Linear Stability Analysis
Hyperbolic conservation law:
$\frac{\partial U}{\partial t} + \frac{\partial F}{\partial x} + \frac{\partial G}{\partial y} + \frac{\partial H}{\partial z} = 0$
$t, x, y, z =$ basis
$U =$ state vector
$F, G, H =$ flux in $x,y,z$ direction
Rewriting fluxes in terms of $U$:
$\frac{\partial U}{\partial t} + \frac{\partial F}{\partial U} \frac{\partial U}{\partial x} + \frac{\partial G}{\partial U} \frac{\partial U}{\partial y} + \frac{\partial H}{\partial U} \frac{\partial U}{\partial z} = 0$
Perturbation in $x$ direction:
$U(t,x,y,z) = \mathring{U}(t,x,y,z) + \epsilon exp(i(yk + zm)) \hat{U}(t,x)$
$\mathring{U}(t,x,y,z) =$ base state
$\hat{U}(t,x) =$ perturbation in $t,x$
$k, m = $ wave numbers.
Derivatives:
$\partial_t U = \partial_t \mathring{U} + \epsilon exp(i(yk + zm)) \partial_t \hat{U}$
$\partial_x U = \partial_x \mathring{U} + \epsilon exp(i(yk + zm)) \partial_x \hat{U}$
$\partial_y U = \partial_y \mathring{U} + i k \epsilon exp(i(yk + zm)) \hat{U}$
$\partial_z U = \partial_z \mathring{U} + i m \epsilon exp(i(yk + zm)) \hat{U}$
Solving the hyperbolic conservation law at $\mathring{U}$:
$\frac{\partial \mathring{U}}{\partial t}
+ \frac{\partial F(\mathring{U})}{\partial U} \frac{\partial \mathring{U}}{\partial x}
+ \frac{\partial G(\mathring{U})}{\partial U} \frac{\partial \mathring{U}}{\partial y}
+ \frac{\partial H(\mathring{U})}{\partial U} \frac{\partial \mathring{U}}{\partial z} = 0$
Substitute pertrubed state into hyperbolic conservative law:
$ \partial_t \mathring{U} + \epsilon exp(i(yk + zm)) \partial_t \hat{U}
+ \frac{\partial F(\mathring{U})}{\partial U} (\partial_x \mathring{U} + \epsilon exp(i(yk + zm)) \partial_x \hat{U})
+ \frac{\partial G(\mathring{U})}{\partial U} (\partial_y \mathring{U} + \epsilon exp(i(yk + zm)) (i k \hat{U}))
+ \frac{\partial H(\mathring{U})}{\partial U} (\partial_z \mathring{U} + \epsilon exp(i(yk + zm)) (i m \hat{U}))
= 0$
$ \partial_t \mathring{U}
+ \frac{\partial F(\mathring{U})}{\partial U} \partial_x \mathring{U}
+ \frac{\partial G(\mathring{U})}{\partial U} \partial_y \mathring{U}
+ \frac{\partial H(\mathring{U})}{\partial U} \partial_z \mathring{U}
+ \epsilon exp(i(yk + zm)) (
\partial_t \hat{U}
+ \frac{\partial F(\mathring{U})}{\partial U} \partial_x \hat{U}
+ \frac{\partial G(\mathring{U})}{\partial U} (i k \hat{U})
+ \frac{\partial H(\mathring{U})}{\partial U} (i m \hat{U})
) = 0$
This leaves the perturbation solution:
$ \epsilon exp(i(yk + zm)) (
\partial_t \hat{U}
+ \frac{\partial F(\mathring{U})}{\partial U} \partial_x \hat{U}
+ \frac{\partial G(\mathring{U})}{\partial U} (i k \hat{U})
+ \frac{\partial H(\mathring{U})}{\partial U} (i m \hat{U})
) = 0$
$ \partial_t \hat{U}
+ \frac{\partial F(\mathring{U})}{\partial U} \partial_x \hat{U}
+ i k \frac{\partial G(\mathring{U})}{\partial U} \hat{U}
+ i m \frac{\partial H(\mathring{U})}{\partial U} \hat{U}
= 0$
Considering only the 2D solution of the perturbation (setting $m = 0$):
$ \partial_t \hat{U}
+ \frac{\partial F(\mathring{U})}{\partial U} \partial_x \hat{U}
= -i k \frac{\partial G(\mathring{U})}{\partial U} \hat{U}$
MHD equations
primitives:
$\rho =$ density
$v_i =$ velocity
$B_i =$ magnetic field
$P =$ pressure
$\gamma =$ heat capacity ratio
state vector:
$U_i = \downarrow i \left[\matrix{
\rho \\ \rho v_i \\ B_i \\ E_{total}
}\right]$
$E_{kin} = \frac{1}{2} \rho v^2 =$ kinetic energy
$E_{int} = \frac{P}{\gamma - 1} = $ internal energy
$E_{mag} = \frac{1}{2 \mu_0} =$ magnetic energy
$E_{total} = E_{kin} + E_{int} + E_{mag} = \frac{1}{2} \rho v^2 + \frac{P}{\gamma - 1} + \frac{1}{2 \mu_0} B^2 =$ total energy
Flux in $j$th direction:
$F_{ij} = \downarrow i \left[\matrix{
\rho v_j \\
\rho v_i v_j + \delta_{ij} P_{total} - {1\over\mu_0} B_i B_j \\
B_i v_j - B_j v_i \\
H_{total} v_j - {1\over\mu_0} B_k v_k B_j
}\right]$
$P_{total} = P + \frac{1}{2 \mu_0} B^2 = $ total pressure
$H_{total} = E_{total} + P_{total} = \frac{1}{2} \rho v^2 + \frac{\gamma}{\gamma - 1} P + \frac{1}{\mu_0} B^2 =$ total enthalpy
Derivative of flux wrt conservative variables:
$\frac{\partial F_{ij}}{\partial U_k}
= \downarrow i \overset{\rightarrow k}{ \left[\matrix{
0 &
\delta_{jk} &
0 &
0 \\
-v_i v_j + \frac{1}{2} \delta_{ij} (\gamma-1) v^2 &
\delta_{ik} v_j + \delta_{jk} v_i - \delta_{ij} (\gamma-1) v_k &
\frac{1}{\mu_0}(-\delta_{ij} (\gamma-2) B_k - \delta_{ik} B_j - \delta_{jk} B_i) &
\delta_{ij} (\gamma-1) \\
-\frac{1}{\rho} (B_i v_j - B_j v_i) &
\frac{1}{\rho} (B_i \delta_{jk} - B_j \delta_{ik}) &
\delta_{ik} v_j - \delta_{jk} v_i &
0 \\
v_j (\frac{1}{2} (\gamma-1) v^2 - h_{total}) + \frac{1}{\mu_0\rho} B_j B_m v_m &
-(\gamma-1) v_j v_k + (\delta_{jk} h_{total} - \frac{1}{\mu_0\rho} B_j B_k) &
\frac{1}{\mu_0} (-(\gamma - 2) v_j B_k - v_k B_j - \delta_{jk} v_m B_m) &
\gamma v_j
}\right] }$
$h_{total} = \frac{1}{\rho} H_{total} =$ specific total enthalpy
Derivative of flux wrt conservative variables in x direction:
$\frac{\partial F_{ix}}{\partial U_k}
= \left[\matrix{
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-v_x v_x + \frac{1}{2} (\gamma-1) v^2 &
-(\gamma-3) v_x &
-(\gamma-1) v_y &
-(\gamma-1) v_z &
-\frac{1}{\mu_0} \gamma B_x &
-\frac{1}{\mu_0} (\gamma-2) B_y &
-\frac{1}{\mu_0} (\gamma-2) B_z &
\gamma - 1 \\
-v_y v_x & v_y & v_x & 0 & -\frac{1}{\mu_0} B_y & -\frac{1}{\mu_0} B_x & 0 & 0 \\
-v_z v_x & v_z & 0 & v_x & -\frac{1}{\mu_0} B_z & 0 & -\frac{1}{\mu_0} B_x & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{1}{\rho} (B_y v_x - B_x v_y) & \frac{1}{\rho} B_y & -\frac{1}{\rho} B_x & 0 & -v_y & v_x & 0 & 0 \\
-\frac{1}{\rho} (B_z v_x - B_x v_z) & \frac{1}{\rho} B_z & 0 & -\frac{1}{\rho} B_x & - v_z & 0 & v_x & 0 \\
v_x (\frac{1}{2} (\gamma-1) v^2 - h_{total}) + \frac{1}{\mu_0\rho} B_x B_m v_m &
-(\gamma-1) v_x v_x - \frac{1}{\mu_0\rho} B_x B_x + h_{total} &
-(\gamma-1) v_x v_y - \frac{1}{\mu_0\rho} B_x B_y &
-(\gamma-1) v_x v_z - \frac{1}{\mu_0\rho} B_x B_z &
-\frac{1}{\mu_0} ((\gamma - 2) v_x B_x + v_x B_x + v_m B_m) &
-\frac{1}{\mu_0} ((\gamma - 2) v_x B_y + v_y B_x) &
-\frac{1}{\mu_0} ((\gamma - 2) v_x B_z + v_z B_x) &
\gamma v_x
}\right] $
Derivative of flux wrt conservative variables in y direction:
$\frac{\partial F_{iy}}{\partial U_k}
\left[\matrix{
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
-v_x v_y & v_y & v_x & 0 & -\frac{1}{\mu_0} B_y & -\frac{1}{\mu_0} B_x & 0 & 0 \\
-v_y v_y + \frac{1}{2} (\gamma-1) v^2 &
-(\gamma-1) v_x &
-(\gamma-3) v_y &
-(\gamma-1) v_z &
-\frac{1}{\mu_0} (\gamma-2) B_x &
-\frac{1}{\mu_0} \gamma B_y &
-\frac{1}{\mu_0} (\gamma-2) B_z &
\gamma-1 \\
-v_z v_y & 0 & v_z & v_y & 0 & -\frac{1}{\mu_0} B_z & -\frac{1}{\mu_0} B_y & 0 \\
-\frac{1}{\rho} (B_x v_y - B_y v_x) & -\frac{1}{\rho} B_y & \frac{1}{\rho} B_x & 0 & v_y & -v_x & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{1}{\rho} (B_z v_y - B_y v_z) & 0 & \frac{1}{\rho} B_z & -\frac{1}{\rho} B_y & 0 & -v_z & v_y & 0 \\
v_y (\frac{1}{2} (\gamma-1) v^2 - h_{total}) + \frac{1}{\mu_0\rho} B_y B_m v_m &
-(\gamma-1) v_y v_x - \frac{1}{\mu_0\rho} B_y B_x &
-(\gamma-1) v_y v_y - \frac{1}{\mu_0\rho} B_y B_y + h_{total} &
-(\gamma-1) v_y v_z - \frac{1}{\mu_0\rho} B_y B_z &
-\frac{1}{\mu_0} ((\gamma - 2) v_y B_x + v_x B_y) &
-\frac{1}{\mu_0} ((\gamma - 2) v_y B_y + v_y B_y + v_m B_m) &
-\frac{1}{\mu_0} ((\gamma - 2) v_y B_z + v_z B_y) &
\gamma v_y
}\right] $
MHD 2D perturbations:
$
\downarrow i \left[\matrix{
\partial_t \hat{\rho} \\
\partial_t \hat{(\rho v_i)} \\
\partial_t \hat{B}_i \\
\partial_t \hat{E}_{total}
}\right]
+
\downarrow i \overset{\rightarrow k}{ \left[\matrix{
0 &
\delta_{xk} &
0 &
0 \\
-\mathring{v}_i \mathring{v}_x + \frac{1}{2} \delta_{ix} (\gamma-1) \mathring{v}^2 &
\delta_{ik} \mathring{v}_x + \delta_{xk} \mathring{v}_i - \delta_{ix} (\gamma-1) \mathring{v}_k &
\frac{1}{\mu_0}(-\delta_{ix} (\gamma-2) \mathring{B}_k - \delta_{ik} \mathring{B}_x - \delta_{xk} \mathring{B}_i) &
\delta_{ix} (\gamma-1) \\
-\frac{1}{ \mathring{\rho} } (\mathring{B}_i \mathring{v}_x - \mathring{B}_x \mathring{v}_i) &
\frac{1}{ \mathring{\rho} } (\mathring{B}_i \delta_{xk} - \mathring{B}_x \delta_{ik}) &
\delta_{ik} \mathring{v}_x - \delta_{xk} \mathring{v}_i &
0 \\
\mathring{v}_x (\frac{1}{2} (\gamma-1) \mathring{v}^2 - \mathring{h}_{total}) + \frac{1}{\mu_0 \mathring{\rho} } \mathring{B}_x \mathring{B}_m \mathring{v}_m &
-(\gamma-1) \mathring{v}_x \mathring{v}_k + (\delta_{xk} \mathring{h}_{total} - \frac{1}{\mu_0 \mathring{\rho} } \mathring{B}_x \mathring{B}_k) &
\frac{1}{\mu_0} (-(\gamma - 2) \mathring{v}_x \mathring{B}_k - \mathring{v}_k \mathring{B}_x - \delta_{xk} \mathring{v}_m \mathring{B}_m) &
\gamma \mathring{v}_x
}\right] }
\cdot
\downarrow k \left[\matrix{
\partial_x \hat{\rho} \\
\partial_x \hat{(\rho v_k)} \\
\partial_x \hat{B}_k \\
\partial_x \hat{E}_{total}
}\right]
= -i k
\cdot
\downarrow i \overset{\rightarrow k}{ \left[\matrix{
0 &
\delta_{yk} &
0 &
0 \\
-\mathring{v}_i \mathring{v}_y + \frac{1}{2} \delta_{iy} (\gamma-1) \mathring{v}^2 &
\delta_{ik} \mathring{v}_y + \delta_{yk} \mathring{v}_i - \delta_{iy} (\gamma-1) \mathring{v}_k &
\frac{1}{\mu_0}(-\delta_{iy} (\gamma-2) \mathring{B}_k - \delta_{ik} \mathring{B}_y - \delta_{yk} \mathring{B}_i) &
\delta_{iy} (\gamma-1) \\
-\frac{1}{ \mathring{\rho} } (\mathring{B}_i \mathring{v}_y - \mathring{B}_y \mathring{v}_i) &
\frac{1}{ \mathring{\rho} } (\mathring{B}_i \delta_{yk} - \mathring{B}_y \delta_{ik}) &
\delta_{ik} \mathring{v}_y - \delta_{yk} \mathring{v}_i &
0 \\
\mathring{v}_y (\frac{1}{2} (\gamma-1) \mathring{v}^2 - \mathring{h}_{total}) + \frac{1}{\mu_0 \mathring{\rho} } \mathring{B}_y \mathring{B}_m \mathring{v}_m &
-(\gamma-1) \mathring{v}_y \mathring{v}_k + (\delta_{yk} \mathring{h}_{total} - \frac{1}{\mu_0 \mathring{\rho} } \mathring{B}_y \mathring{B}_k) &
\frac{1}{\mu_0} (-(\gamma - 2) \mathring{v}_y \mathring{B}_k - \mathring{v}_k \mathring{B}_y - \delta_{yk} \mathring{v}_m \mathring{B}_m) &
\gamma \mathring{v}_y
}\right] }
\cdot
\downarrow i \left[\matrix{
\hat{\rho} \\
\hat{(\rho v_i)} \\
\hat{B}_i \\
\hat{E}_{total}
}\right]$
Here I'm representing the flux Jacobians in terms of $\mathring{W}$, but what about $\mathring{W}$ in terms of $\mathring{U}$?
Is that where the linearization and elimination of higher-order terms comes in?
One way to avoid this could be to represent $\frac{\partial F(\mathring{U})}{\partial U}$ in terms of $\mathring{U}$ instead of $\mathring{W}$.
Expand Kronecker deltas:
$
\left[\matrix{
\partial_t \hat{\rho} \\
\partial_t \hat{(\rho v_x)} \\
\partial_t \hat{(\rho v_y)} \\
\partial_t \hat{(\rho v_z)} \\
\partial_t \hat{B}_x \\
\partial_t \hat{B}_y \\
\partial_t \hat{B}_z \\
\partial_t \hat{E}_{total}
}\right]
+
\left[\matrix{
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-\mathring{v}_x \mathring{v}_x + \frac{1}{2} (\gamma-1) \mathring{v}^2 &
-(\gamma-3) \mathring{v}_x &
-(\gamma-1) \mathring{v}_y &
-(\gamma-1) \mathring{v}_z &
-\frac{1}{\mu_0} \gamma \mathring{B}_x &
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_y &
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_z &
\gamma - 1 \\
-\mathring{v}_y \mathring{v}_x & \mathring{v}_y & \mathring{v}_x & 0 & -\frac{1}{\mu_0} \mathring{B}_y & -\frac{1}{\mu_0} \mathring{B}_x & 0 & 0 \\
-\mathring{v}_z \mathring{v}_x & \mathring{v}_z & 0 & \mathring{v}_x & -\frac{1}{\mu_0} \mathring{B}_z & 0 & -\frac{1}{\mu_0} \mathring{B}_x & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{1}{\mathring{\rho}} (\mathring{B}_y \mathring{v}_x - \mathring{B}_x \mathring{v}_y) & \frac{1}{\mathring{\rho}} \mathring{B}_y & -\frac{1}{\mathring{\rho}} \mathring{B}_x & 0 & -\mathring{v}_y & \mathring{v}_x & 0 & 0 \\
-\frac{1}{\mathring{\rho}} (\mathring{B}_z \mathring{v}_x - \mathring{B}_x \mathring{v}_z) & \frac{1}{\mathring{\rho}} \mathring{B}_z & 0 & -\frac{1}{\mathring{\rho}} \mathring{B}_x & - \mathring{v}_z & 0 & \mathring{v}_x & 0 \\
\mathring{v}_x (\frac{1}{2} (\gamma-1) \mathring{v}^2 - \mathring{h}_{total}) + \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_m \mathring{v}_m &
-(\gamma-1) \mathring{v}_x \mathring{v}_x - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_x + \mathring{h}_{total} &
-(\gamma-1) \mathring{v}_x \mathring{v}_y - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_y &
-(\gamma-1) \mathring{v}_x \mathring{v}_z - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_z &
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_x \mathring{B}_x + \mathring{v}_x \mathring{B}_x + \mathring{v}_m \mathring{B}_m) &
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_x \mathring{B}_y + \mathring{v}_y \mathring{B}_x) &
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_x \mathring{B}_z + \mathring{v}_z \mathring{B}_x) &
\gamma \mathring{v}_x
}\right]
\left[\matrix{
\partial_x \hat{\rho} \\
\partial_x \hat{(\rho v_x)} \\
\partial_x \hat{(\rho v_y)} \\
\partial_x \hat{(\rho v_z)} \\
\partial_x \hat{B}_x \\
\partial_x \hat{B}_y \\
\partial_x \hat{B}_z \\
\partial_x \hat{E}_{total}
}\right]
= -i k
\left[\matrix{
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
-\mathring{v}_x \mathring{v}_y & \mathring{v}_y & \mathring{v}_x & 0 & -\frac{1}{\mu_0} \mathring{B}_y & -\frac{1}{\mu_0} \mathring{B}_x & 0 & 0 \\
-\mathring{v}_y \mathring{v}_y + \frac{1}{2} (\gamma-1) \mathring{v}^2 &
-(\gamma-1) \mathring{v}_x &
-(\gamma-3) \mathring{v}_y &
-(\gamma-1) \mathring{v}_z &
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_x &
-\frac{1}{\mu_0} \gamma \mathring{B}_y &
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_z &
\gamma-1 \\
-\mathring{v}_z \mathring{v}_y & 0 & \mathring{v}_z & \mathring{v}_y & 0 & -\frac{1}{\mu_0} \mathring{B}_z & -\frac{1}{\mu_0} \mathring{B}_y & 0 \\
-\frac{1}{\mathring{\rho}} (\mathring{B}_x \mathring{v}_y - \mathring{B}_y \mathring{v}_x) & -\frac{1}{\mathring{\rho}} \mathring{B}_y & \frac{1}{\mathring{\rho}} \mathring{B}_x & 0 & \mathring{v}_y & -\mathring{v}_x & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-\frac{1}{\mathring{\rho}} (\mathring{B}_z \mathring{v}_y - \mathring{B}_y \mathring{v}_z) & 0 & \frac{1}{\mathring{\rho}} \mathring{B}_z & -\frac{1}{\mathring{\rho}} \mathring{B}_y & 0 & -\mathring{v}_z & \mathring{v}_y & 0 \\
\mathring{v}_y (\frac{1}{2} (\gamma-1) \mathring{v}^2 - \mathring{h}_{total}) + \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_m \mathring{v}_m &
-(\gamma-1) \mathring{v}_y \mathring{v}_x - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_x &
-(\gamma-1) \mathring{v}_y \mathring{v}_y - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_y + \mathring{h}_{total} &
-(\gamma-1) \mathring{v}_y \mathring{v}_z - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_z &
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_y \mathring{B}_x + \mathring{v}_x \mathring{B}_y) &
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_y \mathring{B}_y + \mathring{v}_y \mathring{B}_y + \mathring{v}_m \mathring{B}_m) &
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_y \mathring{B}_z + \mathring{v}_z \mathring{B}_y) &
\gamma \mathring{v}_y
}\right]
\left[\matrix{
\hat{\rho} \\
\hat{(\rho v_x)} \\
\hat{(\rho v_y)} \\
\hat{(\rho v_z)} \\
\hat{B}_x \\
\hat{B}_y \\
\hat{B}_z \\
\hat{E}_{total}
}\right]$
Written out:
$\partial_t \hat{\rho} + \partial_x \hat{\rho} = -i k \hat{(\rho v_y)}$
$\partial_t \hat{(\rho v_x)}
+ (-\mathring{v}_x \mathring{v}_x + \frac{1}{2} (\gamma-1) \mathring{v}^2) \partial_x \hat{\rho}
-(\gamma-3) \mathring{v}_x \partial_x \hat{(\rho v_x)}
-(\gamma-1) \mathring{v}_y \partial_x \hat{(\rho v_y)}
-(\gamma-1) \mathring{v}_z \partial_x \hat{(\rho v_z)}
-\frac{1}{\mu_0} \gamma \mathring{B}_x \partial_x \hat{B}_x
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_y \partial_x \hat{B}_y
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_z \partial_x \hat{B}_z
+ (\gamma - 1) \partial_x \hat{E}_{total}
= -i k (
-\mathring{v}_x \mathring{v}_y \hat{\rho}
+\mathring{v}_y \hat{(\rho v_x)}
+\mathring{v}_x \hat{(\rho v_y)}
-\frac{1}{\mu_0} \mathring{B}_y \hat{B}_x
-\frac{1}{\mu_0} \mathring{B}_x \hat{B}_y
)$
$
\partial_t \hat{(\rho v_y)}
-\mathring{v}_y \mathring{v}_x
\partial_x \hat{\rho}
+\mathring{v}_y
\partial_x \hat{(\rho v_x)}
+\mathring{v}_x
\partial_x \hat{(\rho v_y)}
-\frac{1}{\mu_0} \mathring{B}_y
\partial_x \hat{B}_x
-\frac{1}{\mu_0} \mathring{B}_x
\partial_x \hat{B}_y
= -i k (
(-\mathring{v}_y \mathring{v}_y + \frac{1}{2} (\gamma-1) \mathring{v}^2) \hat{\rho}
-(\gamma-1) \mathring{v}_x \hat{(\rho v_x)}
-(\gamma-3) \mathring{v}_y \hat{(\rho v_y)}
-(\gamma-1) \mathring{v}_z \hat{(\rho v_z)}
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_x \hat{B}_x
-\frac{1}{\mu_0} \gamma \mathring{B}_y \hat{B}_y
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_z \hat{B}_z
-(\gamma-1) \hat{E}_{total}
)$
$\partial_t \hat{(\rho v_z)}
-\mathring{v}_z \mathring{v}_x \partial_x \hat{\rho}
+\mathring{v}_z \partial_x \hat{(\rho v_x)}
+\mathring{v}_x \partial_x \hat{(\rho v_z)}
-\frac{1}{\mu_0} \mathring{B}_z \partial_x \hat{B}_x
-\frac{1}{\mu_0} \mathring{B}_x \partial_x \hat{B}_z
= -i k (
-\mathring{v}_z \mathring{v}_y \hat{\rho}
+\mathring{v}_z \hat{(\rho v_y)}
+\mathring{v}_y \hat{(\rho v_z)}
-\frac{1}{\mu_0} \mathring{B}_z \hat{B}_y
-\frac{1}{\mu_0} \mathring{B}_y \hat{B}_z
)$
$\partial_t \hat{B}_x
= -i k (
-\frac{1}{\mathring{\rho}} (\mathring{B}_x \mathring{v}_y - \mathring{B}_y \mathring{v}_x) \hat{\rho}
- \frac{1}{\mathring{\rho}} \mathring{B}_y \hat{(\rho v_x)}
+ \frac{1}{\mathring{\rho}} \mathring{B}_x \hat{(\rho v_y)}
+ \mathring{v}_y \hat{B}_x
- \mathring{v}_x \hat{B}_y
)$
$ \partial_t \hat{B}_y
-\frac{1}{\mathring{\rho}} (\mathring{B}_y \mathring{v}_x - \mathring{B}_x \mathring{v}_y) \partial_x \hat{\rho}
+\frac{1}{\mathring{\rho}} \mathring{B}_y \partial_x \hat{(\rho v_x)}
-\frac{1}{\mathring{\rho}} \mathring{B}_x \partial_x \hat{(\rho v_y)}
-\mathring{v}_y \partial_x \hat{B}_x
+\mathring{v}_x \partial_x \hat{B}_y
= 0$
$ \partial_t \hat{B}_z
-\frac{1}{\mathring{\rho}} (\mathring{B}_z \mathring{v}_x - \mathring{B}_x \mathring{v}_z)
\partial_x \hat{\rho}
+\frac{1}{\mathring{\rho}} \mathring{B}_z
\partial_x \hat{(\rho v_x)}
-\frac{1}{\mathring{\rho}} \mathring{B}_x
\partial_x \hat{(\rho v_z)}
-\mathring{v}_z
\partial_x \hat{B}_x
+\mathring{v}_x
\partial_x \hat{B}_z
= -i k (
-\frac{1}{\mathring{\rho}} (\mathring{B}_z \mathring{v}_y - \mathring{B}_y \mathring{v}_z)
\hat{\rho}
+\frac{1}{\mathring{\rho}} \mathring{B}_z
\hat{(\rho v_y)}
-\frac{1}{\mathring{\rho}} \mathring{B}_y
\hat{(\rho v_z)}
-\mathring{v}_z
\hat{B}_y
+\mathring{v}_y
\hat{B}_z
)$
$ \partial_t \hat{E}_{total}
+(\mathring{v}_x (\frac{1}{2} (\gamma-1) \mathring{v}^2 - \mathring{h}_{total}) + \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_m \mathring{v}_m) \partial_x \hat{\rho}
+(-(\gamma-1) \mathring{v}_x \mathring{v}_x - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_x + \mathring{h}_{total}) \partial_x \hat{(\rho v_x)}
+(-(\gamma-1) \mathring{v}_x \mathring{v}_y - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_y ) \partial_x \hat{(\rho v_y)}
+(-(\gamma-1) \mathring{v}_x \mathring{v}_z - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_z) \partial_x \hat{(\rho v_z)}
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_x \mathring{B}_x + \mathring{v}_x \mathring{B}_x + \mathring{v}_m \mathring{B}_m) \partial_x \hat{B}_x
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_x \mathring{B}_y + \mathring{v}_y \mathring{B}_x) \partial_x \hat{B}_y
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_x \mathring{B}_z + \mathring{v}_z \mathring{B}_x) \partial_x \hat{B}_z
\gamma \mathring{v}_x \partial_x \hat{E}_{total}
= -i k (
(\mathring{v}_y (\frac{1}{2} (\gamma-1) \mathring{v}^2 - \mathring{h}_{total}) + \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_m \mathring{v}_m ) \hat{\rho}
+(-(\gamma-1) \mathring{v}_y \mathring{v}_x - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_x) \hat{(\rho v_x)}
+(-(\gamma-1) \mathring{v}_y \mathring{v}_y - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_y + \mathring{h}_{total}) \hat{(\rho v_y)}
+(-(\gamma-1) \mathring{v}_y \mathring{v}_z - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_z) \hat{(\rho v_z)}
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_y \mathring{B}_x + \mathring{v}_x \mathring{B}_y) \hat{B}_x
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_y \mathring{B}_y + \mathring{v}_y \mathring{B}_y + \mathring{v}_m \mathring{B}_m) \hat{B}_y
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_y \mathring{B}_z + \mathring{v}_z \mathring{B}_y) \hat{B}_z
+\gamma \mathring{v}_y \hat{E}_{total}
)$
Primitive perturbations:
$W_i = \downarrow i \left[\matrix{ \rho \\ v_i \\ B_i \\ P }\right]$
$W_i = \mathring{W}_i + \epsilon exp(i(yk + zm)) \hat{W}_i$
individual terms of MHD primitive perturbations:
$\rho = \mathring{\rho} + \epsilon exp(i(yk + zm)) \hat{\rho}$
$v_i = \mathring{v}_i + \epsilon exp(i(yk + zm)) \hat{v}_i$
$B_i = \mathring{B}_i + \epsilon exp(i(yk + zm)) \hat{B}_i$
$P = \mathring{P} + \epsilon exp(i(yk + zm)) \hat{P}$
individual terms of MHD conservative perturbations:
$\rho = \mathring{\rho} + \epsilon exp(i(yk + zm)) \hat{\rho}$
$\rho v_i = (\mathring{\rho} + \epsilon exp(i(yk + zm)) \hat{\rho}) (\mathring{v_i} + \epsilon exp(i(yk + zm)) \hat{v}_i)
= \mathring{\rho} \mathring{v}_i + \epsilon exp(i(yk + zm)) (\mathring{\rho} \hat{v}_i + \hat{\rho} \mathring{v}_i) + \mathcal{O}(\epsilon^2)
= \mathring{(\rho v_i)} + \epsilon exp(i(yk + zm)) \hat{(\rho v_i)} + \mathcal{O}(\epsilon^2)
$
$B_i = \mathring{B}_i + \epsilon exp(i(yk + zm)) \hat{B}_i$
$E_{total}
= \frac{1}{2} \rho v^2 + \frac{1}{\gamma - 1} P + \frac{1}{2 \mu_0} B^2
= \frac{1}{2} (
\mathring{\rho} + \epsilon exp(i(yk + zm)) \hat{\rho}
) \Sigma_{k=1}^3 (
\mathring{v}_k + \epsilon exp(i(yk + zm)) \hat{v}_k
)^2
+ \frac{1}{\gamma-1} (
\mathring{P} + \epsilon exp(i(yk + zm)) \hat{P}
)
+ \frac{1}{2 \mu_0} \Sigma_{k=1}^3 (
\mathring{B}_k + \epsilon exp(i(yk + zm)) \hat{B}_k
)^2
$
$ = \frac{1}{2} (
\mathring{\rho} + \epsilon exp(i(yk + zm)) \hat{\rho}
) \Sigma_{k=1}^3 (
\mathring{v}_k^2 + 2 \epsilon exp(i(yk + zm)) \mathring{v}_k \hat{v}_k + \mathcal{O}(\epsilon^2)
)
+ \frac{1}{\gamma-1} (
\mathring{P} + \epsilon exp(i(yk + zm)) \hat{P}
)
+ \frac{1}{2 \mu_0} \Sigma_{k=1}^3 (
\mathring{B}_k^2 + 2 \epsilon exp(i(yk + zm)) \mathring{B}_k \hat{B}_k + \mathcal{O}(\epsilon^2)
)
$
$ = \frac{1}{2} \mathring{\rho} \mathring{v}^2
+ \frac{1}{\gamma-1} \mathring{P}
+ \frac{1}{2 \mu_0} \mathring{B}^2
+ \epsilon exp(i(yk + zm)) (
\frac{1}{2} \hat{\rho} \mathring{v}^2
+ \mathring{\rho} \mathring{v}_k \hat{v}_k
+ \frac{1}{\gamma-1} \hat{P}
+ \frac{1}{\mu_0} \mathring{B}_k \hat{B}_k
)
+ \mathcal{O}(\epsilon^2)
$
$ = \mathring{E}_{total} + \epsilon exp(i(yk + zm)) \hat{E}_{total}$
Derived linearized perturbation relations:
$\mathring{U}_i
= \downarrow i \left[\matrix{
\mathring{\rho} \\
\mathring{(\rho v_i)} \\
\mathring{B}_i \\
\mathring{E}_{total}
}\right] + \epsilon exp(i(yk + zm)) \cdot \downarrow i \left[\matrix{
\hat{\rho} \\
\hat{(\rho v_i)} \\
\hat{B}_i \\
\hat{E}_{total}
}\right]
= \downarrow i \left[\matrix{
\mathring{\rho} \\
\mathring{\rho} \mathring{v}_i \\
\mathring{B}_i \\
\frac{1}{2} \mathring{\rho} \mathring{v}^2 + \frac{1}{\gamma - 1} \mathring{P} + \frac{1}{2 \mu_0} \mathring{B}^2
}\right] + \epsilon exp(i(yk + zm)) \cdot \downarrow i \left[\matrix{
\hat{\rho} \\
\mathring{\rho} \hat{v}_i + \hat{\rho} \mathring{v}_i \\
\hat{B}_i \\
\frac{1}{2} \hat{\rho} \mathring{v}^2
+ \mathring{\rho} \mathring{v}_k \hat{v}_k
+ \frac{1}{\gamma-1} \hat{P}
+ \frac{1}{\mu_0} \mathring{B}_k \hat{B}_k
}\right]
$
individually:
$\mathring{(\rho v_i)} = \mathring{\rho} \mathring{v}_i$
$\hat{(\rho v_i)} = \mathring{\rho} \hat{v}_i + \hat{\rho} \mathring{v}_i$
$\mathring{E}_{total} = \frac{1}{2} \mathring{\rho} \mathring{v}^2 + \frac{1}{\gamma-1} \mathring{P} + \frac{1}{2 \mu_0} \mathring{B}^2$
$\hat{E}_{total} = \frac{1}{2} \hat{\rho} \mathring{v}^2 + \mathring{\rho} \mathring{v}_k \hat{v}_k + \frac{1}{\gamma-1} \hat{P} + \frac{1}{\mu_0} \mathring{B}_k \hat{B}_k$
coincidentally, in terms of differentials, things look very similar, for $\hat{x} = \delta \mathring{x}$:
$\delta (\rho v_i) = \delta (\rho v_i) = \rho \delta v_i + v_i \delta \rho$
$\delta E_{total}
= \delta (\frac{1}{2} \rho v^2 + \frac{1}{\gamma-1} P + \frac{1}{2 \mu_0} B^2)
= \frac{1}{2} v^2 \delta \rho + \rho v_k \delta v_k + \frac{1}{\gamma-1} \delta P + \frac{1}{\mu_0} B_k \delta B_k$
Next: plug the $\hat{W}_i$'s into the $\hat{U}$ MHD equations...
$\partial_t \hat{\rho} + \partial_x \hat{\rho} = -i k (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)$
$\partial_t (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
+ (-\mathring{v}_x \mathring{v}_x + \frac{1}{2} (\gamma-1) \mathring{v}^2) \partial_x \hat{\rho}
-(\gamma-3) \mathring{v}_x \partial_x (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
-(\gamma-1) \mathring{v}_y \partial_x (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
-(\gamma-1) \mathring{v}_z \partial_x (\mathring{\rho} \hat{v}_z + \hat{\rho} \mathring{v}_z)
-\frac{1}{\mu_0} \gamma \mathring{B}_x \partial_x \hat{B}_x
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_y \partial_x \hat{B}_y
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_z \partial_x \hat{B}_z
+ (\gamma - 1) \partial_x (\frac{1}{2} \hat{\rho} \mathring{v}^2 + \mathring{\rho} \mathring{v}_q \hat{v}_q + \frac{1}{\gamma-1} \hat{P} + \frac{1}{\mu_0} \mathring{B}_q \hat{B}_q)
= -i k (
-\mathring{v}_x \mathring{v}_y \hat{\rho}
+\mathring{v}_y (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
+\mathring{v}_x (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
-\frac{1}{\mu_0} \mathring{B}_y \hat{B}_x
-\frac{1}{\mu_0} \mathring{B}_x \hat{B}_y
)$
$\partial_t (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
-\mathring{v}_y \mathring{v}_x \partial_x \hat{\rho}
+\mathring{v}_y \partial_x (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
+\mathring{v}_x \partial_x (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
-\frac{1}{\mu_0} \mathring{B}_y \partial_x \hat{B}_x
-\frac{1}{\mu_0} \mathring{B}_x \partial_x \hat{B}_y
= -i k (
(-\mathring{v}_y \mathring{v}_y + \frac{1}{2} (\gamma-1) \mathring{v}^2) \hat{\rho}
-(\gamma-1) \mathring{v}_x (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
-(\gamma-3) \mathring{v}_y (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
-(\gamma-1) \mathring{v}_z (\mathring{\rho} \hat{v}_z + \hat{\rho} \mathring{v}_z)
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_x \hat{B}_x
-\frac{1}{\mu_0} \gamma \mathring{B}_y \hat{B}_y
-\frac{1}{\mu_0} (\gamma-2) \mathring{B}_z \hat{B}_z
-(\gamma-1) (\frac{1}{2} \hat{\rho} \mathring{v}^2 + \mathring{\rho} \mathring{v}_q \hat{v}_q + \frac{1}{\gamma-1} \hat{P} + \frac{1}{\mu_0} \mathring{B}_q \hat{B}_q)
)$
$\partial_t (\mathring{\rho} \hat{v}_z + \hat{\rho} \mathring{v}_z)
-\mathring{v}_z \mathring{v}_x \partial_x \hat{\rho}
+\mathring{v}_z \partial_x (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
+\mathring{v}_x \partial_x (\mathring{\rho} \hat{v}_z + \hat{\rho} \mathring{v}_z)
-\frac{1}{\mu_0} \mathring{B}_z \partial_x \hat{B}_x
-\frac{1}{\mu_0} \mathring{B}_x \partial_x \hat{B}_z
= -i k (
-\mathring{v}_z \mathring{v}_y \hat{\rho}
+\mathring{v}_z (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
+\mathring{v}_y (\mathring{\rho} \hat{v}_z + \hat{\rho} \mathring{v}_z)
-\frac{1}{\mu_0} \mathring{B}_z \hat{B}_y
-\frac{1}{\mu_0} \mathring{B}_y \hat{B}_z
)$
$\partial_t \hat{B}_x
= -i k (
-\frac{1}{\mathring{\rho}} (\mathring{B}_x \mathring{v}_y - \mathring{B}_y \mathring{v}_x) \hat{\rho}
- \frac{1}{\mathring{\rho}} \mathring{B}_y (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
+ \frac{1}{\mathring{\rho}} \mathring{B}_x (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
+ \mathring{v}_y \hat{B}_x
- \mathring{v}_x \hat{B}_y
)$
$ \partial_t \hat{B}_y
-\frac{1}{\mathring{\rho}} (\mathring{B}_y \mathring{v}_x - \mathring{B}_x \mathring{v}_y) \partial_x \hat{\rho}
+\frac{1}{\mathring{\rho}} \mathring{B}_y \partial_x (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
-\frac{1}{\mathring{\rho}} \mathring{B}_x \partial_x (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
-\mathring{v}_y \partial_x \hat{B}_x
+\mathring{v}_x \partial_x \hat{B}_y
= 0$
$ \partial_t \hat{B}_z
-\frac{1}{\mathring{\rho}} (\mathring{B}_z \mathring{v}_x - \mathring{B}_x \mathring{v}_z) \partial_x \hat{\rho}
+\frac{1}{\mathring{\rho}} \mathring{B}_z \partial_x (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
-\frac{1}{\mathring{\rho}} \mathring{B}_x \partial_x (\mathring{\rho} \hat{v}_z + \hat{\rho} \mathring{v}_z)
-\mathring{v}_z \partial_x \hat{B}_x
+\mathring{v}_x \partial_x \hat{B}_z
= -i k (
-\frac{1}{\mathring{\rho}} (\mathring{B}_z \mathring{v}_y - \mathring{B}_y \mathring{v}_z) \hat{\rho}
+\frac{1}{\mathring{\rho}} \mathring{B}_z (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
-\frac{1}{\mathring{\rho}} \mathring{B}_y (\mathring{\rho} \hat{v}_z + \hat{\rho} \mathring{v}_z)
-\mathring{v}_z \hat{B}_y
+\mathring{v}_y \hat{B}_z
)$
$ \partial_t (\frac{1}{2} \hat{\rho} \mathring{v}^2 + \mathring{\rho} \mathring{v}_q \hat{v}_q + \frac{1}{\gamma-1} \hat{P} + \frac{1}{\mu_0} \mathring{B}_q \hat{B}_q)
+(\mathring{v}_x (\frac{1}{2} (\gamma-1) \mathring{v}^2 - \mathring{h}_{total}) + \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_m \mathring{v}_m) \partial_x \hat{\rho}
+(-(\gamma-1) \mathring{v}_x \mathring{v}_x - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_x + \mathring{h}_{total}) \partial_x (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
+(-(\gamma-1) \mathring{v}_x \mathring{v}_y - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_y ) \partial_x (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
+(-(\gamma-1) \mathring{v}_x \mathring{v}_z - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_x \mathring{B}_z) \partial_x (\mathring{\rho} \hat{v}_z + \hat{\rho} \mathring{v}_z)
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_x \mathring{B}_x + \mathring{v}_x \mathring{B}_x + \mathring{v}_m \mathring{B}_m) \partial_x \hat{B}_x
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_x \mathring{B}_y + \mathring{v}_y \mathring{B}_x) \partial_x \hat{B}_y
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_x \mathring{B}_z + \mathring{v}_z \mathring{B}_x) \partial_x \hat{B}_z
\gamma \mathring{v}_x \partial_x (\frac{1}{2} \hat{\rho} \mathring{v}^2 + \mathring{\rho} \mathring{v}_q \hat{v}_q + \frac{1}{\gamma-1} \hat{P} + \frac{1}{\mu_0} \mathring{B}_q \hat{B}_q)
= -i k (
(\mathring{v}_y (\frac{1}{2} (\gamma-1) \mathring{v}^2 - \mathring{h}_{total}) + \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_m \mathring{v}_m ) \hat{\rho}
+(-(\gamma-1) \mathring{v}_y \mathring{v}_x - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_x) (\mathring{\rho} \hat{v}_x + \hat{\rho} \mathring{v}_x)
+(-(\gamma-1) \mathring{v}_y \mathring{v}_y - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_y + \mathring{h}_{total}) (\mathring{\rho} \hat{v}_y + \hat{\rho} \mathring{v}_y)
+(-(\gamma-1) \mathring{v}_y \mathring{v}_z - \frac{1}{\mu_0\mathring{\rho}} \mathring{B}_y \mathring{B}_z) (\mathring{\rho} \hat{v}_z + \hat{\rho} \mathring{v}_z)
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_y \mathring{B}_x + \mathring{v}_x \mathring{B}_y) \hat{B}_x
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_y \mathring{B}_y + \mathring{v}_y \mathring{B}_y + \mathring{v}_m \mathring{B}_m) \hat{B}_y
-\frac{1}{\mu_0} ((\gamma - 2) \mathring{v}_y \mathring{B}_z + \mathring{v}_z \mathring{B}_y) \hat{B}_z
+\gamma \mathring{v}_y (\frac{1}{2} \hat{\rho} \mathring{v}^2 + \mathring{\rho} \mathring{v}_q \hat{v}_q + \frac{1}{\gamma-1} \hat{P} + \frac{1}{\mu_0} \mathring{B}_q \hat{B}_q)
)$
...and simplifying...
$\partial_t \hat{B}_x
= -i k (
- \mathring{B}_y \hat{v}_x
+ \mathring{B}_x \hat{v}_y
+ \mathring{v}_y \hat{B}_x
- \mathring{v}_x \hat{B}_y
)$
EMHD equations
state vector:
$U_i = \downarrow i \left[\matrix{
\rho \\ \rho v_i \\ E_{total} \\ E_i \\ B_i
}\right]$
$E_{total} = E_{kin} + E_{int} + E_{EM}
= \frac{1}{2} \rho v^2 + \frac{P}{\gamma - 1} + \frac{1}{2} ( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 ) =$ total energy
Maxwell equations:
Lorentz force law:
$\rho \partial_t v^i = q (E^i + {\epsilon^i}_{jk} v^j B^k)$
Flux in $j$th direction:
$F_{ij} = \downarrow i \left[\matrix{
\rho v_j \\
\rho v_i v_j + \delta_{ij} P_{total} \\
H_{total} v_j \\
}\right]$