Nonlinear Waves in the Terrestrial Quasiparallel Foreshock

We provide strongly conclusive evidence that the cubic nonlinearity plays an important part in the evolution of the large amplitude magnetic structures in the terrestrial foreshock. Large amplitude nonlinear wave trains at frequencies above the proton cyclotron frequency are identified after nonharmonic slow variations are filtered out by applying the empirical mode decomposition. Numerical solutions of the derivative nonlinear Schrödinger equation, predicted analytically by the use of a pseudopotential approach, are found to be consistent with the observed wave forms. The approximate phase speed of these nonlinear waves, indicated by the parameters of numerical solutions, is of the order of the local Alfvén speed. We suggest that the feedback of the large amplitude fluctuations on background plasma is reflected in the evolution of the pseudopotential.

Figure 1

Summary of Cluster observations for the interval of interest. Panels from top to bottom show: solar wind ion plasma number density, solar wind ion bulk speed, magnetic field magnitude and the omnidirectional proton flux spectrum. Vertical dashed lines mark the start of intervals I1, I2, and I3.

Figure 2

Nonlinear wave forms in the magnitude of the transverse magnetic field fluctuations for interval I1. Panel (a): Original squared magnitude of transverse fluctuations (black) and the trend (red). Panel (b): Normalized and detrended signal with color dots marking the start of nonlinear waves (red), the end of the nonlinear waves or start of small amplitude waves (blue), and the end of small amplitude waves (green). The mean spacecraft frame frequency of the signal (red) is given in the top right corner. Panel (c): Hodograms of the transverse components used in (a). Panel (d): The phase space of the signal shown in (b) with the same color scheme and the equivalent trajectories obtained from the numerical solutions of Eq. (2) for parameters: $C_b=18.9$, $V=5.1$, $C=3.7$ (nonlinear solution, black) and $C_b=3.91$, $V=8$, $C=7.2$ (small amplitude solution, blue).

Figure 3

Same as Fig. 2 for interval I2. Parameters of numerical solutions: $C_b=25.08$, $V=4.3$, $C=3.7$ (nonlinear solution, black) and $C_b=5.31$, $V=8$, $C=7.2$ (small amplitude solution, blue).

Figure 4

Same as Fig. 2 for interval I3. Parameters of numerical solutions: $C_b=20.25$, $V=5.1$, $C=4.8$ (nonlinear solution, black) and $C_b=7.33$, $V=9$, $C=8.4$ (small amplitude solution, blue).

Figure 5

The form of the potential function $U_b$ in canonical representation for I1 (a), I2 (b), and I3(c). In all cases $\mu =1/2$, $\mathrm{\Omega }=1$, and $\alpha =1$ and other parameters are specified in the captions of Figs. 2, 3, 4. Red and green curves correspond to the nonlinear trajectories and the small amplitude fluctuations, respectively, as shown in panel (c) of Figs 2, 3, 4.