We establish a new geometric wave function that combined with a variational principle efficiently describes a system of bosons interacting in a one-dimensional trap. By means of a a combination of the exact wave function solution for contact interact
ions and the asymptotic behaviour of the harmonic potential solution we obtain the ground state energy, probability density and profiles of a few boson system in a harmonic trap. We are able to access all regimes, ranging from the strongly attractive to the strongly repulsive one with an original and simple formulation.
We present the first quantified measure of the rate of energy dissipated per unit volume by high frequency electromagnetic waves in the transition region of the Earths collisionless bow shock using data from the THEMIS spacecraft. Every THEMIS shock
crossing examined with available wave burst data showed both low frequency (< 10 Hz) magnetosonic-whistler waves and high frequency (> 10 Hz) electromagnetic and electrostatic waves throughout the entire transition region and into the magnetosheath. The waves in both frequency ranges had large amplitudes, but the higher frequency waves, which are the focus of this study, showed larger contributions to both the Poynting flux and the energy dissipation rates. The higher frequency waves were identified as combinations of ion-acoustic waves, electron cyclotron drift instability driven waves, electrostatic solitary waves, and whistler mode waves. These waves were found to have: (1) amplitudes capable of exceeding dB ~ 10 nT and dE ~ 300 mV/m, though more typical values were dB ~ 0.1-1.0 nT and dE ~ 10-50 mV/m; (2) energy fluxes in excess of 2000 x 10^(-6) W m^(-2); (3) resistivities > 9000 Ohm m; and (4) energy dissipation rates > 3 x 10^(-6) W m^(-3). The dissipation rates were found to be in excess of four orders of magnitude greater than was necessary to explain the increase in entropy across the shocks. Thus, the waves need only be, at times, < 0.01% efficient to balance the nonlinear wave steepening that produces the shocks. Therefore, these results show for the first time that high frequency electromagnetic and electrostatic waves have the capacity to regulate the global structure of collisionless shocks.
We present waveform observations of electromagnetic lower hybrid and whistler waves with f_ci << f < f_ce downstream of four supercritical interplanetary (IP) shocks using the Wind search coil magnetometer. The whistler waves were observed to have a
weak positive correlation between partialB and normalized heat flux magnitude and an inverse correlation with T_eh/T_ec. All were observed simultaneous with electron distributions satisfying the whistler heat flux instability threshold and most with T_{perp,h}/T_{para,h} > 1.01. Thus, the whistler mode waves appear to be driven by a heat flux instability and cause perpendicular heating of the halo electrons. The lower hybrid waves show a much weaker correlation between partialB and normalized heat flux magnitude and are often observed near magnetic field gradients. A third type of event shows fluctuations consistent with a mixture of both lower hybrid and whistler mode waves. These results suggest that whistler waves may indeed be regulating the electron heat flux and the halo temperature anisotropy, which is important for theories and simulations of electron distribution evolution from the sun to the earth.
