Criterion for interchange instability in a plasma connected to a conducting ionosphere

We have combined two empirical numerical models to estimate the entropy parameter PV5/3 in the plasma sheet, where P is the plasma sheet pressure and V = integral ds/B is the volume of a flux tube containing one unit of magnetic flux. The Tsyganenko and Stern (1996) magnetic field model is used to calculate the flux tube volume, and the Tsyganenko and Mukai (2003) plasma sheet statistical model is used to calculate the plasma sheet pressure. Contour plots for PV5/3 and V in the equatorial plane are presented for various solar wind conditions. These empirical models suggest that, although both PV5/3 and V generally increase tailward, their gradients are generally not parallel or antiparallel to each other, whereas most theoretical discussions of interchange instability assume that the vectors are lined up. The Vasyliunas equation implies that the Birkeland current is proportional to the cross product of del PV5/3 and del V, so that region 1 and region 2 Birkeland current flow between plasma sheet and ionosphere confirms that the gradients of entropy parameter and flux tube volume are not generally parallel. We present analytical calculations to investigate the criterion for interchange instability in a quasi-static, low-beta plasma that is connected to a conducting ionosphere and has del PV5/3 and del V at an arbitrary angle alpha, which implies shear flows. A boundary layer is assumed to separate two regions with different but uniform PV5/3. The main conclusion is that the system is interchange unstable if alpha > pi/2. The results suggest that the statistical average plasma sheet configuration is interchange stable since the statistical plasma sheet models indicate that alpha < pi/2. Our conclusion is quite different from analyses that neglect shear flow since they imply instability unless del PV5/3 and del V are exactly parallel.