Derivation of the Continuity Equation

The continuity equation in JOREK is given by:

\[\begin{equation} \partial_t\rho=-\nabla\cdot(\rho\mathbf{v})+\nabla\cdot(D\nabla\rho) + S_\rho \end{equation}\]

The expression for the velocity $\mathbf{v}$ in the reduced MHD model is:

\[\begin{equation*} \mathbf{v} = -R^2\nabla u \times \nabla\phi + v_{||}\mathbf{B} \end{equation*}\]

Substituting this into the first term of the right hand side of the continuity equation, we get:

\[\begin{equation*}\begin{split} -\nabla\cdot(\rho\mathbf{v}) &= -\nabla\cdot\left[\rho\left(-R^2\nabla u \times \nabla\phi + v_{||}\mathbf{B}\right)\right] \\ &= \nabla\cdot\left[\rho R^2\nabla u \times \nabla\phi\right] - \nabla\cdot\left(\rho v_{||}\mathbf{B}\right) \end{split}\end{equation*}\]

Using the vector identity $\nabla\cdot(\mathbf{A}\times\mathbf{C}) = \mathbf{C}\cdot(\nabla\times\mathbf{A}) - \mathbf{A}\cdot(\nabla\times\mathbf{C})$ and the fact that $\nabla\times\nabla\phi = 0$, the first term becomes:

\[\begin{equation*}\begin{split} \nabla\cdot\left(\rho R^2\nabla u \times \nabla\phi\right) &= \nabla\phi\cdot\left[\nabla\times\left(\rho R^2\nabla u\right)\right] \\ &= \nabla\phi\cdot\left[\nabla(\rho R^2)\times\nabla u + \rho R^2\underbrace{\nabla\times\nabla u}_{=0}\right] \\ &= \nabla\phi\cdot\left[\nabla(\rho R^2)\times\nabla u\right] \\ &= \frac{1}{R}[\rho R^2, u] \end{split}\end{equation*}\]

For the second term, using $\nabla\cdot\mathbf{B} = 0$, we have:

\[\begin{equation*}\begin{split} -\nabla\cdot\left(\rho v_{||}\mathbf{B}\right) &= -\mathbf{B}\cdot\nabla(\rho v_{||}) - \rho v_{||}\underbrace{\nabla\cdot\mathbf{B}}_{=0} \\ &= -\mathbf{B}\cdot\nabla(\rho v_{||}) \\ &= -\frac{F}{R^2}\partial_3(\rho v_{||}) - \frac{1}{R}[\rho v_{||}, \psi] \end{split}\end{equation*}\]

Combining these, the continuity equation in reduced MHD variables is:

\[\begin{equation} \partial_t\rho = \frac{1}{R}[\rho R^2, u] - \frac{1}{R}[\rho v_{||}, \psi] - \frac{F}{R^2}\partial_3(\rho v_{||}) + \nabla\cdot(D\nabla\rho) + S_\rho \end{equation}\]