − η A D b ^ × ( J ⃗ × b ^ ) − η H J ⃗ × b ^ − F ⃗ T ] = 0 ⃗ . (4) \frac{\partial\vec{B}}{\partial t}-\vec{\nabla}\times\left[\vec{v}\times\vec{B}-\eta\vec{J}-\eta_{\mathrm{AD}}\hat{b}\times(\vec{J}\times\hat{b})-\eta_{\mathrm{H}}\vec{J}\times\hat{b}-\vec{F}_{T}\right]=\vec{0}. \tag*{(4)} ∂ t ∂ B − ∇ × [ v × B − η J − η AD b ^ × ( J × b ^ ) − η H J × b ^ − F T ] = 0 . (4) Here, ρ \rho ρ v ⃗ \vec{v} v P P P B ⃗ \vec{B} B b ^ = B ⃗ / ∣ B ⃗ ∣ \hat{b}=\vec{B}/|\vec{B}| b ^ = B /∣ B ∣ J ⃗ = ∇ ⃗ × B ⃗ \vec{J}=\vec{\nabla}\times\vec{B} J = ∇ × B T = T e = T i = P / ρ T=T_{\mathrm{e}}=T_{\mathrm{i}}=P/\rho T = T e = T i = P / ρ η \eta η η A D \eta_{\mathrm{AD}} η AD η H \eta_{\mathrm{H}} η H
The total energy per unit volume is comprised of the internal, kinemtic and magnetic energy
ε = ε i n t + ε k i n + ε m a g , ε i n t = P γ − 1 , ε k i n = 1 2 ρ ∣ v ⃗ ∣ 2 , ε m a g = 1 2 ∣ B ⃗ ∣ 2 . (5) \varepsilon=\varepsilon_{\mathrm{int}}+\varepsilon_{\mathrm{kin}}+\varepsilon_{\mathrm{mag}},\qquad\varepsilon_{\mathrm{int}}=\frac{P}{\gamma-1},\qquad\varepsilon_{\mathrm{kin}}=\frac{1}{2}\rho|\vec{v}|^2,\qquad\varepsilon_{\mathrm{mag}}=\frac{1}{2}|\vec{B}|^2. \tag*{(5)} ε = ε int + ε kin + ε mag , ε int = γ − 1 P , ε kin = 2 1 ρ ∣ v ∣ 2 , ε mag = 2 1 ∣ B ∣ 2 . (5) q ⃗ a n i s o \vec{q}_{\mathrm{aniso}} q aniso [23 ]
q ⃗ a n i s o = ( 1 − x i o n ) q ⃗ n e u + x i o n q ⃗ e , q ⃗ n e u = − κ n ρ ∇ ⃗ [ P / ρ ] , q ⃗ e = q ⃗ ∥ + q ⃗ ⊥ + q ⃗ ∧ , (6) \vec{q}_{\mathrm{aniso}}=(1-x_{\mathrm{ion}})\vec{q}_{\mathrm{neu}}+x_{\mathrm{ion}}\vec{q}_{\mathrm{e}},\qquad\vec{q}_{\mathrm{neu}}=-\kappa_{\mathrm{n}}\rho\vec{\nabla}[P/\rho],\qquad\vec{q}_{\mathrm{e}}=\vec{q}_{\parallel}+\vec{q}_{\perp}+\vec{q}_{\wedge}, \tag*{(6)} q aniso = ( 1 − x ion ) q neu + x ion q e , q neu = − κ n ρ ∇ [ P / ρ ] , q e = q ∥ + q ⊥ + q ∧ , (6) q ⃗ ∥ = − κ ∥ ρ ( b ^ ⋅ ∇ ⃗ [ P / ρ ] ) b ^ , q ⃗ ⊥ = − κ ⊥ ρ ( ∇ ⃗ [ P / ρ ] − ( b ^ ⋅ ∇ ⃗ [ P / ρ ] ) b ^ ) , q ⃗ ∧ = − κ ∧ ρ b ^ × ∇ ⃗ [ P / ρ ] . (7) \vec{q}_{\parallel}=-\kappa_{\parallel}\rho\left(\hat{b}\cdot\vec{\nabla}[P/\rho]\right)\hat{b},\qquad\vec{q}_{\perp}=-\kappa_{\perp}\rho\left(\vec{\nabla}[P/\rho]-\left(\hat{b}\cdot\vec{\nabla}[P/\rho]\right)\hat{b}\right),\qquad\vec{q}_{\wedge}=-\kappa_{\wedge}\rho\hat{b}\times\vec{\nabla}[P/\rho]. \tag*{(7)} q ∥ = − κ ∥ ρ ( b ^ ⋅ ∇ [ P / ρ ] ) b ^ , q ⊥ = − κ ⊥ ρ ( ∇ [ P / ρ ] − ( b ^ ⋅ ∇ [ P / ρ ] ) b ^ ) , q ∧ = − κ ∧ ρ b ^ × ∇ [ P / ρ ] . (7) Where x i o n x_{\mathrm{ion}} x ion κ ∗ \kappa_{*} κ ∗ [27 ] .
Π a n i s o \mathbf{\Pi}_{\mathrm{aniso}} Π aniso [23 ]
Π a n i s o = Π b u l k + ( 1 − x i o n ) Π n e u + x i o n Π i o n , Π b u l k = ζ ( ∇ ⃗ ⋅ v ⃗ ) I , Π n e u = ν ρ W , Π i o n = ∑ m = 0 4 Π ( m ) , (8) \mathbf{\Pi}_{\mathrm{aniso}}=\mathbf{\Pi}_{\mathrm{bulk}}+(1-x_{\mathrm{ion}})\mathbf{\Pi}_{\mathrm{neu}}+x_{\mathrm{ion}}\mathbf{\Pi}_{\mathrm{ion}},\qquad\mathbf{\Pi}_{\mathrm{bulk}}=\zeta(\vec{\nabla}\cdot\vec{v})\mathbf{I},\qquad\mathbf{\Pi}_{\mathrm{neu}}=\nu\rho\mathbf{W},\qquad\mathbf{\Pi}_{\mathrm{ion}}=\sum_{m=0}^{4}\mathbf{\Pi}^{(m)}, \tag*{(8)} Π aniso = Π bulk + ( 1 − x ion ) Π neu + x ion Π ion , Π bulk = ζ ( ∇ ⋅ v ) I , Π neu = ν ρ W , Π ion = m = 0 ∑ 4 Π ( m ) , (8) Π ( 0 ) = 3 2 ρ ν ∥ ( b ^ ⊗ b ^ − 1 3 I ) ( b ^ ⋅ W ⋅ b ^ ) , (9) \mathbf{\Pi}^{(0)}=\frac{3}{2}\rho\nu_{\parallel}\left(\hat{b}\otimes\hat{b}-\frac{1}{3}\mathbf{I}\right)(\hat{b}\cdot\mathbf{W}\cdot\hat{b}), \tag*{(9)} Π ( 0 ) = 2 3 ρ ν ∥ ( b ^ ⊗ b ^ − 3 1 I ) ( b ^ ⋅ W ⋅ b ^ ) , (9) Π ( 1 ) = ρ ν ⊥ 1 [ W ⊥ + 1 2 I ⊥ ( b ^ ⋅ W ⋅ b ^ ) ] (10) \boldsymbol{\Pi}^{(1)}=\rho\nu_{\perp1}\left[W_{\perp}+\frac{1}{2}I_{\perp}(\hat{b}\cdot W\cdot\hat{b})\right] \tag*{(10)} Π ( 1 ) = ρ ν ⊥ 1 [ W ⊥ + 2 1 I ⊥ ( b ^ ⋅ W ⋅ b ^ ) ] (10) Π ( 2 ) = 4 ρ ν ⊥ 2 [ I ⊥ ⋅ W ⋅ ( b ^ ⊗ b ^ ) + ( b ^ ⊗ b ^ ) ⋅ W ⋅ I ⊥ ] (11) \boldsymbol{\Pi}^{(2)}=4\rho\nu_{\perp2}\left[I_{\perp}\cdot W\cdot(\hat{b}\otimes\hat{b})+(\hat{b}\otimes\hat{b})\cdot W\cdot I_{\perp}\right] \tag*{(11)} Π ( 2 ) = 4 ρ ν ⊥ 2 [ I ⊥ ⋅ W ⋅ ( b ^ ⊗ b ^ ) + ( b ^ ⊗ b ^ ) ⋅ W ⋅ I ⊥ ] (11) Π ( 3 ) = − 1 2 ρ ν ∧ 1 [ b ^ × W ⋅ I ⊥ − I ⊥ ⋅ W × b ^ ] (12) \boldsymbol{\Pi}^{(3)}=-\frac{1}{2}\rho\nu_{\wedge1}\left[\hat{b}\times W\cdot I_{\perp}-I_{\perp}\cdot W\times\hat{b}\right] \tag*{(12)} Π ( 3 ) = − 2 1 ρ ν ∧ 1 [ b ^ × W ⋅ I ⊥ − I ⊥ ⋅ W × b ^ ] (12) Π ( 4 ) = − 2 ρ ν ∧ 2 [ b ^ × W ⋅ ( b ^ ⊗ b ^ ) − ( b ^ ⊗ b ^ ) ⋅ W × b ^ ] . (13) \boldsymbol{\Pi}^{(4)}=-2\rho\nu_{\wedge2}\left[\hat{b}\times W\cdot(\hat{b}\otimes\hat{b})-(\hat{b}\otimes\hat{b})\cdot W\times\hat{b}\right]. \tag*{(13)} Π ( 4 ) = − 2 ρ ν ∧ 2 [ b ^ × W ⋅ ( b ^ ⊗ b ^ ) − ( b ^ ⊗ b ^ ) ⋅ W × b ^ ] . (13) Where ζ \zeta ζ ν ∗ \nu_{*} ν ∗ I ⊥ I_{\perp} I ⊥ W W W W ⊥ W_{\perp} W ⊥ [28 ]
I ⊥ = I − b ^ ⊗ b ^ , W = ∇ ⃗ [ v ⃗ ] + ∇ ⃗ [ v ⃗ ] T − 2 3 ( ∇ ⃗ ⋅ v ⃗ ) I , W ⊥ = I ⊥ ⋅ W ⋅ I ⊥ . (14) I_{\perp}=I-\hat{b}\otimes\hat{b},\qquad W=\vec{\nabla}[\vec{v}]+\vec{\nabla}[\vec{v}]^{T}-\frac{2}{3}(\vec{\nabla}\cdot\vec{v})I,\qquad W_{\perp}=I_{\perp}\cdot W\cdot I_{\perp}. \tag*{(14)} I ⊥ = I − b ^ ⊗ b ^ , W = ∇ [ v ] + ∇ [ v ] T − 3 2 ( ∇ ⋅ v ) I , W ⊥ = I ⊥ ⋅ W ⋅ I ⊥ . (14) The parallel (Π ( 0 ) \boldsymbol{\Pi}^{(0)} Π ( 0 ) Π ( 1 ) + Π ( 2 ) \boldsymbol{\Pi}^{(1)}+\boldsymbol{\Pi}^{(2)} Π ( 1 ) + Π ( 2 ) Π ( 3 ) + Π ( 4 ) \boldsymbol{\Pi}^{(3)}+\boldsymbol{\Pi}^{(4)} Π ( 3 ) + Π ( 4 ) [23 ] or the viscous Hall effect [29 ] . The factors of 4 and 2 are retained so that, in the strongly magnetised limit, ν ⊥ 2 = ν ⊥ 1 \nu_{\perp2}=\nu_{\perp1} ν ⊥ 2 = ν ⊥ 1 ν ∧ 2 = ν ∧ 1 \nu_{\wedge2}=\nu_{\wedge1} ν ∧ 2 = ν ∧ 1 [23 ] .
q ⃗ J \vec{q}_{J} q J [23 ] . The parallel and perpendicular parts are the Peltier effect along and across the magnetic field respectively, while the wedge term is the Ettingshausen effect [27 ]
q ⃗ J = x i o n ( q ⃗ J , ∥ + q ⃗ J , ⊥ + q ⃗ J , ∧ ) , q ⃗ J , ∥ = − κ J , ∥ ( b ^ ⋅ J ⃗ ) b ^ , q ⃗ J , ⊥ = − κ J , ⊥ ( J ⃗ − b ^ ( b ^ ⋅ J ⃗ ) ) , q ⃗ J , ∧ = − κ J , ∧ b ^ × J ⃗ , (15) \vec{q}_{J}=x_{\mathrm{ion}}(\vec{q}_{J,\parallel}+\vec{q}_{J,\perp}+\vec{q}_{J,\wedge}),\qquad\vec{q}_{J,\parallel}=-\kappa_{J,\parallel}(\hat{b}\cdot\vec{J})\hat{b},\qquad\vec{q}_{J,\perp}=-\kappa_{J,\perp}(\vec{J}-\hat{b}(\hat{b}\cdot\vec{J})),\qquad\vec{q}_{J,\wedge}=-\kappa_{J,\wedge}\hat{b}\times\vec{J}, \tag*{(15)} q J = x ion ( q J , ∥ + q J , ⊥ + q J , ∧ ) , q J , ∥ = − κ J , ∥ ( b ^ ⋅ J ) b ^ , q J , ⊥ = − κ J , ⊥ ( J − b ^ ( b ^ ⋅ J )) , q J , ∧ = − κ J , ∧ b ^ × J , (15) where the κ J , ∗ \kappa_{J,*} κ J , ∗ T e T β u T \frac{T_{e}}{T} \beta^{uT} T T e β u T
F ⃗ T \vec{F}_{T} F T [23 ]
F ⃗ T = x i o n P / ρ ( F ⃗ T , ∥ + F ⃗ T , ⊥ + F ⃗ T , ∧ ) , (16) \vec{F}_{T}=\frac{x_{\mathrm{ion}}}{P/\rho}(\vec{F}_{T,\parallel}+\vec{F}_{T,\perp}+\vec{F}_{T,\wedge}), \tag*{(16)} F T = P / ρ x ion ( F T , ∥ + F T , ⊥ + F T , ∧ ) , (16) F ⃗ T , ∥ = − κ J , ∥ b ^ ( b ^ ⋅ ∇ ⃗ [ P / ρ ] ) , F ⃗ T , ⊥ = − κ J , ⊥ ( ∇ ⃗ [ P / ρ ] − b ^ ( b ^ ⋅ ∇ ⃗ [ P / ρ ] ) ) , F ⃗ T , ∧ = − κ J , ∧ b ^ × ∇ ⃗ [ P / ρ ] . (17) \vec{F}_{T,\parallel}=-\kappa_{J,\parallel}\hat{b}(\hat{b}\cdot\vec{\nabla}[P/\rho]),\qquad\vec{F}_{T,\perp}=-\kappa_{J,\perp}(\vec{\nabla}[P/\rho]-\hat{b}(\hat{b}\cdot\vec{\nabla}[P/\rho])),\qquad\vec{F}_{T,\wedge}=-\kappa_{J,\wedge}\hat{b}\times\vec{\nabla}[P/\rho]. \tag*{(17)} F T , ∥ = − κ J , ∥ b ^ ( b ^ ⋅ ∇ [ P / ρ ]) , F T , ⊥ = − κ J , ⊥ ( ∇ [ P / ρ ] − b ^ ( b ^ ⋅ ∇ [ P / ρ ])) , F T , ∧ = − κ J , ∧ b ^ × ∇ [ P / ρ ] . (17) The factors 1 P / ρ κ J , ∗ \frac{1}{P/\rho}\kappa_{J,*} P / ρ 1 κ J , ∗ 1 e β u T \frac{1}{e}\beta^{uT} e 1 β u T q ⃗ J \vec{q}_{J} q J [30 , 31 ] . The F ⃗ T \vec{F}_{T} F T κ J , ∗ \kappa_{J,*} κ J , ∗ F ⃗ T \vec{F}_{T} F T q ⃗ J \vec{q}_{J} q J [27 ] .
The κ J , ∗ \kappa_{J,*} κ J , ∗ x e = ω e τ e ∝ B T e 3 / 2 / n e x_{e}=\omega_{e}\tau_{e}\propto BT_{e}^{3/2}/n_{e} x e = ω e τ e ∝ B T e 3/2 / n e [23 ] . The Thomson effect of [27 ] therefore need not be added separately: it arises automatically from ∇ ⃗ ⋅ q ⃗ J \vec{\nabla}\cdot\vec{q}_{J} ∇ ⋅ q J J ⃗ ⋅ F ⃗ T \vec{J}\cdot\vec{F}_{T} J ⋅ F T κ J , ∗ \kappa_{J,*} κ J , ∗
The thermal conduction (neutral and anisotropic), kinematic (neutral and Braginskii) viscosity, and Hall effect coefficients are in dimensions of diffusivity L 2 T − 1 \mathrm{L}^{2}\mathrm{T}^{-1} L 2 T − 1 [ B ] L 2 T − 1 [B]\mathrm{L}^{2}\mathrm{T}^{-1} [ B ] L 2 T − 1 ζ \zeta ζ M L − 1 T − 1 \mathrm{ML}^{-1}\mathrm{T}^{-1} ML − 1 T − 1 L 3 / 2 M 1 / 2 T − 2 \mathrm{L}^{3/2}\mathrm{M}^{1/2}\mathrm{T}^{-2} L 3/2 M 1/2 T − 2 κ J , ∥ = 0.71 k B c e 4 π T = 0.71 m ˉ c e 4 π P ρ \kappa_{J,\parallel}=0.71\frac{k_{\mathrm{B}}c}{e\sqrt{4\pi}}T=0.71\frac{\bar{m}c}{e\sqrt{4\pi}}\frac{P}{\rho} κ J , ∥ = 0.71 e 4 π k B c T = 0.71 e 4 π m ˉ c ρ P m ˉ = μ m \bar{m}=\mu m m ˉ = μ m m ˉ \bar{m} m ˉ μ \mu μ m m m x i o n = 1 x_{\mathrm{ion}}=1 x ion = 1
CFL Conditions for Non-Ideal Terms
Here we perform stability analysis on the new physics implemented. Throughout we write the imaginary unit as − 1 \sqrt{-1} − 1
The timestep limits for the non-ideal terms all follow from a von Neumann analysis of the semi-discrete equations, linearised about a background state with coefficients frozen at their local values [32 ] . Each term is analysed in one dimension with wavevector k ⃗ = ( k x , 0 , 0 ) \vec{k}=(k_x,0,0) k = ( k x , 0 , 0 )
b ^ = ( cos θ , sin θ cos ϕ , sin θ sin ϕ ) , u ≡ cos 2 θ ∈ [ 0 , 1 ] , (18) \hat{b}=(\cos\theta,\sin\theta\cos\phi,\sin\theta\sin\phi),\qquad u\equiv\cos^2\theta\in[0,1], \tag*{(18)} b ^ = ( cos θ , sin θ cos ϕ , sin θ sin ϕ ) , u ≡ cos 2 θ ∈ [ 0 , 1 ] , (18) Substituting the spatial part of the Fourier mode ansatz U ~ i n = U ~ exp ( n σ Δ t ) exp ( − 1 k x i Δ x ) \tilde{U}_i^n=\tilde{U}\exp(n\sigma\Delta t)\exp(\sqrt{-1}k_xi\Delta x) U ~ i n = U ~ exp ( nσ Δ t ) exp ( − 1 k x i Δ x ) [32 ] into the semi-discrete equations replaces the spatial operator by its symbol L ^ ( k , u ) \hat{L}(k,u) L ^ ( k , u ) d U ~ i / d t = L ^ ( k , u ) U ~ i \mathrm{d}\tilde{U}_i/\mathrm{d}t=\hat{L}(k,u)\tilde{U}_i d U ~ i / d t = L ^ ( k , u ) U ~ i L ^ \hat{L} L ^
U i + 1 − 2 U i + U i − 1 Δ x 2 → L ^ = − 4 Δ x 2 sin 2 ( k x Δ x 2 ) , U i + 1 − U i − 1 2 Δ x → L ^ = − 1 Δ x sin ( k x Δ x ) , (19) \frac{U_{i+1}-2U_i+U_{i-1}}{\Delta x^2}\to\hat{L}=-\frac{4}{\Delta x^2}\sin^2\left(\frac{k_x\Delta x}{2}\right),\qquad\frac{U_{i+1}-U_{i-1}}{2\Delta x}\to\hat{L}=\frac{\sqrt{-1}}{\Delta x}\sin(k_x\Delta x), \tag*{(19)} Δ x 2 U i + 1 − 2 U i + U i − 1 → L ^ = − Δ x 2 4 sin 2 ( 2 k x Δ x ) , 2Δ x U i + 1 − U i − 1 → L ^ = Δ x − 1 sin ( k x Δ x ) , (19) whose magnitudes peak at 4 / Δ x 2 4/\Delta x^2 4/Δ x 2 k x Δ x = π k_x\Delta x=\pi k x Δ x = π 1 / Δ x 1/\Delta x 1/Δ x k x Δ x = π / 2 k_x\Delta x=\pi/2 k x Δ x = π /2
An explicit integrator advances each mode as U ~ i n + 1 = R ( Δ t L ^ ) U ~ i n \tilde{U}_i^{n+1}=R(\Delta t\hat{L})\tilde{U}_i^n U ~ i n + 1 = R ( Δ t L ^ ) U ~ i n R ( Z ) R(Z) R ( Z ) s s s s ≤ 4 s\le4 s ≤ 4 R ( Z ) = ∑ m = 0 s Z m / m ! R(Z)=\sum_{m=0}^{s}Z^m/m! R ( Z ) = ∑ m = 0 s Z m / m ! [33 ] . For an eigenvector of L ^ \hat{L} L ^ λ ( k , u ) \lambda(k,u) λ ( k , u )
exp ( σ Δ t ) = R ( Z ) , Z ≡ λ Δ t , (20) \exp(\sigma\Delta t)=R(Z),\qquad Z\equiv\lambda\Delta t, \tag*{(20)} exp ( σ Δ t ) = R ( Z ) , Z ≡ λ Δ t , (20) so stability requires ∣ R ( Z ) ∣ ≤ 1 |R(Z)|\le1 ∣ R ( Z ) ∣ ≤ 1 L ^ \hat{L} L ^ 1 / Δ x 1/\Delta x 1/Δ x 1 / Δ x 2 1/\Delta x^2 1/Δ x 2
Δ t ≤ r R max ( ∣ λ ( k , u ) ∣ ) (dissipative) , Δ t ≤ r I max ( ∣ λ ( k , u ) ∣ ) (non-dissipative) , (21) \Delta t\le\frac{r_R}{\max(|\lambda(k,u)|)}\quad\text{(dissipative)},\qquad\Delta t\le\frac{r_I}{\max(|\lambda(k,u)|)}\quad\text{(non-dissipative)}, \tag*{(21)} Δ t ≤ max ( ∣ λ ( k , u ) ∣ ) r R (dissipative) , Δ t ≤ max ( ∣ λ ( k , u ) ∣ ) r I (non-dissipative) , (21) where r R r_R r R r I r_I r I
We take r R = 2 r_R=2 r R = 2 Table 1 . RK1 and RK2 have r I = 0 r_I=0 r I = 0 r I = 3 r_I=\sqrt{3} r I = 3 ∣ λ ∣ |\lambda| ∣ λ ∣ u = 0 u=0 u = 0 u = 1 u=1 u = 1 u ∗ u^* u ∗ ∂ u ∣ λ ∣ = 0 \partial_u|\lambda|=0 ∂ u ∣ λ ∣ = 0 u ∗ ∈ [ 0 , 1 ] u^*\in[0,1] u ∗ ∈ [ 0 , 1 ]
Table 1. Stability functions and the extents of their stability regions along the negative real (r R r_R r R r I r_I r I s s s [20 ] .
In N D i m N_{\mathrm{Dim}} N Dim [34 pp. 63–64] . For a Laplacian on a uniform grid this gives max ∣ λ N D ∣ = N D i m max ∣ λ 1 D ∣ \max|\lambda_{N\mathrm{D}}|=N_{\mathrm{Dim}}\max|\lambda_{1\mathrm{D}}| max ∣ λ N D ∣ = N Dim max ∣ λ 1 D ∣
We do not include numerical dissipation from the Riemann solver in this analysis as numerical dissipation shifts imaginary eigenvalues into the left half-plane (negative real) and therefore omitting numerical dissipation is conservative for the timestep bound.
Thermal Conduction
For constant ρ \rho ρ b ^ \hat{b} b ^ P = ( γ − 1 ) ε i n t P=(\gamma-1)\varepsilon_{\mathrm{int}} P = ( γ − 1 ) ε int
∂ t ε i n t = ( γ − 1 ) ∇ ⃗ ⋅ [ κ 0 ∇ ⃗ [ ε i n t ] + Δ κ ( b ^ ⋅ ∇ ⃗ [ ε i n t ] ) b ^ + κ w b ^ × ∇ ⃗ [ ε i n t ] ] (22) \partial_t\varepsilon_{\mathrm{int}}=(\gamma-1)\vec{\nabla}\cdot\left[\kappa_0\vec{\nabla}[\varepsilon_{\mathrm{int}}]+\Delta\kappa\left(\hat{b}\cdot\vec{\nabla}[\varepsilon_{\mathrm{int}}]\right)\hat{b}+\kappa_w\hat{b}\times\vec{\nabla}[\varepsilon_{\mathrm{int}}]\right] \tag*{(22)} ∂ t ε int = ( γ − 1 ) ∇ ⋅ [ κ 0 ∇ [ ε int ] + Δ κ ( b ^ ⋅ ∇ [ ε int ] ) b ^ + κ w b ^ × ∇ [ ε int ] ] (22) with κ 0 = ( 1 − x i o n ) κ n + x i o n κ ⊥ \kappa_0=(1-x_{\mathrm{ion}})\kappa_n+x_{\mathrm{ion}}\kappa_\perp κ 0 = ( 1 − x ion ) κ n + x ion κ ⊥ Δ κ = x i o n ( κ ∥ − κ ⊥ ) \Delta\kappa=x_{\mathrm{ion}}(\kappa_\parallel-\kappa_\perp) Δ κ = x ion ( κ ∥ − κ ⊥ ) κ w = x i o n κ ∧ \kappa_w=x_{\mathrm{ion}}\kappa_\wedge κ w = x ion κ ∧ ∇ ⃗ ⋅ ( b ^ × ∇ ⃗ [ ε i n t ] ) = ( ∇ ⃗ × b ^ ) ⋅ ∇ ⃗ [ ε i n t ] \vec{\nabla}\cdot(\hat{b}\times\vec{\nabla}[\varepsilon_{\mathrm{int}}])=(\vec{\nabla}\times\hat{b})\cdot\vec{\nabla}[\varepsilon_{\mathrm{int}}] ∇ ⋅ ( b ^ × ∇ [ ε int ]) = ( ∇ × b ^ ) ⋅ ∇ [ ε int ] κ w \kappa_w κ w b ^ \hat{b} b ^ § 3.2 .
In 1D this reduces to ∂ t ε i n t = ( γ − 1 ) ( κ 0 + Δ κ u ) ∂ x x ε i n t \partial_t\varepsilon_{\mathrm{int}}=(\gamma-1)(\kappa_0+\Delta\kappa u)\partial_{xx}\varepsilon_{\mathrm{int}} ∂ t ε int = ( γ − 1 ) ( κ 0 + Δ κ u ) ∂ xx ε int ( k x Δ x = π ) (k_x\Delta x=\pi) ( k x Δ x = π ) λ = − 4 ( γ − 1 ) ( κ 0 + Δ κ u ) / Δ x 2 \lambda=-4(\gamma-1)(\kappa_0+\Delta\kappa u)/\Delta x^2 λ = − 4 ( γ − 1 ) ( κ 0 + Δ κ u ) /Δ x 2 κ 0 + Δ κ u \kappa_0+\Delta\kappa u κ 0 + Δ κ u u u u u = 0 u=0 u = 0 u = 1 u=1 u = 1 ∣ R ( Z ) R K 1 ∣ ≤ 1 |R(Z)_{\mathrm{RK1}}|\leq1 ∣ R ( Z ) RK1 ∣ ≤ 1
Δ t κ , ∥ ⊥ = C F L min ( Δ x ) 2 2 N D i m ( γ − 1 ) ( ( 1 − x i o n ) κ n + x i o n ( κ ∥ , κ ⊥ ) ) (23) \Delta t_{\kappa,\parallel\perp}=\mathrm{CFL}\frac{\min(\Delta x)^2}{2N_{\mathrm{Dim}}(\gamma-1)\left((1-x_{\mathrm{ion}})\kappa_n+x_{\mathrm{ion}}(\kappa_\parallel,\kappa_\perp)\right)} \tag*{(23)} Δ t κ , ∥⊥ = CFL 2 N Dim ( γ − 1 ) ( ( 1 − x ion ) κ n + x ion ( κ ∥ , κ ⊥ ) ) min ( Δ x ) 2 (23) This reduces to the standard isotropic result when Δ κ = 0 \Delta\kappa=0 Δ κ = 0
Righi–Leduc (Thermal Hall) Effect
Returning to Eq. (3), considering the wedge term of the thermal conduction on its own, and allowing the magnetic field unit vector to vary freely in three dimensions, the one-dimensional operator is
∂ t ε i n t = κ w ( γ − 1 ) ( ∇ ⃗ × b ^ ) x ∂ x ε i n t . (24) \partial_t\varepsilon_{\mathrm{int}}=\kappa_w(\gamma-1)(\vec{\nabla}\times\hat{b})_x\partial_x\varepsilon_{\mathrm{int}}. \tag*{(24)} ∂ t ε int = κ w ( γ − 1 ) ( ∇ × b ^ ) x ∂ x ε int . (24) Unlike the parallel and perpendicular conduction, which contain second order derivatives, this is a first-order advective operator, with an effective phase speed set by the curl of the magnetic unit vector. In our flux implementation in Athena++ the transverse temperature gradients are formed with second-order central differences, so the effective discretisation of ∂ x ε i n t \partial_x\varepsilon_{\mathrm{int}} ∂ x ε int x x x b ^ \hat{b} b ^
∣ ( ∇ ⃗ × b ^ ) x ∣ ≤ 2 Δ y + 2 Δ z = 2 Δ ⊥ , Δ ⊥ = ( 1 Δ y + 1 Δ z ) − 1 . (25) \left|(\vec{\nabla}\times\hat{b})_x\right|\leq\frac{2}{\Delta_y}+\frac{2}{\Delta_z}=\frac{2}{\Delta_\perp},\qquad\Delta_\perp=\left(\frac{1}{\Delta_y}+\frac{1}{\Delta_z}\right)^{-1}. \tag*{(25)} ( ∇ × b ^ ) x ≤ Δ y 2 + Δ z 2 = Δ ⊥ 2 , Δ ⊥ = ( Δ y 1 + Δ z 1 ) − 1 . (25) The stability parameter, Z κ , ∧ = 2 − 1 κ w ( γ − 1 ) sin ( k x Δ x ) Δ t / ( Δ x Δ ⊥ ) Z_{\kappa,\wedge}=2\sqrt{-1}\kappa_w(\gamma-1)\sin(k_x\Delta x)\Delta t/(\Delta x\Delta_\perp) Z κ , ∧ = 2 − 1 κ w ( γ − 1 ) sin ( k x Δ x ) Δ t / ( Δ x Δ ⊥ ) r I > 0 r_I>0 r I > 0 ∣ R ( Z κ , ∧ ) ∣ 2 ≤ 1 |R(Z_{\kappa,\wedge})|^2\leq1 ∣ R ( Z κ , ∧ ) ∣ 2 ≤ 1 Δ t κ , ∧ = C F L 3 Δ x Δ ⊥ / ( 2 ( γ − 1 ) x i o n κ ∧ ) \Delta t_{\kappa,\wedge}=\mathrm{CFL}\sqrt{3}\Delta x\Delta_\perp/(2(\gamma-1)x_{\mathrm{ion}}\kappa_\wedge) Δ t κ , ∧ = CFL 3 Δ x Δ ⊥ / ( 2 ( γ − 1 ) x ion κ ∧ ) d d d Δ ⊥ , d = ( ∑ i ≠ d Δ x i − 1 ) − 1 \Delta_{\perp,d}=(\sum_{i\neq d}\Delta x_i^{-1})^{-1} Δ ⊥ , d = ( ∑ i = d Δ x i − 1 ) − 1 i i i
Δ t κ , ∧ = C F L 3 2 ( γ − 1 ) x i o n κ ∧ ∑ d ( Δ x d Δ ⊥ , d ) − 1 (26) \Delta t_{\kappa,\wedge}=\mathrm{CFL}\frac{\sqrt{3}}{2(\gamma-1)x_{\mathrm{ion}}\kappa_\wedge\sum_d(\Delta x_d\Delta_{\perp,d})^{-1}} \tag*{(26)} Δ t κ , ∧ = CFL 2 ( γ − 1 ) x ion κ ∧ ∑ d ( Δ x d Δ ⊥ , d ) − 1 3 (26) On a uniform Cartesian grid, Δ ⊥ , d = Δ x / 2 \Delta_{\perp,d}=\Delta x/2 Δ ⊥ , d = Δ x /2
Δ t κ , ∧ , x y z = C F L min ( Δ x ) 2 4 3 ( γ − 1 ) N D i m x i o n κ ∧ . (27) \Delta t_{\kappa,\wedge,xyz}=\mathrm{CFL}\frac{\min(\Delta x)^2}{\frac{4}{\sqrt{3}}(\gamma-1)N_{\mathrm{Dim}}x_{\mathrm{ion}}\kappa_\wedge}. \tag*{(27)} Δ t κ , ∧ , x y z = CFL 3 4 ( γ − 1 ) N Dim x ion κ ∧ min ( Δ x ) 2 . (27) If STS is disabled (unsplit thermal conduction), then we simply take the inverse sum of both timesteps as a worst-case bound
Δ t κ = ( 1 / Δ t κ , ∥ ⊥ + 1 / Δ t κ , ∧ ) − 1 . (28) \Delta t_\kappa=\left(1/\Delta t_{\kappa,\parallel\perp}+1/\Delta t_{\kappa,\wedge}\right)^{-1}. \tag*{(28)} Δ t κ = ( 1/Δ t κ , ∥⊥ + 1/Δ t κ , ∧ ) − 1 . (28) Magnetic Diffusion
Considering just the dissipative parts of the non-ideal magnetic terms (Ohmic and ambipolar)
∂ t B ⃗ = η ∇ 2 [ B ⃗ ] + η A D ( ∇ 2 [ B ⃗ ] + ∇ [ b ^ ⋅ ( ∇ × B ⃗ ) ] × b ^ ) . (29) \partial_t \vec{B} = \eta\nabla^2[\vec{B}] + \eta_{AD}\left(\nabla^2[\vec{B}] + \nabla\left[\hat{b} \cdot\left(\nabla\times\vec{B}\right)\right] \times\hat{b}\right). \tag*{(29)} ∂ t B = η ∇ 2 [ B ] + η A D ( ∇ 2 [ B ] + ∇ [ b ^ ⋅ ( ∇ × B ) ] × b ^ ) . (29) Assuming b ^ \hat{b} b ^
∂ t ( B y B z ) = ( ( η + ( 1 − b z 2 ) η A D ) b y b z η A D b y b z η A D ( η + ( 1 − b y 2 ) η A D ) ) ∂ x x ( B y B z ) . (30) \partial_t
\begin{pmatrix}
B_y \\
B_z
\end{pmatrix}
=
\begin{pmatrix}
\left(\eta+ \left(1-b_z^2\right)\eta_{AD}\right) & b_y b_z\eta_{AD} \\
b_y b_z\eta_{AD} & \left(\eta+ \left(1-b_y^2\right)\eta_{AD}\right)
\end{pmatrix}
\partial_{xx}
\begin{pmatrix}
B_y \\
B_z
\end{pmatrix}. \tag*{(30)} ∂ t ( B y B z ) = ( ( η + ( 1 − b z 2 ) η A D ) b y b z η A D b y b z η A D ( η + ( 1 − b y 2 ) η A D ) ) ∂ xx ( B y B z ) . (30) Taking the Fourier space ansatz at the binding mode k x Δ x = π k_x\Delta x = \pi k x Δ x = π λ η , ∥ , ⊥ \lambda_{\eta,\parallel,\perp} λ η , ∥ , ⊥ Z η , ∥ ⊥ = − 4 Δ t λ η , ∥ , ⊥ / Δ x 2 Z_{\eta,\parallel\perp} = -4\Delta t\lambda_{\eta,\parallel,\perp}/\Delta x^2 Z η , ∥⊥ = − 4Δ t λ η , ∥ , ⊥ /Δ x 2
λ η , ∥ , ⊥ , 0 = η + η A D , λ η , ∥ , ⊥ 1 = η + η A D ( 1 − b y 2 − b z 2 ) . (31) \lambda_{\eta,\parallel,\perp,0} = \eta+ \eta_{AD}, \qquad\lambda_{\eta,\parallel,\perp1} = \eta+ \eta_{AD}\left(1-b_y^2-b_z^2\right). \tag*{(31)} λ η , ∥ , ⊥ , 0 = η + η A D , λ η , ∥ , ⊥ 1 = η + η A D ( 1 − b y 2 − b z 2 ) . (31) As 1 − b y 2 − b z 2 = b x 2 = u 1-b_y^2-b_z^2=b_x^2=u 1 − b y 2 − b z 2 = b x 2 = u u u u λ η , ∥ , ⊥ , max = λ η , ∥ , ⊥ , 0 = λ η , ∥ , ⊥ 1 ( 1 ) \lambda_{\eta,\parallel,\perp,\max} = \lambda_{\eta,\parallel,\perp,0} = \lambda_{\eta,\parallel,\perp1}(1) λ η , ∥ , ⊥ , m a x = λ η , ∥ , ⊥ , 0 = λ η , ∥ , ⊥ 1 ( 1 ) λ η , ∥ , ⊥ , max = η + η A D \lambda_{\eta,\parallel,\perp,\max} = \eta+ \eta_{AD} λ η , ∥ , ⊥ , m a x = η + η A D ∣ R ( Z η , ∥ ) R K 1 ∣ 2 ≤ 1 \lvert R(Z_{\eta,\parallel})_{\mathrm{RK1}}\rvert^2 \leq1 ∣ R ( Z η , ∥ ) RK1 ∣ 2 ≤ 1
Δ t η , ∥ ⊥ = C F L min ( Δ x ) 2 2 N D i m ( η + η A D ) . (32) \Delta t_{\eta,\parallel\perp} = \mathrm{CFL}\frac{\min(\Delta x)^2}{2N_{\mathrm{Dim}}(\eta+\eta_{AD})}. \tag*{(32)} Δ t η , ∥⊥ = CFL 2 N Dim ( η + η A D ) min ( Δ x ) 2 . (32) Hall Effect
Considering the Hall term on its own, ∂ t B ⃗ = − ∇ ⃗ × η H ( J ⃗ × b ^ ) \partial_t\vec{B}=-\vec{\nabla}\times\eta_H(\vec{J}\times\hat{b}) ∂ t B = − ∇ × η H ( J × b ^ ) B ⃗ \vec{B} B x x x
∂ t ( B y , B z ) T = η H b x ∂ x x ( B z , − B y ) T . (33) \partial_t\left(B_y,B_z\right)^T = \eta_H b_x\partial_{xx}\left(B_z,-B_y\right)^T. \tag*{(33)} ∂ t ( B y , B z ) T = η H b x ∂ xx ( B z , − B y ) T . (33) In Fourier space the eigenvalues of the coefficient matrix are λ η , ∧ , ± = ± − 1 η H b x \lambda_{\eta,\wedge,\pm}=\pm\sqrt{-1}\eta_Hb_x λ η , ∧ , ± = ± − 1 η H b x Z η , ∧ = ± − 1 η H b x 4 Δ t sin 2 ( k x Δ x / 2 ) / Δ x 2 Z_{\eta,\wedge}=\pm\sqrt{-1}\eta_Hb_x4\Delta t\sin^2(k_x\Delta x/2)/\Delta x^2 Z η , ∧ = ± − 1 η H b x 4Δ t sin 2 ( k x Δ x /2 ) /Δ x 2 k x Δ x = π k_x\Delta x=\pi k x Δ x = π ∣ R ( Z η , ∧ ) ∣ \lvert R(Z_{\eta,\wedge})\rvert ∣ R ( Z η , ∧ )∣
Δ t η , ∧ = C F L min ( Δ x ) 2 4 3 N D i m η H . (34) \Delta t_{\eta,\wedge}=\mathrm{CFL}\frac{\min(\Delta x)^2}{\frac{4}{\sqrt{3}}N_{\mathrm{Dim}}\eta_H}. \tag*{(34)} Δ t η , ∧ = CFL 3 4 N Dim η H min ( Δ x ) 2 . (34) In 1D this is in agreement with [11 ] . This is conservative by a factor N D i m \sqrt{N_{\mathrm{Dim}}} N Dim [12 ] for the edge-centred Hall EMF in Athena++.
Unsplit Magnetic Diffusion
Summing the effects of Ohmic resistivity, ambipolar diffusion, and the Hall effect, the matrix equation becomes
d d t ( B ~ y B ~ z ) = ( − b z 2 η A D + η + η A D b x η H + b y b z η A D − b x η H + b y b z η A D − b y 2 η A D + η + η A D ) − 4 Δ x 2 ( B ~ y B ~ z ) . (35) \frac{d}{dt}
\begin{pmatrix}
\widetilde{B}_y \\
\widetilde{B}_z
\end{pmatrix}
=
\begin{pmatrix}
-b_z^2\eta_{AD}+\eta+\eta_{AD} & b_x\eta_H+b_yb_z\eta_{AD} \\
-b_x\eta_H+b_yb_z\eta_{AD} & -b_y^2\eta_{AD}+\eta+\eta_{AD}
\end{pmatrix}
\frac{-4}{\Delta x^2}
\begin{pmatrix}
\widetilde{B}_y \\
\widetilde{B}_z
\end{pmatrix}. \tag*{(35)} d t d ( B y B z ) = ( − b z 2 η A D + η + η A D − b x η H + b y b z η A D b x η H + b y b z η A D − b y 2 η A D + η + η A D ) Δ x 2 − 4 ( B y B z ) . (35) The eigenvalues of this are
λ η , ± = η + 1 2 η A D ( u + 1 ) ± 1 2 η A D 2 ( u − 1 ) 2 − 4 η H 2 u . (36) \lambda_{\eta,\pm}=\eta+\frac{1}{2}\eta_{AD}(u+1)\pm\frac{1}{2}\sqrt{\eta_{AD}^2(u-1)^2-4\eta_H^2u}. \tag*{(36)} λ η , ± = η + 2 1 η A D ( u + 1 ) ± 2 1 η A D 2 ( u − 1 ) 2 − 4 η H 2 u . (36) This maximises at u = 1 u=1 u = 1 λ η , max = η + η A D ± − 1 η H \lambda_{\eta,\max}=\eta+\eta_{AD}\pm\sqrt{-1}\eta_H λ η , m a x = η + η A D ± − 1 η H ∣ R ( Z η ) ∣ 2 ≤ 1 \lvert R(Z_\eta)\rvert^2\leq1 ∣ R ( Z η ) ∣ 2 ≤ 1
Δ t η , R K 1 = C F L min ( Δ x ) 2 ( η + η A D ) 2 N D i m ( ( η + η A D ) 2 + η H 2 ) . (37) \Delta t_{\eta,\mathrm{RK1}}=\mathrm{CFL}\frac{\min(\Delta x)^2(\eta+\eta_{AD})}{2N_{\mathrm{Dim}}\left((\eta+\eta_{AD})^2+\eta_H^2\right)}. \tag*{(37)} Δ t η , RK1 = CFL 2 N Dim ( ( η + η A D ) 2 + η H 2 ) min ( Δ x ) 2 ( η + η A D ) . (37) In Appendix E we show that this limit is too restrictive to be usable when the Hall term is significant, so an integrator with r I > 0 r_{\mathrm{I}} > 0 r I > 0 Z η Z_{\eta} Z η ∣ R ( Z η ) ∣ 2 ≤ 1 |R(Z_{\eta})|^{2} \le1 ∣ R ( Z η ) ∣ 2 ≤ 1 Δ t \Delta t Δ t
Δ t η , R K 3 = C F L min ( Δ x ) 2 4 3 ( η + η A D ) 2 + η H 2 (38) \Delta t_{\eta,\mathrm{RK3}} = \mathrm{CFL}\,\frac{\min(\Delta x)^{2}}{\frac{4}{\sqrt{3}}\sqrt{(\eta+\eta_{\mathrm{AD}})^{2}+\eta_{\mathrm{H}}^{2}}} \tag*{(38)} Δ t η , RK3 = CFL 3 4 ( η + η AD ) 2 + η H 2 min ( Δ x ) 2 (38) Since Re ( λ η ) ≥ 0 \operatorname{Re}(\lambda_{\eta}) \ge0 Re ( λ η ) ≥ 0 ∣ λ η ∣ ≤ ( η + η A D ) 2 + η H 2 |\lambda_{\eta}| \le\sqrt{(\eta+\eta_{\mathrm{AD}})^{2}+\eta_{\mathrm{H}}^{2}} ∣ λ η ∣ ≤ ( η + η AD ) 2 + η H 2 Z η Z_{\eta} Z η ∣ Z η ∣ ≤ 3 |Z_{\eta}| \le\sqrt{3} ∣ Z η ∣ ≤ 3 R ( Z ) R(Z) R ( Z ) ∣ R ∣ ≤ 1 |R| \le1 ∣ R ∣ ≤ 1 η H > 0 \eta_{\mathrm{H}} > 0 η H > 0 Δ t η , ∥ , ⊥ \Delta t_{\eta,\parallel,\perp} Δ t η , ∥ , ⊥ Δ t η , R K 1 \Delta t_{\eta,\mathrm{RK1}} Δ t η , RK1
Viscosity
The individual components of each tensor, and the scalars in the viscosity equations, are presented in Appendix A. We take the perpendicular coefficient of Π ( 1 ) \boldsymbol{\Pi}^{(1)} Π ( 1 ) ν ⊥ 1 \nu_{\perp1} ν ⊥ 1 Π ( 2 ) \boldsymbol{\Pi}^{(2)} Π ( 2 ) ν ⊥ 2 \nu_{\perp2} ν ⊥ 2 x x x
∂ t v ⃗ = M ( ν , ∥ , ⊥ ) ∂ x x v ⃗ (39) \partial_t\vec{v} = M^{(\nu,\parallel,\perp)}\partial_{xx}\vec{v} \tag*{(39)} ∂ t v = M ( ν , ∥ , ⊥ ) ∂ xx v (39) with the real symmetric matrix M ( ν , ∥ , ⊥ ) M^{(\nu,\parallel,\perp)} M ( ν , ∥ , ⊥ ) k x Δ x = π k_x\Delta x = \pi k x Δ x = π d v ⃗ ~ / d t = − ( 4 / Δ x 2 ) M ( ν , ∥ , ⊥ ) v ⃗ ~ d\tilde{\vec{v}}/dt = -(4/\Delta x^{2})M^{(\nu,\parallel,\perp)}\tilde{\vec{v}} d v ~ / d t = − ( 4/Δ x 2 ) M ( ν , ∥ , ⊥ ) v ~ ϕ \phi ϕ ϕ = 0 \phi= 0 ϕ = 0 ϕ \phi ϕ x ^ \hat{x} x ^ ϕ \phi ϕ u u u ϕ = 0 \phi= 0 ϕ = 0 v z v_z v z b z = 0 b_z = 0 b z = 0 x y xy x y M z z = λ ν , ∥ , ⊥ , z M_{zz} = \lambda_{\nu,\parallel,\perp,z} M z z = λ ν , ∥ , ⊥ , z
Tr ( M x y ( ν , ∥ , ⊥ ) ) = ζ ρ + 7 3 ( 1 − x i o n ) ν n e u + x i o n [ ν ∥ ( u + 1 3 ) + ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 ] (40) \operatorname{Tr}\left(M_{xy}^{(\nu,\parallel,\perp)}\right) = \frac{\zeta}{\rho} + \frac{7}{3}(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}} + x_{\mathrm{ion}}\left[\nu_{\parallel}\left(u+\frac{1}{3}\right)+\nu_{\perp1}(1-u)+4\nu_{\perp2}\right] \tag*{(40)} Tr ( M x y ( ν , ∥ , ⊥ ) ) = ρ ζ + 3 7 ( 1 − x ion ) ν neu + x ion [ ν ∥ ( u + 3 1 ) + ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 ] (40) Det ( M x y ( ν , ∥ , ⊥ ) ) = ζ ρ ( 1 − x i o n ) ν n e u + 4 3 ( 1 − x i o n ) 2 ν n e u 2 + x i o n [ ζ ρ [ ( 3 ν ∥ + ν ⊥ 1 ) u ( 1 − u ) + 4 ν ⊥ 2 ( 2 u − 1 ) 2 ] + ( 1 − x i o n ) ν n e u [ 3 ν ∥ ( u − 2 3 ) 2 + ν ⊥ 1 ( 1 − u ) 2 + 16 ν ⊥ 2 u ( 1 − u ) ] + 4 3 ( 1 − x i o n ) ν n e u [ ( 3 ν ∥ + ν ⊥ 1 ) u ( 1 − u ) + 4 ν ⊥ 2 ( 2 u − 1 ) 2 ] + 4 3 x i o n 2 [ ν ∥ ν ⊥ 1 u ( 1 − u ) + ν ∥ ν ⊥ 2 ( u + 1 ) 2 + 3 ν ⊥ 1 ν ⊥ 2 ( 1 − u ) 2 ] , (41) \begin{aligned}
\operatorname{Det}\left(M_{xy}^{(\nu,\parallel,\perp)}\right) ={}& \frac{\zeta}{\rho}(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}+\frac{4}{3}(1-x_{\mathrm{ion}})^{2}\nu_{\mathrm{neu}}^{2}\\
&+x_{\mathrm{ion}}\left[\frac{\zeta}{\rho}\left[(3\nu_{\parallel}+\nu_{\perp1})u(1-u)+4\nu_{\perp2}(2u-1)^{2}\right]\right.\\
&\left.+(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}\left[3\nu_{\parallel}\left(u-\frac{2}{3}\right)^{2}+\nu_{\perp1}(1-u)^{2}+16\nu_{\perp2}u(1-u)\right]\right.\\
&\left.+\frac{4}{3}(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}\left[(3\nu_{\parallel}+\nu_{\perp1})u(1-u)+4\nu_{\perp2}(2u-1)^{2}\right]\right.\\
&\left.+\frac{4}{3}x_{\mathrm{ion}}^{2}\left[\nu_{\parallel}\nu_{\perp1}u(1-u)+\nu_{\parallel}\nu_{\perp2}(u+1)^{2}+3\nu_{\perp1}\nu_{\perp2}(1-u)^{2}\right]\right., \tag*{(41)}
\end{aligned} Det ( M x y ( ν , ∥ , ⊥ ) ) = ρ ζ ( 1 − x ion ) ν neu + 3 4 ( 1 − x ion ) 2 ν neu 2 + x ion [ ρ ζ [ ( 3 ν ∥ + ν ⊥ 1 ) u ( 1 − u ) + 4 ν ⊥ 2 ( 2 u − 1 ) 2 ] + ( 1 − x ion ) ν neu [ 3 ν ∥ ( u − 3 2 ) 2 + ν ⊥ 1 ( 1 − u ) 2 + 16 ν ⊥ 2 u ( 1 − u ) ] + 3 4 ( 1 − x ion ) ν neu [ ( 3 ν ∥ + ν ⊥ 1 ) u ( 1 − u ) + 4 ν ⊥ 2 ( 2 u − 1 ) 2 ] + 3 4 x ion 2 [ ν ∥ ν ⊥ 1 u ( 1 − u ) + ν ∥ ν ⊥ 2 ( u + 1 ) 2 + 3 ν ⊥ 1 ν ⊥ 2 ( 1 − u ) 2 ] , (41) λ ν , ∥ , ⊥ , z = ( 1 − x i o n ) ν n e u + x i o n ( ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 u ) . (42) \lambda_{\nu,\parallel,\perp,z} = (1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}+x_{\mathrm{ion}}\left(\nu_{\perp1}(1-u)+4\nu_{\perp2}u\right). \tag*{(42)} λ ν , ∥ , ⊥ , z = ( 1 − x ion ) ν neu + x ion ( ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 u ) . (42) The x y xy x y x y xy x y
λ ν ∥ , ⊥ , x y ± = 1 2 [ Tr ( M x y ( ν , ∥ , ⊥ ) ) ± Tr ( M x y ( ν , ∥ , ⊥ ) ) 2 − 4 Det ( M x y ( ν , ∥ , ⊥ ) ) ] (43) \lambda_{\nu\parallel,\perp,xy\pm} = \frac{1}{2}\left[\operatorname{Tr}\left(M_{xy}^{(\nu,\parallel,\perp)}\right) \pm\sqrt{\operatorname{Tr}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)^{2}-4\operatorname{Det}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)}\right] \tag*{(43)} λ ν ∥ , ⊥ , x y ± = 2 1 [ Tr ( M x y ( ν , ∥ , ⊥ ) ) ± Tr ( M x y ( ν , ∥ , ⊥ ) ) 2 − 4 Det ( M x y ( ν , ∥ , ⊥ ) ) ] (43) Since M ( ν , ∥ , ⊥ ) M^{(\nu,\parallel,\perp)} M ( ν , ∥ , ⊥ ) 2 × 2 2 \times2 2 × 2 Tr ( M x y ( ν , ∥ , ⊥ ) ) ≥ 0 \operatorname{Tr}\left(M_{xy}^{(\nu,\parallel,\perp)}\right) \ge0 Tr ( M x y ( ν , ∥ , ⊥ ) ) ≥ 0 Det ( M x y ( ν , ∥ , ⊥ ) ) ≥ 0 \operatorname{Det}(M_{xy}^{(\nu,\parallel,\perp)}) \ge0 Det ( M x y ( ν , ∥ , ⊥ ) ) ≥ 0 x i o n , u ∈ [ 0 , 1 ] x_{\mathrm{ion}}, u \in[0,1] x ion , u ∈ [ 0 , 1 ]
We now maximise λ ν , ∥ , ⊥ \lambda_{\nu,\parallel,\perp} λ ν , ∥ , ⊥ d λ ν , x y + / d u = 0 d\lambda_{\nu,xy+}/du = 0 d λ ν , x y + / d u = 0 Tr \operatorname{Tr} Tr Det \operatorname{Det} Det u u u u u u
u 1 ∗ = ν ∥ + ν ⊥ 1 − 8 ν ⊥ 2 3 ν ∥ + ν ⊥ 1 − 16 ν ⊥ 2 , (44) u_1^* = \frac{\nu_{\parallel}+\nu_{\perp1}-8\nu_{\perp2}}{3\nu_{\parallel}+\nu_{\perp1}-16\nu_{\perp2}}, \tag*{(44)} u 1 ∗ = 3 ν ∥ + ν ⊥ 1 − 16 ν ⊥ 2 ν ∥ + ν ⊥ 1 − 8 ν ⊥ 2 , (44) u 2 ∗ = 2 ( 1 − x i o n ) ν n e u + x i o n ( 3 ν ⊥ 1 − ν ∥ ) + 6 ζ / ρ 4 ( 1 − x i o n ) ν n e u + x i o n ( ν ∥ + 3 ν ⊥ 1 ) + 12 ζ / ρ . (45) u_2^* = \frac{2(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}+x_{\mathrm{ion}}(3\nu_{\perp1}-\nu_{\parallel})+6\zeta/\rho}{4(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}+x_{\mathrm{ion}}(\nu_{\parallel}+3\nu_{\perp1})+12\zeta/\rho}. \tag*{(45)} u 2 ∗ = 4 ( 1 − x ion ) ν neu + x ion ( ν ∥ + 3 ν ⊥ 1 ) + 12 ζ / ρ 2 ( 1 − x ion ) ν neu + x ion ( 3 ν ⊥ 1 − ν ∥ ) + 6 ζ / ρ . (45) Squaring admits critical points of λ ν , x y − \lambda_{\nu,xy-} λ ν , x y − λ ν , x y + \lambda_{\nu,xy+} λ ν , x y + λ ν , x y + \lambda_{\nu,xy+} λ ν , x y + λ ν , x y + \lambda_{\nu,xy+} λ ν , x y +
λ ν , max = max ( λ ν , x y + ( 0 ) , λ ν , x y + ( 1 ) , 1 [ 0 , 1 ] ( u 1 ∗ ) λ ν , x y + ( u 1 ∗ ) , 1 [ 0 , 1 ] ( u 2 ∗ ) λ ν , x y + ( u 2 ∗ ) ) . (46) \lambda_{\nu,\max}=\max\left(\lambda_{\nu,xy+}(0),\ \lambda_{\nu,xy+}(1),\ \mathbf{1}_{[0,1]}(u_1^*)\lambda_{\nu,xy+}(u_1^*),\ \mathbf{1}_{[0,1]}(u_2^*)\lambda_{\nu,xy+}(u_2^*)\right). \tag*{(46)} λ ν , m a x = max ( λ ν , x y + ( 0 ) , λ ν , x y + ( 1 ) , 1 [ 0 , 1 ] ( u 1 ∗ ) λ ν , x y + ( u 1 ∗ ) , 1 [ 0 , 1 ] ( u 2 ∗ ) λ ν , x y + ( u 2 ∗ ) ) . (46) Solving for the RK1 stability polynomial gives the timestep
Δ t ζ , ν , ∥ , ⊥ = CFL min ( Δ x ) 2 2 N D i m λ ν , max . (47) \Delta t_{\zeta,\nu,\parallel,\perp}=\operatorname{CFL}\frac{\min(\Delta x)^2}{2N_{\mathrm{Dim}}\lambda_{\nu,\max}}. \tag*{(47)} Δ t ζ , ν , ∥ , ⊥ = CFL 2 N Dim λ ν , m a x min ( Δ x ) 2 . (47) Gyroviscosity (Viscous Hall Effect)
We now consider the sum of two wedge terms of viscosity. Taking a constant magnetic field unit vector and only the x x x Π ( 3 ) \boldsymbol{\Pi}^{(3)} Π ( 3 ) Π ( 4 ) \boldsymbol{\Pi}^{(4)} Π ( 4 )
∂ t ( v x v y v z ) = ( 0 M x y ( ν , ∧ ) M x z ( ν , ∧ ) − M x y ( ν , ∧ ) 0 M y z ( ν , ∧ ) − M x z ( ν , ∧ ) − M y z ( ν , ∧ ) 0 ) ⋅ ∂ x x ( v x v y v z ) , (48) \partial_t
\begin{pmatrix}
v_x\\
v_y\\
v_z
\end{pmatrix}
=
\begin{pmatrix}
0 & M_{xy}^{(\nu,\wedge)} & M_{xz}^{(\nu,\wedge)}\\
-M_{xy}^{(\nu,\wedge)} & 0 & M_{yz}^{(\nu,\wedge)}\\
-M_{xz}^{(\nu,\wedge)} & -M_{yz}^{(\nu,\wedge)} & 0
\end{pmatrix}
\cdot\partial_{xx}
\begin{pmatrix}
v_x\\
v_y\\
v_z
\end{pmatrix}, \tag*{(48)} ∂ t v x v y v z = 0 − M x y ( ν , ∧ ) − M x z ( ν , ∧ ) M x y ( ν , ∧ ) 0 − M y z ( ν , ∧ ) M x z ( ν , ∧ ) M y z ( ν , ∧ ) 0 ⋅ ∂ xx v x v y v z , (48) with
M x y ( ν , ∧ ) = − M y x ( ν , ∧ ) = x i o n b z [ ν ∧ 1 ( 1 − u ) + 4 ν ∧ 2 u ] , (49) M_{xy}^{(\nu,\wedge)}=-M_{yx}^{(\nu,\wedge)}=x_{\mathrm{ion}}b_z\left[\nu_{\wedge1}(1-u)+4\nu_{\wedge2}u\right], \tag*{(49)} M x y ( ν , ∧ ) = − M y x ( ν , ∧ ) = x ion b z [ ν ∧ 1 ( 1 − u ) + 4 ν ∧ 2 u ] , (49) M x z ( ν , ∧ ) = − M z x ( ν , ∧ ) = − x i o n b y [ ν ∧ 1 ( 1 − u ) + 4 ν ∧ 2 u ] , (50) M_{xz}^{(\nu,\wedge)}=-M_{zx}^{(\nu,\wedge)}=-x_{\mathrm{ion}}b_y\left[\nu_{\wedge1}(1-u)+4\nu_{\wedge2}u\right], \tag*{(50)} M x z ( ν , ∧ ) = − M z x ( ν , ∧ ) = − x ion b y [ ν ∧ 1 ( 1 − u ) + 4 ν ∧ 2 u ] , (50) M y z ( ν , ∧ ) = − M z y ( ν , ∧ ) = x i o n b x [ ν ∧ 1 ( 1 − u ) + 2 ν ∧ 2 ( 2 u − 1 ) ] . (51) M_{yz}^{(\nu,\wedge)}=-M_{zy}^{(\nu,\wedge)}=x_{\mathrm{ion}}b_x\left[\nu_{\wedge1}(1-u)+2\nu_{\wedge2}(2u-1)\right]. \tag*{(51)} M y z ( ν , ∧ ) = − M z y ( ν , ∧ ) = x ion b x [ ν ∧ 1 ( 1 − u ) + 2 ν ∧ 2 ( 2 u − 1 ) ] . (51) The matrix M ( ν , ∧ ) M^{(\nu,\wedge)} M ( ν , ∧ ) ϕ \phi ϕ
λ ν , ∧ , z = 0 , λ ν , ∧ , x y , ± = ± x i o n − 1 ( ν ∧ 1 ( 1 − u ) + 2 ν ∧ 2 u ) 2 + 4 ν ∧ 2 2 u ( 1 − u ) . (52) \lambda_{\nu,\wedge,z}=0,\qquad
\lambda_{\nu,\wedge,xy,\pm}=\pm x_{\mathrm{ion}}\sqrt{-1}\sqrt{\left(\nu_{\wedge1}(1-u)+2\nu_{\wedge2}u\right)^2+4\nu_{\wedge2}^2u(1-u)}. \tag*{(52)} λ ν , ∧ , z = 0 , λ ν , ∧ , x y , ± = ± x ion − 1 ( ν ∧ 1 ( 1 − u ) + 2 ν ∧ 2 u ) 2 + 4 ν ∧ 2 2 u ( 1 − u ) . (52) The magnitude of the positive branch has a single critical point
u ∗ = ν ∧ 1 2 − 2 ν ∧ 1 ν ∧ 2 − 2 ν ∧ 2 2 ν ∧ 1 ( ν ∧ 1 − 4 ν ∧ 2 ) , ∣ λ ν , ∧ + ( u ∗ ) ∣ = ∣ 2 x i o n ν ∧ 2 3 ( 2 ν ∧ 1 + ν ∧ 2 ) ν ∧ 1 ( 4 ν ∧ 2 − ν ∧ 1 ) ∣ . (53) u^*=\frac{\nu_{\wedge1}^2-2\nu_{\wedge1}\nu_{\wedge2}-2\nu_{\wedge2}^2}{\nu_{\wedge1}(\nu_{\wedge1}-4\nu_{\wedge2})},
\qquad
\left|\lambda_{\nu,\wedge+}(u^*)\right|
=
\left|2x_{\mathrm{ion}}\sqrt{\frac{\nu_{\wedge2}^3(2\nu_{\wedge1}+\nu_{\wedge2})}{\nu_{\wedge1}(4\nu_{\wedge2}-\nu_{\wedge1})}}\right|. \tag*{(53)} u ∗ = ν ∧ 1 ( ν ∧ 1 − 4 ν ∧ 2 ) ν ∧ 1 2 − 2 ν ∧ 1 ν ∧ 2 − 2 ν ∧ 2 2 , ∣ λ ν , ∧+ ( u ∗ ) ∣ = 2 x ion ν ∧ 1 ( 4 ν ∧ 2 − ν ∧ 1 ) ν ∧ 2 3 ( 2 ν ∧ 1 + ν ∧ 2 ) . (53) As u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ] u ∗ u^* u ∗ ∣ λ ν , ∧ + ( 0 ) ∣ = x i o n ν ∧ 1 \left|\lambda_{\nu,\wedge+}(0)\right|=x_{\mathrm{ion}}\nu_{\wedge1} ∣ λ ν , ∧+ ( 0 ) ∣ = x ion ν ∧ 1 ∣ λ ν , ∧ + ( 1 ) ∣ = 2 x i o n ν ∧ 2 \left|\lambda_{\nu,\wedge+}(1)\right|=2x_{\mathrm{ion}}\nu_{\wedge2} ∣ λ ν , ∧+ ( 1 ) ∣ = 2 x ion ν ∧ 2
∣ λ ν , ∧ + , max ∣ = max ( x i o n ν ∧ 1 , 2 x i o n ν ∧ 2 , 1 [ 0 , 1 ] ( u ∗ ) ∣ 2 x i o n ν ∧ 2 3 ( 2 ν ∧ 1 + ν ∧ 2 ) ν ∧ 1 ( 4 ν ∧ 2 − ν ∧ 1 ) ∣ ) . (54) \left|\lambda_{\nu,\wedge+, \max}\right|
=
\max\left(
x_{\mathrm{ion}}\nu_{\wedge1},
2x_{\mathrm{ion}}\nu_{\wedge2},
\mathbf{1}_{[0,1]}(u^*)
\left|2x_{\mathrm{ion}}\sqrt{\frac{\nu_{\wedge2}^3(2\nu_{\wedge1}+\nu_{\wedge2})}{\nu_{\wedge1}(4\nu_{\wedge2}-\nu_{\wedge1})}}\right|
\right). \tag*{(54)} ∣ λ ν , ∧+ , m a x ∣ = max ( x ion ν ∧ 1 , 2 x ion ν ∧ 2 , 1 [ 0 , 1 ] ( u ∗ ) 2 x ion ν ∧ 1 ( 4 ν ∧ 2 − ν ∧ 1 ) ν ∧ 2 3 ( 2 ν ∧ 1 + ν ∧ 2 ) ) . (54) The stability parameter, Z ν , ∧ = ± − 1 4 Δ t ∣ λ ν , ∧ ∣ / Δ x 2 Z_{\nu,\wedge}=\pm\sqrt{-1}4\Delta t|\lambda_{\nu,\wedge}|/\Delta x^2 Z ν , ∧ = ± − 1 4Δ t ∣ λ ν , ∧ ∣/Δ x 2
Δ t ν , ∧ = C F L min ( Δ x ) 2 4 3 N D i m ∣ λ ν , ∧ + , max ∣ . (55) \Delta t_{\nu,\wedge}=\mathrm{CFL}\frac{\min(\Delta x)^2}{\frac{4}{\sqrt{3}}N_{\mathrm{Dim}}|\lambda_{\nu,\wedge+,\max}|}. \tag*{(55)} Δ t ν , ∧ = CFL 3 4 N Dim ∣ λ ν , ∧+ , m a x ∣ min ( Δ x ) 2 . (55) If STS is disabled (unsplit viscosity), the operator is the sum of the dissipative and non-dissipative viscosity matrices, M ( ν , ∥ , ⊥ ) + M ( ν , ∧ ) M^{(\nu,\parallel,\perp)}+M^{(\nu,\wedge)} M ( ν , ∥ , ⊥ ) + M ( ν , ∧ )
Δ t ν = ( 1 Δ t ζ , ν , ∥ , ⊥ + 1 Δ t ν , ∧ ) − 1 . (56) \Delta t_\nu=\left(\frac{1}{\Delta t_{\zeta,\nu,\parallel,\perp}}+\frac{1}{\Delta t_{\nu,\wedge}}\right)^{-1}. \tag*{(56)} Δ t ν = ( Δ t ζ , ν , ∥ , ⊥ 1 + Δ t ν , ∧ 1 ) − 1 . (56) In Appendix E we compare this to numerically calculated timesteps with physical diffusion coefficients at RK3 time integration.
Isolated Thermoelectric Effects
Thermoelectric effects are given by q ⃗ J \vec{q}_J q J F ⃗ T \vec{F}_T F T ∂ t ε i n t = ∂ t ε − B ⃗ ⋅ ∂ t B ⃗ \partial_t\varepsilon_{\mathrm{int}}=\partial_t\varepsilon-\vec{B}\cdot\partial_t\vec{B} ∂ t ε int = ∂ t ε − B ⋅ ∂ t B ∂ t ε = ∇ ⋅ q ⃗ J + J ⃗ ⋅ F ⃗ T \partial_t\varepsilon=\nabla\cdot\vec{q}_J+\vec{J}\cdot\vec{F}_T ∂ t ε = ∇ ⋅ q J + J ⋅ F T J ⃗ ⋅ F ⃗ T \vec{J}\cdot\vec{F}_T J ⋅ F T
By linearising, only q ⃗ J \vec{q}_J q J κ J , ∗ / ( P / ρ ) \kappa_{J,*}/(P/\rho) κ J , ∗ / ( P / ρ ) F ⃗ T \vec{F}_T F T k x Δ x = π k_x\Delta x=\pi k x Δ x = π
d d t ( ε ~ i n t , B ~ y , B ~ z ) T = − ( 4 / Δ x 2 ) M ( J ) ( ε ~ i n t , B ~ y , B ~ z ) T , (57) \frac{d}{dt}(\widetilde{\varepsilon}_{\mathrm{int}},\widetilde{B}_y,\widetilde{B}_z)^{T}=-(4/\Delta x^2)M^{(J)}(\widetilde{\varepsilon}_{\mathrm{int}},\widetilde{B}_y,\widetilde{B}_z)^{T}, \tag*{(57)} d t d ( ε int , B y , B z ) T = − ( 4/Δ x 2 ) M ( J ) ( ε int , B y , B z ) T , (57) with
M ( J ) = ( 0 κ J , w b y − Δ κ J b x b z κ J , w b z + Δ κ J b x b y κ J , w b y + Δ κ J b x b z ε i n t 0 0 κ J , w b z − Δ κ J b x b y ε i n t 0 0 ) . (58) M^{(J)}=
\begin{pmatrix}
0 & \kappa_{J,w}b_y-\Delta\kappa_J b_xb_z & \kappa_{J,w}b_z+\Delta\kappa_J b_xb_y \\
\frac{\kappa_{J,w}b_y+\Delta\kappa_J b_xb_z}{\varepsilon_{\mathrm{int}}} & 0 & 0 \\
\frac{\kappa_{J,w}b_z-\Delta\kappa_J b_xb_y}{\varepsilon_{\mathrm{int}}} & 0 & 0
\end{pmatrix}. \tag*{(58)} M ( J ) = 0 ε int κ J , w b y + Δ κ J b x b z ε int κ J , w b z − Δ κ J b x b y κ J , w b y − Δ κ J b x b z 0 0 κ J , w b z + Δ κ J b x b y 0 0 . (58) Where Δ κ J = x i o n ( κ J , ⊥ − κ J , ∥ ) \Delta\kappa_J=x_{\mathrm{ion}}(\kappa_{J,\perp}-\kappa_{J,\parallel}) Δ κ J = x ion ( κ J , ⊥ − κ J , ∥ ) κ J , w = x i o n κ J , ∧ \kappa_{J,w}=x_{\mathrm{ion}}\kappa_{J,\wedge} κ J , w = x ion κ J , ∧ M ( J ) M^{(J)} M ( J ) λ T E \lambda_{\mathrm{TE}} λ TE Z T E = − 4 Δ t λ T E / Δ x 2 Z_{\mathrm{TE}}=-4\Delta t\lambda_{\mathrm{TE}}/\Delta x^2 Z TE = − 4Δ t λ TE /Δ x 2
− λ T E ( λ T E 2 − M ε y ( J ) M y ε ( J ) − M ε z ( J ) M z ε ( J ) ) = 0 , (59) -\lambda_{\mathrm{TE}}\left(\lambda_{\mathrm{TE}}^2-M^{(J)}_{\varepsilon y}M^{(J)}_{y\varepsilon}-M^{(J)}_{\varepsilon z}M^{(J)}_{z\varepsilon}\right)=0, \tag*{(59)} − λ TE ( λ TE 2 − M ε y ( J ) M y ε ( J ) − M ε z ( J ) M z ε ( J ) ) = 0 , (59) where both products are differences of squares, e.g. M ε y ( J ) M y ε ( J ) = ( κ J , w 2 b y 2 − Δ κ J 2 b x 2 b z 2 ) / ε i n t M^{(J)}_{\varepsilon y}M^{(J)}_{y\varepsilon}=(\kappa_{J,w}^2b_y^2-\Delta\kappa_J^2b_x^2b_z^2)/\varepsilon_{\mathrm{int}} M ε y ( J ) M y ε ( J ) = ( κ J , w 2 b y 2 − Δ κ J 2 b x 2 b z 2 ) / ε int b y b_y b y b z b_z b z b y 2 + b z 2 = 1 − u b_y^2+b_z^2=1-u b y 2 + b z 2 = 1 − u ϕ \phi ϕ
λ T E , 0 = 0 , λ T E , ± = ± 1 ε i n t ( κ J , w 2 − Δ κ J 2 u ) ( 1 − u ) . (60) \lambda_{\mathrm{TE},0}=0,\qquad\lambda_{\mathrm{TE},\pm}=\pm\frac{1}{\sqrt{\varepsilon_{\mathrm{int}}}}\sqrt{\left(\kappa_{J,w}^2-\Delta\kappa_J^2u\right)(1-u)}. \tag*{(60)} λ TE , 0 = 0 , λ TE , ± = ± ε int 1 ( κ J , w 2 − Δ κ J 2 u ) ( 1 − u ) . (60) The radicand is quadratic in u u u
u ∗ 1 = Δ κ J 2 + κ J , w 2 2 Δ κ J 2 , (61) u_*^1=\frac{\Delta\kappa_J^2+\kappa_{J,w}^2}{2\Delta\kappa_J^2}, \tag*{(61)} u ∗ 1 = 2Δ κ J 2 Δ κ J 2 + κ J , w 2 , (61) and it vanishes at u = 1 u=1 u = 1 u = κ J , w 2 / Δ κ J 2 u=\kappa_{J,w}^2/\Delta\kappa_J^2 u = κ J , w 2 /Δ κ J 2 λ T E , ± \lambda_{\mathrm{TE},\pm} λ TE , ± λ T E , + \lambda_{\mathrm{TE},+} λ TE , + λ T E , + ( 0 ) = κ J , w / ε i n t \lambda_{\mathrm{TE},+}(0)=\kappa_{J,w}/\sqrt{\varepsilon_{\mathrm{int}}} λ TE , + ( 0 ) = κ J , w / ε int λ T E , + ( u ∗ 1 ) = − 1 ∣ Δ κ J 2 − \lambda_{\mathrm{TE},+}(u_*^1)=\sqrt{-1}|\Delta\kappa_J^2- λ TE , + ( u ∗ 1 ) = − 1 ∣Δ κ J 2 − κ J , w 2 / ( 2 ∣ Δ κ J ∣ ε i n t ) \kappa_{J,w}^{2}/(2|\Delta\kappa_J|\sqrt{\varepsilon_{\mathrm{int}}}) κ J , w 2 / ( 2∣Δ κ J ∣ ε int ) λ T E , + ( 1 ) = λ T E , + ( κ J , w 2 / Δ κ J 2 ) = 0 \lambda_{\mathrm{TE},+(1)}=\lambda_{\mathrm{TE},+}(\kappa_{J,w}^{2}/\Delta\kappa_J^{2})=0 λ TE , + ( 1 ) = λ TE , + ( κ J , w 2 /Δ κ J 2 ) = 0
∣ λ T E , m a x ∣ = 1 ε i n t max ( κ J , w , 1 [ 0 , 1 ] ( u 1 ∗ ) Δ κ J 2 − κ J , w 2 2 ∣ Δ κ J ∣ ) , (62) |\lambda_{\mathrm{TE,max}}|=\frac{1}{\sqrt{\varepsilon_{\mathrm{int}}}}\max\left(\kappa_{J,w},\mathbb{1}_{[0,1]}(u_1^*)\frac{\Delta\kappa_J^2-\kappa_{J,w}^2}{2|\Delta\kappa_J|}\right), \tag*{(62)} ∣ λ TE , max ∣ = ε int 1 max ( κ J , w , 1 [ 0 , 1 ] ( u 1 ∗ ) 2∣Δ κ J ∣ Δ κ J 2 − κ J , w 2 ) , (62) with Δ κ J \Delta\kappa_J Δ κ J κ J , w \kappa_{J,w} κ J , w κ J , ∧ = 0 \kappa_{J,\wedge}=0 κ J , ∧ = 0 r I > 0 r_I>0 r I > 0 κ J , ∧ > 0 \kappa_{J,\wedge}>0 κ J , ∧ > 0
In a scenario where thermoelectric effects are modelled, we assume that a user likely has anisotropic thermal conduction and magnetic diffusion present. These terms will allow the combined system to be well-posed (which we will show in the next subsection). We take the isolated thermoelectric limit to be
Δ t T E = C F L min ( Δ x ) 2 4 3 N D i m ∣ λ T E , m a x ∣ , (63) \Delta t_{\mathrm{TE}}=\mathrm{CFL}\frac{\min(\Delta x)^2}{\frac{4}{\sqrt{3}}N_{\mathrm{Dim}}|\lambda_{\mathrm{TE,max}}|}, \tag*{(63)} Δ t TE = CFL 3 4 N Dim ∣ λ TE , max ∣ min ( Δ x ) 2 , (63) which is a valid stability limit only when κ J , ∧ = 0 \kappa_{J,\wedge}=0 κ J , ∧ = 0
Well-Posed Criterion for Thermoelectric Effects
We now combine the full expressions for thermal conduction, magnetic diffusion, and thermoelectric effects into a single system. We will refer to this as the thermo-magnetic system for brevity. Its linearised operator M ( κ , η , J ) M^{(\kappa,\eta,J)} M ( κ , η , J ) ( ε ~ i n t , B ~ y , B ~ z ) (\tilde{\varepsilon}_{\mathrm{int}},\tilde{B}_y,\tilde{B}_z) ( ε ~ int , B ~ y , B ~ z ) ψ = k x Δ x / 2 \psi=k_x\Delta x/2 ψ = k x Δ x /2 § 12 . Deriving the CFL condition directly from this system, by solving the cubic and maximising over the three angles, is infeasible even with computer algebra. We instead find a sufficient criterion for the system to be well-posed. If every eigenvalue of M ( κ , η , J ) M^{(\kappa,\eta,J)} M ( κ , η , J ) Δ t / Δ x 2 \Delta t/\Delta x^2 Δ t /Δ x 2 Δ x → 0 \Delta x\to0 Δ x → 0 Δ t \Delta t Δ t
We make use of the eigenvalues of two similar matrices being invariant. For square matrices A A A B B B P P P B = P − 1 ⋅ A ⋅ P B=P^{-1}\cdot A\cdot P B = P − 1 ⋅ A ⋅ P A A A B B B [35 chapter 6, theorem 12] . By choosing P = diag ( 1 , ε i n t − 1 / 2 , ε i n t − 1 / 2 ) P=\operatorname{diag}(1,\varepsilon_{\mathrm{int}}^{-1/2},\varepsilon_{\mathrm{int}}^{-1/2}) P = diag ( 1 , ε int − 1/2 , ε int − 1/2 ) κ J , ∧ \kappa_{J,\wedge} κ J , ∧ M ~ ( κ , η , J ) = P − 1 ⋅ M ( κ , η , J ) ⋅ P \tilde{M}^{(\kappa,\eta,J)}=P^{-1}\cdot M^{(\kappa,\eta,J)}\cdot P M ~ ( κ , η , J ) = P − 1 ⋅ M ( κ , η , J ) ⋅ P Δ κ J \Delta\kappa_J Δ κ J
We now bound the real parts of the eigenvalues of M ~ ( κ , η , J ) \tilde{M}^{(\kappa,\eta,J)} M ~ ( κ , η , J ) [36 ] ,
H M ~ = 1 2 ( M ~ ( κ , η , J ) + M ~ ( κ , η , J ) † ) . (64) H_{\tilde{M}}=\frac{1}{2}\left(\tilde{M}^{(\kappa,\eta,J)}+\tilde{M}^{(\kappa,\eta,J)\dagger}\right). \tag*{(64)} H M ~ = 2 1 ( M ~ ( κ , η , J ) + M ~ ( κ , η , J ) † ) . (64) For a unit eigenvector x ⃗ \vec{x} x λ \lambda λ λ = x ⃗ † M ~ ( κ , η , J ) x ⃗ \lambda=\vec{x}^{\dagger}\tilde{M}^{(\kappa,\eta,J)}\vec{x} λ = x † M ~ ( κ , η , J ) x Re ( λ ) = x ⃗ † H M ~ x ⃗ ∈ [ λ min ( H M ~ ) , λ max ( H M ~ ) ] \operatorname{Re}(\lambda)=\vec{x}^{\dagger}H_{\tilde{M}}\vec{x}\in[\lambda_{\min}(H_{\tilde{M}}),\lambda_{\max}(H_{\tilde{M}})] Re ( λ ) = x † H M ~ x ∈ [ λ m i n ( H M ~ ) , λ m a x ( H M ~ )] H M ~ H_{\tilde{M}} H M ~ ∣ ε ~ i n t ∣ 2 + ε i n t ( ∣ B ~ y ∣ 2 + ∣ B ~ z ∣ 2 ) |\tilde{\varepsilon}_{\mathrm{int}}|^2+\varepsilon_{\mathrm{int}}(|\tilde{B}_y|^2+|\tilde{B}_z|^2) ∣ ε ~ int ∣ 2 + ε int ( ∣ B ~ y ∣ 2 + ∣ B ~ z ∣ 2 ) M ( κ , η , J ) M^{(\kappa,\eta,J)} M ( κ , η , J )
The Hermitian part of M ~ ( κ , η , J ) \tilde{M}^{(\kappa,\eta,J)} M ~ ( κ , η , J ) M T ( κ , η , J ) M_T^{(\kappa,\eta,J)} M T ( κ , η , J ) sin 2 ( ψ ) = 1 \sin^2(\psi)=1 sin 2 ( ψ ) = 1 § 12 . To eliminate the ϕ \phi ϕ Q Q Q
H F ~ , r o t = Q − 1 ⋅ H M ~ ⋅ Q = − 4 Δ t Δ x 2 ( ( γ − 1 ) ( Δ κ cos 2 ( θ ) + κ 0 ) κ J , w sin ( θ ) ε i n t 0 κ J , w sin ( θ ) ε i n t η + η A D 0 0 0 η + η A D cos 2 ( θ ) ) , (65) H_{\tilde{F},\mathrm{rot}}=Q^{-1}\cdot H_{\tilde{M}}\cdot Q=-\frac{4\Delta t}{\Delta x^2}
\begin{pmatrix}
(\gamma-1)(\Delta\kappa\cos^2(\theta)+\kappa_0) & \dfrac{\kappa_{J,w}\sin(\theta)}{\sqrt{\varepsilon_{\mathrm{int}}}} & 0 \\
\dfrac{\kappa_{J,w}\sin(\theta)}{\sqrt{\varepsilon_{\mathrm{int}}}} & \eta+\eta_{AD} & 0 \\
0 & 0 & \eta+\eta_{AD}\cos^2(\theta)
\end{pmatrix}, \tag*{(65)} H F ~ , rot = Q − 1 ⋅ H M ~ ⋅ Q = − Δ x 2 4Δ t ( γ − 1 ) ( Δ κ cos 2 ( θ ) + κ 0 ) ε int κ J , w sin ( θ ) 0 ε int κ J , w sin ( θ ) η + η A D 0 0 0 η + η A D cos 2 ( θ ) , (65) we retrieve our criterion by solving λ H M ~ , r o t , max < 0 \lambda_{H_{\tilde{M}},\mathrm{rot},\max}<0 λ H M ~ , rot , m a x < 0
λ H M ~ 1 = − 4 Δ t Δ x 2 ( η + η A D cos 2 ( θ ) ) , (66) \lambda_{H_{\tilde{M}1}}=-\frac{4\Delta t}{\Delta x^2}\left(\eta+\eta_{AD}\cos^2(\theta)\right), \tag*{(66)} λ H M ~ 1 = − Δ x 2 4Δ t ( η + η A D cos 2 ( θ ) ) , (66) λ H M ~ 2 , 3 = − 2 Δ t Δ x 2 ( ( γ − 1 ) Δ κ cos 2 ( θ ) + η + η A D + ( γ − 1 ) κ 0 ± ( ( γ − 1 ) κ 0 + ( γ − 1 ) Δ κ cos 2 ( θ ) − η − η A D ) 2 + 4 κ J , w 2 ε i n t sin 2 ( θ ) ) , (67) \lambda_{H_{\tilde{M}2,3}}=-2\frac{\Delta t}{\Delta x^2}\left((\gamma-1)\Delta\kappa\cos^2(\theta)+\eta+\eta_{AD}+(\gamma-1)\kappa_0\pm\sqrt{\left((\gamma-1)\kappa_0+(\gamma-1)\Delta\kappa\cos^2(\theta)-\eta-\eta_{AD}\right)^2+4\frac{\kappa_{J,w}^2}{\varepsilon_{\mathrm{int}}}\sin^2(\theta)}\right), \tag*{(67)} λ H M ~ 2 , 3 = − 2 Δ x 2 Δ t ( γ − 1 ) Δ κ cos 2 ( θ ) + η + η A D + ( γ − 1 ) κ 0 ± ( ( γ − 1 ) κ 0 + ( γ − 1 ) Δ κ cos 2 ( θ ) − η − η A D ) 2 + 4 ε int κ J , w 2 sin 2 ( θ ) , (67) with the negative sign for λ H M ~ 2 \lambda_{H_{\tilde{M}2}} λ H M ~ 2 λ H M ~ 3 \lambda_{H_{\tilde{M}3}} λ H M ~ 3 λ H M ~ 3 \lambda_{H_{\tilde{M}3}} λ H M ~ 3
( γ − 1 ) ( κ 0 + Δ κ cos 2 ( θ ) ) + η + η A D > ( ( γ − 1 ) κ 0 + ( γ − 1 ) Δ κ cos 2 ( θ ) − η − η A D ) 2 + 4 κ J , w 2 ε i n t sin 2 ( θ ) . (68) (\gamma-1)\left(\kappa_0+\Delta\kappa\cos^2(\theta)\right)+\eta+\eta_{AD}>\sqrt{\left((\gamma-1)\kappa_0+(\gamma-1)\Delta\kappa\cos^2(\theta)-\eta-\eta_{AD}\right)^2+4\frac{\kappa_{J,w}^2}{\varepsilon_{\mathrm{int}}}\sin^2(\theta)}. \tag*{(68)} ( γ − 1 ) ( κ 0 + Δ κ cos 2 ( θ ) ) + η + η A D > ( ( γ − 1 ) κ 0 + ( γ − 1 ) Δ κ cos 2 ( θ ) − η − η A D ) 2 + 4 ε int κ J , w 2 sin 2 ( θ ) . (68) This is linear in u u u u = 1 u=1 u = 1 ( γ − 1 ) ( κ 0 + Δ κ ) ( η + η A D ) ≥ 0 (\gamma-1)(\kappa_0+\Delta\kappa)(\eta+\eta_{AD})\geq0 ( γ − 1 ) ( κ 0 + Δ κ ) ( η + η A D ) ≥ 0 u = 0 u=0 u = 0
( γ − 1 ) κ 0 ( η + η A D ) > κ J , w 2 ε i n t . (69) (\gamma-1)\kappa_0(\eta+\eta_{AD})>\frac{\kappa_{J,w}^2}{\varepsilon_{\mathrm{int}}}. \tag*{(69)} ( γ − 1 ) κ 0 ( η + η A D ) > ε int κ J , w 2 . (69) Expanding to original coefficients gives
( γ − 1 ) ( ( 1 − x i o n ) κ n + x i o n κ ⊥ ) ( η + η A D ) > x i o n 2 κ J , Δ 2 ε i n t . (70) (\gamma-1)\left((1-x_{\mathrm{ion}})\kappa_n+x_{\mathrm{ion}}\kappa_\perp\right)(\eta+\eta_{AD})>\frac{x_{\mathrm{ion}}^2\kappa_{J,\Delta}^2}{\varepsilon_{\mathrm{int}}}. \tag*{(70)} ( γ − 1 ) ( ( 1 − x ion ) κ n + x ion κ ⊥ ) ( η + η A D ) > ε int x ion 2 κ J , Δ 2 . (70) This is a sufficient condition for well-posedness. When the Righi–Leduc, Hall, and Peltier/Seebeck terms are absent, M ~ ( κ , η , J ) \tilde{M}^{(\kappa,\eta,J)} M ~ ( κ , η , J ) H M ~ = M ~ ( κ , η , J ) H_{\tilde{M}}=\tilde{M}^{(\kappa,\eta,J)} H M ~ = M ~ ( κ , η , J ) § 15 we test this criterion against physically determined coefficients and find that it always holds.
We take a conservative worst-case bound for the timestep using the inverse sum of each subsystem’s timestep
Δ t κ , η , J = ( 1 Δ t κ , ∥ ⊥ + 1 Δ t κ , ∧ + 1 Δ t η , R K 3 + 1 Δ t T E ) − 1 . (71) \Delta t_{\kappa,\eta,J}=\left(\frac{1}{\Delta t_{\kappa,\parallel\perp}}+\frac{1}{\Delta t_{\kappa,\wedge}}+\frac{1}{\Delta t_{\eta,\mathrm{RK3}}}+\frac{1}{\Delta t_{\mathrm{TE}}}\right)^{-1}. \tag*{(71)} Δ t κ , η , J = ( Δ t κ , ∥⊥ 1 + Δ t κ , ∧ 1 + Δ t η , RK3 1 + Δ t TE 1 ) − 1 . (71) In Appendix § 15 we test this bound with RK3 time integration against physical diffusion coefficients.
Dissipative Thermo-Magnetic System
The Hermitian part of the scaled thermo-magnetic system keeps thermal conduction, magnetic diffusion and the Nernst/Ettingshausen coupling. After the scaling P P P H M ~ H_{\tilde{M}} H M ~ § 3.9 , so its stability parameters are the eigenvalues λ H M ~ 1 , 2 , 3 \lambda_{H_{\tilde{M}1,2,3}} λ H M ~ 1 , 2 , 3 λ H M ~ 2 = − 4 Δ t Δ x 2 λ T M , R , + \lambda_{H_{\tilde{M}2}}=-4\frac{\Delta t}{\Delta x^2}\lambda_{\mathrm{TM,R,+}} λ H M ~ 2 = − 4 Δ x 2 Δ t λ TM , R , +
λ T M , R , + = 1 2 [ ( γ − 1 ) ( κ 0 + Δ κ u ) + η + η A D ] + 1 4 [ ( γ − 1 ) ( κ 0 + Δ κ u ) − η − η A D ] 2 + κ J , w 2 ε i n t ( 1 − u ) . (72) \lambda_{\mathrm{TM,R,+}}=\frac{1}{2}\left[(\gamma-1)(\kappa_0+\Delta\kappa u)+\eta+\eta_{AD}\right]+\sqrt{\frac{1}{4}\left[(\gamma-1)(\kappa_0+\Delta\kappa u)-\eta-\eta_{AD}\right]^2+\frac{\kappa_{J,w}^2}{\varepsilon_{\mathrm{int}}}(1-u)}. \tag*{(72)} λ TM , R , + = 2 1 [ ( γ − 1 ) ( κ 0 + Δ κ u ) + η + η A D ] + 4 1 [ ( γ − 1 ) ( κ 0 + Δ κ u ) − η − η A D ] 2 + ε int κ J , w 2 ( 1 − u ) . (72) Setting d λ T M , R , + / d u = 0 d\lambda_{\mathrm{TM,R,+}}/du=0 d λ TM , R , + / d u = 0 u u u λ T M , R , + \lambda_{\mathrm{TM,R,+}} λ TM , R , + u u u λ T M , R , + ( 1 ) = max ( ( γ − 1 ) ( κ 0 + Δ κ ) , η + η A D ) \lambda_{\mathrm{TM,R,+}}(1)=\max\left((\gamma-1)(\kappa_0+\Delta\kappa),\eta+\eta_{AD}\right) λ TM , R , + ( 1 ) = max ( ( γ − 1 ) ( κ 0 + Δ κ ) , η + η A D )
λ T M , R , + ( 0 ) = 1 2 [ ( γ − 1 ) κ 0 + η + η A D ] + 1 4 [ ( γ − 1 ) κ 0 − η − η A D ] 2 + κ J , w 2 ε i n t . (73) \lambda_{\mathrm{TM,R,+}}(0)=\frac{1}{2}\left[(\gamma-1)\kappa_0+\eta+\eta_{AD}\right]+\sqrt{\frac{1}{4}\left[(\gamma-1)\kappa_0-\eta-\eta_{AD}\right]^2+\frac{\kappa_{J,w}^2}{\varepsilon_{\mathrm{int}}}}. \tag*{(73)} λ TM , R , + ( 0 ) = 2 1 [ ( γ − 1 ) κ 0 + η + η A D ] + 4 1 [ ( γ − 1 ) κ 0 − η − η A D ] 2 + ε int κ J , w 2 . (73) The dissipative thermo-magnetic timestep is
Δ t T M , R = C F L min ( Δ x ) 2 2 N D i m max ( λ T M , R , + ( 0 ) , λ T M , R , + ( 1 ) ) . (74) \Delta t_{\mathrm{TM,R}}=\frac{\mathrm{CFL}\min(\Delta x)^2}{2N_{\mathrm{Dim}}\max\left(\lambda_{\mathrm{TM,R,+}}(0),\lambda_{\mathrm{TM,R,+}}(1)\right)}. \tag*{(74)} Δ t TM , R = 2 N Dim max ( λ TM , R , + ( 0 ) , λ TM , R , + ( 1 ) ) CFL min ( Δ x ) 2 . (74) Non-Dissipative Thermo-Magnetic System
The skew-Hermitian part of the scaled thermo-magnetic system contains the Righi–Leduc, Hall, and Peltier/Seebeck terms. We first treat the Hall and Peltier/Seebeck terms, and combine them with the Righi–Leduc effect, which has a different Fourier factor, at the end. At the binding mode, scaling by P \mathbf{P} P
M ~ ( H J ) = − 4 Δ t Δ x 2 cos θ ( 0 − Δ κ J sin θ sin ϕ / ε i n t Δ κ J sin θ cos ϕ / ε i n t Δ κ J sin θ sin ϕ / ε i n t 0 η H − Δ κ J sin θ cos ϕ / ε i n t − η H 0 ) . (75) \widetilde{\mathbf{M}}^{(\mathrm{HJ})}=-4\frac{\Delta t}{\Delta x^{2}}\cos\theta\begin{pmatrix}0 & -\Delta\kappa_J\sin\theta\sin\phi/\sqrt{\varepsilon_{\mathrm{int}}} & \Delta\kappa_J\sin\theta\cos\phi/\sqrt{\varepsilon_{\mathrm{int}}} \\ \Delta\kappa_J\sin\theta\sin\phi/\sqrt{\varepsilon_{\mathrm{int}}} & 0 & \eta_H \\ -\Delta\kappa_J\sin\theta\cos\phi/\sqrt{\varepsilon_{\mathrm{int}}} & -\eta_H & 0\end{pmatrix}. \tag*{(75)} M ( HJ ) = − 4 Δ x 2 Δ t cos θ 0 Δ κ J sin θ sin ϕ / ε int − Δ κ J sin θ cos ϕ / ε int − Δ κ J sin θ sin ϕ / ε int 0 − η H Δ κ J sin θ cos ϕ / ε int η H 0 . (75) This is real and antisymmetric, so its eigenvalues are 0 and ± − 1 \pm\sqrt{-1} ± − 1 sin ϕ \sin\phi sin ϕ cos ϕ \cos\phi cos ϕ sin 2 ϕ + cos 2 ϕ \sin^{2}\phi+\cos^{2}\phi sin 2 ϕ + cos 2 ϕ ϕ \phi ϕ − 4 Δ t / Δ x 2 -4\Delta t/\Delta x^{2} − 4Δ t /Δ x 2
λ H J , 0 = 0 , λ H J , ± = ± − 1 u η H 2 + ( 1 − u ) Δ κ J 2 ε i n t . (76) \lambda_{\mathrm{HJ},0}=0,\qquad\lambda_{\mathrm{HJ},\pm}=\pm\sqrt{-1}\sqrt{u}\sqrt{\eta_H^{2}+(1-u)\frac{\Delta\kappa_J^{2}}{\varepsilon_{\mathrm{int}}}}. \tag*{(76)} λ HJ , 0 = 0 , λ HJ , ± = ± − 1 u η H 2 + ( 1 − u ) ε int Δ κ J 2 . (76) The stability parameter Z H J = − 4 Δ t λ H J , ± / Δ x 2 Z_{\mathrm{HJ}}=-4\Delta t\lambda_{\mathrm{HJ},\pm}/\Delta x^{2} Z HJ = − 4Δ t λ HJ , ± /Δ x 2 r I > 0 r_{\mathrm{I}}>0 r I > 0 ∣ λ H J ( u ) ∣ = u η H 2 + u ( 1 − u ) Δ κ J 2 / ε i n t |\lambda_{\mathrm{HJ}}(u)|=\sqrt{u\eta_H^{2}+u(1-u)\Delta\kappa_J^{2}/\varepsilon_{\mathrm{int}}} ∣ λ HJ ( u ) ∣ = u η H 2 + u ( 1 − u ) Δ κ J 2 / ε int u u u η H 2 + ( 1 − 2 u ) Δ κ J 2 / ε i n t \eta_H^{2}+(1-2u)\Delta\kappa_J^{2}/\varepsilon_{\mathrm{int}} η H 2 + ( 1 − 2 u ) Δ κ J 2 / ε int u ∗ = 1 / 2 + η H 2 ε i n t / ( 2 Δ κ J 2 ) u^{*}=1/2+\eta_H^{2}\varepsilon_{\mathrm{int}}/(2\Delta\kappa_J^{2}) u ∗ = 1/2 + η H 2 ε int / ( 2Δ κ J 2 ) ∣ λ H J ( 0 ) ∣ = 0 |\lambda_{\mathrm{HJ}}(0)|=0 ∣ λ HJ ( 0 ) ∣ = 0 ∣ λ H J ( 1 ) ∣ = η H |\lambda_{\mathrm{HJ}}(1)|=\eta_H ∣ λ HJ ( 1 ) ∣ = η H
∣ λ η , J , H ∣ = max ( η H , 1 2 1 [ 0 , 1 ] ( u ∗ ) ( η H 2 ε i n t ∣ Δ κ J ∣ + ∣ Δ κ J ∣ ε i n t ) ) , (77) |\lambda_{\eta,J,H}|=\max\left(\eta_H,\frac{1}{2}\mathbf{1}_{[0,1]}(u^{*})\left(\frac{\eta_H^{2}\sqrt{\varepsilon_{\mathrm{int}}}}{|\Delta\kappa_J|}+\frac{|\Delta\kappa_J|}{\sqrt{\varepsilon_{\mathrm{int}}}}\right)\right), \tag*{(77)} ∣ λ η , J , H ∣ = max ( η H , 2 1 1 [ 0 , 1 ] ( u ∗ ) ( ∣Δ κ J ∣ η H 2 ε int + ε int ∣Δ κ J ∣ ) ) , (77) where the second argument is ∣ λ H J ( u ∗ ) ∣ |\lambda_{\mathrm{HJ}}(u^{*})| ∣ λ HJ ( u ∗ ) ∣ ∣ Δ κ J ∣ / ( 2 ε i n t ) |\Delta\kappa_J|/(2\sqrt{\varepsilon_{\mathrm{int}}}) ∣Δ κ J ∣/ ( 2 ε int ) η H \eta_H η H
Δ t H J = C F L min ( Δ x ) 2 4 3 N D i m ∣ λ H J , max ∣ , Δ t T M , I = ( 1 Δ t κ , ∧ + 1 Δ t H J ) − 1 . (78) \Delta t_{\mathrm{HJ}}=\mathrm{CFL}\frac{\min(\Delta x)^{2}}{\frac{4}{\sqrt{3}}N_{\mathrm{Dim}}|\lambda_{\mathrm{HJ},\max}|},\qquad\Delta t_{\mathrm{TM,I}}=\left(\frac{1}{\Delta t_{\kappa,\wedge}}+\frac{1}{\Delta t_{\mathrm{HJ}}}\right)^{-1}. \tag*{(78)} Δ t HJ = CFL 3 4 N Dim ∣ λ HJ , m a x ∣ min ( Δ x ) 2 , Δ t TM , I = ( Δ t κ , ∧ 1 + Δ t HJ 1 ) − 1 . (78) Relationship with Ideal Timestep
Linearising the ideal induction equation with the Hall term gives the whistler dispersion relation, whose phase speed at the grid scale can exceed the fast-magnetosonic speed c f c_f c f c w ( k ) = 1 2 η H k + 1 4 η H 2 k 2 + c A 2 ≤ c f + η H k c_w(k)=\frac{1}{2}\eta_H k+\sqrt{\frac{1}{4}\eta_H^{2}k^{2}+c_A^{2}}\leq c_f+\eta_H k c w ( k ) = 2 1 η H k + 4 1 η H 2 k 2 + c A 2 ≤ c f + η H k Δ t i d e a l \Delta t_{\mathrm{ideal}} Δ t ideal Δ t = min ( Δ t i d e a l , Δ t d i f f ) \Delta t=\min(\Delta t_{\mathrm{ideal}},\Delta t_{\mathrm{diff}}) Δ t = min ( Δ t ideal , Δ t diff )
After linearisation the viscous operator acts only on v ⃗ \vec{v} v ( ε i n t , B ⃗ ) (\varepsilon_{\mathrm{int}},\vec{B}) ( ε int , B )
When STS is enabled, the dissipative parts are super-time-stepped with the parabolic limit Δ t p a r = min ( Δ t v , ∥ , ⊥ , Δ t T M , R ) \Delta t_{\mathrm{par}}=\min(\Delta t_{\mathrm{v},\parallel,\perp},\Delta t_{\mathrm{TM,R}}) Δ t par = min ( Δ t v , ∥ , ⊥ , Δ t TM , R )
Δ t h y p = ( 1 / Δ t i d e a l + 1 / Δ t κ , ∧ + max ( 1 / Δ t v , ∧ , 1 / Δ t H J ) ) − 1 . (79) \Delta t_{\mathrm{hyp}}=\left(1/\Delta t_{\mathrm{ideal}}+1/\Delta t_{\kappa,\wedge}+\max(1/\Delta t_{\mathrm{v},\wedge},1/\Delta t_{\mathrm{HJ}})\right)^{-1}. \tag*{(79)} Δ t hyp = ( 1/Δ t ideal + 1/Δ t κ , ∧ + max ( 1/Δ t v , ∧ , 1/Δ t HJ ) ) − 1 . (79) When STS is disabled we take the inverse sum of the ideal timestep, and the most restrictive unsplit timestep
Δ t u n s p l i t = ( 1 / Δ t i d e a l + max ( 1 / Δ t ζ , v , ∥ , ⊥ + 1 / Δ t v , ∧ , 1 / Δ t κ , ∥ , ⊥ + 1 / Δ t κ , ∧ + 1 / Δ t η + 1 / Δ t E ) ) − 1 . (80) \Delta t_{\mathrm{unsplit}}=\left(1/\Delta t_{\mathrm{ideal}}+\max(1/\Delta t_{\zeta,\mathrm{v},\parallel,\perp}+1/\Delta t_{\mathrm{v},\wedge},1/\Delta t_{\kappa,\parallel,\perp}+1/\Delta t_{\kappa,\wedge}+1/\Delta t_{\eta}+1/\Delta t_E)\right)^{-1}. \tag*{(80)} Δ t unsplit = ( 1/Δ t ideal + max ( 1/Δ t ζ , v , ∥ , ⊥ + 1/Δ t v , ∧ , 1/Δ t κ , ∥ , ⊥ + 1/Δ t κ , ∧ + 1/Δ t η + 1/Δ t E ) ) − 1 . (80) The ideal timestep is given by the standard result
Δ t i d e a l = C F L min i ∈ x , y , z Δ i N d i m ( v i + c f ) . (81) \Delta t_{\mathrm{ideal}} = \mathrm{CFL}\min_{i\in x,y,z}\frac{\Delta_i}{N_{\mathrm{dim}}(v_i+c_f)}. \tag*{(81)} Δ t ideal = CFL i ∈ x , y , z min N dim ( v i + c f ) Δ i . (81) HLLE-Mod Riemann Solver
The Hall effect suffers from non-linear grid-scale instabilities even at RK3 time integration [12 ] . One solution to this is Hall-HLLE-Mod, whereby the fastest whistler wave (which scales ∝ 1 / Δ x \propto1/\Delta x ∝ 1/Δ x [17 ] . We therefore opt to implement the same logic for every other non-dissipative term and its appropriate variable in the HLLE Riemann solver.
Throughout we write u = cos 2 θ u=\cos^2\theta u = cos 2 θ E = η H 2 k x 2 E=\eta_H^2 k_x^2 E = η H 2 k x 2 D = ( Δ κ J k x ) 2 ε i n t = ( γ − 1 ) ( Δ κ J k x ) 2 ρ 0 c i s o 2 D=\frac{(\Delta\kappa_J k_x)^2}{\varepsilon_{\mathrm{int}}}=\frac{(\gamma-1)(\Delta\kappa_J k_x)^2}{\rho_0c_{\mathrm{iso}}^2} D = ε int ( Δ κ J k x ) 2 = ρ 0 c iso 2 ( γ − 1 ) ( Δ κ J k x ) 2 c i s o = c s / γ c_{\mathrm{iso}}=c_s/\sqrt{\gamma} c iso = c s / γ c f 2 = c a 2 + c s 2 c_f^2=c_a^2+c_s^2 c f 2 = c a 2 + c s 2 − λ 2 / k x 2 = Ω -\lambda^2/k_x^2=\Omega − λ 2 / k x 2 = Ω Ω \sqrt{\Omega} Ω Ω \Omega Ω k x = 1 / Δ x k_x=1/\Delta x k x = 1/Δ x [16 ] , which [12 ] found sufficient to suppress the Hall instability. For each derivation we convert to the primitive variables W ⃗ = ( ρ , v x , v y , v z , ε i n t , B x , B y , B z ) \vec{W}=(\rho,v_x,v_y,v_z,\varepsilon_{\mathrm{int}},B_x,B_y,B_z) W = ( ρ , v x , v y , v z , ε int , B x , B y , B z ) ( ρ 0 , 0 ⃗ , ε 0 , B ⃗ ) (\rho_0,\vec{0},\varepsilon_0,\vec{B}) ( ρ 0 , 0 , ε 0 , B ) W ⃗ \vec{W} W δ W ⃗ = δ W ⃗ ~ exp ( − 1 ( k x x − ω t ) ) \delta\vec{W}=\delta\widetilde{\vec{W}}\exp(\sqrt{-1}(k_xx-\omega t)) δ W = δ W exp ( − 1 ( k x x − ω t )) u u u F ( Ω ) = Ω 3 + A 2 Ω 2 + A 1 Ω + A 0 F(\Omega)=\Omega^3+A_2\Omega^2+A_1\Omega+A_0 F ( Ω ) = Ω 3 + A 2 Ω 2 + A 1 Ω + A 0 0 ≤ r 1 ≤ r 2 ≤ r 3 0\le r_1\le r_2\le r_3 0 ≤ r 1 ≤ r 2 ≤ r 3 Λ \Lambda Λ F ( Λ ) ≥ 0 F(\Lambda)\ge0 F ( Λ ) ≥ 0 2 Λ ≥ r 1 + r 2 + r 3 = − A 2 2\Lambda\ge r_1+r_2+r_3=-A_2 2Λ ≥ r 1 + r 2 + r 3 = − A 2 r 3 ≤ Λ r_3\le\Lambda r 3 ≤ Λ F ( Λ ) ≥ 0 F(\Lambda)\ge0 F ( Λ ) ≥ 0 Λ < r 3 \Lambda<r_3 Λ < r 3 r 1 ≤ Λ ≤ r 2 r_1\le\Lambda\le r_2 r 1 ≤ Λ ≤ r 2 r 2 + r 3 > 2 Λ ≥ r 1 + r 2 + r 3 ≥ r 2 + r 3 r_2+r_3>2\Lambda\ge r_1+r_2+r_3\ge r_2+r_3 r 2 + r 3 > 2Λ ≥ r 1 + r 2 + r 3 ≥ r 2 + r 3
Hall Effect (Whistler Waves)
First we derive the fastest whistler wave phase speed. We consider Eq. (1) with only the ideal and Hall terms
∂ t δ W ⃗ = M ( I , η H ) δ ∂ x x W ⃗ . (82) \partial_t\delta\vec{W}=M^{(I,\eta_H)}\delta\partial_{xx}\vec{W}. \tag*{(82)} ∂ t δ W = M ( I , η H ) δ ∂ xx W . (82) The full matrix is given in Appendix § 13 (the B x B_x B x λ I , η H \lambda_{I,\eta_H} λ I , η H λ I , η H = 0 \lambda_{I,\eta_H}=0 λ I , η H = 0 λ I , η H \lambda_{I,\eta_H} λ I , η H Ω \Omega Ω
F ( Ω , u ) = − c a 4 c s 2 u 2 + u ( − Ω 2 ( E + c a 2 ) + Ω ( E c s 2 + c a 4 + 2 c a 2 c s 2 ) ) + Ω 2 ( Ω − c a 2 − c s 2 ) . (83) F(\Omega,u)=-c_a^4c_s^2u^2+u\left(-\Omega^2(E+c_a^2)+\Omega(Ec_s^2+c_a^4+2c_a^2c_s^2)\right)+\Omega^2(\Omega-c_a^2-c_s^2). \tag*{(83)} F ( Ω , u ) = − c a 4 c s 2 u 2 + u ( − Ω 2 ( E + c a 2 ) + Ω ( E c s 2 + c a 4 + 2 c a 2 c s 2 ) ) + Ω 2 ( Ω − c a 2 − c s 2 ) . (83) Here, the dependence ϕ \phi ϕ b y 2 + b z 2 = 1 − u b_y^2+b_z^2=1-u b y 2 + b z 2 = 1 − u ∂ 2 F / ∂ u 2 = − 2 c s 2 c a 4 < 0 \partial^2F/\partial u^2=-2c_s^2c_a^4<0 ∂ 2 F / ∂ u 2 = − 2 c s 2 c a 4 < 0 F ( Ω , u ) F(\Omega,u) F ( Ω , u ) u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] Ω \Omega Ω u u u
F ( Ω , u ) = ( 1 − u ) F ( Ω , 0 ) + u F ( Ω , 1 ) + c a 4 c s 2 u ( 1 − u ) . (84) F(\Omega,u)=(1-u)F(\Omega,0)+uF(\Omega,1)+c_a^4c_s^2u(1-u). \tag*{(84)} F ( Ω , u ) = ( 1 − u ) F ( Ω , 0 ) + u F ( Ω , 1 ) + c a 4 c s 2 u ( 1 − u ) . (84) Let Ω 3 ( u ) \Omega_3(u) Ω 3 ( u ) F ( Ω , u ) F(\Omega,u) F ( Ω , u ) Ω 3 , max = max ( Ω 3 ( 0 ) , Ω 3 ( 1 ) ) \Omega_{3,\max}=\max(\Omega_3(0),\Omega_3(1)) Ω 3 , m a x = max ( Ω 3 ( 0 ) , Ω 3 ( 1 )) F F F Ω \Omega Ω Ω ≥ Ω 3 , max \Omega\ge\Omega_{3,\max} Ω ≥ Ω 3 , m a x F ( Ω , 0 ) ≥ 0 F(\Omega,0)\ge0 F ( Ω , 0 ) ≥ 0 F ( Ω , 1 ) ≥ 0 F(\Omega,1)\ge0 F ( Ω , 1 ) ≥ 0 (84) then gives F ( Ω , u ) > 0 F(\Omega,u)>0 F ( Ω , u ) > 0 u ∈ ( 0 , 1 ) u\in(0,1) u ∈ ( 0 , 1 ) F ( Ω , u ) F(\Omega,u) F ( Ω , u ) [ Ω 3 , max , ∞ ) [\Omega_{3,\max},\infty) [ Ω 3 , m a x , ∞ ) u u u max u ∈ [ 0 , 1 ] Ω 3 ( u ) = max ( Ω 3 ( 0 ) , Ω 3 ( 1 ) ) \max_{u\in[0,1]}\Omega_3(u)=\max(\Omega_3(0),\Omega_3(1)) max u ∈ [ 0 , 1 ] Ω 3 ( u ) = max ( Ω 3 ( 0 ) , Ω 3 ( 1 ))
The square roots of the roots of F F F u = 0 u=0 u = 0 c n H , 0 = 0 c_{\mathrm{nH},0}=0 c nH , 0 = 0 c n H , 1 = c s 2 + c a 2 = c f c_{\mathrm{nH},1}=\sqrt{c_s^2+c_a^2}=c_f c nH , 1 = c s 2 + c a 2 = c f b ^ ⊥ x ^ \hat{b}\perp\hat{x} b ^ ⊥ x ^ u = 1 u=1 u = 1
c n H , 2 = c s , c n H , 3 , 4 = ∓ ∣ η H ∣ 2 Δ x + η H 2 / Δ x 2 + 4 c a 2 2 . (85) c_{\mathrm{nH},2}=c_s,\qquad c_{\mathrm{nH},3,4}=\mp\frac{|\eta_H|}{2\Delta x}+\frac{\sqrt{\eta_H^2/\Delta x^2+4c_a^2}}{2}. \tag*{(85)} c nH , 2 = c s , c nH , 3 , 4 = ∓ 2Δ x ∣ η H ∣ + 2 η H 2 /Δ x 2 + 4 c a 2 . (85) These are the adiabatic sound speed, ion-cyclotron speed, and whistler phase speed, with c n H , 3 c n H , 4 = c a 2 c_{\mathrm{nH},3}c_{\mathrm{nH},4}=c_a^2 c nH , 3 c nH , 4 = c a 2 c w ≡ c n H , 4 c_w\equiv c_{\mathrm{nH},4} c w ≡ c nH , 4 c f ≥ c s c_f\ge c_s c f ≥ c s max u Ω 3 ( u ) = max ( c f 2 , c w 2 ) \max_u\Omega_3(u)=\max(c_f^2,c_w^2) max u Ω 3 ( u ) = max ( c f 2 , c w 2 ) max ( c f , c w ) \max(c_f,c_w) max ( c f , c w )
Peltier/Seebeck (Thermoelectric Wave)
We now consider the Peltier/Seebeck effects with ideal terms to retrieve the thermoelectric phase speed. We take eq:1 with only the ideal and Peltier/Seebeck terms (q ⃗ J , ∥ \vec{q}_{J,\parallel} q J , ∥ q ⃗ J , ⊥ \vec{q}_{J,\perp} q J , ⊥ F ⃗ T , ∥ \vec{F}_{T,\parallel} F T , ∥ F ⃗ T , ⊥ \vec{F}_{T,\perp} F T , ⊥
∂ t δ W ⃗ = M ( I , J ) δ ∂ x x W ⃗ . (86) \partial_t \delta\vec{W} = M^{(\mathrm{I,J})}\delta\partial_{xx}\vec{W}. \tag*{(86)} ∂ t δ W = M ( I , J ) δ ∂ xx W . (86) The entries for matrix M ( I , J ) M^{(\mathrm{I,J})} M ( I , J )
The characteristic polynomial is
F ( Ω , u ) = ( Ω − c a 2 u ) [ Ω 2 − c f 2 Ω + c a 2 c s 2 u ] − D u ( 1 − u ) [ Ω 2 − ( c a 2 + c i s o 2 ) Ω + c a 2 c i s o 2 u ] . (87) F(\Omega,u) = (\Omega-c_a^2u)\left[\Omega^2-c_f^2\Omega+c_a^2c_s^2u\right]-Du(1-u)\left[\Omega^2-(c_a^2+c_{\mathrm{iso}}^2)\Omega+c_a^2c_{\mathrm{iso}}^2u\right]. \tag*{(87)} F ( Ω , u ) = ( Ω − c a 2 u ) [ Ω 2 − c f 2 Ω + c a 2 c s 2 u ] − D u ( 1 − u ) [ Ω 2 − ( c a 2 + c iso 2 ) Ω + c a 2 c iso 2 u ] . (87) The first term describes ideal MHD and is adiabatic; substituting D = 0 D=0 D = 0 u = 0 u=0 u = 0 u = 1 u=1 u = 1
This cubic is solvable only through the general cubic formula, and maximising over u u u c f c_f c f
The roots of F F F S S S
δ W ⃗ † S δ W ⃗ = ρ 0 ∣ δ v ⃗ ∣ 2 + ∣ δ B ⃗ ∣ 2 + γ − 1 γ ∣ δ ε i n t ∣ 2 ε 0 + ε 0 γ ∣ δ ε i n t ε 0 − γ δ ρ ρ 0 ∣ 2 , (88) \delta\vec{W}^{\dagger}S\delta\vec{W}=\rho_0|\delta\vec{v}|^2+|\delta\vec{B}|^2+\frac{\gamma-1}{\gamma}\frac{|\delta\varepsilon_{\mathrm{int}}|^2}{\varepsilon_0}+\frac{\varepsilon_0}{\gamma}\left|\frac{\delta\varepsilon_{\mathrm{int}}}{\varepsilon_0}-\gamma\frac{\delta\rho}{\rho_0}\right|^2, \tag*{(88)} δ W † S δ W = ρ 0 ∣ δ v ∣ 2 + ∣ δ B ∣ 2 + γ γ − 1 ε 0 ∣ δ ε int ∣ 2 + γ ε 0 ε 0 δ ε int − γ ρ 0 δ ρ 2 , (88) the linearised wave energy (kinetic, magnetic, compressional and entropy parts) is positive definite for γ > 1 \gamma>1 γ > 1 M ( I , J ) M^{(\mathrm{I,J})} M ( I , J ) S M + M † S = 0 SM+M^{\dagger}S=0 S M + M † S = 0 S ε ε M ε B y ( J ) = − M ε B y ( J ) S_{\varepsilon\varepsilon}M_{\varepsilon B_y}^{(\mathrm{J})}=-M_{\varepsilon B_y}^{(\mathrm{J})} S εε M ε B y ( J ) = − M ε B y ( J ) S ρ ε M ρ B y ( J ) = − M ρ B y ( J ) S_{\rho\varepsilon}M_{\rho B_y}^{(\mathrm{J})}=-M_{\rho B_y}^{(\mathrm{J})} S ρε M ρ B y ( J ) = − M ρ B y ( J ) B z B_z B z S 1 / 2 M S − 1 / 2 S^{1/2}MS^{-1/2} S 1/2 M S − 1/2 ω \omega ω F F F
Before bounding the full cubic we isolate the thermoelectric contribution alone. Setting c a = c s = 0 c_a=c_s=0 c a = c s = 0
F ( Ω , u ) ∣ c a = c s = 0 = Ω 2 ( Ω − D u ( 1 − u ) ) . (89) F(\Omega,u)\big|_{c_a=c_s=0}=\Omega^2\left(\Omega-Du(1-u)\right). \tag*{(89)} F ( Ω , u ) c a = c s = 0 = Ω 2 ( Ω − D u ( 1 − u ) ) . (89) The Peltier/Seebeck thermoelectric phase speed is
c κ J = D u ( 1 − u ) ≤ D / 2. (90) c_{\kappa_J}=\sqrt{Du(1-u)}\le\sqrt{D}/2. \tag*{(90)} c κ J = D u ( 1 − u ) ≤ D /2. (90) This is maximised at u = 1 / 2 u=1/2 u = 1/2 u ( 1 − u ) = 1 / 4 u(1-u)=1/4 u ( 1 − u ) = 1/4 c κ J , max = ∣ Δ κ J ∣ k x / ( 2 ε i n t ) c_{\kappa_J,\max}=|\Delta\kappa_J|k_x/(2\sqrt{\varepsilon_{\mathrm{int}}}) c κ J , m a x = ∣Δ κ J ∣ k x / ( 2 ε int ) u = 0 u=0 u = 0 c f c_f c f u ( 1 − u ) u(1-u) u ( 1 − u )
Λ ( u ) = c f 2 + D u ( 1 − u ) , c T E ( u ) = Λ ( u ) , (91) \Lambda(u)=c_f^2+Du(1-u),\qquad c_{\mathrm{TE}}(u)=\sqrt{\Lambda(u)}, \tag*{(91)} Λ ( u ) = c f 2 + D u ( 1 − u ) , c TE ( u ) = Λ ( u ) , (91) as the candidate bound. We show that no root of F F F Λ ( u ) \Lambda(u) Λ ( u )
The Peltier/Seebeck cubic has − A 2 = c f 2 + c a 2 u + D u ( 1 − u ) ≤ 2 Λ ( u ) -A_2=c_f^2+c_a^2u+Du(1-u)\le2\Lambda(u) − A 2 = c f 2 + c a 2 u + D u ( 1 − u ) ≤ 2Λ ( u ) c a 2 u ≤ c f 2 c_a^2u\le c_f^2 c a 2 u ≤ c f 2
F ( Λ ) = D u ( 1 − u ) Λ [ c a 2 ( 1 − u ) + c i s o 2 ] + c a 2 c s 2 u [ c f 2 − c a 2 u + D u ( 1 − u ) γ − 1 γ ] ≥ 0 , (92) F(\Lambda)=Du(1-u)\Lambda\left[c_a^2(1-u)+c_{\mathrm{iso}}^2\right]+c_a^2c_s^2u\left[c_f^2-c_a^2u+Du(1-u)\frac{\gamma-1}{\gamma}\right]\ge0, \tag*{(92)} F ( Λ ) = D u ( 1 − u ) Λ [ c a 2 ( 1 − u ) + c iso 2 ] + c a 2 c s 2 u [ c f 2 − c a 2 u + D u ( 1 − u ) γ γ − 1 ] ≥ 0 , (92) which holds term by term, so no root of F F F Λ ( u ) \Lambda(u) Λ ( u ) u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ]
Maximising u ( 1 − u ) = 1 / 4 u(1-u)=1/4 u ( 1 − u ) = 1/4 k x = 1 / Δ x k_x=1/\Delta x k x = 1/Δ x
c T E = c f 2 + Δ κ J 2 4 ε i n t Δ x 2 . (93) c_{\mathrm{TE}}=\sqrt{c_f^2+\frac{\Delta\kappa_J^2}{4\varepsilon_{\mathrm{int}}\Delta x^2}}. \tag*{(93)} c TE = c f 2 + 4 ε int Δ x 2 Δ κ J 2 . (93) Since c T E ≥ c f c_{\mathrm{TE}}\ge c_f c TE ≥ c f Δ κ J = 0 \Delta\kappa_J=0 Δ κ J = 0 max ( c f , c T E ) = c T E \max(c_f,c_{\mathrm{TE}})=c_{\mathrm{TE}} max ( c f , c TE ) = c TE [17 ] , we apply it to the internal energy and magnetic field, the variables on which the Peltier/Seebeck terms act directly.
Combined Hall and Peltier/Seebeck Wave
Taking Eq. (1) now with the ideal, Hall, and Peltier/Seebeck terms, with a 1D gradient, converting to primitives and linearising, M ( I , η H , J ) = M ( I ) + M ( η H ) + M ( J ) M^{(\mathrm{I},\eta_H,J)} = M^{(\mathrm{I})} + M^{(\eta_H)} + M^{(J)} M ( I , η H , J ) = M ( I ) + M ( η H ) + M ( J ) F ( Ω , u ) = Ω 3 + A 2 Ω 2 + A 1 Ω + A 0 F(\Omega,u)=\Omega^3+A_2\Omega^2+A_1\Omega+A_0 F ( Ω , u ) = Ω 3 + A 2 Ω 2 + A 1 Ω + A 0
A 2 = − [ c f 2 + u ( c a 2 + E ) + D u ( 1 − u ) ] ≤ 0 , (94) A_2=-\left[c_f^2+u\left(c_a^2+E\right)+Du(1-u)\right]\leq0, \tag*{(94)} A 2 = − [ c f 2 + u ( c a 2 + E ) + D u ( 1 − u ) ] ≤ 0 , (94) A 1 = c a 2 u ( c a 2 + 2 c s 2 ) + E c s 2 u + u ( 1 − u ) [ D ( c a 2 + c i s o 2 ) + 2 s c a c i s o ( γ − 1 ) D E ] ≥ 0 , (95) A_1=c_a^2u\left(c_a^2+2c_s^2\right)+Ec_s^2u+u(1-u)\left[D\left(c_a^2+c_{\mathrm{iso}}^2\right)+2sc_ac_{\mathrm{iso}}\sqrt{(\gamma-1)DE}\right]\geq0, \tag*{(95)} A 1 = c a 2 u ( c a 2 + 2 c s 2 ) + E c s 2 u + u ( 1 − u ) [ D ( c a 2 + c iso 2 ) + 2 s c a c iso ( γ − 1 ) D E ] ≥ 0 , (95) A 0 = − c a 2 c i s o 2 u 2 [ γ c a 2 + D ( 1 − u ) ] ≤ 0 , (96) A_0=-c_a^2c_{\mathrm{iso}}^2u^2\left[\gamma c_a^2+D(1-u)\right]\leq0, \tag*{(96)} A 0 = − c a 2 c iso 2 u 2 [ γ c a 2 + D ( 1 − u ) ] ≤ 0 , (96) where s = − sgn ( η H Δ κ J ) = ± 1 s=-\operatorname{sgn}(\eta_H\Delta\kappa_J)=\pm1 s = − sgn ( η H Δ κ J ) = ± 1 η H \eta_H η H Δ κ J \Delta\kappa_J Δ κ J (88) . Collecting terms,
F ( Ω , u ) = ( Ω − c a 2 u ) [ Ω 2 − c f 2 Ω + c a 2 c s 2 u ] − E u Ω [ Ω − c s 2 ] − D u ( 1 − u ) [ Ω 2 − ( c a 2 + c i s o 2 ) Ω + c a 2 c i s o 2 u ] + 2 s c a c i s o ( γ − 1 ) D E u ( 1 − u ) Ω . (97) \begin{aligned}
F(\Omega,u)={}&(\Omega-c_a^2u)\left[\Omega^2-c_f^2\Omega+c_a^2c_s^2u\right]\\
&-Eu\Omega\left[\Omega-c_s^2\right]\\
&-Du(1-u)\left[\Omega^2-\left(c_a^2+c_{\mathrm{iso}}^2\right)\Omega+c_a^2c_{\mathrm{iso}}^2u\right]\\
&+2sc_ac_{\mathrm{iso}}\sqrt{(\gamma-1)DE}u(1-u)\Omega. \tag*{(97)}
\end{aligned} F ( Ω , u ) = ( Ω − c a 2 u ) [ Ω 2 − c f 2 Ω + c a 2 c s 2 u ] − E u Ω [ Ω − c s 2 ] − D u ( 1 − u ) [ Ω 2 − ( c a 2 + c iso 2 ) Ω + c a 2 c iso 2 u ] + 2 s c a c iso ( γ − 1 ) D E u ( 1 − u ) Ω. (97) The term carrying neither D D D E E E D D D E E E D E \sqrt{DE} D E
F ( Ω , u ) = G ( Ω , u ) − u ( 1 − u ) T ( Ω , u ) , (98) F(\Omega,u)=G(\Omega,u)-u(1-u)T(\Omega,u), \tag*{(98)} F ( Ω , u ) = G ( Ω , u ) − u ( 1 − u ) T ( Ω , u ) , (98) where G G G (83) and T T T D D D D D D u ( 1 − u ) u(1-u) u ( 1 − u ) u = 0 u=0 u = 0 u = 1 u=1 u = 1 u = 1 / 2 u=1/2 u = 1/2
This polynomial is solvable only through the general cubic formula, and maximising its largest root over u u u c f c_f c f D = E = 0 D=E=0 D = E = 0 max ( c f , c w ) \max(c_f,c_w) max ( c f , c w ) D = 0 D=0 D = 0 c T E c_{\mathrm{TE}} c TE E = 0 E=0 E = 0 Ψ = max ( c f , c w ) \Psi=\max(c_f,c_w) Ψ = max ( c f , c w )
c T E w 2 = Ψ 2 + D 4 + D E 2 . (99) c_{\mathrm{TEw}}^2=\Psi^2+\frac{D}{4}+\frac{\sqrt{DE}}{2}. \tag*{(99)} c TEw 2 = Ψ 2 + 4 D + 2 D E . (99) This reduces correctly in all three limits, since Ψ = c f \Psi=c_f Ψ = c f E = 0 E=0 E = 0 D D D D = 0 D=0 D = 0 (91) , the Ψ 2 \Psi^2 Ψ 2 D / 4 D/4 D /4 u = 0 u=0 u = 0 u = 1 u=1 u = 1 (84) , where the u ( 1 − u ) u(1-u) u ( 1 − u ) (98) vanishes. In Appendix C we show that − A 2 ≤ 2 c T E w 2 -A_2\leq2c_{\mathrm{TEw}}^2 − A 2 ≤ 2 c TEw 2 F ( c T E w 2 , u ) ≥ 0 F(c_{\mathrm{TEw}}^2,u)\geq0 F ( c TEw 2 , u ) ≥ 0 u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] s s s γ > 1 \gamma>1 γ > 1 F F F c T E w 2 c_{\mathrm{TEw}}^2 c TEw 2 D , E ≥ 0 D,E\geq0 D , E ≥ 0 k x = 1 / Δ x k_x=1/\Delta x k x = 1/Δ x
c T E w = max ( c f , c w ) 2 + Δ κ J 2 4 ε i n t Δ x 2 + ∣ η H Δ κ J ∣ 2 Δ x 2 ε i n t . (100) c_{\mathrm{TEw}}=\sqrt{\max(c_f,c_w)^2+\frac{\Delta\kappa_J^2}{4\varepsilon_{\mathrm{int}}\Delta x^2}+\frac{|\eta_H\Delta\kappa_J|}{2\Delta x^2\sqrt{\varepsilon_{\mathrm{int}}}}}. \tag*{(100)} c TEw = max ( c f , c w ) 2 + 4 ε int Δ x 2 Δ κ J 2 + 2Δ x 2 ε int ∣ η H Δ κ J ∣ . (100) Gyro-Viscous Waves
Starting from the ideal MHD equations and adding the gyro-viscous tensors Π ( 3 ) \boldsymbol{\Pi}^{(3)} Π ( 3 ) Π ( 4 ) \boldsymbol{\Pi}^{(4)} Π ( 4 ) M ( I , ν ∧ ) \boldsymbol{M}^{(\mathrm{I},\nu_\wedge)} M ( I , ν ∧ )
g 1 = x i o n [ ν ∧ 1 ( 1 − u ) + 4 ν ∧ 2 u ] , g 2 = g 1 − 2 x i o n ν ∧ 2 . (101) g_1 = x_{\mathrm{ion}}\left[\nu_{\wedge1}(1-u)+4\nu_{\wedge2}u\right], \qquad g_2 = g_1-2x_{\mathrm{ion}}\nu_{\wedge2}. \tag*{(101)} g 1 = x ion [ ν ∧ 1 ( 1 − u ) + 4 ν ∧ 2 u ] , g 2 = g 1 − 2 x ion ν ∧ 2 . (101) The gyro-viscous entries act only on the velocity, ∂ t δ v ⃗ = k x 2 w ⃗ × δ v ⃗ \partial_t\delta\vec{v}=k_x^2\vec{w}\times\delta\vec{v} ∂ t δ v = k x 2 w × δ v w ⃗ = ( g 2 b x , g 1 b y , g 1 b z ) \vec{w}=(g_2b_x,g_1b_y,g_1b_z) w = ( g 2 b x , g 1 b y , g 1 b z )
G ν ( u ) ≡ ∣ w ⃗ ∣ 2 = g 1 2 ( 1 − u ) + g 2 2 u = x i o n 2 [ ν ∧ 1 2 ( 1 − u ) 2 + 4 ν ∧ 1 ν ∧ 2 u ( 1 − u ) + 4 ν ∧ 2 2 u 2 ] . (102) G_\nu(u)\equiv\lvert\vec{w}\rvert^2=g_1^2(1-u)+g_2^2u=x_{\mathrm{ion}}^2\left[\nu_{\wedge1}^2(1-u)^2+4\nu_{\wedge1}\nu_{\wedge2}u(1-u)+4\nu_{\wedge2}^2u^2\right]. \tag*{(102)} G ν ( u ) ≡ ∣ w ∣ 2 = g 1 2 ( 1 − u ) + g 2 2 u = x ion 2 [ ν ∧ 1 2 ( 1 − u ) 2 + 4 ν ∧ 1 ν ∧ 2 u ( 1 − u ) + 4 ν ∧ 2 2 u 2 ] . (102) As before, ϕ \phi ϕ F ( Ω , u ) = Ω 3 + A 2 Ω 2 + A 1 Ω + A 0 F(\Omega,u)=\Omega^3+A_2\Omega^2+A_1\Omega+A_0 F ( Ω , u ) = Ω 3 + A 2 Ω 2 + A 1 Ω + A 0
A 2 = − [ c f 2 + c a 2 u + k x 2 G ν ( u ) ] , (103) A_2=-\left[c_f^2+c_a^2u+k_x^2G_\nu(u)\right], \tag*{(103)} A 2 = − [ c f 2 + c a 2 u + k x 2 G ν ( u ) ] , (103) A 1 = c a 2 u ( c a 2 + 2 c s 2 ) + x i o n 2 k x 2 u [ 4 ν ∧ 2 2 c a 2 ( 1 − u ) + c s 2 ( ν ∧ 1 ( 1 − u ) + 2 ν ∧ 2 ( 2 u − 1 ) ) 2 ] , (104) A_1=c_a^2u\left(c_a^2+2c_s^2\right)+x_{\mathrm{ion}}^2k_x^2u\left[4\nu_{\wedge2}^2c_a^2(1-u)+c_s^2\left(\nu_{\wedge1}(1-u)+2\nu_{\wedge2}(2u-1)\right)^2\right], \tag*{(104)} A 1 = c a 2 u ( c a 2 + 2 c s 2 ) + x ion 2 k x 2 u [ 4 ν ∧ 2 2 c a 2 ( 1 − u ) + c s 2 ( ν ∧ 1 ( 1 − u ) + 2 ν ∧ 2 ( 2 u − 1 ) ) 2 ] , (104) A 0 = − c a 4 c s 2 u 2 . (105) A_0=-c_a^4c_s^2u^2. \tag*{(105)} A 0 = − c a 4 c s 2 u 2 . (105) At the endpoints the characteristic polynomial factorises,
F ( Ω , 0 ) = Ω 2 ( Ω − c f 2 − x i o n 2 ν ∧ 1 2 k x 2 ) , (106) F(\Omega,0)=\Omega^2\left(\Omega-c_f^2-x_{\mathrm{ion}}^2\nu_{\wedge1}^2k_x^2\right), \tag*{(106)} F ( Ω , 0 ) = Ω 2 ( Ω − c f 2 − x ion 2 ν ∧ 1 2 k x 2 ) , (106) F ( Ω , 1 ) = ( Ω − c s 2 ) [ Ω 2 − 2 ( c a 2 + 2 x i o n 2 ν ∧ 2 2 k x 2 ) Ω + c a 4 ] , (107) F(\Omega,1)=\left(\Omega-c_s^2\right)\left[\Omega^2-2\left(c_a^2+2x_{\mathrm{ion}}^2\nu_{\wedge2}^2k_x^2\right)\Omega+c_a^4\right], \tag*{(107)} F ( Ω , 1 ) = ( Ω − c s 2 ) [ Ω 2 − 2 ( c a 2 + 2 x ion 2 ν ∧ 2 2 k x 2 ) Ω + c a 4 ] , (107) giving the phase speeds
c g v , 0 ( 0 ) = 0 , c g v , 0 ( 1 ) = c s (108) c_{gv,0}(0)=0,\qquad c_{gv,0}(1)=c_s \tag*{(108)} c g v , 0 ( 0 ) = 0 , c g v , 0 ( 1 ) = c s (108) c g v , 1 ( 0 ) = c f 2 + x i o n 2 ν ∧ 1 2 k x 2 , c g v , 1 , 2 ( 1 ) = x i o n 2 ν ∧ 2 2 k x 2 + c a 2 + x i o n ∣ ν ∧ 2 ∣ k x . (109) c_{gv,1}(0)=\sqrt{c_f^2+x_{\mathrm{ion}}^2\nu_{\wedge1}^2k_x^2},\qquad c_{gv,1,2}(1)=\sqrt{x_{\mathrm{ion}}^2\nu_{\wedge2}^2k_x^2+c_a^2}+x_{\mathrm{ion}}\lvert\nu_{\wedge2}\rvert k_x. \tag*{(109)} c g v , 1 ( 0 ) = c f 2 + x ion 2 ν ∧ 1 2 k x 2 , c g v , 1 , 2 ( 1 ) = x ion 2 ν ∧ 2 2 k x 2 + c a 2 + x ion ∣ ν ∧ 2 ∣ k x . (109) Apart from the Alfvén and slow speeds, which vanish at u = 0 u=0 u = 0 u = 1 u=1 u = 1 u = 0 u=0 u = 0 c T E c_{\mathrm{TE}} c TE u = 1 u=1 u = 1 c a 2 c_a^2 c a 2
Unlike the thermoelectric case, a root-sum-of-squares bound fails here: at u = 1 u=1 u = 1 c s = 0 c_s=0 c s = 0 c g v , 2 2 ( 1 ) = c a 2 + 2 ( x i o n ν ∧ 2 k x ) 2 + 2 x i o n ν ∧ 2 k x ( x i o n ν ∧ 2 k x ) 2 + c a 2 c_{gv,2}^2(1)=c_a^2+2(x_{\mathrm{ion}}\nu_{\wedge2}k_x)^2+2x_{\mathrm{ion}}\nu_{\wedge2}k_x\sqrt{(x_{\mathrm{ion}}\nu_{\wedge2}k_x)^2+c_a^2} c g v , 2 2 ( 1 ) = c a 2 + 2 ( x ion ν ∧ 2 k x ) 2 + 2 x ion ν ∧ 2 k x ( x ion ν ∧ 2 k x ) 2 + c a 2 c f 2 + k x 2 G ν ( 1 ) = c a 2 + 4 ( x i o n ν ∧ 2 k x ) 2 c_f^2+k_x^2G_\nu(1)=c_a^2+4(x_{\mathrm{ion}}\nu_{\wedge2}k_x)^2 c f 2 + k x 2 G ν ( 1 ) = c a 2 + 4 ( x ion ν ∧ 2 k x ) 2
We have therefore opted to use the Weyl bound. The ideal and gyroviscous parts of M ( I , ν ∧ ) M^{(\mathrm{I},\nu_\wedge)} M ( I , ν ∧ ) S M + M † S = 0 SM+M^\dagger S=0 S M + M † S = 0 S S S (88) , so S 1 / 2 M ( I , ν ∧ ) S − 1 / 2 S^{1/2}M^{(\mathrm{I},\nu_\wedge)}S^{-1/2} S 1/2 M ( I , ν ∧ ) S − 1/2 c f c_f c f k x ∣ w ⃗ ∣ = k x G ν ( u ) k_x\lvert\vec{w}\rvert=k_x\sqrt{G_\nu(u)} k x ∣ w ∣ = k x G ν ( u ) c ≤ c f + k x G ν ( u ) c\le c_f+k_x\sqrt{G_\nu(u)} c ≤ c f + k x G ν ( u ) k x = 1 / Δ x k_x=1/\Delta x k x = 1/Δ x
c g v = c f + ∣ λ ν , ∧ , max ∣ Δ x , (110) c_{\mathrm{gv}}=c_f+\frac{\lvert\lambda_{\nu,\wedge,\max}\rvert}{\Delta x}, \tag*{(110)} c gv = c f + Δ x ∣ λ ν , ∧ , m a x ∣ , (110) where λ ν , ∧ , max \lambda_{\nu,\wedge,\max} λ ν , ∧ , m a x § 3.7 . Since c g v ≥ c f c_{\mathrm{gv}}\ge c_f c gv ≥ c f max ( c f , c g v ) = c g v \max(c_f,c_{\mathrm{gv}})=c_{\mathrm{gv}} max ( c f , c gv ) = c gv [17 ] , we apply to the momentum, on which the gyroviscous terms act.
Righi–Leduc Velocity
The Righi–Leduc effect is a first-order advective operator acting on internal energy. We therefore simply add this to the phase speed for the internal energy, with the characteristic speed being
c κ ∧ = γ − 1 ρ ∣ [ ∇ ⃗ × ( x i o n ρ κ ∧ b ^ ) ] ⋅ n ^ ∣ . (111) c_{\kappa\wedge}=\frac{\gamma-1}{\rho}\left\lvert\left[\vec{\nabla}\times\left(x_{\mathrm{ion}}\rho\kappa_\wedge\hat{b}\right)\right]\cdot\hat{n}\right\rvert. \tag*{(111)} c κ ∧ = ρ γ − 1 [ ∇ × ( x ion ρ κ ∧ b ^ ) ] ⋅ n ^ . (111) Instead of upper bounding the curl of the unit vector in the normal direction (as we do with the CFL condition), we evaluate it with centred differences at the cell centres on either side of the face, including the metric factors of curvilinear coordinates. Along an unresolved direction the field is uniform and only the derivative of the scale factor contributes, which retains, for example, the cot θ ϕ / r \cot\theta_\phi/r cot θ ϕ / r
HLLE-Mod Signal Speed Summary
Following [17 ] , each variable is assigned the fastest phase speed of the modes in which it participates, and the signal speeds are formed as S L = min ( v ~ x , L − c ~ L , ∗ , v x , L − c L , ∗ ) S_L = \min(\tilde{v}_{x,L} - \tilde{c}_{L,*}, v_{x,L} - c_{L,*}) S L = min ( v ~ x , L − c ~ L , ∗ , v x , L − c L , ∗ ) S R = max ( v ~ x , R + c ~ R , ∗ , v x , R + c R , ∗ ) S_R = \max(\tilde{v}_{x,R} + \tilde{c}_{R,*}, v_{x,R} + c_{R,*}) S R = max ( v ~ x , R + c ~ R , ∗ , v x , R + c R , ∗ )
c ρ = c f , c v = c g v , c ε i n t = c κ ∧ + c T E , c B = c T E w . (112) c_{\rho} = c_f,\qquad c_v = c_{\mathrm{gv}},\qquad c_{\varepsilon_{\mathrm{int}}} = c_{\kappa\wedge} + c_{\mathrm{TE}},\qquad c_B = c_{\mathrm{TEw}}. \tag*{(112)} c ρ = c f , c v = c gv , c ε int = c κ ∧ + c TE , c B = c TEw . (112) Each speed reduces to c f c_f c f c T E c_{\mathrm{TE}} c TE c T E w c_{\mathrm{TEw}} c TEw ε i n t \varepsilon_{\mathrm{int}} ε int
c f c_f c f c ρ c_{\rho} c ρ c v c_v c v c ε i n t c_{\varepsilon_{\mathrm{int}}} c ε int u u u 3 × 10 5 3 \times10^5 3 × 1 0 5 c w c_w c w c T E w c_{\mathrm{TEw}} c TEw c B c_B c B c f 2 = ( γ P + ∣ B ⃗ ∣ 2 ) / ρ c_f^2 = (\gamma P + |\vec{B}|^2)/\rho c f 2 = ( γ P + ∣ B ∣ 2 ) / ρ c a = ∣ B ⃗ ∣ / ρ c_a = |\vec{B}|/\sqrt{\rho} c a = ∣ B ∣/ ρ
The Riemann solver in Athena++ evolves the total energy rather than the internal energy, so we split the ideal energy flux into internal, kinetic, and magnetic parts,
F i n t = ( ε i n t + P ) v x , F k i n = 1 2 ρ ∣ v ⃗ ∣ 2 v x , F m a g = ∣ B ⃗ ∣ 2 v x − B x ( B ⃗ ⋅ v ⃗ ) , (113) \mathcal{F}_{\mathrm{int}} = (\varepsilon_{\mathrm{int}} + P)v_x,\qquad\mathcal{F}_{\mathrm{kin}} = \frac{1}{2}\rho|\vec{v}|^2v_x,\qquad\mathcal{F}_{\mathrm{mag}} = |\vec{B}|^2v_x - B_x(\vec{B}\cdot\vec{v}), \tag*{(113)} F int = ( ε int + P ) v x , F kin = 2 1 ρ ∣ v ∣ 2 v x , F mag = ∣ B ∣ 2 v x − B x ( B ⋅ v ) , (113) which sum to the ideal energy flux, with densities ε i n t \varepsilon_{\mathrm{int}} ε int ε k i n = 1 2 ρ ∣ v ⃗ ∣ 2 \varepsilon_{\mathrm{kin}} = \frac{1}{2}\rho|\vec{v}|^2 ε kin = 2 1 ρ ∣ v ∣ 2 ε m a g = 1 2 ∣ B ⃗ ∣ 2 \varepsilon_{\mathrm{mag}} = \frac{1}{2}|\vec{B}|^2 ε mag = 2 1 ∣ B ∣ 2
F H L L ( ε ) = F H L L ( ε i n t ; S ε i n t ) + F H L L ( ε k i n ; S v ) + F H L L ( ε m a g ; S B ) , (114) \mathcal{F}^{\mathrm{HLL}}(\varepsilon) = \mathcal{F}^{\mathrm{HLL}}(\varepsilon_{\mathrm{int}};S_{\varepsilon_{\mathrm{int}}}) + \mathcal{F}^{\mathrm{HLL}}(\varepsilon_{\mathrm{kin}};S_v) + \mathcal{F}^{\mathrm{HLL}}(\varepsilon_{\mathrm{mag}};S_B), \tag*{(114)} F HLL ( ε ) = F HLL ( ε int ; S ε int ) + F HLL ( ε kin ; S v ) + F HLL ( ε mag ; S B ) , (114) with
F H L L ( q ; S ) = S R F q , L − S L F q , R + S L S R ( q R − q L ) S R − S L . (115) \mathcal{F}^{\mathrm{HLL}}(q;S) = \frac{S_R F_{q,L} - S_L F_{q,R} + S_LS_R(q_R-q_L)}{S_R-S_L}. \tag*{(115)} F HLL ( q ; S ) = S R − S L S R F q , L − S L F q , R + S L S R ( q R − q L ) . (115) The non-ideal fluxes are computed separately and added at the faces. Each sub-flux reduces to its physical flux for equal left and right states, so the total is consistent and the total energy remains conserved. Since the HLL flux is linear in ( q , F q ) (q,F_q) ( q , F q )
Implementation
Following standard conventions, cell centred quantities carry integer indices, and a subscript i − 1 / 2 i - 1/2 i − 1/2 x x x i − 1 i - 1 i − 1 i i i Δ x i \Delta x_i Δ x i i i i d x i ≡ x i + 1 − x i dx_i \equiv x_{i+1} - x_i d x i ≡ x i + 1 − x i d x i − 1 dx_{i-1} d x i − 1 i − 1 i-1 i − 1 i i i ρ i − 1 / 2 , j , k = 1 2 ( ρ i , j , k + ρ i − 1 , j , k ) \rho_{i-1/2,j,k} = \frac{1}{2}(\rho_{i,j,k} + \rho_{i-1,j,k}) ρ i − 1/2 , j , k = 2 1 ( ρ i , j , k + ρ i − 1 , j , k ) B x , i , j , k = 1 2 ( B x , i + 1 / 2 , j , k + B x , i − 1 / 2 , j , k ) B_{x,i,j,k} = \frac{1}{2}(B_{x,i+1/2,j,k} + B_{x,i-1/2,j,k}) B x , i , j , k = 2 1 ( B x , i + 1/2 , j , k + B x , i − 1/2 , j , k ) B y B_y B y B z B_z B z
Thermal Conduction
The spatial derivatives of the temperature are taken as the central difference of adjacent cell centred values. For the x x x
∂ x T i − 1 / 2 , j , k = ( T i , j , k − T i − 1 , j , k ) / d x i − 1 . (116) \partial_x T_{i-1/2,j,k} = (T_{i,j,k} - T_{i-1,j,k})/dx_{i-1}. \tag*{(116)} ∂ x T i − 1/2 , j , k = ( T i , j , k − T i − 1 , j , k ) / d x i − 1 . (116) For the y y y
∂ y T i − 1 / 2 , j , k = 1 2 ( ( T i , j + 1 , k + T i − 1 , j + 1 , k ) − ( T i , j − 1 , k + T i − 1 , j − 1 , k ) ) / ( d y j − 1 + d y j ) , (117) \partial_y T_{i-1/2,j,k} = \frac{1}{2}\left((T_{i,j+1,k} + T_{i-1,j+1,k}) - (T_{i,j-1,k} + T_{i-1,j-1,k})\right)/(dy_{j-1} + dy_j), \tag*{(117)} ∂ y T i − 1/2 , j , k = 2 1 ( ( T i , j + 1 , k + T i − 1 , j + 1 , k ) − ( T i , j − 1 , k + T i − 1 , j − 1 , k ) ) / ( d y j − 1 + d y j ) , (117) and likewise for ∂ z T \partial_z T ∂ z T
In non-Cartesian coordinates, the appropriate metric factors are present to allow for curvilinear coordinates. The fluxes are then calculated directly at ( i − 1 / 2 , j , k ) (i - 1/2, j, k) ( i − 1/2 , j , k )
To prevent heat from flowing from cold cells into hot cells in highly anisotropic regimes (which would violate the second law of thermodynamics) we limit the transverse temperature gradients following the asymmetric scheme of [37 ] . On the x x x ( i − 1 / 2 , j , k ) (i - 1/2, j, k) ( i − 1/2 , j , k )
∂ y T i − 1 / 2 , j , k = MC [ MC ( ∂ y + T i , j , k , ∂ y − T i , j , k ) , MC ( ∂ y + T i − 1 , j , k , ∂ y − T i − 1 , j , k ) ] , (118) \partial_y T_{i-1/2,j,k} = \operatorname{MC}\left[\operatorname{MC}\left(\partial_y^{+}T_{i,j,k},\partial_y^{-}T_{i,j,k}\right),\operatorname{MC}\left(\partial_y^{+}T_{i-1,j,k},\partial_y^{-}T_{i-1,j,k}\right)\right], \tag*{(118)} ∂ y T i − 1/2 , j , k = MC [ MC ( ∂ y + T i , j , k , ∂ y − T i , j , k ) , MC ( ∂ y + T i − 1 , j , k , ∂ y − T i − 1 , j , k ) ] , (118) and likewise for ∂ z T i − 1 / 2 , j , k \partial_z T_{i-1/2,j,k} ∂ z T i − 1/2 , j , k
∂ y + T i , j , k = T i , j + 1 , k − T i , j , k d y j , ∂ y − T i , j , k = T i , j , k − T i , j − 1 , k d y j − 1 , (119) \partial_y^{+}T_{i,j,k} = \frac{T_{i,j+1,k} - T_{i,j,k}}{dy_j}, \qquad\partial_y^{-}T_{i,j,k} = \frac{T_{i,j,k} - T_{i,j-1,k}}{dy_{j-1}}, \tag*{(119)} ∂ y + T i , j , k = d y j T i , j + 1 , k − T i , j , k , ∂ y − T i , j , k = d y j − 1 T i , j , k − T i , j − 1 , k , (119) and likewise for z z z d z k dz_k d z k d z k − 1 dz_{k-1} d z k − 1
MC ( a , b ) = { sign ( a ) min ( 2 ∣ a ∣ , 2 ∣ b ∣ , 1 2 ∣ a + b ∣ ) a b > 0 0 a b ≤ 0 . (120) \operatorname{MC}(a,b) =
\begin{cases}
\operatorname{sign}(a)\min\left(2|a|,2|b|,\frac{1}{2}|a+b|\right) & ab > 0 \\
0 & ab \le0
\end{cases}. \tag*{(120)} MC ( a , b ) = { sign ( a ) min ( 2∣ a ∣ , 2∣ b ∣ , 2 1 ∣ a + b ∣ ) 0 ab > 0 ab ≤ 0 . (120) The normal gradient, ∂ x T i − 1 / 2 , j , k = ( T i , j , k − T i − 1 , j , k ) / d x i − 1 \partial_x T_{i-1/2,j,k} = (T_{i,j,k} - T_{i-1,j,k})/dx_{i-1} ∂ x T i − 1/2 , j , k = ( T i , j , k − T i − 1 , j , k ) / d x i − 1
By default (though the user can turn it off) we also apply the saturation flux limiter of [3 ] . The saturated flux is
q s a t = 5 ϕ s a t ρ c i s o 3 , ϕ s a t = 0.3 , (121) q_{\mathrm{sat}} = 5\phi_{\mathrm{sat}}\rho c_{\mathrm{iso}}^{3}, \qquad\phi_{\mathrm{sat}} = 0.3, \tag*{(121)} q sat = 5 ϕ sat ρ c iso 3 , ϕ sat = 0.3 , (121) evaluated in the cell on the upwind side of the face, i.e. the side the heat flows from. The parallel and perpendicular fluxes are each multiplied by their own factor, q ⃗ ∥ → f s a t , ∥ q ⃗ ∥ \vec{q}_{\parallel} \to f_{\mathrm{sat},\parallel}\vec{q}_{\parallel} q ∥ → f sat , ∥ q ∥ q ⃗ ⊥ → f s a t , ⊥ q ⃗ ⊥ \vec{q}_{\perp} \to f_{\mathrm{sat},\perp}\vec{q}_{\perp} q ⊥ → f sat , ⊥ q ⊥
f s a t , ∥ = q s a t q s a t + κ ∥ ρ ∣ b ^ ⋅ ∇ T ∣ , f s a t , ⊥ = q s a t q s a t + κ ⊥ ρ ∣ ∇ T − b ^ ( b ^ ⋅ ∇ T ) ∣ . (122) f_{\mathrm{sat},\parallel} = \frac{q_{\mathrm{sat}}}{q_{\mathrm{sat}} + \kappa_{\parallel}\rho\left|\hat{b} \cdot\nabla T\right|}, \qquad f_{\mathrm{sat},\perp} = \frac{q_{\mathrm{sat}}}{q_{\mathrm{sat}} + \kappa_{\perp}\rho\left|\nabla T - \hat{b}\left(\hat{b} \cdot\nabla T\right)\right|}. \tag*{(122)} f sat , ∥ = q sat + κ ∥ ρ b ^ ⋅ ∇ T q sat , f sat , ⊥ = q sat + κ ⊥ ρ ∇ T − b ^ ( b ^ ⋅ ∇ T ) q sat . (122) Viscosity
The primitive variables and viscosity coefficients at the interface are taken as the arithmetic mean of the adjacent cell centres, as is done with thermal conduction. The velocity derivatives at the x x x T T T v m v_m v m
The fluxes at face ( i − 1 / 2 , j , k ) (i - 1/2, j, k) ( i − 1/2 , j , k )
We then limit the parallel viscosity by capping the pressure anisotropy at the firehose and mirror instability thresholds, following [38 ] . The pressure anisotropy produced by the parallel viscosity is
Δ P = P ⊥ − P ∥ = 3 ρ x i o n ν ∥ ( b ^ ⊗ b ^ − 1 3 I ) : ∇ ⃗ [ v ⃗ ] , (123) \Delta P = P_{\perp} - P_{\parallel} = 3\rho x_{\mathrm{ion}}\nu_{\parallel}\left(\hat{b}\otimes\hat{b}-\frac{1}{3}I\right):\vec{\nabla}[\vec{v}], \tag*{(123)} Δ P = P ⊥ − P ∥ = 3 ρ x ion ν ∥ ( b ^ ⊗ b ^ − 3 1 I ) : ∇ [ v ] , (123) which we clip to the range in which neither instability is excited, − B 2 ≤ Δ P ≤ B 2 / 2 -B^2 \le\Delta P \le B^2/2 − B 2 ≤ Δ P ≤ B 2 /2 B 2 / 2 B^2/2 B 2 /2 P ∥ − P ⊥ = B 2 P_{\parallel}-P_{\perp}=B^2 P ∥ − P ⊥ = B 2 P ⊥ − P ∥ = B 2 / 2 P_{\perp}-P_{\parallel}=B^2/2 P ⊥ − P ∥ = B 2 /2 Δ P \Delta P Δ P Π ( 0 ) \boldsymbol{\Pi}^{(0)} Π ( 0 )
Anisotropic viscosity can also transport momentum in the wrong direction across steep velocity gradients, analogous to heat flowing from cold to hot cells in anisotropic thermal conduction. Following [39 ] , who limited the anisotropic viscous fluxes with the MC limiter, we limit the transverse velocity derivatives in the dissipative stresses Π ( 0 ) \boldsymbol{\Pi}^{(0)} Π ( 0 ) Π ( 2 ) \boldsymbol{\Pi}^{(2)} Π ( 2 ) x x x ( i − 1 / 2 , j , k ) (i-1/2,j,k) ( i − 1/2 , j , k ) ∂ y v m \partial_y v_m ∂ y v m ∂ z v m \partial_z v_m ∂ z v m m = x , y , z m=x,y,z m = x , y , z (118) –Eq. (120) with T T T v m v_m v m ∂ x v m \partial_x v_m ∂ x v m Π ( 3 ) \boldsymbol{\Pi}^{(3)} Π ( 3 ) Π ( 4 ) \boldsymbol{\Pi}^{(4)} Π ( 4 )
Π ( 3 ) \boldsymbol{\Pi}^{(3)} Π ( 3 ) Π ( 4 ) \boldsymbol{\Pi}^{(4)} Π ( 4 ) Π ( 0 ) \boldsymbol{\Pi}^{(0)} Π ( 0 ) Π ( 1 ) \boldsymbol{\Pi}^{(1)} Π ( 1 ) Π ( 2 ) \boldsymbol{\Pi}^{(2)} Π ( 2 )
Hall Effect
The edge-centred current density is calculated from the face centred magnetic field in the follow way
J x , i , j − 1 / 2 , k − 1 / 2 = B z , i , j , k − B z , i , j − 1 , k d y j − 1 − B y , i , j , k − B y , i , j , k − 1 d z k − 1 , (124) J_{x,i,j-1/2,k-1/2}=\frac{B_{z,i,j,k}-B_{z,i,j-1,k}}{d y_{j-1}}-\frac{B_{y,i,j,k}-B_{y,i,j,k-1}}{d z_{k-1}}, \tag*{(124)} J x , i , j − 1/2 , k − 1/2 = d y j − 1 B z , i , j , k − B z , i , j − 1 , k − d z k − 1 B y , i , j , k − B y , i , j , k − 1 , (124) and similarly for J y J_y J y J z J_z J z x x x
J y , i , j − 1 / 2 , k − 1 / 2 = 1 4 ( J y , i − 1 / 2 , j , k − 1 / 2 + J y , i + 1 / 2 , j , k − 1 / 2 + J y , i − 1 / 2 , j − 1 , k − 1 / 2 + J y , i + 1 / 2 , j − 1 , k − 1 / 2 ) . (125) J_{y,i,j-1/2,k-1/2}=\frac{1}{4}\left(J_{y,i-1/2,j,k-1/2}+J_{y,i+1/2,j,k-1/2}+J_{y,i-1/2,j-1,k-1/2}+J_{y,i+1/2,j-1,k-1/2}\right). \tag*{(125)} J y , i , j − 1/2 , k − 1/2 = 4 1 ( J y , i − 1/2 , j , k − 1/2 + J y , i + 1/2 , j , k − 1/2 + J y , i − 1/2 , j − 1 , k − 1/2 + J y , i + 1/2 , j − 1 , k − 1/2 ) . (125) The same logic is applied to calculate the edge magnetic fields and the Hall coefficient: the normal field B x B_x B x η H \eta_H η H
B x , i , j − 1 / 2 , k − 1 / 2 = 1 4 ( B x , i , j , k + B x , i , j − 1 , k + B x , i , j , k − 1 + B x , i , j − 1 , k − 1 ) , (126) B_{x,i,j-1/2,k-1/2}=\frac{1}{4}\left(B_{x,i,j,k}+B_{x,i,j-1,k}+B_{x,i,j,k-1}+B_{x,i,j-1,k-1}\right), \tag*{(126)} B x , i , j − 1/2 , k − 1/2 = 4 1 ( B x , i , j , k + B x , i , j − 1 , k + B x , i , j , k − 1 + B x , i , j − 1 , k − 1 ) , (126) and the transverse fields over the two faces that share it
B y , i , j − 1 / 2 , k − 1 / 2 = 1 2 ( B y , i , j − 1 / 2 , k + B y , i , j − 1 / 2 , k − 1 ) . (127) B_{y,i,j-1/2,k-1/2}=\frac{1}{2}\left(B_{y,i,j-1/2,k}+B_{y,i,j-1/2,k-1}\right). \tag*{(127)} B y , i , j − 1/2 , k − 1/2 = 2 1 ( B y , i , j − 1/2 , k + B y , i , j − 1/2 , k − 1 ) . (127) The Hall EMF at ( i , j − 1 / 2 , k − 1 / 2 ) (i,j-1/2,k-1/2) ( i , j − 1/2 , k − 1/2 )
E ⃗ = − η H b ^ × J ⃗ . (128) \vec{\mathcal{E}}=-\eta_H\hat{b}\times\vec{J}. \tag*{(128)} E = − η H b ^ × J . (128) Thermoelectric Effects
For what we term thermoelectric conduction; the Peltier and Ettingshausen effects ( q ⃗ J ) (\vec{q}_J) ( q J )
J x , i , j , k = 1 4 ( J x , i , j − 1 / 2 , k − 1 / 2 + J x , i , j + 1 / 2 , k − 1 / 2 + J x , i , j − 1 / 2 , k + 1 / 2 + J x , i , j + 1 / 2 , k + 1 / 2 ) , (129) J_{x,i,j,k}=\frac{1}{4}\left(J_{x,i,j-1/2,k-1/2}+J_{x,i,j+1/2,k-1/2}+J_{x,i,j-1/2,k+1/2}+J_{x,i,j+1/2,k+1/2}\right), \tag*{(129)} J x , i , j , k = 4 1 ( J x , i , j − 1/2 , k − 1/2 + J x , i , j + 1/2 , k − 1/2 + J x , i , j − 1/2 , k + 1/2 + J x , i , j + 1/2 , k + 1/2 ) , (129) and likewise for J y J_y J y J z J_z J z J ⃗ i − 1 / 2 , j , k = 1 2 ( J ⃗ i , j , k + J ⃗ i − 1 , j , k ) \vec{J}_{i-1/2,j,k} = \frac{1}{2}(\vec{J}_{i,j,k} + \vec{J}_{i-1,j,k}) J i − 1/2 , j , k = 2 1 ( J i , j , k + J i − 1 , j , k ) (129) applies to the transverse current densities an additional 1 4 ( 1 , 2 , 1 ) \frac{1}{4}(1,2,1) 4 1 ( 1 , 2 , 1 ) ( i − 1 / 2 , j , k ) (i-1/2,j,k) ( i − 1/2 , j , k )
For the thermoelectric force - the Seebeck and Nernst effects, the edge centred fields require the edge-centred temperature, which are taken as the arithmetic mean of the four surrounding cell centres
T i , j − 1 / 2 , k − 1 / 2 = 1 4 ( T i , j , k + T i , j , k − 1 + T i , j − 1 , k + T i , j − 1 , k − 1 ) . (130) T_{i,j-1/2,k-1/2} = \frac{1}{4}(T_{i,j,k} + T_{i,j,k-1} + T_{i,j-1,k} + T_{i,j-1,k-1}). \tag*{(130)} T i , j − 1/2 , k − 1/2 = 4 1 ( T i , j , k + T i , j , k − 1 + T i , j − 1 , k + T i , j − 1 , k − 1 ) . (130) At the x x x
∂ x T i , j − 1 / 2 , k − 1 / 2 = T i + 1 , j − 1 / 2 , k − 1 / 2 − T i − 1 , j − 1 / 2 , k − 1 / 2 d x i − 1 + d x i . (131) \partial_x T_{i,j-1/2,k-1/2} = \frac{T_{i+1,j-1/2,k-1/2} - T_{i-1,j-1/2,k-1/2}}{d x_{i-1} + d x_i}. \tag*{(131)} ∂ x T i , j − 1/2 , k − 1/2 = d x i − 1 + d x i T i + 1 , j − 1/2 , k − 1/2 − T i − 1 , j − 1/2 , k − 1/2 . (131) The transverse temperature gradient is
∂ y T i , j − 1 / 2 , k − 1 / 2 = T i , j + 1 / 2 , k − 1 / 2 − T i , j − 3 / 2 , k − 1 / 2 Δ y j − 1 + Δ y j , (132) \partial_y T_{i,j-1/2,k-1/2} = \frac{T_{i,j+1/2,k-1/2} - T_{i,j-3/2,k-1/2}}{\Delta y_{j-1} + \Delta y_j}, \tag*{(132)} ∂ y T i , j − 1/2 , k − 1/2 = Δ y j − 1 + Δ y j T i , j + 1/2 , k − 1/2 − T i , j − 3/2 , k − 1/2 , (132) and likewise for ∂ z T \partial_z T ∂ z T
The edge centred magnetic fields are then calculated in the same way as Eq. (126) , and the thermoelectric EMF is calculated using Eq. (7).
The perpendicular factor f s a t ⊥ f_{\mathrm{sat}\perp} f sat ⊥ (122) is applied to the Nernst EMF F ⃗ T , ∧ \vec{F}_{T,\wedge} F T , ∧ q ⃗ J , ∧ \vec{q}_{J,\wedge} q J , ∧ [40 ] recommend limiting Nernst advection by the same factor as the perpendicular heat flux. We apply the same factor to the Ettingshausen flux. The Ettingshausen flux through the x x x
f s a t ⊥ , i − 1 / 2 , j , k = q s a t q s a t + κ ⊥ , i − 1 / 2 , j , k ρ i − 1 / 2 , j , k ∣ ∇ T ∣ i − 1 / 2 , j , k 2 − ( b ^ ⋅ ∇ T ) i − 1 / 2 , j , k 2 . (133) f_{\mathrm{sat}\perp,i-1/2,j,k} = \frac{q_{\mathrm{sat}}}{q_{\mathrm{sat}} + \kappa_{\perp,i-1/2,j,k}\rho_{i-1/2,j,k}\sqrt{\lvert\nabla T\rvert^2_{i-1/2,j,k} - \left(\hat{b}\cdot\nabla T\right)^2_{i-1/2,j,k}}}. \tag*{(133)} f sat ⊥ , i − 1/2 , j , k = q sat + κ ⊥ , i − 1/2 , j , k ρ i − 1/2 , j , k ∣ ∇ T ∣ i − 1/2 , j , k 2 − ( b ^ ⋅ ∇ T ) i − 1/2 , j , k 2 q sat . (133) The x x x x x x ( i , j − 1 / 2 , k − 1 / 2 ) (i,j-1/2,k-1/2) ( i , j − 1/2 , k − 1/2 )
f s a t ⊥ , i , j − 1 / 2 , k − 1 / 2 = q ‾ s a t q ‾ s a t + κ ‾ ⊥ ρ ‾ ∣ ∇ T ∣ i , j − 1 / 2 , k − 1 / 2 2 − ( b ^ ⋅ ∇ T ) i , j − 1 / 2 , k − 1 / 2 2 . (134) f_{\mathrm{sat}\perp,i,j-1/2,k-1/2} = \frac{\overline{q}_{\mathrm{sat}}}{\overline{q}_{\mathrm{sat}} + \overline{\kappa}_{\perp}\overline{\rho}\sqrt{\lvert\nabla T\rvert^2_{i,j-1/2,k-1/2} - \left(\hat{b}\cdot\nabla T\right)^2_{i,j-1/2,k-1/2}}}. \tag*{(134)} f sat ⊥ , i , j − 1/2 , k − 1/2 = q sat + κ ⊥ ρ ∣ ∇ T ∣ i , j − 1/2 , k − 1/2 2 − ( b ^ ⋅ ∇ T ) i , j − 1/2 , k − 1/2 2 q sat . (134) Limiting κ ⊥ \kappa_{\perp} κ ⊥ κ J , ∧ \kappa_{J,\wedge} κ J , ∧ f ≤ 1 f \le1 f ≤ 1 § 3.9 , since its left-hand side decreases at most linearly in f f f f 2 f^2 f 2
Tests
For all tests we use piecewise-linear reconstruction and the HLLE Riemann solver (HLLE-Mod where stated), with a CFL number of 0.9. The dissipative tests use RK1; the non-dissipative tests are run with RK1, RK2, and RK3. With STS enabled, the dissipative terms are integrated with RKL1. With STS the super-step is capped at 50 parabolic timesteps, otherwise STS runs finish in a single ideal hydro timestep. We have modified Athena++’s ideal timestep so that the number of dimensions is accounted for before the CFL number is applied for consistency with the non-ideal limits. Unless stated otherwise, the background state is ρ 0 = 1 \rho_0 = 1 ρ 0 = 1 P 0 = 1 P_0 = 1 P 0 = 1 B ⃗ 0 = ( 1 , 0 , 0 ) \vec{B}_0 = (1, 0, 0) B 0 = ( 1 , 0 , 0 ) 16 3 16^3 1 6 3 32 3 32^3 3 2 3 64 3 64^3 6 4 3 [ − 10 , 10 ] 3 [-10, 10]^3 [ − 10 , 10 ] 3 8 3 8^3 8 3 16 3 16^3 1 6 3 32 3 32^3 3 2 3 x i o n = 0 x_{\mathrm{ion}} = 0 x ion = 0
The analytical solutions are derived for frozen flow, retaining only the non-ideal terms under test. The simulations evolve the full MHD equations, so we choose coefficients for which the non-ideal timescales are much shorter than the sound and Alfvén crossing times, making the ideal dynamics negligible over each test. The saturation and pressure-anisotropy are disabled, while the slope limiters are enabled. We measure convergence with
∥ e N ( t ) ∥ 2 = ( 1 V ∑ i e i 2 ( t ) Δ V i ) 1 / 2 , E N = max t ∥ e N ( t ) ∥ 2 , p = log 2 E N / 2 E N . (135) \left\|e_N(t)\right\|_2 = \left(\frac{1}{V}\sum_i e_i^2(t)\Delta V_i\right)^{1/2}, \qquad E_N = \max_t \left\|e_N(t)\right\|_2, \qquad p = \log_2\frac{E_{N/2}}{E_N}. \tag*{(135)} ∥ e N ( t ) ∥ 2 = ( V 1 i ∑ e i 2 ( t ) Δ V i ) 1/2 , E N = t max ∥ e N ( t ) ∥ 2 , p = log 2 E N E N /2 . (135) The convergence orders of all tests are collected in Table 2 and Table 3 .
Table 2. Convergence order of the dissipative tests (and the Nernst/Ettingshausen tests) for unsplit RK1 and STS RKL1: p 32 p_{32} p 32 p 64 p_{64} p 64 32 3 32^3 3 2 3 64 3 64^3 6 4 3 t × t_{\times} t × t t t t = 0.128 t = 0.128 t = 0.128 16 3 16^3 1 6 3 32 3 32^3 3 2 3 64 3 64^3 6 4 3 p p p t = 0.008 t = 0.008 t = 0.008
Table 3. Convergence order of the non-dissipative tests for RK1, RK2, RK3, and RK3 with HLLE-Mod: p 32 p_{32} p 32 p 64 p_{64} p 64 32 3 32^3 3 2 3 64 3 64^3 6 4 3 t × t_{\times} t × t t t
Thermal Conduction
To test the dissipative thermal conduction we initialise a temperature perturbation given by
T ( 0 , x , y , z ) = T 0 ( 1 + A T ( 0 ) cos ( k x x + k y y ) ) , (136) T(0,x,y,z) = T_0\left(1 + A_T(0)\cos(k_xx + k_yy)\right), \tag*{(136)} T ( 0 , x , y , z ) = T 0 ( 1 + A T ( 0 ) cos ( k x x + k y y ) ) , (136) with k x = 2 π N x / L x k_x = 2\pi N_x/L_x k x = 2 π N x / L x k y = 2 π N y / L y k_y = 2\pi N_y/L_y k y = 2 π N y / L y k z = 2 π N z / L z k_z = 2\pi N_z/L_z k z = 2 π N z / L z x x x y y y
For our anisotropic thermal conduction implementation, this has the analytical solution
T ( t , x , y , z ) = T 0 ( A T ( 0 ) exp ( − Γ κ t ) cos ( k x x + k y y + k z z ) + 1 ) , (137) T(t,x,y,z) = T_0\left(A_T(0)\exp(-\Gamma_\kappa t)\cos(k_xx + k_yy + k_zz) + 1\right), \tag*{(137)} T ( t , x , y , z ) = T 0 ( A T ( 0 ) exp ( − Γ κ t ) cos ( k x x + k y y + k z z ) + 1 ) , (137) Γ κ = ( γ − 1 ) ( k ∥ 2 ( x i o n κ ∥ + ( 1 − x i o n ) κ n ) + k ⊥ 2 ( x i o n κ ⊥ + ( 1 − x i o n ) κ n ) ) , (138) \Gamma_\kappa= (\gamma- 1)\left(k_\parallel^2\left(x_{\mathrm{ion}}\kappa_\parallel+ (1-x_{\mathrm{ion}})\kappa_n\right) + k_\perp^2\left(x_{\mathrm{ion}}\kappa_\perp+ (1-x_{\mathrm{ion}})\kappa_n\right)\right), \tag*{(138)} Γ κ = ( γ − 1 ) ( k ∥ 2 ( x ion κ ∥ + ( 1 − x ion ) κ n ) + k ⊥ 2 ( x ion κ ⊥ + ( 1 − x ion ) κ n ) ) , (138) with parallel and perpendicular wavenumbers k ⃗ ∥ = ( k ⃗ ⋅ b ^ ) b ^ \vec{k}_\parallel= (\vec{k}\cdot\hat{b})\hat{b} k ∥ = ( k ⋅ b ^ ) b ^ k ⃗ ⊥ = k ⃗ − k ⃗ ∥ \vec{k}_\perp= \vec{k}-\vec{k}_\parallel k ⊥ = k − k ∥ k x = 2 π / L x k_x = 2\pi/L_x k x = 2 π / L x k y = 2 π / L y k_y = 2\pi/L_y k y = 2 π / L y k z = 0 k_z = 0 k z = 0 A T ( 0 ) = 10 − 6 A_T(0) = 10^{-6} A T ( 0 ) = 1 0 − 6
Figure 1 shows the dissipative thermal conduction L2 error over time for RK1 and RKL1 time integration. The error peaks between t ≈ 1.1 × 10 − 3 t \approx1.1 \times10^{-3} t ≈ 1.1 × 1 0 − 3 t ≈ 1.5 × 10 − 3 t \approx1.5 \times10^{-3} t ≈ 1.5 × 1 0 − 3 Γ κ − 1 ≈ 1.4 × 10 − 3 \Gamma_{\kappa}^{-1} \approx1.4 \times10^{-3} Γ κ − 1 ≈ 1.4 × 1 0 − 3 64 3 64^3 6 4 3 10 − 12 10^{-12} 1 0 − 12
Figure 1. Thermal conduction test: combined neutral, parallel, and perpendicular. κ n = κ ∥ = κ ⊥ = 5500 \kappa_n = \kappa_\parallel= \kappa_\perp= 5500 κ n = κ ∥ = κ ⊥ = 5500 x i o n = 0.5 x_{\mathrm{ion}} = 0.5 x ion = 0.5
Viscosity
We initialise our velocity perturbation with a wavevector k ⃗ = ( k x , 0 , 0 ) \vec{k} = (k_x, 0, 0) k = ( k x , 0 , 0 ) b ^ = ( 0 , cos ( θ ) , sin ( θ ) ) \hat{b} = (0, \cos(\theta), \sin(\theta)) b ^ = ( 0 , cos ( θ ) , sin ( θ )) θ = π / 3 \theta= \pi/3 θ = π /3
v ⃗ ( x , y , z , 0 ) = ( v 0 a 1 ( 0 ) sin ( k x x ) v 0 ( a 2 ( 0 ) cos ( θ ) − a 3 ( 0 ) sin ( θ ) ) sin ( k x x ) v 0 ( a 2 ( 0 ) sin ( θ ) + a 3 ( 0 ) cos ( θ ) ) sin ( k x x ) ) (139) \vec{v}(x,y,z,0) =
\begin{pmatrix}
v_0a_1(0)\sin(k_xx) \\
v_0\left(a_2(0)\cos(\theta)-a_3(0)\sin(\theta)\right)\sin(k_xx) \\
v_0\left(a_2(0)\sin(\theta)+a_3(0)\cos(\theta)\right)\sin(k_xx)
\end{pmatrix} \tag*{(139)} v ( x , y , z , 0 ) = v 0 a 1 ( 0 ) sin ( k x x ) v 0 ( a 2 ( 0 ) cos ( θ ) − a 3 ( 0 ) sin ( θ ) ) sin ( k x x ) v 0 ( a 2 ( 0 ) sin ( θ ) + a 3 ( 0 ) cos ( θ ) ) sin ( k x x ) (139) This has the analytical solution
v ⃗ ( x , y , z , t ) = v 0 sin ( k x x ) ( a 1 ( 0 ) exp ( − Γ 1 t ) a 2 ( 0 ) exp ( − Γ 2 t ) cos ( θ ) − a 3 ( 0 ) exp ( − Γ 3 t ) sin ( θ ) a 2 ( 0 ) exp ( − Γ 2 t ) sin ( θ ) + a 3 ( 0 ) exp ( − Γ 3 t ) cos ( θ ) ) (140) \vec{v}(x,y,z,t) = v_0\sin(k_xx)
\begin{pmatrix}
a_1(0)\exp(-\Gamma_1t) \\
a_2(0)\exp(-\Gamma_2t)\cos(\theta)-a_3(0)\exp(-\Gamma_3t)\sin(\theta) \\
a_2(0)\exp(-\Gamma_2t)\sin(\theta)+a_3(0)\exp(-\Gamma_3t)\cos(\theta)
\end{pmatrix} \tag*{(140)} v ( x , y , z , t ) = v 0 sin ( k x x ) a 1 ( 0 ) exp ( − Γ 1 t ) a 2 ( 0 ) exp ( − Γ 2 t ) cos ( θ ) − a 3 ( 0 ) exp ( − Γ 3 t ) sin ( θ ) a 2 ( 0 ) exp ( − Γ 2 t ) sin ( θ ) + a 3 ( 0 ) exp ( − Γ 3 t ) cos ( θ ) (140) where
Γ 1 = k x 2 ( − 4 ν n ( x i o n − 1 ) 3 + x i o n ( ν ∥ + 3 ν ⊥ 1 ) 3 + ζ ρ 0 ) (141) \Gamma_1 = k_x^2\left(-\frac{4\nu_n(x_{\mathrm{ion}}-1)}{3}+\frac{x_{\mathrm{ion}}(\nu_{\parallel}+3\nu_{\perp1})}{3}+\frac{\zeta}{\rho_0}\right) \tag*{(141)} Γ 1 = k x 2 ( − 3 4 ν n ( x ion − 1 ) + 3 x ion ( ν ∥ + 3 ν ⊥ 1 ) + ρ 0 ζ ) (141) Γ 2 = k x 2 ( 4 ν ⊥ 2 x i o n + ν n ( 1 − x i o n ) ) (142) \Gamma_2 = k_x^2\left(4\nu_{\perp2}x_{\mathrm{ion}}+\nu_n(1-x_{\mathrm{ion}})\right) \tag*{(142)} Γ 2 = k x 2 ( 4 ν ⊥ 2 x ion + ν n ( 1 − x ion ) ) (142) Γ 3 = k x 2 ( ν ⊥ 1 x i o n + ν n ( 1 − x i o n ) ) . (143) \Gamma_3 = k_x^2\left(\nu_{\perp1}x_{\mathrm{ion}}+\nu_n(1-x_{\mathrm{ion}})\right). \tag*{(143)} Γ 3 = k x 2 ( ν ⊥ 1 x ion + ν n ( 1 − x ion ) ) . (143) We set ∣ B 0 ∣ = 1 |B_0| = 1 ∣ B 0 ∣ = 1 v 0 = 10 − 6 v_0 = 10^{-6} v 0 = 1 0 − 6 a 1 ( 0 ) = a 2 ( 0 ) = a 3 ( 0 ) = 1 a_1(0) = a_2(0) = a_3(0) = 1 a 1 ( 0 ) = a 2 ( 0 ) = a 3 ( 0 ) = 1 a 1 a_1 a 1 a 1 a_1 a 1 a 2 a_2 a 2 a 3 a_3 a 3 a 1 a_1 a 1 a 3 a_3 a 3 a 2 a_2 a 2
Figure 2 shows the L2 error in dissipative viscosity over time for RK1 and RKL1 time integration. The RK1 error peaks at t ≈ 2 × 10 − 3 t \approx2 \times10^{-3} t ≈ 2 × 1 0 − 3 0 ⃗ \vec{0} 0 t ≈ 0.025 t \approx0.025 t ≈ 0.025 32 3 32^3 3 2 3 64 3 64^3 6 4 3 ≈ 7 × 10 − 13 \approx7 \times10^{-13} ≈ 7 × 1 0 − 13 a 1 a_1 a 1
Figure 2. Viscosity test: combined bulk, neutral, parallel (Π ( 0 ) \Pi^{(0)} Π ( 0 ) Π ( 1 ) + Π ( 2 ) \Pi^{(1)}+\Pi^{(2)} Π ( 1 ) + Π ( 2 ) ζ = 5200 \zeta= 5200 ζ = 5200 ν n = ν ∥ = ν ⊥ 1 = ν ⊥ 2 = 3900 \nu_n = \nu_{\parallel} = \nu_{\perp1} = \nu_{\perp2} = 3900 ν n = ν ∥ = ν ⊥ 1 = ν ⊥ 2 = 3900 x i o n = 0.5 x_{\mathrm{ion}} = 0.5 x ion = 0.5
Hall Effect (+ Magnetic Diffusion)
We consider the induction equation with frozen flow. Only Ohmic, ambipolar, and Hall terms are present. We take our field to be ∣ B ⃗ 0 ∣ = 1 |\vec{B}_0| = 1 ∣ B 0 ∣ = 1 b ^ = ( cos ( θ ) , sin ( θ ) , 0 ) \hat{b} = (\cos(\theta), \sin(\theta), 0) b ^ = ( cos ( θ ) , sin ( θ ) , 0 ) θ = π / 6 \theta= \pi/6 θ = π /6 B ⃗ ( x , y , z , 0 ) = ( B 0 cos θ , B 0 sin θ , δ B 0 cos ( k x x ) ) \vec{B}(x,y,z,0) = (B_0 \cos\theta, B_0 \sin\theta, \delta B_0 \cos(k_xx)) B ( x , y , z , 0 ) = ( B 0 cos θ , B 0 sin θ , δ B 0 cos ( k x x )) δ B 0 = 10 − 6 \delta B_0 = 10^{-6} δ B 0 = 1 0 − 6
B ⃗ ( x , y , z , t ) = ( B 0 cos ( θ ) B 0 sin ( θ ) − δ B 0 D exp ( − a ˉ k x 2 t ) sinh ( D k x 2 t ) cos ( k x x ) δ B 0 ( cosh ( D k x 2 t ) + δ a D sinh ( D k x 2 t ) ) exp ( − a ˉ k x 2 t ) cos ( k x x ) ) (144) \vec{B}(x,y,z,t) =
\begin{pmatrix}
B_0\cos(\theta) \\
B_0\sin(\theta) - \frac{\delta B_0}{D}\exp(-\bar{a}k_x^2t)\sinh(Dk_x^2t)\cos(k_xx) \\
\delta B_0\left(\cosh(Dk_x^2t)+\frac{\delta_a}{D}\sinh(Dk_x^2t)\right)\exp(-\bar{a}k_x^2t)\cos(k_xx)
\end{pmatrix} \tag*{(144)} B ( x , y , z , t ) = B 0 cos ( θ ) B 0 sin ( θ ) − D δ B 0 exp ( − a ˉ k x 2 t ) sinh ( D k x 2 t ) cos ( k x x ) δ B 0 ( cosh ( D k x 2 t ) + D δ a sinh ( D k x 2 t ) ) exp ( − a ˉ k x 2 t ) cos ( k x x ) (144) with
a ˉ = η + 1 2 η A D ( 1 + cos 2 ( θ ) ) , δ a = 1 2 η A D sin 2 ( θ ) , β = η H cos ( θ ) , D 2 = δ a 2 − β 2 . (145) \bar{a} = \eta+ \frac{1}{2}\eta_{AD}(1+\cos^2(\theta)), \qquad
\delta_a = \frac{1}{2}\eta_{AD}\sin^2(\theta), \qquad
\beta= \eta_H\cos(\theta), \qquad
D^2 = \delta_a^2-\beta^2. \tag*{(145)} a ˉ = η + 2 1 η A D ( 1 + cos 2 ( θ )) , δ a = 2 1 η A D sin 2 ( θ ) , β = η H cos ( θ ) , D 2 = δ a 2 − β 2 . (145) Figure 3. Hall effect test. η H = 6000 \eta_H = 6000 η H = 6000
Figure 3 shows the magnetic field test L2 errors with only the Hall effect active (non-dissipative), across RK1, RK2, and RK3 (with the standard and modified HLLE Riemann solver) time integration. The Hall effect fails completely at RK1. Mass, momentum, and energy conservation fail, the timestep becomes zero, and infinities and NaNs propagate throughout the grid. The blow-up time decreases with resolution, scaling as O ( Δ x 2 ) O(\Delta x^2) O ( Δ x 2 ) 32 3 32^3 3 2 3
Nernst/Ettingshausen Effects
We consider the ideal MHD equations with only neutral and perpendicular thermal conduction; Ohmic and ambipolar magnetic diffusion; and the Nernst/Ettingshausen effects. We take B 0 = 1 B_0 = 1 B 0 = 1 ρ 0 = 1 \rho_0 = 1 ρ 0 = 1 b ^ = ( 0 , 1 , 0 ) \hat{b} = (0,1,0) b ^ = ( 0 , 1 , 0 ) T ( x , y , z , 0 ) = T 0 + δ T 0 cos ( k x x ) T(x,y,z,0) = T_0 + \delta T_0 \cos(k_xx) T ( x , y , z , 0 ) = T 0 + δ T 0 cos ( k x x ) B ⃗ ( x , y , z , 0 ) = ( 0 , B 0 , B 0 cos ( k x x ) ) \vec{B}(x,y,z,0) = (0,B_0, B_0\cos(k_xx)) B ( x , y , z , 0 ) = ( 0 , B 0 , B 0 cos ( k x x )) δ T 0 = δ B 0 = 10 − 6 \delta T_0 = \delta B_0 = 10^{-6} δ T 0 = δ B 0 = 1 0 − 6
T ( x , y , z , t ) = T 0 + ( δ T 0 cosh ( D k x 2 t ) − δ B 0 D T B + δ T 0 D sinh ( D k x 2 t ) ) exp ( − a ˉ k x 2 t ) cos ( k x x ) , (146) T(x,y,z,t) = T_0 + \left(\delta T_0\cosh(Dk_x^2t) - \frac{\delta B_0D_{TB}+\delta T_0}{D}\sinh(Dk_x^2t)\right)\exp(-\bar{a}k_x^2t)\cos(k_xx), \tag*{(146)} T ( x , y , z , t ) = T 0 + ( δ T 0 cosh ( D k x 2 t ) − D δ B 0 D T B + δ T 0 sinh ( D k x 2 t ) ) exp ( − a ˉ k x 2 t ) cos ( k x x ) , (146) B ⃗ ( x , y , z , t ) = ( 0 B 0 + ( δ B 0 cosh ( D k x 2 t ) + d δ B 0 − δ T 0 D B T D sinh ( D k x 2 t ) ) exp ( − a ˉ k x 2 t ) cos ( k x x ) 0 ) (147) \vec{B}(x,y,z,t)=
\begin{pmatrix}
0\\
B_0+\left(\delta B_0\cosh(Dk_x^2t)+\frac{d\delta B_0-\delta T_0D_{BT}}{D}\sinh(Dk_x^2t)\right)\exp(-\bar{a}k_x^2t)\cos(k_xx)\\
0
\end{pmatrix} \tag*{(147)} B ( x , y , z , t ) = 0 B 0 + ( δ B 0 cosh ( D k x 2 t ) + D d δ B 0 − δ T 0 D B T sinh ( D k x 2 t ) ) exp ( − a ˉ k x 2 t ) cos ( k x x ) 0 (147) With
D T = ( γ − 1 ) ( ( 1 − x i o n ) κ n + x i o n κ ⊥ ) , D B = η + η A D , D T B = ( γ − 1 ) x i o n κ J Λ ρ 0 , (148) D_T=(\gamma-1)((1-x_{\mathrm{ion}})\kappa_n+x_{\mathrm{ion}}\kappa_{\perp}),\qquad
D_B=\eta+\eta_{AD},\qquad
D_{TB}=\frac{(\gamma-1)x_{\mathrm{ion}}\kappa_{J\Lambda}}{\rho_0}, \tag*{(148)} D T = ( γ − 1 ) (( 1 − x ion ) κ n + x ion κ ⊥ ) , D B = η + η A D , D T B = ρ 0 ( γ − 1 ) x ion κ J Λ , (148) D B T = x i o n κ J Λ T 0 , a ˉ = 1 2 ( D T + D B ) , d = 1 2 ( D T − D B ) , (149) D_{BT}=\frac{x_{\mathrm{ion}}\kappa_{J\Lambda}}{T_0},\qquad
\bar{a}=\frac{1}{2}(D_T+D_B),\qquad
d=\frac{1}{2}(D_T-D_B), \tag*{(149)} D B T = T 0 x ion κ J Λ , a ˉ = 2 1 ( D T + D B ) , d = 2 1 ( D T − D B ) , (149) D 2 = D T B D B T + d 2 . (150) D^2=D_{TB}D_{BT}+d^2. \tag*{(150)} D 2 = D T B D B T + d 2 . (150) This also demonstrates the well-posedness criterion in a model: when D > a ˉ D>\bar{a} D > a ˉ cosh ( D k x 2 t ) exp ( − a ˉ k x 2 t ) \cosh(Dk_x^2t)\exp(-\bar{a}k_x^2t) cosh ( D k x 2 t ) exp ( − a ˉ k x 2 t ) sinh ( D k x 2 t ) exp ( − a ˉ k x 2 t ) \sinh(Dk_x^2t)\exp(-\bar{a}k_x^2t) sinh ( D k x 2 t ) exp ( − a ˉ k x 2 t ) exp ( ( D − a ˉ ) k x 2 t ) \exp((D-\bar{a})k_x^2t) exp (( D − a ˉ ) k x 2 t ) a ˉ 2 − d 2 = D T D B \bar{a}^2-d^2=D_TD_B a ˉ 2 − d 2 = D T D B D T B D B T > D T D B D_{TB}D_{BT}>D_TD_B D T B D B T > D T D B k x 2 k_x^2 k x 2
Figure 4 shows the dissipative thermo-magnetic system (perpendicular thermal conduction, ambipolar diffusion, and Nernst/Ettingshausen effects) L 2 L2 L 2 § 3.9 holds and fails, respectively. The time axis is scaled by ( N / 64 ) 2 (N/64)^2 ( N /64 ) 2 O ( Δ x 2 ) O(\Delta x^2) O ( Δ x 2 )
Figure 4. (a) Well-posed thermo-magnetic system. κ ⊥ = 7800 \kappa_{\perp}=7800 κ ⊥ = 7800 η A D = 5200 \eta_{AD}=5200 η A D = 5200 κ J Λ = 6070 \kappa_{J\Lambda}=6070 κ J Λ = 6070 x i o n = 1 x_{\mathrm{ion}}=1 x ion = 1 κ ⊥ = 7800 \kappa_{\perp}=7800 κ ⊥ = 7800 η A D = 5200 \eta_{AD}=5200 η A D = 5200 κ J Λ = 9010 \kappa_{J\Lambda}=9010 κ J Λ = 9010 x i o n = 1 x_{\mathrm{ion}}=1 x ion = 1
Righi–Leduc Effect
We set up a Beltrami magnetic field with ∣ B 0 ∣ = 10 − 8 |\mathbf{B}_0|=10^{-8} ∣ B 0 ∣ = 1 0 − 8 b ^ = ( cos ( k B z ) , sin ( k B z ) , 0 ) \hat{\mathbf{b}}=(\cos(k_B z),\sin(k_B z),0) b ^ = ( cos ( k B z ) , sin ( k B z ) , 0 )
T ( x , y , z , 0 ) = T 0 + δ T 0 cos ( k T x ) . (151) T(x,y,z,0)=T_0+\delta T_0\cos(k_T x). \tag*{(151)} T ( x , y , z , 0 ) = T 0 + δ T 0 cos ( k T x ) . (151) This has the analytical solution
T ( x , y , z , t ) = T 0 + δ T cos ( D Λ k B k T cos ( k B z ) t − k T x ) (152) T(x,y,z,t)=T_0+\delta T\cos\left(D_{\Lambda}k_B k_T\cos(k_B z)t-k_T x\right) \tag*{(152)} T ( x , y , z , t ) = T 0 + δ T cos ( D Λ k B k T cos ( k B z ) t − k T x ) (152) with D Λ = ( γ − 1 ) x i o n κ Λ D_{\Lambda}=(\gamma-1)x_{\mathrm{ion}}\kappa_{\Lambda} D Λ = ( γ − 1 ) x ion κ Λ T 0 = 1 T_0=1 T 0 = 1 δ T = 10 − 6 \delta T=10^{-6} δ T = 1 0 − 6 k B = 2 π / L z k_B=2\pi/L_z k B = 2 π / L z k T = 2 π / L x k_T=2\pi/L_x k T = 2 π / L x
Figure 5 shows the Righi–Leduc L 2 L_2 L 2 ∣ ∇ ⃗ × b ⃗ ^ ∣ \left|\vec{\nabla}\times\hat{\vec{b}}\right| ∇ × b ^
Figure 5. Righi–Leduc effect.
Gyroviscosity
We take ideal MHD with only the Π ( 3 ) \mathbf{\Pi}^{(3)} Π ( 3 ) Π ( 4 ) \mathbf{\Pi}^{(4)} Π ( 4 ) z z z b ^ = ( 0 , 0 , 1 ) \hat{b} = (0,0,1) b ^ = ( 0 , 0 , 1 ) ∣ B 0 ∣ = 1 \lvert B_0\rvert= 1 ∣ B 0 ∣ = 1 v ⃗ ( x , y , z , 0 ) = v 0 ( sin ( k z z ) , sin ( k x x ) , 0 ) \vec{v}(x,y,z,0) = v_0(\sin(k_z z),\sin(k_x x),0) v ( x , y , z , 0 ) = v 0 ( sin ( k z z ) , sin ( k x x ) , 0 ) v 0 = 10 − 6 v_0 = 10^{-6} v 0 = 1 0 − 6
v ⃗ ( x , y , z , t ) = v 0 ( − sin ( x i o n ν ∧ 1 k x 2 t ) sin ( k x x ) + sin ( k z z ) cos ( 2 x i o n ν ∧ 2 k x 2 t ) sin ( 2 x i o n ν ∧ 2 k x 2 t ) sin ( k z z ) + sin ( k x x ) cos ( x i o n ν ∧ 1 k x 2 t ) 0 ) . (153) \vec{v}(x,y,z,t) = v_0
\begin{pmatrix}
-\sin\left(x_{\mathrm{ion}}\nu_{\wedge1}k_x^2t\right)\sin(k_xx) + \sin(k_zz)\cos\left(2x_{\mathrm{ion}}\nu_{\wedge2}k_x^2t\right) \\
\sin\left(2x_{\mathrm{ion}}\nu_{\wedge2}k_x^2t\right)\sin(k_zz) + \sin(k_xx)\cos\left(x_{\mathrm{ion}}\nu_{\wedge1}k_x^2t\right) \\
0
\end{pmatrix}. \tag*{(153)} v ( x , y , z , t ) = v 0 − sin ( x ion ν ∧ 1 k x 2 t ) sin ( k x x ) + sin ( k z z ) cos ( 2 x ion ν ∧ 2 k x 2 t ) sin ( 2 x ion ν ∧ 2 k x 2 t ) sin ( k z z ) + sin ( k x x ) cos ( x ion ν ∧ 1 k x 2 t ) 0 . (153) Figure 6 shows the gyroviscosity (Π ( 3 ) + Π ( 4 ) \mathbf{\Pi}^{(3)} + \mathbf{\Pi}^{(4)} Π ( 3 ) + Π ( 4 ) O ( Δ x 2 ) O(\Delta x^2) O ( Δ x 2 )
Figure 6. Gyro-viscosity test (Π ( 3 ) + Π ( 4 ) \mathbf{\Pi}^{(3)} + \mathbf{\Pi}^{(4)} Π ( 3 ) + Π ( 4 ) ν ∧ 1 = ν ∧ 2 = 5200 \nu_{\wedge1} = \nu_{\wedge2} = 5200 ν ∧ 1 = ν ∧ 2 = 5200 x i o n = 1 x_{\mathrm{ion}} = 1 x ion = 1
Peltier/Seebeck Effects (with Hall Effect)
We initialise the background temperature and magnetic field as T ( x , y , z , 0 ) = T 0 + δ T ( 0 ) cos ( k x x ) T(x,y,z,0) = T_0 + \delta T(0)\cos(k_xx) T ( x , y , z , 0 ) = T 0 + δ T ( 0 ) cos ( k x x ) B ⃗ ( x , y , z , 0 ) = ∣ B 0 ∣ ( cos ( θ ) , sin ( θ ) , 0 ) \vec{B}(x,y,z,0) = \lvert B_0\rvert(\cos(\theta),\sin(\theta),0) B ( x , y , z , 0 ) = ∣ B 0 ∣ ( cos ( θ ) , sin ( θ ) , 0 ) T 0 = 1 T_0 = 1 T 0 = 1 ∣ B 0 ∣ = 1 \lvert B_0\rvert= 1 ∣ B 0 ∣ = 1 δ T ( 0 ) = 10 − 6 \delta T(0) = 10^{-6} δ T ( 0 ) = 1 0 − 6 θ = π / 4 \theta= \pi/4 θ = π /4
With only the Hall effect and the parallel and perpendicular Peltier/Seebeck effects, this has the analytical solution
T ( x , y , z , t ) = T 0 + ( δ T 0 + 1 T 0 Ω 2 ρ 0 ( δ T 0 x i o n 2 ( γ − 1 ) ( κ J ∥ − κ J ⊥ ) 2 ( cos ( Ω k x 2 t ) − 1 ) sin 2 ( θ ) cos 2 ( θ ) ) ) cos ( k x x ) , (154) T(x,y,z,t) = T_0 + \left(\delta T_0 + \frac{1}{T_0\Omega^2\rho_0}\left(\delta T_0x_{\mathrm{ion}}^2(\gamma-1)(\kappa_{J\parallel}-\kappa_{J\perp})^2\left(\cos(\Omega k_x^2t)-1\right)\sin^2(\theta)\cos^2(\theta)\right)\right)\cos(k_xx), \tag*{(154)} T ( x , y , z , t ) = T 0 + ( δ T 0 + T 0 Ω 2 ρ 0 1 ( δ T 0 x ion 2 ( γ − 1 ) ( κ J ∥ − κ J ⊥ ) 2 ( cos ( Ω k x 2 t ) − 1 ) sin 2 ( θ ) cos 2 ( θ ) ) ) cos ( k x x ) , (154) B ⃗ ( x , y , z , t ) = ( B 0 cos ( θ ) B 0 sin ( θ ) − 1 T 0 Ω 2 ( δ T 0 η H x i o n ( κ J ∥ − κ J ⊥ ) ( cos ( Ω k x 2 t ) − 1 ) sin ( θ ) cos 2 ( θ ) cos ( k x x ) ) − 1 T 0 Ω 2 ( δ T 0 x i o n ( κ J ∥ − κ J ⊥ ) sin ( θ ) sin ( Ω k x 2 t ) cos ( θ ) cos ( k x x ) ) ) , (155) \vec{B}(x,y,z,t) =
\begin{pmatrix}
B_0\cos(\theta) \\
B_0\sin(\theta) - \frac{1}{T_0\Omega^2}\left(\delta T_0\eta_Hx_{\mathrm{ion}}(\kappa_{J\parallel}-\kappa_{J\perp})\left(\cos(\Omega k_x^2t)-1\right)\sin(\theta)\cos^2(\theta)\cos(k_xx)\right) \\
-\frac{1}{T_0\Omega^2}\left(\delta T_0x_{\mathrm{ion}}(\kappa_{J\parallel}-\kappa_{J\perp})\sin(\theta)\sin(\Omega k_x^2t)\cos(\theta)\cos(k_xx)\right)
\end{pmatrix}, \tag*{(155)} B ( x , y , z , t ) = B 0 cos ( θ ) B 0 sin ( θ ) − T 0 Ω 2 1 ( δ T 0 η H x ion ( κ J ∥ − κ J ⊥ ) ( cos ( Ω k x 2 t ) − 1 ) sin ( θ ) cos 2 ( θ ) cos ( k x x ) ) − T 0 Ω 2 1 ( δ T 0 x ion ( κ J ∥ − κ J ⊥ ) sin ( θ ) sin ( Ω k x 2 t ) cos ( θ ) cos ( k x x ) ) , (155) with
Ω 2 = cos 2 ( θ ) T 0 ρ 0 ( T 0 η H 2 ρ 0 + γ κ J ∥ 2 x i o n 2 sin 2 ( θ ) − 2 γ κ J ∥ κ J ⊥ x i o n 2 sin 2 ( θ ) + γ κ J ⊥ 2 x i o n 2 sin 2 ( θ ) − κ J ∥ 2 x i o n 2 sin 2 ( θ ) + 2 κ J ∥ κ J ⊥ x i o n 2 sin 2 ( θ ) − κ J ⊥ 2 x i o n 2 sin 2 ( θ ) ) . (156) \Omega^2 = \frac{\cos^2(\theta)}{T_0\rho_0}\left(T_0\eta_H^2\rho_0+\gamma\kappa_{J\parallel}^2x_{\mathrm{ion}}^2\sin^2(\theta)-2\gamma\kappa_{J\parallel}\kappa_{J\perp}x_{\mathrm{ion}}^2\sin^2(\theta)+\gamma\kappa_{J\perp}^2x_{\mathrm{ion}}^2\sin^2(\theta)-\kappa_{J\parallel}^2x_{\mathrm{ion}}^2\sin^2(\theta)+2\kappa_{J\parallel}\kappa_{J\perp}x_{\mathrm{ion}}^2\sin^2(\theta)-\kappa_{J\perp}^2x_{\mathrm{ion}}^2\sin^2(\theta)\right). \tag*{(156)} Ω 2 = T 0 ρ 0 cos 2 ( θ ) ( T 0 η H 2 ρ 0 + γ κ J ∥ 2 x ion 2 sin 2 ( θ ) − 2 γ κ J ∥ κ J ⊥ x ion 2 sin 2 ( θ ) + γ κ J ⊥ 2 x ion 2 sin 2 ( θ ) − κ J ∥ 2 x ion 2 sin 2 ( θ ) + 2 κ J ∥ κ J ⊥ x ion 2 sin 2 ( θ ) − κ J ⊥ 2 x ion 2 sin 2 ( θ ) ) . (156) We run with and without the Hall effect enabled to verify the CFL condition holds at RK3, and to confirm the Peltier/Seebeck physics alone is correctly implemented.
Figure 7 shows the Peltier/Seebeck (+Hall) L2 errors across RK1 (without the Hall effect only), RK2, and RK3 (with the standard and modified HLLE Riemann solver) time integration. Subfigures 7a and 7b are without and with the Hall effect, respectively. The Peltier/Seebeck tests fail at RK1 as expected, blow-up times scaling with O ( Δ x 2 ) O(\Delta x^{2}) O ( Δ x 2 ) η H = 0 \eta_{H}=0 η H = 0 T = T 0 + δ T 0 cos ( Ω k x 2 t ) cos ( k x x ) T=T_{0}+\delta T_{0}\cos(\Omega k_{x}^{2}t)\cos(k_{x}x) T = T 0 + δ T 0 cos ( Ω k x 2 t ) cos ( k x x ) Ω 2 = ( γ − 1 ) x i o n 2 Δ κ J 2 sin 2 θ cos 2 θ / ( ρ 0 T 0 ) \Omega^{2}=(\gamma-1)x_{\mathrm{ion}}^{2}\Delta\kappa_{J}^{2}\sin^{2}\theta\cos^{2}\theta/(\rho_{0}T_{0}) Ω 2 = ( γ − 1 ) x ion 2 Δ κ J 2 sin 2 θ cos 2 θ / ( ρ 0 T 0 ) 64 3 64^3 6 4 3 ∣ B ⃗ ∣ \lvert\vec{B} \rvert ∣ B ∣ ≈ 1 \approx1 ≈ 1
Figure 7. Peltier/Seebeck (+Hall) tests.
The RK1 runs are omitted from Peltier/Seebeck + Hall as the timestep collapses after the first few steps, causing the simulation to effectively stall.
Table 2 summarises the convergence rates across all dissipative physics tests. For well-posed tests, RK1 runs converge at second order for all configurations. Using RKL1 the convergence rate is initially greater than two on coarser runs, but approaches two as resolution increases. The super-step is capped at 50 parabolic timesteps, it therefore scales as Δ x 2 \Delta x^2 Δ x 2 Δ x \Delta x Δ x
Table 3 shows the convergence and blow-up times for the non-dissipative tests. As discussed previously, several tests appear to show RK2 convergence, but this is an artefact of the test problem and is not general. The growing oscillations are far smaller at RK2 than at RK1, allowing the simulations to appear stable over the duration of the test. We are also able to capture the failure of RK2 for the Hall effect via its negative convergence rate. HLLE-Mod can lower the order of convergence down to one due to the increased numerical dissipation: the modified signal speeds scale as 1 / Δ x 1/\Delta x 1/Δ x [15 ] . We expect on longer runs, for the convergence rates to eventually increase and approach order 3 once the spatial discretisation error and numerical diffusion are subdominant.
Our convergence orders match those reported for similar tests against analytical solutions in other codes. [41 ] tested anisotropic (parallel) thermal conduction with a diffusing Gaussian in PLUTO, finding second-order convergence of the L1 error with RKL2, but first-order convergence with RKL1. This is because the super-step was scaled with Δ x \Delta x Δ x Δ x 2 \Delta x^2 Δ x 2 [6 ] found second-order convergence of the root-mean-square difference between the simulated and analytical internal energy for Braginskii (parallel) viscosity in AREPO with RKL2.
Tests with discontinuous initial conditions converge more slowly. [37 ] tested anisotropic thermal conduction (parallel and perpendicular) with a hot patch diffusing along circular field lines. They report L2 errors that fall only from 0.025–0.063 at 50 2 50^2 5 0 2 400 2 400^2 40 0 2 [5 ] find L1 convergence orders of 0.41–0.49 for a similar test with parallel thermal conduction.
For the Hall effect, [15 ] found relative L1 errors of 0.42, 0.085, 0.019, and 0.0052 after one whistler wave period with 16, 32, 64, and 128 cells per wavelength respectively. They used a second-order Runge–Kutta scheme with the MC limiter. Our RK3 Hall test converges at a similar rate, with an error of about 10 − 2 10^{-2} 1 0 − 2 64 3 64^3 6 4 3
In Athena++ with RK3, [12 ] found second-order convergence of the L2 error of linear waves of relative amplitude 10 − 6 10^{-6} 1 0 − 6
We are not aware of comparable published tests for the thermoelectric, Righi–Leduc, or gyroviscous terms.
HLLE-Mod Tests
To test HLLE-mod we initialise a white noise background for the relevant variables: temperature for Righi–Leduc, field for Hall, field and temperature for Peltier/Seebeck (+ Hall), velocity for gyroviscosity, and all variables for a test with all effects active.
We perform a similar white noise test to [12 ] . We seed Gaussian white noise in the temperature, velocity, and magnetic field, and compare RK3 with the standard HLLE solver against RK3 with HLLE-mod.
Each noise field is generated from independent Gaussian deviates in every cell, filtered in Fourier space, and scaled to an RMS amplitude; the magnetic noise is the discrete curl of a filtered vector potential. The procedure is given in full in Appendix § 14 .
We use a periodic box of side L = 4 L = 4 L = 4 N = 32 3 N = 32^3 N = 3 2 3 Δ x = 1 / 8 \Delta x = 1/8 Δ x = 1/8 ρ 0 = 1 \rho_0 = 1 ρ 0 = 1 P 0 = 1 / γ P_0 = 1/\gamma P 0 = 1/ γ c s = 1 c_s = 1 c s = 1 B 0 = 0 \mathbf{B}_0 = \mathbf{0} B 0 = 0
We set the effective dispersivity of every term to α e f f = 10 4 \alpha_{\rm eff} = 10^4 α eff = 1 0 4 x i o n = 1 x_{\rm ion} = 1 x ion = 1 η H = 10 4 \eta_H = 10^4 η H = 1 0 4 ( γ − 1 ) κ ∧ = 10 4 (\gamma- 1)\kappa_\wedge= 10^4 ( γ − 1 ) κ ∧ = 1 0 4 ∣ κ J ∥ − κ J ⊥ ∣ / ( 2 ϵ i n t ) = 10 4 \lvert\kappa_{J\parallel} - \kappa_{J\perp}\rvert/(2\sqrt{\epsilon_{\rm int}}) = 10^4 ∣ κ J ∥ − κ J ⊥ ∣ / ( 2 ϵ int ) = 1 0 4 ν ∧ 1 = ν ∧ 2 \nu_{\wedge1} = \nu_{\wedge2} ν ∧ 1 = ν ∧ 2 2 ν ∧ 2 = 10 4 2\nu_{\wedge2} = 10^4 2 ν ∧ 2 = 1 0 4 α e f f \alpha_{\rm eff} α eff
We run to 20 t w 20t_w 20 t w t w = 2 Δ x 2 / ( π α e f f ) ≈ 9.95 × 10 − 7 t_w = 2\Delta x^2/(\pi\alpha_{\rm eff}) \approx9.95 \times10^{-7} t w = 2Δ x 2 / ( π α eff ) ≈ 9.95 × 1 0 − 7 ω = α e f f k 2 \omega= \alpha_{\rm eff}k^2 ω = α eff k 2 σ B = 10 − 6 \sigma_B = 10^{-6} σ B = 1 0 − 6 σ B = 1 \sigma_B = 1 σ B = 1 [12 ] , since only the field is seeded and measured there. We set σ T = 10 − 6 \sigma_T = 10^{-6} σ T = 1 0 − 6 σ v = 10 − 6 \sigma_v = 10^{-6} σ v = 1 0 − 6
Figure 8 (PDF p. 33). (e) Peltier/Seebeck + Hall. (f) All effects. Panel (e) legend: RK3, T T T ∣ B ˉ ∣ |\bar{B}| ∣ B ˉ ∣ T T T ∣ B ˉ ∣ |\bar{B}| ∣ B ˉ ∣ T T T ∣ v ˉ ∣ |\bar{v}| ∣ v ˉ ∣ ∣ B ˉ ∣ |\bar{B}| ∣ B ˉ ∣ T T T ∣ v ˉ ∣ |\bar{v}| ∣ v ˉ ∣ ∣ B ˉ ∣ |\bar{B}| ∣ B ˉ ∣
Figure 8 shows the RMS of each seeded variable divided by its initial value. Using the standard HLLE Riemann solver the noise grows in every run under RK3. The Righi–Leduc, Peltier/Seebeck, and all-effects runs blow up between 17 t w 17t_w 17 t w 19 t w 19t_w 19 t w 2.5 t w 2.5t_w 2.5 t w 20 t w 20t_w 20 t w § 3 , which treat b ^ \hat{b} b ^ b ^ \hat{b} b ^ 20 t w 20t_w 20 t w
Figure 8. RMS of variables for each non-dissipative effect. (a) Righi–Leduc. (b) Gyroviscosity. (c) Hall Effect. (d) Peltier/Seebeck. The axes are t / t w t/t_w t / t w T T T T T T ∣ v ∣ \lvert\mathbf{v}\rvert ∣ v ∣ ∣ v ∣ \lvert\mathbf{v}\rvert ∣ v ∣ ∣ B ∣ \lvert\mathbf{B}\rvert ∣ B ∣ ∣ B ∣ \lvert\mathbf{B}\rvert ∣ B ∣ T T T ∣ B ∣ \lvert\mathbf{B}\rvert ∣ B ∣ T T T ∣ B ∣ \lvert\mathbf{B}\rvert ∣ B ∣
Figure 9 shows the z = 0 z = 0 z = 0 x y xy x y t = 0 t = 0 t = 0 17 t w 17t_w 17 t w t = 20 t w t = 20t_w t = 20 t w δ T \delta T δ T δ v x \delta v_x δ v x δ B x \delta B_x δ B x 20 t w 20t_w 20 t w
Figure 9. Slices through the noise fields at t = 0 t = 0 t = 0 t = 20 t w t = 20t_w t = 20 t w § 14 .
Conclusions
We have implemented anisotropic thermal conduction, bulk and anisotropic viscosity, the Hall effect, and the thermoelectric effects of the Braginskii closure in Athena++. The thermal conduction comprises an isotropic neutral component and the parallel, perpendicular, and Righi–Leduc components of the ionised gas. The viscosity comprises the molecular bulk viscosity, an isotropic neutral component, and all five Braginskii ion viscosity tensors. In the neutral and ionised parts are weighted by the ionisation fraction. The thermoelectric terms are the Peltier, Seebeck, Nernst, and Ettingshausen effects. All terms enter through the face fluxes and the edge-centred electric fields of Athena++, and the dissipative parts can be integrated with super-time-stepping.
We have derived timestep limits for every term, valid for any Runge–Kutta integrator through the extent of its stability region along the real and imaginary axes. The dissipative terms (thermal conduction, viscosity, and magnetic diffusion) need only the negative real axis, so forward Euler suffices. The non-dissipative terms (the Hall, Righi–Leduc, gyroviscous, and Peltier/Seebeck terms) require a stability region that includes part of the imaginary axis, such as that of RK3. The Nernst/Ettingshausen terms are ill-posed on their own. We have derived the criterion for well-posedness: ( γ − 1 ) ( ( 1 − x i o n ) κ n + x i o n κ ⊥ ) ( η + η A D ) ≥ x i o n 2 κ ∥ 2 / ε i n t (\gamma-1)\left((1-x_{\mathrm{ion}})\kappa_n+x_{\mathrm{ion}}\kappa_\perp\right)(\eta+\eta_{\mathrm{AD}}) \ge x_{\mathrm{ion}}^2\kappa_\parallel^2/\varepsilon_{\mathrm{int}} ( γ − 1 ) ( ( 1 − x ion ) κ n + x ion κ ⊥ ) ( η + η AD ) ≥ x ion 2 κ ∥ 2 / ε int
The new signal speeds in HLLE-Mod keep runs with grid-scale noise bounded where RK3 with the standard solver fails. The cost is excess numerical dissipation when the non-dissipative terms are strong, which lowers the convergence order to between 1.1 and 1.9. To reduce this without resorting to hyper-resistivity, the approach could be extended to lower-dissipation HLL-type solvers, such as HLLD or LDHLLD [42 ] . This would require a derivation for the intermediate states of a Riemann fan containing the new non-dissipative waves.
The Biermann battery, the one Ohm’s-law term of the single-fluid Braginskii closure we have not implemented, is harder to treat than the others. First, a direct discretisation generates spurious field at shocks that grows with resolution, the ‘Biermann catastrophe’ [43 ] . Graziani et al. obtained convergence by writing the term in terms of the electron temperature, which is continuous through a shock only when it is evolved separately from the ion temperature. Second, the battery term linearised on its own is nilpotent, with only zero eigenvalues, so its stability limit comes from its second-order coupling to the internal-energy equation. Once coupled, the term supports thermal–magnetic waves whose speed depends on the background gradients, requiring new signal speeds and a further stability analysis. A single-fluid, single-temperature implementation exists [44 ] ; adapting it to Athena++ is left to future work.
Code availability
Athena-Aniso is available at https://github.com/AlexandraAlbery/Athena-Aniso DOI: 10.5281/zen-odo.23887593, and the repository documentation describes the input parameters and the interface for supplying spatially varying coefficients.
The scripts used to verify the analytical and numerical results in this paper are available at https://github.com/AlexandraAlbery/athena-aniso-python-scripts
Data availability
Data files for the tests of Section 6 are available upon a reasonable request to the corresponding author.
CRediT Authorship Contribution Statement
Alexandra F. Albery: Conceptualisation, Methodology, Software, Validation, Formal analysis, Investigation, Writing - Original Draft, Visualisation.
Julian M. Pittard: Writing - Review & Editing, Supervision, Project administration, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the United Kingdom Research and Innovation (UKRI) Science and Technology Facilities Council (STFC) grant ST/Y509711/1.
Declaration of Generative AI and AI-assisted Technologies in the Manuscript Preparation Process
During the preparation of this work, the authors used Claude Opus 5.5 (Anthropic) for proof reading and formatting of LaTeX. After using this service, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.
1D Viscosity Terms
W = ( 4 3 ∂ x v x ∂ x v y ∂ x v z ∂ x v y − 2 3 ∂ x v x 0 ∂ x v z 0 − 2 3 ∂ x v x ) (157) W =
\begin{pmatrix}
\frac{4}{3}\partial_x v_x & \partial_x v_y & \partial_x v_z \\
\partial_x v_y & -\frac{2}{3}\partial_x v_x & 0 \\
\partial_x v_z & 0 & -\frac{2}{3}\partial_x v_x
\end{pmatrix} \tag*{(157)} W = 3 4 ∂ x v x ∂ x v y ∂ x v z ∂ x v y − 3 2 ∂ x v x 0 ∂ x v z 0 − 3 2 ∂ x v x (157) b ^ ⋅ W ⋅ b ^ = b x [ 4 3 b x ∂ x v x + b y ∂ x v y + b z ∂ x v z ] + b y [ b x ∂ x v y − 2 3 b y ∂ x v x ] + b z [ b x ∂ x v z − 2 3 b z ∂ x v x ] (158) \hat{b}\cdot W\cdot\hat{b}
= b_x\left[\frac{4}{3}b_x\partial_x v_x+b_y\partial_x v_y+b_z\partial_x v_z\right]
+b_y\left[b_x\partial_x v_y-\frac{2}{3}b_y\partial_x v_x\right]
+b_z\left[b_x\partial_x v_z-\frac{2}{3}b_z\partial_x v_x\right] \tag*{(158)} b ^ ⋅ W ⋅ b ^ = b x [ 3 4 b x ∂ x v x + b y ∂ x v y + b z ∂ x v z ] + b y [ b x ∂ x v y − 3 2 b y ∂ x v x ] + b z [ b x ∂ x v z − 3 2 b z ∂ x v x ] (158) b ^ ⊗ b ^ − 1 3 I = ( b x 2 − 1 3 b x b y b x b z b y b x b y 2 − 1 3 b y b z b z b x b z b y b z 2 − 1 3 ) (159) \hat{b}\otimes\hat{b}-\frac{1}{3}I
=
\begin{pmatrix}
b_x^2-\frac{1}{3} & b_xb_y & b_xb_z \\
b_yb_x & b_y^2-\frac{1}{3} & b_yb_z \\
b_zb_x & b_zb_y & b_z^2-\frac{1}{3}
\end{pmatrix} \tag*{(159)} b ^ ⊗ b ^ − 3 1 I = b x 2 − 3 1 b y b x b z b x b x b y b y 2 − 3 1 b z b y b x b z b y b z b z 2 − 3 1 (159) I ⊥ = ( 1 − b x 2 − b x b y − b x b z − b y b x 1 − b y 2 − b y b z − b z b x − b z b y 1 − b z 2 ) (160) I_{\perp}=
\begin{pmatrix}
1-b_x^2 & -b_xb_y & -b_xb_z \\
-b_yb_x & 1-b_y^2 & -b_yb_z \\
-b_zb_x & -b_zb_y & 1-b_z^2
\end{pmatrix} \tag*{(160)} I ⊥ = 1 − b x 2 − b y b x − b z b x − b x b y 1 − b y 2 − b z b y − b x b z − b y b z 1 − b z 2 (160) ( W ⊥ ) x x = ( 1 − b x 2 ) ( 2 3 ( 2 − 3 b x 2 ) ∂ x v x − 2 b x b y ∂ x v y − 2 b x b z ∂ x v z ) (161) (W_{\perp})_{xx}
=(1-b_x^2)\left(\frac{2}{3}(2-3b_x^2)\partial_xv_x-2b_xb_y\partial_xv_y-2b_xb_z\partial_xv_z\right) \tag*{(161)} ( W ⊥ ) xx = ( 1 − b x 2 ) ( 3 2 ( 2 − 3 b x 2 ) ∂ x v x − 2 b x b y ∂ x v y − 2 b x b z ∂ x v z ) (161) ( W ⊥ ) y y = [ 2 b x 2 b y 2 − 2 3 ( 1 − b y 2 ) ] ∂ x v x − 2 b x b y ( 1 − b y 2 ) ∂ x v y + 2 b x b y 2 b z ∂ x v z (162) (W_{\perp})_{yy}
=\left[2b_x^2b_y^2-\frac{2}{3}(1-b_y^2)\right]\partial_xv_x-2b_xb_y(1-b_y^2)\partial_xv_y+2b_xb_y^2b_z\partial_xv_z \tag*{(162)} ( W ⊥ ) y y = [ 2 b x 2 b y 2 − 3 2 ( 1 − b y 2 ) ] ∂ x v x − 2 b x b y ( 1 − b y 2 ) ∂ x v y + 2 b x b y 2 b z ∂ x v z (162) ( W ⊥ ) z z = [ 2 b x 2 b z 2 − 2 3 ( 1 − b z 2 ) ] ∂ x v x + 2 b x b y b z 2 ∂ x v y − 2 b x b z ( 1 − b z 2 ) ∂ x v z (163) (W_{\perp})_{zz}
=\left[2b_x^2b_z^2-\frac{2}{3}(1-b_z^2)\right]\partial_xv_x+2b_xb_yb_z^2\partial_xv_y-2b_xb_z(1-b_z^2)\partial_xv_z \tag*{(163)} ( W ⊥ ) z z = [ 2 b x 2 b z 2 − 3 2 ( 1 − b z 2 ) ] ∂ x v x + 2 b x b y b z 2 ∂ x v y − 2 b x b z ( 1 − b z 2 ) ∂ x v z (163) ( W ⊥ ) x y = ( W ⊥ ) y x = 2 b x b y ( b x 2 − 2 3 ) ∂ x v x + ( b z 2 + 2 b x 2 b y 2 ) ∂ x v y + b y b z ( 2 b x 2 − 1 ) ∂ x v z (164) (W_{\perp})_{xy}=(W_{\perp})_{yx}
=2b_xb_y\left(b_x^2-\frac{2}{3}\right)\partial_xv_x+(b_z^2+2b_x^2b_y^2)\partial_xv_y+b_yb_z(2b_x^2-1)\partial_xv_z \tag*{(164)} ( W ⊥ ) x y = ( W ⊥ ) y x = 2 b x b y ( b x 2 − 3 2 ) ∂ x v x + ( b z 2 + 2 b x 2 b y 2 ) ∂ x v y + b y b z ( 2 b x 2 − 1 ) ∂ x v z (164) ( W ⊥ ) x z = ( W ⊥ ) z x = 2 b x b z ( b x 2 − 2 3 ) ∂ x v x + b y b z ( 2 b x 2 − 1 ) ∂ x v y + ( b y 2 + 2 b x 2 b z 2 ) ∂ x v z (165) (W_{\perp})_{xz}=(W_{\perp})_{zx}
=2b_xb_z\left(b_x^2-\frac{2}{3}\right)\partial_xv_x+b_yb_z(2b_x^2-1)\partial_xv_y+(b_y^2+2b_x^2b_z^2)\partial_xv_z \tag*{(165)} ( W ⊥ ) x z = ( W ⊥ ) z x = 2 b x b z ( b x 2 − 3 2 ) ∂ x v x + b y b z ( 2 b x 2 − 1 ) ∂ x v y + ( b y 2 + 2 b x 2 b z 2 ) ∂ x v z (165) ( W ⊥ ) y z = ( W ⊥ ) z y = 2 b y b z ( b x 2 + 1 3 ) ∂ x v x + b x b z ( 2 b y 2 − 1 ) ∂ x v y + b x b y ( 2 b z 2 − 1 ) ∂ x v z (166) (W_{\perp})_{yz}=(W_{\perp})_{zy}
=2b_yb_z\left(b_x^2+\frac{1}{3}\right)\partial_xv_x+b_xb_z(2b_y^2-1)\partial_xv_y+b_xb_y(2b_z^2-1)\partial_xv_z \tag*{(166)} ( W ⊥ ) y z = ( W ⊥ ) z y = 2 b y b z ( b x 2 + 3 1 ) ∂ x v x + b x b z ( 2 b y 2 − 1 ) ∂ x v y + b x b y ( 2 b z 2 − 1 ) ∂ x v z (166) ( M 1 ) x x = 4 b x 2 ( 1 − b x 2 ) ∂ x v x + 2 b x b y ( 1 − 2 b x 2 ) ∂ x v y + 2 b x b z ( 1 − 2 b x 2 ) ∂ x v z (167) (M_1)_{xx}=4b_x^2(1-b_x^2)\partial_xv_x+2b_xb_y(1-2b_x^2)\partial_xv_y+2b_xb_z(1-2b_x^2)\partial_xv_z \tag*{(167)} ( M 1 ) xx = 4 b x 2 ( 1 − b x 2 ) ∂ x v x + 2 b x b y ( 1 − 2 b x 2 ) ∂ x v y + 2 b x b z ( 1 − 2 b x 2 ) ∂ x v z (167) ( M 1 ) y y = − 4 b x 2 b y 2 ∂ x v x + 2 b x b y ( 1 − 2 b y 2 ) ∂ x v y − 4 b x b y 2 b z ∂ x v z (168) (M_1)_{yy}=-4b_x^2b_y^2\partial_xv_x+2b_xb_y(1-2b_y^2)\partial_xv_y-4b_xb_y^2b_z\partial_xv_z \tag*{(168)} ( M 1 ) y y = − 4 b x 2 b y 2 ∂ x v x + 2 b x b y ( 1 − 2 b y 2 ) ∂ x v y − 4 b x b y 2 b z ∂ x v z (168) ( M 1 ) z z = − 4 b x 2 b z 2 ∂ x v x − 4 b x b y b z 2 ∂ x v y + 2 b x b z ( 1 − 2 b z 2 ) ∂ x v z (169) (M_1)_{zz}=-4b_x^2b_z^2\partial_xv_x-4b_xb_yb_z^2\partial_xv_y+2b_xb_z(1-2b_z^2)\partial_xv_z \tag*{(169)} ( M 1 ) z z = − 4 b x 2 b z 2 ∂ x v x − 4 b x b y b z 2 ∂ x v y + 2 b x b z ( 1 − 2 b z 2 ) ∂ x v z (169) ( M 1 ) x y = ( M 1 ) y x = 2 b x b y ( 1 − 2 b x 2 ) ∂ x v x + ( b x 2 + b y 2 − 4 b x 2 b y 2 ) ∂ x v y + b y b z ( 1 − 4 b x 2 ) ∂ x v z (170) (M_1)_{xy}=(M_1)_{yx}
=2b_xb_y(1-2b_x^2)\partial_xv_x+(b_x^2+b_y^2-4b_x^2b_y^2)\partial_xv_y+b_yb_z(1-4b_x^2)\partial_xv_z \tag*{(170)} ( M 1 ) x y = ( M 1 ) y x = 2 b x b y ( 1 − 2 b x 2 ) ∂ x v x + ( b x 2 + b y 2 − 4 b x 2 b y 2 ) ∂ x v y + b y b z ( 1 − 4 b x 2 ) ∂ x v z (170) ( M 1 ) x z = ( M 1 ) z x = 2 b x b z ( 1 − 2 b x 2 ) ∂ x v x + b y b z ( 1 − 4 b x 2 ) ∂ x v y + ( b x 2 + b z 2 − 4 b x 2 b z 2 ) ∂ x v z (171) (M_1)_{xz}=(M_1)_{zx}
=2b_xb_z(1-2b_x^2)\partial_xv_x+b_yb_z(1-4b_x^2)\partial_xv_y+(b_x^2+b_z^2-4b_x^2b_z^2)\partial_xv_z \tag*{(171)} ( M 1 ) x z = ( M 1 ) z x = 2 b x b z ( 1 − 2 b x 2 ) ∂ x v x + b y b z ( 1 − 4 b x 2 ) ∂ x v y + ( b x 2 + b z 2 − 4 b x 2 b z 2 ) ∂ x v z (171) ( M 1 ) y z = ( M 1 ) z y = − 4 b x 2 b y b z ∂ x v x + b x b z ( 1 − 4 b y 2 ) ∂ x v y + b x b y ( 1 − 4 b z 2 ) ∂ x v z (172) (M_1)_{yz}=(M_1)_{zy}
=-4b_x^2b_yb_z\partial_xv_x+b_xb_z(1-4b_y^2)\partial_xv_y+b_xb_y(1-4b_z^2)\partial_xv_z \tag*{(172)} ( M 1 ) y z = ( M 1 ) z y = − 4 b x 2 b y b z ∂ x v x + b x b z ( 1 − 4 b y 2 ) ∂ x v y + b x b y ( 1 − 4 b z 2 ) ∂ x v z (172) where: I ⊥ ⋅ W ⋅ ( b ^ ⊗ b ^ ) + ( b ^ ⊗ b ^ ) ⋅ W ⋅ I ⊥ = M I_{\perp}\cdot W\cdot(\hat{b}\otimes\hat{b})+(\hat{b}\otimes\hat{b})\cdot W\cdot I_{\perp}=M I ⊥ ⋅ W ⋅ ( b ^ ⊗ b ^ ) + ( b ^ ⊗ b ^ ) ⋅ W ⋅ I ⊥ = M
( W ⋅ I ⊥ ) x x = 4 3 ∂ x v x ( 1 − b x 2 ) − b x b y ∂ x v y − b x b z ∂ x v z , (173) (\mathbf{W}\cdot\mathbf{I}_{\perp})_{xx}=\frac{4}{3}\partial_xv_x(1-b_x^2)-b_xb_y\partial_xv_y-b_xb_z\partial_xv_z, \tag*{(173)} ( W ⋅ I ⊥ ) xx = 3 4 ∂ x v x ( 1 − b x 2 ) − b x b y ∂ x v y − b x b z ∂ x v z , (173) ( W ⋅ I ⊥ ) y y = − 2 3 ( 1 − b y 2 ) ∂ x v x − b x b y ∂ x v y , (174) (\mathbf{W}\cdot\mathbf{I}_{\perp})_{yy}=-\frac{2}{3}(1-b_y^2)\partial_xv_x-b_xb_y\partial_xv_y, \tag*{(174)} ( W ⋅ I ⊥ ) y y = − 3 2 ( 1 − b y 2 ) ∂ x v x − b x b y ∂ x v y , (174) ( W ⋅ I ⊥ ) z z = − 2 3 ( 1 − b z 2 ) ∂ x v x − b x b z ∂ x v z , (175) (\mathbf{W}\cdot\mathbf{I}_{\perp})_{zz}=-\frac{2}{3}(1-b_z^2)\partial_xv_x-b_xb_z\partial_xv_z, \tag*{(175)} ( W ⋅ I ⊥ ) z z = − 3 2 ( 1 − b z 2 ) ∂ x v x − b x b z ∂ x v z , (175) ( W ⋅ I ⊥ ) x y = − 4 3 b x b y ∂ x v x + ( 1 − b y 2 ) ∂ x v y − b y b z ∂ x v z , (176) (\mathbf{W}\cdot\mathbf{I}_{\perp})_{xy}=-\frac{4}{3}b_xb_y\partial_xv_x+(1-b_y^2)\partial_xv_y-b_yb_z\partial_xv_z, \tag*{(176)} ( W ⋅ I ⊥ ) x y = − 3 4 b x b y ∂ x v x + ( 1 − b y 2 ) ∂ x v y − b y b z ∂ x v z , (176) ( W ⋅ I ⊥ ) x z = − 4 3 ∂ x v x b x b z − b y b z ∂ x v y + ( 1 − b z 2 ) ∂ x v z , (177) (\mathbf{W}\cdot\mathbf{I}_{\perp})_{xz}=-\frac{4}{3}\partial_xv_xb_xb_z-b_yb_z\partial_xv_y+(1-b_z^2)\partial_xv_z, \tag*{(177)} ( W ⋅ I ⊥ ) x z = − 3 4 ∂ x v x b x b z − b y b z ∂ x v y + ( 1 − b z 2 ) ∂ x v z , (177) ( W ⋅ I ⊥ ) y x = 2 3 b x b y ∂ x v x + ( 1 − b x 2 ) ∂ x v y , (178) (\mathbf{W}\cdot\mathbf{I}_{\perp})_{yx}=\frac{2}{3}b_xb_y\partial_xv_x+(1-b_x^2)\partial_xv_y, \tag*{(178)} ( W ⋅ I ⊥ ) y x = 3 2 b x b y ∂ x v x + ( 1 − b x 2 ) ∂ x v y , (178) ( W ⋅ I ⊥ ) y z = 2 3 b y b z ∂ x v x − b x b z ∂ x v y , (179) (\mathbf{W}\cdot\mathbf{I}_{\perp})_{yz}=\frac{2}{3}b_yb_z\partial_xv_x-b_xb_z\partial_xv_y, \tag*{(179)} ( W ⋅ I ⊥ ) y z = 3 2 b y b z ∂ x v x − b x b z ∂ x v y , (179) ( W ⋅ I ⊥ ) z x = 2 3 b x b z ∂ x v x + ( 1 − b x 2 ) ∂ x v z , (180) (\mathbf{W}\cdot\mathbf{I}_{\perp})_{zx}=\frac{2}{3}b_xb_z\partial_xv_x+(1-b_x^2)\partial_xv_z, \tag*{(180)} ( W ⋅ I ⊥ ) z x = 3 2 b x b z ∂ x v x + ( 1 − b x 2 ) ∂ x v z , (180) ( W ⋅ I ⊥ ) z y = 2 3 b y b z ∂ x v x − b x b y ∂ x v z . (181) (\mathbf{W}\cdot\mathbf{I}_{\perp})_{zy}=\frac{2}{3}b_yb_z\partial_xv_x-b_xb_y\partial_xv_z. \tag*{(181)} ( W ⋅ I ⊥ ) z y = 3 2 b y b z ∂ x v x − b x b y ∂ x v z . (181) ( I ⊥ ⋅ W ) x x = 4 3 ∂ x v x ( 1 − b x 2 ) − b x b y ∂ x v y − b x b z ∂ x v z , (182) (\mathbf{I}_{\perp}\cdot\mathbf{W})_{xx}=\frac{4}{3}\partial_xv_x(1-b_x^2)-b_xb_y\partial_xv_y-b_xb_z\partial_xv_z, \tag*{(182)} ( I ⊥ ⋅ W ) xx = 3 4 ∂ x v x ( 1 − b x 2 ) − b x b y ∂ x v y − b x b z ∂ x v z , (182) ( I ⊥ ⋅ W ) y y = − 2 3 ( 1 − b y 2 ) ∂ x v x − b x b y ∂ x v y , (183) (\mathbf{I}_{\perp}\cdot\mathbf{W})_{yy}=-\frac{2}{3}(1-b_y^2)\partial_xv_x-b_xb_y\partial_xv_y, \tag*{(183)} ( I ⊥ ⋅ W ) y y = − 3 2 ( 1 − b y 2 ) ∂ x v x − b x b y ∂ x v y , (183) ( I ⊥ ⋅ W ) z z = − 2 3 ( 1 − b z 2 ) ∂ x v x − b x b z ∂ x v z , (184) (\mathbf{I}_{\perp}\cdot\mathbf{W})_{zz}=-\frac{2}{3}(1-b_z^2)\partial_xv_x-b_xb_z\partial_xv_z, \tag*{(184)} ( I ⊥ ⋅ W ) z z = − 3 2 ( 1 − b z 2 ) ∂ x v x − b x b z ∂ x v z , (184) ( I ⊥ ⋅ W ) x y = 2 3 b x b y ∂ x v x + ( 1 − b x 2 ) ∂ x v y , (185) (\mathbf{I}_{\perp}\cdot\mathbf{W})_{xy}=\frac{2}{3}b_xb_y\partial_xv_x+(1-b_x^2)\partial_xv_y, \tag*{(185)} ( I ⊥ ⋅ W ) x y = 3 2 b x b y ∂ x v x + ( 1 − b x 2 ) ∂ x v y , (185) ( I ⊥ ⋅ W ) x z = 2 3 b x b z ∂ x v x + ( 1 − b x 2 ) ∂ x v z , (186) (\mathbf{I}_{\perp}\cdot\mathbf{W})_{xz}=\frac{2}{3}b_xb_z\partial_xv_x+(1-b_x^2)\partial_xv_z, \tag*{(186)} ( I ⊥ ⋅ W ) x z = 3 2 b x b z ∂ x v x + ( 1 − b x 2 ) ∂ x v z , (186) ( I ⊥ ⋅ W ) y x = − 4 3 b x b y ∂ x v x + ( 1 − b y 2 ) ∂ x v y − b y b z ∂ x v z , (187) (\mathbf{I}_{\perp}\cdot\mathbf{W})_{yx}=-\frac{4}{3}b_xb_y\partial_xv_x+(1-b_y^2)\partial_xv_y-b_yb_z\partial_xv_z, \tag*{(187)} ( I ⊥ ⋅ W ) y x = − 3 4 b x b y ∂ x v x + ( 1 − b y 2 ) ∂ x v y − b y b z ∂ x v z , (187) ( I ⊥ ⋅ W ) y z = 2 3 b y b z ∂ x v x − b x b y ∂ x v z , (188) (\mathbf{I}_{\perp}\cdot\mathbf{W})_{yz}=\frac{2}{3}b_yb_z\partial_xv_x-b_xb_y\partial_xv_z, \tag*{(188)} ( I ⊥ ⋅ W ) y z = 3 2 b y b z ∂ x v x − b x b y ∂ x v z , (188) ( I ⊥ ⋅ W ) z x = − 4 3 ∂ x v x b x b z − b y b z ∂ x v y + ( 1 − b z 2 ) ∂ x v z , (189) (\mathbf{I}_{\perp}\cdot\mathbf{W})_{zx}=-\frac{4}{3}\partial_xv_xb_xb_z-b_yb_z\partial_xv_y+(1-b_z^2)\partial_xv_z, \tag*{(189)} ( I ⊥ ⋅ W ) z x = − 3 4 ∂ x v x b x b z − b y b z ∂ x v y + ( 1 − b z 2 ) ∂ x v z , (189) ( I ⊥ ⋅ W ) z y = 2 3 b y b z ∂ x v x − b x b z ∂ x v y . (190) (\mathbf{I}_{\perp}\cdot\mathbf{W})_{zy}=\frac{2}{3}b_yb_z\partial_xv_x-b_xb_z\partial_xv_y. \tag*{(190)} ( I ⊥ ⋅ W ) z y = 3 2 b y b z ∂ x v x − b x b z ∂ x v y . (190) ( b ^ × W ⋅ I ⊥ ) x x = ( 1 − b x 2 ) ( b y ∂ x v z − b z ∂ x v y ) , (191) (\hat{b} \times\mathbf{W} \cdot I_{\perp})_{xx} = (1-b_x^2)(b_y\partial_xv_z-b_z\partial_xv_y), \tag*{(191)} ( b ^ × W ⋅ I ⊥ ) xx = ( 1 − b x 2 ) ( b y ∂ x v z − b z ∂ x v y ) , (191) ( b ^ × W ⋅ I ⊥ ) x y = 2 3 b z ∂ x v x + b x b y b z ∂ x v y − b x b y 2 ∂ x v z , (192) (\hat{b} \times\mathbf{W} \cdot I_{\perp})_{xy} = \frac{2}{3}b_z\partial_xv_x+b_xb_yb_z\partial_xv_y-b_xb_y^2\partial_xv_z, \tag*{(192)} ( b ^ × W ⋅ I ⊥ ) x y = 3 2 b z ∂ x v x + b x b y b z ∂ x v y − b x b y 2 ∂ x v z , (192) ( b ^ × W ⋅ I ⊥ ) x z = − 2 3 b y ∂ x v x + b x b z 2 ∂ x v y − b x b y b z ∂ x v z , (193) (\hat{b} \times\mathbf{W} \cdot I_{\perp})_{xz} = -\frac{2}{3}b_y\partial_xv_x+b_xb_z^2\partial_xv_y-b_xb_yb_z\partial_xv_z, \tag*{(193)} ( b ^ × W ⋅ I ⊥ ) x z = − 3 2 b y ∂ x v x + b x b z 2 ∂ x v y − b x b y b z ∂ x v z , (193) ( b ^ × W ⋅ I ⊥ ) y x = ( 4 3 − 2 b z 2 ) b z ∂ x v x − b x b y b z ∂ x v y − b x ( b y 2 + 2 b z 2 ) ∂ x v z , (194) (\hat{b} \times\mathbf{W} \cdot I_{\perp})_{yx} = \left(\frac{4}{3}-2b_z^2\right)b_z\partial_xv_x-b_xb_yb_z\partial_xv_y-b_x(b_y^2+2b_z^2)\partial_xv_z, \tag*{(194)} ( b ^ × W ⋅ I ⊥ ) y x = ( 3 4 − 2 b z 2 ) b z ∂ x v x − b x b y b z ∂ x v y − b x ( b y 2 + 2 b z 2 ) ∂ x v z , (194) ( b ^ × W ⋅ I ⊥ ) y y = − 2 b x b y b z ∂ x v x + ( 1 − b y 2 ) b z ∂ x v y + ( b x 2 − b z 2 ) b y ∂ x v z , (195) (\hat{b} \times\mathbf{W} \cdot I_{\perp})_{yy} = -2b_xb_yb_z\partial_xv_x+(1-b_y^2)b_z\partial_xv_y+(b_x^2-b_z^2)b_y\partial_xv_z, \tag*{(195)} ( b ^ × W ⋅ I ⊥ ) y y = − 2 b x b y b z ∂ x v x + ( 1 − b y 2 ) b z ∂ x v y + ( b x 2 − b z 2 ) b y ∂ x v z , (195) ( b ^ × W ⋅ I ⊥ ) y z = ( 2 3 − 2 b z 2 ) b x ∂ x v x − b y b z 2 ∂ x v y + ( 2 b x 2 + b y 2 ) b z ∂ x v z , (196) (\hat{b} \times\mathbf{W} \cdot I_{\perp})_{yz} = \left(\frac{2}{3}-2b_z^2\right)b_x\partial_xv_x-b_yb_z^2\partial_xv_y+(2b_x^2+b_y^2)b_z\partial_xv_z, \tag*{(196)} ( b ^ × W ⋅ I ⊥ ) y z = ( 3 2 − 2 b z 2 ) b x ∂ x v x − b y b z 2 ∂ x v y + ( 2 b x 2 + b y 2 ) b z ∂ x v z , (196) ( b ^ × W ⋅ I ⊥ ) z x = − ( 4 3 − 2 b z 2 ) b y ∂ x v x + b x ( 2 b y 2 + b z 2 ) ∂ x v y + b x b y b z ∂ x v z , (197) (\hat{b} \times\mathbf{W} \cdot I_{\perp})_{zx} = -\left(\frac{4}{3}-2b_z^2\right)b_y\partial_xv_x+b_x(2b_y^2+b_z^2)\partial_xv_y+b_xb_yb_z\partial_xv_z, \tag*{(197)} ( b ^ × W ⋅ I ⊥ ) z x = − ( 3 4 − 2 b z 2 ) b y ∂ x v x + b x ( 2 b y 2 + b z 2 ) ∂ x v y + b x b y b z ∂ x v z , (197) ( b ^ × W ⋅ I ⊥ ) z y = ( 2 b y 2 − 2 3 ) b x ∂ x v x − b y ( 2 b x 2 + b z 2 ) ∂ x v y + b y 2 b z ∂ x v z , (198) (\hat{b} \times\mathbf{W} \cdot I_{\perp})_{zy} = \left(2b_y^2-\frac{2}{3}\right)b_x\partial_xv_x-b_y(2b_x^2+b_z^2)\partial_xv_y+b_y^2b_z\partial_xv_z, \tag*{(198)} ( b ^ × W ⋅ I ⊥ ) z y = ( 2 b y 2 − 3 2 ) b x ∂ x v x − b y ( 2 b x 2 + b z 2 ) ∂ x v y + b y 2 b z ∂ x v z , (198) ( b ^ × W ⋅ I ⊥ ) z z = 2 b x b y b z ∂ x v x + ( b y 2 − b x 2 ) b z ∂ x v y − ( 1 − b z 2 ) b y ∂ x v z . (199) (\hat{b} \times\mathbf{W} \cdot I_{\perp})_{zz} = 2b_xb_yb_z\partial_xv_x+(b_y^2-b_x^2)b_z\partial_xv_y-(1-b_z^2)b_y\partial_xv_z. \tag*{(199)} ( b ^ × W ⋅ I ⊥ ) z z = 2 b x b y b z ∂ x v x + ( b y 2 − b x 2 ) b z ∂ x v y − ( 1 − b z 2 ) b y ∂ x v z . (199) ( I ⊥ ⋅ W × b ^ ) x x = b y 2 b z ∂ x v z − b z 2 b y ∂ x v y − b y ∂ x v z + b z ∂ x v y , (200) (I_{\perp}\cdot\mathbf{W}\times\hat{b})_{xx}=b_y^2b_z\partial_xv_z-b_z^2b_y\partial_xv_y-b_y\partial_xv_z+b_z\partial_xv_y, \tag*{(200)} ( I ⊥ ⋅ W × b ^ ) xx = b y 2 b z ∂ x v z − b z 2 b y ∂ x v y − b y ∂ x v z + b z ∂ x v y , (200) ( I ⊥ ⋅ W × b ^ ) x y = − b x 3 ∂ x v z + 2 b x 2 b z ∂ x v x + b x b y b z ∂ x v y + b x b z 2 ∂ x v z + b x ∂ x v z − 4 b z ∂ x v x 3 , (201) (I_{\perp}\cdot\mathbf{W}\times\hat{b})_{xy}=-b_x^3\partial_xv_z+2b_x^2b_z\partial_xv_x+b_xb_yb_z\partial_xv_y+b_xb_z^2\partial_xv_z+b_x\partial_xv_z-\frac{4b_z\partial_xv_x}{3}, \tag*{(201)} ( I ⊥ ⋅ W × b ^ ) x y = − b x 3 ∂ x v z + 2 b x 2 b z ∂ x v x + b x b y b z ∂ x v y + b x b z 2 ∂ x v z + b x ∂ x v z − 3 4 b z ∂ x v x , (201) ( I ⊥ ⋅ W × b ^ ) x z = b x 3 ∂ x v y − 2 b x 2 b y ∂ x v x − b x b y 2 ∂ x v y − b x b y b z ∂ x v z − b x ∂ x v y + 4 b y ∂ x v x 3 , (202) (I_{\perp}\cdot\mathbf{W}\times\hat{b})_{xz}=b_x^3\partial_xv_y-2b_x^2b_y\partial_xv_x-b_xb_y^2\partial_xv_y-b_xb_yb_z\partial_xv_z-b_x\partial_xv_y+\frac{4b_y\partial_xv_x}{3}, \tag*{(202)} ( I ⊥ ⋅ W × b ^ ) x z = b x 3 ∂ x v y − 2 b x 2 b y ∂ x v x − b x b y 2 ∂ x v y − b x b y b z ∂ x v z − b x ∂ x v y + 3 4 b y ∂ x v x , (202) ( I ⊥ ⋅ W × b ^ ) y x = b x b y 2 ∂ x v z − b x b y b z ∂ x v y − 2 b z ∂ x v x 3 , (203) (I_{\perp}\cdot\mathbf{W}\times\hat{b})_{yx}=b_xb_y^2\partial_xv_z-b_xb_yb_z\partial_xv_y-\frac{2b_z\partial_xv_x}{3}, \tag*{(203)} ( I ⊥ ⋅ W × b ^ ) y x = b x b y 2 ∂ x v z − b x b y b z ∂ x v y − 3 2 b z ∂ x v x , (203) ( I ⊥ ⋅ W × b ^ ) y y = − b x 2 b y ∂ x v z + 2 b x b y b z ∂ x v x + b y 2 b z ∂ x v y + b y b z 2 ∂ x v z − b z ∂ x v y , (204) (I_{\perp}\cdot\mathbf{W}\times\hat{b})_{yy}=-b_x^2b_y\partial_xv_z+2b_xb_yb_z\partial_xv_x+b_y^2b_z\partial_xv_y+b_yb_z^2\partial_xv_z-b_z\partial_xv_y, \tag*{(204)} ( I ⊥ ⋅ W × b ^ ) y y = − b x 2 b y ∂ x v z + 2 b x b y b z ∂ x v x + b y 2 b z ∂ x v y + b y b z 2 ∂ x v z − b z ∂ x v y , (204) ( I ⊥ ⋅ W × b ^ ) y z = b x 2 b y ∂ x v y − 2 b x b y 2 ∂ x v x + 2 b x ∂ x v x 3 − b y 3 ∂ x v y − b y 2 b z ∂ x v z + b y ∂ x v y , (205) (I_{\perp}\cdot\mathbf{W}\times\hat{b})_{yz}=b_x^2b_y\partial_xv_y-2b_xb_y^2\partial_xv_x+\frac{2b_x\partial_xv_x}{3}-b_y^3\partial_xv_y-b_y^2b_z\partial_xv_z+b_y\partial_xv_y, \tag*{(205)} ( I ⊥ ⋅ W × b ^ ) y z = b x 2 b y ∂ x v y − 2 b x b y 2 ∂ x v x + 3 2 b x ∂ x v x − b y 3 ∂ x v y − b y 2 b z ∂ x v z + b y ∂ x v y , (205) ( I ⊥ ⋅ W × b ^ ) z x = b x b y b z ∂ x v z − b x b z 2 ∂ x v y + 2 b y ∂ x v x 3 , (206) (I_{\perp}\cdot\mathbf{W}\times\hat{b})_{zx}=b_xb_yb_z\partial_xv_z-b_xb_z^2\partial_xv_y+\frac{2b_y\partial_xv_x}{3}, \tag*{(206)} ( I ⊥ ⋅ W × b ^ ) z x = b x b y b z ∂ x v z − b x b z 2 ∂ x v y + 3 2 b y ∂ x v x , (206) ( I ⊥ ⋅ W × b ^ ) z y = − b x 2 b z ∂ x v z + 2 b x b z 2 ∂ x v x − 2 b x ∂ x v x 3 + b y b z 2 ∂ x v y + b z 3 ∂ x v z − b z ∂ x v z , (207) (I_{\perp}\cdot\mathbf{W}\times\hat{b})_{zy}=-b_x^2b_z\partial_xv_z+2b_xb_z^2\partial_xv_x-\frac{2b_x\partial_xv_x}{3}+b_yb_z^2\partial_xv_y+b_z^3\partial_xv_z-b_z\partial_xv_z, \tag*{(207)} ( I ⊥ ⋅ W × b ^ ) z y = − b x 2 b z ∂ x v z + 2 b x b z 2 ∂ x v x − 3 2 b x ∂ x v x + b y b z 2 ∂ x v y + b z 3 ∂ x v z − b z ∂ x v z , (207) ( I ⊥ ⋅ W × b ^ ) z z = b x 2 b z ∂ x v y − 2 b x b y b z ∂ x v x − b y 2 b z ∂ x v y − b y b z 2 ∂ x v z + b y ∂ x v z . (208) (I_{\perp}\cdot\mathbf{W}\times\hat{b})_{zz}=b_x^2b_z\partial_xv_y-2b_xb_yb_z\partial_xv_x-b_y^2b_z\partial_xv_y-b_yb_z^2\partial_xv_z+b_y\partial_xv_z. \tag*{(208)} ( I ⊥ ⋅ W × b ^ ) z z = b x 2 b z ∂ x v y − 2 b x b y b z ∂ x v x − b y 2 b z ∂ x v y − b y b z 2 ∂ x v z + b y ∂ x v z . (208) Details of the Eigenvalue Derivations
Viscosity
With only x x x § 3.6 reduce to
∂ t v ⃗ = ( M x x ( ν , ∥ , ⊥ ) M x y ( ν , ∥ , ⊥ ) M x z ( ν , ∥ , ⊥ ) M x y ( ν , ∥ , ⊥ ) M y y ( ν , ∥ , ⊥ ) M y z ( ν , ∥ , ⊥ ) M x z ( ν , ∥ , ⊥ ) M y z ( ν , ∥ , ⊥ ) M z z ( ν , ∥ , ⊥ ) ) ∂ x x v ⃗ , (209) \partial_t\vec{v} =
\begin{pmatrix}
M_{xx}^{(\nu,\parallel,\perp)} & M_{xy}^{(\nu,\parallel,\perp)} & M_{xz}^{(\nu,\parallel,\perp)} \\
M_{xy}^{(\nu,\parallel,\perp)} & M_{yy}^{(\nu,\parallel,\perp)} & M_{yz}^{(\nu,\parallel,\perp)} \\
M_{xz}^{(\nu,\parallel,\perp)} & M_{yz}^{(\nu,\parallel,\perp)} & M_{zz}^{(\nu,\parallel,\perp)}
\end{pmatrix}\partial_{xx}\vec{v}, \tag*{(209)} ∂ t v = M xx ( ν , ∥ , ⊥ ) M x y ( ν , ∥ , ⊥ ) M x z ( ν , ∥ , ⊥ ) M x y ( ν , ∥ , ⊥ ) M y y ( ν , ∥ , ⊥ ) M y z ( ν , ∥ , ⊥ ) M x z ( ν , ∥ , ⊥ ) M y z ( ν , ∥ , ⊥ ) M z z ( ν , ∥ , ⊥ ) ∂ xx v , (209) with
M x x ( ν , ∥ , ⊥ ) = ζ ρ + 4 3 ( 1 − x i o n ) ν n e u + x i o n [ 3 ν ∥ ( u − 1 3 ) 2 + ν ⊥ 1 ( 1 − u ) 2 + 16 ν ⊥ 2 u ( 1 − u ) ] , (210) M_{xx}^{(\nu,\parallel,\perp)}=\frac{\zeta}{\rho}+\frac{4}{3}(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}+x_{\mathrm{ion}}\left[3\nu_{\parallel}\left(u-\frac{1}{3}\right)^2+\nu_{\perp1}(1-u)^2+16\nu_{\perp2}u(1-u)\right], \tag*{(210)} M xx ( ν , ∥ , ⊥ ) = ρ ζ + 3 4 ( 1 − x ion ) ν neu + x ion [ 3 ν ∥ ( u − 3 1 ) 2 + ν ⊥ 1 ( 1 − u ) 2 + 16 ν ⊥ 2 u ( 1 − u ) ] , (210) M y y ( ν , ∥ , ⊥ ) = ( 1 − x i o n ) ν n e u + x i o n [ ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 u ] + x i o n b y 2 [ 3 ν ∥ u − ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 ( 1 − 4 u ) ] , (211) M_{yy}^{(\nu,\parallel,\perp)}=(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}+x_{\mathrm{ion}}\left[\nu_{\perp1}(1-u)+4\nu_{\perp2}u\right]+x_{\mathrm{ion}}b_y^2\left[3\nu_{\parallel}u-\nu_{\perp1}(1-u)+4\nu_{\perp2}(1-4u)\right], \tag*{(211)} M y y ( ν , ∥ , ⊥ ) = ( 1 − x ion ) ν neu + x ion [ ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 u ] + x ion b y 2 [ 3 ν ∥ u − ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 ( 1 − 4 u ) ] , (211) M z z ( ν , ∥ , ⊥ ) = ( 1 − x i o n ) ν n e u + x i o n [ ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 u ] + x i o n b z 2 [ 3 ν ∥ u − ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 ( 1 − 4 u ) ] , (212) M_{zz}^{(\nu,\parallel,\perp)}=(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}+x_{\mathrm{ion}}\left[\nu_{\perp1}(1-u)+4\nu_{\perp2}u\right]+x_{\mathrm{ion}}b_z^2\left[3\nu_{\parallel}u-\nu_{\perp1}(1-u)+4\nu_{\perp2}(1-4u)\right], \tag*{(212)} M z z ( ν , ∥ , ⊥ ) = ( 1 − x ion ) ν neu + x ion [ ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 u ] + x ion b z 2 [ 3 ν ∥ u − ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 ( 1 − 4 u ) ] , (212) M x y ( ν , ∥ , ⊥ ) = M y x ( ν , ∥ , ⊥ ) = x i o n b x b y [ 3 ν ∥ ( u − 1 3 ) − ν ⊥ 1 ( 1 − u ) + 8 ν ⊥ 2 ( 1 − 2 u ) ] , (213) M_{xy}^{(\nu,\parallel,\perp)}=M_{yx}^{(\nu,\parallel,\perp)}=x_{\mathrm{ion}}b_xb_y\left[3\nu_{\parallel}\left(u-\frac{1}{3}\right)-\nu_{\perp1}(1-u)+8\nu_{\perp2}(1-2u)\right], \tag*{(213)} M x y ( ν , ∥ , ⊥ ) = M y x ( ν , ∥ , ⊥ ) = x ion b x b y [ 3 ν ∥ ( u − 3 1 ) − ν ⊥ 1 ( 1 − u ) + 8 ν ⊥ 2 ( 1 − 2 u ) ] , (213) M x z ( ν , ∥ , ⊥ ) = M z x ( ν , ∥ , ⊥ ) = x i o n b x b z [ 3 ν ∥ ( u − 1 3 ) − ν ⊥ 1 ( 1 − u ) + 8 ν ⊥ 2 ( 1 − 2 u ) ] , (214) M_{xz}^{(\nu,\parallel,\perp)}=M_{zx}^{(\nu,\parallel,\perp)}=x_{\mathrm{ion}}b_xb_z\left[3\nu_{\parallel}\left(u-\frac{1}{3}\right)-\nu_{\perp1}(1-u)+8\nu_{\perp2}(1-2u)\right], \tag*{(214)} M x z ( ν , ∥ , ⊥ ) = M z x ( ν , ∥ , ⊥ ) = x ion b x b z [ 3 ν ∥ ( u − 3 1 ) − ν ⊥ 1 ( 1 − u ) + 8 ν ⊥ 2 ( 1 − 2 u ) ] , (214) M y z ( ν , ∥ , ⊥ ) = M z y ( ν , ∥ , ⊥ ) = x i o n b y b z [ 3 ν ∥ u − ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 ( 1 − 4 u ) ] . (215) M_{yz}^{(\nu,\parallel,\perp)}=M_{zy}^{(\nu,\parallel,\perp)}=x_{\mathrm{ion}}b_yb_z\left[3\nu_{\parallel}u-\nu_{\perp1}(1-u)+4\nu_{\perp2}(1-4u)\right]. \tag*{(215)} M y z ( ν , ∥ , ⊥ ) = M z y ( ν , ∥ , ⊥ ) = x ion b y b z [ 3 ν ∥ u − ν ⊥ 1 ( 1 − u ) + 4 ν ⊥ 2 ( 1 − 4 u ) ] . (215) The eigenvalues of this system form the characteristic polynomial
− λ ν , ∥ , ⊥ 3 + Tr ( M ( ν , ∥ , ⊥ ) ) λ ν , ∥ , ⊥ 2 − 1 2 ( Tr ( M ( ν , ∥ , ⊥ ) ) 2 − Tr ( M ( ν ) 2 ) ) λ ν , ∥ , ⊥ + det ( M ( ν , ∥ , ⊥ ) ) = 0. (216) -\lambda_{\nu,\parallel,\perp}^{3}+\operatorname{Tr}\left(M^{(\nu,\parallel,\perp)}\right)\lambda_{\nu,\parallel,\perp}^{2}-\frac{1}{2}\left(\operatorname{Tr}\left(M^{(\nu,\parallel,\perp)}\right)^{2}-\operatorname{Tr}\left(M^{(\nu)2}\right)\right)\lambda_{\nu,\parallel,\perp}+\det\left(M^{(\nu,\parallel,\perp)}\right)=0. \tag*{(216)} − λ ν , ∥ , ⊥ 3 + Tr ( M ( ν , ∥ , ⊥ ) ) λ ν , ∥ , ⊥ 2 − 2 1 ( Tr ( M ( ν , ∥ , ⊥ ) ) 2 − Tr ( M ( ν ) 2 ) ) λ ν , ∥ , ⊥ + det ( M ( ν , ∥ , ⊥ ) ) = 0. (216) Substituting the trigonometric form of the magnetic unit vector, the coefficients of this polynomial are lengthy, but the ϕ \phi ϕ ϕ = 0 \phi=0 ϕ = 0 ϕ \phi ϕ x ^ \hat{x} x ^
M ( ν , ∥ , ⊥ ) ( θ , ϕ ) = Q ( ϕ ) M ( ν , ∥ , ⊥ ) ( θ , 0 ) Q ( ϕ ) T , Q ( ϕ ) = ( 1 0 0 0 cos ϕ − sin ϕ 0 sin ϕ cos ϕ ) . (217) M^{(\nu,\parallel,\perp)}(\theta,\phi)=Q(\phi)M^{(\nu,\parallel,\perp)}(\theta,0)Q(\phi)^{T},\qquad Q(\phi)=\begin{pmatrix}1&0&0\\0&\cos\phi&-\sin\phi\\0&\sin\phi&\cos\phi\end{pmatrix}. \tag*{(217)} M ( ν , ∥ , ⊥ ) ( θ , ϕ ) = Q ( ϕ ) M ( ν , ∥ , ⊥ ) ( θ , 0 ) Q ( ϕ ) T , Q ( ϕ ) = 1 0 0 0 cos ϕ sin ϕ 0 − sin ϕ cos ϕ . (217) The characteristic polynomial, and hence every eigenvalue, is independent of ϕ \phi ϕ u u u ϕ = 0 \phi=0 ϕ = 0 v z v_z v z b z = 0 b_z=0 b z = 0 x y xy x y M z z = λ ν , ∥ , ⊥ , z M_{zz}=\lambda_{\nu,\parallel,\perp,z} M z z = λ ν , ∥ , ⊥ , z
( λ ν , ∥ , ⊥ − λ ν , ∥ , ⊥ , z ) ( − λ ν , ∥ , ⊥ 2 + Tr ( M x y ( ν , ∥ , ⊥ ) ) λ ν , ∥ , ⊥ − Det ( M x y ( ν , ∥ , ⊥ ) ) ) = 0. (218) \left(\lambda_{\nu,\parallel,\perp}-\lambda_{\nu,\parallel,\perp,z}\right)\left(-\lambda_{\nu,\parallel,\perp}^{2}+\operatorname{Tr}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)\lambda_{\nu,\parallel,\perp}-\operatorname{Det}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)\right)=0. \tag*{(218)} ( λ ν , ∥ , ⊥ − λ ν , ∥ , ⊥ , z ) ( − λ ν , ∥ , ⊥ 2 + Tr ( M x y ( ν , ∥ , ⊥ ) ) λ ν , ∥ , ⊥ − Det ( M x y ( ν , ∥ , ⊥ ) ) ) = 0. (218) The coefficients are given in § 3.6 . The x y xy x y w ^ = ( sin θ , − cos θ ) \hat{w}=(\sin\theta,-\cos\theta) w ^ = ( sin θ , − cos θ ) b ^ \hat{b} b ^
λ ν , ∥ , ⊥ , x y + ≥ w ^ ⋅ M x y ( ν , ∥ , ⊥ ) ⋅ w ^ = λ ν , ∥ , ⊥ , z + ( 1 − u ) [ ζ ρ + ( 1 − x i o n ) ν n e u + x i o n ν ∥ 3 ] ≥ λ ν , ∥ , ⊥ , z . (219) \lambda_{\nu,\parallel,\perp,xy+}\geq\hat{w}\cdot M_{xy}^{(\nu,\parallel,\perp)}\cdot\hat{w}=\lambda_{\nu,\parallel,\perp,z}+(1-u)\left[\frac{\zeta}{\rho}+\frac{(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}+x_{\mathrm{ion}}\nu_{\parallel}}{3}\right]\geq\lambda_{\nu,\parallel,\perp,z}. \tag*{(219)} λ ν , ∥ , ⊥ , x y + ≥ w ^ ⋅ M x y ( ν , ∥ , ⊥ ) ⋅ w ^ = λ ν , ∥ , ⊥ , z + ( 1 − u ) [ ρ ζ + 3 ( 1 − x ion ) ν neu + x ion ν ∥ ] ≥ λ ν , ∥ , ⊥ , z . (219) Finding the critical point of the positive branch, we first obtain
d d u [ λ ν , ∥ , ⊥ , x y + ( u ) ] = 1 2 Tr ( M x y ( ν , ∥ , ⊥ ) ) d d u [ Tr ( M x y ( ν , ∥ , ⊥ ) ) ] − d d u [ Det ( M x y ( ν , ∥ , ⊥ ) ) ] Tr ( M x y ( ν , ∥ , ⊥ ) ) 2 − 4 Det ( M x y ( ν , ∥ , ⊥ ) ) + 1 2 d d u [ Tr ( M x y ( ν , ∥ , ⊥ ) ) ] . (220) \frac{d}{du}\left[\lambda_{\nu,\parallel,\perp,xy+}(u)\right]=\frac{\frac{1}{2}\operatorname{Tr}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)\frac{d}{du}\left[\operatorname{Tr}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)\right]-\frac{d}{du}\left[\operatorname{Det}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)\right]}{\sqrt{\operatorname{Tr}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)^2-4\operatorname{Det}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)}}+\frac{1}{2}\frac{d}{du}\left[\operatorname{Tr}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)\right]. \tag*{(220)} d u d [ λ ν , ∥ , ⊥ , x y + ( u ) ] = Tr ( M x y ( ν , ∥ , ⊥ ) ) 2 − 4 Det ( M x y ( ν , ∥ , ⊥ ) ) 2 1 Tr ( M x y ( ν , ∥ , ⊥ ) ) d u d [ Tr ( M x y ( ν , ∥ , ⊥ ) ) ] − d u d [ Det ( M x y ( ν , ∥ , ⊥ ) ) ] + 2 1 d u d [ Tr ( M x y ( ν , ∥ , ⊥ ) ) ] . (220) The derivatives of the two components are
d d u [ Tr ( M x y ( ν , ∥ , ⊥ ) ) ] = x i o n ( ν ∥ − ν ⊥ 1 ) , (221) \frac{d}{du}\left[\operatorname{Tr}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)\right]=x_{\mathrm{ion}}(\nu_{\parallel}-\nu_{\perp1}), \tag*{(221)} d u d [ Tr ( M x y ( ν , ∥ , ⊥ ) ) ] = x ion ( ν ∥ − ν ⊥ 1 ) , (221) d d u [ Det ( M x y ( ν , ∥ , ⊥ ) ) ] = x i o n ( 1 − 2 u ) ( 3 ν ∥ + ν ⊥ 1 − 16 ν ⊥ 2 ) [ ζ ρ + 4 3 ( 1 − x i o n ) ν n e u ] + 2 x i o n ( 1 − x i o n ) ν n e u [ ν ∥ ( 3 u − 1 ) − ν ⊥ 1 ( 1 − u ) + 8 ν ⊥ 2 ( 1 − 2 u ) ] + 4 3 x i o n 2 [ ν ∥ ν ⊥ 1 ( 1 − 2 u ) + 2 ν ∥ ν ⊥ 2 ( u + 1 ) − 6 ν ⊥ 1 ν ⊥ 2 ( 1 − u ) ] . (222) \begin{aligned}
\frac{d}{du}\left[\operatorname{Det}\left(M_{xy}^{(\nu,\parallel,\perp)}\right)\right]={}&x_{\mathrm{ion}}(1-2u)(3\nu_{\parallel}+\nu_{\perp1}-16\nu_{\perp2})\left[\frac{\zeta}{\rho}+\frac{4}{3}(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}\right]\\
&+2x_{\mathrm{ion}}(1-x_{\mathrm{ion}})\nu_{\mathrm{neu}}\left[\nu_{\parallel}(3u-1)-\nu_{\perp1}(1-u)+8\nu_{\perp2}(1-2u)\right]\\
&+\frac{4}{3}x_{\mathrm{ion}}^2\left[\nu_{\parallel}\nu_{\perp1}(1-2u)+2\nu_{\parallel}\nu_{\perp2}(u+1)-6\nu_{\perp1}\nu_{\perp2}(1-u)\right]. \tag*{(222)}
\end{aligned} d u d [ Det ( M x y ( ν , ∥ , ⊥ ) ) ] = x ion ( 1 − 2 u ) ( 3 ν ∥ + ν ⊥ 1 − 16 ν ⊥ 2 ) [ ρ ζ + 3 4 ( 1 − x ion ) ν neu ] + 2 x ion ( 1 − x ion ) ν neu [ ν ∥ ( 3 u − 1 ) − ν ⊥ 1 ( 1 − u ) + 8 ν ⊥ 2 ( 1 − 2 u ) ] + 3 4 x ion 2 [ ν ∥ ν ⊥ 1 ( 1 − 2 u ) + 2 ν ∥ ν ⊥ 2 ( u + 1 ) − 6 ν ⊥ 1 ν ⊥ 2 ( 1 − u ) ] . (222) Setting d λ ν , x y + / d u = 0 d_{\lambda_{\nu,xy+}}/du=0 d λ ν , x y + / d u = 0
d d u [ Det 2 ] − Tr d d u [ Tr ] d d u [ Det ] + d d u [ Tr ] 2 Det = 0. (223) \frac{d}{du}\left[\operatorname{Det}^{2}\right]-\operatorname{Tr}\frac{d}{du}[\operatorname{Tr}]\frac{d}{du}[\operatorname{Det}]+\frac{d}{du}[\operatorname{Tr}]^{2}\operatorname{Det}=0. \tag*{(223)} d u d [ Det 2 ] − Tr d u d [ Tr ] d u d [ Det ] + d u d [ Tr ] 2 Det = 0. (223) As Tr \operatorname{Tr} Tr Det \operatorname{Det} Det u u u u u u u 1 ∗ u_{1}^{*} u 1 ∗ u 2 ∗ u_{2}^{*} u 2 ∗ § 3.6 .
Gyroviscosity
In Fourier space, maximising over the binding mode, Eq. (48) becomes
d d t ( v ~ x v ~ y v ~ z ) = ( 0 M x y ( ν , ∧ ) M x z ( ν , ∧ ) − M x y ( ν , ∧ ) 0 M y z ( ν , ∧ ) − M x z ( ν , ∧ ) − M y z ( ν , ∧ ) 0 ) ⋅ − 4 Δ x 2 ( v ~ x v ~ y v ~ z ) , (224) \frac{d}{dt}
\begin{pmatrix}
\widetilde{v}_{x}\\
\widetilde{v}_{y}\\
\widetilde{v}_{z}
\end{pmatrix}
=
\begin{pmatrix}
0 & M_{xy}^{(\nu,\wedge)} & M_{xz}^{(\nu,\wedge)}\\
-M_{xy}^{(\nu,\wedge)} & 0 & M_{yz}^{(\nu,\wedge)}\\
-M_{xz}^{(\nu,\wedge)} & -M_{yz}^{(\nu,\wedge)} & 0
\end{pmatrix}
\cdot\frac{-4}{\Delta x^{2}}
\begin{pmatrix}
\widetilde{v}_{x}\\
\widetilde{v}_{y}\\
\widetilde{v}_{z}
\end{pmatrix}, \tag*{(224)} d t d v x v y v z = 0 − M x y ( ν , ∧ ) − M x z ( ν , ∧ ) M x y ( ν , ∧ ) 0 − M y z ( ν , ∧ ) M x z ( ν , ∧ ) M y z ( ν , ∧ ) 0 ⋅ Δ x 2 − 4 v x v y v z , (224) The matrix M ( ν , ∧ ) M^{(\nu,\wedge)} M ( ν , ∧ ) ϕ \phi ϕ
λ ν , ∧ ( − λ ν , ∧ 2 + Δ ( M x y ( ν , ∧ ) ) ) = 0 , (225) \lambda_{\nu,\wedge}\left(-\lambda_{\nu,\wedge}^{2}+\Delta\left(M_{xy}^{(\nu,\wedge)}\right)\right)=0, \tag*{(225)} λ ν , ∧ ( − λ ν , ∧ 2 + Δ ( M x y ( ν , ∧ ) ) ) = 0 , (225) where
Δ ( M x y ( ν , ∧ ) ) = x i o n 2 ( − ν ∧ 1 2 u 2 + 2 ν ∧ 1 2 u − ν ∧ 1 2 + 4 ν ∧ 1 ν ∧ 2 u 2 − 4 ν ∧ 1 ν ∧ 2 u − 4 ν ∧ 2 2 u ) . (226) \Delta\left(M_{xy}^{(\nu,\wedge)}\right)=x_{\mathrm{ion}}^{2}\left(-\nu_{\wedge1}^{2}u^{2}+2\nu_{\wedge1}^{2}u-\nu_{\wedge1}^{2}+4\nu_{\wedge1}\nu_{\wedge2}u^{2}-4\nu_{\wedge1}\nu_{\wedge2}u-4\nu_{\wedge2}^{2}u\right). \tag*{(226)} Δ ( M x y ( ν , ∧ ) ) = x ion 2 ( − ν ∧ 1 2 u 2 + 2 ν ∧ 1 2 u − ν ∧ 1 2 + 4 ν ∧ 1 ν ∧ 2 u 2 − 4 ν ∧ 1 ν ∧ 2 u − 4 ν ∧ 2 2 u ) . (226) Considering the magnitude of the positive branch eigenvalue, the u u u
d d u [ ∣ λ ν , ∧ + ( u ) ∣ ] = x i o n ( ν ∧ 1 2 u − ν ∧ 1 2 − 4 ν ∧ 1 ν ∧ 2 u + 2 ν ∧ 1 ν ∧ 2 + 2 ν ∧ 2 2 ) ( ν ∧ 1 ( 1 − u ) + 2 ν ∧ 2 u ) 2 + 4 ν ∧ 2 2 u ( 1 − u ) . (227) \frac{d}{du}\left[|\lambda_{\nu,\wedge+}(u)|\right]=\frac{x_{\mathrm{ion}}\left(\nu_{\wedge1}^{2}u-\nu_{\wedge1}^{2}-4\nu_{\wedge1}\nu_{\wedge2}u+2\nu_{\wedge1}\nu_{\wedge2}+2\nu_{\wedge2}^{2}\right)}{\sqrt{\left(\nu_{\wedge1}(1-u)+2\nu_{\wedge2}u\right)^{2}+4\nu_{\wedge2}^{2}u(1-u)}}. \tag*{(227)} d u d [ ∣ λ ν , ∧+ ( u ) ∣ ] = ( ν ∧ 1 ( 1 − u ) + 2 ν ∧ 2 u ) 2 + 4 ν ∧ 2 2 u ( 1 − u ) x ion ( ν ∧ 1 2 u − ν ∧ 1 2 − 4 ν ∧ 1 ν ∧ 2 u + 2 ν ∧ 1 ν ∧ 2 + 2 ν ∧ 2 2 ) . (227) This vanishes at the critical point u ∗ u^{*} u ∗ § 3.7 .
Isolated thermoelectric effects
For the eigenvalues λ T E , ± \lambda_{\mathrm{TE},\pm} λ TE , ± § 3.8 , where ( κ J , w 2 − Δ κ J 2 u ) ( 1 − u ) > 0 (\kappa_{J,w}^{2}-\Delta\kappa_{J}^{2}u)(1-u)>0 ( κ J , w 2 − Δ κ J 2 u ) ( 1 − u ) > 0
d d u [ λ T E , + ] = 2 Δ κ J 2 u − Δ κ J 2 − κ J , w 2 2 ε i n t ( κ J , w 2 − Δ κ J 2 u ) ( 1 − u ) , (228) \frac{d}{du}\left[\lambda_{\mathrm{TE},+}\right]=\frac{2\Delta\kappa_{J}^{2}u-\Delta\kappa_{J}^{2}-\kappa_{J,w}^{2}}{2\sqrt{\varepsilon_{\mathrm{int}}}\sqrt{(\kappa_{J,w}^{2}-\Delta\kappa_{J}^{2}u)(1-u)}}, \tag*{(228)} d u d [ λ TE , + ] = 2 ε int ( κ J , w 2 − Δ κ J 2 u ) ( 1 − u ) 2Δ κ J 2 u − Δ κ J 2 − κ J , w 2 , (228) and where ( κ J , w 2 − Δ κ J 2 u ) ( 1 − u ) < 0 (\kappa_{J,w}^{2}-\Delta\kappa_{J}^{2}u)(1-u)<0 ( κ J , w 2 − Δ κ J 2 u ) ( 1 − u ) < 0
d d u [ ∣ λ T E , ± ∣ ] = − 2 Δ κ J 2 u − Δ κ J 2 − κ J , w 2 2 ε i n t − ( κ J , w 2 − Δ κ J 2 u ) ( 1 − u ) . (229) \frac{d}{du}\left[|\lambda_{\mathrm{TE},\pm}|\right]=-\frac{2\Delta\kappa_{J}^{2}u-\Delta\kappa_{J}^{2}-\kappa_{J,w}^{2}}{2\sqrt{\varepsilon_{\mathrm{int}}}\sqrt{-(\kappa_{J,w}^{2}-\Delta\kappa_{J}^{2}u)(1-u)}}. \tag*{(229)} d u d [ ∣ λ TE , ± ∣ ] = − 2 ε int − ( κ J , w 2 − Δ κ J 2 u ) ( 1 − u ) 2Δ κ J 2 u − Δ κ J 2 − κ J , w 2 . (229) Both vanish at the stationary point u 1 ∗ u_{1}^{*} u 1 ∗
Thermo-magnetic system
For the thermo-magnetic system of § 3.9 we use k x Δ x / 2 = ψ k_x\Delta x/2=\psi k x Δ x /2 = ψ
M T ( κ , η , J ) = − 4 ( γ − 1 ) Δ t Δ x 2 ( κ 0 + Δ κ cos 2 ( θ ) ) sin 2 ( ψ ) , (230) M_T^{(\kappa,\eta,J)}=-4(\gamma-1)\frac{\Delta t}{\Delta x^2}\left(\kappa_0+\Delta\kappa\cos^2(\theta)\right)\sin^2(\psi), \tag*{(230)} M T ( κ , η , J ) = − 4 ( γ − 1 ) Δ x 2 Δ t ( κ 0 + Δ κ cos 2 ( θ ) ) sin 2 ( ψ ) , (230) M T , Λ ( κ , η , J ) = 4 ( γ − 1 ) − 1 Δ t Δ ⊥ Δ x κ w sin ( ψ ) cos ( ψ ) , (231) M_{T,\Lambda}^{(\kappa,\eta,J)}=4(\gamma-1)\sqrt{-1}\frac{\Delta t}{\Delta_\perp\Delta x}\kappa_w\sin(\psi)\cos(\psi), \tag*{(231)} M T , Λ ( κ , η , J ) = 4 ( γ − 1 ) − 1 Δ ⊥ Δ x Δ t κ w sin ( ψ ) cos ( ψ ) , (231) M M , y ( κ , η , J ) = − 4 Δ t Δ x 2 ( η + η A D ( 1 − sin 2 ( θ ) sin 2 ( ϕ ) ) ) sin 2 ( ψ ) , (232) M_{M,y}^{(\kappa,\eta,J)}=-4\frac{\Delta t}{\Delta x^2}\left(\eta+\eta_{AD}\left(1-\sin^2(\theta)\sin^2(\phi)\right)\right)\sin^2(\psi), \tag*{(232)} M M , y ( κ , η , J ) = − 4 Δ x 2 Δ t ( η + η A D ( 1 − sin 2 ( θ ) sin 2 ( ϕ ) ) ) sin 2 ( ψ ) , (232) M M , z ( κ , η , J ) = − 4 Δ t Δ x 2 ( η + η A D ( 1 − sin 2 ( θ ) cos 2 ( ϕ ) ) ) sin 2 ( ψ ) , (233) M_{M,z}^{(\kappa,\eta,J)}=-4\frac{\Delta t}{\Delta x^2}\left(\eta+\eta_{AD}\left(1-\sin^2(\theta)\cos^2(\phi)\right)\right)\sin^2(\psi), \tag*{(233)} M M , z ( κ , η , J ) = − 4 Δ x 2 Δ t ( η + η A D ( 1 − sin 2 ( θ ) cos 2 ( ϕ ) ) ) sin 2 ( ψ ) , (233) M M , A D ( κ , η , J ) = − 4 Δ t Δ x 2 η A D sin 2 ( θ ) cos ( ϕ ) sin ( ϕ ) sin 2 ( ψ ) , (234) M_{M,AD}^{(\kappa,\eta,J)}=-4\frac{\Delta t}{\Delta x^2}\eta_{AD}\sin^2(\theta)\cos(\phi)\sin(\phi)\sin^2(\psi), \tag*{(234)} M M , A D ( κ , η , J ) = − 4 Δ x 2 Δ t η A D sin 2 ( θ ) cos ( ϕ ) sin ( ϕ ) sin 2 ( ψ ) , (234) M H ( κ , η , J ) = 4 Δ t Δ x 2 η H cos ( θ ) sin 2 ( ψ ) , (235) M_H^{(\kappa,\eta,J)}=4\frac{\Delta t}{\Delta x^2}\eta_H\cos(\theta)\sin^2(\psi), \tag*{(235)} M H ( κ , η , J ) = 4 Δ x 2 Δ t η H cos ( θ ) sin 2 ( ψ ) , (235) M T E , ε y ( κ , η , J ) = − 4 Δ t Δ x 2 ( − Δ κ J sin ( ϕ ) cos ( θ ) sin ( θ ) + κ J , w cos ( ϕ ) sin ( θ ) ) sin 2 ( ψ ) , (236) M_{TE,\varepsilon_y}^{(\kappa,\eta,J)}=-4\frac{\Delta t}{\Delta x^2}\left(-\Delta\kappa_J\sin(\phi)\cos(\theta)\sin(\theta)+\kappa_{J,w}\cos(\phi)\sin(\theta)\right)\sin^2(\psi), \tag*{(236)} M T E , ε y ( κ , η , J ) = − 4 Δ x 2 Δ t ( − Δ κ J sin ( ϕ ) cos ( θ ) sin ( θ ) + κ J , w cos ( ϕ ) sin ( θ ) ) sin 2 ( ψ ) , (236) M T E , ε z ( κ , η , J ) = − 4 Δ t Δ x 2 ( Δ κ J cos ( ϕ ) cos ( θ ) sin ( θ ) + κ J , w sin ( ϕ ) sin ( θ ) ) sin 2 ( ψ ) , (237) M_{TE,\varepsilon_z}^{(\kappa,\eta,J)}=-4\frac{\Delta t}{\Delta x^2}\left(\Delta\kappa_J\cos(\phi)\cos(\theta)\sin(\theta)+\kappa_{J,w}\sin(\phi)\sin(\theta)\right)\sin^2(\psi), \tag*{(237)} M T E , ε z ( κ , η , J ) = − 4 Δ x 2 Δ t ( Δ κ J cos ( ϕ ) cos ( θ ) sin ( θ ) + κ J , w sin ( ϕ ) sin ( θ ) ) sin 2 ( ψ ) , (237) M T E , B y ( κ , η , J ) = − 4 Δ t Δ x 2 1 ε i n t ( Δ κ J cos ( θ ) sin ( ϕ ) sin ( θ ) + κ J , w cos ( ϕ ) sin ( θ ) ) sin 2 ( ψ ) , (238) M_{TE,B_y}^{(\kappa,\eta,J)}=-4\frac{\Delta t}{\Delta x^2}\frac{1}{\varepsilon_{\mathrm{int}}}\left(\Delta\kappa_J\cos(\theta)\sin(\phi)\sin(\theta)+\kappa_{J,w}\cos(\phi)\sin(\theta)\right)\sin^2(\psi), \tag*{(238)} M T E , B y ( κ , η , J ) = − 4 Δ x 2 Δ t ε int 1 ( Δ κ J cos ( θ ) sin ( ϕ ) sin ( θ ) + κ J , w cos ( ϕ ) sin ( θ ) ) sin 2 ( ψ ) , (238) M T E , B z ( κ , η , J ) = − 4 Δ t Δ x 2 1 ε i n t ( − Δ κ J cos ( ϕ ) cos ( θ ) sin ( θ ) + κ J , w sin ( ϕ ) sin ( θ ) ) sin 2 ( ψ ) . (239) M_{TE,B_z}^{(\kappa,\eta,J)}=-4\frac{\Delta t}{\Delta x^2}\frac{1}{\varepsilon_{\mathrm{int}}}\left(-\Delta\kappa_J\cos(\phi)\cos(\theta)\sin(\theta)+\kappa_{J,w}\sin(\phi)\sin(\theta)\right)\sin^2(\psi). \tag*{(239)} M T E , B z ( κ , η , J ) = − 4 Δ x 2 Δ t ε int 1 ( − Δ κ J cos ( ϕ ) cos ( θ ) sin ( θ ) + κ J , w sin ( ϕ ) sin ( θ ) ) sin 2 ( ψ ) . (239) The matrix equation then becomes
d d t ( ε ~ i n t B ~ y B ~ z ) = ( M T ( κ , η , J ) + M T , Λ ( κ , η , J ) M T E , ε y ( κ , η , J ) M T E , ε z ( κ , η , J ) M T E , B y ( κ , η , J ) M M , y ( κ , η , J ) M M , A D ( κ , η , J ) − M H ( κ , η , J ) M T E , B z ( κ , η , J ) M M , A D ( κ , η , J ) + M H ( κ , η , J ) M M , z ( κ , η , J ) ) ( ε ~ i n t B ~ y B ~ z ) . (240) \frac{d}{dt}
\begin{pmatrix}
\widetilde{\varepsilon}_{\mathrm{int}}\\
\widetilde{B}_y\\
\widetilde{B}_z
\end{pmatrix}
=
\begin{pmatrix}
M_T^{(\kappa,\eta,J)}+M_{T,\Lambda}^{(\kappa,\eta,J)} & M_{TE,\varepsilon_y}^{(\kappa,\eta,J)} & M_{TE,\varepsilon_z}^{(\kappa,\eta,J)}\\
M_{TE,B_y}^{(\kappa,\eta,J)} & M_{M,y}^{(\kappa,\eta,J)} & M_{M,AD}^{(\kappa,\eta,J)}-M_H^{(\kappa,\eta,J)}\\
M_{TE,B_z}^{(\kappa,\eta,J)} & M_{M,AD}^{(\kappa,\eta,J)}+M_H^{(\kappa,\eta,J)} & M_{M,z}^{(\kappa,\eta,J)}
\end{pmatrix}
\begin{pmatrix}
\widetilde{\varepsilon}_{\mathrm{int}}\\
\widetilde{B}_y\\
\widetilde{B}_z
\end{pmatrix}. \tag*{(240)} d t d ε int B y B z = M T ( κ , η , J ) + M T , Λ ( κ , η , J ) M T E , B y ( κ , η , J ) M T E , B z ( κ , η , J ) M T E , ε y ( κ , η , J ) M M , y ( κ , η , J ) M M , A D ( κ , η , J ) + M H ( κ , η , J ) M T E , ε z ( κ , η , J ) M M , A D ( κ , η , J ) − M H ( κ , η , J ) M M , z ( κ , η , J ) ε int B y B z . (240) With P P P
P = ( 1 0 0 0 1 / ε i n t 0 0 0 1 / ε i n t ) , (241) P=
\begin{pmatrix}
1&0&0\\
0&1/\sqrt{\varepsilon_{\mathrm{int}}}&0\\
0&0&1/\sqrt{\varepsilon_{\mathrm{int}}}
\end{pmatrix}, \tag*{(241)} P = 1 0 0 0 1/ ε int 0 0 0 1/ ε int , (241) we obtain the scaled matrix
M ~ ( κ , η , J ) = P − 1 ⋅ M ( κ , η , J ) ⋅ P (242) \widetilde{M}^{(\kappa,\eta,J)}=P^{-1}\cdot M^{(\kappa,\eta,J)}\cdot P \tag*{(242)} M ( κ , η , J ) = P − 1 ⋅ M ( κ , η , J ) ⋅ P (242) = ( M T ( κ , η , J ) + M T , Λ ( κ , η , J ) M T E , ε y ( κ , η , J ) / ε i n t M T E , ε z ( κ , η , J ) / ε i n t M T E , B y ( κ , η , J ) ε i n t M M , y ( κ , η , J ) M M , A D ( κ , η , J ) − M H ( κ , η , J ) M T E , B z ( κ , η , J ) ε i n t M M , A D ( κ , η , J ) + M H ( κ , η , J ) M M , z ( κ , η , J ) ) . (243) =
\begin{pmatrix}
M_T^{(\kappa,\eta,J)}+M_{T,\Lambda}^{(\kappa,\eta,J)}&
M_{TE,\varepsilon_y}^{(\kappa,\eta,J)}/\sqrt{\varepsilon_{\mathrm{int}}}&
M_{TE,\varepsilon_z}^{(\kappa,\eta,J)}/\sqrt{\varepsilon_{\mathrm{int}}}\\
M_{TE,B_y}^{(\kappa,\eta,J)}\sqrt{\varepsilon_{\mathrm{int}}}&
M_{M,y}^{(\kappa,\eta,J)}&
M_{M,AD}^{(\kappa,\eta,J)}-M_H^{(\kappa,\eta,J)}\\
M_{TE,B_z}^{(\kappa,\eta,J)}\sqrt{\varepsilon_{\mathrm{int}}}&
M_{M,AD}^{(\kappa,\eta,J)}+M_H^{(\kappa,\eta,J)}&
M_{M,z}^{(\kappa,\eta,J)}
\end{pmatrix}. \tag*{(243)} = M T ( κ , η , J ) + M T , Λ ( κ , η , J ) M T E , B y ( κ , η , J ) ε int M T E , B z ( κ , η , J ) ε int M T E , ε y ( κ , η , J ) / ε int M M , y ( κ , η , J ) M M , A D ( κ , η , J ) + M H ( κ , η , J ) M T E , ε z ( κ , η , J ) / ε int M M , A D ( κ , η , J ) − M H ( κ , η , J ) M M , z ( κ , η , J ) . (243) This allows for the κ J , Λ \kappa_{J,\Lambda} κ J , Λ Δ κ J \Delta\kappa_J Δ κ J M ~ ( κ , η , J ) \widetilde{M}^{(\kappa,\eta,J)} M ( κ , η , J ) M T , Λ ( κ , η , J ) M_{T,\Lambda}^{(\kappa,\eta,J)} M T , Λ ( κ , η , J ) sin 2 ( ψ ) = 1 \sin^2(\psi)=1 sin 2 ( ψ ) = 1
H M ~ = − 4 Δ t Δ x 2 ( H ε ε H ε y H ε z H ε y H y y H y z H ε z H y z H z z ) . (244) H_{\widetilde{M}}=-\frac{4\Delta t}{\Delta x^2}
\begin{pmatrix}
H_{\varepsilon\varepsilon}&H_{\varepsilon y}&H_{\varepsilon z}\\
H_{\varepsilon y}&H_{yy}&H_{yz}\\
H_{\varepsilon z}&H_{yz}&H_{zz}
\end{pmatrix}. \tag*{(244)} H M = − Δ x 2 4Δ t H εε H ε y H ε z H ε y H y y H y z H ε z H y z H z z . (244) with
H ξ ξ = ( γ − 1 ) ( κ 0 + Δ κ cos 2 θ ) , H ξ y = κ J , w sin θ cos ϕ / ε i n t , H ξ z = κ J , w sin θ sin ϕ / ε i n t , H y y = η + η A D ( 1 − sin 2 θ sin 2 ϕ ) , H z z = η + η A D ( 1 − sin 2 θ cos 2 ϕ ) , H y z = η A D sin 2 θ sin ϕ cos ϕ . (245) \begin{aligned}
H_{\xi\xi} &= (\gamma-1)(\kappa_0+\Delta\kappa\cos^2\theta),\\
H_{\xi y} &= \kappa_{J,w}\sin\theta\cos\phi/\sqrt{\varepsilon_{\mathrm{int}}},\\
H_{\xi z} &= \kappa_{J,w}\sin\theta\sin\phi/\sqrt{\varepsilon_{\mathrm{int}}},\\
H_{yy} &= \eta+\eta_{AD}(1-\sin^2\theta\sin^2\phi),\\
H_{zz} &= \eta+\eta_{AD}(1-\sin^2\theta\cos^2\phi),\\
H_{yz} &= \eta_{AD}\sin^2\theta\sin\phi\cos\phi. \tag*{(245)}
\end{aligned} H ξ ξ H ξ y H ξ z H y y H z z H y z = ( γ − 1 ) ( κ 0 + Δ κ cos 2 θ ) , = κ J , w sin θ cos ϕ / ε int , = κ J , w sin θ sin ϕ / ε int , = η + η A D ( 1 − sin 2 θ sin 2 ϕ ) , = η + η A D ( 1 − sin 2 θ cos 2 ϕ ) , = η A D sin 2 θ sin ϕ cos ϕ . (245) To eliminate the ϕ \phi ϕ
Q = ( 1 0 0 0 cos ( ϕ ) sin ( ϕ ) 0 sin ( ϕ ) − cos ( ϕ ) ) . (246) \mathbf{Q}=
\begin{pmatrix}
1 & 0 & 0\\
0 & \cos(\phi) & \sin(\phi)\\
0 & \sin(\phi) & -\cos(\phi)
\end{pmatrix}. \tag*{(246)} Q = 1 0 0 0 cos ( ϕ ) sin ( ϕ ) 0 sin ( ϕ ) − cos ( ϕ ) . (246) This gives the rotated matrix H F ~ , r o t \mathbf{H}_{\tilde{F},\mathrm{rot}} H F ~ , rot
Details of the HLLE-Mod derivations
Linearised Operators
For the ideal and Hall terms (Section 4.1), the linearised operator acting on the perturbation δ W \delta\mathbf{W} δ W
M ( I , η H ) = − 1 k x ( 0 − ρ 0 0 0 0 0 0 0 0 0 0 − γ − 1 ρ 0 − B 0 b y ρ 0 − B 0 b z ρ 0 0 0 0 0 0 B 0 b x ρ 0 0 0 0 0 0 0 0 B 0 b x ρ 0 0 − γ ϵ 0 0 0 0 0 0 0 − B 0 b y B 0 b x 0 0 0 − 1 η H k x b x 0 − B 0 b z 0 B 0 b x 0 − − 1 η H k x b x 0 ) . (247) M^{(\mathrm{I},\eta_H)}=\sqrt{-1}k_x
\begin{pmatrix}
0 & -\rho_0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & -\frac{\gamma-1}{\rho_0} & -\frac{B_0b_y}{\rho_0} & -\frac{B_0b_z}{\rho_0}\\
0 & 0 & 0 & 0 & 0 & \frac{B_0b_x}{\rho_0} & 0\\
0 & 0 & 0 & 0 & 0 & 0 & \frac{B_0b_x}{\rho_0}\\
0 & -\gamma\epsilon_0 & 0 & 0 & 0 & 0 & 0\\
0 & -B_0b_y & B_0b_x & 0 & 0 & 0 & \sqrt{-1}\eta_H k_xb_x\\
0 & -B_0b_z & 0 & B_0b_x & 0 & -\sqrt{-1}\eta_H k_xb_x & 0
\end{pmatrix}. \tag*{(247)} M ( I , η H ) = − 1 k x 0 0 0 0 0 0 0 − ρ 0 0 0 0 − γ ϵ 0 − B 0 b y − B 0 b z 0 0 0 0 0 B 0 b x 0 0 0 0 0 0 0 B 0 b x 0 − ρ 0 γ − 1 0 0 0 0 0 0 − ρ 0 B 0 b y ρ 0 B 0 b x 0 0 0 − − 1 η H k x b x 0 − ρ 0 B 0 b z 0 ρ 0 B 0 b x 0 − 1 η H k x b x 0 . (247) For the ideal and Peltier/Seebeck terms (Section 4.2) it is
M ( I , J ) = − 1 k x ( 0 − ρ 0 0 0 0 0 0 0 0 0 0 − γ − 1 ρ 0 − B 0 b y ρ 0 − B 0 b z ρ 0 0 0 0 0 0 B 0 b x ρ 0 0 0 0 0 0 0 0 B 0 b x ρ 0 0 − γ ϵ 0 0 0 0 − − 1 K b z − 1 K b y − − 1 K b z ρ 0 − B 0 b y B 0 b x 0 − 1 K b z ϵ 0 0 0 − 1 K b y ρ 0 − B 0 b z 0 B 0 b x − − 1 K b y ϵ 0 0 0 ) , (248) M^{(\mathrm{I},J)}=\sqrt{-1}k_x
\begin{pmatrix}
0 & -\rho_0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & -\frac{\gamma-1}{\rho_0} & -\frac{B_0b_y}{\rho_0} & -\frac{B_0b_z}{\rho_0}\\
0 & 0 & 0 & 0 & 0 & \frac{B_0b_x}{\rho_0} & 0\\
0 & 0 & 0 & 0 & 0 & 0 & \frac{B_0b_x}{\rho_0}\\
0 & -\gamma\epsilon_0 & 0 & 0 & 0 & -\sqrt{-1}K b_z & \sqrt{-1}K b_y\\
-\frac{\sqrt{-1}K b_z}{\rho_0} & -B_0b_y & B_0b_x & 0 & \frac{\sqrt{-1}K b_z}{\epsilon_0} & 0 & 0\\
\frac{\sqrt{-1}K b_y}{\rho_0} & -B_0b_z & 0 & B_0b_x & -\frac{\sqrt{-1}K b_y}{\epsilon_0} & 0 & 0
\end{pmatrix}, \tag*{(248)} M ( I , J ) = − 1 k x 0 0 0 0 0 − ρ 0 − 1 K b z ρ 0 − 1 K b y − ρ 0 0 0 0 − γ ϵ 0 − B 0 b y − B 0 b z 0 0 0 0 0 B 0 b x 0 0 0 0 0 0 0 B 0 b x 0 − ρ 0 γ − 1 0 0 0 ϵ 0 − 1 K b z − ϵ 0 − 1 K b y 0 − ρ 0 B 0 b y ρ 0 B 0 b x 0 − − 1 K b z 0 0 0 − ρ 0 B 0 b z 0 ρ 0 B 0 b x − 1 K b y 0 0 , (248) where K ≡ Δ κ k x k x b x K\equiv\Delta\kappa k_xk_xb_x K ≡ Δ κ k x k x b x g 1 g_1 g 1 g 2 g_2 g 2
M ( I , ν Λ ) = − 1 k x ( 0 − ρ 0 0 0 0 0 0 0 0 − 1 k x g 1 b z − − 1 k x g 1 b y − γ − 1 ρ 0 − B 0 b y ρ 0 − B 0 b z ρ 0 0 − − 1 k x g 1 b z 0 − 1 k x g 2 b x 0 B 0 b x ρ 0 0 0 − 1 k x g 1 b y − − 1 k x g 2 b x 0 0 0 B 0 b x ρ 0 0 − γ ϵ 0 0 0 0 0 0 0 − B 0 b y B 0 b x 0 0 0 0 0 − B 0 b z 0 B 0 b x 0 0 0 ) , (249) M^{(\mathrm{I},\nu_\Lambda)}=\sqrt{-1}k_x
\begin{pmatrix}
0 & -\rho_0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & \sqrt{-1}k_xg_1b_z & -\sqrt{-1}k_xg_1b_y & -\frac{\gamma-1}{\rho_0} & -\frac{B_0b_y}{\rho_0} & -\frac{B_0b_z}{\rho_0}\\
0 & -\sqrt{-1}k_xg_1b_z & 0 & \sqrt{-1}k_xg_2b_x & 0 & \frac{B_0b_x}{\rho_0} & 0\\
0 & \sqrt{-1}k_xg_1b_y & -\sqrt{-1}k_xg_2b_x & 0 & 0 & 0 & \frac{B_0b_x}{\rho_0}\\
0 & -\gamma\epsilon_0 & 0 & 0 & 0 & 0 & 0\\
0 & -B_0b_y & B_0b_x & 0 & 0 & 0 & 0\\
0 & -B_0b_z & 0 & B_0b_x & 0 & 0 & 0
\end{pmatrix}, \tag*{(249)} M ( I , ν Λ ) = − 1 k x 0 0 0 0 0 0 0 − ρ 0 0 − − 1 k x g 1 b z − 1 k x g 1 b y − γ ϵ 0 − B 0 b y − B 0 b z 0 − 1 k x g 1 b z 0 − − 1 k x g 2 b x 0 B 0 b x 0 0 − − 1 k x g 1 b y − 1 k x g 2 b x 0 0 0 B 0 b x 0 − ρ 0 γ − 1 0 0 0 0 0 0 − ρ 0 B 0 b y ρ 0 B 0 b x 0 0 0 0 0 − ρ 0 B 0 b z 0 ρ 0 B 0 b x 0 0 0 , (249) Combined Hall and Peltier/Seebeck Bound
We show that no root of F ( Ω , u ) F(\Omega,u) F ( Ω , u )
c T E w 2 = Ψ 2 + 1 4 D + 1 2 D E , Ψ = max ( c f , c w ) , (250) c_{\mathrm{TEw}}^{2}=\Psi^{2}+\frac{1}{4}D+\frac{1}{2}\sqrt{DE},\qquad\Psi=\max(c_f,c_w), \tag*{(250)} c TEw 2 = Ψ 2 + 4 1 D + 2 1 D E , Ψ = max ( c f , c w ) , (250) for any u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] s s s γ > 1 \gamma>1 γ > 1 D , E ≥ 0 D,E\geq0 D , E ≥ 0 F = G − u ( 1 − u ) T F=G-u(1-u)T F = G − u ( 1 − u ) T (98) are
R ( Ω , u ) = Ω 2 − c f 2 Ω + c a 2 c s 2 u , (251) R(\Omega,u)=\Omega^{2}-c_f^{2}\Omega+c_a^{2}c_s^{2}u, \tag*{(251)} R ( Ω , u ) = Ω 2 − c f 2 Ω + c a 2 c s 2 u , (251) G ( Ω , u ) = ( Ω − c a 2 u ) R ( Ω , u ) − E u Ω ( Ω − c s 2 ) , (252) G(\Omega,u)=(\Omega-c_a^{2}u)R(\Omega,u)-Eu\Omega(\Omega-c_s^{2}), \tag*{(252)} G ( Ω , u ) = ( Ω − c a 2 u ) R ( Ω , u ) − E u Ω ( Ω − c s 2 ) , (252) T ( Ω , u ) = D [ Ω 2 − ( c a 2 + c i s o 2 ) Ω + c a 2 c i s o 2 u ] − 2 s c a c i s o γ − 1 D E Ω , (253) T(\Omega,u)=D\left[\Omega^{2}-(c_a^{2}+c_{\mathrm{iso}}^{2})\Omega+c_a^{2}c_{\mathrm{iso}}^{2}u\right]-2sc_ac_{\mathrm{iso}}\sqrt{\gamma-1}DE\Omega, \tag*{(253)} T ( Ω , u ) = D [ Ω 2 − ( c a 2 + c iso 2 ) Ω + c a 2 c iso 2 u ] − 2 s c a c iso γ − 1 D E Ω , (253) where G G G (83) , c f 2 = c a 2 + c s 2 c_f^{2}=c_a^{2}+c_s^{2} c f 2 = c a 2 + c s 2 c s 2 = γ c i s o 2 c_s^{2}=\gamma c_{\mathrm{iso}}^{2} c s 2 = γ c iso 2
δ f = Ψ 2 − c f 2 , δ E = Ψ 2 − c a 2 − E , δ w = ( Ψ 2 − c a 2 ) 2 − E Ψ 2 , (254) \delta_f=\Psi^{2}-c_f^{2},\qquad\delta_E=\Psi^{2}-c_a^{2}-E,\qquad\delta_w=(\Psi^{2}-c_a^{2})^{2}-E\Psi^{2}, \tag*{(254)} δ f = Ψ 2 − c f 2 , δ E = Ψ 2 − c a 2 − E , δ w = ( Ψ 2 − c a 2 ) 2 − E Ψ 2 , (254) which are all positive. The first is because Ψ ≥ c f \Psi\geq c_f Ψ ≥ c f c w 2 = c a 2 + 1 2 E + 1 2 E ( E + 4 c a 2 ) c_w^{2}=c_a^{2}+\frac{1}{2}E+\frac{1}{2}\sqrt{E(E+4c_a^{2})} c w 2 = c a 2 + 2 1 E + 2 1 E ( E + 4 c a 2 ) ( Ω − c a 2 ) 2 − E Ω (\Omega-c_a^{2})^{2}-E\Omega ( Ω − c a 2 ) 2 − E Ω (85) ), and Ψ 2 ≥ c w 2 \Psi^{2}\geq c_w^{2} Ψ 2 ≥ c w 2 E ( E + 4 c a 2 ) ≥ E \sqrt{E(E+4c_a^{2})}\geq E E ( E + 4 c a 2 ) ≥ E c w 2 ≥ c a 2 + E c_w^{2}\geq c_a^{2}+E c w 2 ≥ c a 2 + E δ E ≥ 0 \delta_E\geq0 δ E ≥ 0 δ w ≥ 0 \delta_w\geq0 δ w ≥ 0
Ψ 2 − c a 2 = δ f + c s 2 , Ψ 2 − c s 2 = δ f + c a 2 , Ψ 2 − 2 c a c s = δ f + ( c a − c s ) 2 , (255) \Psi^{2}-c_a^{2}=\delta_f+c_s^{2},\qquad\Psi^{2}-c_s^{2}=\delta_f+c_a^{2},\qquad\Psi^{2}-2c_ac_s=\delta_f+(c_a-c_s)^{2}, \tag*{(255)} Ψ 2 − c a 2 = δ f + c s 2 , Ψ 2 − c s 2 = δ f + c a 2 , Ψ 2 − 2 c a c s = δ f + ( c a − c s ) 2 , (255) which are also positive.
The coefficient A 2 A_2 A 2 Ω 2 \Omega^{2} Ω 2 F F F 2 c T E w 2 2c_{\mathrm{TEw}}^{2} 2 c TEw 2
A 2 = − [ c f 2 + ( c a 2 + E ) u + D u ( 1 − u ) ] , (256) A_2=-\left[c_f^{2}+(c_a^{2}+E)u+Du(1-u)\right], \tag*{(256)} A 2 = − [ c f 2 + ( c a 2 + E ) u + D u ( 1 − u ) ] , (256) 2 c T E w 2 + A 2 = δ f + δ E + ( c a 2 + E ) ( 1 − u ) + 1 4 D [ 1 + ( 1 − 2 u ) 2 ] + D E ≥ 0. (257) 2c_{\mathrm{TEw}}^{2}+A_2=\delta_f+\delta_E+(c_a^{2}+E)(1-u)+\frac{1}{4}D\left[1+(1-2u)^{2}\right]+\sqrt{DE}\geq0. \tag*{(257)} 2 c TEw 2 + A 2 = δ f + δ E + ( c a 2 + E ) ( 1 − u ) + 4 1 D [ 1 + ( 1 − 2 u ) 2 ] + D E ≥ 0. (257) By the test of Section 4, it is therefore enough to show that F ( c T E w 2 , u ) ≥ 0 F(c_{\mathrm{TEw}}^{2},u)\geq0 F ( c TEw 2 , u ) ≥ 0
The sign s s s γ \gamma γ F F F T T T c i s o 2 c_{\mathrm{iso}}^{2} c iso 2 s c i s o γ − 1 = s c s 1 − 1 / γ sc_{\mathrm{iso}}\sqrt{\gamma-1}=sc_s\sqrt{1-1/\gamma} s c iso γ − 1 = s c s 1 − 1/ γ − c s -c_s − c s c s c_s c s F ^ \hat{F} F ^ F F F s = − 1 s=-1 s = − 1 γ → ∞ \gamma\to\infty γ → ∞ c s c_s c s c i s o → 0 c_{\mathrm{iso}}\to0 c iso → 0 c i s o γ − 1 → c s c_{\mathrm{iso}}\sqrt{\gamma-1}\to c_s c iso γ − 1 → c s
F ^ ( Ω , u ) = G ( Ω , u ) − u ( 1 − u ) Ω [ D ( Ω − c a 2 ) + 2 c a c s D E ] . (258) \hat{F}(\Omega,u)=G(\Omega,u)-u(1-u)\Omega\left[D(\Omega-c_a^{2})+2c_ac_s\sqrt{DE}\right]. \tag*{(258)} F ^ ( Ω , u ) = G ( Ω , u ) − u ( 1 − u ) Ω [ D ( Ω − c a 2 ) + 2 c a c s D E ] . (258) For any s s s γ \gamma γ
F ( Ω , u ) − F ^ ( Ω , u ) = u ( 1 − u ) { D c i s o 2 ( Ω − c a 2 u ) + 2 c a D E Ω [ c s + s c i s o γ − 1 ] } , (259) F(\Omega,u)-\hat{F}(\Omega,u)=u(1-u)\left\{Dc_{\mathrm{iso}}^{2}(\Omega-c_a^{2}u)+2c_a\sqrt{DE}\Omega\left[c_s+sc_{\mathrm{iso}}\sqrt{\gamma-1}\right]\right\}, \tag*{(259)} F ( Ω , u ) − F ^ ( Ω , u ) = u ( 1 − u ) { D c iso 2 ( Ω − c a 2 u ) + 2 c a D E Ω [ c s + s c iso γ − 1 ] } , (259) and both terms are positive when Ω ≥ c a 2 u \Omega\geq c_a^{2}u Ω ≥ c a 2 u Ω = c T E w 2 \Omega=c_{\mathrm{TEw}}^{2} Ω = c TEw 2 c T E w 2 ≥ Ψ 2 ≥ c a 2 u c_{\mathrm{TEw}}^{2}\geq\Psi^{2}\geq c_a^{2}u c TEw 2 ≥ Ψ 2 ≥ c a 2 u F ( c T E w 2 , u ) ≥ F ^ ( c T E w 2 , u ) F(c_{\mathrm{TEw}}^{2},u)\geq\hat{F}(c_{\mathrm{TEw}}^{2},u) F ( c TEw 2 , u ) ≥ F ^ ( c TEw 2 , u ) F ^ ( c T E w 2 , u ) ≥ 0 \hat{F}(c_{\mathrm{TEw}}^{2},u)\geq0 F ^ ( c TEw 2 , u ) ≥ 0
Since G G G Ω \Omega Ω Ψ 2 \Psi^{2} Ψ 2 h 3 h^{3} h 3 h = 1 4 D + 1 2 D E h=\frac{1}{4}D+\frac{1}{2}\sqrt{DE} h = 4 1 D + 2 1 D E G ( Ψ 2 + h , u ) = g 0 + g 1 h + g 2 h 2 + h 3 G(\Psi^{2}+h,u)=g_0+g_1h+g_2h^{2}+h^{3} G ( Ψ 2 + h , u ) = g 0 + g 1 h + g 2 h 2 + h 3 g n = ∂ Ω n G ( Ψ 2 , u ) / n ! g_n=\partial_{\Omega}^{n}G(\Psi^{2},u)/n! g n = ∂ Ω n G ( Ψ 2 , u ) / n !
g 0 = G ( Ψ 2 , u ) , (260) g_0=G(\Psi^{2},u), \tag*{(260)} g 0 = G ( Ψ 2 , u ) , (260) g 1 = 3 Ψ 4 − 2 [ c f 2 + ( c a 2 + E ) u ] Ψ 2 + [ c a 2 ( c f 2 + c s 2 ) + E c s 2 ] u , (261) g_1=3\Psi^{4}-2\left[c_f^{2}+(c_a^{2}+E)u\right]\Psi^{2}+\left[c_a^{2}(c_f^{2}+c_s^{2})+Ec_s^{2}\right]u, \tag*{(261)} g 1 = 3 Ψ 4 − 2 [ c f 2 + ( c a 2 + E ) u ] Ψ 2 + [ c a 2 ( c f 2 + c s 2 ) + E c s 2 ] u , (261) g 2 = 3 Ψ 2 − c f 2 − ( c a 2 + E ) u . (262) g_2=3\Psi^{2}-c_f^{2}-(c_a^{2}+E)u. \tag*{(262)} g 2 = 3 Ψ 2 − c f 2 − ( c a 2 + E ) u . (262) Substituting into F ^ \hat{F} F ^ D \sqrt{D} D
F ^ ( c T E w 2 , u ) = ∑ k = 0 6 a k D k / 2 , (263) \hat{F}(c_{\mathrm{TEw}}^{2},u)=\sum_{k=0}^{6}a_kD^{k/2}, \tag*{(263)} F ^ ( c TEw 2 , u ) = k = 0 ∑ 6 a k D k /2 , (263) with
a 0 = g 0 , (264) a_0 = g_0, \tag*{(264)} a 0 = g 0 , (264) a 1 = 1 2 E [ g 1 − 4 c a c s Ψ 2 u ( 1 − u ) ] , (265) a_1 = \frac{1}{2}\sqrt{E}\left[g_1 - 4c_a c_s \Psi^2 u(1-u)\right], \tag*{(265)} a 1 = 2 1 E [ g 1 − 4 c a c s Ψ 2 u ( 1 − u ) ] , (265) a 2 = 1 4 [ g 1 + E g 2 − 4 u ( 1 − u ) ( Ψ 2 ( Ψ 2 − c a 2 ) + c a c s E ) ] , (266) a_2 = \frac{1}{4}\left[g_1 + E g_2 - 4u(1-u)\left(\Psi^2\left(\Psi^2-c_a^2\right)+c_a c_s E\right)\right], \tag*{(266)} a 2 = 4 1 [ g 1 + E g 2 − 4 u ( 1 − u ) ( Ψ 2 ( Ψ 2 − c a 2 ) + c a c s E ) ] , (266) a 3 = 1 8 E [ 2 g 2 + E − 4 u ( 1 − u ) ( 2 Ψ 2 − c a 2 + c a c s ) ] , (267) a_3 = \frac{1}{8}\sqrt{E}\left[2g_2+E-4u(1-u)\left(2\Psi^2-c_a^2+c_a c_s\right)\right], \tag*{(267)} a 3 = 8 1 E [ 2 g 2 + E − 4 u ( 1 − u ) ( 2 Ψ 2 − c a 2 + c a c s ) ] , (267) a 4 = 1 16 [ g 2 + 3 E − 4 u ( 1 − u ) ( 2 Ψ 2 − c a 2 + E ) ] , (268) a_4 = \frac{1}{16}\left[g_2+3E-4u(1-u)\left(2\Psi^2-c_a^2+E\right)\right], \tag*{(268)} a 4 = 16 1 [ g 2 + 3 E − 4 u ( 1 − u ) ( 2 Ψ 2 − c a 2 + E ) ] , (268) a 5 = 1 32 E [ 3 − 8 u ( 1 − u ) ] , (269) a_5 = \frac{1}{32}\sqrt{E}\left[3-8u(1-u)\right], \tag*{(269)} a 5 = 32 1 E [ 3 − 8 u ( 1 − u ) ] , (269) a 6 = 1 64 ( 1 − 2 u ) 2 . (270) a_6 = \frac{1}{64}(1-2u)^2. \tag*{(270)} a 6 = 64 1 ( 1 − 2 u ) 2 . (270) As D ≥ 0 D \ge0 D ≥ 0 a k ≥ 0 a_k \ge0 a k ≥ 0 u 2 u^2 u 2 G G G − c a 2 c s 2 -c_a^2c_s^2 − c a 2 c s 2 Ω \Omega Ω g 1 g_1 g 1 g 2 g_2 g 2 u u u
g 1 ∣ u = 0 = Ψ 4 + 2 Ψ 2 δ f , g 1 ∣ u = 1 = c a 2 c s 2 + δ w + c a 2 ( δ f + δ E ) + δ f ( c s 2 + δ f + δ E ) , ( g 1 + E g 2 ) ∣ u = 0 = Ψ 2 ( Ψ 2 + 2 E ) + δ f ( 2 Ψ 2 + E ) , ( g 1 + E g 2 ) ∣ u = 1 = Ψ 2 ( Ψ 2 − c a 2 ) + 2 δ f ( Ψ 2 − c a 2 ) + ( c a 2 + E ) δ E , ( 2 g 2 + E ) ∣ u = 0 = 4 Ψ 2 + 2 δ f + E , ( 2 g 2 + E ) ∣ u = 1 = 2 Ψ 2 + 3 δ f + δ E + c s 2 . (271) \begin{aligned}
g_1\big|_{u=0} &= \Psi^4+2\Psi^2\delta_f,\\
g_1\big|_{u=1} &= c_a^2c_s^2+\delta_w+c_a^2(\delta_f+\delta_E)+\delta_f(c_s^2+\delta_f+\delta_E),\\
(g_1+Eg_2)\big|_{u=0} &= \Psi^2(\Psi^2+2E)+\delta_f(2\Psi^2+E),\\
(g_1+Eg_2)\big|_{u=1} &= \Psi^2(\Psi^2-c_a^2)+2\delta_f(\Psi^2-c_a^2)+(c_a^2+E)\delta_E,\\
(2g_2+E)\big|_{u=0} &= 4\Psi^2+2\delta_f+E,\\
(2g_2+E)\big|_{u=1} &= 2\Psi^2+3\delta_f+\delta_E+c_s^2. \tag*{(271)}
\end{aligned} g 1 u = 0 g 1 u = 1 ( g 1 + E g 2 ) u = 0 ( g 1 + E g 2 ) u = 1 ( 2 g 2 + E ) u = 0 ( 2 g 2 + E ) u = 1 = Ψ 4 + 2 Ψ 2 δ f , = c a 2 c s 2 + δ w + c a 2 ( δ f + δ E ) + δ f ( c s 2 + δ f + δ E ) , = Ψ 2 ( Ψ 2 + 2 E ) + δ f ( 2 Ψ 2 + E ) , = Ψ 2 ( Ψ 2 − c a 2 ) + 2 δ f ( Ψ 2 − c a 2 ) + ( c a 2 + E ) δ E , = 4 Ψ 2 + 2 δ f + E , = 2 Ψ 2 + 3 δ f + δ E + c s 2 . (271) Every term after the first is positive, so dropping them gives, for u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ]
g 1 ≥ ( 1 − u ) Ψ 4 + u c a 2 c s 2 , (272) g_1 \ge(1-u)\Psi^4+u c_a^2c_s^2, \tag*{(272)} g 1 ≥ ( 1 − u ) Ψ 4 + u c a 2 c s 2 , (272) g 1 + E g 2 ≥ ( 1 − u ) Ψ 2 ( Ψ 2 + 2 E ) + u Ψ 2 ( Ψ 2 − c a 2 ) , (273) g_1+Eg_2 \ge(1-u)\Psi^2(\Psi^2+2E)+u\Psi^2(\Psi^2-c_a^2), \tag*{(273)} g 1 + E g 2 ≥ ( 1 − u ) Ψ 2 ( Ψ 2 + 2 E ) + u Ψ 2 ( Ψ 2 − c a 2 ) , (273) 2 g 2 + E ≥ 4 ( 1 − u ) Ψ 2 + 2 u Ψ 2 . (274) 2g_2+E \ge4(1-u)\Psi^2+2u\Psi^2. \tag*{(274)} 2 g 2 + E ≥ 4 ( 1 − u ) Ψ 2 + 2 u Ψ 2 . (274) Each a k a_k a k a 0 a_0 a 0 (84) at Ω = Ψ 2 \Omega=\Psi^2 Ω = Ψ 2 G ( Ω , 0 ) = Ω 2 ( Ω − c f 2 ) G(\Omega,0)=\Omega^2(\Omega-c_f^2) G ( Ω , 0 ) = Ω 2 ( Ω − c f 2 ) G ( Ω , 1 ) = ( Ω − c s 2 ) [ ( Ω − c a 2 ) 2 − E Ω ] G(\Omega,1)=(\Omega-c_s^2)[(\Omega-c_a^2)^2-E\Omega] G ( Ω , 1 ) = ( Ω − c s 2 ) [( Ω − c a 2 ) 2 − E Ω ] a 1 a_1 a 1 a 2 a_2 a 2 a 3 a_3 a 3 c a c s ≤ 1 2 Ψ 2 c_a c_s \le\frac{1}{2}\Psi^2 c a c s ≤ 2 1 Ψ 2 a 3 a_3 a 3 a 4 a_4 a 4 4 u ( 1 − u ) = 1 − ( 1 − 2 u ) 2 4u(1-u)=1-(1-2u)^2 4 u ( 1 − u ) = 1 − ( 1 − 2 u ) 2
a 0 = ( 1 − u ) Ψ 4 δ f + u ( Ψ 2 − c s 2 ) δ w + c a 2 c s 2 u ( 1 − u ) . (275) a_0=(1-u)\Psi^4\delta_f+u(\Psi^2-c_s^2)\delta_w+c_a^2c_s^2u(1-u). \tag*{(275)} a 0 = ( 1 − u ) Ψ 4 δ f + u ( Ψ 2 − c s 2 ) δ w + c a 2 c s 2 u ( 1 − u ) . (275) a 1 ≥ 1 2 E [ ( 1 − u ) Ψ 4 + u c a 2 c s 2 − 4 c a c s Ψ 2 u ( 1 − u ) ] = 1 2 E ( [ ( 1 − u ) Ψ 2 − u c a c s ] 2 + u ( 1 − u ) ( Ψ 2 − c a c s ) 2 ) , (276) \begin{aligned}
a_1 &\ge\frac{1}{2}\sqrt{E}\left[(1-u)\Psi^4+uc_a^2c_s^2-4c_ac_s\Psi^2u(1-u)\right]\\
&=\frac{1}{2}\sqrt{E}\left(\left[(1-u)\Psi^2-uc_ac_s\right]^2+u(1-u)(\Psi^2-c_ac_s)^2\right), \tag*{(276)}
\end{aligned} a 1 ≥ 2 1 E [ ( 1 − u ) Ψ 4 + u c a 2 c s 2 − 4 c a c s Ψ 2 u ( 1 − u ) ] = 2 1 E ( [ ( 1 − u ) Ψ 2 − u c a c s ] 2 + u ( 1 − u ) ( Ψ 2 − c a c s ) 2 ) , (276) 4 a 2 ≥ ( 1 − u ) Ψ 2 ( Ψ 2 + 2 E ) + u Ψ 2 ( Ψ 2 − c a 2 ) − 4 u ( 1 − u ) [ Ψ 2 ( Ψ 2 − c a 2 ) + c a c s E ] = ( 1 − 2 u ) 2 Ψ 2 ( Ψ 2 − c a 2 ) + ( 1 − u ) 2 Ψ 2 ( c a 2 + 2 E ) + u ( 1 − u ) [ c a 2 Ψ 2 + 2 E ( Ψ 2 − 2 c a c s ) ] , (277) \begin{aligned}
4a_2 &\ge(1-u)\Psi^2(\Psi^2+2E)+u\Psi^2(\Psi^2-c_a^2)-4u(1-u)\left[\Psi^2(\Psi^2-c_a^2)+c_ac_sE\right]\\
&=(1-2u)^2\Psi^2(\Psi^2-c_a^2)+(1-u)^2\Psi^2(c_a^2+2E)\\
&\quad+u(1-u)\left[c_a^2\Psi^2+2E(\Psi^2-2c_ac_s)\right], \tag*{(277)}
\end{aligned} 4 a 2 ≥ ( 1 − u ) Ψ 2 ( Ψ 2 + 2 E ) + u Ψ 2 ( Ψ 2 − c a 2 ) − 4 u ( 1 − u ) [ Ψ 2 ( Ψ 2 − c a 2 ) + c a c s E ] = ( 1 − 2 u ) 2 Ψ 2 ( Ψ 2 − c a 2 ) + ( 1 − u ) 2 Ψ 2 ( c a 2 + 2 E ) + u ( 1 − u ) [ c a 2 Ψ 2 + 2 E ( Ψ 2 − 2 c a c s ) ] , (277) a 3 ≥ 1 8 E [ 4 ( 1 − u ) Ψ 2 + 2 u Ψ 2 − 10 u ( 1 − u ) Ψ 2 ] = 1 4 E Ψ 2 [ ( 1 − 2 u ) 2 + ( 1 − u ) 2 ] , (278) \begin{aligned}
a_3 &\ge\frac{1}{8}\sqrt{E}\left[4(1-u)\Psi^2+2u\Psi^2-10u(1-u)\Psi^2\right]\\
&=\frac{1}{4}\sqrt{E}\Psi^2\left[(1-2u)^2+(1-u)^2\right], \tag*{(278)}
\end{aligned} a 3 ≥ 8 1 E [ 4 ( 1 − u ) Ψ 2 + 2 u Ψ 2 − 10 u ( 1 − u ) Ψ 2 ] = 4 1 E Ψ 2 [ ( 1 − 2 u ) 2 + ( 1 − u ) 2 ] , (278) 16 a 4 = δ f + c a 2 ( 1 − u ) + E ( 2 − u ) + ( 1 − 2 u ) 2 ( Ψ 2 + δ f + c s 2 + E ) , (279) 16a_4=\delta_f+c_a^2(1-u)+E(2-u)+(1-2u)^2(\Psi^2+\delta_f+c_s^2+E), \tag*{(279)} 16 a 4 = δ f + c a 2 ( 1 − u ) + E ( 2 − u ) + ( 1 − 2 u ) 2 ( Ψ 2 + δ f + c s 2 + E ) , (279) a 5 = 1 32 E [ 1 + 2 ( 1 − 2 u ) 2 ] , (280) a_5=\frac{1}{32}\sqrt{E}\left[1+2(1-2u)^2\right], \tag*{(280)} a 5 = 32 1 E [ 1 + 2 ( 1 − 2 u ) 2 ] , (280) a 6 = 1 64 ( 1 − 2 u ) 2 . (281) a_6=\frac{1}{64}(1-2u)^2. \tag*{(281)} a 6 = 64 1 ( 1 − 2 u ) 2 . (281) Every term on the right is a product of positive factors, so every a k ≥ 0 a_k \ge0 a k ≥ 0 F ( c T E w 2 , u ) ≥ F ^ ( c T E w 2 , u ) ≥ 0 F(c_{\mathrm{TEw}}^2,u)\ge\widehat{F}(c_{\mathrm{TEw}}^2,u)\ge0 F ( c TEw 2 , u ) ≥ F ( c TEw 2 , u ) ≥ 0 u ∈ [ 0 , 1 ] u\in[0,1] u ∈ [ 0 , 1 ] s s s γ > 1 \gamma>1 γ > 1 F F F c T E w 2 c_{\mathrm{TEw}}^2 c TEw 2 D , E ≥ 0 D,E\ge0 D , E ≥ 0
HLLE-Mod Noise Tests
Noise Initial Conditions
Each noise field is generated over the whole mesh in two steps. First, every cell is given an independent Gaussian deviate of unit variance by the Box–Muller transform [45 ] ,
g = − 2 ln u 1 cos ( 2 π u 2 ) , (282) g=\sqrt{-2\ln u_1}\cos(2\pi u_2), \tag*{(282)} g = − 2 ln u 1 cos ( 2 π u 2 ) , (282) where u 1 u_1 u 1 u 2 u_2 u 2 ( 0 , 1 ) (0,1) ( 0 , 1 ) [46 ] . Each field and component receives a different seed. We then take the discrete Fourier transform [47 ] , multiply each mode by a filter W ( k ⃗ ) W(\vec{k}) W ( k ) W ( 0 ) = 0 W(0)=0 W ( 0 ) = 0 W = 1 W=1 W = 1 A ⃗ \vec{A} A x x x
δ B x ( i − 1 2 , j , k ) = A z ( i − 1 2 , j + 1 2 , k ) − A z ( i − 1 2 , j − 1 2 , k ) Δ y i , j , k − A y ( i − 1 2 , j , k + 1 2 ) − A y ( i − 1 2 , j , k − 1 2 ) Δ z i , j , k . (283) \begin{aligned}
\delta B_x\left(i-\frac{1}{2},j,k\right)
&=\frac{A_z\left(i-\frac{1}{2},j+\frac{1}{2},k\right)-A_z\left(i-\frac{1}{2},j-\frac{1}{2},k\right)}{\Delta y_{i,j,k}}\\
&\quad-\frac{A_y\left(i-\frac{1}{2},j,k+\frac{1}{2}\right)-A_y\left(i-\frac{1}{2},j,k-\frac{1}{2}\right)}{\Delta z_{i,j,k}}. \tag*{(283)}
\end{aligned} δ B x ( i − 2 1 , j , k ) = Δ y i , j , k A z ( i − 2 1 , j + 2 1 , k ) − A z ( i − 2 1 , j − 2 1 , k ) − Δ z i , j , k A y ( i − 2 1 , j , k + 2 1 ) − A y ( i − 2 1 , j , k − 2 1 ) . (283) The filter for the magnetic field is
W ( k ⃗ ) = 1 ∣ k ⃗ ~ ∣ , ∣ k ⃗ ~ ∣ 2 = ∑ d = 1 3 ( 2 Δ x d sin ( k d Δ x d 2 ) ) 2 , (284) W(\vec{k})=\frac{1}{\lvert\tilde{\vec{k}}\rvert},\qquad
\lvert\tilde{\vec{k}}\rvert^2=\sum_{d=1}^{3}\left(\frac{2}{\Delta x_d}\sin\left(\frac{k_d\Delta x_d}{2}\right)\right)^2, \tag*{(284)} W ( k ) = ∣ k ~ ∣ 1 , ∣ k ~ ∣ 2 = d = 1 ∑ 3 ( Δ x d 2 sin ( 2 k d Δ x d ) ) 2 , (284) for each component of the magnetic vector potential A ⃗ \vec{A} A k d = 2 π m d / L d k_d=2\pi m_d/L_d k d = 2 π m d / L d
Finally, each noise field is multiplied by a constant so that its RMS over the cells is the amplitude,
⟨ δ T 2 ⟩ 1 / 2 = σ T , ⟨ ∣ v ⃗ ∣ 2 ⟩ 1 / 2 = σ v , ⟨ ∣ δ B ⃗ c c ∣ 2 ⟩ 1 / 2 = σ B , (285) \langle\delta T^2 \rangle^{1/2} = \sigma_T,\qquad\langle\lvert\vec{v} \rvert^2 \rangle^{1/2} = \sigma_v,\qquad\langle\lvert\delta\vec{B}_{\mathrm{cc}} \rvert^2 \rangle^{1/2} = \sigma_B, \tag*{(285)} ⟨ δ T 2 ⟩ 1/2 = σ T , ⟨∣ v ∣ 2 ⟩ 1/2 = σ v , ⟨∣ δ B cc ∣ 2 ⟩ 1/2 = σ B , (285) where δ B ⃗ c c \delta\vec{B}_{\mathrm{cc}} δ B cc ρ = ρ 0 \rho= \rho_0 ρ = ρ 0 P = ρ 0 ( T 0 + δ T ) P = \rho_0(T_0 + \delta T) P = ρ 0 ( T 0 + δ T ) v ⃗ \vec{v} v B ⃗ = B ⃗ 0 + δ B ⃗ \vec{B} = \vec{B}_0 + \delta\vec{B} B = B 0 + δ B
Noise Slices for Individual Effects
Figure 10. Slices through the noise fields at t = 0 t = 0 t = 0 t = 20 t w t = 20t_w t = 20 t w
Figure 10 (PDF p. 49). Slices through the noise fields at t = 0 t = 0 t = 0 t = 20 t w t = 20t_w t = 20 t w
Figure 10 shows the same slices as Figure 9 for each effect on its own. With the standard solver the Righi–Leduc, Hall, and Peltier/Seebeck runs blow up at 18 t w 18t_w 18 t w 2.5 t w 2.5t_w 2.5 t w 19 t w 19t_w 19 t w 20 t w 20t_w 20 t w v x v_x v x δ T \delta T δ T δ B x \delta B_x δ B x σ B = 1 \sigma_B = 1 σ B = 1 10 − 6 10^{-6} 1 0 − 6 ∼ 14 \sim14 ∼ 14
Bounded Timestep and Well-Posed Numerical Checks
Physical Diffusion Coefficients
For neutral gas, the Parker conductivity of neutral hydrogen of [48 ] is used
κ P = 2.5 × 10 5 T / 10 4 e r g s − 1 c m − 1 K − 1 . (286) \kappa_P = 2.5 \times10^5 \sqrt{T/10^4}\ \mathrm{erg\ s^{-1}\ cm^{-1}\ K^{-1}}. \tag*{(286)} κ P = 2.5 × 1 0 5 T /1 0 4 erg s − 1 c m − 1 K − 1 . (286) For ionised gas the Spitzer conductivity for ionised hydrogen is used [49 , 50 ]
κ s = 1.70 × 10 11 ( T / 10 7 ) 5 / 2 1 + 0.029 ln ( T 10 7 ( 10 − 2 n e ) − 1 / 2 ) e r g s − 1 c m − 1 K − 1 . (287) \kappa_s = \frac{1.70 \times10^{11}(T/10^7)^{5/2}}{1 + 0.029 \ln\left(\frac{T}{10^7}\left(\frac{10^{-2}}{n_e}\right)^{-1/2}\right)}\ \mathrm{erg\ s^{-1}\ cm^{-1}\ K^{-1}}. \tag*{(287)} κ s = 1 + 0.029 ln ( 1 0 7 T ( n e 1 0 − 2 ) − 1/2 ) 1.70 × 1 0 11 ( T /1 0 7 ) 5/2 erg s − 1 c m − 1 K − 1 . (287) We take n e = max ( 1.2 x i o n , 10 − 6 ) n H n_e = \max(1.2x_{\mathrm{ion}},10^{-6})n_H n e = max ( 1.2 x ion , 1 0 − 6 ) n H n e n_e n e n H n_H n H n i = x i o n n H n_i = x_{\mathrm{ion}}n_H n i = x ion n H n n = ( 1 − x i o n ) n H n_n = (1-x_{\mathrm{ion}})n_H n n = ( 1 − x ion ) n H κ p h y s \kappa_{\mathrm{phys}} κ phys κ c o d e = κ p h y s m ~ / ( ρ k B ) \kappa_{\mathrm{code}} = \kappa_{\mathrm{phys}}\tilde{m}/(\rho k_B) κ code = κ phys m ~ / ( ρ k B ) m ~ = μ ( T ) m P \tilde{m} = \mu(T)m_P m ~ = μ ( T ) m P μ ( T ) \mu(T) μ ( T ) m P m_P m P
The anisotropic thermal conduction is constructed as [23 ]
κ n = κ P , κ ∥ = κ s , (288) \kappa_n = \kappa_P,\qquad\kappa_{\parallel} = \kappa_s, \tag*{(288)} κ n = κ P , κ ∥ = κ s , (288) κ ⊥ = κ s 3.1616 4.664 β e 2 + 11.92 β e 4 + 14.79 β e 2 + 3.7703 , (289) \kappa_{\perp} = \frac{\kappa_s}{3.1616}\frac{4.664\beta_e^2 + 11.92}{\beta_e^4 + 14.79\beta_e^2 + 3.7703}, \tag*{(289)} κ ⊥ = 3.1616 κ s β e 4 + 14.79 β e 2 + 3.7703 4.664 β e 2 + 11.92 , (289) κ ∧ = κ s β e 3.1616 5 / 2 β e 2 + 21.67 β e 4 + 14.79 β e 2 + 3.7703 , (290) \kappa_{\wedge} = \frac{\kappa_s\beta_e}{3.1616}\frac{5/2\beta_e^2 + 21.67}{\beta_e^4 + 14.79\beta_e^2 + 3.7703}, \tag*{(290)} κ ∧ = 3.1616 κ s β e β e 4 + 14.79 β e 2 + 3.7703 5/2 β e 2 + 21.67 , (290) where β e \beta_e β e
β e = ω c e ν e i + ν e n , (291) \beta_e = \frac{\omega_{ce}}{\nu_{ei}+\nu_{en}}, \tag*{(291)} β e = ν e i + ν e n ω ce , (291) and ω c e \omega_{ce} ω ce
ω c e = e ∣ B ∣ m e c l i g h t , (292) \omega_{ce} = \frac{e|\mathbf{B}|}{m_e c_{\mathrm{light}}}, \tag*{(292)} ω ce = m e c light e ∣ B ∣ , (292) with ∣ B ∣ |\mathbf{B}| ∣ B ∣ 4 π \sqrt{4\pi} 4 π ν e i \nu_{ei} ν e i ν e n \nu_{en} ν e n [51 , 52 ]
ν e i = 2.91 × 10 − 6 n e ln Λ T e V − 3 / 2 s − 1 , (293) \nu_{ei} = 2.91 \times10^{-6}n_e \ln\Lambda T_{\mathrm{eV}}^{-3/2}\ \mathrm{s^{-1}}, \tag*{(293)} ν e i = 2.91 × 1 0 − 6 n e ln Λ T eV − 3/2 s − 1 , (293) ν e n = 10 − 9 n n T 1 / 2 ( 2.841 + 0.093 log 10 ( T ) + 0.245 log 10 2 ( T ) − 0.089 log 10 3 ( T ) ) s − 1 , (294) \nu_{en} = 10^{-9}n_nT^{1/2}\left(2.841 + 0.093\log_{10}(T) + 0.245\log_{10}^{2}(T) - 0.089\log_{10}^{3}(T)\right)\ \mathrm{s^{-1}}, \tag*{(294)} ν e n = 1 0 − 9 n n T 1/2 ( 2.841 + 0.093 log 10 ( T ) + 0.245 log 10 2 ( T ) − 0.089 log 10 3 ( T ) ) s − 1 , (294) where T e V = k B T / ( 1 e V ) = 8.617 × 10 − 5 T T_{\mathrm{eV}} = k_BT/(1\ \mathrm{eV}) = 8.617 \times10^{-5}T T eV = k B T / ( 1 eV ) = 8.617 × 1 0 − 5 T ν e n \nu_{en} ν e n T ≈ 3.4 × 10 4 K T \approx3.4 \times10^4\ \mathrm{K} T ≈ 3.4 × 1 0 4 K 10 4 K 10^4\ \mathrm{K} 1 0 4 K 10 4 K 10^4\ \mathrm{K} 1 0 4 K
We take the bulk viscosity to come entirely from H2 _2 2 2 _2 2 x H 2 x_{\mathrm{H}_2} x H 2 2 _2 2 [53 ]
ζ = x H 2 ( ζ r o t + ζ v i b ) , (295) \zeta= x_{\mathrm{H}_2}\left(\zeta_{\mathrm{rot}}+\zeta_{\mathrm{vib}}\right), \tag*{(295)} ζ = x H 2 ( ζ rot + ζ vib ) , (295) with
ζ r o t = ( γ H 2 − 1 ) 2 c r o t R P τ r o t , (296) \zeta_{\mathrm{rot}}=(\gamma_{\mathrm{H}_2}-1)^2\frac{c_{\mathrm{rot}}}{R}P\tau_{\mathrm{rot}}, \tag*{(296)} ζ rot = ( γ H 2 − 1 ) 2 R c rot P τ rot , (296) ζ v i b = ( γ H 2 − 1 ) 2 c v i b R P τ v i b . (297) \zeta_{\mathrm{vib}}=(\gamma_{\mathrm{H}_2}-1)^2\frac{c_{\mathrm{vib}}}{R}P\tau_{\mathrm{vib}}. \tag*{(297)} ζ vib = ( γ H 2 − 1 ) 2 R c vib P τ vib . (297) Here γ H 2 ( T ) \gamma_{\mathrm{H}_2}(T) γ H 2 ( T ) 2 _2 2 [54 ]
γ H 2 − 1 = 1 3 / 2 + c r o t / R + c v i b / R . (298) \gamma_{\mathrm{H}_2}-1=\frac{1}{3/2+c_{\mathrm{rot}}/R+c_{\mathrm{vib}}/R}. \tag*{(298)} γ H 2 − 1 = 3/2 + c rot / R + c vib / R 1 . (298) c r o t c_{\mathrm{rot}} c rot c v i b c_{\mathrm{vib}} c vib [55 , 56 ]
c r o t R = ∑ s ∈ { p , o } x s ⟨ E 2 ⟩ s − ⟨ E ⟩ s 2 ( k B T ) 2 , (299) \frac{c_{\mathrm{rot}}}{R}=\sum_{s\in\{p,o\}}x_s\frac{\langle E^2\rangle_s-\langle E\rangle_s^2}{(k_{\mathrm{B}}T)^2}, \tag*{(299)} R c rot = s ∈ { p , o } ∑ x s ( k B T ) 2 ⟨ E 2 ⟩ s − ⟨ E ⟩ s 2 , (299) c v i b R = ( θ v i b T ) 2 exp ( − θ v i b / T ) ( 1 − exp ( − θ v i b / T ) ) 2 , (300) \frac{c_{\mathrm{vib}}}{R}=\left(\frac{\theta_{\mathrm{vib}}}{T}\right)^2\frac{\exp(-\theta_{\mathrm{vib}}/T)}{\left(1-\exp(-\theta_{\mathrm{vib}}/T)\right)^2}, \tag*{(300)} R c vib = ( T θ vib ) 2 ( 1 − exp ( − θ vib / T ) ) 2 exp ( − θ vib / T ) , (300) and R R R
⟨ E n ⟩ s = ∑ J ∈ J s ( 2 J + 1 ) ε J n exp ( − ε J / k B T ) ∑ J ∈ J s ( 2 J + 1 ) exp ( − ε J / k B T ) , (301) \langle E^n\rangle_s=\frac{\sum_{J\in\mathcal{J}_s}(2J+1)\varepsilon_J^n\exp(-\varepsilon_J/k_{\mathrm{B}}T)}{\sum_{J\in\mathcal{J}_s}(2J+1)\exp(-\varepsilon_J/k_{\mathrm{B}}T)}, \tag*{(301)} ⟨ E n ⟩ s = ∑ J ∈ J s ( 2 J + 1 ) exp ( − ε J / k B T ) ∑ J ∈ J s ( 2 J + 1 ) ε J n exp ( − ε J / k B T ) , (301) where
ε J = h c l i g h t B r o t J ( J + 1 ) , (302) \varepsilon_J=hc_{\mathrm{light}}B_{\mathrm{rot}}J(J+1), \tag*{(302)} ε J = h c light B rot J ( J + 1 ) , (302) are the rigid-rotor levels and h h h s = p s=p s = p J J J s = o s=o s = o J J J x p = 1 / 4 x_p=1/4 x p = 1/4 x o = 3 / 4 x_o=3/4 x o = 3/4 B r o t = 59.32 c m − 1 B_{\mathrm{rot}}=59.32\ \mathrm{cm}^{-1} B rot = 59.32 cm − 1 θ v i b = 5984.0 K \theta_{\mathrm{vib}}=5984.0\ \mathrm{K} θ vib = 5984.0 K [57 p. 250] .
We then take our relaxation times as [58 , 59 , 60 ]
P τ r o t = ( 1.822 × 10 − 4 + 4.64 × 10 − 6 T ) P a s , (303) P\tau_{\mathrm{rot}}=\left(1.822\times10^{-4}+4.64\times10^{-6}T\right)\ \mathrm{Pa\ s}, \tag*{(303)} P τ rot = ( 1.822 × 1 0 − 4 + 4.64 × 1 0 − 6 T ) Pa s , (303) P τ v i b = 10 34.74 T − 1 / 3 − 8.686 a t m s , (304) P\tau_{\mathrm{vib}}=10^{34.74T^{-1/3}-8.686}\ \mathrm{atm\ s}, \tag*{(304)} P τ vib = 1 0 34.74 T − 1/3 − 8.686 atm s , (304) where 1 P a s = 10 d y n s c m − 2 1\ \mathrm{Pa\ s}=10\ \mathrm{dyn\ s\ cm}^{-2} 1 Pa s = 10 dyn s cm − 2 1 a t m s = 1.01325 × 10 6 d y n s c m − 2 1\ \mathrm{atm\ s}=1.01325\times10^6\ \mathrm{dyn\ s\ cm}^{-2} 1 atm s = 1.01325 × 1 0 6 dyn s cm − 2
For neutral gas we use the Sutherland viscosity for hydrogen [61 ]
ν n e u = 1 ρ 8.76 × 10 − 5 ( T 300 ) 3 / 2 372 T + 72 c m 2 s − 1 . (305) \nu_{\mathrm{neu}}=\frac{1}{\rho}8.76\times10^{-5}\left(\frac{T}{300}\right)^{3/2}\frac{372}{T+72}\ \mathrm{cm}^2\ \mathrm{s}^{-1}. \tag*{(305)} ν neu = ρ 1 8.76 × 1 0 − 5 ( 300 T ) 3/2 T + 72 372 cm 2 s − 1 . (305) For ionised plasma the Spitzer model is used [49, Eq. 5-54]
ν i o n = 1 ρ 2.21 × 10 − 15 T 5 / 2 ln Λ c m 2 s − 1 , (306) \nu_{\mathrm{ion}}=\frac{1}{\rho}2.21\times10^{-15}\frac{T^{5/2}}{\ln\Lambda}\ \mathrm{cm}^2\ \mathrm{s}^{-1}, \tag*{(306)} ν ion = ρ 1 2.21 × 1 0 − 15 ln Λ T 5/2 cm 2 s − 1 , (306) which is Braginskii’s η 0 = 0.96 n i k B T τ i / ρ = ν ∥ \eta_0=0.96n_i k_{\mathrm{B}}T\tau_i/\rho=\nu_{\parallel} η 0 = 0.96 n i k B T τ i / ρ = ν ∥ [23 ]
ν ∥ = ν i o n , (307) \nu_{\parallel}=\nu_{\mathrm{ion}}, \tag*{(307)} ν ∥ = ν ion , (307) ν ⊥ 1 = ν i o n 0.96 4.8 β i 2 + 2.23 16 β i 4 + 16.12 β i 2 + 2.33 , ν ⊥ 2 = ν i o n 0.96 1.2 β i 2 + 2.23 4 ( β i 4 + 4.03 β i 2 + 2.33 ) (308) \nu_{\perp1}=\frac{\nu_{\mathrm{ion}}}{0.96}\frac{4.8\beta_i^{2}+2.23}{16\beta_i^{4}+16.12\beta_i^{2}+2.33},\qquad
\nu_{\perp2}=\frac{\nu_{\mathrm{ion}}}{0.96}\frac{1.2\beta_i^{2}+2.23}{4\left(\beta_i^{4}+4.03\beta_i^{2}+2.33\right)} \tag*{(308)} ν ⊥ 1 = 0.96 ν ion 16 β i 4 + 16.12 β i 2 + 2.33 4.8 β i 2 + 2.23 , ν ⊥ 2 = 0.96 ν ion 4 ( β i 4 + 4.03 β i 2 + 2.33 ) 1.2 β i 2 + 2.23 (308) ν ∧ 1 = ν i o n 0.96 β i ( 8 β i 2 + 4.76 ) 16 β i 4 + 16.12 β i 2 + 2.33 , ν ∧ 2 = ν i o n 0.96 β i ( β i 2 + 2.38 ) 2 ( β i 4 + 4.03 β i 2 + 2.33 ) . (309) \nu_{\wedge1}=\frac{\nu_{\mathrm{ion}}}{0.96}\frac{\beta_i\left(8\beta_i^{2}+4.76\right)}{16\beta_i^{4}+16.12\beta_i^{2}+2.33},\qquad
\nu_{\wedge2}=\frac{\nu_{\mathrm{ion}}}{0.96}\frac{\beta_i\left(\beta_i^{2}+2.38\right)}{2\left(\beta_i^{4}+4.03\beta_i^{2}+2.33\right)}. \tag*{(309)} ν ∧ 1 = 0.96 ν ion 16 β i 4 + 16.12 β i 2 + 2.33 β i ( 8 β i 2 + 4.76 ) , ν ∧ 2 = 0.96 ν ion 2 ( β i 4 + 4.03 β i 2 + 2.33 ) β i ( β i 2 + 2.38 ) . (309) These are Braginskii’s η 1 \eta_1 η 1 η 2 / 4 \eta_2/4 η 2 /4 η 3 \eta_3 η 3 η 4 / 2 \eta_4/2 η 4 /2 ρ \rho ρ η 1 \eta_1 η 1 η 3 \eta_3 η 3 η 2 \eta_2 η 2 η 4 \eta_4 η 4 2 β i 2\beta_i 2 β i Π ( 2 ) \boldsymbol{\Pi}^{(2)} Π ( 2 ) Π ( 4 ) \boldsymbol{\Pi}^{(4)} Π ( 4 ) ν ⊥ 1 = ν ⊥ 2 \nu_{\perp1}=\nu_{\perp2} ν ⊥ 1 = ν ⊥ 2 ν ∧ 1 = ν ∧ 2 \nu_{\wedge1}=\nu_{\wedge2} ν ∧ 1 = ν ∧ 2 β i \beta_i β i
β i = ω c i ν i i + ν i n , (310) \beta_i=\frac{\omega_{ci}}{\nu_{ii}+\nu_{in}}, \tag*{(310)} β i = ν ii + ν in ω c i , (310) and ω c i \omega_{ci} ω c i
ω c i = e ∣ B ∣ m p c l i g h t . (311) \omega_{ci}=\frac{e|\mathbf{B}|}{m_p c_{\mathrm{light}}}. \tag*{(311)} ω c i = m p c light e ∣ B ∣ . (311) ν i i \nu_{ii} ν ii ν i n \nu_{in} ν in [51 , 52 ]
ν i i = 4.80 × 10 − 8 n i ln Λ T e V − 3 / 2 s − 1 , (312) \nu_{ii}=4.80\times10^{-8}n_i\ln\Lambda T_{\mathrm{eV}}^{-3/2}\ \mathrm{s}^{-1}, \tag*{(312)} ν ii = 4.80 × 1 0 − 8 n i ln Λ T eV − 3/2 s − 1 , (312) ν i n = m n m i + m n 0.649 × 10 − 9 n n T 0.375 s − 1 , (313) \nu_{in}=\frac{m_n}{m_i+m_n}0.649\times10^{-9}n_nT^{0.375}\ \mathrm{s}^{-1}, \tag*{(313)} ν in = m i + m n m n 0.649 × 1 0 − 9 n n T 0.375 s − 1 , (313) where m n / ( m i + m n ) = 1 / 2 m_n/(m_i+m_n)=1/2 m n / ( m i + m n ) = 1/2 + ^{+} + [51 ] to a momentum-transfer rate. ln Λ \ln\Lambda ln Λ [23 ]
ln Λ = { 23.4 − 1.15 log 10 n e + 3.45 log 10 T e V , T e V < 50 , 25.3 − 1.15 log 10 n e + 2.30 log 10 T e V , T e V ≥ 50. (314) \ln\Lambda=
\begin{cases}
23.4-1.15\log_{10}n_e+3.45\log_{10}T_{\mathrm{eV}}, & T_{\mathrm{eV}}<50,\\
25.3-1.15\log_{10}n_e+2.30\log_{10}T_{\mathrm{eV}}, & T_{\mathrm{eV}}\ge50.
\end{cases} \tag*{(314)} ln Λ = { 23.4 − 1.15 log 10 n e + 3.45 log 10 T eV , 25.3 − 1.15 log 10 n e + 2.30 log 10 T eV , T eV < 50 , T eV ≥ 50. (314) Our magnetic diffusivities are
η = c l i g h t 2 m e ( ν e i + ν e n ) 4 π e 2 n e , η A D = ∣ B ∣ 2 ρ n 2 4 π ρ 2 ρ i ν i n , η H = c l i g h t ∣ B ∣ 4 π e n e . (315) \eta=\frac{c_{\mathrm{light}}^{2}m_e(\nu_{ei}+\nu_{en})}{4\pi e^{2}n_e},\qquad
\eta_{\mathrm{AD}}=\frac{|\mathbf{B}|^{2}\rho_n^{2}}{4\pi\rho^{2}\rho_i\nu_{in}},\qquad
\eta_{\mathrm{H}}=\frac{c_{\mathrm{light}}|\mathbf{B}|}{4\pi e n_e}. \tag*{(315)} η = 4 π e 2 n e c light 2 m e ( ν e i + ν e n ) , η AD = 4 π ρ 2 ρ i ν in ∣ B ∣ 2 ρ n 2 , η H = 4 π e n e c light ∣ B ∣ . (315) Here, ρ n \rho_n ρ n ρ i \rho_i ρ i
Our thermoelectric coefficients are [23 ]
κ J , ∥ = 0.7110 c l i g h t k B T 4 π e , (316) \kappa_{J,\parallel}=0.7110\frac{c_{\mathrm{light}}k_{\mathrm{B}}T}{\sqrt{4\pi e}}, \tag*{(316)} κ J , ∥ = 0.7110 4 π e c light k B T , (316) κ J , ⊥ = c l i g h t k B T 4 π e 5.101 β e 2 + 2.681 β e 4 + 14.79 β e 2 + 3.7703 , (317) \kappa_{J,\perp}=\frac{c_{\mathrm{light}}k_{\mathrm{B}}T}{\sqrt{4\pi e}}\frac{5.101\beta_e^{2}+2.681}{\beta_e^{4}+14.79\beta_e^{2}+3.7703}, \tag*{(317)} κ J , ⊥ = 4 π e c light k B T β e 4 + 14.79 β e 2 + 3.7703 5.101 β e 2 + 2.681 , (317) κ J , ∧ = c l i g h t k B T 4 π e β e ( 1.5 β e 2 + 3.053 ) β e 4 + 14.79 β e 2 + 3.7703 . (318) \kappa_{J,\wedge}=\frac{c_{\mathrm{light}}k_{\mathrm{B}}T}{\sqrt{4\pi e}}\frac{\beta_e\left(1.5\beta_e^{2}+3.053\right)}{\beta_e^{4}+14.79\beta_e^{2}+3.7703}. \tag*{(318)} κ J , ∧ = 4 π e c light k B T β e 4 + 14.79 β e 2 + 3.7703 β e ( 1.5 β e 2 + 3.053 ) . (318) We retrieve our neutral/ionised mass fractions and thermodynamic variables from a linearly interpolated lookup table [62 ] .
For all plots we exclude the domain where single fluid non-relativistic MHD breaks down (these are the grey areas). These regimes are either due to the single fluid model for ambipolar diffusion breaking down at
η A D ≥ c A Δ x , (319) \eta_{\mathrm{AD}}\ge c_A\Delta x, \tag*{(319)} η AD ≥ c A Δ x , (319) where c A c_A c A
k B T m e c l i g h t 2 ≥ 0.03 , (320) \frac{k_{\mathrm{B}}T}{m_{\mathrm{e}}c_{\mathrm{light}}^{2}} \ge0.03, \tag*{(320)} m e c light 2 k B T ≥ 0.03 , (320) c A ≥ 0.3 c l i g h t . (321) c_A \ge0.3c_{\mathrm{light}}. \tag*{(321)} c A ≥ 0.3 c light . (321) Magnetic system
Figure 11 shows the ratio Δ t R K 3 / Δ t R K 1 \Delta t_{\mathrm{RK3}}/\Delta t_{\mathrm{RK1}} Δ t RK3 /Δ t RK1 ≈ 0.87 \approx0.87 ≈ 0.87 2 / 2.51 ≈ 0.80 2/2.51 \approx0.80 2/2.51 ≈ 0.80
Figure 11. Ratio of RK3 and Forward Euler magnetic timesteps. From left to right, top to bottom the mass density ranges from 10 − 30 10^{-30} 1 0 − 30 − 3 ^{-3} − 3 10 − 16 10^{-16} 1 0 − 16 − 3 ^{-3} − 3 ψ = π / 2 \psi= \pi/2 ψ = π /2 θ = π / 2 \theta= \pi/2 θ = π /2 ϕ = 0 \phi= 0 ϕ = 0
Our most restrictive ratio is ∼ 10 18 \sim10^{18} ∼ 1 0 18 Δ t η , R K 1 ∝ 1 / η H 2 \Delta t_{\eta,\mathrm{RK1}} \propto1/\eta_{\mathrm{H}}^2 Δ t η , RK1 ∝ 1/ η H 2 Δ t η , R K 3 ∝ 1 / ( η + η A D + η H ) \Delta t_{\eta,\mathrm{RK3}} \propto1/(\eta+\eta_{\mathrm{AD}}+\eta_{\mathrm{H}}) Δ t η , RK3 ∝ 1/ ( η + η AD + η H ) ∝ 1 / ( η + η A D ) \propto1/(\eta+\eta_{\mathrm{AD}}) ∝ 1/ ( η + η AD )
Thermo-Magnetic System
Here we present our numerical verification of our our bounded timestep and well-posedness criterion for the coupled thermo-magnetic system. We take Δ x = 1 c m \Delta x = 1\ \mathrm{cm} Δ x = 1 cm Δ ⊥ = 1 / 2 c m \Delta_{\perp} = 1/2\ \mathrm{cm} Δ ⊥ = 1/2 cm
Figure 12 shows the timestep ratio for the parallel b x = 1 b_x=1 b x = 1 2 / 2.51 ≈ 0.80 2/2.51 \approx0.80 2/2.51 ≈ 0.80 β e ≈ 0.5 \beta_e \approx0.5 β e ≈ 0.5 ψ = π / 2 \psi=\pi/2 ψ = π /2 ψ = π / 4 \psi=\pi/4 ψ = π /4
Figure 12. Thermo-magnetic timestep bound for RK3 using physical diffusion coefficients setting b y = b z = 0 b_y=b_z=0 b y = b z = 0 b y = x b_y=x b y = x 10 − 30 g c m − 3 10^{-30}\,\mathrm{g\,cm^{-3}} 1 0 − 30 g c m − 3 10 − 16 g c m − 3 10^{-16}\,\mathrm{g\,cm^{-3}} 1 0 − 16 g c m − 3 ψ = π / 2 \psi=\pi/2 ψ = π /2 θ = 0 \theta=0 θ = 0 ϕ = π / 2 \phi=\pi/2 ϕ = π /2
Figure 13 shows our numerically determined largest real component eigenvalue for the thermo-magnetic system (we take the negative for ease of logarithmic plotting). Taking b y = 1 b_y = 1 b y = 1
Figure 13. Numerically determined largest magnitude eigenvalue for the thermo-magnetic system using physical diffusion coefficients. b x = b z = 0 b_x = b_z = 0 b x = b z = 0 b y = 1 b_y = 1 b y = 1 10 − 30 g c m − 3 10^{-30}\ \mathrm{g\ cm^{-3}} 1 0 − 30 g c m − 3 10 − 16 g c m − 3 10^{-16}\ \mathrm{g\ cm^{-3}} 1 0 − 16 g c m − 3 ψ = π / 2 \psi= \pi/2 ψ = π /2 θ = π / 2 \theta= \pi/2 θ = π /2 ϕ = 0 \phi= 0 ϕ = 0
Figure 14 shows the logarithm of the well-posedness criterion expressed as a ratio. Our worst-case scenario gives an actual ratio of ∼ 2.02 \sim2.02 ∼ 2.02
Figure 14. Analytically determined logarithm of the well-posedness criterion for the thermo-magnetic system using physical diffusion coefficients. b x = b z = 0 b_x = b_z = 0 b x = b z = 0 b y = 1 b_y = 1 b y = 1 10 − 30 10^{-30} 1 0 − 30 − 3 ^{-3} − 3 10 − 16 10^{-16} 1 0 − 16 − 3 ^{-3} − 3 ψ = π / 2 \psi= \pi/2 ψ = π /2 θ = π / 2 \theta= \pi/2 θ = π /2 ϕ = 0 \phi= 0 ϕ = 0
Viscosity
Figure 15 shows the ratio between the numerically determined and inverse sum bounded timestep for the unsplit viscosity system in the parallel case. Our least conservative ratios are in regimes when a single dissipative term dominates 2 / 2.51 ≈ 0.80 2/2.51 \approx0.80 2/2.51 ≈ 0.80 ≈ 0.55 \approx0.55 ≈ 0.55 ω c i τ i ∼ 1 \omega_{ci}\tau_i \sim1 ω c i τ i ∼ 1 0.7 0.7 0.7 0.08 ≲ ω c i τ i ≲ 6 0.08 \lesssim\omega_{ci}\tau_i \lesssim6 0.08 ≲ ω c i τ i ≲ 6
Figure 15. Ratio of numerically determined unsplit viscosity timestep and inverse sum bound for b x = 1 b_x = 1 b x = 1 10 − 30 g c m − 3 10^{-30}\ \mathrm{g\ cm^{-3}} 1 0 − 30 g c m − 3 10 − 16 g c m − 3 10^{-16}\ \mathrm{g\ cm^{-3}} 1 0 − 16 g c m − 3 ψ = π / 2 \psi= \pi/2 ψ = π /2 θ = 0 \theta= 0 θ = 0 ϕ = 0 \phi= 0 ϕ = 0
