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[1] In space plasma missions, measurements of the three-dimensional particle velocity distribution are used to examine the dynamical evolution of plasmas. The first velocity moment of the measured particle velocity distribution is often regarded as the bulk flow of the population. Accurate determination of bulk flows is crucial if the frozen-in-condition is judged by E + V × B = 0, where E is the electric field, V is the bulk flow, and B is the magnetic field. In this paper, we show that this first velocity moment computed from the measured particle velocity distribution can deviate substantially from the bulk motion of the particle population when a significant pressure gradient exists near the measurement location. The discrepancy, which arises from the diamagnetic drift with a finite Larmor radius effect, increases with increasing energy range used in the computation. The result calls for caution in the proper interpretation of this velocity moment and suggests a means to avoid error in the determination of bulk flow due to pressure gradient effects.

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[2] In the early days of space plasma research, fluid parameters were used predominantly to describe the state of the measured plasma population and to investigate the observed dynamic phenomena. These efforts are extremely useful to gain an overall description of the space environment, providing insights into distinct plasma regimes. For instance, the global characteristics of the Earth's magnetosphere and its gross interaction with the solar wind can be ascertained with the fluid approach.

[3] It has become increasingly clear as space plasma research becomes more mature as a discipline that there are kinetic processes involved in some of these dynamic phenomena. Identification of the physical processes responsible for these dynamic phenomena requires knowledge on parameters beyond the fluid description.

[4] Charged particle detectors used in measuring plasma properties in space nowadays measure particle intensities in three dimensions with high enough time resolution to enable one to probe the dynamic evolution of the particle velocity distribution. This capability provides a powerful clue to probe these dynamic processes.

[5] Previously, studies have been made to investigate the particle velocity distribution and point out the deficiencies in just relying on the fluid parameters used commonly in magnetohydrodynamic (MHD) treatment when multiple components are observed [Ueno et al., 2001; Lui and Hori, 2006]. Importance of particle velocity distribution in plasma diagnostics has also been discussed by Parks et al. [2001]. Extraction of plasma parameters related to particle source from careful analysis of the particle velocity distribution has also been demonstrated [Lui, 2006].

[6] In this paper, we model the effect of a pressure gradient on the particle velocity distribution sampled by a particle detector in a non-uniform plasma. We show that calculation of the first velocity moment of the measured particle velocity distribution gives significant discrepancies from the actual bulk motion of the population when measurements are made in the vicinity of a pressure gradient region. The difference between the computed first velocity component of the particle velocity distribution and the actual bulk flow, i.e., error arising from the pressure gradient, increases with increasing energy coverage in the calculation. This difference is basically due to the gyromotions of the particles, a kinetic aspect not considered by the MHD theory.

2. An Equilibrium Model With a Pressure Gradient

[7] Let us consider a simple case in which the pressure gradient arises from a gradient directed in the dawn-dusk direction taken to be in the y-axis. The magnetic field is directed in the z-axis. Figure 1a shows the profiles of the plasma pressure, the magnetic pressure, and the total pressure from combining the magnetic field and the plasma across the gradient region. This equilibrium model is described by the following expressions for the plasma pressure p(y) and the corresponding magnetic field B_{z}(y):

where p_{b} is the background minimum plasma pressure, p_{0} defines the range in the change of plasma pressure, B_{0} is the ambient magnetic field at the location of maximum plasma pressure (p = p_{0} + p_{b}), and L is the gradient scale in plasma pressure. In this equilibrium model, we use p_{b} = 0.05 keV cm^{−3}, p_{0} = 5 keV cm^{−3}, B_{0} = 2 nT, and L = 500 km. As shown in Figure 1a, the total pressure is constant across the gradient, satisfying the condition for an equilibrium configuration. In the following, we shall examine the first velocity moment of the particle velocity distribution as a satellite crosses this pressure gradient.

3. Particle Velocity Distribution in a Pressure Gradient Region

[8] In order to model the particle velocity distribution (PVD) as a satellite passes through the model pressure gradient, we numerically track the trajectory of a charged particle back in time in the above equilibrium model using the Boris algorithm [Birdsall and Langdon, 1985]. The changing magnitude of the magnetic field at each time step is taken into account. The phase space density (PSD) contribution for that particle is determined by the PSD corresponding to that energy at the gyrocenter of that particle, which is determined at the turning point of the particle's motion in the x-direction, i.e., the gyrocenter of the particle at the point where v_{x} = 0. 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. The PSD of the PVD is given by

where n_{0}(y) is the number density and w(y) is the thermal speed. The dependence of n_{0}(y) and w(y) on the y-location is such that p(y) = 0.5m_{i}n(y)w^{2}(y)κ/(κ − 1.5), where m_{i} is the ion mass. For simplicity, we shall model the plasma pressure gradient by first the number density gradient and second the temperature gradient.

[9]Figure 1b shows 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) in the XY-plane as the satellite crosses the pressure gradient region. The dotted, dashed, and solid lines represent, respectively, the values by calculating 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. This calculation uses n_{o} = 1 cm^{−3} and T = 5 keV to produce p_{0} = 5 keV cm^{−3}. 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. The profiles are skewed with respect to y/L = 0. The error increases with increasing coverage of the energy used in the summation. The maximum error for each profile occurs close to y/L = −1.0, at the values of ∼148, 295, and 355 km/s for the short, medium, and large energy ranges, respectively. It may also be noticed that the profile is asymmetric with respect to its peak. On the positive side of y, M_{x} remains at a significant value for a long distance from the gradient location in comparison to the negative side of y. This arises from the fact that the magnetic field at the high plasma pressure side is weak such that a significant portion of the particle population senses the gradient in their gyromotion. This is the finite Larmor radius effect of this diamagnetic drift that cannot be reproduced by the MHD theory.

[10]Figure 1c shows the parameter M_{x} for a population with a bulk flow of 500 km/s, i.e., u_{x} = 500 km/s, u_{y} = u_{z} = 0 km/s. Again, the calculation uses n_{o} = 1 cm^{−3} and T = 5 keV. Similar to Figure 1b, the profiles are also skewed with respect to y/L = 0, with deviations from M_{x} = 500 km/s increasing with larger energy range used in the summation. The maximum values of M_{x} are ∼ 424, 752, and 861 km/s for the three energy ranges and occur also close to y/L = −1.0. The reason for M_{x} to be below 500 km/s for the short energy range is because a substantial portion of the PVD is excluded if the energy coverage only goes up to 10 keV when the bulk flow of the population is 500 km/s.

[11] A better understanding on the cause of the deviation of M_{x} from the bulk flow can be gleaned from examining the PSD. Figure 2 (top) shows the one-dimensional cut of the three-dimensional PSD along the x-direction for the two cases of different bulk flows. These cuts are made at location y/L = −1 for L = 500 km. 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. Note also that the peaks of the PVDs remain close to V_{x} = 0 km/s. Figure 2 (bottom) shows the corresponding two-dimensional cuts of the three-dimensional PSD on the V_{x}V_{y}-plane for these two cases. The skew of the PSD about the V_{x} = 0 km/s axis for the stationary population is also evident in this two-dimensional cut. There is no such skew about the V_{y} = 0 axis, as expected. For the population with the bulk flow of 500 km/s, the skew of the PSD occurs with respect to the V_{x} = 500 km/s axis.

[12] These results indicate that if one interprets the first velocity moment of the PVD as the bulk motion of the population, then the error increases with increasing energy coverage. Furthermore, these plots show a means to avoid making the wrong interpretation of the first velocity moment. If the bulk velocity were determined from the peak of the PVD, then the error introduced by the pressure gradient would be minimized or eliminated.

[13] The discrepancies shown in Figures 1 depend on the assumed thermal energies and gradient scale lengths. Figure 3 shows the effects from different values of thermal energy and gradient scale length for a particle population with no bulk flow. The profiles of M_{x} for gradient scale lengths of 500, 1000, 2000, and 4000 km are shown. It is evident from Figures 3a–3d that the discrepancy decreases with a larger value for the scale length L. The asymmetry of the profile with respect to its peak also decreases with a larger value for L. This tendency can be understood by knowing that the asymmetry is due to the encounter of the gradient during the gyromotion of some particles in the population. The larger the scale length, the less is the number of particles encountering the gradient.

[14] The effect from different thermal energies is shown by comparing M_{x} in Figure 3b, where T = 10 keV, from that in Figure 3c, where T = 2 keV. The corresponding number density is adjusted to produce p_{0} = 5 keV cm^{−3}. As expected, the discrepancy is higher for higher thermal energy. Distinct from the density gradient effect, Figure 3d shows the profile for a pressure gradient arising from a temperature gradient. The thermal energy increases from 2 keV on the negative side of y to 10 keV on the positive side. The general trend in the dependence on the scale length still holds. However, the peak value is close to the location y/L = 0 for L = 500 km and gradually shifts to the positive side of y with increasing value of L. In addition, the asymmetry is more pronounced than that in density gradient case. Again, these features arise from the finite Larmor radius effect in the diamagnetic drift.

4. Summary and Discussion

[15] We model the expected particle velocity distribution measured by a particle detector or detectors on a satellite passing through a pressure gradient region. From this modeling, we show the effect of pressure gradient on the measured particle velocity distribution and the computed velocity moment. It is shown that the computed first velocity moment of the particle velocity distribution can deviate substantially from the bulk flow due to the diamagnetic drift associated with the pressure gradient, revealing some features arising from the finite Larmor radius effect. The discrepancy increases with increasing coverage of the energy due to the asymmetry in the measured particle velocity distribution from the pressure gradient. The result indicates that caution is needed to give the proper interpretation of the first velocity moment of the particle velocity distribution. In order to avoid the error introduced by the pressure gradient, the bulk motion of the population is better determined by the peak of the three-dimensional particle velocity distribution than by relying on the computed first velocity moment.

[16] We have examined the situation when the pressure gradient is in the y-direction and the magnetic field is in the z-direction. In the case of a thin cross-tail current sheet when the pressure gradient is in the z-direction and the magnetic field is in the x-direction, then the velocity discrepancy occurs in the y-direction.

[17] One might think that the diamagnetic drift is included correctly in the MHD definition of the bulk flow and so the larger velocity range included in the computation, the more accurate would the calculated bulk flow be. However, note that MHD is a fluid theory and does not take into account guiding center locations. The bulk velocity of the fluid is defined by the first velocity moment of the phase space density at a fixed location, treating the spatial and velocity coordinates as independent variables. Therefore, when the gyroradii of all particles are small in comparison with the pressure gradient scale, one may obtain the correct value for the bulk flow of the plasma from the first velocity moment. However, when the gyroradii of energetic particles are significant in comparison with the pressure gradient scale, the measured phase space density is comprised of particles with gyrocenters at different locations rather than at a fixed location. In other words, the spatial and velocity coordinates are not independent variables. This introduces the discrepancy between the actual bulk flow of the particle population and the first velocity moment of the measured phase space density.

[18] Large plasma pressure gradients are often seen in the magnetotail during active times. For example, Nakamura et al. [2006] reported Cluster encountering a thin current sheet with a full width of 780 km, corresponding to L = 390 km. Large magnetic and plasma pressure variations were seen during the encounter (see their Figure 2) with the first velocity moment of u_{x} reaching 1000 km/s. If there were substantial pressure gradient in the y-direction, the gradient effect could have contributed partially to this high value.

[19] Proper interpretation of the first velocity moment of the measured particle velocity distribution is very important in plasma diagnostics. One common practice in determining energy exchange between fields and particles is the evaluation of E′ = E + V × B, where E′ is the electric field in the plasma rest frame, E is the electric field in the observer frame, V is the bulk flow, and B is the magnetic field. The condition E′ = 0 is the frozen-in-field condition. A non-zero E′ indicates energy exchange between fields and particles locally. Therefore, error caused by pressure gradient would lead to an erroneous evaluation of E′ and misidentification of locations where energy exchange occurs.

Acknowledgments

[20] This work was supported by the NSF grant ATM-0630912 and NASA grant NNX07AU74G to The Johns Hopkins University Applied Physics Laboratory.