How to include diamagnetic drift in the viscosity term
- When is this useful? When using diamagnetic terms with low resistivity.
- Implemented for: Models 303, 307, 333, 500, and 600.
- How is it activated? Set
Wdia = .true.in the namelist input file. The contribution also requires nonzerotauICandviscovalues.
Viscous diamagnetic term derivation
In the current equations, we have the viscous term as
\[-\int \mu \nabla v \cdot \nabla W \, dV,\]so that the actual physics term, before integration by parts, is
\[+\int v \, \nabla \cdot \left(\mu \nabla W\right)\, dV.\]That is, the viscosity is inside the divergence.
For the viscous diamagnetic term, in weak form, we start with
\[\int v \, \nabla \cdot \left(\mu \nabla W_{\mathrm{dia}}\right)\, dV.\]Integration by parts gives
\[-\int \mu \nabla v \cdot \nabla W_{\mathrm{dia}}\, dV + \int \nabla \cdot \left( \mu v \nabla W_{\mathrm{dia}} \right)\, dV.\]Assuming the boundary term vanishes,
\[-\int \mu \nabla v \cdot \nabla W_{\mathrm{dia}}\, dV.\]A second integration by parts gives
\[+\int \left[ \nabla \cdot \left(\mu \nabla v\right) \right] W_{\mathrm{dia}}\, dV - \int \nabla \cdot \left( \mu \nabla v \, W_{\mathrm{dia}} \right)\, dV.\]Again assuming the boundary term vanishes, the final form is
\[+\int W_{\mathrm{dia}} \left[ \nabla \cdot \left(\mu \nabla v\right) \right] \, dV.\]Expanding the divergence,
\[+\int W_{\mathrm{dia}} \nabla\mu \cdot \nabla v \, dV + \int \mu W_{\mathrm{dia}} \nabla^2 v \, dV.\]If $\mu$ depends on temperature,
\[\nabla \mu = \frac{\partial \mu}{\partial T}\nabla T,\]and therefore
\[+\int W_{\mathrm{dia}} \frac{\partial\mu}{\partial T} \nabla T \cdot \nabla v \, dV + \int \mu W_{\mathrm{dia}} \nabla^2 v \, dV.\]JOREK implementation
In JOREK, with
\[dV = R\,\mathrm{xjac},\]the corresponding terms are
\[+ R \frac{\partial\mu}{\partial T} W_{\mathrm{dia}} \left( v_x T_x + v_y T_y \right) \mathrm{xjac},\]and
\[+ R \mu W_{\mathrm{dia}} \left( v_{xx} + \frac{v_x}{R} + v_{yy} \right) \mathrm{xjac}.\]Here,
\[W_{\mathrm{dia}} = \frac{\tau}{\rho} \left( p_{xx} + \frac{p_x}{R} + p_{yy} \right) - \frac{\tau}{\rho^2} \left( \rho_x p_x + \rho_y p_y \right).\]