Crockett L. Grabbe

Analytic MHD theory for Earth's bow shock at low Mach numbers

A previous MHD theory for the density jump at the Earths bow shock, which assumed the Alfven (MA) and sonic (Ms) Mach numbers are both ≫ 1, is reanalyzed and generalized. It is shown that the MHD jump equation can be analytically solved much more directly using perturbation theory, with the ordering determined by MA and Ms, and that the first-order perturbation solution is identical to the solution found in the earlier theory. The second-order perturbation solution is calculated, whereas the earlier approach cannot be used to obtain it. The second-order terms generally are important over most of the range of MA and Ms in the solar wind when the angle θ between the normal to the bow shock and magnetic field is not close to 0° or 180° (the solutions are symmetric about 90°). This new perturbation solution is generally accurate under most solar wind conditions at 1 AU, with the exception of low Mach numbers when θ is close to 90°. In this exceptional case the new solution does not improve on the first-order solutions obtained earlier, and the predicted density ratio can vary by 10–20% from the exact numerical MHD solutions. For θ ∼ 90° another perturbation solution is derived that predicts the density ratio much more accurately. This second solution is typically accurate for quasi-perpendicular conditions. Taken together, these two analytical solutions are generally accurate for the Earths bow shock, except in the rare circumstance that MA ≤ 2. MHD and gasdynamic simulations have produced empirical models in which the shocks standoff distance as, is linearly related to the density jump ratio X at the subsolar point. Using an empirical relationship between as and X obtained from MHD simulations, as values predicted using the MHD solutions for X are compared with the predictions of phenomenological models commonly used for modeling observational data, and with the predictions of a modified phenomenological model proposed recently. The similarities and differences between these results are illustrated using plots of X and as predicted for the Earths bow shock. The plots show that the new analytic solutions agree very well with the exact numerical MHD solutions and that these MHD solutions should replace the the corresponding phenomenological relations in comparisons with data. Furthermore, significant differences exist between the standoff distances predicted at low MA using the MHD models versus those predicted by the new modified phenomenological model. These differences should be amenable to observational testing.

Trapped-electron solitary wave structures in a magnetized plasma

The classic one-dimensional (1D) BGK-mode (trapped-particle) analysis is extended to 3D plasmas in which a background magnetic field is present. A theoretical analysis is made of such nonlinear states from the coupled plasma equations for a magnetized plasma, which yields a generalized BGK-like equation for trapped-particle distributions that produce this structure in the guiding-center approximation, and the resulting conditional requirement for electron trapping as a function of the electric-field components, and magnetic gradient and polarization drifts. This condition shows that the amount of trapping decreases as the angle of the electric field with respect to the background magnetic field increases, often ceasing at a critical finite angle. The full solution to the generalized BGK-type equation for the trapped-electron distribution is obtained and reduced to the extent possible, thereby extending the classic 1D solution for electron holes to that for the 3D magnetized plasmas for cylindrical symmetr...

Broadband electrostatic wave observations in the auroral region on Polar and comparisons with theory

[i] Broadband electrostatic wave (BEN) data gathered from Polar for passes through the aurora are analyzed and compared with theoretical predictions for the origin of magnetized solitary waves in BEN. Two passes exhibiting solitary waves were chosen for comparison, and times were selected that show numerous such waves. Examination of the data shows that even for these cases the nonsolitary form of BEN occurs much more commonly than the solitary form that is being particularly examined here. The first pass was for the period of 1430-1550 UT on 7 April 1996 at distances near 7R E , and the second pass was for the period of 0600-0800 UT on 10 May 1996 at distances near 8.8 R E . Four short intervals were chosen from the passes that show interesting solitary waves, each at slightly different radial distance, and the plasma parameters measured are used in developed theory to give predictions for the minimum and maximum electric field for propagation ofBGK solitary waves, which have been the focus of theory for the nonlinear form of BEN for over a decade. This predicted minimum is compared to observations for solitary waves observed in these spacecraft time intervals along with parameters of the plasma environment at the time of observation, identified from the HYDRA data for these passes. The actual electric field is found to lie below the predicted minimum for all cases. Our conclusion is that these solitary waves are clearly not BGK waves in their observed form and may possibly have been produced by means other than electron trapping processes for BGK solitary waves. Feasible alternative theories for the solitary waves are discussed.

Asymptotics of the modified plasma dispersion function generalized to real kappa

The modified plasma dispersion function (MPDF), an extension to κ particle distributions of the standard plasma dispersion function (PDF), has been recently generalized to real κ. An analysis of the general form of the MPDF is made to obtain its asymptotic solution for real κ≥2. A recursion relation is derived and applied to calculate the asymptotic expression directly in terms of the PDF. A numerical analysis is made of its error relative to the exact value for a range of integer values of κ from 2 to 5, and the error is found to typically be about 8% for κ=2, 2% for κ=3, 0.4% for κ=4, and 0.2% for κ=5. The precision increases as κ increases, and generally the first four to five terms of the expansion give accurate results. Potential applications for analyzing processes in a variety of space and astrophysical plasmas are discussed.

The polytropic index for the solar wind at Earth's bow shock

Analysis of spacecraft shock-crossing data on the magnetosheath thickness to determine the polytropic index γ of the solar wind is shown to overestimate the value for γ when the gasdynamic (GD) equations are used for the shock jump. This is done by comparing GD estimates for γ extrapolated from observed data points to values obtained when more appropriate MHD equations are used. As a result of this bias a shock-crossing data analysis carried out recently using the GD equations appears to have obtained a value for γ that is over 0.5 too large. This is in part associated with the breakdown of GD with respect to nonzero angle between the bow-shock normal and the magnetic field for finite Alfven Mach numbers. The magnetosonic phenomenological model (sonic Mach number replaced with the magnetosonic Mach number in the GD equations), which was also recently used to determine γ, is found to reduce this bias by 70–90%. This analysis also strongly suggests that the phenomenological model inadequately treats variations in the angle between the bow-shock normal and the magnetic field, which may also produce systematic errors in the experimental determination of γ using this model.

MHD theory of Earth's magnetosheath for an axisymmetric model

An analysis is made of the Earths magnetosheath along the Sun-Earth line under conditions that the IMF (Interplanetary Magnetic Field) is nearly parallel or anti-parallel to the solar wind flow. MHD conservation equations in temporally-averaged steady-state form for the mass, momentum and energy density are combined with the magnetic divergence and induction equations, a hard conducting-sphere model for the magnetopause, and an adiabatic equation of state in the magnetosheath. The equations are integrated from the nose of the bow shock to the magnetopause and reduced to a set of nonlinear-coupled equations for the magnetosheath thickness and average magnetosheath parameters, which are then used to obtain a new equation for the thickness of the magnetosheath. This is the first analytical equation for the magnetosheath thickness that has been derived, and it exhibits an interesting functional dependence on Alfven and sonic Mach number MA and Ms,the angle θ0 between the bow shock normal and the IMF, the Chapman-Ferraro constant k0 at the magnetopause, the polytropic index γ, and the thermal conductivity Qr at the bow shock. The thickness is found to decrease for decreasing Ms, but increase with decreasing MA It exhibits the qualitative feature found in both gasdynamic and θ0 ≥ 45° MHD simulations of an approximate linear variation of the magnetosheath thickness with the density jump ratio X across the bow shock, but it also exhibits a unique negative slope and an offset that is a function both of MA and of K0.

Low Mach number predictions in an extended axially symmetric MHD theory of the magnetosheath

A recent axisymmetric magnetohydrodynamic (MHD) model for the Earths magnetosheath near the Sun-Earth line, with boundary conditions given by Rankine-Hugoniot (RH) equations at the bow shock and the Chapman-Ferraro (CF) condition at the magnetopause, is extended by including smaller terms in the CF condition that become important at low Mach numbers. Three conclusions are predicted for the magnetosheath. (1) The magnetosheath thickness Δms decreases with decreasing Alfven Mach number MA for MA > M* (in which only the “gasdynamic” root of the RH equations for the bow shock boundary is physical), where M* ∼ 2 is the root-transition value for the RH equations at which all 3 roots coalesce. When MA < M*switch-on shock roots (purely MHD) are also physical, and Δms associated with those roots shows a rapid increase to a large value, followed by a rapid decrease back to a very small value as the decreasing MA approaches 1. Thus Δms associated with these switch-on roots may give more distant bow shocks, and may rapidly vary with small MA changes. (2) Δms is very sensitive to the value for the CF constant kCF at the magnetopause. For larger MA and sonic Mach numbers Ms the ratio of Δms to the geocentric radial distance amp of the magnetopause varies from 0.44 for 90% coupling efficiency of the solar wind momentum density to the magnetopause down to 0.18 for 60% coupling efficiency. This suggests a balance between the solar wind coupling efficiency to the magnetopause and the thickness of the magnetosheath, in which an increase (decrease) in the coupling efficiency increases (decreases) the thickness of the magnetosheath to counter that efficiency by moving the bow shock further upstream (downstream), hence minimizing the variation in time of that coupling efficiency. (3) The linear relation between Δms and the density ratio ρsw/ρbs observed in simulations breaks down for MA or Ms ≲ 2. The slope steadily drops as Ms → 1, but increases as MA → 1.

Relaxed states of an ideal MHD plasma with external magnetic field

We study an ideal MHD plasma with the non-vanishing invariants energy, cross-helicity and magnetic helicity, confined in a cylinder with infinitely conducting walls and an externally applied magnetic field B 0 . The magnetic and velocity fields are expanded in base vector fields, satisfying ⊇ x B λ = B λ . Boundary conditions are imposed to make the curl a self-adjoint operator. The three invariants depend on the time-dependent coefficients of the base vector fields, and are used to construct the partition function to gather statistical information about the equlibrium thermodynamic state to which the plasma relaxes after a turbulent transition. For zero external magnetic field but large magnetic helicity, the energy resides preferentially in magnetic field fluctuations. A sizeable fraction of the kinetic energy initially present is transformed into magnetic energy. The energy condenses via an inverse cascade predominantly to the lowest energy eigenstate, in agreement with results obtained by Taylor. However, since we consider the whole spectrum of eigenstates, the energy does not exclusively occupy the lowest eigenstate. If the eigenvalues are densely spaced (as in a thin torus), the higher eigenmodes also contain appreciable amounts of energy, resulting in a finite pressure of the plasma. For constant and finite external magnetic field, the average induced magnetic field exactly cancels the external field. This indicates that, on a statistical average, the plasma is diamagnetic or superconducting. Superimposed on the average statistical state are fluctuations that may become large if the magnetic helicity is large.

Resonance cone structure in a warm inhomogeneous bounded plasma with lower-hybrid resonance layers

The problem of the wave fields excited by a gap source at the edge of an inhomogeneous, magnetized plasma with a pair of lower-hybrid resonance layers present and bounded by conducting walls is solved. The approach used is that of a solution as a sum of multiply-reflected extraordinary mode and ion-thermal resonance cones as an alternative to the guided-wave mode approach. This is achieved by dividing up the plasma into cross-sections where WKB solutions are valid, and into lower-hybrid resonance layers where asymptotic methods are employed and connexion coefficients between each region are obtained. A diagrammatic scheme for writing the solution is introduced which can in principle be used to write down the solution for any problem of this general type once the connexion coefficients across a resonance layer have been determined via asymptotic analysis. This allows a determination in great detail of the structure and properties of the resonance cones in our model and the way they transform across the back-to-back hybrid layers. Evanescent resonance cones are found to exist in the high-density region between the hybrid resonance layers and tunnel through to the other side, maintaining their general structure if the layer is relatively thin.