We present a method for estimating the portion of the ionospheric high-latitude potential that maps to the magnetospheric boundary layer during steady northward IMF and global ionospheric 4-cell convection patterns associated with lobe reconnection, together with the results of a statistical study based on DMSP F13 data from 1996–2004. In comparison with a previous study for steady southward IMF by K. Å. T. Sundberg et al. (2008), the results show significantly larger boundary layer potentials, with a mean value of 10 kV for the 271 events studied, corresponding to roughly 30–35% of the potential generated by the solar wind interaction. In a statistical analysis, the boundary layer potential is also shown to depend significantly on viscous parameters such as the solar wind velocity, density and pressure.
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 The solar wind flow past the Earth is continuously transferring both energy and plasma into the terrestrial magnetosphere, and to understand the nature of this transfer and the factors controlling it is fundamental to solar-terrestrial science. Magnetic reconnection has long been regarded as the principal means for how the energy transfer takes place. The idea was first suggested by Dungey , and a significant amount of work has since then been carried out in order to understand the details of the reconnection mechanism; see, for example, Sibeck et al.  or Lui et al.  for extensive reviews on the topic. Reconnection is considered especially effective when the interplanetary magnetic field (IMF) is directed southward, anti-parallel to the dayside magnetospheric field. During these circumstances, the main reconnection site is located on the dayside magnetopause. For northward IMF, reconnection can take place either as lobe reconnection tailward of the cusp [e.g., Dungey, 1963; Crooker, 1988, 1992; Gosling et al., 1991], or as component reconnection on the dayside [Cowley, 1976; Cowley and Owen, 1989]. The energy transfer due to either of these reconnection topologies is significantly lower than during southward IMF conditions.
 Several other dynamos have been proposed to contribute to the energy transfer for both southward and northward IMF, but their efficiency remains to be established. A group of such processes that may supply the low-latitude boundary layer (LLBL) with energetic magnetosheath plasma are referred to as viscous interaction. The fundamental idea is that the magnetosheath shear causes instabilities and vortices on the flanks of the magnetopause, which in turn can lead to a local breakdown of the frozen-in flux condition and allow plasma to diffuse into the magnetosphere or field lines to reconnect locally inside the turbulent region. The main source treated in the literature and observed in spacecraft data is the Kelvin-Helmholtz instability [e.g., Fairfield et al., 2000; Nykyri and Otto, 2001; Hasegawa et al., 2004, 2006] although instabilities such as small-scale drift-kinetic Alfvén vortices [Johnson and Cheng, 1997; Sundkvist et al., 2005] and the lower hybrid drift instability [Gary and Eastman, 1979; Treumann et al., 1992] are other possible sources. It is still unclear however how important these instabilities are for the total energy transfer.
 The 4-cell convection pattern [e.g., Burke et al., 1979; Cumnock et al., 1995] is a general ionospheric signature of a global lobe reconnection topology where magnetic reconnection poleward of the cusp drives sunward convection over the polar cap. According to the model presented by Crooker , the reconnection driven sunward flow is connected to two return flow regions, one on each side at lower latitudes, creating two separate high-latitude reverse convection cells, assumed to be on open field lines. In addition, the low-latitude dynamos may still drive anti-sunward convection in the boundary layer, creating a low-latitude cell on the dawn and dusk flank respectively, thus completing the 4-cell pattern. In order to determine the potential associated with each of these dynamos, we need in principle to find the polar cap boundary (the open-closed field line boundary). The use of particle precipitation data for determining the polar cap boundary is inherently difficult for northward IMF, high electron fluxes in the form of polar showers are often present in the polar cap, sometimes accompanied by ion precipitation [Shinohara and Kokubun, 1996]. The ion population at the polar cap boundary in general show a gradual increase in number and energy flux, making trustworthy definitions based on particle precipitation characteristics impossible.
 However, assuming the Crooker  model, we can divide the potential distribution into a few measurable regions, as shown in Figure 1. The reverse convection potential, ΦRC, taken as the potential difference over the central region of sunward flow, functions as a good measure of the present energy input due to lobe reconnection. The end-points are defined by the local minimum or maximum potentials found at the convection reversals. In a steady state configuration, we can assume that the potential over the return flow regions connected to the reverse convection, ΦRCRF, should be equal to the reverse convection potential: ΦRC = ΦRCRF. Furthermore, we can determine the potential difference over the two regions of anti-sunward convection, ΦASC, each defined from the boundaries of the reverse convection region to the first significant occurrence of sunward convection on each flank. Given the previous assumption of steady state, we can then determine the total of the dawn and dusk side boundary layer potentials, ΦBL, and thus an estimate of the low-latitude energy transfer, as the remaining potential in the anti-sunward convection region not covered by the reverse convection return flow, i.e. ΦBL = ΦASC − ΦRCRF. All the potentials above are defined positive.
 This method for determining the boundary layer potential may not be a trustworthy procedure for treating data on a case-by-case basis, as the values are derived with a high degree of uncertainty due to time variations and deviations from the ideal case in both the convection pattern and the satellite's trajectory. When treating larger datasets, however, the method should be fairly accurate for detecting trends and for determining the order of magnitude, as we do not expect any systematic bias in the measurements.
3. Data Set
 The data used in this study are from DMSP F13, an ionospheric satellite in a sun-synchronous polar orbit at 850 km altitude, fixed in local time approximately in the dawn-dusk meridian. The along-track electric field data are derived from the cross-track plasma velocity, under the assumption that the plasma motion is exclusively due to the × drift. The data set used is the same as the one derived by Sundberg et al. , consisting of 271 lobe reconnection events selected during the period 1996–2004. The selection criteria were the following: (1) the IMF should be strictly northward with Bz > ∣By∣ during at least 2 hours preceding the event, (2) the plasma convection should follow a large-scale 4-cell pattern typical for lobe reconnection, and (3) the orbit should pass on the dayside, above 80 degrees MLAT. This will in general give trustworthy lower-limit estimates of the reverse convection and boundary layer potentials.
 The algorithm used to detect 4-cell convection patterns requires a clear region of sunward convection near the maximum latitude passage of the satellite, surrounded by two regions of anti-sunward convection. The potential profile along the orbit requires at least two local maxima and two local minima, corresponding to the main convection reversals. Additional local maxima or minima may be acceptable if they do not alter the overall convection pattern significantly. In addition, each of the three regions are required to have a spatial extent exceeding 100 km in order to make sure that the regions are well defined and that the global topology is easily interpretable.
 All solar wind data presented are from the OMNI 5 minute averaged data set provided by NASA's Space Physics Data Facility (SPDF), and the parameters given in the statistical analysis are mean values taken over one hour periods. The potential data have been corrected for instrumental bias and time variation effects by forcing the potential to zero at 65° MLAT, and any offset over the interval has been symmetrically removed. It is worth noting that not all events with steady northward IMF result in 4-cell convection patterns. Their occurrence is strongly favored for low solar zenith angles, and thus for higher ionization levels in the high-latitude ionosphere, see Sundberg et al.  for details.
 The results from a study of linear regression of the estimated boundary layer potential on a set of solar wind variables is displayed in Table 1. The variables included in the study were the solar wind velocity, v, number density, n, pressure, P, the magnetic field components, BX, BY, BZ, and BT = BY2 + BZ2), and the electric field in the transverse plane, E = vBT. Among the simple variables tested, the solar wind pressure shows the best correlation with the derived boundary layer potentials, with the ratio of explained variance to total variance R2 = 16.5%. If we add an E-dependence to the model, i.e. ΦBL = c0 + c1P + c2E, where c0, c1, and c2 are regression coefficients, the R2 value only increases with 1%, a fairly insignificant increase. These results indicate that most of the correlation between ΦBL and E is likely due to a co-dependence on v and that no significant direct relation exists between the two. It is of importance to note that the results above do not necessarily indicate a direct (causal) dependence between ΦBL and P. It should rather be considered as an interplay of the different viscous parameters, i.e., the solar wind velocity, density and/or the pressure. Other viscous-like functions give similar or better results, such as the v2 function provided by Newell et al. , which gives an R2 value of 27% (corresponding to a correlation coefficient of 0.52), although the direct physical significance of this function is unclear. It is important to note that even though the R2 values reported are rather low, they are clearly statistically significant. These findings thus bring credibility to the method used for deriving the boundary layer potential, which leads us to believe that the potentials are on average reliable estimates and that the method is adequate for this type of analysis. When individual cases are considered however, they are still to be treated only as rough estimates. The analysis assumes that no dayside merging cells are present, as in the model by Burch et al. . Such cells are not likely to form for the IMF Bz conditions imposed on the data set [Reiff and Burch, 1985], and any larger contribution from such cells would be reflected in a stronger E or By dependence in the regression analysis.
Table 1. R2 Value for Linear Regressions of the Boundary Layer Potential on Different Solar Wind Parametersa
Solar Wind Parameter
The R2 value is the ratio of explained variance to total variance. The combined function of P and E result is given by a multiple linear regression model.
P and E
 With the general acceptance of the boundary layer potential estimates, we can proceed to investigate their properties and distribution, shown in a histogram in Figure 2. Both the mean and the median values of the distribution are around 9–10 kV. If instead the ratio of the boundary layer potential to that of the total dynamo related potential is considered, i.e., ϕBL/(ϕBL + ϕRC), the mean and median values average to around 30–35%.
 Previous estimates of the boundary layer potential, based either on case studies or less extensive datasets [e.g., Sonnerup, 1980; Mozer, 1984; Newell et al., 1991; Lu et al., 1994], have in general derived values of the order of 5–10 kV, with the exception of a study by Blomberg et al.  which indicated the existence of higher boundary layer potentials (the selected events were all from active times likely related to southward IMF). None of the studies above have treated IMF related effects to any large extent. The 28 crossings of the dusk side boundary layer studied by Mozer  did show a slight tendency for larger potentials during northward IMF, and the few events which contained a negative potential drop (sunward convection in the boundary layer) all occurred during southward IMF conditions but the differences were not regarded as significant. In a comprehensive statistical study by Sundberg et al.  treating 2-cell convection patterns during steady southward IMF, the low-latitude dynamo was shown to be less important than previous predictions and estimates, contributing merely 1–2 kV to the total potential drop in the average case. For close to half of the events in that data set the low-latitude dynamo contributed less than 1 kV to the total potential drop. In comparison, the estimates presented here show the presence of a significantly larger boundary layer potential for northward IMF conditions, which indicates that the IMF direction is a vital factor in the formation of the boundary layer. Both studies show a tail in the potential distribution ranging as far out as 25 kV, however the higher potentials are more common in the northward IMF data set.
 The increased potential during steady northward IMF is likely consistent with previous high altitude studies of the boundary layer by Mitchell et al.  and Hasegawa et al. , where the former concluded that the boundary layer is in general thicker for northward IMF conditions, and the latter that a dense and stagnant boundary layer appears during northward IMF. Another indication that the formation of the boundary layer is less effective during southward IMF was presented in a study of 235 magnetopause crossings without a boundary layer by Eastman et al. . These were found to occur significantly more frequently for southward than northward IMF, with a 60% compared to 15% occurrence frequency. Statistical studies of the properties of the plasma sheet [Terasawa et al., 1997] also indicate an increased inflow of cold and dense solar wind plasma when the IMF is northward, accredited to a diffusive-like transport.
 The results of our statistical analysis indicate that the main entry mechanisms is of viscous-type, which agrees well with previous studies of the Kelvin-Helmholtz instability: Hasegawa et al.  reported an increase of rolled-up Kelvin-Helmholtz instabilities for steady northward IMF, to some extent associated with plasma transport into the LLBL. Fairfield et al.  and Otto and Fairfield  have also reported that the formation of strong Kelvin-Helmholtz instabilities may be favored by northward IMF.
 One may also consider the option of a boundary layer formed by lobe reconnection in both hemispheres as proposed by Song and Russell  and Song et al. , a process which would create closed field lines on the dayside containing solar wind plasma. However, such a convection cell would likely close in the reverse convection return flow region [Song et al., 2000], and would not would be detectable with the method presented here. This would allow for an extra contribution to the boundary layer, possibly making the values presented here lower-limit estimates of the true boundary layer potential.
 The method presented here for deriving the boundary layer potential contribution for 4-cell convection patterns should on average give reliable estimates, although with a large error variance on a case-by-case basis. The method is validated by a statistical analysis showing a clear dependence between ΦBL and the viscous parameters in the solar wind (velocity, density and pressure), whereas the dependence on the interplanetary electric field is insignificant. The data show significantly larger boundary layer potentials compared to a previous study for southward IMF conditions [Sundberg et al., 2008], on average contributing approximately 10 kV (30-35%) to the total potential generated by the solar wind interaction. The results thus indicate that the IMF direction is a vital factor in the formation of the boundary layer.
 Work at the Royal Institute of Technology was supported by the Swedish Research Council. Work at the University of Texas at Dallas was supported by NSF grant ATM0536868. The authors thank the ACE, IMP-8, and Wind instrument teams for providing magnetometer and plasma data through the OMNIWeb data explorer. The DMSP thermal ion data were obtained from the Center for Space Sciences at the University of Texas at Dallas. We thank Rod Heelis, Marc Hairston, and Robin Coley for its use.