Adiabatic behavior of an elliptical vortex in a time-dependent external strain flow

Abstract

The adiabatic behavior of two-dimensional (2D) fluid vortices subject to slowly varying, time-dependent external strain flows is studied theoretically, and experimentally using magnetized non-neutral electron plasmas. Here the E × B drift dynamics of the electrons perpendicular to the magnetic field are analogous to the motion of fluid vorticity under the 2D Euler equations describing ideal fluids. A low-dimensional elliptical vortex patch model is used along with a WKB approximation to derive a formula for the breaking of an adiabatic invariant due to the changing external flow. The invariant is interpreted as the amplitude of a perturbative oscillation about a stable fixed point corresponding to a steady elliptical vortex. Smooth, hyperbolic tangent and piecewise linear ramp functions are considered for the external strain time dependence. Standard exponential breaking is observed in the former case, whereas the latter exhibits a power-law breaking curve with periodic modulation. It is found that a driving term in the equations of motion contributes the majority of the breaking whereas the frequency variation plays a weaker, but significant role. The experimental data agree closely with the theoretical model. The most significant deviation is due to inviscid damping behavior associated with the smooth edges of the experimental vortices, which tends to reduce the amplitude of the oscillation. Finally, the results are compared and contrasted with other related experimental, numerical, and theoretical work.