To improve theoretical understanding of the braking oscillations observed in Earth's inner plasma sheet, we have derived a theoretical model that describes k ∥ = 0 magnetohydrodynamic waves in an idealized magnetospheric configuration that consists of a 2-D wedge with circular-arc field lines. The low-frequency, short-perpendicular-wavelength mode obeys a differential equation that is often used to describe buoyancy oscillations in a neutral atmosphere, so we call those waves "buoyancy waves," though the magnetospheric buoyancy force results from magnetic tension rather than gravity. Propagation of the wave is governed mainly by a position-dependent frequency ω b , the "buoyancy frequency," which is a fundamental property of the magnetosphere. The waves propagate if ω b > ω but otherwise evanesce. In the wedge magnetosphere, ω b turns out to be exactly the fundamental oscillation frequency for poloidal oscillations of a thin magnetic filament, and we assume that the same is true for the real magnetosphere. Observable properties of buoyancy oscillations are discussed, but propagation characteristics vary considerably with the state of the magnetosphere. For a given event, the buoyancy frequency and propagation characteristics can be determined from pressure and density profiles and a magnetic field model, and these characteristics have been worked out for one typical configuration. A localized disturbance that initially resembles a dipolarizing flux bundle spreads east-west and also penetrates into the plasmasphere to some extent. The calculated amplitude near the center of the original wave packet decays in a few oscillation periods, even though our calculation includes no dissipation.Plain Language Summary Plasma in the near-Earth region of space exhibits many kinds of ultralow-frequency waves. The present paper deals with a specific class of space plasma ultralow-frequency waves that have unique properties and have not been much studied. They can be called "buoyancy waves," because they are mathematically equivalent to buoyancy waves in a neutral atmosphere, but the buoyancy force in near-Earth space plasmas is due to magnetic tension rather than gravity. This paper develops a theory of near-Earth space buoyancy waves, by considering a simplified geometry for which the wave equations can be solved analytically. The frequency and propagation characteristics of the waves are determined mainly by a parameter called the "buoyancy frequency," which can be calculated from a mathematical model of near-Earth space. Transient bursts of very rapid flow in Earth's magnetotail cause buoyancy waves in near-Earth space. When one of these bursts encounters the strong magnetic field near the Earth, the flow brakes and oscillates a few times before coming to rest. That process generates buoyancy waves that spread out through a region of space, like a raindrop generates ripples in a pond. However, the magnetosphere is highly nonuniform, causing refraction and reflection.

We derive a coupled pair of differential equations that describe linear oscillations of a thin magnetic filament that slides without friction through a stationary medium that is in equilibrium. Background field lines are assumed to be in the xz plane, and filament motion is confined to that plane. Sample eigenfunctions and eigenfrequencies are computed for a numerical equilibrium that approximately represents the average magnetosphere but departs from exact equilibrium due to finite grid spacing and other numerical inaccuracy. The most important result of the calculation is the value of the eigenfrequency of the lowest even mode for filaments that cross the equatorial plane at y = 0 and −18 < x < −2 R E . The characteristics of the lowest even mode depend on geocentric distances of the field line. For field lines that extend out in the plasma sheet, that mode is characterized as a buoyancy wave but, for the inner magnetosphere, it is best characterized as a long-wavelength slow mode.Plain Language Summary Bursty bulk flows often move sunward through the Earth's plasma sheet, coming to rest in the inner plasma sheet. Before coming to rest, they often exhibit damped oscillations with periods of a few minutes. These oscillations are very similar to oscillations of Earth's neutral atmosphere; if a small parcel of air is displaced downward a small distance, gravity causes the parcel to bob up and down, executing what atmospheric scientists call "buoyancy oscillations." The oscillatory motion observed in the plasma sheet is very similar to buoyancy oscillations in the neutral atmosphere, except that buoyancy force in the magnetosphere is caused by the curvature of magnetic fields rather than gravity. The theoretical analogy to the small parcel of air is a thin magnetic filament in the magnetosphere. This paper works out the theory of small amplitude oscillations of a thin magnetospheric filament and calculates the frequency lowest even mode for different field lines in an average magnetosphere. Out in the plasma sheet, these oscillations are best described as buoyancy waves, but in the inner magnetosphere it is a long-wavelength slow mode.