Oblique Whistler-Mode Waves in the Earth’s Inner Magnetosphere: Energy Distribution, Origins, and Role in Radiation Belt Dynamics

Abstract

In this paper we review recent spacecraft observations of oblique whistler-mode waves in the Earth’s inner magnetosphere as well as the various consequences of the presence of such waves for electron scattering and acceleration. In particular, we survey the statistics of occurrences and intensity of oblique chorus waves in the region of the outer radiation belt, comprised between the plasmapause and geostationary orbit, and discuss how their actual distribution may be explained by a combination of linear and non-linear generation, propagation, and damping processes. We further examine how such oblique wave populations can be included into both quasi-linear diffusion models and fully nonlinear models of wave-particle interaction. On this basis, we demonstrate that varying amounts of oblique waves can significantly change the rates of particle scattering, acceleration, and precipitation into the atmosphere during quiet times as well as in the course of a storm. Finally, we discuss possible generation mechanisms for such oblique waves in the radiation belts. We demonstrate that oblique whistler-mode chorus waves can be considered as an important ingredient of the radiation belt system and can play a key role in many aspects of wave-particle resonant interactions.

Appendix: Analytical Estimates of Trapped Electron Lifetimes

Let us consider first quasi-parallel waves. In this case, it has been shown (e.g., see Mourenas et al. 2014b) that when $p\varOmega_{pe0}\omega^{1/2}/\varOmega_{ce0}^{3/2} > 1.5$ (a condition roughly equivalent to $E \geq 100~\mbox{keV}$ for $L \sim 5$), trapped electron lifetimes $\tau_{L}$ can be estimated as

$$ \tau_{L,\theta< 45^{\circ}}~[\mbox{s}] \approx \frac{{220[\text{pT}^{2}\cdot\text{s}^{2}/\text{rad}] }}{{B_{w}^{2}}} \frac{{p^{14/9} \gamma \omega _{m}^{7/9} \varOmega _{pe0}^{14/9} }}{{\varOmega _{ce0}^{12/9} }} $$

(75)

where from now on bounce-averaged RMS wave amplitude $B_{w}$ is in pT, angular frequencies (plasma frequency $\varOmega_{pe}$, electron cyclotron frequency $\varOmega_{ce}$, and the mean frequency of the wave ensemble $\omega_{m}$) are in rad/s, and $p=(\gamma^{2}-1)^{1/2}$ in the normalized momentum. At lower energy (typically for $E < 100~\mbox{keV}$), previous analytical estimates of pitch-angle diffusion rates (Mourenas et al. 2012b) must be sensibly modified, especially as concerns the effective latitudinal width $\Delta\lambda_{R}$ of the domain where resonance exists. Such a modification stems from the progressive disappearance of cyclotron resonance at the lowest frequencies as electron energy diminishes, which decreases $\Delta\lambda_{R}$. Let us assume an upper frequency cutoff for lower-band chorus waves at $\omega_{UC} \sim \omega_{m}+\delta\omega \sim \varOmega_{ce0}/2$, where $\omega_{m}$ and $\delta\omega$ are the mean value and half-width of the gaussian frequency distribution. The latitude of resonance $\lambda_{R, \omega}$ at a given frequency $\omega$ can be estimated by combining first cyclotron resonance and first adiabatic invariant (Artemyev et al. 2013b): it is given by Eq. (27) for $\omega = \omega_{m}$. The minimum frequency such that first cyclotron resonance is possible can be written approximately as $\omega_{MIN} \sim \max(\omega_{LC}, \varOmega_{ce0}/(p^{2}\omega_{pe0}^{2}/\varOmega_{ce0}^{2} + 2\gamma))$ in the useful range $\omega_{MIN} \sim (0.15-0.4)\varOmega_{ce0}$, using $\omega_{LC} \simeq \omega_{m}-\delta\omega$. Further accounting for a possible confinement of the waves below a certain latitude $\lambda^{+}$, it leads to a rough estimate of $\Delta\lambda_{R}$:

$$ \Delta\lambda_{R} \sim \max\bigl(\min\bigl(\lambda^{+}, \lambda_{R,\omega_{UC}}\bigr) - \lambda_{R,\omega_{MIN}}, 0\bigr) $$

(76)

superseding the formula provided in the work by Artemyev et al. (2013b), which was valid mostly for $E \geq 100~\mbox{keV}$. It gives finally ratio of trapped electron lifetimes calculated for energies $E$ and $E_{0}$ given in MeVs

$$ \frac{{\tau_{L,\theta< 45^{\circ}}(E)}}{{\tau_{L,\theta< 45^{\circ}}(E_{0})}} \simeq \frac{{((1 + 2E)^{2} -1)^{7/9} (1 + 2E) \Delta\lambda_{R}(E_{0}) }}{{((1 + 2E_{0})^{2} -1)^{7/9} (1 + 2E_{0}) \Delta\lambda_{R}(E) }} $$

(77)

over the range $E \approx 5\mbox{--}5000~\mbox{keV}$ where cyclotron resonance exists at small $\alpha_{0}$ in the outer radiation belt.

Let us turn now to very oblique lower-band chorus waves propagating near the resonance cone angle. In this case, one important factor is the level of Landau damping due to 100–500 eV suprathermal electrons: it should impose an upper-bound $N_{\max}$ on the value of the wave refractive index $N$ expected to increase with latitude (see Sect. 4.2 and Horne and Sazhin 1990; Mourenas et al. 2014b; Li et al. 2014a). Accordingly, assuming a low level of wave obliqueness at low latitudes $|\lambda| < 15^{\circ}$ and a much higher level of wave obliqueness at higher latitudes (as often observed on board Cluster, see Artemyev et al. 2013b; Mourenas et al. 2014b; Agapitov et al. 2015a), electron lifetimes will be mainly determined by the inverse of the pitch-angle scattering rate at low equatorial pitch-angle $\alpha_{0} < 30^{\circ}$, since strong scattering of electrons by very oblique waves occurs mainly for $\sin\alpha_{0} < 2\omega/\varOmega_{ce0}$ (Mourenas et al. 2012b). One important factor is that many cyclotron $n$-resonances can contribute to scattering in the case of very oblique waves as compared with parallel waves for which only the fundamental resonance exists at low $\alpha_{0} \ll 90^{\circ}- \theta_{\max}$ (see estimates of the number $\mathcal {N}_{res}$ of contributing resonances in Sect. 2, and Mourenas et al. 2012b).

Rough analytical estimates of $\tau_{L}$ in the presence of very oblique chorus waves have been derived by Mourenas et al. (2012b, 2014b):

$$ \tau_{L,\theta>60^{\circ}}~[\mbox{s}] \approx \frac{{40~[\text{pT}^{2}\cdot\text{s}^{2}/\text{rad}] \gamma p \varOmega_{pe0} }}{{B_{w}^{2}}}, $$

(78)

which are valid typically for $E \geq 100~\mbox{keV}$, provided that $N_{\max} \approx N_{limit}$ (see Sects. 2 and 4.2) and for a scaling $N_{\max} \propto \varOmega_{pe}/\omega$ as used in previous papers (e.g., Mourenas et al. 2014b). For a different scaling of $N_{\max}$, the preceding expression would need to be modified.

Moreover, at low energy, one must take into account three important facts: (i) at low enough energy, there will remain only two or three cyclotron resonances, (ii) the $|n| = 1$ resonance contribution to scattering, proportional to $J_{0}^{2} \sim 1$, as well as the $|n| = 2$ resonance contribution proportional to $J_{1}(x)^{2} \approx 2(1+0.81/2)/(\pi x)$, are both larger by a factor $\sim 3/2$ than the asymptotic value $\simeq 2/(\max(1,|n|)\pi x)$ assumed for commodity in our previous estimates, and (iii) the relative width $\Delta x/x$ of the peak of $J_{|n|-1}^{2}(x)$ is also larger at the smallest $|n|$, varying approximately like $1/\max(1,(|n|-1)^{1/2})$. A larger width $\Delta x/x$ corresponds to a larger range of strong wave-particle coupling $\Delta\lambda$ ($\delta\omega$ or $\Delta\theta$) for fixed frequency (latitude), likely leading to a stronger scattering (Mourenas et al. 2012b).

Thus, for low enough electron energy such that the number of contributing resonances for oblique waves $\mathcal{N}_{res,Obl} < 4$ at $\alpha_{0} \approx 7^{\circ}-20^{\circ}$, the scattering rate should be multiplied by a factor $\approx (3/2)\cdot 2 \sim 3$, reducing lifetimes by a similar amount (assuming that wave obliqueness is very small at low latitudes). This finally gives a rough multiplicative factor $C_{LowE}$ to expression (78) for $\tau_{L,\theta>60^{\circ}}$ with

$$ C_{LowE} \approx \frac{{50 \varOmega_{pe0}\omega_{m}^{1/3}}}{{N_{\max} \varOmega_{ce0}^{4/3}}} \frac{{1}}{{1 + 2 \min(1, C^{5}) }} $$

(79)

where $C=60(\varOmega_{ce0}/\omega_{m})^{2/3}/(pN_{\max})$ and we used $\mathcal{N}_{res,Obl}$ at $\alpha_{0} \approx 10^{\circ}$ while $N_{\max}$ should be evaluated at $\lambda \sim 30^{\circ}\mbox{--}35^{\circ}$ where oblique wave coupling with particles is then stronger (Artemyev et al. 2013b; Mourenas et al. 2014b; Li et al. 2014a). Lifetime’s variation with $E$ is consequently given approximately by

$$ \frac{{\tau_{L,\theta>60^{\circ}}(E)}}{{\tau_{L,\theta>60^{\circ}}(E_{0})}} \simeq \frac{{\sqrt{(1 + 2E)^{2} -1}(1 + 2E) C_{LowE}(E) }}{{\sqrt{(1 + 2E_{0})^{2} -1}(1 + 2E_{0}) C_{LowE}(E_{0}) }} $$

(80)

over the range $E \approx 5\mbox{--}5000~\mbox{keV}$ in the outer radiation belt.