Knowledge of the three-dimensional particle velocity distribution (PVD) in space plasmas is vital to probe the physical processes. Space plasmas often are not homogenous and pressure gradients are often encountered. Proper interpretation on the velocity moments of the measured PVD near pressure gradient region is therefore important in assessing the plasma dynamics.

Modeling of Measured Particle Velocity Distribution Near A Pressure Gradient

Let us consider a pressure gradient region, illustrated in Figure 1, where the plasma pressure p(y) and the corresponding magnetic field B_z(y) are given analytically by:

p(y) = p₀[1 + tanh(y/L)]/2 + p_b,
B_z(y) = [B₀² + 2μ₀(p₀ + p_b – p(y))]^1/2.

Here p_b is the background minimum plasma pressure, p₀ defines the range in the change of plasma pressure, B₀ is the ambient magnetic field at the location of maximum plasma pressure, and L is the gradient scale in plasma pressure. We numerically track the trajectory of a charged particle back in time in the above model using the Boris algorithm, taking into account the changing magnitude of the magnetic field at each time step. The phase space density (PSD) contribution for that particle is determined by the PSD corresponding to that energy at the gyrocenter of that particle. The particle population is assumed to have a bulk flow (u_x, u_y, u_z) and have a kappa velocity distribution with kappa κ = 5, i.e., the PSD is given by

f(y, vx, vy, vz) =	n0(y)Γ(κ+1)		[1+	(vx-ux)2+(vy -uy)2+ (vz-uz)2	]-(x+1)
	(πκ)3/2Γ(κ-0.5)w3			κw2(y),

where n₀(y) is the number density and w(y) is the thermal speed.

Figure 1. A schematic diagram of a pressure gradient region showing more particle flux at the high-pressure side than at the low- pressure side.

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Assuming that ions make the dominate contribution to the bulk flow and pressure, we show in Figure 2 the profile of the x-component of the first velocity moment of the PVD M_x for a stationary population (i.e., u_x = u_y = u_z = 0 km/s) as a satellite crosses the pressure gradient region where L = 500 km. This calculation uses n₀ = 1 cm^-3 and T = 5 keV and the pressure gradient arises from density gradient. The dotted, dashed, and solid lines represent, respectively, the profiles of the first velocity moment of the PVD M_x sampled for the energy ranges of 10 eV - 10 keV, 10 eV - 40 keV, and 10 eV - 800 keV. The second energy range corresponds to the typical energy range of a plasma detector and the third corresponds to an energy range by combining a plasma detector with an energetic particle detector. It can be seen that M_x takes on a non-zero value as the satellite crosses the pressure gradient region for all energy ranges. The deviation of M_x from zero represents the error introduced by the pressure gradient if this first velocity moment of PVD were taken as the bulk flow.

Figure 2. The first velocity moment of a particle velocity distribution without bulk flow across the plasma pressure gradient computed by using three different energy ranges.

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Figure 3a shows the one-dimensional cut of the measured three-dimensional PVD along the x-direction. This cut is made at location y/L = –1. It can be seen that the PVD is skewed from V_x = 0 due to the anisotropy from the pressure gradient. Naturally, an increase in the coverage of the velocity space for the computation of the first velocity moment would lead to a larger discrepancy from the true value of the bulk flow even though the peak of the PVD remains close to V_x = 0 km/s. Figure 3b shows the corresponding two-dimensional cut on the V_xV_y-plane. The asymmetry of the PVD in the V_x-direction is quite evident. These plots show the deviations of measured PVD from the actual PSD at a fixed location in space, which has no asymmetry with respect to its peak.

Figure 3. (a) one-dimensional cut at Vx = Vz = 0 of the particle velocity distribution at y/L = –1 for the cases of no bulk flow; (b) the corresponding two-dimensional cut at Vz = 0 of the particle velocity distribution at y/L = –1.

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Conclusion

We show that the computed first velocity moment of the measured particle velocity distribution can deviate substantially from the bulk flow of the particle population near a pressure gradient region. The discrepancy increases with increasing energy coverage. Caution is needed to give the proper interpretation of the first velocity moment of the particle velocity distribution when it is measured near a pressure gradient. This result is very important in plasma diagnostics. One common practice in determining the breakdown of the frozen-in-condition is by computing E + V x B. An error in V obviously affects the outcome of this determination.

Biographical Note

Anthony T. Y. Lui is a Co-Investigator of the THEMIS Project at The Johns Hopkins University Applied Physics Laboratory in Laurel, MD. His research interests focus on magnetospheric phenomena and their physical processes, particularly on the topic of substorms.