Fluid-Kinetic Variations in the Storm-Time Inner Magnetosphere.

Storm-time broadband electromagnetic field variations along the interface between the dipolar field of the Earth's inner-magnetosphere and the stretched fields of the plasma-sheet are decomposed as a superposition of fluid-kinetic modes. Using model eigen-vectors operating on the full set of Van Allen Probes fields measurements it is shown how these variations are composed of a broad spectrum of dispersive Alfvén waves with significant spectral energy densities in the fast and slow modes over scales extending into the kinetic range. These modes occupy volumes in k -space that define the field variations observed at each spacecraft frame frequency ( f s c ). They are in aggregate not necessarily planar and often comprise filamentary structures with no distinct propagation direction in the perpendicular plane. Within these volumes the characteristic parallel phase speeds of the fast and Alfvénic modes coincide over a broad range of f s c suggestive of coupling/conversion between modes.

Introduction

The collision of fast Earthward flows through the plasma sheet with the dipolar fields of the inner magnetosphere drive field and plasma compressions, shears, particle energization and plasma heating (Gabrielse et al., 2014; Liu et al., 2016, 2018; Runov et al., 2013). The electromagnetic energy transport and conversion processes at the center of these dynamics are mediated by the normal modes of the plasma. These modes describe the average relationship between the variations in the electromagnetic fields and bulk particle distributions (Kivelson and Russell, 1995). They characterize the energy transport through Poynting flux into and out of those regions where Earthward flows deposit energy as their inward motion is arrested (Angelopoulos et al., 2002). Identifying the nature of these variations in terms of the constituent plasma modes is therefore necessary to characterize the flow of energy from the plasma sheet into the inner magnetosphere, ionosphere, and upper atmosphere (Dombeck et al., 2005; Hull et al., 2020; Keiling et al., 2002; Tian et al., 2021; Wygant et al., 2000).

Observations from the Van Allen Probes have demonstrated the prevalence of broad‐spectrum electromagnetic fluctuations along the interface between the dipolar field‐lines of the inner magnetosphere and tail‐like field‐lines extending outward into the plasma sheet (Chaston et al., 2014, 2015; Hull et al., 2019; Malaspina et al., 2015; Moya et al., 2015). An illustrative example recorded during the geomagnetic storm of 1 June 2013, observed from Van Allen Probe A is shown in Figure 1. Here, broad‐spectrum fluctuations in electromagnetic fields (Figures 1b–1e), particularly during storm main phase (Figure 1a), dominate the spectrum from 0.01 to 10 Hz in the spacecraft frame. The variation in the spectral energy density as a function of time, or spacecraft location, is synchronous with changes in the energetic fluxes in both ions and electrons shown in Figures 1f and 1g respectively. This synchronicity is symptomatic of the coupling of large scale Magnetohydrodynamics (MHD) plasma compressions and flows to intermediate and kinetic scale broad‐band field variations. Figures 1h–1l capture these variations in a reduced set of parameters to be analyzed in the main body of this report.

Figure 1

Storm‐time observations from Van Allen Probe A, 1 June 2013. (a) Dst (Disturbance Storm Time) and AE (Auroral Electrojet) indices. (b–e) low frequency electric and magnetic field spectrograms from EFW and EMFISIS—components are in spacecraft coordinates as defined in the text. (f, g) Proton and electron omni‐directional differential energy flux spectrograms from MAGEIS and HOPE. (h–j) Density, average ion mass, and average temperatures from HOPE. (k, l) magnetic and electric field (Earth's frame) vectors in spacecraft coordinates.

The broad‐band fluctuations along this interface have been characterized as part of an Alfvénic cascade delivering energy to kinetic scales to drive ion and electron energization and scattering (Chaston et al., 2015; Hull et al., 2020). This characterization is based on the relative magnitude of spectral energy densities in transverse electric and magnetic fields that follow the expected scale dependent impedance relations for dispersive Alfvén waves. While this model is consistent with the transverse variations observed, it is often the case that simultaneous compressive variations are observed on intermediate scales that are not necessarily in agreement with a purely Alfvénic description (Chaston et al., 2015; Malaspina et al., 2015). Indeed, flows in the near‐Earth plasma sheet are thought to launch fast mode waves Earthward (Kepko et al., 2001; Saito et al., 2008) while dipolarization fronts have a slow‐mode‐like character (Runov et al., 2011) and fluctuations within flows trailing these fronts have been interpreted as slow mode waves (Wang et al., 2016). Plasma structures here have a diamagnetic character suggestive of interchange/ballooning instability in the generation of turbulence in the near‐Earth plasma sheet (Cheng & Zaharia, 2004; Miura et al., 1989; Panov et al., 2012; Panov & Pritchett, 2018; Pritchett & Coroniti, 2010; Roux et al., 1991). Furthermore, inhomogeneities along the interface between inner magnetospheric and stretched plasma‐sheet field‐lines during disturbed times infer that mode coupling and conversion should occur here (Allan & Wright, 2000; Lysak et al., 2009). These considerations demand additional analyses of field and plasma variations in this critical region of near‐Earth space to characterize the spectral composition more completely.

Here, a technique that decomposes the full set of observed field variables from the Van Allen Probes into the constituent normal modes of the plasma at frequencies below the ion cyclotron frequency () is applied. This is performed over a range of scales extending from those of MHD deep into the kinetic range. The analysis is implemented using a background plasma model that varies with the scale of the fluctuations enabling the characterization of 3‐D time variant structures on plasma gradients as the superposition of linear modes populating volumes in ‐space.

Fluid‐Kinetic Analysis Technique

The basis for performing an analysis of the broad‐spectrum electromagnetic fluctuations observed from the Van Allen Probes is a plasma model that appropriately describes the fluctuations over the observed range of spatial and temporal scales. Following previous analyses of these observations (Chaston et al., 2014), fluctuations for which are considered and described using a fluid‐kinetic model (Hollwegg, 1999) on scales extending from MHD down to sub ion‐gyro radii. In this model kinetic effects due to finite perpendicular scale are included in a “fluid” formalism that provides wave properties in agreement with those derived from a full kinetic treatment with the omission of wave damping (Cheng & Johnson, 1999; Lysak & Lotko, 1996). Analysis of the model equations for an isotropic plasma provides the dispersion relation, where is the wave frequency, is the Alfvén speed, and and are the ion acoustic and ion gyro‐radii respectively. Here and are the slowly varying plasma density and total magnetic field strength, and are the electron and ion temperatures and is the ion mass. Note that contributions due to electron inertia are ignored since locally . The wavevector in field‐aligned coordinates is with along . At each location in ‐space Equation 1 provides 6 solutions corresponding to the Alfvén, fast and slow modes propagating along (forward, > 0) and opposite to (backward, < 0) . The normalized phase speeds () of these modes as function of wave‐normal angle () and for a proton‐electron plasma representative of those observed is shown in Figures 2a–2c. These panels apply to both forward and backward modes. On large transverse scales (small ) these solutions describe the classical non‐dispersive MHD modes that transition into the corresponding kinetic modes as increases. The field components of these modes facilitate the formation of eigenvectors which can be used to “filter” the observed fluctuations and estimate the contribution of each mode to the total spectral energy density () observed. This approach has been described for MHD fluctuations by Glassmeier et al. (1995).

Figure 2

(a–c) Normalized phase speeds () of the Alfvén, fast and slow modes for the plasma model indicated as a function of wave‐normal angle () and normalized perpendicular wavenumber (). Inner product of eigenvectors as a function of , and the azimuthal angle, , for the modes indicated. (g–i) within volume for each mode at the and time indicated.

Over the frequency range of interest, the Van Allen Probes provide the electric field vector in the spacecraft spin plane () and the full magnetic field vector () (Kletzing et al., 2013; Wygant et al., 2013). Here is normal to the spacecraft spin plane and points toward the Sun, lies along the projection of the ecliptic plane into the spacecraft spin plane and retrograde to planetary motion, while completes the right‐handed set. In addition, density moments are provided by the plasma instrument onboard the Van Allen Probes (Funsten et al., 2013), however these are not of sufficient cadence to provide the fluctuating density (nor fluctuations in the flow) for this study. Higher cadence density measurements could be estimated from the spacecraft potential (Jahn et al., 2020), however, close to apogee during storm times interpretation of these voltages as density fluctuations requires verification. The variations at each time () and spacecraft frame frequency () are therefore described by a reduced state vector defined in spacecraft coordinates as, where is the background mass‐density. The corresponding partial eigenvectors () for each mode are derived from the solutions of Equation 1 using the relations for and defined in the Supporting Information. These model‐field vectors are defined in the field‐aligned coordinate system for a given and rotated into spacecraft coordinates using the projection of into this system () to form . The fraction of attributed to each mode () is then provided by the matrix equation, where is the full observed spectral matrix, , is the model‐defined complex unit partial eigenvector of the mode, and and respectively denote Hermitian adjoint and trace. If is known for each mode at each then can in principle be evaluated from Equation 3.

The omission of the variations in particle pressure in the formulation of reduces the capacity to distinguish modes in the plasma. To quantify “leakage” between modes Figures 2d–2f show the inner product evaluated using normalized partial eigenvectors corresponding to the solutions shown in Figures 2a–2c in spherical coordinates and ). The azimuthal angle () is measured in a right‐hand sense around between the ‐direction in field‐aligned (FAC) coordinates and the direction of . in the system lies along the vector, , with along . Here, is taken to lie in the plane and inclined 45° to both coordinate axes to represent orientations during storm‐times near apogee and local midnight from the Van Allen Probes. Alternate orientations provide comparable results. Values of zero in Figures 2d–2f indicate orthogonality with no “leakage” between modes, while values of one indicate that the modes cannot be distinguished using the state vector alone at that location in ‐space. Based on this calculation it would seem possible to distinguish the Alfvén mode from the fast and slow modes over most of the volume shown with additional constraints required to distinguish the fast from slow mode.

The additional constraints are supplied by the differences in the mode phase speeds apparent in Figures 2a–2c. These speeds determine at which a mode at a given and may appear. This frequency is, where is the velocity of the spacecraft through the plasma. For the broad spectra and flows measured from the Van Allen Probes, at each and will have contributions over the range of and that satisfy Equation 4. For this case the integrated composition can be defined as, where , is for a logarithmic grid in , and the integral is performed over the volume () in ‐space where Equation 4 is satisfied for each wave mode (). The normalized composition is then .

Within , a characteristic and may be identified at that location in ‐space where is largest. This can be located by searching for minima of the function, much in the manner of ‐filtering (Motschmann et al., 1998) with the “filter” defined by the mode eigen‐vector . Here it is noted that for the broad‐spectrum fluctuations observed, of the characteristic mode will not necessarily coincide with that returned by a priori assuming a single plane wave at each and and deriving using Faraday's law (e.g., Santolik et al., 2003).

Storm Time Composition

The measurements required to perform this analysis over the “main phase” of the storm of Figure 1 are presented in Figures 1h–1l. Estimates for the spectral coefficients at each required to populate are derived using Morlet wavelets (Torrence & Compo, 1998) applied to the full magnetic field vector and the and electric field components in the rest frame of Earth (Figures 1k and 1l). The background plasma parameters () at each wavelet scale are defined by low‐pass filtering as where is the wavelet e‐folding time. Values of provide nearly identical results and here is used. In advance, it is also noted that variations in the background parameters of up to 50% cause less than ∼10% variation in composition (Chaston, Bonnell, Bale, et al., 2020). The spacecraft spin plane electric field measurements along with the axial component () given by the approximation are used to estimate . For demonstration purposes, Figure 1l includes derived in this manner with defined by low pass filtering the magnetic field vector to Hz. The velocity of the spacecraft through the plasma is then where is the orbital velocity. In practice, is derived at each wavelet scale and time for use in conjunction with the dispersion results to define via Equation 4 for each mode at each and . Each bin in has width , and all those ‐space grid locations that satisfy Equation 4 over the range /2 are included in .

To illustrate how the analysis proceeds, Figures 2g–2i show within for each of the forward propagating modes at and time 04:26:02 UT. Figure 2g shows that for the Alfvén mode is broadly distributed in and balanced in with peaks at ∼180° and ∼360° and “soft” maxima occurring at and . The distribution of in ‐space for the other modes is more localized, however multiple maxima with similar peak values are apparent. These distributions indicate that the fluctuations are not well described as a single Fourier mode at a specific wavenumber. This non‐specificity is replicated at each and throughout this storm interval. Integration of these distributions, as defined by Equation 5, provides the corresponding mode composition results presented in Figure 3.

Figure 3

Fluid‐kinetic mode composition. (a, b) Morlet wavelet spectrograms of the low frequency electric and magnetic field variations for the components indicated. (c–k) (normalized total composition) and for each mode.

Figures 3a and 3b show the power spectra of the magnetic and electric field variations. The black regions in Figure 3b correspond to the digitization noise floor of the fluxgate magnetometer that constrains the frequency range for analysis. The normalized integrated composition over this range, , as defined by Equation 5 and the ratio of forward to backward composition, , is presented in Figures 3c–3k. Note that these results provide continuous coverage from the MHD range through to kinetic scales where .

Comparing Figures 3c–3k indicates a composition dominated by Alfvénic modes (Figures 3c and 3d) with a preference for propagation in the forward sense represented by the preponderance of red shading in Figure 3e. For the magnetic field orientation and the location of the Van Allen Probes at this time, this result corresponds to propagation, or energy transport, in the Alfvén mode along primarily toward the northern ionosphere. There is however a significant flux in backward propagating Alfvén modes whose composition is at times anti‐correlated with the forward mode in and . The simultaneous presence of spectral energy densities in both the forward and backward modes is indicative of counter propagation and/or consistent with the presence of field‐line eigenmodes identified from the Van Allen Probes in previous studies (Chaston et al., 2014). Significantly, the contribution from the backward mode at the lowest frequencies becomes largest after 0550 UT following the dipolarization of the magnetic field (Figure 1k).

Figures 3f and 3g show significant contributions from the fast mode throughout the interval at . There is a preference for propagation in the forward direction along (Figure 3h). The fast mode effectively disappears from the composition for over which range the compressional field variations are dominated by the slow mode (Figures 3i and 3j). In fact, comparison of Figures 3f and 3g to Figures 3i and 3j shows that slow and fast mode spectral energy densities are anti‐correlated. For the slow mode shows no persistent preference for forward or backward propagation along , but above this frequency this mode is preferentially found in the backward sense (Figure 3k). This suggests a source located closer to the northern ionosphere. After the field dipolarization there is a progressive decrease and increase in the slow mode and fast mode composition respectively.

Wave Properties

While the composition results described above are defined by integration over , it is instructive to consider the characteristic wave properties at the minima in within at each and . Figure 4 shows these properties for the forward modes, noting that the results for the backward modes are qualitatively similar. For context the average ion temperature (∼6 keV), mass (∼10−26 kg) and magnetic field (∼300 nT) provide a thermal ion gyro‐radius of ∼90 km so that at . Values of at, and above, this wavenumber, correspond to the kinetic range and are represented in Figures 4a, 4e and 4i by yellow to red shading.

Figure 4

(a–l) Wavevector magnitude (, Wavenormal Angle (), wave frequency normalized to the proton gyrofrequency () and azimuthal angle () at minima of for the forward Alfvén, fast and slow modes respectively. (m) Average of each mode as a function of . (n) Filamentation in the magnetic field morphology.

Figure 4 shows that the characteristic wavevectors (Figure 4a) for the Alfvén mode span the range from MHD scales well into the kinetic range at largest . These wavevectors are highly oblique with wave‐normal angles (Figure 4b) approaching 90° over the frequency range shown. This result is consistent with those returned via fits of impedance relations for kinetic Alfvén waves to the observed electromagnetic wave spectra where (Chaston et al., 2015). Plasma frame wave frequencies ( (Figure 4c) increase with with values normalized by the ion gyrofrequency () generally much less than one. There also appears to be no definitive orientation for ; varies erratically over the full 2 radians in the perpendicular plane (Figure 4d).

In contrast, the same approach shows that the fast mode is generally not found in the kinetic range (Figure 4e). There are however impulsive intervals where kinetic scales are present particularly after ∼0550 UT. At the lowest frequencies surveyed () Figure 4 shows no fast mode solutions satisfying Equation 4 exist. Wave normal angles (Figure 4f) just above these frequencies are nearly field aligned, become progressively oblique with increasing , and are nearly perpendicular at the largest . Like in the Alfvén mode, increases with but more closely approaches at large (Figure 4g). Again, no clearly organized propagation direction in the plane perpendicular to is found (Figure 4h).

The slow mode, like the Alfvén mode, becomes kinetic at the largest with wavenumbers extending well into the range (Figure 4i). Here there appears to be a background of large‐scale fluctuations with impulsive intervals of highly kinetic fluctuations. The impulsive intervals coincide with enhanced spectral energy densities where analyses above the magnetometer noise floor are sometimes possible up to . At the lowest the slow mode is nearly field‐aligned, almost perpendicular at intermediate , and moderately oblique at the highest (Figure 4j). Wave frequencies increase with similar to that in the Alfvén mode (Figure 4k). The distribution of again indicates no clear propagation direction in the perpendicular plane (Figure 4l).

Discussion and Conclusion

The analyses performed above indicate that in addition to the documented Alfvénic nature of broadband electromagnetic fluctuations outside the plasmapause during storm times, a significant fraction of observed spectral energy density may also reside in fast and slow mode variations. This description provides an explanation for the compressional structures observed at large non‐kinetic and intermediate scales (Chaston et al., 2012, 2015) on which the Alfvén mode is mostly transverse. These structures have the properties of diamagnetic cavities often comprising localized regions of energetic plasmas. They are decomposed in this analysis as the superposition of Alfvénic and slow mode wavelets and appear in Figure 3 at frequencies below 0.01 Hz. These Alfvénic/slow mode features progressively disappear after dipolarization of the geomagnetic field with reduction in slow mode composition as the spacecraft transitions to lower L‐shells and passes through magnetic midnight. During this transition, the fast mode composition above 0.01 Hz intensifies.

The coexistence of spectral energy densities in all three modes is suggestive of the operation of mode coupling. This may be a consequence of inhomogeneities in the plasma (Allan & Wright, 1998, 2000; Hasegawa, 1976; Johnson et al., 2001; Lysak et al., 2009) and/or non‐linear processes such as parametric decay (Chen & Zonca, 2011; Hasegawa & Chen, 1976). Figure 4m shows the average parallel phase speeds, , of each mode at minima in as a function of . The similarity of in the fast and Alfvén mode over much of the range shown introduces the possibility of energy transfer between these modes. While a definitive demonstration of energy transfer remains to be presented, this result suggests that such a process may have been prevalent during this storm main phase.

The lack of a distinct direction of propagation in the plane perpendicular to and the broad distributions of for the Alfvénic mode may be understood by considering the morphology of the fluctuations. Figure 4n shows the “filamentation” of the field variations in the perpendicular plane based on the relationship between the corresponding magnetic field components (Chaston, Bonnell, Wygant, et al., 2020). For planar features consistent with a single plane wave at each and the value of the filamentation should approach zero. However, Figure 4 shows the filamentation is often larger than zero, and sometimes approaches one. This represents elliptical/filamentary features that require the superposition of modes over a range in to account for the associated distribution of at each and . For such features there is no clearly defined direction of propagation through the plasma in the plane perpendicular to .

In closing, a full account of the spectral energy in the broadband electromagnetic fluctuations during the storm of 1 June 2013, requires not only a superposition of modes at each and but also contributions over a range of wave‐vectors in each mode. This is required by the observed non‐planar field morphology and is symptomatic of the operation of the non‐linear processes presumably driving the broad ‐spectra. The Alfvén mode, when integrated over a range of wavevectors capable of contributing to at a particular , is generally the largest contributor to . However, this mode is associated with, and likely coupled to, fast and/or slow mode field variations by processes yet to be identified. While this decomposition analysis has been performed for a single storm, statistical studies of comparable events (Chaston et al., 2015, Hull et al., 2019, 2020) suggest that the properties described above are likely pervasive outside the plasmapause during geomagnetically disturbed intervals.

Supporting Information S1

Click here for additional data file.