In the nightside region of Earth’s magnetosphere, buoyancy modes have been associated with low entropy bubbles. These bubbles form in the plasma sheet, particularly during substorm expansion, and move rapidly earthward and come to rest in the inner plasma sheet or inner magnetosphere. They often exhibit damped oscillations with periods of a few minutes and have been associated with Pi2 pulsations. In previous work, we used the thin filament approximation to compare the frequencies and modes of buoyancy waves using three approaches: magnetohydrodynamic (MHD) ballooning theory, classic interchange theory, and an idealized formula. Interchange oscillations differ from the more general MHD oscillations in that they assume a constant pressure on each magnetic field line. It was determined that the buoyancy and interchange modes are very similar for field lines that extend into the plasma sheet but differ for field lines that map to the inner magnetosphere. In this paper, we create a small region of entropy depletion in an otherwise stable entropy background profile of the magnetotail to represent the presence of a plasma bubble and determine the properties of the buoyancy modes using the same 3 approaches. In the bubble region, we find that in some regions the interchange and buoyancy modes overlap resulting in frequencies that are much lower than the background. In other regions within the bubble, we find interchange unstable modes while in other locations MHD normal mode predicts an MHD slow mode wave solution which is not found in the pure interchange solution.

This paper describes magnetospheric waves of very long wavelength in thin magnetic filaments. We consider an average magnetospheric configuration with zero ionospheric conductance and calculate waves using two different formulations: classic interchange theory and ideal MHD. Classic interchange theory, which is developed in detail in this paper, is basically analytic and is relatively straightforward to determine computationally, but it can’t offer very high accuracy.The two formalisms agree well for the plasma sheet and also for the inner magnetosphere. The eigenfrequencies range over about a factor of seven, but the formulations generally agree with a root-mean-square difference of the $log_{10}$ of the ratio of the interchange to MHD frequencies to be $\sim 0.054$. The pressure perturbations in the classic interchange theory are assumed constant along each field line, but the MHD computed pressure perturbations along the field line vary in a range $\sim 30 \%$ in the plasma sheet but are larger in the inner magnetosphere. The parallel and perpendicular displacements, which are very different in the plasma sheet and inner magnetosphere, show good qualitative agreement between the two approaches. In the plasma sheet, the perpendicular displacements are strongly concentrated in the equatorial plane, whereas the parallel displacements are spread through most of the plasma sheet away from the equatorial plane; and can be regarded as buoyancy waves. In the inner magnetosphere, the displacements are more sinusoidal and are more like conventional slow modes. The different forms of the waves are best characterized by the flux tube entropy $PV^\gamma$.

The shear flow-interchange instability is proposed as the initiating mechanism behind substorm onset. ULF waves occurring within minutes of substorm onset are observed in the magnetotail at frequencies similar to those of the auroral beads, which are a result of a near-earth magnetospheric instability initiating current disruption in the plasma sheet. Growth rates were statistically determined as a function of wavenumber by Kalmoni et al. (2015) using ASI data from a set of substorm events. The RCM-E provides growth phase-evolved runs of background fields for stability analysis of a magnetospheric wave equation for shear flow-interchange modes derived in Derr et al. (2019), from which growth rates and dispersion relations can be calculated for comparison with the statistically-determined growth rates and frequencies of the beads. In the plasma sheet, interchange and shear flow represent a competition between Kelvin-Helmholtz instability and overall interchange stability. On average, flux-entropy increases with radial distance. As the growth phase proceeds, the middle plasma sheet becomes nearly interchange stable, but flux-entropy decreases sharply at the inner edge. Destabilizing shear is weak in the middle of the sheet but quite strong in the SAPS region, earthward of the inner edge. We examine the conditions under which shear can overwhelm interchange stability to trigger instability. Instability phenomenology will be discussed in detail, including discussion of Doppler-resonance structure and a dimensionless parameter W* for characterizing stability domains. Mapping spatial properties to the ionosphere along field lines allows for comparison of instability wavelengths with those of the auroral beads. All substorms terminate in relaxation, either because higher order nonlinearities ultimately suppress growth or due to external conditions which alter the background fields to suppress nonlinear growth. If higher order amplitude expansion terms contribute negatively at some order, then nonlinear relaxation occurs, and a method for determining field saturation values is established.