ULF Wave Applications - Magnetoseismology and Radiation Belt Electron Dynamics #

Equatorial mass density inferred from ground-based magnetometers. Noon is at the top of each figure and midnight is at the bottom. The figures show the percentage difference in mass densities when assuming the toroidal mode compared with the poloidal mode. (a) Quiet magnetic conditions (b) Active conditions.

Ultra low frequency (ULF) Alfven waves, which are often excited by the action of the solar wind on the magnetosphere, are observed as geomagnetic perturbations in ground-based magnetometers. As discussed in one of the links below, they can be used to remote-sense the distribution of equatorial cold plasma mass density in an application known as "Magnetoseismology". The figure on the right shows estimates of mass density based on cross-phase spectral measurements of ULF waves in the CARISMA magnetometer chain in Northern Canada. ULF waves also provide one of the mechanisms for accelerating radiation belt electrons that have the potential to damage satellites at or near geosynchronous orbit. As discussed in the links below, their effectiveness in accelerating particles via "drift resonance" depends on both the wave frequency and polarization.

Techniques appropriate for analysis of the wave characteristics in arbitrary magnetic fields have been developed in FDAM and applied to both empirical (Tsyganenko) and self-consistent (Global MHD) magnetic field models. ULF waves in application against magnetoseismology and compressed dpole fields are discussed in this section of the website.

ULF wave cross-phase Analysis#

The magnetized cold plasma in Earth's magnetosphere supports two magnetohydrodynamic (MHD) wave modes: the fast and shear modes. The fast mode propagates isotropically across geomagnetic field lines, whereas the shear Alfven mode is guided along the magnetic field to form standing waves between the conjugate ionospheres. At locations where the fast mode frequency matches the natural field line eigenfrequency, the fast mode is converted to the shear mode. These field line resonances (FLRs) have frequencies that are determined by the magnetic field topology and plasma mass loading on the field line. If the FLR frequency at a given latitude can be obtained, the equatorial plasma mass density can be found given the magnetic field and field-aligned mass distribution, and an appropriate wave model. Since FLR frequencies are largely determined by mass density at the equatorial plane, they provide a means for monitoring how cold plasma evolves in response to changing solar wind conditions.

FLR frequencies can be obtained using ground magnetometer data from the Canadian Auroral Network for the OPEN program Unified Study (CANOPUS) magnetometer array Samson et al., 1992. Coordinate details for the relevant sites are shown in Table 1. Typically, the magnetometer data are bandpass filtered using a fourth order Butterworth digital filter with cutoff frequencies at 0.5 and 20 mHz respectively. The FLRs for latitudes between consecutive latitudinal magnetometer pairs can then be obtained using the cross phase method described by Waters et al., 1991, 1996.

The location of the Churchill line of magnetometers used for cross-phase analysis of ULF waves.

An example of FLR frequency determination based on CANOPUS observations and cross-phase analysis is shown in the figure below. The figure shows cross phase spectra and the change of FLR frequencies with time and latitude from the CANOPUS magnetometer data for 9 Feb. 1995.

Fig. 1. Cross phase spectra showing the change of FLR frequencies with time and latitude from CANOPUS magnetometer data for 9 Feb. 1995.

References:

1. Samson, J. C., D.D. Wallis, T.J. Hughes, F. Creutzberg, J. M., Ruohoniemi, and R. A. Greenwald, Substorm Intensifications and field line resonances in the nightside magnetosphere, J. Geophys. Res., 97, 8495, 1992.
2. Waters, C. L., F. W. Menk, and B.J. Fraser, The resonance structure of low latitude Pc 3 geomagnetic pulsations, Geophys. Res. Lett., 18, 2293, 1991.
3. Waters C. L., J. C. Samson, and E. F. Donovan, Variation of plasmatrough density derived from magnetospheric field line resonances, J. Geophys. Res., 101, 24737, 1996.
4. Singer H. J., D. J. Southwood, R. J. Walker, and M. G. Kivelson, Alfven wave resonances in a realistic magnetospheric magnetic field geometry, J. Geophys Res., 86, 4589, 1981.

Magnetoseismology#

Based on results of cross-phase analysis, "Magnetoseismology" refers to the actual reconstruction of equatorial magnetospheric density from ground-based magnetometer measurements of ULF waves. This involves cross-correlating measurements made using pairs of magnetometers in a meridional chain, in order to determine the natural Alfvenic (or FLR) frequencies of individual field lines. The FLR frequencies depend on the magnetic field variation along the field line and the density distribution along field lines in the magnetosphere. By using a magnetic field model (either empirical, or self-consistent results of a global simulation) and a ULF wave model it becomes possible to remotely estimate the equatorial plasma density. This technique in its essence is very similar to seismic analysis as applied in geophysics to study Earth's lithosphere; hence the name, magnetosismology. In practice, magnetosismology makes it possible to monitor the plasma content of the magnetosphere on a longer-term basis than is currently possible with in-situ satellite measurements.

An example of magnetoseismology applied to CANOPUS ULF cross-phase frequency spectra is shown on the figure below. The gap close to midnight is typical, as the cross-phase technique applied to this region is problematic due to the very active state of the magnetosphere on the nightside.

Two-dimensional reconstruction of the equatorial plasma density in the magnetosphere for February 9, 1995 using CANOPUS magnetometer data.

References:

1. Alfvenic field line resonances in arbitrary magnetic field topology, R. Rankin, K. Kabin, and R. Marchand, Advances in Space Research, 38, 1720-1729, 2006.
2. Different eigenproblem models for field line resonances in cold plasma: effect on magnetospheric density estimates. K. Kabin, R. Rankin, C. L. Waters, R. Marchand, E. F. Donovan, J. C. Samson, Planetary and Space Sciences, 55(6), 820-828, 2007.
3. ULF wave polarization and magnetic field model effects on the estimation of proton number densities in the magnetosphere using field line resonances, C.L. Waters, R. Rankin, K. Kabin, E. Donovan J. C. Samson Planetary and Space Sciences, 55(6), 809-819, 2007.

Compressed dipole fields#

The description of ULF waves excited in Earth's magnetic field is complicated by the distortion of the field due to the action of the solar wind. FDAM scientists have developed mathematical techniques that are appropriate for analysis of wave characteristics in arbitrary magnetic fields. These have been applied to both empirical (Tsyganenko) and self-consistent (Global MHD) magnetic field models. The results, however, become particularly simple in so-called "compressed dipole" field topology, which is a simplified magnetospheric field model based on Euler potentials. This model has somewhat realistic compression of the magnetic field on the day side and stretching on the night side. As the mathematical theory is quite complicated, it is not presented here, but references are provided at the bottom of this page for the interested reader.

The figures below show magnetic field lines in compressed dipole magnetic field topology. The amount of compression (stretching) on the dayside (nightside) is an adjustable parameter. Thus, this model can approximate actual solar wind conditions affecting the topology of the geomagnetic field in the inner magnetosphere.

Fig. 1. Field lines of a compressed dipole in the equatorial (upper panel) and noon-midnight meridional planes for B_0=31000 nT, b_1=10 nT, and b_2=8. These parameters define the amount of compression (stretching) of the field on the dayside (nightside). They are defined in reference 2 below.

As an application of the compressed dipole field, FDAM has developed a model for ULF waves supported by such a topology. The two sets of figures below show the periods of wave modes, and their corresponding polarizations. One interesting result is that the polarization of ULF waves changes significantly along the expected drift path of energetic electrons, which implies that acceleration through "drift-resonance" between electrons and ULF waves will occur only in certain MLT sectors. The results lead to the expectation that typically an electron will experience interaction with a broad-spectrum wave packet along its path, rather than with a wave of a single frequency. This contradicts previous results on particle acceleration in non-axisymmetric fields, and provides an important first step to more detailed analysis of drift-resonance acceleration mechanisms in the future.

Fig. 2. Periods of the two different Alfven modes for B_0=31000 nT, b_1=10 nT b_2=8. Left panel: mode with radial electric field polarization at midnight; right panel -- with azimuthal electric field at midnight. Thick solid lines show the contours of constant B initiated at r=2, 4, and 6 at noon. Dashed lines are circles with r= 2, 4, and 6, respectively.

Fig. 3. Electric field polarizations in the equatorial plane for the two Alfvenic eigenmodes. Left panel: mode with radial electric field polarization at midnight; right panel -- with azimuthal electric field at midnight. Thick solid lines show the contours of constant B initiated at r=3, 5, and 7 at noon. Dashed lines are circles of constant radius with r=3, 5, and 7.

References:

1. Alfvenic field line resonances in arbitrary magnetic field topology, R. Rankin, K. Kabin, and R. Marchand, Advances in Space Research, 38, 1720-1729, 2006.
2. Polarization properties of standing shear Alfven waves in non-axisymmetric background magnetic fields. K. Kabin, R. Rankin, I. R. Mann, A. W. Degeling, R. Marchand, Annales Geophysicae, 25(3), 815-822, 2007.