A suggestion that two-dimensional turbulence contributes to polar cap convection for Bz north



[1] Recently, a linear dependence has been found between the polar cap potential and the log of ULF power in the solar wind for Bz north. We argue here that this is very strong evidence that Alfvén waves propagating from the solar wind to the polar cap drive two-dimensional turbulence in agreement with other work along these lines. In brief, two-dimensional (2-D) turbulence is characterized by an inverse cascade of energy from intermediate to large scales, eventually leading to two large counter-rotating vortices filling the available volume and, we think, creating two of the four vortices in the Bz north polar cap. In this model, the Alfvén waves create velocity shear at the 500 or so kilometer scale associated with polar cap arcs. This energy/vorticity then inverse cascades to fill the polar cap.

1. Introduction

[2] Recently, Kim et al. [2011] reported that, for Bz north in the solar wind, the polar cap potential is linearly dependent on the log of ULF power in the solar wind. Figure 1 shows their main result, which was determined using fluctuations in the interplanetary magnetic field (IMF).

Figure 1.

Bin averages of all the data points for northward IMF, i.e., data points with IMF Bz > 1.5 nT. The triangles represent the bin averages with bin width of 0.2 ULF index, and the bars indicate the standard deviation of the data points in each bin. The 95% confidence interval for each bin average is indicated by the horizontal ticks on the bars.

[3] The ULF power is due to either magnetohydrodynamic (MHD) turbulence in the solar wind or to Alfvén waves generated in the solar wind. Either way, these fluctuations will propagate to the polar cap on field lines connected to the solar wind as Alfvén waves. The link between the solar wind and the polar cap for Bz north is presumably due to connections between the interplanetary magnetic field (IMF) and the earth's field in the polar cusp [Kelley, 2009]. This connection has long been linked to the two polar cap convection cells. The mechanism discussed here may contribute to these cells.

[4] The energy flux from the solar wind to the magnetosphere has long been studied for a steady solar wind velocity and various IMF geometries and quantified by Weimer [2005]. Fluctuations in the velocity and the IMF are well-documented. The solar wind area in magnetic contact with the polar cap may act like an aperture antenna launching Alfvén waves toward the earth. This presumably occurs for any IMF geometry but during Bz south, dayside convection dominates the polar cap potential (PCP). We argue here that, for Bz north, the mechanism discussed here competes for control of the PCP equally with cusp connections and, at lower latitudes, with viscous interaction [Axford and Hines, 1961].

[5] There are two aurora characteristics in the Bz-north polar cap. The most well-known is the theta aurora [Frank et al., 1986]. This auroral form splits the polar cap and is collocated with an upward field-aligned current. Cumnock et al. [2002] showed that this arc erupts out of the auroral oval and migrates across the polar cap to its theta-bar location. This was verified in a rocket experiment in which consecutive rockets crossed an arc emerging out of the oval. Leaving sunward convection equatorward of the arc [Berg et al., 1994], the arc itself propagated at a speed higher than a realistic convection speed. This is similar to the rapid poleward (apparent) motion of substorm-related aurora.

[6] The second type of polar cap aurora was reported by Lassen and Danielsen [1978] and is shown in Figure 2. These multiple arcs were synthesized from all-sky camera data over many years. The more northward Bz is, the more polar cap arcs exist. But multiple arcs are seen on a given night, not just a single theta aurora [Rich et al., 1990]. Figure 3 shows an example of multiple reversals in the electric field, each one of which has a field-aligned current associated with it. In fact, since ∇ × V = Jz (upward in northern hemisphere) [Kelley, 2009], even small variations in V may have subvisual arcs.

Figure 2.

Mass plots of arcs in corrected geomagnetic coordinates for different values of Bz. Clearly, the polar cap has many arcs for Bz north and none for Bz south [after Lassen and Danielsen, 1978].

Figure 3.

Coincident measurements by ACE (−Vsw × Bz), Jicamarca vertical drift, AE, and Sym-H [after Kelley and Dao, 2009].

[7] The reciprocal of the intrinsic impedance of the Alfvén (ηA) wave is comparable to the height-integrated Pedersen conductivity (ΣP) of the polar cap, so energy will be absorbed or reflected, depending on the season and other factors. However, when a reflected wave returns to the solar wind, again, some of the energy will be reflected back to the ionosphere. Thus, a constant source of Poynting flux exists. Using transmission line theory [Inan and Inan, 2000], the reflection coefficient is,

display math

[8] Large fluctuations in the solar wind are often observed, in particular during high speed streams, as shown in Figure 3 where the fluctuations in the interplanetary electric field (IEFy) are large and the lowest frequencies penetrate to the equatorial ionosphere (panel 2) [Kelley and Dao, 2009]. In the polar cap, the fluctuations are also the order of the mean [Kelley, 2009]. The fluctuations in the winter polar cap are much larger than in the summer polar cap [Heelis and Hanson, 1980; see also Kelley, 2009, Figure 8.8b]. The high conductivity of the summer cap will reflect more solar wind energy than the winter cap. Thus, larger fluctuation electric fields are expected in winter, as observed.

[9] For Bz north, polar cap arcs are ubiquitous, as shown in Figure 2, and common in the winter polar cap. The tapering of the electric field from the solar wind to the ionosphere goes as (BSW/BI)1/2 [Mozer, 1970]. Using BSW = 5 nT and BI = 50,000 nT, 8 × 105 km in the solar wind becomes 80 km in the ionosphere. The vortices are even smaller than this, as shown in Figure 4, indicating vortices of 10 km in the polar cap for Bz north. We believe that 2-D turbulence theory explains vortex expansion from 10 km to the two large vortices as being due to inverse cascade, which characterizes this type of turbulence [Kraichnan, 1967].

Figure 4.

High-latitude plasma drift velocity in the ionosphere shown for a northward IMF. The flow is extremely structured and does not indicate a simple two-cell convection pattern, as is often the case during southward IMF [after Heelis and Hanson, 1980].

[10] Earle and Kelley [1993] pointed out that electric field data on rockets and satellites [Kelley and Kintner, 1978; Boehm et al., 1995] indicate a break in the slope of polar cap electric-field spectral density around 1 km scale with a dependence of k−5/3 at smaller k and k−3 at larger k where k denotes a wave number. These results are predicted by Kraichnan [1967] and hence are evidence for 2-D turbulence.

2. Discussion

[11] Many geophysical phenomena have been discussed in light of the theory of 2-D turbulence, including ocean dynamics [Kelley, 1984] and atmospheric phenomena [Larsen et al., 1982]. This was first described by Kraichnan [1967], who found that if energy is input at some scale, it cascades to larger scales, unlike 3-D turbulence in which energy only cascades to smaller scales. In addition, angular momentum, called entropy in this context, cascades to smaller scales. The 3-D and 2-D spectra vary as k−5/3 and k−3, respectively, for large k. These are exactly what Earle and Kelley [1993] and Kelley and Kintner [1978] have reported.

[12] It has been established that auroral arcs are collocated with velocity shears [Carlson and Kelley, 1977; Weber et al., 1989] and Alfvén waves [Earle and Kelley, 1993; Boehm et al., 1995]. Velocity shear and the corresponding vorticity are such that upward field-aligned currents and associated precipitating electrons occur for positive vorticity. We believe that polar cap arcs, which fill the polar cap for Bz north [Lassen and Danielsen, 1978], are the signature of positive vorticity (W) since Wz = (∇ × Vz) = J|| [Kelley, 2009]. Thus, the arcs show the input scales for 2-D turbulence, which, as argued next, contribute to the PCP for Bz north.

[13] Energy cascading proceeds until there are two counter-rotating vortices filling the available space, a result confirmed using both a computer simulation [Seyler et al., 1975] for the Navier-Stokes equation in 2-D and a guiding center plasma, which obey the same equations.

[14] An intermediate state was, perhaps, detected by the AE-C satellite, as shown in Figure 4. Here, nearly 100 small shears were detected and a dozen larger-scale shears during northward Bz. Each region with negative vorticity corresponds to an upward field-aligned current and to precipitating electrons [Kelley, 2009]. It is easy to see that further coalescence of vortices might occur.

[15] We propose here that Alfvén waves input energy into the polar cap at a relatively small scale, say, less than 10 km [Weber et al., 1989]. An inverse cascade then occurs, eventually creating two of the four vortices observed by Burke et al. [1979]. The other two vortices likely are due to viscous interaction with the solar wind.

3. Conclusion

[16] We propose a new explanation for polar cap convection, or at least an idea competing with cusp connection, when Bz is northward. Kim et al. [2011] showed that the polar cap potential connection scales with the ULF index. These waves, we argue, may create a stirring scale for 2-D turbulence, supporting an inverse cascade that may explain two of the scales in the polar cap. The lower latitude cells may be due to viscous interaction.


[17] Work at Cornell was sponsored by the National Science Foundation under grant ATM-0551107 and at UCLA by NASA grant NNX09AJ72G.

[18] The Editor thanks Frederick Rich and an anonymous reviewer for their assistance in evaluating this paper.