The dynamics of magnetosphere‐ionosphere coupling has been investigated by means of a two‐dimensional two—fluid MHD model including anomalous resistivity. When field‐aligned current is generated on auroral field lines, the disturbance propagates toward the ionosphere in the form of a kinetic Alfvén wave. When the current exceeds a critical value, microscopic turbulence is produced, which modifies the propagation of the Alfvén wave. This process is modeled by a nonlinear collision frequency, which increases with the excess of the drift velocity over the critical value. The system evolves toward an electrostatic structure, with the perpendicular electric field having a shorter scale than the field‐aligned current. The approach to a steady state is strongly dependent on the presence or absence of the turbulence and on the boundary conditions imposed in the generator. As current is increased or scale size is decreased, the turbulent region reflects and absorbs most of the Alfvén wave energy, decoupling the generator from the ionosphere.

A stability analysis which directly uses particle distribution functions determined from experiments or transport theory, rather than model distributions, is carried out. The features of distribution functions relevant to whistlers, ion cyclotron waves, including their low-frequency extensions for propagation along the magnetic field, and to ion-acoustic waves are analyzed in detail. The dependence of wave growth on the precise shape of the distributions and the numerical feasibility of the method is demonstrated by the use of measured solar wind distributions.

It is shown that the broadening of wave-particle resonances by the random motion of particles in a turbulent electric field may determine the saturation level of a variety of high-frequency instabilities. Secular changes of the guiding center positions, cyclotron radii, and phase angles give rise to resonance broadening and diffusion, similar to that produced by collisional scattering. The field dependent broadening is expressed in terms of resonance functions which replace the familiar resonant denominators of the linear theory. Resonance functions are derived in a simple manner from the solution of a Brownian motion problem, leading to an expression in terms of diffusion coefficients. The close resemblance of the theory to quasilinear theory and the linear theory including collisions allows one to start from a linear stability analysis and then assess the importance of nonlinear effects. This method is illustrated by the determination of the saturation level of cyclotron instabilities from the condition of vanishing nonlinear growth rate.

Turbulent heating and stabilization of the ion sound instability is investigated by twodimensional computer simulation. Quasilinear rather than nonlinear effects determine the evolution of the instability. The instability is quenched by flattening of the electron distribution and the formation of a high-energy ion tail.Numerous stabilization mechanisms have been proposed for the current-driven ion sound instability. 1 "* 5 We have done extensive simulation studies in order to provide a test for these basic predictions. 6 The two-dimensional code has been described previously. 7 Specifically for the purpose of testing nonlinear theories of stabilization, we have made runs in which the ratio between drift velocity and electron thermal velocity was kept constant, in addition to runs with constant current. In the same vein we discuss the case of a current perpendicular to a weak magnetic field (fy/o> e = 0.04). The magnetic field (perpendicular to the plane of computation) has a very small effect on wave dispersion, but keeps the electron distribution isotropic. (In the case of a current along a magnetic field, further complicated dynamical effects are added by the formation of an electron runaway tail.)We find that for a wide range of initial parameters the growth phase of the instability is followed by the decay of the wave energy W , the return of the fluctuation level W/nT e to the thermal level, 8 and termination of heating in typically (100-200)a) i " 1 . Clearly, in the case of constant current, the growth phase of the instability must terminate at the latest when the phase velocity reaches the drift velocity, w«c s = (T e /M) 1/2 , The runs with constant u/c s , however, show quench-ing in much the same way; see Fig. 1. It is seen that in this case the plasma enters a regime in which the macroscopic parameters remain constant.Nonlinear theories of stabilization generally determine a quasisteady fluctuation level W/nT e as a function of m/M, u/v ey and T e /T t from the condition that the nonlinear damping just balances the linear growth rate, y = y L + y NL = 0. Actually, for (*/dt)ln(W/nT e ) = W/W-T e /T, = 0 9 W/W=2y must be balanced by the electron heat-FIG. 1. Wave energy W, fluctuation level W/nT e , T e , and T i /T e for a typical run. M/m=100, (T^T^ = 0.02, w/v e = 0.75. 1231

When a medium is dissipative, the classic expression for the group velocity, dω/dk, is complex with an imaginary part often being far from negligible. To clarify the role of this imaginary term, the motion of a wave packet in a dissipative, homogeneous medium is examined. The integral representation of the packet is analyzed by means of a saddle-point method. It is shown that in a moving frame attached to its maximum the packet looks self-similar. A Gaussian packet keeps its Gaussian identity, as is familiar for the case of a nondissipative medium. However, the central wave number of the packet slowly changes because of a differential damping among the Fourier components: Im(dω/dk)=dγ/dk≠0, where ω≡ωr+iγ. The packet height can be computed self-consistently as integrated damping (or growth). The real group velocity becomes a time-dependent combination of Re(dω/dk) and Im(dω/dk). Only where the medium is both homogeneous and loss free, does the group velocity remain constant. Simple ‘‘ray-tracing equations’’ are derived to follow the packet centers in coordinate and Fourier spaces. The analysis is illustrated with a comparison to geometric optics, and by two applications: the case of a medium with some resonant damping (or growth) and the propagation of whistler waves in a collisional plasma.

The role played by nonlinear scattering during the relaxation of a warm electron beam is investigated with the help of a numerical code. The code is based on kinetic equations and includes the quasilinear wave–electron interaction as well as wave–wave scattering off ion clouds. Both mechanisms have been observed to play key roles in a recent particle simulation with a large number of modes. It is found that (1) ions with velocity 2vi (vi being the ion thermal velocity) are the most efficient to scatter the Langmuir waves off their polarization clouds. As a result, the transfer rate of the spectrum out of resonance with the beam is larger by a factor 3 compared to the usual estimates in the literature, which assume a static ion response. The predicted wave number k of the secondary spectrum differs also substantially. (2) If the beam density nb, drift Ub, and width vb satisfy the condition nb/n0>4.2(ve/Ub)2 ×(vb/Ub)3, the changes brought to the dispersion relation by the presence of the beam electrons dramatically alter the characteristics of the secondary spectrum. Forward propagating waves may grow where the conventional picture expects backward propagating waves. Most strikingly, in a late phase the classic condensate about k∼0 is depleted with the formation of a new condensate in resonance with the flat-topped beam distribution. This contradicts the commonly assumed cascade in wave numbers, but follows simply from the fact that the mere presence of the beam electrons creates a minimum in the frequency–wave-number relation. There is no contradiction with a cascade toward lower frequencies driven by an isotropic ion distribution. For strong and slow beams (nb/n0∼10−2, Ub∼10ve) the predictions of this code can be compared with the results obtained in the particle simulation. The agreement is excellent if one uses a dispersion relation that includes the beam. Complete plateau formation by resonant diffusion and late formation of a secondary spectrum are observed. Time scales and spectral characteristics compare well. For faster and weaker beams, it is demonstrated that the nonlinear wave scattering may intervene before complete quasilinear relaxation. Once the beam top has been erased by diffusion, a wave condensate forms, which inhibits further relaxation toward lower velocities. Modes in resonance with the positive slope at the low-velocity front of the flat-topped beam are stabilized by a fast transfer of their energy into the condensate.

The kinetic equation for electrons including Coulomb collisions and scattering by ion sound and related spectra is reduced to a system of equations for the energy distribution and the anisotropic part. The energy distribution is obtained for the cases where Coulomb collisions, runaway or turbulent heating dominates. Resistivity, heating rate, and the dispersion relation are significantly modified for the self-consistent non-Maxwellian distribution. Applications to turbulent heating by ion sound are made.

Two-dimensional simulations of the electron-beam plasma instability with large system size are carried out and are compared with recent one-dimensional simulations for plasma parameters appropriate to the electron foreshock. It is found that wave propagation and diffusion perpendicular to the beam drift are significant at all times. Because a plateau cannot be maintained in this case, the wave level decreases much more rapidly than in one-dimensional simulations. The nonlinear wave scattering process which occurs at late times also differs in that it generates a broad secondary spectrum rather than a condensate. The two-dimensional model, in addition, allows the investigation of the effects of increasing magnetic field strength (e.g., along auroral field lines). For intermediate magnetic fields Langmuir waves and a highly oblique spectrum belonging to the lower hybrid branch are simultaneously excited. The oblique wave spectrum for strong magnetic fields can be explained by mapping from the magnetic field direction.