Estimating quasi-linear diffusion coefficients for varying values of density ratio

Abstract

This paper considers a method for estimating bounce-averaged quasi-linear diffusion coefficients due to whistler-mode waves for a specified ratio of plasma frequency to gyrofrequency, , using values precomputed for a different value of that ratio. This approach was recently introduced to facilitate calculations associated with the “POES technique,” generalized to infer both wave intensity and cold plasma density from measurements of particle fluxes near the loss cone. The original derivation was justified on the basis of parallel-propagating waves but applied to calculations with much more general models of the waves. Here, we justify the estimates, which are based on equating resonant frequencies for differing values of and energy, for wide ranges of wave normal angle, resonance number, energy, and equatorial pitch angle. Refinements of the original estimates are obtained and tested numerically against full calculations of the diffusion coefficients for representative wave models. The estimated diffusion coefficients can be calculated rapidly and generally give useful estimates for energies in the 30-keV–300-keV range, especially when both relevant values of the ratio are large.

1 Introduction

Cyclotron-resonant wave–particle interactions are a crucial aspect in magnetospheric dynamics, especially in radiation belts, and there is a vast tradition of simulating the process as quasi-linear diffusion of phase space density by a broad-band spectrum of small, incoherent waves, following the pioneering work of Kennel and Engelmann (1966) and Lerche (1968), with further development by Lyons et al. (1971) and Lyons (1974a). Modern formulations were given by Horne et al. (2005), Glauert and Horne (2005), and Albert (2005).

Known wave parameters can be used to calculate quasi-linear diffusion coefficients, which help determine decay lifetimes and levels of trapped and precipitating particle fluxes. Conversely, measurements of particle flux can be used to estimate the wave intensity, using a process often referred to as the “POES technique” (Li et al., 2013; Ni et al., 2014), which exploits the ability of low-altitude POES satellites to resolve the loss cone. Essentially, diffusion coefficients are calculated and used to estimate the ratio of trapped to precipitating flux, using assumed values for parameters including the wave amplitude, wave frequency, wave normal angle distributions, and cold plasma density; this is repeated for different wave amplitudes until an agreement with measurements is obtained. The wave amplitude is a specifically convenient, as well as important, parameter to vary because the diffusion coefficients simply scale with .

Longley et al. (2022) recently generalized this procedure to treat cold plasma density as an additional free parameter, using the additional constraint that the wave amplitude calculated from two POES energy channels remains the same. In principle, this requires extensive, time-consuming recalculations of diffusion coefficients for different values of the ratio of electron plasma frequency to gyrofrequency, , referred to here as the density ratio. Repeated, rapid evaluation or approximation of diffusion coefficients is also required in many other settings, such as diffusion simulations using event-specific wave parameters (Watt et al., 2021; Yu et al., 2022) or when used to modify test particle trajectories to account for subscale wave–particle interactions in the evolving conditions of global MHD simulations (Chan et al., 2023; Michael et al., 2024).

One expedient for rapidly estimating diffusion coefficients is the “mean value approximation,” which replaces integration over the wave normal angle with evaluations at a few carefully chosen values (Albert, 2007b; 2008). Another approach, introduced by Longley et al. (2022), estimates diffusion coefficients for particle energy and pitch angle and a chosen value of from a table of values at a reference density value , for which the calculations are already done. This was done by finding adjusted values and that equate an approximate expression for the resonant frequency for the two sets of parameters, so thatIn Section 2, we consider the approximations underlying Equation 1. Longley et al. (2022) only considered parallel-propagating waves (for which only the primary resonance contributes to diffusion) and characterized the validity of Equation 1 in terms of particle energy. In this study, constraints are developed in terms of wave frequency, justifying its use for a wide range of wave normal angle, particle energy, and resonant harmonic number. In Section 3, Equation 1 is used to estimate the diffusion coefficients, e.g.,obtaining differing relativistic modifications for , , and and depending on whether or . Finally, in Section 4, we numerically compare calculated and estimated diffusion coefficients, using two very simple chorus wave models, for several representative combinations of and .

2 Resonant frequencies

The condition for resonance between a gyrating charged particle and a plane wave iswhere the particle has parallel velocity , relativistic factor , and gyrofrequency (here, is the sign of the particle charge, with all frequencies unsigned), and the wave has a frequency and parallel wave number . The “harmonic number” can be any integer; indicates Landau resonance. After squaring, Equation 3 becomesThe left-hand side is the inverse-squared index of refraction of the wave, and the particle pitch angle and wave normal angle are given by and , respectively. Albert (1999, 2004, 2005, 2007a) noted the usefulness of this form for both analysis and approximation.

2.1 Full cold plasma refractive index

For completeness, we formulate the full cold plasma refractive index for R-mode (whistler) waves aswhere is the sign of (typically negative), and the standard plasma quantities , , and for a single-component plasma (Stix, 1992) can be written as

Here, is the electron plasma frequency , and the mass ratio is .

To determine the resonant frequency as a function of , Horne et al. (2005) and Glauert and Horne (2005) expanded these expressions into a order polynomial in (or order in if ) and retained only the valid real roots. Ions heavier than are ignored here, although treated by Albert (2003) for electromagnetic ion cyclotron (EMIC) waves. Albert (2005) and references therein developed criteria to isolate the roots of Equation 4 within restricted intervals of , allowing the efficient use of real-valued, one-dimensional root finding and frequently reducing or even eliminating the consideration of irrelevant ranges of and . Figure 1 shows the quantities on both sides of Equation 4 as functions of at fixed , as well as various approximations of each, as discussed below.

FIGURE 1

2.2 Quasi-longitudinal approximation

From Equation 5, the quasi-longitudinal approximation neglects the terms proportional to and , and further considers and . The well-known result iswhich is illustrated in Figure 1. With , Equation 7 is the same as Equation A13 of Longley et al. (2022). Using this in the resonance condition, Equation 4, gives a cubic equation in , namely,with

With (for which only contributes to diffusion), these coefficients reduce to Equation 17 of Longley et al. (2022), who noted that entered only in the expression for , which would be left unchanged by compensating changes in . Indeed, changing also affects and , but this effect was declared minor in the “weakly relativistic limit” such that is approximately unchanged. By implication, is also approximately unchanged, and further justification is provided by small values of . To this extent, the resonant value of depends on mainly through the combination , or the square of , where .

With , Equations 8, 9 reduce to the quadratic equation

If , the bracketed expression in Equation 10 is , regardless of energy, which suggests holding constant, as in the case . Alternatively, the approximation applies if either is small or is small (of course neither can exceed 1), which motivates considering constant. The two prescriptions coincide if both and are small (since then and are approximately equal). Thus, holding constant rather than constant is favored only if both and . Since the resonance condition for is , this case implies , which is an atypically low value, as illustrated in Figure 1.

2.3 High-density approximation

Lyons (1974b) and Albert (1999) used “high-density approximation,” which amounts to neglecting the leading term 1 relative to the terms proportional to in the definitions of , , and . Then, Equation 5 reduces towhich is illustrated in Figure 1. A similar form applies to L-mode EMIC waves. Equation 11 should be accurate for a wider range of and than Equation 7, although for a narrower range of . Indeed, it was introduced by Lyons and Thorne (1970) to study the magnetospheric reflection of whistler waves, which involves passing through . The full, quasi-longitudinal (QL), and high-density (HD) versions of the refractive index were compared for several combinations of and in Figure 4 of Albert (2005), where it was also shown analytically (in Appendix A) that . Combining with the resonance condition gives a order polynomial in , but Albert (1999) showed that it yields at most three roots in the whistler range.

Setting in Equation 11 (which requires to be large compared to the lower hybrid frequency, ) yields

which was used by Albert (2017) in a treatment of highly oblique whistler waves. It can also be regarded as a simplified version of quasi-longitudinal approximation. Combining it with the resonance condition yields a cubic equation of the form of Equation 8 with

This set of coefficients agrees exactly with Equation 9 for and , and approximately for and if . Changing and adjusting to maintain the value of again modifies and .

2.4 Approximate resonance condition

Lyons et al. (1972) and Albert (1994) considered in the resonance condition (Equation 4) for to obtain

and further approximated the refractive index (Equation 12) as

Both of these approximations are illustrated in Figure 1. At the risk of discarding two potential “anomalous” resonances with (Albert, 2005), they lead to the explicit expressions

However, combining Equation 14 with Equation 11, instead of the cruder approximation Equation 15, replaces Equation 16 with a quadratic equation for (for ), whose coefficients still depend on only through the combination . For , Equation 11 combined with the full resonance condition (Equation 4) gives a quadratic equation for whose coefficients depend on only through the combination .

2.5 Compensating changes in R and energy

The various expressions in the preceding sections all suggest that the resonant frequency can be approximated as , without restrictions on , if and are chosen such that

The scaling for is consistent with Equations 23, 25 (although not the reversed notation in Equations 24, 26) of Longley et al. (2022).

Equation 17 might further suggest that both the local pitch angle and (or ) be considered free variables that are chosen to satisfy Equation 1. However, this would introduce undesirable dependence on latitude in the relationship between the corresponding values of the equatorial pitch angle, . For example, the choice in Equation 17 gives or, converting from the local pitch angle to the equatorial pitch angle ,

The dependence on in Equation 18 is not compatible with the use of a table of bounce-averaged diffusion coefficients evaluated on a grid of values, as intended by Longley et al. (2022). Similarly, the approach hinges on being able to choose or in a manner independent of and , which are also integrated or summed over in the calculation of diffusion coefficients.

Thus, with and using the relations , where is the rest energy , Equation 17 yields

Longley et al. (2022) used this expression with to estimate

For , the implicit restriction in Equation 19 that cannot be too large for a given value of just reflects the requirement in Equation 17 that . Figure 2 shows as a function of for two different choices of and . It is evident that with , the value is less than and may require the table based on to extend to lower energy than the actual range of interest. Similarly, if , the table may be required to extend to quite high energy, and, for , Equation 19 for may become singular.

FIGURE 2

3 Diffusion coefficients

Even when can be adjusted to yield the same resonant frequency for two different values of , it does not follow that the quasi-linear coefficients are the same since they are not functions of alone. However, approximate values of the corresponding values of can be related simply, as explained herein.

Using the expressions and notation of Albert (2005) and Albert (2007b) (which are exactly equivalent to those in Glauert and Horne (2005)), the local pitch angle diffusion coefficient iswhere has dimensions of , andFurthermore, from Lyons (1974a),Note that in Equation 22 does not depend on , , or , but depends on through . Using Equation 12 gives (Albert, 2017), so in this approximation.

To characterize , substituting Equation 12 in Equation 6 of Albert (2005) yields

so the factor of involving is typically . Similarly,

The factor is given in Equation 9 of Lyons (1974b). Using Equation 12, the only dependence on is through the argument of the Bessel functions, expressed by Albert (1999) as . This is zero for parallel-propagating waves, and the dependence on is otherwise neglected for both and . Thus, for , and for .

The overall scaling of Equation 21 is therefore,

Thus, if and are chosen in accordance with Equation 17, the corresponding values of will be approximately related by

which refines Equation 20. Similarly, using Equations 23, 25, taking , converting from local pitch angle to equatorial pitch angle , and bounce averaging (which does not change any of the scaling) extend Equation 27 to

Mourenas and Ripoll (2012) and Artemyev et al. (2013), relying on the refractive index of Equation 15 and approximations of , also obtained detailed analytical estimates for the total bounce-averaged pitch angle diffusion coefficient. With determined by Equation 19 and with , the scalings are consistent with those obtained here.

4 Numerical tests

Bounce-averaged quasi-linear diffusion coefficients were calculated using the simple chorus wave models of Glauert and Horne (2005) and Horne et al. (2005), namely,

The parameter values used in Equation 29 are and = at . The “equatorial” model restricts the latitude to , while the “mid-latitude” model differs only in taking . Cold plasma density was taken as a constant along dipole magnetic field lines, with chosen representative values of . For these comparisons, only the range to 5 was included and, for concreteness, the wave amplitude was set to pT. For each wave model and value, diffusion coefficients were calculated for 80 (logarithmically spaced) values of between 100 eV and 10 MeV and 89 values of between and .

The top panel of Figure 3 shows values of vs. energy, at a fixed equatorial pitch angle , for the equatorial model. Values calculated for are shown by dashed curves, values calculated for are shown by solid curves, and symbols show values for estimated using the values calculated for . Results for and are shown separately, in black and magenta, respectively. Estimates based on Equation 20, denoted , are indicated by asterisks, while estimates based on Equation 28, denoted , are marked by diamonds. It is seen that the estimates are qualitatively close to the calculated values for much of the energy range, although less so for than for , and that the factors of in Equation 28 noticeably improve the agreement above a few hundred keV. Note that, from Figure 2 (left), below 1 keV would require below 100 eV, outside the range of the calculated table (100 eV–10 MeV).

FIGURE 3

The comparison is made clearer in the bottom panel of Figure 3, which shows the ratios of estimated to calculated values. For , especially with the factors of included, the agreement is quite good between 30 keV and 300 keV, which is the range covered by the POES satellite and considered by Longley et al. (2022). However, the estimates are much less reliable for energy outside that range, giving underestimates at low and very high energy and overestimates in a broad range at approximately 1 MeV. Cases for which estimated values are zero (due to lack of resonances) but calculated values are nonzero are shown by crosses at the bottom of the panel, and crosses at the top of the panel indicate nonzero estimates where calculations result in zero.

The results of conversely using values calculated for to estimate values for are shown in Figure 4. For this combination of and , the estimated values of are within a factor of approximately 10 for energy below approximately 2 MeV, with underestimates below 1 MeV and overestimates above 1 MeV. From Figure 2 (right), values of are outside the range of the calculated table for approximately MeV for and keV for .

FIGURE 4

Figures 5, 6 repeat Figures 3, 4 using the mid-latitude chorus model. With and , the estimates for are within a factor of 10 in the range 100 keV to 2 MeV (if the relativistic factors are included), but they are lower at lower energy and higher at higher energy. For , the estimates are with a factor of 10 between 20 keV and roughly 150 keV but predict nonzero values up to approximately 600 keV, while the full calculations indicate they should be zero. With and , the estimates for are fairly reliable between 10 keV and 2 MeV but fail above 2 MeV because (as shown in Figure 2) they require values beyond the computed table. For , the estimates are fairly good between 2 keV and 10 keV but fail above 10 keV because, again, Equation 19 requires values of beyond the calculated table.

FIGURE 5

FIGURE 6

Figure 7 shows equatorial pitch angle diffusion coefficients for the equatorial chorus model as a function of both energy and pitch angle. Calculated values of , for , as shown in the plots along the main diagonal of the figure, while the ratios of estimated-to-calculated values, , are shown otherwise. For the values of and used, it can be seen that the ratios often fall between 0.10 and 10 but frequently exceed that range. Naturally, and are closest when and are closest. The estimates are closer when and are both larger than when they are both small. Generally, is more likely to be an underestimate of (blue regions) when and an overestimate (red regions) when . The best agreement occurs roughly at approximately 100 keV at a low pitch angle and a few hundred keV for a larger pitch angle, with deteriorating agreement for energy above or below those values. Finally, similar trends are seen in Figure 8 for the mid-latitude chorus model.

FIGURE 7

FIGURE 8

5 Summary

This paper explored a method for quickly and easily estimating bounce-averaged quasi-linear diffusion coefficients for one value of “density ratio” using an existing set of values computed for a different value of . It was introduced by Longley et al. (2022) for the specific application of generalizing the “POES technique” to infer both wave intensity and cold plasma density from measurements of particle flux near the loss cone. In principle, that the procedure could be done without such approximations, although it would require repeated, time-consuming calculation of diffusion coefficients.

Although the original derivation was justified on the basis of parallel-propagating waves (wave normal angle ), for which only resonances with contribute, the results were applied to diffusion coefficient calculations with much more general models of the waves. Here, we have justified the estimate of by , (Equation 2), based on equating the resonant frequency (Equation 1), for wide ranges of , , and the equatorial pitch angle , drawing on both the quasi-longitudinal and high-density approximations of the whistler-mode refractive index. Modifications accounting for dependence of on energy beyond that captured by the dependence on were obtained (Equation 28) and found to improve the agreement of the estimates with full calculations using two simple, idealized wave models.

The resulting agreement is far from perfect, however, especially when and differ greatly. For the tests done, the estimates were typically within a factor of 10 for energy between 10s of keV and 1 MeV for resonances with but frequently worse for greater energy. For diffusion driven by Landau resonance, , the estimates for were reliable only over limited ranges of energy below 100 keV. These results depend on the wave models: Landau resonance typically occurs near the particle mirror point, so it might be expected that values for the “equatorial” wave model, restricted to below latitude, are difficult to estimate.

We note that all parameters specifying the wave frequency and wave normal angle distributions have been held fixed throughout. Sensitivity to these parameters was considered by Albert (2012). Furthermore, the quasi-linear diffusion paradigm itself has well-known caveats and limitations, e.g., Allanson et al. (2024). For use with the POES technique, which has additional sources of inaccuracy and uncertainty, the estimates here are probably acceptable. If nothing else, the wave amplitude and density values obtained can be used in more refined calculations. More generally, the tradeoff of accuracy for computational convenience should be carefully assessed for each intended application. One possible avenue toward improvement is to precompute several tables of diffusion coefficients, with different values of , and use the approach described here to scale from the nearest match to required values. This is identical to the interpolation in but in a manner that respects the (approximate) analytical form of the underlying expressions.

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