A generalized Hasegawa–Mima equation in curved magnetic fields

Abstract

We derive a model equation describing electrostatic plasma turbulence in general (inhomogeneous and curved) magnetic fields by analysing the effect of curved geometry on the ion fluid polarization drift velocity. The derived nonlinear equation generalizes the Hasegawa–Mima equation governing drift wave turbulence in a straight homogeneous magnetic field, and may serve as a toy model for the description of turbulent plasmas. The equation is most appropriate for configurations with a small $\boldsymbol {E}\times \boldsymbol {B}$ drift velocity divergence, or a mild spatial change in $\boldsymbol {E}\times \boldsymbol {B}$ drift velocity. We identify the conserved energy of the system, and obtain conditions on magnetic field topology for conservation of generalized enstrophy. Through numerical examples, we further show how the curvature of the magnetic field reshapes self-organized steady turbulent states.

Turbulence-driven vortex-flow around a magnetic island

“…Note that here and in equation ( 2), the standard approximation that ψ h only affects B through the parallel gradient is used. For the role of equilibrium density and magnetic-field inhomogeneity in the standard Hasegawa-Mima model, see [19].…”

An Extended Hasegawa-Mima Equation For Nonlinear Drift Wave Turbulence in General Magnetic Configurations

“…( 1) can be obtained by modeling an ion-electron plasma as a two-fluid system. The derivation follows the same steps of section 2 in [23] up to equation (2.26) therein. These steps are therefore omitted here, and can be summarized as the expansion to second order of the ion momentum and…”

“…Nevertheless, it is natural to ask whether the HM equation can be generalized to allow strong magnetic field and density inhomogeneities while maintaining a single governing equation for the electric potential 𝜑. In [23] this question has been answered positively, and a generalized Hasegawa-Mima (GHM) equation has been obtained from a two-fluid model [24] of an ion-electron plasma in the form below:…”

“…A two-fluid plasma model represents a physical system where both the ion and electron components are governed by fluid equations. The ordering used in [23] to derive the GHM equation ( 4) only involves one ordering condition on the spatial derivatives of the magnetic field and the electron spatial density, effectively extending the range of the HM equation to general magnetic field geometries. In Section 2 we will see that this remaining ordering condition on spatial derivatives of magnetic field and spatial density used to obtain the GHM Eq.…”

“…In Section 2 we will see that this remaining ordering condition on spatial derivatives of magnetic field and spatial density used to obtain the GHM Eq. ( 4) in [23] can be removed at the price of replacing the electrostatic potential 𝜑 with the reduced charged particle energy 𝜒 = 𝜑 + 𝜎 2 𝒗 2 𝑬 as dynamical variable. The resulting governing equation is the HMGM Eq.…”