Force balance equation in tokamak plasmas: Grad-Shafranov equation

Next, we consider what constraints the force balance imposes on the axisymmetric magnetic field discussed above. The momentum equation of plasmas is given by

$$\rho \left( \frac{\partial \mathbf{u}}{\partial t} +\mathbf{u} \c... ...ht) = \rho_q \mathbf{E}- \nabla \cdot \mathbbm{P}+\mathbf{J} \times \mathbf{B},$$ (42)

where $\rho$, $\rho_q$, $\mathbbm{P}$, $\mathbf{J}$, $\mathbf{E}$, and $\mathbf {B}$ are mass density, charge density, thermal pressure tensor, current density, electric field, and magnetic field, respectively. The electric field force $\rho_q \mathbf{E}$ is usually ignored due to either $\rho_q = 0$ or $\mathbf{E}= 0$. Further assuming there is no plasma flow and the plasma pressure is isotropic, the steady state momentum equation (force balance equation) is written

$$\mathbf{J} \times \mathbf{B}= \nabla P,$$ (43)

where $P$ is the scalar plasma pressure.