from 2023 Warnecke et al - Numerical evidence for a small-scale dynamo approaching solar magnetic Prandtl numbers

paper says:
$\frac{Du}{Dt} = -c_s^2 \nabla ln \rho + J \times B / \rho + \nabla \cdot (2 \rho \nu S) / \rho + f$
$\frac{\partial A}{\partial t} = u \times B + \eta \nabla^2 A$
$\frac{D \rho}{D t} = - \nabla \cdot (\rho u)$

after simplifying becomes...
$\rho \partial_t u_i + \rho u^j \partial_j u_i = -c_s^2 \partial_i \rho + \epsilon_{ijk} J_j B_k + \partial_j (2 \rho \nu S_{ij}) + \rho f_i$
$\partial_t A_i = \epsilon_{ijk} u_j B_k + \eta \partial^j \partial_j A_i$
$\partial_t \rho + \rho \partial_j u_j + 2 u_j \partial_j \rho = 0$

... this is different from the source 2000 Brandenburg, and different from all other MHD / compressible fluid dynamics literature
. It has an extra term. Maybe if the $D_t$ was replaced with a $\partial_t$ it'd match.
... while the source says ...
2000 Brandenburg "The inverse cascade and nonlinear alpha-effect in simulations of isotropic helical hydromagnetic turbulence":

$\partial_t ln \rho + u^j \partial_j ln \rho = -\partial_j u_j$
$\frac{1}{\rho} \partial_t \rho + \frac{1}{\rho} u^j \partial_j \rho = -\partial_j u_j$
$\partial_t \rho + u^j \partial_j \rho = -\rho \partial_j u_j$
$\partial_t \rho + u^j \partial_j \rho + \rho \partial_j u_j = 0$
$\partial_t \rho + \partial_j (\rho u_j) = 0$
...which is correct compressible Euler equations.
so I'll use this for the 3rd IVP equation going forward.

2023 also says $B = \nabla \times A$ and $J = \nabla \times B / \mu_0$

$\rho \partial_t u_i + \rho u^j \partial_j u_i = -c_s^2 \partial_i \rho + \epsilon_{ijk} J_j B_k + \partial_j (2 \rho \nu S_{ij}) + \rho f_i$
$\partial_t A_i = \epsilon_{ijk} u_j B_k + \eta \partial^j \partial_j A_i$
$\partial_t \rho + \rho \partial_j u_j + 2 u_j \partial_j \rho = 0$

$\rho \partial_t u_i + \rho u^j \partial_j u_i + c_s^2 \partial_i \rho - \frac{1}{\mu_0} \epsilon_{ijk} B_k \epsilon_{jmn} \partial_m B_n - \partial_j (2 \rho \nu S_{ij}) = \rho f_i$
$\partial_t A_i - \epsilon_{ijk} u_j B_k - \eta \partial^j \partial_j A_i = 0$
$\partial_t \rho + \rho \partial_j u_j + u_j \partial_j \rho = 0$

$ \partial_t (\rho u_i) + \partial_j (\rho u_i u_j) + c_s^2 \partial_i \rho + \frac{1}{\mu_0} B_j \partial_i B_j - \frac{1}{\mu_0} B_j \partial_j B_i - \partial_j (2 \rho \nu S_{ij}) = \rho f_i$
$\partial_t A_i - \epsilon_{ijk} u_j B_k - \eta \partial^j \partial_j A_i = 0$
$\partial_t \rho + \partial_j (u_j \rho) = 0$