Abstract.The

We present theory and numerical simulations of strong nonlinear effects in standing shear Alfv6n waves (SAWs) in the Earth's magnetosphere, which is modeled as a finite size box with straight magnetic lines and (partially) reflecting boundaries. In a low/• plasma it is shown that the ponderomotive force can lead to large-amplitude SAW spatial harmonic generation due to nonlinear coupling between the SAW and a slow magnetosonic wave. The nonlinear coupling leads to secularly growing frequency shifts, and in the case of driven systems, nonlinear dephasing can lead to saturation of the driven wave fields. The results are discussed in the context of their possible relevance to the theory of standing ionospheric cavity wave modes and field line resonances in the high-latitude magnetosphere. decay instabilities [Lashmore-Davies, 1976; Sakai and $onnerup, 1983; Wong and Goldstein, 1986]. Boehm et al. [1990] have observed very large amplitude Alfv6n waves at altitudes of 1000 km or so in the auroral ionosphere. The observed frequencies of the waves are typically greater than 1 Hz, and their electric fields correspond to roughly 200 mV/m. These Alfv•n waves may be a manifestation of the standing iono-•On leave from P. N. Lebedev Physics Institute, Russian Academy of Sciences, Moscow. spheric cavity wave modes proposed by Trakhtengertz and Feldstein [1984] and Lysak [1991], who suggested that the wave modes might reach large enough amplitudes for nonlinear effects to become important. Li and Ternerin [1993] and Boehm et al. [1990] have suggested that spatial gradients in the envelopes of the wave fields can lead to ponderomotive forces that produce density enhancements and depletions that are large enough to account for those seen in the magnetosphere. ULF (1-5 mHz) shear Alfv•n field line resonances (FLRs) are also commonly observed in the Earth's magnetosphere and in the auroral ionosphere [Ruohoniemi et al., 1991; Samson et al., 1992]. The equatorial velocity fields of these FLRs can be as large as 200 km/s [,V[itchell et al., 1990]. Samson et al. [1992] and Rankin et al. [1993a] have shown that these large amplitude FLRs should be nonlinearly unstable to the Kelvin-Helmholtz instability in the equatorial plane. The importance of ponderomotive forces in standing Alfv•n waves has recently been recognized by Allan [1993] and Li and Ternerin [1993], who attributed them to a mechanism for the differential acceleration of auroral particles and species redistribution. Alfv•n waves can show manifestations of two distinct nonlinear phenomena, harmonic generation and instabilities, which may distort the wave fields and lead to saturation of the amplitudes of the waves. The first of these effects is generally much stronger in standing Alfv•n waves than in propagating waves because of larger spatial gradients in the direction of the ambient magnetic field. The second class of nonlinearity includes instabilities such as the Kelvin-Helmholtz instability [Samson et al., 1992, Rankin et al., 1993a] and the nonlinear tearing instabilit...

A model is presented which describes the nonlinear interaction of dispersive shear Alfvén wave (SAW) field line resonances (FLRs) and ion acoustic waves (IAWs), with applications to the Earth’s magnetosphere. Two limits are considered: In low-β plasma (β<me/mi), dispersion is dominated by electron inertia (EI), while for higher β it is dominated by the electron thermal effect. In each case, the ponderomotive force steepens the SAW in the radial direction, taken as earthward in the equatorial plane. Following the time of nonlinear steepening, the dynamics strongly depends on dispersion. In the EI case, standing SAWs excited in FLRs exhibit a parametric decay instability (PDI) into secondary SAWs and IAWs. Nonlinearity and dispersion broaden the FLR in the radial direction, leading to rapid density and parallel electric field fluctuations and scale lengths comparable to the EI length. In warm plasmas, SAWs are stable to the PDI, and in this case the FLR emits short perpendicular scale SAW-IAW solitons in the anti-earthward direction. Observational consequences of both scenarios are discussed.

The nonlinear evolution of driven standing shear Alfvén waves is investigated by virtue of a model which includes the interaction of the waves with density perturbations excited by the ponderomotive force. It is shown that the plasma density perturbations take the form of a slow magnetosonic wave which nonlinearly shifts the frequency of the shear Alfvén wave and decouples it from the external source. This results in a complicated and sometimes chaotic temporal behavior of the amplitude of the excited waves which depends strongly on the plasma pressure, driver strength, and the frequency mismatch between the driver and shear Alfvén eigenmode. The results are discussed in the context of ultra low frequency (ULF) field line resonances in the Earth’s magnetosphere and the excitation of waves in the Alfvénic wave resonator near to the polar ionosphere.

Magnetohydrodynamic, field line resonances in the Earth's magnetosphere can have very large velocity shears and field‐aligned currents. Auroral radar measurements of high‐latitude resonances indicate that the velocities associated with the resonances in the E and F regions are often substantially greater than 1 km/s, and that the frequencies are in the interval from 1 to 4 mHz. Assuming that these resonances are oscillating at the fundamental mode frequency, and mapping these velocity fields along magnetic field lines to the equatorial plane shows that the velocity shears in the equatorial plane are of the order of 200 km/s over a radial distance of less than 2000 km (the amplitude of the velocity fluctuations is 100 km/s). Using a three‐dimensional magnetohydrodynamic computer simulation code, we show that the resonances evolve through the development of Kelvin‐Helmholtz instabilities near the equatorial plane. Within this framework, the instability is taking place on dipole magnetic field lines, and the resonances form a standing shear Alfvén wave field due to the boundary conditions which must be satisfied at the polar ionospheres. We find that the nonlinear evolution of the Kelvin‐Helmholtz instability leads to the propagation of vorticity from the equatorial plane to the polar ionosphere and that the vorticity leads ultimately to the dissipation of the resonance. This occurs within a quarter wave period of the shear Alfvén field associated with the resonances.