UCLA BPPL - Properties of Nonlinear Whistlers

The properties of whistler modes with wave magnetic fields larger than the ambient field have been investigated. Such waves are solutions of a nonlinear partial differential equation, derived from Maxwell's equations and an Ohm's law of an EMHD plasma. Since a nonlinear equation cannot be Fourier analyzed, there exists no dispersion relation like for linear whistlers. A monochromatic source generates a spectrum of waves. The propagation depends on amplitude and topology. For given initial and boundary conditions the field evolution has to be solved (numerically) for each individual case. Experimentally, we have chosen a simple axisymmetric source, a loop antenna, and applied a weakly damped oscillatory current in the low frequency whistler regime, based upon the ambient uniform dc magnetic field. Two wave modes are excited: whistler spheromaks and whistler mirrors. The total magnetic field lines for a whistler spheromak are shown below.

Schematic

Field lines of a whistler spheromak.

Such a magnetic vortex propagates in the whistler mode along the ambient magnetic field. It does not propagate along the total magnetic field, otherwise its structure would not be maintained. The propagation velocity is amplitude dependent: The larger the spheromak field, the slower it propagates. The reverse holds for whistler mirrors as is shown below. The amplitude-dependent propagation speed can give rise to various nonlinear wave phenomena such as wave steepening and shock formation or wave-wave collisions between slow spheromaks and fast mirrors. Only a few of such effects, such a spheromak collisions, have been investigated.

Another interesting phenomenon is nonlinear wave damping. Comparing the spatial damping of a small and large amplitude whistler mode, much stronger damping is observed for a field-reversed topology than for a linear wave which has no magnetic nulls. The stronger damping is thought to arise from the free acceleration of electrons in the toroidal null line by a toroidal electric field. Wave damping arises from the finite transit time of electrons through the wave packet which propagates slower than the electron thermal speed. In a linear whistler mode or a whistler mirror the electrons perform E×B drifts which do not energize electrons.

Nonlinear propagation speed (B_y,B_z) vectors, oscillatory current

Nonlinear propagation speed of whistler spheromaks and mirrors.

Spatial damping of linear and nonlinear whistler spheromaks.

The enhanced wave damping and large current densities in whistler spheromaks leads to strong electron heating near magnetic null points. The electron energization is directly measured with Langmuir probes and indirectly observed from light excitation in the dark afterglow plasma. Both bulk heating and energetic tails are observed in spheromaks but none in mirrors. The source of heat/light travels axially at the spead of the spheromak, which is smaller than the electron thermal speed. Electrons must be scattered during the acceleration process since a laminar flow of electrons would produce currents and fields, but would not dissipate the net magnetic energy of the spheromak.

Space-time profiles of electron temperature and plasma potential have been mapped with probes. As shown below, the heating is localized radially and temporally to the whistler spheromak rather than the antenna. The energy source for heating the electrons is the free magnetic energy in the nonlinear wave. In whistler mirrors the energy is mainly convected, in spheromaks it is dissipated.

. (B_y,B_z) vectors, oscillatory current

Langmuir probe traces in a whistler spheromak showing electron heating and non-Maxwellian distributions.

Light emission vs time and position, indicating that the source of hot electrons lies in the propagating whistler spheromaks.

The nonlinear wave also strongly modifies the plasma potential. Space charge electric fields are the essence of Electron MHD phenomena. In whistler mirrors the plasma potential is highly positive (yellow contours) so that the radial electric field and axial magnetic field produce a toroidal Hall current responsible for the axial mirror magnetic field. In whistler spheromaks the potential is approximately negative (blue contours), but closer inspection shows a delay/displacement of heat and potential contours. The asymmetry results in a parallel electric field inside the spheromak, consistent with the axial (poloidal) current that produces the toroidal magnetic field of the spheromak.

Schematic

Electron temperature and plasma potential vs radial position y and time t for a sinusoidal antenna current.

Finally, we show one example of nonlinear wave-wave interactions, the collision of two counter-propagating whistler spheromaks excited by two identical loop antennas. To recall, the collision of linear whistler vortices has been studied earlier and the results were not surprising: Two linear whistler waves propagate through each other without interaction. A different result is obtained for whistler spheromaks: The fields collide inelastically and merge into a stationary Field-Reversed Configuration (FRC). The magnetic energy is converted into electron heat. The process is demonstrated by a series of pictures of magnetic field lines and contours of magnetic energy density:

Schematic

Field lines and free energy density of colliding counter-helicity whistler spheromaks. A continuous display can be seen in a MOVIE of field lines and current densities .

During the collision the FRC tilts back and forth, which is the projection of a 3D FRC precession studied earlier.

References