The coupling of quasi-linear pitch angle and energy diffusion

Albert, J. M.

doi:10.1016/j.jastp.2008.11.014

Cited by 19 publications

(26 citation statements)

References 20 publications

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“…We further assume that the evolution of the trapped electron distribution function F can be described by a Fokker‐Planck diffusion equation with quasi‐linear bounce‐averaged isotropic energy and pitch angle diffusion coefficients [ Horne et al , ]:

\frac{∂F}{∂t} = \frac{\partial}{∂E} ((), A (E) (〈〉, D_{EE}) \frac{\partial}{∂E} \frac{F}{A (E)}) - \frac{F}{τ_{L}}

with

A (E) = (E + m_{e} c^{2}) \sqrt{(E + 2 m_{e} c^{2}) E}

. Although mixed diffusion can be important for particle energization [e.g., Albert , ], we omit this effect to consider solutions proposed by Balikhin et al [] for an initially cold distribution without high‐energy electrons. The approximation of an initially cold electron distribution is supported by observations of high‐energy electron evacuation from the outer radiation belt region at the beginning of storms [ Turner et al , ; Baker et al , ].…”

Section: Electron Lifetimes and Energization As A Function Of Parameterssupporting

confidence: 82%

Storm‐induced energization of radiation belt electrons: Effect of wave obliquity

Artemyev

Agapitov

Mourenas

et al. 2013

Geophysical Research Letters

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[1] New Cluster statistics allow us to determine for the first time the variations of both the obliquity and intensity of lower-band chorus waves as functions of latitude and geomagnetic activity near L 5. The portion of wave power in very oblique waves decreases during highly disturbed periods, consistent with increased Landau damping by inward-penetrating suprathermal electrons. Simple analytical considerations as well as full numerical calculations of quasi-linear diffusion rates demonstrate that early-time electron acceleration occurs in a regime of loss-limited energization. In this regime, the average wave obliquity plays a critical role in mitigating lifetime reduction as wave intensity increases with geomagnetic activity, suggesting that much larger energization levels should be reached during the early recovery phase of storms than during quiet time or moderate disturbances, the latter corresponding to stronger losses. These new effects should be included in realistic radiation belt simulations. Lett., 40,[4138][4139][4140][4141][4142][4143]

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\frac{∂F}{∂t} = \frac{\partial}{∂E} ((), A (E) (〈〉, D_{EE}) \frac{\partial}{∂E} \frac{F}{A (E)}) - \frac{F}{τ_{L}}

with

A (E) = (E + m_{e} c^{2}) \sqrt{(E + 2 m_{e} c^{2}) E}

Section: Electron Lifetimes and Energization As A Function Of Parameterssupporting

confidence: 82%

Storm‐induced energization of radiation belt electrons: Effect of wave obliquity

Artemyev

Agapitov

Mourenas

et al. 2013

Geophysical Research Letters

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“…Neglecting mixed diffusion (which can be important too) [see Albert , ] and assuming an initially cold distribution without high‐energy electrons (for instance, just after dropouts) [e.g., see Turner et al , , and references therein], the energy broadening of the electron distribution F ( E , t ) in the presence of quasi‐linear energy diffusion by quasi‐parallel whistler mode waves is given approximately by [ Horne et al , ; Balikhin et al , ]

\frac{∂F}{∂t} = \frac{\partial}{∂E} (A (E) 〈 D_{EE} 〉 \frac{\partial (F / A (E))}{∂E}) - \frac{F}{τ_{L}}

where

A (E) \sim (E + 0.511) \sqrt{E (E + 1)}

with E in MeV.…”

Section: Analytical Expressions Of the Trapped Electron Distributionmentioning

confidence: 99%

“…However, a more complete diffusion equation must take into account both pitch angle and energy diffusion coefficients [ Glauert and Horne , ; Albert , ], giving for the full electron distribution f :

\begin{matrix} alignleft \\ align-1 \frac{∂f}{∂t} & align-2 = \frac{\partial}{∂E} (A (E) 〈 D_{EE} 〉 \frac{\partial (f / A (E))}{∂E}) \\ align-1 & align-2 + \frac{1}{T_{b} \sin 2 α_{0}} \frac{\partial}{\partial α_{0}} (〈 D_{αα} 〉 T_{b} \sin 2 α_{0} \frac{∂f}{\partial α_{0}}) \end{matrix}

…”

Section: Analytical Expressions Of the Trapped Electron Distributionmentioning

confidence: 99%

Approximate analytical solutions for the trapped electron distribution due to quasi‐linear diffusion by whistler mode waves

et al. 2014

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The distribution of trapped energetic electrons inside the Earth's radiation belts is the focus of intense studies aiming at better describing the evolution of the space environment in the presence of various disturbances induced by the solar wind or by an enhanced lightning activity. Such studies are usually performed by means of comparisons with full numerical simulations solving the Fokker‐Planck quasi‐linear diffusion equation for the particle distribution function. Here we present for the first time approximate but realistic analytical solutions for the electron distribution, which are shown to be in good agreement with exact numerical solutions in situations where resonant scattering of energetic electrons by whistler mode hiss, lightning‐generated or chorus waves, is the dominant process. Quiet time distributions are well recovered, as well as the evolution of energized relativistic electron distributions during disturbed geomagnetic conditions. It is further shown that careful comparisons between the analytical solutions and measured distributions may allow to infer important bounce‐ and drift‐averaged wave characteristics (such as wave amplitude). It could also help to improve the global understanding of underlying physical phenomena.

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“…However, cross diffusion, which expresses the physical relationship between resonant changes in α 0 and p , can have significant consequences, since typically > > D pp . Thus, it is preferable to retain it despite the numerical difficulties it presents to straightforward finite differencing in ( α 0 , E , L ) [ Albert , 2004, 2009]. These difficulties may be overcome in a number of ways [ Albert and Young , 2005; Tao et al , 2008, 2009; Xiao et al , 2009].…”

Section: Diffusion Equationmentioning

confidence: 99%

Three‐dimensional diffusion simulation of outer radiation belt electrons during the 9 October 1990 magnetic storm

2009

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[1] Relativistic (>1 MeV) electron flux increases in the Earth's radiation belts are significantly underestimated by models that only include transport and loss processes, suggesting that some additional acceleration process is required. Here we use a new, three-dimensional code that includes radial diffusion and quasi-linear pitch angle and energy diffusion due to chorus waves, including cross terms, to simulate the 9 October 1990 magnetic storm. The diffusion coefficients are activity dependent, and time-dependent boundary conditions are imposed on all six boundary faces, taken from fits to CRRES Medium Electrons A electron data. Although the main phase dropout is not fully captured, the persistent phase space density peaks observed during the recovery phase are well explained, but this requires both chorus wave acceleration and radial diffusion.

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The coupling of quasi-linear pitch angle and energy diffusion

Cited by 19 publications

References 20 publications

Storm‐induced energization of radiation belt electrons: Effect of wave obliquity

Storm‐induced energization of radiation belt electrons: Effect of wave obliquity

Approximate analytical solutions for the trapped electron distribution due to quasi‐linear diffusion by whistler mode waves

Three‐dimensional diffusion simulation of outer radiation belt electrons during the 9 October 1990 magnetic storm

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