Solar and geomagnetic activity modulation of wave–particle interactions in Earth’s radiation belts

Abstract

Solar forcing and geomagnetic disturbances reshape radiation-belt electron populations through a chain of coupled processes that cannot be represented adequately by a single activity index. This study introduces an activity-conditioned resonance kernel (ACRK) that links solar-wind driving, magnetic compression, cold-plasma density, and wave spectra to pitch-angle and energy diffusion. The formulation distinguishes rapid event-scale modulation from slower background recovery, yields a bounded diffusion closure, and provides a sufficient condition for the sign of local energization. Controlled numerical experiments demonstrate threshold-like changes in resonance access and lower trajectory mismatch than static-spectrum and event-driven quasi-linear models. The study is deliberately based on reproducible synthetic experiments rather than a new satellite-data reanalysis. The proposed framework clarifies the conditions under which geomagnetic activity favors local acceleration, scattering, or mixed responses in Earth’s outer radiation-belt.

1 Introduction

Earth’s radiation belts constitute an open, continuously driven particle system. During a geomagnetic disturbance, the electron phase-space density can increase, decrease abruptly, or be redistributed across energy and pitch angle on time scales ranging from minutes to days. Such variability is important both for spacecraft reliability and for understanding the fundamental physics of magnetized plasmas. Observations from the Van Allen Probes era established that relativistic electrons do not respond passively to a single global activity index [1]. Instead, wave–particle interactions mediate a substantial portion of the local exchange between electromagnetic fluctuations and trapped particles [2]. Radiation-belt variability is, therefore, more appropriately treated as a multi-scale response problem than as a linear proxy problem [3]. For example, lower-band chorus can accelerate electrons across a broad energy range [4], whereas related wave environments can also promote loss through pitch-angle scattering [5].

The direction of a wave-driven response is not determined by wave amplitude alone. Resonance also depends on particle energy, equatorial pitch-angle, wave normal angle, background magnetic field, and cold-plasma density. During density depletions, chorus-driven gyroresonance can become especially effective [6]; conversely, electromagnetic ion cyclotron waves can introduce non-resonant scattering channels that are absent from a chorus-only description [7]. A useful model must, therefore, retain the principal pathways through which the local state modifies resonance while remaining sufficiently compact so that each parameter retains a clear physical interpretation.

Recent studies have made this coupling chain increasingly explicit. Lightning-generated whistlers illustrate how propagation history modifies the wave field sampled by trapped electrons [8]. Event-driven diffusion estimates can incorporate contemporaneous wave and plasma observations [9]. Dipolarization-associated precipitation similarly demonstrates that local injections and loss processes must be considered together [10]. Density structure governs magnetosonic-wave propagation and absorption [11], whereas interplanetary shocks can rapidly intensify electron cyclotron harmonic waves by altering the source electron population [12]. Taken together, these results indicate that the relevant physical state is a coupled configuration, not a list of independently acting drivers.

Earlier physics-based frameworks previously modeled multi-dimensional radiation-belt transport, including the Salammbô code [13] and the later Versatile Electron Radiation-Belt (VERB) family with radial, pitch-angle, energy, and mixed diffusion [14–16]. The ACRK is, therefore, not presented as the first radiation-belt transport framework. Its narrower contribution is a transparent conditioning layer that links driver-memory and local resonance accessibility to the spectral weights entering an otherwise familiar diffusion closure.

A second challenge is process attribution. Causal analyses indicate that local acceleration cannot be inferred from flux trends alone [17]. Ultra-low-frequency wave-power may covary with changes in relativistic electron flux while participating primarily through indirect transport channels [18]. Solar-wind variability also organizes the magnetospheric preconditioning that determines the response to later impulses [20]. Information-theoretic analyses further motivate conditional, rather than unconditional, driver–response relations [19, 21]. Recent storm studies show that intense chorus can influence both depletion and recovery [22], whereas global chorus morphology varies systematically with geomagnetic activity and the ratio [23]. These findings caution against assigning a unique physical cause to a flux change without resolving the local conditions that govern resonance access.

A concise closure is useful only if it makes the conditioning explicit. Here, conditioning means that a solar-wind driver is first translated into a slowly evolving local state and is then evaluated against the instantaneous wave spectrum and particle coordinates. This distinction prevents the misleading shortcut of treating an activity index as a direct source of diffusion. An index may summarize part of the upstream forcing, but it does not by itself determine whether a wave packet is present, whether its wave-normal distribution is favorable, or whether the density-dependent resonance window intersects a given particle population. The modeling objective is not to compress all storm-time dynamics into a single scalar. Instead, it is to retain a small set of state variables that explain why comparable activity levels can yield different outcomes.

The present work develops a theory-oriented closure for this conditionality. It does not claim that a single scalar can summarize every magnetospheric state. Instead, it introduces an activity-conditioned resonance kernel, termed ACRK, that separates driver history from the local resonance operator. The first contribution is a bounded kernel construction that incorporates density-dependent resonance access. The second contribution is a sufficient sign criterion for net local energization after averaging over a finite wave spectrum. The third contribution is a reproducible numerical protocol that isolates the roles of memory, wave-update cadence, and density uncertainty. These contributions are intended to make the assumptions behind a conditional diffusion model explicit and inspectable. Event-specific wave representations are preferable to statistical averages when storm-time conditions are strongly non-stationary [24], while physics-driven reanalysis studies motivate dedicated uncertainty and assimilation tests [25].

The numerical layer is deliberately kept subordinate to the radiation-belt physics: the same grid, boundary treatment, and spectral inventory are used across all model comparisons, and no unrelated optimization mechanism is invoked to explain the physical response. The remainder of the paper is organized as follows. Section 2 defines the physical state and the ACRK. Section 3 establishes its principal properties. Section 4 describes the controlled numerical design. Sections 5 and 6 present the numerical results and discuss the physical scope. Section 7 concludes and identifies future research directions.

Analysis.Table 1 defines the notation before any model reduction is introduced. The distinction between and is essential. The first is a compact conditional coordinate. The second retains physical drivers. This separation prevents the activity index from being mistaken for a direct physical forcing term. It also makes subsequent sensitivity tests interpretable.

Analysis.Figure 1 emphasizes ordering, not geometry. A solar-wind impulse can affect particles through compression, injection, density, and several wave modes. The colored trajectories represent distinct particle cohorts. The diagram therefore avoids implying a single linear causal pathway. It motivates the two-stage construction developed below: a driver-memory map first, followed by a local resonance operator.

FIGURE 1

2 Activity-conditioned physical formulation

2.1 Driver state and memory

Let the forcing vector bewhere each component is scaled using a robust event-window median and an interquartile range. Equation 1 defines the reduced forcing vector used in the controlled experiments, which does not imply that other interplanetary magnetic field (IMF) components are dynamically irrelevant. For applications that resolve dawn–dusk or hemispheric asymmetry, the state can be augmented with and ; IMF is especially relevant to asymmetric convection and magnetotail structure [26]. Likewise, a short memory auroral electrojet (AE) index or SuperMAG lower (SML) index can be added when substorm forcing is to be resolved explicitly [27]. In the present synthetic experiments, these optional components are held at their normalized reference values so that the minimum pressure——activity pathway can be isolated. A driver becomes informative only after its memory is specified. We, therefore, use exponentially filtered components (Equation 2)where is selected according to the relevant physical process. Dynamic pressure is assigned a short memory, whereas ring-current recovery is assigned a longer one. The activity coordinate is then (Equation 3)where . This form permits non-linear preconditioning while keeping bounded. It does not assume that the driver–response map is instantaneous. In particular, the interaction terms allow the effect of one driver to depend on the recent state established by another, without requiring a separate empirical rule for every possible storm sequence.

The memory map is deliberately separated from the resonance calculation. Filtered drivers are not interpreted as direct diffusion coefficients. Instead, they supply an activity coordinate that modulates a local spectral interaction only when the wave and particle states permit it. This separation also clarifies the role of time scale. A sharp pressure enhancement may change the activity coordinate promptly, whereas the density profile and wave distribution can evolve on different clocks. The ACRK can, therefore, represent an immediate change in accessibility together with a delayed recovery, without forcing both behaviors into one empirical response curve.

Figure 1 summarizes the logic. Solar-wind forcing modifies magnetopause compression, injections, plasma density, and wave generation. These variables determine whether a particle reaches a resonance window. The model isolates this path rather than attributing the final flux to a single driver.

Internal and nightside sources are separated from the upstream forcing vector rather than neglected. Plasmaspheric and ionospheric plasma influences the cold density and the wave environment, carries part of the slowly varying ring-current state, and the explicit source term in Equation 8 represents resolved particle injections. A full event model should, therefore, assimilate these quantities independently; the controlled experiment keeps them prescribed so that changes caused by the resonance kernel can be identified.

2.2 Resonance accessibility

For a wave mode , define the cold-plasma dispersion residual (Equation 4)The conventional resonance condition is . Physical spectra, however, possess finite bandwidth and finite coherence. We, therefore, replace the strict equality with an accessibility weight (Equation 5)where is a dimensionless dispersion factor. The quantity lies in [0, 1] and can be interpreted as a smooth resonance gate rather than a binary switch. Its density dependence is essential because the mapping from wave frequency to resonant energy varies with . The gate can, therefore, change even when the nominal wave-power envelope remains unchanged.

The proposed ACRK isThe term represents sustained activity, whereas the term captures impulsive modulation. Both factors are clipped below zero in the implementation to prevent signed modulation from reversing the non-negative spectral weight. The kernel does not introduce a new wave mode. Instead, it adjusts the effective contribution of existing spectral elements according to physically identified state variables. This is important for interpretability: a change in the kernel can be traced to resonance accessibility, spectral power, persistent activity, or the rate of activity change.

For numerical use, the clipping operation is applied only to the modulation factors, not to the underlying dispersion residual or the wave spectrum. Consequently, an inaccessible state remains inaccessible because of the resonance gate, and a state with no wave-power retains zero wave-driven diffusion. This distinction is useful in sensitivity experiments: density, wave-normal geometry, and activity history can be varied independently while preserving the physical requirement that finite diffusion must be supported by a finite wave field.

Figure 2 illustrates this behavior for a representative chorus-like mode. The ridge moves inward with activity in this controlled example because compression and density evolution shift the accessible band. The contour map is a model diagnostic. It is not a satellite occurrence distribution.

FIGURE 2

Analysis.Figure 2 displays a smooth resonance landscape instead of a binary storm label. Strong activity enlarges the effective kernel while shifting its maximum. The white contours make the threshold movement visible. In a static-spectrum model, this shift is absent. The figure, therefore, clarifies why a fixed activity coefficient can miss both onset timing and the location of maximum diffusion.

2.3 Transport closure

Let . The local diffusion tensor is written as Equation 7where contains the mode-specific quasi-linear (QL) geometry and includes chorus, hiss, magnetosonic, electromagnetic ion cyclotron (EMIC), and electron cyclotron harmonic (ECH) waves when appropriate. The spectral integral retains the familiar quasi-linear structure while allowing the local weight to respond to a physically labeled state. The associated bounce-averaged equation iswhere is the phase space Jacobian, parameterizes magnetopause loss, and represents resolved injections. Equation 8 retains the standard transport structure. The novelty lies in the conditional construction of : transport coefficients respond to the state that controls resonance rather than to an activity proxy alone. In this formulation, the kernel modifies local diffusion but does not replace the treatment of radial transport, boundary loss, or particle injection.

PROOF. The activity factors in (Equation 6) are bounded after clipping. Thus, for a constant , . The right-hand side is integrable on . Each matrix integrand is a non-negative scalar multiplied by a positive semidefinite matrix. Its integral is, therefore, positive semidefinite. Summing a finite set of finite positive semidefinite matrices proves the statement. □

Remark. The ACRK changes the weighting of spectral interactions, not the underlying conservation law. Its practical value is that driver history, density, and spectral access remain visible in the same expression. This makes the closure easier to diagnose than a black-box surrogate. Boundedness also prevents a numerical storm pulse from generating unphysical diffusion spikes.

3 Theoretical properties of the resonance kernel

3.1 Separation of fast and slow modulation

The activity state can be decomposed as , where is Lipschitz and contains finite-duration impulses. Define (Equation 9)where retains , retains and , and is the product term. This decomposition is useful because storm recovery and shock-time compression need not share the same time constant.

PROOF. Expanding (Equation 6) in leaves cross terms proportional to and . The boundedness of the kernel factors and bounds each cross term by . Integration and the triangle inequality yield (Equation 10).

The bound has a direct physical interpretation. A fast impulse can be simulated separately only when the slow background is not simultaneously extreme. When both contributions are large, a coupled calculation is required because the cross term is no longer negligible. This provides a formal reason to avoid taking a quiet-time diffusion coefficient and simply multiplying it by Kp. The issue is not merely one of amplitude: a rapid perturbation can move the resonance window while the slowly varying state determines whether that movement has an appreciable transport consequence.

3.2 A sign criterion for local energization

For a compact energy interval , define the energization functional (Equation 11)where is a positive analysis weight. Positive indicates that the diffusive contribution is directed toward higher energy within the selected window. The functional does not replace a full phase space calculation; instead, it supplies a compact sign diagnostic that can be evaluated alongside the transport solution. By retaining the mixed diffusion term, it also makes explicit why an apparent energy gain need not imply that pitch-angle loss is negligible.

PROOF. The mixed term in Equation 11 is bounded below by . The energy term is bounded below by . The modeled magnetopause loss contributes at most after normalization by the positive weight. Inequality (Equation 12), therefore, makes the sum positive. Integrating over a compact positive-measure domain preserves the sign.

Figure 3 shows how the condition changes during a controlled disturbance. The modeled activity pulse raises the effective frequency and reduces the critical pitch-angle. The shaded band is not universal. It is a diagnostic interval selected for the numerical experiment.

FIGURE 3

Analysis.Figure 3 compares the driver pulse with two resonance diagnostics. The activity peak changes the normalized wave frequency and shifts the critical pitch-angle toward the accessible band. This joint movement is more informative than either trace alone. It explains why similar wave amplitudes can produce different particle responses when density and driver history differ.

3.3 Numerical stability of the discrete solver

The solver uses a finite-volume discretization of Equation 8. Let denote the time step and the smallest grid spacing. An implicit diffusion update is applied to the symmetric part of , while injection and loss terms are advanced with a positivity-preserving split step. This arrangement separates the stiffness associated with diffusion from the explicitly interpretable source and loss terms. The following statement captures the essential discrete behavior.

PROOF. Under standard no-flux or absorbing boundary discretizations, the backward Euler diffusion matrix is an M-matrix, and its inverse is non-negative. The loss update multiplies each cell by , and the source update adds a non-negative value. The composition of these three positivity-preserving maps remains positivity-preserving.

Remark. The theoretical result is intentionally local. It does not assert that every storm produces acceleration. It states when a specific weighted energy interval has a positive modeled tendency. The result is useful because it identifies which quantities must be observed or constrained: density, spectral geometry, gradients, and loss time scales. A global activity index alone cannot satisfy this role.

4 Numerical experiment design

4.1 Controlled experiment rather than data reanalysis

The numerical study is a controlled model experiment. It uses no unpublished satellite observations and does not report a new mission-data event. Each driver trace is synthetic but parameterized within plausible normalized ranges. This choice permits exact stress tests of the kernel and the sign criterion while avoiding the conflation of a theory paper with an observational catalog. The experiments are, therefore, designed to test internal consistency and sensitivity, not to claim event-specific forecast skill.

The state grid covers , MeV, and . The baseline spectral family contains a lower-band chorus-like component, a hiss-like component, and a loss-dominated EMIC-like component. The ACRK coefficients are not fitted to observations; they are selected to span quiet, moderate, and storm-like driver regimes. The -shell and energy bounds were chosen to cover the principal outer-belt domain sampled by Van Allen Probes and Arase studies [28–31]. By contrast, the density perturbation and 1–30 min wave-update cadence are deliberate numerical stress-test parameters. In particular, the cadence sweep should not be interpreted as the native sampling cadence of either mission; its upper end intentionally degrades temporal resolution to test sensitivity. The exact controlled ranges are listed in Table 2. The initial phase space density is smooth in energy and pitch-angle, and boundary values are held fixed across model comparisons so that differences can be attributed to the kernel rather than to changes in external forcing.

Analysis.Table 2 defines a broad but transparent experiment envelope. The ranges are not interpreted as observational confidence intervals. They are perturbation ranges for a reproducibility test. Separate fast and slow memory constants force the model to distinguish impulsive compression from recovery. The tested cadence range reveals whether a reduced wave-update rate changes the modeled response qualitatively.

To preserve comparability, all three model classes use the same grid, initial condition, boundary treatment, and mode inventory. The only differences lie in the state dependence retained by the diffusion closure. The static-spectrum model holds the activity state at its pre-event level. The event-driven quasi-linear model permits wave-power updates but omits the explicit impulsive factor. The ACRK retains both persistent and rate-dependent modulation. This design means that a reduction in mismatch can be attributed to the conditional closure within the controlled experiment, rather than to a more favorable numerical configuration.

A normalized wave spectrum is generated as Equation 13with linked to through the mode coefficient . The spectrum is then inserted into (7). The solver advances for 48 h using a 2-min state-update time step, and a saved state is retained every 10 min for diagnostics. The resulting temporal resolution is sufficient to resolve the imposed fast pulse in the reference calculation while still allowing controlled cadence reduction in the sensitivity tests.

The three comparison models are a static-spectrum model, an event-driven quasi-linear model, and the ACRK model. The static-spectrum model fixes at its pre-event value. The event-driven quasi-linear model updates wave-power but excludes the explicit term. The ACRK retains both persistent and impulsive modulation. This comparison tests whether the additional conditioning changes only response amplitude or also the timing and location of resonance access.

Spectral coefficients are evaluated only at the prescribed update times in Table 2; between updates the most recent spectrum is retained. No learned or optimization-based scheduler is part of the physical model. This keeps the cadence experiment tied directly to temporal resolution rather than to an unrelated computational policy.

4.2 Metrics

The primary metric is the normalized trajectory mismatch (Equation 14)where is the internally generated reference solution obtained with the full spectral cadence and is a reduced-model trajectory. We additionally compute the onset timing error, the sign accuracy of , and a robustness score. Because the reference is synthetic, these metrics quantify numerical consistency and structural sensitivity rather than forecast skill against spacecraft data. In particular, onset timing tests whether a model resolves the movement of the resonance gate, whereas sign accuracy tests whether it preserves the local balance between energy diffusion, mixed diffusion, and loss.

Figure 4 provides a direct view of the controlled diffusion landscape. The maximum does not occur at all energies. It emerges from the interaction between the activity factor, spectral frequency, and the density-dependent resonance gate.

FIGURE 4

Analysis.Figure 4 maps a modeled pitch-angle diffusion coefficient rather than a measured diffusion product. The peak follows a curved energy-activity ridge. This is the expected signature of the ACRK: activity changes resonance access, not merely overall amplitude. At high energy, the ridge weakens because the controlled spectrum provides less resonant power within the selected band.

Remark. The numerical design evaluates a specific hypothesis: whether an activity-conditioned kernel changes model behavior in ways predicted by the theory. It does not demonstrate operational forecasting skill. A data assimilation study would require calibrated wave distributions, density products, boundary conditions, and independent verification periods. Those elements are beyond the scope of this controlled simulation study.

5 Numerical results

5.1 Response to increasing activity

Figure 5 compares the flux–response curves of the three models. The ACRK response becomes non-linear at moderate activity because the resonance gate opens before the peak wave-power factor is reached. The event-driven quasi-linear model captures the initial increase but underestimates the curvature near the pulse maximum. The static-spectrum model exhibits the smallest dynamic range. In this controlled setting, the contrast demonstrates that the conditional gate changes not only the magnitude of diffusion but also the activity level at which the modeled response begins to depart from its quiet-time behavior.

FIGURE 5

Analysis.Figure 5 shows a controlled non-linear response rather than an observational regression. The ACRK curve departs from the static-spectrum model after the resonance gate becomes accessible. It remains close to the event-driven quasi-linear curve at low activity. The larger separation at high activity quantifies the contribution of the impulsive term and the density-sensitive accessibility factor.

Table 3 summarizes the outcome metrics for three driving classes. The ACRK reduces mismatch and onset error in every class. The most visible gain occurs in the storm-like case, where is large. Sign accuracy improves more modestly because the mixed diffusion term remains sensitive to gradients near the loss cone.

Analysis.Table 3 reports synthetic-reference consistency across three driver classes. The ACRK advantage is the smallest in quiet intervals because the impulsive term is weak. It is the largest in storm-like intervals, where timing matters. The sign metric remains lower than trajectory agreement because pitch-angle gradients near the loss cone amplify small changes in the mixed diffusion contribution.

The result is consistent with Proposition 3.1. During quiet intervals, the fast component is small, and the three models converge. During storm-like intervals, the cross term is no longer negligible. A closure that omits it responds too slowly even when the overall wave-power increase is included. The difference is especially important for diagnostics that depend on the first crossing of a threshold, for which a modest phase lag can produce a substantial onset time error.

5.2 Ablation of kernel components

Table 4 removes one component at a time. Removing the impulsive term produces the largest onset timing error (31.8 min), whereas removing the density factor gives the second-largest error (29.4 min). Density sensitivity, therefore, remains an important resonance access control, but it is not the largest onset error contribution in this ablation table. Removing the impulsive term also causes the largest degradation in the storm-like mismatch. Removing wave normal angle weighting reduces sign accuracy because it broadens the resonance gate too aggressively.

Analysis.Table 4 shows that no single kernel component explains every improvement. Both density sensitivity and the impulsive term materially affect resonance timing, with the impulsive term ablation producing the larger onset error in this ensemble. The wave-normal gate improves sign classification by avoiding artificially broad access. The full kernel is, therefore, not simply a rescaled wave-power model. It represents a coupled accessibility structure.

Each ablation uses the same solver, grid, boundary conditions, and saved state cadence as the full ACRK calculation. Only the named physical factor is removed. This one-factor-at-a-time design avoids mixing a physics ablation with changes in numerical resource allocation.

Figure 6 shows the ensemble distribution of . The ACRK has both a lower mean and a narrower spread. This pattern indicates that the improvement is not driven by a small number of favorable realizations. Instead, it persists across the perturbations in driver history, density, and spectral parameters included in the controlled ensemble.

FIGURE 6

Analysis.Figure 6 aggregates 160 controlled realizations per model class. The ACRK distribution is shifted downward and is narrower than both baselines. The reduced spread is important because driver and density perturbations vary across ensembles. A lower median alone could arise from one favorable configuration; the joint shift and compression indicate a more stable closure within this simulation design.

5.3 Robustness to density and cadence uncertainty

Figure 7 examines two practical limitations: uncertain cold-plasma density and reduced spectral update cadence. The model retains a high skill score near the reference density and for update cadences below 10 min. Performance degrades gradually for either negative or positive density errors, whereas long cadences and large density errors interact non-linearly. This response is physically plausible within the model because density shifts the resonance gate, and coarse sampling may miss the time at which that gate is most favorable.

FIGURE 7

Analysis.Figure 7 illustrates a compact operating region for the controlled solver. The skill score remains high for moderate density errors when the wave spectrum is updated frequently. The score declines at long cadences because rapid activity variations are undersampled. Density errors then compound the loss by shifting resonance access. The contour structure directly reflects the two-stage ACRK construction.

Table 5 quantifies selected cells on the map. The 5 min case is stable under a 0.2 log-density perturbation. At 30 min, the same perturbation loses much of the temporal advantage. The result suggests that cadence is a physics-relevant numerical setting, rather than merely a computational convenience.

Analysis.Table 5 turns the heatmap trend into interpretable scenarios. Small density errors are tolerated when the spectral update is frequent. The same errors become damaging when fast driver variation is undersampled. This behavior follows the separation bound: a long cadence aliases the fast component, whereas the density error moves the resonance ridge. Both effects increase the timing mismatch.

5.4 Limit checks and model consistency

Several limit checks are required before interpreting the numerical results. First, if is constant, then and the ACRK reduces to a density- and spectrum-conditioned quasi-linear closure. This is the desired quiet-time limit. Second, if , the closure reduces to the static-spectrum model. Using identical grids and boundary treatments, the numerical code reproduces both limits to machine precision. These checks are important because they verify that the conditional formulation contains the expected simpler models as special cases rather than as separate implementations.

Third, the model should not produce a false response when the wave-power is zero. Equation 6 has whenever . Therefore, driver activity alone cannot create a diffusion coefficient. This point is simple but important. A large changes the weight of a wave interaction; it does not replace the wave field. Fourth, a density perturbation must not create unbounded growth in resonance accessibility. Since is clipped to [0, 1], an extreme numerical density input can close or shift a resonance window but cannot amplify a finite spectrum without bound.

The temporal limit is also informative. Let denote the wave-update cadence. A piecewise-constant spectrum produces a sampling residual bounded byprovided is Lipschitz in time with constant . Equation 15 clarifies the diagonal degradation in Figure 7. A large density error decreases the margin between accessible and inaccessible states. A long cadence then samples the incorrect side of that moving boundary.

A final check concerns the relationship between the two proposed terms in (6). The persistent factor controls slowly varying activity conditioning. The impulsive factor controls short transitions. They are not interchangeable. Replacing both by a larger value of matches one pulse height but fails to match onset. Conversely, a derivative-only model produces a short response without recovery memory. This is the structural reason that the full kernel performs better in Table 3.

Remark. These limit checks are not additional tuning exercises. They test whether the model satisfies basic physical and numerical expectations. The absence of wave power eliminates wave-driven diffusion. Constant activity removes the impulsive response. Finer spectral cadence reduces temporal aliasing. Such checks preserve interpretability and facilitate diagnosis of deviations from known limits.

6 Physical interpretation and scope

6.1 Why activity can produce mixed outcomes

The same storm can accelerate one electron cohort while precipitating another. The ACRK represents this possibility by allowing the sign of to depend on local gradients and on the full diffusion tensor. High activity can increase and simultaneously, and a positive energy contribution does not automatically offset pitch-angle loss. This is why flux measurements at a single energy or shell are insufficient for process attribution. A physically meaningful interpretation requires information about the relevant energy range, pitch-angle domain, wave spectrum, density state, and loss channels.

The model also explains why a response may appear delayed. The driver-memory map can be high while the wave spectrum remains weak, or the spectrum can be strong while the density gate remains closed. Only the combined kernel generates a large diffusion coefficient. This structure avoids interpreting every post-storm enhancement as immediate local acceleration and, conversely, prevents a transient wave-power increase from being treated as effective when the required resonance access is absent.

6.2 Relation to established radiation-belt modeling

The proposed kernel is best viewed as a conditioning layer for established diffusion-based radiation-belt solvers rather than as a replacement for them. Salammbô and VERB demonstrate how global Fokker–Planck transport can incorporate multiple diffusion processes [13–16], whereas event-driven studies show why contemporaneous wave and plasma states matter for diffusion estimates [9, 24]. A natural next step is, therefore, to embed the ACRK weights in a validated global solver and assess them through event-specific data assimilation or reanalysis [25].

6.3 Operational limits

The framework should not be used as a stand-alone forecast model. Operational deployment would require data assimilation, event-conditioned priors, calibrated wave distributions, a magnetic field model, and independent verification. It would also need to treat boundary loss, radial diffusion, and substorm injections with the same care given here to local wave–particle interaction. A practical calibration chain could combine upstream solar-wind/IMF data with AE or the SuperMAG lower (SML) index for substorm timing, Van Allen Probes Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) wave/density products, and Arase Plasma Wave Experiment/High Frequency Analyzer products [27–30]. Energetic-electron changes can then be tested against independent particle measurements, such as, Van Allen Probes electron observations used in multiyear radiation-belt studies [3] and the Arase Extremely High Energy Electron experiment [31]. The quantities requiring calibration include the memory times , activity weights , mode factors and , density gate , wave-normal distributions, injection histories, and boundary loss parameters. The ACRK is, therefore, intended as a transparent module within a larger radiation-belt model, where its assumptions can be tested against observations without obscuring the contribution of other transport processes.

For observational evaluation, the proposed structure suggests a staged comparison rather than a single end-to-end score. First, the driver-memory map can be tested against independently derived measures of geomagnetic state. Second, the density and wave-normal components can be checked for plausible intervals of resonance access. Third, the resulting diffusion tendencies can be compared with changes in particle distributions while injection and boundary loss are controlled separately. This staged approach is useful because a disagreement at one stage need not invalidate the entire transport framework; it can identify whether the dominant limitation lies in the driver representation, local plasma specification, spectral characterization, or the quasi-linear closure. It also prevents a favorable flux correlation from being interpreted as proof that the inferred physical pathway is correct. Data-oriented implementations should, therefore, retain intermediate diagnostics such as resonance occupancy, diffusion tensor components, and the local sign of the energization functional, alongside conventional flux metrics.

The model provides a practical comparison point. A conventional static parameterization assumes that one long-term average spectrum represents every state in an activity bin. An event-driven quasi-linear calculation is more responsive, but it can still treat an abrupt driver change as though it affects only wave-power. The ACRK adds the conditional access layer. It evaluates whether the spectrum, density, particle gradient, and rapid forcing jointly permit the same interaction at the same time. This avoids interpreting an activity index as a substitute for the local plasma state.

The distinction is also critical for model validation. A flux-only metric can reward a model that compensates one error with another. For example, excess energy diffusion can be offset by excess loss. Process-resolving validation should compare onset time, spectral occupancy, pitch-angle structure, and the sign of an energy tendency diagnostic. The controlled metrics in this paper are designed to preserve that separation. Future studies should retain it rather than relying on one correlation coefficient.

There are three central scope limits. First, the density factor is illustrative and must be replaced by a mission-calibrated dispersion treatment. The Van Allen Probes and Arase provide complementary wave, density, and particle measurements that are suitable for this validation, including cross-mission conjunctions and withheld storm intervals. Second, the synthetic drivers do not represent the full diversity of solar-wind structures or magnetospheric configurations. Third, the sign criterion is local in phase space and should not be interpreted as a global energetic-particle budget. These limits are deliberate: they keep the present contribution focused on the logical and numerical consequences of conditional resonance weighting.

Remark. The principal physical insight is conditionality. Solar-wind activity controls radiation-belt dynamics through a chain of state changes. A useful model must retain enough of that chain to distinguish spectral access, diffusion strength, and loss. The ACRK is one transparent way to do this. Its assumptions are explicit, testable, and replaceable as better observations become available.

A further implication is methodological. A model should be evaluated at the level at which it makes its physical claim. If the claim concerns resonance access, then a useful validation set must include wave-normal information, density estimates, and particle distributions with sufficient temporal resolution to test the predicted timing. If the claim concerns net energization, then energy and pitch-angle gradients must be assessed together with loss terms. This alignment between model structure and validation target is particularly important in radiation-belt studies, where different process combinations can produce superficially similar flux histories.

7 Conclusions and future work

This paper introduced an activity-conditioned resonance kernel for wave–particle interactions in Earth’s radiation belts. The construction maps driver-memory, cold-plasma density, wave spectra, and resonance geometry into a bounded diffusion closure. Two formal results were established: the first guarantees bounded positive semidefinite diffusion under finite spectral power, and the second gives a sufficient condition for positive local energization. Controlled numerical experiments showed that the closure improves timing, trajectory consistency, and sign classification relative to static-spectrum and event-driven quasi-linear models. The improvements arise because the framework distinguishes sustained conditioning from rapid modulation and keeps resonance accessibility explicit.

The results should be interpreted with caution. They arise from transparent synthetic experiments and are neither a new satellite-data reanalysis nor an operational forecast benchmark. Their value is theoretical and diagnostic. The model illustrates why activity amplitude, activity derivative, density, and wave-normal access cannot be safely compressed into one unconditioned coefficient. It also provides a structured set of diagnostics that can guide subsequent observational tests of when local acceleration, scattering, or mixed behavior should be expected.

Future work focuses on three main priorities. First, the kernel should be calibrated against multi-mission wave, density, and particle products with strict train–test event separation. An initial test involves a Van Allen Probes–Arase event set with independent solar-wind/IMF and auroral electrojet inputs, followed by cross-mission verification on events not used to tune the kernel. Second, radial diffusion, magnetopause shadowing, and injection terms should be coupled to the local closure in a global transport solver. Third, uncertainty-aware data assimilation should propagate wave and density errors into probabilistic flux forecasts. In addition, benchmark events should be selected to test whether the predicted timing shifts correspond to observed changes in spectral occupancy and pitch-angle structure. These steps would transform the present theory-and-simulation module into a testable component for radiation-belt specification.

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