Field Line Resonances and Discrete Arcs#

The figure shows the electric field, field-aligned current and velocity in a Field Line Resonance. The indicated velocity shear is unstable to Kelvin Helmholtz instabilities that stir up the aurora

The visible aurora is modulated by plasma waves covering a wide range of frequencies. The Pc5 range refers to waves with frequencies of a few mHz. When near monochromatic "fast mode" Alfven waves encounter geomagnetic field lines, they resonantly excite Pc5 shear Alfven waves (SAWs) of the same frequency. Since SAWs drive electron currents parallel to the geomagnetic field, they are a potential source of electron precipitation. Normally, the electron current supported by the wave is small. However, if the SAW is latitudinally narrow (on the order of the electron skin depth at the ionosphere), it can form a "parallel electric field" that accelerates electrons to much higher energy.

An important question is whether parallel electric fields in SAWs are large enough to explain discrete auroral arcs. An example of a discrete arc excited by "field line resonance" (FLR) is shown on the panels on the right. The all-sky camera observation can be explained using two-fluid MHD models that determine the parallel electric field and other wave parameters. This aspect of the problem is discussed in the following web pages.

Motivation and objective#

Field Line resonances (FLRs) are standing shear Alfven waves (SAWs) that form on closed geomagnetic field lines in Earth's magnetosphere. They are commonly observed in the nightside magnetosphere and auroral ionosphere [e.g., Ruohoniemi et al., 1991; Samson et al., 1991, 1992], and are narrow in the direction perpendicular to the magnetic field. The scale size at the ionosphere can be as small as a few km, but often extends to several 10's of kilometers in overall latitudinal width. The discrete arcs embedded within FLRs can appear as sub-structure with perpendicular scales approaching a km or so, i.e., comparable to the electron skin depth in the ionosphere. Observations suggest that electron precipitation into the auroral zone by FLRs is associated with temporally modulated discrete auroral arcs. An example of an FLR is shown in the figures below. The left panel shows a sequene of arcs along with a keogram that indicates the intensity of auroral emissions as a function of latitude and time. The range of latitudes is indicated by the red line drawn in the top-left sequence of arcs at 0111. The animation in the right panel below is constructed from the time sequence of arcs observed by the imager at Rankin Inlet.

All-sky camera images (top-left) and "keogram" (lower-left) showing the temporal evolution of a high-latitude FLR. The lower-left panel shows periodic poleward motion expected of an FLR, and associated phase mixing. The right panel shows animation of the auroral arc.

When FLRs narrow to the width of the ion gyroradius (ion acoustic gyroradius) or electron skin depth, parallel electric fields form in them. These "dispersive" wave fields produce electron acceleration and might explain why auroral arcs form in association with FLRs. It is a remarkable fact that narrow perpendicular scales form naturally in FLRs due to a process called phase mixing. This also allows the background plasma to develop sharp density gradients that further decrease the perpendicular scale of the waves and the magnitude of the parallel electric field. The nonlinear steepening of the density gradient comes from the SAW ponderomotive force, which enhances wave dispersion and hence the auroral electric field. Nonlinear effects salso act to tabilize FLR growth by detuning the FLR frequency from the fast mode wave that is exciting it. Evidence for this scenario is provided by ground-based observations of FLRs, and from in-situ measurements that reveal the presence of auroral plasma density cavities, inside which FLRs are trapped. Some of these ideas are explored further in this part of the website.

References:

1. Rankin, R., J.Y. Lu, R. Marchand, and E.F. Donovan, Spatiotemporal characteristics of ultra-low frequency dispersive scale shear Alfven waves in Earth's magnetosphere, Phys. Plasmas.,11(4), 1268, doi:10.1063/1.1647138, 2004.
2. Rankin, R., R. Marchand, J. Y. Lu, K. Kabin, and V.T. Tikhonchuk, Theory of Dispersive Shear Alfven Wave Focusing in Earth's Magnetosphere, Geophys. Res. Letts.., 32, L05102, doi: 10.1029/2004GL021831, 2005.
3. Ruohoniemi, J. M., R. A. Greenwald, K. B. Baker, and J. C. Samson, HF radar observations of Pc 5 field resonances in a midnight/early morning MLT sector, J. Geophys. Res., 96, 15,697, 1991.
4. 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.

Model and Approach#

The nonlinear interaction of shear Alfven Field Line Resonances (FLRs) by compressional waves propagating isotropically through warm inhomogeneous magnetospheric plasma is governed by the set of magnetohydrodynamic (MHD) equations written below:

The first two equations define conservation of plasma fluid mass and momentum, respectively. The next two equations define Faraday's and Ampere's law, while the last is the generalized Ohm's law. FDAM scientists have developed one class of Algorithm that solves this set of nonlinear partial differential equations in complicated geometries [Lu, et al., 2007].

The dynamics of dispersive Alfven waves can also be investigated within the framework of reduced MHD [Lu, et al., 2003a]. The definition of the reduced MHD equations is based on the following assumptions: (1) wave perturbations are characterized by a perpendicular scale length that is much smaller than the parallel scale length, (2) the characteristic time for the evolution of the wave is much longer than the ion gyro period, (3) the ion gyroradius is small compared with the characteristic perpendicular scale, and (4) the plasma remains quasi-neutral. Introducing wave potentials defined for reduced-MHD, this set of equations can be written as,

To solve this last set of equations, the two-dimensional finite element code TOPO, developed by Marchand Marchand and Simard, 1997, is used. This model is chosen because it has the following advantages:

1. All terms can be treated implicitly except for the convection term (semi-implicitly).
2. The Algorithm is able to solve complicated equations in a standardized way.
3. It allows us to study dispersive waves in arbitrary magnetic field geometries.

References:

1. Lu J. Y., R. Rankin, R. Marchand, I. J. Rae, W. Wang, S. C. Solomon, and J. Lei, Electrodynamics of magnetosphere-ionosphere coupling and feedback on magnetospheric field line resonances, J. Geophys. Res.., submitted, 2007.
2. Lu, J.Y., R. Rankin, R. Marchand, V.T. Tikhonchuk, and J. Wanliss, Finite element modelling of nonlinear dispersive field line resonances: trapped shear Alfven waves inside field-aligned density structures, J. Geophys. Res., 108(A11), 1394, doi:10.1029/2003JA010035, 2003a.
3. Marchand, R., and M. Simard, Finite element modelling of Tdev edge and divertor with E _ B drifts, Nucl. Fusion, 37, 1629, 1997.

Nonlinear FLRs#

Fig. 1. Spatial and temporal evolution of the perpendicular electric field in a dispersive Alfven wave in the equatorial plane. See Lu et al. [2003a].

When shear Alfven waves (SAWs) are excited, nonlinear wave ponderomotive forces excite sound waves that form a density channel along geomagnetic field lines [Lu et al., 2003a]. Figures 1 and 2 show the spatial and temporal development of the electric field amplitude and relative density perturbation in a dipolar magnetic field. The ponderomotive force causes plasma to move along resonant field lines, resulting in a density bump (red color in Fig.2) at the equator and a cavity (deep blue) at high latitudes close to the ionosphere.

The density redistribution in Fig. 2 steepens the local Alfven speed gradient and significantly affects the dynamic evolution of FLRs [Lu et al., 2003a, 2003b]. Fig.1 shows a comparison of the linear and nonlinear evolution of the polarization electric field in a dispersive FLR in the equatorial plane. It can be seen that fast temporal and short spatial-scale variations are apparent only in Fig 1b and that the resonance location moves Earthward in the nonlinear case. The associated density depletions near the ionosphere are on the order of 10~15% of the ambient density (Fig.2), but are comparable to the ambient density in the equatorial plane.

Once a density channel is formed, the timescale of the dynamics of the FLRs is very rapid. Time dependent dispersion and steepening lead to the enhancement of dispersive effects and to localization of the FLR within the ionospheric density cavity. Consequently, short scale waves are confined or trapped within the density cavity. Detailed analysis and theory can be found in Lu et al., 2003b; Rankin, et al., 2005. The FLR model successfully explains the formation of density cavities and may point to a common process for density cavity formation that is consistent with many observations [e.g., Chaston et al., 2006].

References:

1. Chaston C.C., et al., Ionospheric erosion by Alfven waves, J. Geophys. Res., 111, A03206, doi:10.1029/2005JA011367, 2006.
2. Lu, J.Y., R. Rankin, R. Marchand, V.T. Tikhonchuk, and J. Wanliss, Finite element modelling of nonlinear dispersive field line resonances: trapped shear Alfven waves inside field-aligned density structures, J. Geophys. Res., 108(A11), 1394, doi:10.1029/2003JA010035, 2003a.
3. Lu, J.Y, R. Rankin, R. Marchand, and V.T. Tikhonchuk, Nonlinear Acceleration of Dispersive Effects in Field Line Resonances, Geophys. Res. Lett., 30(10), 1540, doi: 10.1029/2003GL016929, 2003b.
4. Rankin, R., K. Kabin, J. Y. Lu, I. R. Mann, R. Marchand, and I. J. Rae, V.T. Tikhonchuk, and E. Donovan, Magnetospheric field line resonances: ground-based observations and modeling, J. Geophys. Res.., 110, A10S09, doi: 10.1029/2004JA01919, 2005.
5. Rankin, R., R. Marchand, J. Y. Lu, K. Kabin, and V.T. Tikhonchuk, Theory of Dispersive Shear Alfven Wave Focusing in Earth's Magnetosphere, Geophys. Res. Letts.., 32, L05102, doi: 10.1029/2004GL021831, 2005.

Ionospheric Electron Heating#

Geomagnetic field-aligned-currents (FACs) excited in field line resonances (FLRs) close through perpendicular Pedersen currents in the ionosphere. Through collisional and other processes, these currents produce intense heating of the ionosphere, leading to changes in the underlying ionospheric conductivity. The parallel electric fields excited in FLRs also modulate electron precipitation from the magnetosphere to the ionosphere. This also modifies the conductivity of the ionosphere, which in turn affects current closure. In order to account for these processes, FDAM scientists have developed a model that describes changes to the ionospheric conductivity. The main ingredients for this model are discussed below. It provides a boundary condition for the dispersive wave models discussed earlier in this part of the website.

Consider a reference system in which neutrals are at rest. The electron energy balance equations at the ionosphere are then governed by [Lu et al., 2005b]

where k = 1.38x10^16 erg/K, s_pe is the electron Pedersen conductivity, S_Le is the sum of all the inelastic cooling rates (rotational and vibrational excitation of N2 and O2, and O fine structure excitation).

The electron production in the ionosphere is derived from auroral precipitation, photoionization and/or electron impact ionization and from chemistry, while electron losses are mainly due to ion neutral reactions and electron-ion recombination. The ionization in the E-layer is produced by inelastic collisions of heated electrons and external sources such as cosmic rays. If chemical production and losses are ignored, ionization is balanced by recombination, as defined by the electron density continuity equation

where ghot is the production rate due to hot electron precipitation, and nioniz is the average ionization rate. In the stationary approximation, and neglecting electron precipitation and convection losses, one can find a critical Pedersen current [Lu et al., 2005b]

Fig.1. Pedersen conductivity as a function of ionospheric current

The figure above shows the Pedersen conductivity as a function of the ionospheric wave current for an initial conductivity 0.5, 1 and 2 S. When parallel currents in the wave exceed the critical value defined above, it is found that that the electron temperature increase leads to ionization and enhancement of the ionospheric Pedersen conductivity [Lu et al., 2005a, 2005b]. This effect is more significant for smaller initial ambient conductivities (<1 S), suggesting perhaps that discrete arcs associated with latitudinally narrow FLRs may have their birth in regions of low background conductivity.

References:

1. Lu, J. Y., R. Rankin, R. Marchand, and V.T. Tikhonchuk, Nonlinear electron heating by resonant shear Alfven waves in the ionosphere, Geophys. Res. Letts.., 32, L01106, doi:10.1029/2004GL021830, 2005a.
2. Lu, J. Y., R. Rankin, R. Marchand, and V.T. Tikhonchuk, Reply to comment on " Nonlinear electron heating by resonant shear Alfven waves in the ionosphere" by J.-P. St.-Maurice, Geophys. Res. Letts.., 32, L13103, doi:10.1029/2005GL023149, 2005b.

Ionospheric Feedback#

Magnetospheric field-aligned-currents (FACs) are strongly affected by the conducting ionosphere. Ionospheric ionization processes change the conductivity, which in turn feeds back on the FACs. and the associated wave fields The electron heating mechanism ( Joule dissipation) discussed in the previous section provides additional ionization for large amplitude current systems [Lu et al., 2005a; 2005b]. Another more direct source of ionization in the ionosphere is precipitating electrons from the magnetosphere [Atkinson, 1970]. This type of ionospheric feedback has been studied in [Prakash et al., 2003]. In their calculation, Pedersen conductivity enhancements by precipitating electrons are described by an empirical formula. We have developed a quantitative model that provides an improved estimate of ionospheric conductivity in dipolar and stretched magnetic field lines [Lu et al., 2007]. The improved ionospheric conductivity model, GLOW, considers the effects of field-aligned potential drops and the magnetic mirror force, and calculates the average energy and number-flux of electron precipitation from FACs using the Knight-relationship.

Fig. 1. Comparison of the wave field amplitudes between perfect conductivity (dash dotted), a fixed conductivity (dashed), and the interactive precipitation conductivity model (solid). See Lu et al. 2007.

Figure 1 shows the wave field amplitude at the ionosphere. When auroral electron precipitation is included (solid), the Pedersen conductivity increases, and the ionospheric electric field and SAW dissipation are reduced in comparison to the case with fixed conductivity (dashed). The field-aligned current increases from 0.5 to 3 mA/m2, while the perpendicular electric field is reduced from 37 mV/m to 19 mV/m.

Fig. 2. Comparison of the wave field amplitudes between perfect conductivity (dash dotted), a fixed conductivity (dashed), and the interactive precipitation conductivity model (solid). See Lu et al. 2007.

Figure 2 shows the density perturbation in a typical stretched magnetic field topology approximated by the Tsyganenko T96 model. Along the resonant field line, the plasma moves from high latitudes to the equator, resulting in a density cavity at high latitudes and a density bump at the equator. However, the density perturbation near the equator experiences significant movement across the field lines, whereas in results obtained using reduced-MHD equations for low frequency plasma the density perturbation is mainly along the field lines [Lu et al., 2003b]. This may be the result of field line stretching that brings the equatorial plasma beta close to unity. It has been demonstrated that interchange or ballooning modes can arise when the plasma thermal pressure becomes important in the magnetotail [Liu, 1997].

References:

1. Atkinson, G., Auroral arcs: Result of the Interaction of a Dynamic Magnetosphere with the Ionosphere, J. Geophys. Res., 75(25), 4746, 1970.
2. Liu, W. W., Physics of the explosive growth phase: Ballooning instability revisited, J. Geophys. Res., 102, 4927, 1997.
3. Lu J. Y., R. Rankin, R. Marchand, I. J. Rae, W. Wang, S. C. Solomon, and J. Lei, Electrodynamics of magnetosphere-ionosphere coupling and feedback on magnetospheric field line resonances, J. Geophys. Res.., submitted, 2007.
4. Lu, J. Y., R. Rankin, R. Marchand, and V.T. Tikhonchuk, Nonlinear electron heating by resonant shear Alfven waves in the ionosphere, Geophys. Res. Letts.., 32, L01106, doi:10.1029/2004GL021830, 2005a.
5. Lu, J. Y., R. Rankin, R. Marchand, and V.T. Tikhonchuk, Reply to comment on " Nonlinear electron heating by resonant shear Alfven waves in the ionosphere" by J.-P. St.-Maurice, Geophys. Res. Letts.., 32, L13103, doi:10.1029/2005GL023149, 2005b.
6. Lu, J.Y, R. Rankin, R. Marchand, and V.T. Tikhonchuk, Nonlinear Acceleration of Dispersive Effects in Field Line Resonances, Geophys. Res. Lett., 30(10), 1540, doi: 10.1029/2003GL016929, 2003b.
7. Prakash, K., R. Rankin, and V. T. Tikhonchuk, Precipitation and nonlinear effects in geomagnetic field line resonances, J. Geophys. Res., 108(A4), 8014, doi:10.1029/2002JA009383, 2003.

Real Event Simulation#

We apply our theoretical model to a FLR observation in the nightside magnetosphere at 0426 UT of 31 January 1997 Lu et al., 2007. This observation (Figure 1) was made using magnetometer and meridian scanning photometer (MSP) data from ground-based instruments of the CANOPUS array, and the All-Sky Imager (ASI) from the NORSTAR optical imaging array operated by the University of Calgary. It was associated with wave power at a frequency 1.3 - 1.4 mHz, and corresponding solar wind conditions By = 3 nT, Bz = 1 nT, n = 3.5cm-3, P = 2.0 nPa, and Dst = 15. The length of the field line projected above the observation point on the ground is 24.6 RE. The maximum radial extension of the field line is 10.97 RE, and the minimum geomagnetic field strength along the field line is 18.98 nT Rankin et al., 2005. Here, we use T96 model and the above parameters to approximate the geomagnetic field for this event. The length of the field line is 22 RE, while the equatorial density is chosen so as to give a frequency of 1.3 mHz that matches the observation seen from the ground. The initial ambient Pedersen conductance at conjugate ionosphere is 2.5 S.

Figure 2 shows the azimuthal magnetic field, field-aligned current, and perpendicular electric field as a function of the coordinate perpendicular to the field line at an altitude of 3000 km. The maximum wave magnetic field is about 65 nT, and the maximum parallel current and perpendicular electric field saturate at roughly 2 mA/m2 and 19 mV, respectively. The half-width of the arc is roughly 35 km, which is quite comparable to the perpendicular scale for this observation. The maximum wave magnetic field along the field line (Figure 3) is 85 nT with amplitudes of 10 to 25 nT at most of the locations, while the maximum electric field is 33 mV and the maximum parallel current 3 mA/m2. These amplitude scales are comparable to the observed values for this observation. Along the field line, the maximum relative density perturbation is 20%, while the maximum density accumulation (which occurs at the equatorial plane) is 11% compared with the background value.

Fig. 2. Field aligned current, perpendicular electric field, and perturbed magnetic field at the ionosphere computed from a MHD FRL model for the 31 January 1997 observations. See Lu et al. 2006.

Fig. 3. From top to bottowm is shown the azimuthal magnetic field, perpendicular electric field, FAC current, and relative density perturbation as a function of distance along the field line from one ionosphere to the other. See Lu et al. 2006.

References:

1. Lu J. Y., R. Rankin, R. Marchand, I. J. Rae, W. Wang, S. C. Solomon, and J. Lei, Electrodynamics of magnetosphere-ionosphere coupling and feedback on magnetospheric field line resonances, J. Geophys. Res.., submitted, 2007.
2. Rankin, R., K. Kabin, J. Y. Lu, I. R. Mann, R. Marchand, and I. J. Rae, V.T. Tikhonchuk, and E. Donovan, Magnetospheric field line resonances: ground-based observations and modeling, J. Geophys. Res.., 110, A10S09, doi: 10.1029/2004JA01919, 2005.

References#

1. Atkinson, G., Auroral arcs: Result of the Interaction of a Dynamic Magnetosphere with the Ionosphere, J. Geophys. Res., 75(25), 4746, 1970.
2. Chaston C.C., et al., Ionospheric erosion by Alfven waves, J. Geophys. Res., 111, A03206, doi:10.1029/2005JA011367, 2006.
3. Liu, W. W., Physics of the explosive growth phase: Ballooning instability revisited, J. Geophys. Res., 102, 4927, 1997.
4. Lu J. Y., R. Rankin, R. Marchand, I. J. Rae, W. Wang, S. C. Solomon, and J. Lei, Electrodynamics of magnetosphere-ionosphere coupling and feedback on magnetospheric field line resonances, J. Geophys. Res.., submitted, 2007.
5. Lu, J. Y., R. Rankin, R. Marchand, and V.T. Tikhonchuk, Nonlinear electron heating by resonant shear Alfven waves in the ionosphere, Geophys. Res. Letts.., 32, L01106, doi:10.1029/2004GL021830, 2005a.
6. Lu, J. Y., R. Rankin, R. Marchand, and V.T. Tikhonchuk, Reply to comment on " Nonlinear electron heating by resonant shear Alfven waves in the ionosphere" by J.-P. St.-Maurice, Geophys. Res. Letts.., 32, L13103, doi:10.1029/2005GL023149, 2005b.
7. Lu, J.Y., R. Rankin, R. Marchand, V.T. Tikhonchuk, and J. Wanliss, Finite element modelling of nonlinear dispersive field line resonances: trapped shear Alfven waves inside field-aligned density structures, J. Geophys. Res., 108(A11), 1394, doi:10.1029/2003JA010035, 2003a.
8. Lu, J.Y, R. Rankin, R. Marchand, and V.T. Tikhonchuk, Nonlinear Acceleration of Dispersive Effects in Field Line Resonances, Geophys. Res. Lett., 30(10), 1540, doi: 10.1029/2003GL016929, 2003b.
9. Marchand, R., and M. Simard, Finite element modelling of Tdev edge and divertor with E _ B drifts, Nucl. Fusion, 37, 1629, 1997.
10. Prakash, K., R. Rankin, and V. T. Tikhonchuk, Precipitation and nonlinear effects in geomagnetic field line resonances, J. Geophys. Res., 108(A4), 8014, doi:10.1029/2002JA009383, 2003.
11. Rankin, R., J. C. Samson, and V. T. Tikhonchuk, Parallel electric fields in dispersive shear Alfv\'en waves in the dipolar megnetophere, Geophys. Res. Lett., 26, 3601, 1999.
12. Rankin, R., K. Kabin, J. Y. Lu, I. R. Mann, R. Marchand, and I. J. Rae, V.T. Tikhonchuk, and E. Donovan, Magnetospheric field line resonances: ground-based observations and modeling, J. Geophys. Res.., 110, A10S09, doi: 10.1029/2004JA01919, 2005.
13. Rankin, R., R. Marchand, J. Y. Lu, K. Kabin, and V.T. Tikhonchuk, Theory of Dispersive Shear Alfven Wave Focusing in Earth's Magnetosphere, Geophys. Res. Letts.., 32, L05102, doi: 10.1029/2004GL021831, 2005.
14. Ruohoniemi, J. M., R. A. Greenwald, K. B. Baker, and J. C. Samson, HF radar observations of Pc 5 field resonances in a midnight/early morning MLT sector, J. Geophys. Res., 96, 15,697, 1991.
15. 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.
16. Samson, J. C., T. J. Hughes, F. Creutzberg, D. D. Wallis, R. A. Greenwald, and J. M. Ruohoniemi, Observations of detached, discrete arc in association with field line resonances, J. Geophys. Res., 96, 15,683, 1991.