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Division of Plasma Physics, Naval Research Laboratory, Washington, D. C., USA

Corresponding author: C. Crabtree, Division of Plasma Physics, Naval Research Laboratory, 4555 Overlook Ave., SW, Washington, DC 20375, USA. (chris.crabtree@nrl.navy.mil)

Corresponding author: C. Crabtree, Division of Plasma Physics, Naval Research Laboratory, 4555 Overlook Ave., SW, Washington, DC 20375, USA. (chris.crabtree@nrl.navy.mil)

Abstract

[1] In this letter we consider the dissipation of magnetospherically reflecting whistler waves from both ionospheric sources and from sources outside the plasmasphere such as chorus. We use a simple spatially dependent model of the suprathermal electron population, a standard cold plasma density model based on a diffusive equilibrium, and a range of plasmaspheric temperatures to demonstrate that the often-ignored (electron-ion) collisional damping is usually at least the same order of magnitude and often an order of magnitude larger than the dissipation due to collisionless damping (Landau and transit-time). Furthermore the collisional damping is sensitive to the cold plasmaspheric temperature, which depends on location (night/day-side) and solar conditions. These results indicate that accurate spatially dependent models of plasmaspheric temperatures as well as suprathermal electron fluxes are necessary for modeling the dissipation of magnetospherically reflecting whistler waves.

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[2] Understanding the dissipation mechanisms and accurately evaluating the damping of magnetospherically reflecting whistler waves is important for predicting the amplitude of waves in the radiation belts and thus for evaluating the pitch-angle scattering rates and the lifetime of energetic electrons. It is also important for assessing the threshold for non-linear effects on VLF (very-low frequency) wave propagation [Crabtree et al., 2012].

[3] It is often assumed that the plasmasphere is a collisionless environment, and indeed it is collisionless in many respects. For example, the typical electron-Hydrogen collision frequencyν^{e/H} is of the order of 1 per second, so that this frequency is much smaller than any typical frequency of interest, e.g. cyclotron frequencies, plasma frequencies, VLF frequencies, etc. Thus collisions are not expected to play a substantial role in the real part of the dispersion relation that determines the frequency of the wave, and thus the phase velocities, group velocities and ray trajectories. However, it can be important in the imaginary part that determines the growth/damping rates. There are two physical linear mechanisms that can contribute to growth/damping, (1) resonant collisionless dissipation or growth and (2) collisional dissipation. Because the phase velocity of magnetospherically reflecting whistler waves is much larger than the thermal velocity of the cold dense electrons, collisionless damping will occur due to a suprathermal population of electrons. The average number density of suprathermal electrons depends on L shell [Bell et al., 2002; Li et al., 2010] among other conditions, however inside the plasmasphere the density of suprathermal electrons is about 6 orders of magnitude smaller than the thermal density, it turns out that collisional damping can therefore be much larger.

2. Theoretical Formulation

[4] In general the power dissipated by a given wave is

P=E·J=ω8πE*·χa·E

where χ_{a}is the anti-Hermitian part of the susceptibility,E is the wave electric field, J is the wave generated current, and ωis the real part of the angular frequency. For the power dissipated to be explicitly evaluated, in addition to the susceptibility the wave fields are needed. Since the damping rate is much less than the real frequency it is reasonable to consider the wave fields to be unmodified by the dissipation. Thus we determine the wave fields using cold plasma linear theory with the background magnetic field in the z-direction and the wavevector lying in the x-z plane,i.e.

S−n∥2iDn∥n⊥−iDS−n⊥20n∥n⊥0P−n2ExEyEz=0,

where n = ck/ω, and S, D, and P are the usual cold plasma parameters which in the limit applicable to VLF waves in the plasmasphere (Ω_{i} ≪ ω ≪ |Ω_{e}| and ω_{pe} ≫ |Ω_{e}| ) become

S≃ωpe2Ωe21−ωLH2ω2D≃ωpe2Ωe2ΩeωP≃−ωpe2ω2.

In the above equation we have defined ωLH2=(ωpe2/Ωe2)∑ionsωpi2 where ω_{pe,i} are the electron/ion plasma frequencies, and Ω_{e} is the electron cyclotron frequency. Furthermore, if we restrict our analytical consideration to the whistler limit with me/mi<k¯<1 and k¯⊥<1 then the dispersion relation becomes ω¯2≃k¯2k¯2 and the fields may be simply expressed in terms of each other by,

Ez≃k¯∥k¯⊥ExEx≃−ik¯k¯∥EyEy≃Ωeωpek¯∥k¯k¯⊥Bz

where k¯∥,k¯⊥)=(c/ωpe)(k∥,k⊥, ω=Ωeω¯ and k¯2=k¯∥2+k¯⊥2. In all of the subsequent numerical calculations, including ray tracing and evaluation of damping rates, we use the full cold plasma dispersion relation. We use the approximate expressions only for analytical explanation.

[5] For the suprathermal population of electrons the maximum kinetic energy is usually in the KeV range. The electron energy in cyclotron resonance with VLF (Very Low Frequency) waves is in the MeV range [Thorne and Horne, 1994] so that the anti-Hermitian part of the susceptibility tensor [seeStix, 1992] is given by the Landau term (n = 0) only,

where J_{0} are the zeroth order Bessel functions with argument k_{⊥}v_{⊥}/Ω_{e}. The zz term in equation (5) represents the usual Landau term due to the parallel electric field. Recalling that E_{y} is related to B_{z} (see equation (4)) the yy term in equation (5) can be understood as arising from the force on the electron due to the mirror force (μ∇B_{z}) of the wave field, i.e. transit-time damping. For whistler waves these forces are in opposition and act to reduce the collisionless damping.

[6] To evaluate the power dissipated by the suprathermal electrons with energies between 0.1 and 9 KeV we need a model for the distribution function. Li et al. [2010, Figure 5] analyzed THEMIS data and published differential electron flux values as a function of L shell in five different energy channels during moderately disturbed times (100 ≤ AE* ≤ 300 nT). We used this figure to create a simple spatially dependent model of the suprathermal electron distribution function inside the plasmasphere. Specifically, following Bortnik et al. [2011a], we assumed the suprathermal electrons to have a power-law distributionf(v)= A/v^{n}. We read off the largest differential electron flux values from Li et al. [2010, Figure 5] for each of the five energy channels and at five values of L (L = 2.5,3.0,3.5,4,4.5) and converted these to phase space densities. At each L value we performed a fit to a power law distribution to get an estimate of A(L) and n(L). We then fit these data to the functions A = exp(A_{2}L^{2} + A_{1}L + A_{0}) and n = n_{2}L^{2} + n_{1}L + n_{0} to find the parameters A_{2} = 2.9, A_{1} = −10, A_{0} = −4, n_{2} = −0.14, n_{1} = 0.54, and n_{0} = −3.3.

[7] With this distribution function a simplified expression for the power dissipated by the suprathermal electrons may be written in the whistler limit,

where b=k¯⊥k¯ and x = k_{∥}v_{⊥}/ω. For 2 KeV suprathermal particles k_{⊥}ρ_{e} ≪ 1 such that the Bessel functions may be expanded for small argument (though this expansion must be done with care for power-law distributions since the integral becomes divergent). At L = 2.5 the suprathermal index n = −2.825 and the integral inequation (7) may be approximated in the whistler limit as 0.4 b^{0.825}. The minus sign inside the integral is due to the fact that the transit-time damping and Landau damping are acting in opposition to each other. The dimensionless factornAc^{n + 3}/n_{e} in front of the integral is the ratio of the number of suprathermal electrons to the number of cold electrons modified by the steepness of the suprathermal index. This factor is of the order of 10^{−6} at L = 2.5.

[8] For collisional damping the anti-Hermitian part of the susceptibility due to collisions by speciesα on species β was computed from the Vlasov equation with a Landau collision integral in the cold limit by Brambilla [1995] as

where χ_{cold}^{α,β} is the cold plasma susceptibility of species α and β, T_{α,β} = 1/2m_{α,β}v_{thα,β}^{2} is the temperature and thermal velocity, ν^{α/β} = 4πΛ_{c}Z_{α}^{2}Z_{β}^{2}e^{4}n_{β}/(m_{α}^{2}v_{thα}^{3}) ∝ n_{β}T_{α}^{−3/2} is the collision frequency of species α on species β, and Λ_{c} is the Coulomb logarithm which is about 16 (see Huba [2011] for explicit formula). The collision frequency depends inversely on the cold plasma temperature to the 3/2 power, making the collisional dissipation sensitive to temperature. Equation (8) shows that collisions between the same species do not contribute to dissipation. By investigating terms in equation (8)it can be seen that collisions of electrons on Hydrogen contributes the dominant dissipation for typical magnetospheric parameters. For completeness in numerical computations we include dissipation due to collisions between all species, which includes e-, H+, He+, and O+ for the model plasmasphere used in this letter. In the whistler limit we may derive an approximate expression for the power dissipated due to collisions as,

Pcol≃−23π3/2νe/Hk¯4k¯⊥21−2k¯2Bz2

where the electron on hydrogen collision frequency at L = 2.5 is about 6 × 10^{−6}Ω_{e}, for T_{e} = 0.14 eV and n_{H} = 1.6 × 10^{3}/cm^{3}. We can now compare the power dissipated due to collisions to the power dissipated due to collisionless processes in the whistler limit by dividing equation (9) by equation (7) at L = 2.5,

from which we can see that although the collision frequency is almost five orders of magnitude smaller than the electron cyclotron frequency the effective density of suprathermal electrons is six orders of magnitude smaller than the background density of cold electrons, and for the whistler regime (i.e. me/mi<k¯<1 and k¯⊥<1) the quantity in brackets is about 0.5, leaving the collisional damping to be about two times larger. As it turns out this estimate in the whistler regime leads to the smallest ratio of collisional dissipation to suprathermal dissipation. The collisional damping is much larger otherwise.

[9] In Figure 1 we compare the numerically computed power dissipated due to collisional damping divided by the power dissipated due to collisionless processes. For collisional damping we use equations (7) and (1) with a temperature for each species of 0.14 eV and with cold plasma densities computed using the model described below. For collisionless damping we use equations (5) and (1)with the model suprathermal electrons as described above with numerical integrations over the perpendicular velocity space. We compute this ratio at L = 2.5 for parameters in the equatorial plane as a function of the perpendicular and parallel components of the wave-vector. Specifically, the plasma densities are (n_{e}, n_{H}, n_{He}, n_{O}) = (1.6 × 10^{3}, 1.6 × 10^{3}, 0.5, 0)/cm^{3}, the magnetic field is 0.02 G, all species temperatures are 0.14 eV, and the model suprathermal electron parameters are A = 9.7 × 10^{−6} (cm/s)^{n−3}/cm^{3}and n = −2.8. We find that the collisional dissipation is at least two times greater than the collisionless dissipation and is often 10–100 times greater for larger wave-normal angles. Similar figures could be made for different values of density, temperature, and suprathermal electron flux, which would generally affect the magnitude of this ratio but not the dependence on the wave normal angle.

[10] One can calculate the damping rate of waves from the power dissipated simply by dividing by the energy density of the wave, γ = −P/W, where in general,

W=116πB*·B+E*·∂∂ωϵh·E

where ϵ_{h} is the Hermitian part of the dielectric tensor. Since the energy in the wave is independent of the form of dissipation Figure 1 is also the ratio of damping rates.

3. Ray Tracing Results

[11] For our ray tracing results we use the full cold plasma dispersion relation in dipole coordinates to derive the components of the equations,

dxdt=∂G∂k∂G∂ω−1dkdt=−∂G∂x∂G∂ω−1

where Gis the cold plasma dispersion relation and is equal to zero. A standard Runge-Kutta algorithm is used to integrate the ray trajectories. Along with the ray-tracing equations we integrate the change in energy density due to a linear damping mechanism,i.e. dW/dt = 2γW which can be integrated immediately to yield a reduction factor R

R=exp2∫0tγ(k(t′))dt′

We use a dipole magnetic field model and a cold density model as described in Bortnik et al. [2011a]. The density model is based on the diffusive equilibrium of Angerami and Thomas [1964]with a number of field-aligned density structures added followingBortnik et al. [2011a]. We amend the density model with a simple two-temperature model [Crabtree et al., 2012] with an ionospheric temperature of 0.14 eV in all cases (consistent with the parameters used in the diffusive equilibrium model) that transitions near 1100 km altitude to a magnetospheric temperature which we take to be different values. Plasmaspheric temperatures can range from 0.1 eV on the night-side to 0.5 eV on the day-side with variations dependent on solar conditions, latitude, etc. For ionospheric sources of magnetospherically reflecting whistlers [Thorne and Horne, 1994; Bell et al., 2002], such as lightning, we launch waves from 500 km altitude at different latitudes with vertical wave normals.

[12] In Figure 2we show an example ray trajectory for a wave at 3 kHz, injected into the magnetosphere at 40°S along with its trajectory through k-space (demonstrating our choice of wave-vectors used inFigure 1). In Figure 3 we show the R factor (equation (13)) computed in several ways and for three different plasmaspheric temperatures. In the red curves the damping is assumed to be due to the suprathermal electrons only. In the blue curves the damping is assumed to be collisional. And in the black curve the total damping rate is shown. The top panel shows the case where the plasmasphere is cool (night-side) at 0.14 eV and we see that for all times the total damping rate is dominated by the collisional damping alone. Notice that the wave loses almost a third of its power very quickly due to collisional damping in the dense oxygen dominated ionosphere included in the cold density model. The middle panel shows the case where the plasmaspheric temperature is 0.34 eV. In this case after exiting the ionosphere the collisionless damping is slightly larger than the collisional damping up until about 5 seconds. After 5 seconds the wave-normal angle is very large and consistent withFigure 1the collisional damping becomes larger. The bottom panel shows the case where the plasmaspheric temperature is 0.54 eV, where again we find that collisional damping becomes larger at later times when the wave-normal angle is large. InTable 1 we compute the time for a wave packet to lose 6 dB of power for different frequencies, launching latitudes, and magnetospheric temperatures (T_{Mag}). This table should be compared to Bell et al. [2002] where for example a 3 kHz wave launched from 40°S was expected to live for 4.7 seconds before losing 6 dB of power. Our result for collisionless damping alone predicts a lifetime of similar order as Bell et al. [2002].

Table 1. Calculated Time Before Wave Launched From the Ionosphere Has Lost 6 dB of Power

[13] As a final example we consider a ray trajectory launched at L = 6 in the equatorial plane (well outside the plasmapause) in Figure 4, intended to represent chorus waves [Bortnik et al., 2011a]. This wave had a frequency of 1/10th the electron cyclotron frequency or about 400 Hz. The wave-normal angle was −46.15° so that the ray trajectory entered the plasmasphere after about 0.9 seconds. Inside the plasmasphere this wave packet spends more time at higher L-shells and so encounters a greater flux and spectral steepness of suprathermal electrons. Once the wave-packet entered the plasmasphere damping was turned on. Outside the plasmapause the cold densities are relatively small and the suprathermal electron flux is larger so that collisionless damping is expected to dominate. InFigure 5the relative power is plotted as a function of time and as for ionospheric sources we find that for wave-packets inside the plasmapause collisional damping is much larger for the case with a low-temperature plasmasphere. For higher temperature cases the collisional damping becomes important only as the wave-normal angle gets larger.

4. Conclusions

[14] We have calculated both the collisionless and collisional damping of whistler waves propagating in the plasmasphere in the presence of suprathermal electrons and demonstrated that collisional damping can be of the order of or greater than collisionless damping. Recent ray-tracing studies that focus on Landau damping by suprathermal electrons omit collisional damping, howeverKimura [1966] included collisional but not collisionless damping. Figures 3 and 5demonstrate that for accurate modeling both dissipation mechanisms must be included and that the collisional damping is sensitive to plasmasphere temperatures and wave-normal angles of wave packets. Recently,Bortnik et al. [2011b]found from ray-tracing studies of chorus generated hiss that the distribution of wave-normal angles can be bimodal with one peak in wave power at large wave-normal angles. Our results suggest that this peak will experience preferential collisional damping.

[15] For regions in the plasmasphere with low temperature, likely to occur on the night-side, collisional damping can become quite large. Therefore, in order to explain long-lived whistler waves it may be necessary to invoke wave growth due to cyclotron resonant amplification with anisotropic electron distributions [Kennel and Petschek, 1966]. Recently, this type of amplification has been included in studies of chorus generated whistler-mode hiss [Chen et al., 2012] and ionospherically launched whistlers [Ganguli et al., 2012]. Chen et al. [2012]found that to match observed amplitudes of hiss in the plasmasphere amplification was necessary, however they did not include collisional damping. This indicates that perhaps either a larger amplification is necessary or the plasmasphere was relatively warm so that the collision frequency was small. Further modeling and data-model comparisons are necessary to resolve this issue.

[16] Future models of whistler wave dissipation must include both collisional and collisionless dissipation. The total dissipation is sensitive to both the suprathermal electron flux and the electron temperature. In this work we have taken observations of suprathermal electrons that are near the upper range of flux, meaning we may overestimate the collisionless damping. It is important to point out that the population of suprathermal electrons is, to some degree, sporadic, and may be much reduced at certain times and for certain regions of the plasmasphere. Part of the sporadic observations may be due to diffusion of energy into the loss cone of these suprathermal electrons by whistler waves. Measurements of the cold plasmaspheric temperatures are difficult to make because of spacecraft charging issues, and, existing measurements have a large variation and are dependent on magnetospheric conditions. There are global self-consistent physics based models of both density and temperature in the plasmasphere, such as the SAMI3 model (SAMI3 is Also a Model of the Ionosphere), and future studies should use these to evaluate collisional damping of whistler waves.

Acknowledgments

[17] This work is supported by the Naval Research Laboratory Base program.

[18] The Editor thanks Peter Gary and an anonymous reviewer for their assistance in evaluating this paper.