Derivation of the Diamagnetic Vorticity

In JOREK models that include diamagnetic effects, the total vorticity $W$ is defined as the toroidal component of the curl of the ion velocity $\mathbf{v}_i$:

\[\begin{equation*} W \equiv (\nabla \times \mathbf{v}_i) \cdot \nabla \phi \end{equation*}\]

The ion velocity is given by:

\[\begin{equation*} \mathbf{v}_i = \mathbf{v}_{E \times B} + \mathbf{v}_{dia,i} + v_{||} \mathbf{B} \end{equation*}\]

Substituting the expressions for the $E \times B$ and diamagnetic velocities:

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

The toroidal vorticity $W$ can be split into two main parts:

\[\begin{equation*} W = \omega + \omega^* \end{equation*}\]

where $\omega$ is the standard vorticity related to the $E \times B$ flow:

\[\begin{equation*} \omega = \nabla \cdot \nabla_{pol} u \end{equation*}\]

and $\omega^*$ is the part related to the diamagnetic flow and the parallel velocity term:

\[\begin{equation*} \omega^* = \nabla \cdot \left( \frac{\tau_{IC}}{\rho} \nabla_{pol} p_i - v_{||} \frac{1}{R^2} \nabla_{pol} \psi \right) \end{equation*}\]

The full derivation of these terms, including the cancellation of terms with the gyro-viscous tensor, can be found in the Vorticity Derivation and Reduced MHD Models pages.