Conservative Form of MHD:
Geometry variables:
$n^i =$ flux surface normal, units of [1].
$g_{ij} = $ metric tensor, in units of $[1]$.
Hydrodynamics Variables:
$\rho =$ density, in units of $\frac{kg}{m^3}$
$v^i =$ velocity, in units of $\frac{m}{s}$
$v^2 = g_{ij} v^i v^j$, in units of $\frac{m^2}{s^2}$
$p^i = \rho v^i = $ momentum density, in units of $\frac{kg}{m^2 \cdot s}$
$\gamma =$ heat capacity ratio, in units of $[1]$
$\tilde\gamma = \gamma - 1$
$P = \tilde\gamma \rho e_{int} = \tilde\gamma E_{int} = \tilde\gamma (E_{hydro} - E_{kin}) = $ pressure, in units of $\frac{kg}{m \cdot s^2}$
$e_{kin} = \frac{1}{2} v^2 = $ specific kinetic energy, in units of $\frac{m^2}{s^2}$
$E_{kin} = \rho e_{kin} = \frac{1}{2} \rho v^2$ densitized kinetic energy, in units of $\frac{kg}{m \cdot s^2}$
$e_{int} = \frac{P}{\tilde\gamma \rho} =$ specific internal energy, in units of $\frac{m^2}{s^2}$
$E_{int} = \rho e_{int} = \frac{P}{\tilde\gamma} = $ densitized internal energy, in units of $\frac{kg}{m \cdot s^2}$
$e_{hydro} = e_{kin} + e_{int} =$ specific hydro energy, in units of $\frac{kg}{m \cdot s^2}$
$E_{hydro} = E_{kin} + E_{int} = \frac{1}{2} \rho v^2 + \frac{1}{\tilde\gamma} P =$ hydro energy, in units of $\frac{kg}{m \cdot s^2}$
$H_{hydro} = E_{hydro} + P = \frac{1}{2} \rho v^2 + \frac{\gamma}{\tilde\gamma} P$
$ = E_{hydro} + \tilde\gamma E_{hydro} + \tilde\gamma E_{kin} = \gamma E_{hydro} + \tilde\gamma E_{kin}$
= total hydro enthalpy, in units of $\frac{kg}{m \cdot s^2}$
MHD variables:
$\mu_0 = $ permeability of free space, in $\frac{kg \cdot m}{C^2}$
$B^i =$ magnetic field, in units of $\frac{kg}{C \cdot s}$
$B^2 = g_{ij} B^i B^j$, in units of $\frac{kg^2}{C^2 \cdot s^2}$
$P_{mag} = \frac{1}{2 \mu_0} B^2 =$ is pressure due to magnetic field, is uncannily similar to energy due to magnetic field... in units of $\frac{kg}{m \cdot s^2}$
$P_{total} = P + P_{mag} =$ total MHD pressure from fluid and magnetic field, in units of $\frac{kg}{m \cdot s^2}$
$E_{mag} = \frac{1}{2 \mu_0} B^2$ = magnetic field energy, in units of $\frac{kg}{m \cdot s^2}$
$E_{total} = E_{hydro} + E_{mag} = \frac{1}{2} \rho v^2 + \frac{1}{\tilde\gamma} P + \frac{1}{2 \mu_0} B^2 = $, total MHD energy, in units of $\frac{kg}{m \cdot s^2}$
$H_{total} = E_{total} + P_{total} = \frac{1}{2} \rho v^2 + \frac{\gamma}{\tilde\gamma} P + \frac{1}{\mu_0} B^2 = $ total MHD enthalpy in units of $\frac{kg}{m \cdot s^2}$
Let capital indexes $IJK...$ span all primitive/conserved quantity variables $\{ \rho, v^i, B^i, E_{total} \}$
$\partial_t \rho + \nabla_j (\rho v^j) = 0$
$\partial_t (\rho v^i) + \nabla_j (\rho v^i v^j + g^{ij} P_{total} - \frac{1}{\mu_0} B^i B^j) = 0$
$\partial_t B_i + \nabla_j (B^i v^j - B^j v^i) = 0$
$\partial_t E_{total} + \nabla_j (H_{total} v^j - \frac{1}{\mu_0} B^k v^k B^j) = 0$
Conservation Law Form:
$\int_\Omega (\partial_t U^I + \nabla_n F^I) dV = 0$
Primitive variables:
$W^I = \left[\begin{matrix}
\rho \\ v^i \\ B^i \\ P
\end{matrix}\right]$
in units of $\left[\begin{matrix}
\frac{kg}{m^3} \\
\frac{m}{s} \\
\frac{kg}{C \cdot s} \\
\frac{kg}{m \cdot s^2}
\end{matrix}\right]$
Conservative variables:
$U^I = \left[\begin{matrix}
\rho \\ \rho v^i \\ B^i \\ E_{total}
\end{matrix}\right]$
in units of $\left[\begin{matrix}
\frac{kg}{m^3} \\
\frac{kg}{m^2 \cdot s} \\
\frac{kg}{C \cdot s} \\
\frac{kg}{m \cdot s^2}
\end{matrix}\right]$
$\partial_\rho E_{total}$
$= \partial_\rho E_{kin} + \partial_\rho E_{int} + \partial_\rho E_{mag}$
$= \partial_\rho (\frac{1}{2} \rho v^2)
+ \partial_\rho (\frac{1}{\tilde\gamma} P)
+ \partial_\rho (\frac{1}{2 \mu_0} B^2)$
$= \frac{1}{2} v^2$
$\partial_{v^k} E_{total}$
$= \partial_{v^k} E_{kin} + \partial_{v^k} E_{int} + \partial_{v^k} E_{mag}$
$= \partial_{v^k} (\frac{1}{2} \rho g_{ij} v^i v^j)
+ \partial_{v^k} (\frac{1}{\tilde\gamma} P)
+ \partial_{v^k} (\frac{1}{2 \mu_0} B^2)$
$=
\frac{1}{2} \partial_{v^k} \rho g_{ij} v^i v^j
+ \frac{1}{2} \rho \partial_{v^k} g_{ij} v^i v^j
+ \frac{1}{2} \rho g_{ij} \partial_{v^k} v^i v^j
+ \frac{1}{2} \rho g_{ij} v^i \partial_{v^k} v^j
$
$=
\frac{1}{2} \rho g_{ij} \delta^i_k v^j
+ \frac{1}{2} \rho g_{ij} v^i \delta^j_k
$
$=
\frac{1}{2} \rho v_k
+ \frac{1}{2} \rho v_k
$
$= \rho v_k$
$\partial_{B^k} E_{total}$
$= \partial_{B^k} E_{kin} + \partial_{B^k} E_{int} + \partial_{B^k} E_{mag}$
$= \partial_{B^k} (\frac{1}{2} \rho v^2)
+ \partial_{B^k} (\frac{1}{\tilde\gamma} P)
+ \partial_{B^k} (\frac{1}{2 \mu_0} B^2)$
$= \frac{1}{\mu_0} B_k$
$\partial_P E_{total}$
$= \partial_P E_{kin} + \partial_P E_{int} + \partial_P E_{mag}$
$= \partial_P (\frac{1}{2} \rho v^2)
+ \partial_P (\frac{1}{\tilde\gamma} P)
+ \partial_P (\frac{1}{2 \mu_0} B^2)$
$= \frac{1}{\tilde\gamma} $
Partial of conservative with respect to primitive:
$\frac{\partial U^I}{\partial W^J} = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix}
1 & 0 & 0 & 0 \\
v^i & \rho \delta^i_j & 0 & 0 \\
0 & 0 & \delta^i_j & 0 \\
\frac{1}{2} v^2 & \rho v_j & \frac{1}{\mu_0} B_j & \frac{1}{\tilde{\gamma}}
\end{matrix}\right] }$
Partial of primitive with respect to conservative:
$\frac{\partial W^I}{\partial U^J} = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix}
1 & 0 & 0 & 0 \\
-\frac{1}{\rho} v^i & \frac{1}{\rho} \delta^i_j & 0 & 0 \\
0 & 0 & \delta^i_j & 0 \\
\frac{1}{2} \tilde{\gamma} v^2 & -\tilde{\gamma} v_j & -\tilde{\gamma} \frac{1}{\mu_0} B_j & \tilde{\gamma}
\end{matrix}\right] }$
Flux:
$F^I(n) = \left[\begin{matrix}
\rho v^l n_l \\
\rho v^i v^l n_l + n^i P_{total} - \frac{1}{\mu_0} B^i B^l n_l \\
B^i v^l n_l - v^i B^l n_l \\
H_{total} v^l n_l - \frac{1}{\mu_0} B^k v_k B^l n_l
\end{matrix}\right]$
$= \left[\begin{matrix}
\rho v^l n_l \\
\rho v^i v^l n_l + n^i (P + \frac{1}{2 \mu_0} B^2) - \frac{1}{\mu_0} B^i B^l n_l \\
B^i v^l n_l - v^i B^l n_l \\
(\frac{1}{2} \rho v^2 + \frac{\gamma}{\tilde\gamma} P) v^l n_l + \frac{1}{\mu_0} B^2 v^l n_l - \frac{1}{\mu_0} B^k v_k B^l n_l
\end{matrix}\right]$
$= \left[\begin{matrix}
p^l n_l \\
\frac{1}{\rho} (p^i p^l n_l
- \tilde{\gamma} p^2 n^i
)
+ \frac{1}{\mu_0} (
\frac{1}{2} B^2 (2 - \gamma) n^i
- B^i B^l n_l
)
+ \tilde{\gamma} E_{total} n^i
\\
B^i v^l n_l - v^i B^l n_l
\\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} p^2 p^l n_l
+ \frac{1}{\mu_0 \rho} (
(1 - \frac{1}{2} \gamma) B^2 p^l
- B^k p_k B^l
) n_l
+ \gamma \frac{1}{\rho} E_{total} p^l n_l
\end{matrix}\right]$
in units of
$\left[\begin{matrix}
\frac{kg}{m^2 \cdot s} \\
\frac{kg}{m \cdot s^2} \\
\frac{kg}{C \cdot s^2} \\
\frac{kg}{s^3}
\end{matrix}\right]$
Flux derivative with respect to primitive variables:
$\frac{\partial F^I}{\partial W^J} = \downarrow I \overset{\rightarrow K}{ \left[\begin{matrix}
v^l n_l &
\rho n_j &
0 &
0 \\
v^i v^l n_l &
\rho (v^l n_l \delta^i_j + v_i n_j) &
\frac{1}{\mu_0} (B_j n^i + B^i n_j - \delta^i_j B^l n_l) &
n^i \\
0 &
B^i n_j - \delta^i_j B^l n_l &
\delta^i_j v^l n_l - v^i n_j &
0 \\
\frac{1}{2} v^2 v^l n_l &
n_j H_{total} + n_l (\rho v_j v^l - \frac{1}{\mu_0} B_j B^l ) &
\frac{1}{\mu_0} (2 B_j v^l n_l - v_j B^l n_l - B^k v_k n_j) &
\frac{\gamma}{\tilde{\gamma}} v^l n_l
\end{matrix}\right] }$
TODO here on down, I need to rewrite flux in terms of a normal vector.
Follow CFD/Euler Fluid Equations - Curved Geometry - Contravariant.html.
Derivative of flux vector with respect to conservative variables:
$\frac{\partial F^I}{\partial U^J}
= \frac{\partial F^I}{\partial W^K} \frac{\partial W^K}{\partial U^J}$
$= \downarrow I \overset{\rightarrow L}{ \left[\begin{matrix}
v_j &
\rho g^{jl} &
0 &
0 \\
v_i v_j &
\rho (g^{il} v_j + v_i g^{jl}) &
\frac{1}{\mu_0} (g^{ij} B_l - g^{il} B_j - g^{jl} B_i) &
g^{ij} \\
0 &
B_i g^{jl} - B_j g^{il} &
g^{il} v_j - g^{jl} v_i &
0 \\
\frac{1}{2} v^2 v_j &
\rho v_j v_l + (E_{hydro} + \frac{1}{\mu_0} B^2 + P) g^{jl} - \frac{1}{\mu_0} B_j B_l &
\frac{1}{\mu_0} (2 B_l v_j - v_l B_j - g^{jl} v_m B_m) &
\frac{\gamma}{\tilde{\gamma}} v_j
\end{matrix}\right] } \cdot \downarrow L \overset{\rightarrow K}{ \left[\begin{matrix}
1 & 0 & 0 & 0 \\
-\frac{1}{\rho} v_l & \frac{1}{\rho} g^{kl} & 0 & 0 \\
0 & 0 & g^{kl} & 0 \\
\frac{1}{2} \tilde{\gamma} v^2 & -\tilde{\gamma} v_k & -\tilde{\gamma} \frac{1}{\mu_0} B_k & \tilde{\gamma}
\end{matrix}\right] }$
$= \downarrow I \overset{\rightarrow K}{ \left[\begin{matrix}
0 &
g^{jk} &
0 &
0 \\
-v_i v_j + \frac{1}{2} g^{ij} \tilde{\gamma} v^2 &
g^{ik} v_j + g^{jk} v_i - g^{ij} \tilde{\gamma} v_k &
\frac{1}{\mu_0}(-g^{ij} (\gamma-2) B_k - g^{ik} B_j - g^{jk} B_i) &
g^{ij} \tilde{\gamma} \\
-\frac{1}{\rho} (B_i v_j - B_j v_i) &
\frac{1}{\rho} (B_i g^{jk} - B_j g^{ik}) &
g^{ik} v_j - g^{jk} v_i &
0 \\
v_j (\frac{1}{2} \tilde{\gamma} v^2 - h_{total}) + \frac{1}{\mu_0\rho} B_j B_m v_m &
-\tilde{\gamma} v_j v_k + (g^{jk} h_{total} - \frac{1}{\mu_0\rho} B_j B_k) &
\frac{1}{\mu_0} (-(\gamma - 2) v_j B_k - v_k B_j - g^{jk} v_m B_m) &
\gamma v_j
\end{matrix}\right] }$
for $h_{total} = (E_{total} + P_{total}) / \rho$
Curved Space Partial of Conserved Quantities wrt Primitives
$U^I = \left[\begin{matrix}
\rho \\
m^i \\
B^i \\
E_{total}
\end{matrix}\right] = \left[\begin{matrix}
\rho \\
\rho v^i \\
B^i \\
\tilde\gamma^{-1} P + \frac{1}{2} \rho v^2 + \frac{1}{2} \frac{1}{\mu_0} B^2
\end{matrix}\right]$
$E_{total} = E_{int} + E_{kin} + E_{mag}$
$= \tilde\gamma^{-1} P + \frac{1}{2} \rho v^2 + \frac{1}{2} \frac{1}{\mu_0} B^2$
$P = \tilde\gamma (E_{total} - \frac{1}{2} \rho v^2 - \frac{1}{2} \frac{1}{\mu_0} B^2)$
$P_{total} = P + P_{mag} = \tilde\gamma (E_{total} - \frac{1}{2} \rho v^2 - \frac{1}{2} \frac{1}{\mu_0} B^2) + \frac{1}{2} \frac{1}{\mu_0} B^2$
$= \tilde\gamma E_{total}
- \frac{1}{2} \tilde\gamma \rho v^2
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2
$
$H_{total} = E_{total} + P_{total}$
$= \gamma E_{total}
- \frac{1}{2} \tilde\gamma \rho v^2
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2
$
$F^{Ij}$
$= \left[\begin{matrix}
\rho v^j \\
\rho v^i v^j + g^{ij} P_{total} - \frac{1}{\mu_0} B^i B^j \\
B^i v^j - B^j v^i \\
H_{total} v^j - \frac{1}{\mu_0} B^k v_k B^j
\end{matrix}\right]$
$= \left[\begin{matrix}
m^j \\
\frac{1}{\rho} m^i m^j + g^{ij} (\tilde\gamma (E_{total} - \frac{1}{2} \frac{1}{\rho} m^2 - \frac{1}{2} \frac{1}{\mu_0} B^2) + \frac{1}{2} \frac{1}{\mu_0} B^2) - \frac{1}{\mu_0} B^i B^j \\
\frac{1}{\rho} (B^i m^j - B^j m^i) \\
\frac{1}{\rho} (E_{total} + \tilde\gamma (E_{total} - \frac{1}{2} \frac{1}{\rho} m^2 - \frac{1}{2} \frac{1}{\mu_0} B^2) + \frac{1}{2} \frac{1}{\mu_0} B^2) m^j - \frac{1}{\mu_0} \frac{1}{\rho} B^k m_k B^j
\end{matrix}\right]$
$= \left[\begin{matrix}
m^j
\\
+ \frac{1}{\rho} m^i m^j
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{ij}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{ij}
- \frac{1}{\mu_0} B^i B^j
+ \tilde\gamma g^{ij} E_{total}
\\
+ \frac{1}{\rho} B^i m^j
- \frac{1}{\rho} B^j m^i
\\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^2
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^2
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l g_{kl} B^j
+ \frac{1}{\rho} \gamma m^j E_{total}
\end{matrix}\right]$
Flux partial derivative:
${F^{Ij}}_{,j}$
$= \left[\begin{matrix}
m^j
\\
+ \frac{1}{\rho} m^i m^j
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{ij}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{ij}
- \frac{1}{\mu_0} B^i B^j
+ \tilde\gamma g^{ij} E_{total}
\\
+ \frac{1}{\rho} B^i m^j
- \frac{1}{\rho} B^j m^i
\\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^2
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^2
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l g_{kl} B^j
+ \frac{1}{\rho} \gamma m^j E_{total}
\end{matrix}\right]_{,j}$
$= \left[\begin{matrix}
{m^j}_{,j}
\\
- \frac{1}{\rho^2} \rho_{,j} m^i m^j
+ \frac{1}{\rho} {m^i}_{,j} m^j
+ \frac{1}{\rho} m^i {m^j}_{,j}
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} \rho_{,j} m^2 g^{ij}
- \tilde\gamma \frac{1}{\rho} {m^k}_{,j} m_k g^{ij}
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^k m^l g_{kl,j} g^{ij}
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 {g^{ij}}_{,j}
+ (2 - \gamma) \frac{1}{\mu_0} {B^k}_{,j} B_k g^{ij}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^k B^l g_{kl,j} g^{ij}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 {g^{ij}}_{,j}
- \frac{1}{\mu_0} {B^i}_{,j} B^j
- \frac{1}{\mu_0} B^i {B^j}_{,j}
+ \tilde\gamma {g^{ij}}_{,j} E_{total}
+ \tilde\gamma g^{ij} E_{total,j}
\\
- \frac{1}{\rho^2} \rho_{,j} B^i m^j
+ \frac{1}{\rho} {B^i}_{,j} m^j
+ \frac{1}{\rho} B^i {m^j}_{,j}
+ \frac{1}{\rho^2} \rho_{,j} B^j m^i
- \frac{1}{\rho} {B^j}_{,j} m^i
- \frac{1}{\rho} B^j {m^i}_{,j}
\\
+ \tilde\gamma \frac{1}{\rho^3} \rho_{,j} m^j m^2
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} {m^j}_{,j} m^2
- \tilde\gamma \frac{1}{\rho^2} m^j {m^k}_{,j} m_k
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l g_{kl,j}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho^2} \rho_{,j} m^j B^2
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} {m^j}_{,j} B^2
+ (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j {B^k}_{,j} B_k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l g_{kl,j}
+ \frac{1}{\mu_0} \frac{1}{\rho^2} \rho_{,j} B^k m^l g_{kl} B^j
- \frac{1}{\mu_0} \frac{1}{\rho} {B^k}_{,j} m^l g_{kl} B^j
- \frac{1}{\mu_0} \frac{1}{\rho} B^k {m^l}_{,j} g_{kl} B^j
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l g_{kl,j} B^j
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l g_{kl} {B^j}_{,j}
- \gamma \frac{1}{\rho^2} \rho_{,j} m^j E_{total}
+ \gamma \frac{1}{\rho} {m^j}_{,j} E_{total}
+ \gamma \frac{1}{\rho} m^j E_{total,j}
\end{matrix}\right]$
$= \left[\begin{matrix}
0 &
\delta^j_k &
0 &
0 \\
\frac{1}{\rho^2} (
- m^i m^j
+ \frac{1}{2} \tilde\gamma m^2 g^{ij}
) &
\frac{1}{\rho} (
m^j \delta^i_k
+ m^i \delta^j_k
- \tilde\gamma m_k g^{ij}
) &
\frac{1}{\mu_0} (
(2 - \gamma) B_k g^{ij}
- B^j \delta^i_k
- B^i \delta^j_k
) &
\tilde\gamma g^{ij}
\\
\frac{1}{\rho^2} (
- B^i m^j
+ B^j m^i
) &
\frac{1}{\rho} (
B^i \delta^j_k
- B^j \delta^i_k
) &
\frac{1}{\rho} (
m^j \delta^i_k
- m^i \delta^j_k
) &
0
\\
\frac{1}{\rho^2} (
\tilde\gamma \frac{1}{\rho} m^j m^2
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} m^j B^2
+ \frac{1}{\mu_0} B^k m_k B^j
- \gamma m^j E_{total}
) &
\frac{1}{\rho} (
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 \delta^j_k
- \tilde\gamma \frac{1}{\rho} m^j m_k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 \delta^j_k
- \frac{1}{\mu_0} B_k B^j
+ \gamma E_{total} \delta^j_k
) &
\frac{1}{\mu_0} \frac{1}{\rho} (
(2 - \gamma) m^j B_k
- m_k B^j
- B^l m_l \delta^j_k
) &
\gamma \frac{1}{\rho} m^j
\end{matrix}\right] \left[\begin{matrix}
\rho \\
m^k \\
B^k \\
E_{total}
\end{matrix}\right]_{,j} + g_{kl,j}\left[\begin{matrix}
0 \\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^k m^l g^{ij}
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{ik} g^{lj}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^k B^l g^{ij}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{ik} g^{lj}
- \tilde\gamma E_{total} g^{ik} g^{lj}
\\
0 \\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l B^j
\end{matrix}\right]$
Flux covariant derivative:
${F^{Ij}}_{;j}$
$= \left[\begin{matrix}
0 &
\delta^j_k &
0 &
0 \\
\frac{1}{\rho^2} (
- m^i m^j
+ \frac{1}{2} \tilde\gamma m^2 g^{ij}
) &
\frac{1}{\rho} (
m^j \delta^i_k
+ m^i \delta^j_k
- \tilde\gamma m_k g^{ij}
) &
\frac{1}{\mu_0} (
(2 - \gamma) B_k g^{ij}
- B^j \delta^i_k
- B^i \delta^j_k
) &
\tilde\gamma g^{ij}
\\
\frac{1}{\rho^2} (
- B^i m^j
+ B^j m^i
) &
\frac{1}{\rho} (
B^i \delta^j_k
- B^j \delta^i_k
) &
\frac{1}{\rho} (
m^j \delta^i_k
- m^i \delta^j_k
) &
0
\\
\frac{1}{\rho^2} (
\tilde\gamma \frac{1}{\rho} m^j m^2
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} m^j B^2
+ \frac{1}{\mu_0} B^k m_k B^j
- \gamma m^j E_{total}
) &
\frac{1}{\rho} (
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 \delta^j_k
- \tilde\gamma \frac{1}{\rho} m^j m_k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 \delta^j_k
- \frac{1}{\mu_0} B_k B^j
+ \gamma E_{total} \delta^j_k
) &
\frac{1}{\mu_0} \frac{1}{\rho} (
(2 - \gamma) m^j B_k
- m_k B^j
- B^m m_m \delta^j_k
) &
\gamma \frac{1}{\rho} m^j
\end{matrix}\right] \left[\begin{matrix}
\rho_{,j} \\
{m^k}_{,j} + {\Gamma^k}_{lj} m^l \\
{B^k}_{,j} + {\Gamma^k}_{lj} B^l \\
E_{total,j}
\end{matrix}\right]$
difference:
${F^{Ij}}_{;j}$
$= {F^{Ij}}_{,j}
+ \left[\begin{matrix}
0 &
\delta^j_k &
0 &
0 \\
\frac{1}{\rho^2} (
- m^i m^j
+ \frac{1}{2} \tilde\gamma m^2 g^{ij}
) &
\frac{1}{\rho} (
m^j \delta^i_k
+ m^i \delta^j_k
- \tilde\gamma m_k g^{ij}
) &
\frac{1}{\mu_0} (
(2 - \gamma) B_k g^{ij}
- B^j \delta^i_k
- B^i \delta^j_k
) &
\tilde\gamma g^{ij}
\\
\frac{1}{\rho^2} (
- B^i m^j
+ B^j m^i
) &
\frac{1}{\rho} (
B^i \delta^j_k
- B^j \delta^i_k
) &
\frac{1}{\rho} (
m^j \delta^i_k
- m^i \delta^j_k
) &
0
\\
\frac{1}{\rho^2} (
\tilde\gamma \frac{1}{\rho} m^j m^2
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} m^j B^2
+ \frac{1}{\mu_0} B^k m_k B^j
- \gamma m^j E_{total}
) &
\frac{1}{\rho} (
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 \delta^j_k
- \tilde\gamma \frac{1}{\rho} m^j m_k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 \delta^j_k
- \frac{1}{\mu_0} B_k B^j
+ \gamma E_{total} \delta^j_k
) &
\frac{1}{\mu_0} \frac{1}{\rho} (
(2 - \gamma) m^j B_k
- m_k B^j
- B^m m_m \delta^j_k
) &
\gamma \frac{1}{\rho} m^j
\end{matrix}\right] \left[\begin{matrix}
0 \\
{\Gamma^k}_{lj} m^l \\
{\Gamma^k}_{lj} B^l \\
0
\end{matrix}\right]
- g_{kl,j}\left[\begin{matrix}
0 \\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^k m^l g^{ij}
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{ik} g^{lj}
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^k B^l g^{ij}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{ik} g^{lj}
- \tilde\gamma E_{total} g^{ik} g^{lj}
\\
0 \\
- \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l
- \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l B^j
\end{matrix}\right]
$
$= {F^{Ij}}_{,j}
+ \left[\begin{matrix}
\delta^j_k {\Gamma^k}_{lj} m^l \\
\frac{1}{\rho} (
m^j \delta^i_k
+ m^i \delta^j_k
- \tilde\gamma m_k g^{ij}
) {\Gamma^k}_{lj} m^l
+ \frac{1}{\mu_0} (
(2 - \gamma) B_k g^{ij}
- B^j \delta^i_k
- B^i \delta^j_k
) {\Gamma^k}_{lj} B^l
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^k m^l g^{ij} (\Gamma_{klj} + \Gamma_{lkj})
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{ik} g^{lj} (\Gamma_{klj} + \Gamma_{lkj})
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^k B^l g^{ij} (\Gamma_{klj} + \Gamma_{lkj})
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{ik} g^{lj} (\Gamma_{klj} + \Gamma_{lkj})
+ \tilde\gamma E_{total} g^{ik} g^{lj} (\Gamma_{klj} + \Gamma_{lkj})
\\
\frac{1}{\rho} (
B^i \delta^j_k
- B^j \delta^i_k
) {\Gamma^k}_{lj} m^l
+ \frac{1}{\rho} (
m^j \delta^i_k
- m^i \delta^j_k
) {\Gamma^k}_{lj} B^l
\\
\frac{1}{\rho} (
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 \delta^j_k
- \tilde\gamma \frac{1}{\rho} m^j m_k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 \delta^j_k
- \frac{1}{\mu_0} B_k B^j
+ \gamma E_{total} \delta^j_k
) {\Gamma^k}_{lj} m^l
+ \frac{1}{\mu_0} \frac{1}{\rho} (
(2 - \gamma) m^j B_k
- m_k B^j
- B^m m_m \delta^j_k
) {\Gamma^k}_{lj} B^l
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l (\Gamma_{klj} + \Gamma_{lkj})
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l (\Gamma_{klj} + \Gamma_{lkj})
+ \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l B^j (\Gamma_{klj} + \Gamma_{lkj})
\end{matrix}\right]
$
$= {F^{Ij}}_{,j}
+ {\Gamma^k}_{jk} F^{Ij}
+ \left[\begin{matrix}
0 \\
\frac{1}{\rho} (
{\Gamma^i}_{jk} m^j m^k
- \tilde\gamma {\Gamma_{jk}}^i m^j m^k
)
+ \frac{1}{\mu_0} (
(2 - \gamma) {\Gamma_{jk}}^i B^j B^k
- {\Gamma^i}_{jk} B^j B^k
)
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} {\Gamma_{jk}}^i m^j m^k
- \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^2 g^{jk} {\Gamma^i}_{jk}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} {\Gamma_{jk}}^i B^j B^k
+ \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^2 g^{jk} {\Gamma^i}_{jk}
+ \tilde\gamma E_{total} {\Gamma^i}_{jk} g^{jk}
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho} m^j m^k {\Gamma_{jk}}^i
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} B^j B^k {\Gamma_{jk}}^i
\\
0 \\
\frac{1}{\rho} (
- \tilde\gamma \frac{1}{\rho} m^j m_k {\Gamma^k}_{lj} m^l
- \frac{1}{\mu_0} B_k B^j {\Gamma^k}_{lj} m^l
)
+ \frac{1}{\mu_0} \frac{1}{\rho} (
(2 - \gamma) m^j B_k {\Gamma^k}_{lj} B^l
- m_k B^j {\Gamma^k}_{lj} B^l
)
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l \Gamma_{klj}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l \Gamma_{klj}
+ \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l B^j \Gamma_{klj}
+ \frac{1}{2} \tilde\gamma \frac{1}{\rho^2} m^j m^k m^l \Gamma_{lkj}
- \frac{1}{2} (2 - \gamma) \frac{1}{\mu_0} \frac{1}{\rho} m^j B^k B^l \Gamma_{lkj}
+ \frac{1}{\mu_0} \frac{1}{\rho} B^k m^l B^j \Gamma_{lkj}
\end{matrix}\right]$
$= \frac{1}{\sqrt{g}} ( \sqrt{g} F^{Ij} )_{,j}
+ \left[\begin{matrix}
0 \\
{\Gamma^i}_{jk} (
\frac{1}{\rho} m^j m^k
+ P_{total} g^{jk}
- \frac{1}{\mu_0} B^j B^k
)
\\
0 \\
0
\end{matrix}\right]$