Radiation Belt Radial Diffusion at Earth and Beyond


The year 2019 marks the 60th anniversary of the concept of radial diffusion in magnetospheric research. This makes it one of the oldest research topics in radiation belt science. While first introduced to account for the existence of the Earth’s outer belt, radial diffusion is now applied to the radiation belts of all strongly magnetized planets.

But for all its study and application, radial diffusion remains an elusive process. As the theoretical picture evolved over time, so, too, did the definitions of various related concepts, such as the notion of radial transport. Whether data is scarce or not, doubts in the efficacy of the process remain due to the use of various unchecked assumptions. As a result, quantifying radial diffusion still represents a major challenge to tackle in order to advance our understanding of and ability to model radiation belt dynamics.

The core objective of this review is to address the confusion that emerges from the coexistence of various definitions of radial diffusion, and to highlight the complexity and subtleties of the problem. To contextualize, we provide a historical perspective on radial diffusion research: why and how the concept of radial diffusion was introduced at Earth, how it evolved, and how it was transposed to the radiation belts of the giant planets. Then, we discuss the necessary theoretical tools to unify the evolving image of radial diffusion, describe radiation belt drift dynamics, and carry out contemporary radial diffusion research.

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\(\alpha \) :

local pitch angle

\(\alpha _{eq}\) :

pitch angle at the magnetic equator

\(\boldsymbol{A}\) :

magnetic vector potential

\(A\) :

proportionality coefficient for the asymmetry of the disturbance magnetic field \(\boldsymbol{b}\)

\(\boldsymbol{b}\) :

disturbance magnetic field

\(\mathscr{b}\) :

geocentric stand-off distance to the subsolar point on the magnetopause

\(\boldsymbol{B}\) :

magnetic field

\(\Delta B\) :

asymmetric perturbation of the dipole field, in the model of Fei et al. (2006)

\(B_{E}\), \(B_{P}\) :

magnetic equatorial field at the surface of the Earth (\(E\)) or the planet (\(P\))

\(B_{d}\) :

amplitude of the dipole field

\(B_{m}\) :

magnetic field at the mirror point

\(c\) :

speed of light in vacuum

\(D_{1}, D_{2}, D_{ij}\) :

Fokker–Planck coefficients

\(D_{LL}\) :

radial diffusion coefficient

\(D_{LL,m} \) :

\(D_{LL}\) due to magnetic fluctuations, including the effect of the induced electric fields

\(D_{LL,b} \) :

\(D_{LL}\) due to magnetic fluctuations, in the absence of any kind of electric field

\(D_{LL,e} \) :

\(D_{LL}\) due to electric potential fluctuations

\(D_{LL,\epsilon } \) :

\(D_{LL}\) due to electric field fluctuations, regardless of their nature

\(ds\) :

infinitesimal displacement along a field line

\(dl\) :

infinitesimal displacement along a guiding drift contour (\(\varGamma \))

\(\varepsilon \) :

total energy of the guiding center (kinetic and potential)

\(E_{o}\) :

rest mass energy (511 keV for an electron, 938 MeV for a proton)

\(\boldsymbol{E}\) :

electric field

\(\boldsymbol{E}_{\mathit{ind}}\) :

induced rotational electric field

\(\eta \) :

flux tube content per magnetic flux

\(f, f_{o},F\) :

drift-averaged distribution functions; different notations correspond to different sets of variables: \(f( J_{1}, J_{2}, J_{3},t)\); \(f_{o} (M,J, L,t)\); \(F(M,J, \varPhi ,t)\)

\(\varphi \) :

magnetic local time

\(\varPhi \) :

magnetic flux through a particle drift shell; proportional to \(J_{3}\)

\(\gamma \) :

Lorentz factor

\(\varGamma \) :

guiding drift contour

\(\varGamma ( \alpha _{eq} )\) :

pitch angle factor for \(D_{LL,m}\) (\(\varGamma ( \alpha _{eq} ) = D_{LL,m} / D_{LL,m,eq} \))

\(H\) :

Hamiltonian function

\(I\) :

geometric integral (\(=J/2p\))

\(J\) :

second adiabatic invariant

\(J_{3}\) :

third adiabatic invariant

(\(J_{i}, \varphi _{i} \)):

action-angle variables associated with the \(i\)th quasi-periodic motion (1st: gyration; 2nd: bounce; 3rd: drift)

\(K\) :

adiabatic constant (\(=I \sqrt{B_{m}}\))

\(Kp\) :

3-hour geomagnetic activity index

\(\varLambda \) :

quantity approx. conserved in case of strong pitch angle scattering (\(= p^{3} \oint ds/B\))

\(L\) :

normalized equatorial radial distance

\(L^{*}\) :

Roederer’s parameter (proportional to \(1/\varPhi \))

\(M\) :

first adiabatic invariant

\(m_{o}\) :

particle rest mass

\(N\), \(d \mathcal{N}\) :

number of particles

\(n\) :

particle number density

\(r\) :

radial distance

\(r_{0}\) :

unperturbed equatorial radius of a drift contour

\(\nu \) :

drift frequency (\(=\varOmega /2\pi \))

\(\varOmega \) :

angular drift velocity

\(\boldsymbol{p} \) :

particle momentum

\(\boldsymbol{p}_{{\bot }}, p_{\parallel } \) :

\(\boldsymbol{p}\) components perpendicular (\({\bot }\)) and parallel (∥) to the magnetic field direction

\(P\) :

transition probability—for example from \(J_{3}\) to \(J_{3} + \Delta J_{3}\)

\(P_{X}\) :

power spectrum of the signal \(X\)

\(\varPi \) :


\(q\) :

electric charge of a particle

\(R_{E}\), \(R_{P}\) :

Earth/planetary equatorial radius

\(S\) :

proportionality coefficient for the symmetry of the disturbance magnetic field \(\boldsymbol{b}\)

\(\varSigma \) :

height-integrated Pedersen conductivity

\(\theta \) :

magnetic colatitude

\(t\), \(\Delta t\) :

time, time interval

\(\tau _{C}\) :

characteristic time for the variation of the fields

\(\tau _{G}\) :

gyration period

\(\tau _{B}\) :

bounce period

\(\tau _{D}\) :

drift period

\(T\), \(E\), \(W\) :

kinetic energy of the guiding center

\(U\) :

electrostatic potential

\(\boldsymbol{V}_{\boldsymbol{D}}\) :

bounce-averaged drift velocity

\(V_{L}\) :

\({dL^{*}} / {dt}\): bounce-averaged Lagrangian velocity of the guiding center in \(L^{*}\)

\(\mbox{[ ]}\) :

square brackets = expected value (average value) of the bracketed quantity

\(\langle\ \rangle \) :

angle brackets = average change per unit time of the bracketed quantity


symbol for “approximately equal”


symbol for “directly proportional”


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Contributions  The authors acknowledge the contributions of S.N. Bentley, B. Mauk, A. Osmane and E. Roussos. Sarah N. Bentley helped improve the overall quality of a preliminary manuscript on radial diffusion at Earth. She provided careful proofreading, together with detailed comments and insightful suggestions. Adnane Osmane proofread and commented on a preliminary manuscript on radial diffusion at Earth. He also authored the paragraph entitled, “A brief discussion on the general concept of diffusion in planetary radiation belts,” Sect. 5.3.2. Elias Roussos helped proofread the sections on planetary magnetospheres. Barry Mauk helped improve the overall quality of the manuscript.

S.L. thanks T.P. O’Brien for providing feedback on the paragraphs devoted to space weather.

Both authors thank the anonymous reviewer for their help improving the quality of the manuscript.

General Acknowledgments  S.L. thanks the scientists who, over the years, invited her to give seminars and to discuss radial diffusion within the radiation belt community, in particular J.F. Ripoll and A.Y. Ukhorskiy. She is also particularly grateful to R.B. Horne and the British Antarctic Survey for the visits and the many fruitful discussions. The sum of all these interactions ultimately lead to this ambitious project. S.L. would also like to thank the community of scientists who provided advice regarding the publication of a scientific review, namely N. Ganushkina, M. Liemohn, M. Oka, C.T. Russell, Y. Shprits, and M. Thomsen. She is grateful to F.S. Mozer and J.G. Roederer for their continuous support and encouragement.

Funding  The work of S.L. was performed under JHU/APL Contract No. 922613 (RBSP-EFW) and NASA Grant Award 80NSSC18K1223. P.K. was partially supported by the NASA Office of Space Science under task order 003 of contract NAS5-97271 between NASA/GSFC and JHU.

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Appendix: Derivation for the Instantaneous Rate of Change of the Third Adiabatic Invariant

Appendix: Derivation for the Instantaneous Rate of Change of the Third Adiabatic Invariant

In this section, we present two different ways to derive the analytic formulation of \({d L^{*}} / {dt}\) that was used in Sect. 5.2. Both proofs provide complementary physical insights on the process at play. The results are then reformulated in more compact ways.

A.1 Theoretical Framework and Working Hypotheses

In the following proofs, it is assumed that:

  • the frozen-in condition applies;

  • all three adiabatic invariants of the population are well-defined and meaningful (no open drift shells, and the Lamor radius is small compared to field gradients, etc.);

  • the first two adiabatic invariants are conserved;

  • the characteristic time for the variation of the field, \(\tau _{C}\), is very long in comparison with the bounce period of the population considered \(\tau _{B}\), and very short in comparison with the drift period \(\tau _{D}\):

    $$\begin{aligned} \tau _{G} \ll \tau _{B} \ll \tau _{C} \ll \tau _{D} \end{aligned}$$

    where \(\tau _{G}, \tau _{B}, \tau _{D}\) are respectively the gyration, bounce, and drift periods of the particle considered, and \(\tau _{C}\) is the characteristic time for the variation of the field.

We use an infinitesimal time step, \(dt\), adapted to this ordering:

$$\begin{aligned} \tau _{G} \ll \tau _{B} \ll dt\approx \tau _{C} \ll \tau _{D} \end{aligned}$$

so that we can track the bounce-averaged drift motions of the particle guiding centers.

In a time-varying field, the guiding center drift velocity \(\boldsymbol{V}_{\boldsymbol{D}}\) is:

$$\begin{aligned} \boldsymbol{V}_{\boldsymbol{D}} = \frac{\boldsymbol{B}}{qB^{2}} \times \biggl( -q \boldsymbol{E} + \frac{m}{2B} \bigl( v_{\bot }^{2} +2 v_{\parallel }^{2} \bigr) \boldsymbol{\nabla }_{{\bot }} B+m \frac{d \boldsymbol{V}_{\boldsymbol{D}}}{dt} \biggr) \end{aligned}$$

where \(m\) is the mass of the particle, \({q}\) is the electric charge, and \(v_{\bot }\) and \(v_{\parallel }\) correspond to the particle velocities perpendicular and parallel to the magnetic field direction (e.g., Roederer 1970).

The order of magnitude of the inertia term (last term in the brackets of Eq. (A.3)) is very small:

$$\begin{aligned} \frac{ \vert \frac{m \boldsymbol{B}}{qB^{2}} \times \frac{d \boldsymbol{V}_{\boldsymbol{D}}}{dt} \vert }{ \vert \boldsymbol{V}_{\boldsymbol{D}} \vert } = \biggl\vert \frac{m}{qB} \biggr\vert \cdot \frac{ \vert \frac{d V_{D}}{dt} \vert }{ \vert V_{D} \vert } = \frac{\tau _{G}}{\tau _{C}} \ll 1 \end{aligned}$$

Thus, the inertia term is omitted and the drift velocity is equal to its bounce-averaged expression at the magnetic equator for every time step:

$$\begin{aligned} \boldsymbol{V}_{\boldsymbol{D}} = \frac{2p \boldsymbol{\nabla }_{\boldsymbol{o}} I\times \boldsymbol{e}_{\boldsymbol{o}}}{q \tau _{B} B_{o}} + \frac{\boldsymbol{E}_{\boldsymbol{o}} \times \boldsymbol{e}_{\boldsymbol{o}}}{B_{o}} \end{aligned}$$

where \(p\) is the particle momentum, \(\boldsymbol{e}_{\boldsymbol{o}} = \boldsymbol{B}_{\boldsymbol{o}} / B_{o}\), \(\boldsymbol{B}_{ \boldsymbol{o}}\) is the magnetic field at the magnetic equator (minimum \(B\) surface), \(\boldsymbol{E}_{\boldsymbol{o}}\) is the equatorial electric field (with both induced and electrostatic components), \(I= \int _{s _{m}}^{s_{m} '} \sqrt{1-B(s)/ B_{m}} ds\) is the integral function of \(B_{m}\) between the mirror points \(s_{m}\) and \(s_{m} '\), and \(\boldsymbol{\nabla }_{\boldsymbol{o}} I\) is the equatorial gradient of the quantity, \(I\), determined at constant magnetic field intensity, \(B_{m}\), at the mirror points (e.g., Roederer 1970).

Finally, all variations will be expressed as first-order approximations in \(dt\), and the total rate of change of the third invariant during \(dt\) will be merged with its instantaneous rate of change:

$$\begin{aligned} d L^{*} = \biggl( \frac{d L^{*}}{dt} \biggr) dt \end{aligned}$$

The objective is to compute the rate of change of the magnetic flux encompassed by the drift contour of an equatorial particle in a time-varying magnetic field, in the absence of electrostatic fields.

A.2 Proof #1

Let us track the drift motion of an equatorial particle trapped in a magnetic field. At time, \(t\), the three adiabatic invariants are \((M, J=0, L^{*} )\), and the particle’s guiding center is located at \(\boldsymbol{r}_{\boldsymbol{o}}\) along its drift contour \(\varGamma (r _{o} )\). The magnetic field changes during an infinitesimal time step, \(dt\). Due to the magnetic field variation and the resulting induced electric fields, the drift velocity is altered, and the guiding center moves away from its initial drift contour. At \(t+dt\), the guiding center is located at \(\boldsymbol{r}_{\boldsymbol{o}} +\boldsymbol{d r}_{ \boldsymbol{o}}\). The equatorial magnetic field intensity along the new drift contour \(\varGamma (r_{o} +d r_{o} ) \) is a constant equal to \(B( \boldsymbol{r}_{\boldsymbol{o}} + \boldsymbol{d r}_{\boldsymbol{o}},t+dt)\).

The objective of this demonstration is to quantify the difference, \(d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t )\), between the magnetic flux, \(\varPhi ( \boldsymbol{r}_{\boldsymbol{o}} +\boldsymbol{d r}_{\boldsymbol{o}}, t+dt )\), encompassed by the drift contour, \(\varGamma (r_{o} +d r_{o} )\), at time, \(t+dt\), and the magnetic flux, \(\varPhi ( \boldsymbol{r}_{\boldsymbol{o}}, t )\), encompassed by the drift contour, \(\varGamma (r_{o} )\), at time, \(t\).

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) =&\varPhi ( \boldsymbol{r}_{\boldsymbol{o}} +\boldsymbol{d r}_{\boldsymbol{o}}, t+dt ) - \varPhi ( \boldsymbol{r}_{\boldsymbol{o}}, t ) \\ =& \iint _{S ( r_{o} +d r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} \end{aligned}$$

where \(S ( r_{o} +d r_{o} )\) indicates the area encompassed by \(\varGamma (r_{o} +d r_{o} ) \) at time, \(t+dt\), and \(S ( r_{o} )\) indicates the area encompassed by \(\varGamma (r_{o} )\) at time, \(t\). They are represented in Fig. 14.

Fig. 14

Representation of the drift contours, \(\varGamma (r_{o} )\), at time, \(t\) (dark purple line), and, \(\varGamma (r_{o} +d r_{o} )\), at time, \(t+dt\) (dark red line), and the associated integrating surface areas, \(S(r_{o} )\), at time, \(t\) (purple area), and, \(S(r_{o} +d r_{o} ) \), at time, \(t+dt\) (red area)

By adding and subtracting the quantity \(\iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS}\) to Eq. (A.7), the variation of the magnetic flux associated with the guiding center initially located at \(\boldsymbol{r}_{\boldsymbol{o}}\) can be interpreted as the sum of a spatial contribution and a temporal contribution:

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) = &\biggl( \iint _{S ( r_{o} +d r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} \biggr) \\ &{} + \biggl( \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} \biggr) \end{aligned}$$

The spatial contribution is:

$$\begin{aligned} d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \iint _{S ( r_{o} +d r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} \end{aligned}$$

It corresponds to the magnetic flux at time, \(t+dt\), through the strip, \(A(r_{o} )\), between \(\varGamma (r_{o} )\) and \(\varGamma (r_{o} +d r_{o} )\). The strip is represented in green in Fig. 15.

Fig. 15

Definition of the integrating surfaces: the strip \(A(r_{o} )\) is in green, and the initial integrating surface area, \(S(r_{o} )\), is in blue. The width of the strip, \(A(r_{o} )\), starting from a location, \(\boldsymbol{r}\), along \(\varGamma (r_{o} )\) is \(dh( \boldsymbol{r}_{o}, \boldsymbol{r} )\)

The temporal contribution is:

$$\begin{aligned} d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} \end{aligned}$$

This contribution corresponds to the variation of the magnetic field through the initial integrating surface area \(S(r_{o} )\). It results that:

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) =d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) +d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) \end{aligned}$$

Let us quantify each component individually.

For the spatial component:

$$\begin{aligned} d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \iint _{A ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} = \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \bigl( \boldsymbol{dh}(\boldsymbol{r}_{\boldsymbol{o}},\boldsymbol{r})\times \boldsymbol{dl} \bigr) \end{aligned}$$

For all points along \(\varGamma (r_{o} )\), the width of the strip, \(dh( \boldsymbol{r}_{\boldsymbol{o}}, \boldsymbol{r} )\), is such that

$$\begin{aligned} B ( \boldsymbol{r},t+dt ) - \bigl\vert \nabla B ( \boldsymbol{r},t+dt ) \bigr\vert dh( \boldsymbol{r}_{\boldsymbol{o}}, \boldsymbol{r} )= B( \boldsymbol{r}_{\boldsymbol{o}} + \boldsymbol{d r}_{\boldsymbol{o}},t+dt) \end{aligned}$$

In addition, for all points along \(\varGamma (r_{o} ),B ( \boldsymbol{r},t ) =B ( \boldsymbol{r}_{\boldsymbol{o}},t )\).

Thus, we have:

$$\begin{aligned} B ( \boldsymbol{r},t+dt ) =B ( \boldsymbol{r}_{\boldsymbol{o}},t ) + \frac{\partial B}{\partial t} ( \boldsymbol{r},t ) dt \end{aligned}$$

As a result, for all points \(\boldsymbol{r}\) along \(\varGamma (r_{o} )\)

$$\begin{aligned} dh( \boldsymbol{r}_{\boldsymbol{o}}, \boldsymbol{r} )= \frac{dt}{ \vert \nabla B ( \boldsymbol{r},t+dt ) \vert } \biggl( \frac{\partial B}{\partial t} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) \end{aligned}$$

Consequently, the spatial component is, to the first order in \(dt\):

$$\begin{aligned} d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) =dt \oint _{\varGamma ( r_{o} )} \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \cdot \biggl( \frac{\partial B}{\partial t} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) dl \end{aligned}$$

For the temporal contribution, one can write that:

$$\begin{aligned} d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) =& \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} \\ =&dt \iint _{S ( r_{o} )} \frac{\partial \boldsymbol{B} ( \boldsymbol{r},t )}{\partial t} \cdot \boldsymbol{dS} \end{aligned}$$

Thus, using the integral form of the Maxwell-Faraday equation:

$$\begin{aligned} d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) =-dt \oint _{\varGamma ( r_{o} )} \boldsymbol{E}_{\mathit{ind}} ( \boldsymbol{r},t) \cdot d \boldsymbol{l} \end{aligned}$$

In addition, the projection of the electric field vector, \(\boldsymbol{E}_{\mathit{ind}}\), onto the local direction of the initial guiding drift contour is related to the drift velocity, \(\boldsymbol{V}_{\boldsymbol{D}} = -M \boldsymbol{\nabla } B\times \boldsymbol{B} /\gamma q B^{2} + \boldsymbol{E}_{\mathit{ind}} \times \boldsymbol{B} / B^{2}\), by the relation:

$$\begin{aligned} \boldsymbol{E}_{\mathit{ind}} ( \boldsymbol{r},t ) \cdot d \boldsymbol{l}=- \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) \cdot \boldsymbol{\nabla } B ( \boldsymbol{r},t ) dl \end{aligned}$$


$$\begin{aligned} d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) =dt \oint _{\varGamma ( r_{o} )} \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \boldsymbol{V}_{\boldsymbol{D}} \cdot \boldsymbol{\nabla } B ( \boldsymbol{r},t ) dl \end{aligned}$$

Finally, let us note that for all points along \(\varGamma (r_{o} )\)

$$\begin{aligned} \frac{dB}{dt} ( \boldsymbol{r},t ) = \frac{\partial B}{\partial t} ( \boldsymbol{r},t ) + \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) \cdot \boldsymbol{\nabla } B ( \boldsymbol{r},t ) \end{aligned}$$

As a result, the sum of the spatial and temporal contributions to the variation of the magnetic flux is

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) =&d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) +d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) \\ =&dt \oint _{\varGamma ( r_{o} )} \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \biggl( \frac{dB}{dt} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) dl \end{aligned}$$


$$\begin{aligned} \frac{d\varPhi }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \oint _{\varGamma ( r_{o} )} \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \biggl( \frac{dB}{dt} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) dl \end{aligned}$$


$$\begin{aligned} \frac{dL^{*}}{L^{*2}} = \frac{d\varPhi }{2\pi B_{E} R_{E}^{2}} \end{aligned}$$

we obtain

$$\begin{aligned} \frac{dL^{*}}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \frac{L^{*2}}{2\pi B_{E} R_{E}^{2}} \oint _{\varGamma ( r_{o} )} \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \biggl( \frac{dB}{dt} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) dl \end{aligned}$$

A.3 Proof #2

The second proof consists of tracking the drift motions over all guiding center locations along the same drift contour, \(\varGamma ( r_{o} )\). All guiding centers have initially the same three adiabatic invariants (\(M, J=0, L^{*}\)), but they have different drift phases at the time of the perturbation.

This second proof relies on the fact that the magnetic flux, through a closed curve moving at \(( \boldsymbol{E}_{\mathit{ind}} \times \boldsymbol{B} ) / B^{2}\) is conserved, which is what we will demonstrate as a first step.

A.3.1 Conservation of the Magnetic Flux Through a Closed Curve Moving at \(( \boldsymbol{E}_{\mathit{ind}} \times \boldsymbol{B} ) / B^{2}\)

Let us consider at time, \(t+dt\), the closed curve, \(\tilde{\varGamma }\), formed by all the new guiding center locations (see also Fig. 16).

Fig. 16

Definition of the closed curve, \(\tilde{\varGamma }\), formed by all the new guiding center locations. Because the equatorial magnetic field intensity along \(\tilde{\varGamma }\) is not necessarily constant, \(\tilde{\varGamma }\) is not necessarily a drift contour. Yet, because \(( \boldsymbol{E}_{\mathit{ind}} \times \boldsymbol{B} ) / B^{2}\) is flux-preserving, the flux encompassed by \(\tilde{\varGamma }\) is equal to the initial magnetic flux of the population considered

Because the equatorial magnetic field intensity along \(\tilde{\varGamma }\) is not necessarily constant, \(\tilde{\varGamma }\) is not necessarily a drift contour. Yet, it is interesting to note that the magnetic flux, \(\tilde{\varPhi }\), encompassed by \(\tilde{\varGamma }\) is equal to the initial magnetic flux through \(\varGamma ( r_{o} )\). Indeed:

$$\begin{aligned} \tilde{\varPhi } (t+dt)= \iint _{S( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot d \boldsymbol{S} + \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \bigl( \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) dt\times d \boldsymbol{l} \bigr) \end{aligned}$$


$$\begin{aligned} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \bigl( \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) \times d \boldsymbol{l} \bigr) =& \bigl( \boldsymbol{B} ( \boldsymbol{r},t+dt ) \times \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) \bigr) \cdot d \boldsymbol{l} \\ =& \boldsymbol{E}_{\mathit{ind}} ( \boldsymbol{r},t ) \cdot d \boldsymbol{l} \end{aligned}$$

it results that

$$\begin{aligned} \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \bigl( \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) dt\times d \boldsymbol{l} \bigr) =dt \oint _{\varGamma ( r_{o} )} \boldsymbol{E}_{\mathit{ind}} ( \boldsymbol{r},t ) \cdot d \boldsymbol{l} \end{aligned}$$

Using the integral form of the Maxwell-Faraday equation:

$$\begin{aligned} dt \oint _{\varGamma ( r_{o} )} \boldsymbol{E}_{\mathit{ind}} ( \boldsymbol{r},t ) \cdot d \boldsymbol{l} =&-dt \iint _{S ( r_{o} )} \frac{\partial \boldsymbol{B} ( \boldsymbol{r},t )}{\partial t} \cdot \boldsymbol{dS} \\ =& \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} \end{aligned}$$


$$\begin{aligned} \tilde{\varPhi } (t+dt) =& \iint _{S( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot d \boldsymbol{S} \\ &{}+ \biggl( \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} \biggr) \end{aligned}$$

We conclude that for all guiding center locations, \(\boldsymbol{r} _{\boldsymbol{o}}\), initially along \(\varGamma ( r_{o} )\):

$$\begin{aligned} \varPhi ( \boldsymbol{r}_{\boldsymbol{o}}, t ) = \tilde{\varPhi } (t+dt) \end{aligned}$$

In other words, the drift contour distorts to conserve the magnetic flux. This is due to the fact that \(( \boldsymbol{E}_{ \mathit{ind}} \times \boldsymbol{B} ) / B^{2}\) is flux-preserving (Newcomb 1958).

A.3.2 Reformulation for the variation of the magnetic flux

We reformulate the variation of the magnetic flux (Eq. (A.7)), using the fact that the magnetic flux encompassed by the closed curve \(\tilde{\varGamma }\) at \(t+dt \) is equal to the initial flux (Eq. (A.31)) (see also Fig. 17)

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) =\varPhi ( \boldsymbol{r}_{\boldsymbol{o}} +\boldsymbol{d r}_{\boldsymbol{o}}, t+dt ) - \varPhi ( \boldsymbol{r}_{\boldsymbol{o}}, t ) =\varPhi ( \boldsymbol{r}_{\boldsymbol{o}} +\boldsymbol{d r}_{\boldsymbol{o}}, t+dt ) - \tilde{\varPhi } (t+dt) \end{aligned}$$
Fig. 17

Representation of the variation of the magnetic flux as the difference between the magnetic flux encompassed by the drift contour, \(\varGamma ( r_{o} +d r_{o} )\), at \(t+dt\) and the magnetic flux encompassed by the distorted contour \(\tilde{\varGamma }\)

Combining Eqs. (A.9) and (A.26), we have

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) =d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) - \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \bigl( \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) dt\times d \boldsymbol{l} \bigr) \end{aligned}$$

From Eq. (A.12), we obtain that the variation of the magnetic flux is, to the first order in \(dt\)

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \bigl( \bigl( \boldsymbol{dh}(\boldsymbol{r}_{\boldsymbol{o}},\boldsymbol{r})- \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) dt \bigr) \times \boldsymbol{dl} \bigr) \end{aligned}$$

This expression is also:

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \bigl( \bigl( \boldsymbol{dh}(\boldsymbol{r}_{\boldsymbol{o}},\boldsymbol{r})- \boldsymbol{dh}(\boldsymbol{r},\boldsymbol{r}) \bigr) \times \boldsymbol{dl} \bigr) \end{aligned}$$

Using Eq. (A.15), this result is equivalent to Eq. (A.25). A geometric definition for the variation of the magnetic flux according to Eq. (A.35) is represented in Fig. 18.

Fig. 18

Geometric interpretation of the variation of the magnetic flux

A.4 Reformulation in Terms of Deviation from the Average

Noticing that the drift velocity of a guiding center trapped in a magnetic field in stationary conditions in the absence of electric fields is:

$$\begin{aligned} \boldsymbol{V}_{\boldsymbol{D},\boldsymbol{s}} ( \boldsymbol{r},t ) =- \frac{M}{\gamma q} \frac{\nabla B ( \boldsymbol{r},t ) \times \boldsymbol{e}_{\boldsymbol{o}}}{B ( \boldsymbol{r},t )} \end{aligned}$$

and introducing the infinitesimal time step spent along the drift contour, \(d\tau \), such that

$$\begin{aligned} \vert d\tau \vert = \frac{dl}{ \vert \boldsymbol{V}_{\boldsymbol{D},\boldsymbol{s}} ( \boldsymbol{r},t ) \vert } \end{aligned}$$

Equation (A.25) becomes:

$$\begin{aligned} \frac{d\varPhi }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \int _{0}^{\tau _{D}} \frac{M}{\gamma q} \biggl( \frac{dB}{dt} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) d\tau \end{aligned}$$

Let us introduce the linear operator \([\ ]_{D}\) to denote the spatial drift average along the guiding drift contour, \(\varGamma \). It is defined by

$$\begin{aligned}{} [ f ]_{D} ( t ) = \frac{1}{\tau _{D}} \int _{0}^{\tau _{D}} f\bigl( \boldsymbol{r} ( \tau ),t \bigr)d\tau \end{aligned}$$

This operation determines the spatial average of the quantity, \(f\), along the drift contour, \(\varGamma \), weighted by the time spent drifting through each location under stationary conditions.


$$\begin{aligned} \frac{d\varPhi }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \frac{\tau _{D}}{q} \biggl( \biggl[ \frac{M}{\gamma } \frac{dB}{dt} \biggr]_{D} ( t ) - \frac{M}{\gamma } \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) \end{aligned}$$

In the case of an equatorial guiding center trapped in a magnetic field in the absence of electrostatic fields

$$\begin{aligned} \frac{M}{\gamma } \frac{dB}{dt} = \frac{d\varepsilon }{dt} \end{aligned}$$

where \(\varepsilon \) is the total energy of the guiding center. Thus, we obtain that

$$\begin{aligned} \frac{d\varPhi }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \frac{\tau _{D}}{q} \biggl( \biggl[ \frac{d\varepsilon }{dt} \biggr]_{D} ( t ) - \frac{d\varepsilon }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) \biggr) \end{aligned}$$

This expression is identical to the one derived by Northrop (1963). It is valid in the most general case (e.g., Cary and Brizard 2009; Lejosne et al. 2012; Lejosne 2013). As a result,

$$\begin{aligned} \frac{dL^{*}}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \frac{L^{*2}}{q\varOmega B_{E} R_{E}^{2}} \biggl( \biggl[ \frac{d\varepsilon }{dt} \biggr]_{D} ( t ) - \frac{d\varepsilon }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) \biggr) \end{aligned}$$

where \(\varOmega = {2\pi } / {\tau _{D}}\) is the population drift frequency.

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Lejosne, S., Kollmann, P. Radiation Belt Radial Diffusion at Earth and Beyond. Space Sci Rev 216, 19 (2020). https://doi.org/10.1007/s11214-020-0642-6

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  • Radiation belts
  • Radial diffusion
  • Drift
  • Particle acceleration
  • Adiabatic invariants
  • Earth
  • Jupiter
  • Saturn