Multiscale Finite-Beta Gyrokinetics

Finite-$\beta$ effects on microinstabilities have been investigated by gyrokinetic particles codes using the new double split scheme [Startsev, Lee and Wang, manuscript in preparation]. The scheme requires that $ F=(1 + \psi) F_0+\int dx_{||} {\kappa}_e\cdot (\nabla A_{||}\times {\bf b}_0)+\delta g,\label{eq1} $ so that a new full density gradient, which is set up by the fast electrons transverse to the direction of the full field, is consistent with the condition of $ {\bf b} \cdot \nabla (F_0+\int dx_{||}{\kappa}_e\cdot (\nabla A_{||}\times {\bf b}_0))=0$, where ${\bf b}={\bf b}_0+\delta {\bf B}/B_0$. Here $\phi$ is normalized by $T_e/e$, $A_\parallel$ by $cT_e/e c_s$, $F$ is the total distribution function, $F_0$ is the background distribution function, $\psi = \phi + \int (\partial A_\parallel / \partial t) d x_\parallel/c$, and $\phi$ and $A_\parallel$ are the perturbed potentials. The finite-$\beta$ effects on microinstabilities are found to be related to the multiscale equation of the form, $ ({\rho_s / \lambda_D})^2 \left [\nabla^2 \psi - {\psi / \delta_e^2} \right ]= -4 \pi q {v} \langle \delta({\bf r}) \rangle_\varphi $ where $\varphi$ is the gyro angle based on the ion gyroradius, $\rho_i$, and $\delta_e$ is the electron skin depth, which can be an order of magnitude smaller in the tokamak core. To the lowest order, this equation gives rise to the shielding effect as $ \psi = (q / r) (\rho_s / \lambda_D) exp({-r/\delta_e}). $ Thus, we have the presence of two distinct spatial scales in the problem. A numerical scheme based on the concept of singular perturbation methods is used and the resulting finite-$\beta$ effects on drift instabilities, ion temperature gradient drift modes and electron temperature gradient drift modes will be reported.