Theory and observations of upward field-aligned currents at the magnetopause boundary layer

Abstract

The dependence of the upward field-aligned current density (J_‖) at the dayside magnetopause boundary layer is well described by a simple analytic model based on a velocity shear generator. A previous observational survey confirmed that the scaling properties predicted by the analytical model are applicable between 11 and 17 MLT. We utilize the analytic model to predict field-aligned currents using solar wind and ionospheric parameters and compare with direct observations. The calculated and observed parallel currents are in excellent agreement, suggesting that the model may be useful to infer boundary layer structures. However, near noon, where velocity shear is small, the kinetic pressure gradients and thermal currents, which are not included in the model, could make a small but significant contribution to J_‖. Excluding data from noon, our least squares fit returns log(J_{‖,max_cal}) = (0.96 ± 0.04) log(J_{‖_obs}) + (0.03 ± 0.01) where J_{‖,max_cal} = calculated J_‖,max and J_{‖_obs} = observed J_‖.

1. Introduction

Velocity shears exist at the low-latitude boundary layer where on one side of the magnetospheric boundary the magnetosheath flow is strongly antisunward and on the other side, the magnetospheric flow is weakly sunward. The velocity shear establishes an electric potential drop across the boundary, which drives field-aligned current (FAC) [e.g., Sonnerup, 1980; Stern, 1983; Siscoe et al., 1991]. Lyons [1980] notes that near the boundary layer on the duskside, the convective electric field converges (∇ · E < 0), which can lead to large-scale upward field-aligned current. Echim et al. [2007, 2008] recently developed a 1-D kinetic model describing the current-voltage relationship in the upward FAC region. The model uses the Vlasov equation [Roth et al., 1996; Echim and Lemaire, 2005] to obtain the self-consistent electric potential drop in the current generator region constrained by boundary conditions imposed at the magnetosheath and magnetospheric boundaries. The boundary layer model is coupled to the ionosphere by continuity of FACs, which are determined from the nonlinear Knight relation [Knight, 1973], and ionospheric currents, which are controlled by the height-integrated Pedersen conductivity (Σ_p).

Wing et al. [2011] examines the dependence of FAC density (J_‖), peak electron energy (as a proxy for the field-aligned potential drop (Δφ_‖) [e.g., Lyons, 1980]), and electron energy flux (ε) on solar wind velocity (V_sw) and solar wind density (n_sw) in the afternoon upward FAC region located at the boundary layer or open field lines. The study finds that J_‖ increases with increasing n_sw and V_sw, consistent with the model of Echim et al. [2008]. As V_sw increases, velocity shear at the boundary increases, which leads to higher potential drop across the boundary, which in turn drives larger J_‖. An increase in n_sw increases the number of electron current carriers, which tends to increase J_‖, assuming all other parameters remain the same. On the other hand, as n_sw, the number of current carriers, decreases, a parallel potential drop develops to draw more electrons downward to carry the current. Consequently, it is expected that Δφ_‖ increases as n_sw decreases. This inverse relation is seen in DMSP observations [Wing et al., 2011] as well as in the model [Echim et al., 2008]. In both the model and DMSP observations, Δφ_‖ increases with increasing V_sw because larger V_sw drives larger current, J_‖, which, in turn, requires larger Δφ_‖, assuming other parameters remain the same.

The good comparisons of the observations presented in Wing et al. [2011] with the model of Echim et al. [2008] suggest that the model captures the general dependencies of these dayside magnetosphere-ionosphere (M-I) coupling parameters on V_sw and n_sw and that much of the dynamics of these parameters could be understood from the model. Motivated by Echim et al. [2008] and Wing et al. [2011], Johnson and Wing [2015] derives simple analytic expressions that capture the dependence of the upward field-aligned current and its spatial scale on solar wind and ionospheric parameters. The study verifies the analytical results through comparison with the rigorous approach of Echim et al. [2008] and with DMSP observations in the upward FAC region that is located at the boundary layer and open field lines.

Lyons [1980] derives an expression for upward J_‖ at a sharp velocity shear at the boundary layer, but it does not include velocity shear layer thickness, Δ_m. Johnson and Wing [2015] considers a similar approach as Lyons [1980], but it includes Δ_m. As shown in Johnson and Wing [2015] and summarized below, Δ_m is an important parameter for estimating J_‖. For a given velocity shear and magnetic field (or electric field) at the boundary layer, the potential drop across the boundary and J_‖ vary inversely with Δ_m. Moreover, FAC latitudinal thickness (Λ) scales with Δ_m. The dependence of J_‖ on Δ_m also makes it possible to infer boundary layer structure from measurements of ionospheric and solar wind parameters, which is a significant advance not possible using the model of Lyons [1980].

Johnson and Wing [2015] provides a theory and then compares the scaling relationships of J_‖ as a function of n_sw, V_sw, and Δ_m. The study also examines scaling relationships of FAC latitudinal width (Λ) versus n_sw and Δ_m versus V_sw. The general trends identified in that analysis are consistent with the theory, but there is substantial spread in the data because of the variance in the unconstrained parameters. More recently, Wing et al. [2015] finds the parallel electric potential predicted by the Johnson and Wing [2015] theory is in fairly good agreement with observations. The present study compares the observed J_‖ with that predicted from Johnson and Wing [2015] theory.

2. Analytical Theory for Upward FAC at the Magnetopause Boundary

Johnson and Wing [2015] considers a velocity shear layer with a width Δ_m at the magnetopause

$$V_y(x)=\frac{V_0}{2}(1+\text{tanh}(x/{\mathrm{\Delta }}_m))$$ (1)

that sustains a system of field-aligned currents flowing into and through the ionosphere as described by the continuity equation

$$\frac{\mathrm{d}}{\mathrm{d}x}{\sum }_p\frac{\mathrm{d}{\varphi }_i}{\mathrm{d}x}=J_{\Vert }({\varphi }_m,{\varphi }_i)$$ (2)

where the field-aligned current into the ionosphere is determined from the potential drop between the driver and the ionosphere assuming a linear Knight relation [Knight, 1973]

$$J_{\Vert }=\kappa ({\varphi }_i-{\varphi }_m)$$ (3)

where V₀ = velocity difference across the boundary layer, ϕ_i = electric potential in the ionosphere, ϕ_m = electric potential in the magnetosphere, κ = n_ee²/(2πm_ek_bT_e)^0.5, n_e = electron number density, e = electron charge, m_e = electron mass, T_e = electron temperature, and k_b = Boltzman’s constant. The assumption of a linear current-voltage relationship in equation (2) ignores thermal current, nonlinear saturation, and restricts the magnetospheric electron distribution function to a Maxwellian. The remarkable similarity of the analytical scaling relations with observations [e.g., Wing et al., 2011; Johnson and Wing, 2015] and to the numerical solutions of Echim et al. [2008] (which include a self-consistent treatment of the nonuniform magnetopause boundary layer including pressure gradients, nonlinear Pedersen conductivity, and nonlinear Knight relation) suggest that the simple relations probably capture the most important physical dependencies on the solar wind and ionospheric parameters.

Johnson and Wing [2015] obtain an exact analytical solution of the field-aligned current in the ionosphere

$$J_{\Vert }(x)=\kappa \frac{V_0{B}_m{\mathrm{\Delta }}_m}{2}(\frac{\pi }{2}\frac{e^{-\left|x\right|/L}}{\text{sin}(\pi \alpha )}+\sum _{n=1}^{\infty }{(-1)}^n\frac{n{e}^{-2\left|x\right|/{\mathrm{\Delta }}_i}}{n^2-{\alpha }^2})$$ (4)

where ${\mathrm{\Delta }}_i={\mathrm{\Delta }}_m/\sqrt{b}$ is the boundary layer thickness mapped to the ionosphere using the magnetic field ratio, b ≡ B_i/B₀, of the ionosphere field, B_i to the magnetospheric field, B₀. The parameter α = Δ_i/2L, which is the ratio of the mapped boundary layer thickness to the electrostatic auroral scale height, $L=\sqrt{{\sum }_p/\kappa }$, determines whether the magnetospheric potential maps to the ionosphere. The maximum current can be approximated by the simple expression,

$$J_{\Vert ,\text{max}}=\frac{{\sum }_p{V}_0{B}_{0}b}{2({\mathrm{\Delta }}_m+\sqrt{b}L)}$$ (5)

relating J_‖,max to observable solar wind, magnetospheric, and ionospheric parameters.

One of the parameters in equation (5), the width of the boundary layer, Δ_m, can be obtained directly from high-altitude satellite observations, but it can also be inferred from ionospheric observations and considerations of ionosphere-magnetosphere mapping scale length. From the full width at half maximum of the parallel current, Λ, Johnson and Wing [2015] derives an equation that relates the observable quantity Λ (the latitudinal width of the FAC current sheet) to Δ_m

$$1.0224{\alpha }^2+(3.5255-0.29\frac{\Lambda }{L})\alpha -(\frac{\Lambda }{L}-1.3863)=0$$ (6)

The quadratic equation can be solved for α to obtain ${\mathrm{\Delta }}_i={\mathrm{\Delta }}_m/\sqrt{b}=2\alpha L$.

3. Data and Methodology

Johnson and Wing [2015] presents theory-data comparisons for the dependencies of several magnetosphere-ionosphere (M-I) coupling parameters, as mentioned in section 1. For the present study, we compare the J_‖ calculated from equation (5) with J_‖ obtained observationally in region 1 (R1) afternoon upward FAC that is located at boundary layer and open field lines. The present study uses observations from DMSP Special Sensor Precipitating Electron and Ion Spectrometer (SSJ4/5) [Hardy et al., 1984] and magnetometer data [Rich et al., 1985] from 1983 to 2006. The data treatment has been described in Wing et al. [2010, 2011] and Johnson and Wing [2015]. Hence, only a brief summary is given here. We identify large-scale field-aligned currents using the Higuchi and Ohtani [2000] algorithm, which uses the first-order B-spline to fit line segments to the azimuthal component of the DMSP magnetic field data. Only data that have R_fit < 10 are selected (R_fit gives a measure of the goodness of fit). For large-scale (>1°) FAC, an infinite sheet approximation holds and each line segment corresponds to a crossing of a FAC sheet. The field-aligned current source region is determined by simultaneous particle precipitation signature. Newell and Meng [1988] and Newell et al. [1991a, 1991b, 1991c] developed an automatic algorithm to determine the origins of the precipitating particles. This algorithm is used to determine whether the FAC occurs on boundary layer, open or closed field lines.

Solar wind observations come from ACE, WIND, IMP8, ISEE1, and ISEE3 spacecraft. Using the minimum variance technique [Weimer et al., 2003], solar wind is propagated to GSM (X, Y, Z) = (17, 0, 0) R_E. From there, the solar wind is propagated to the ionosphere with a constant propagation time of 10 min (~2 min from X = 17 R_E to magnetopause location at X = 10 R_E with V_sw = 450 km/s + 5–6 min for a delay in the magnetosheath [e.g., Lockwood et al., 1989; Spreiter and Stahara, 1985] +2–3 min for the magnetopause to the ionosphere [Keller et al., 2002; Lockwood et al., 1989; Rostoker et al., 1982]). Thirty minute averages of solar wind parameters centered on the time of the DMSP encounters, the equatormost boundary of FAC, are calculated. Due to these rather long time averages, the exact solar wind propagation time is not crucial. The 30 min averages of solar wind parameters are assigned for the entire FAC encounter.

We calculate J_‖,max using equation (5) and parameters that are estimated observationally and through the use of empirical formulas, as described herein. We do not have measurements along the boundary layer. For simplicity, we use the approximation V₀ = 0.15 V_sw and B₀ = 30 nT. These values are similar to observations of the velocity shear and magnetic field at low-latitude boundary layer between noon and the dusk flank [e.g., Fujimoto et al., 1998; Vaisberg et al., 2001]. The magnetic field ratio at the DMSP altitude and magneto-pause is taken to be b = B_i/B₀ ~ 1000. L = (Σ_p/κ)^0.5 is obtained using empirical formulas for Σ_p and solar wind parameters to infer κ. κ is computed using n_e = n_sw [e.g., Scudder et al., 1973; Phan and Paschman, 1996] and T_e = 1 × 10⁶ K [e.g., Phan and Paschmann, 1996].

As in Johnson and Wing [2015], we assume Σ_p = Σ_p,s + Σ_p,e where Σ_p,s = 0.88(S_a cos χ)^0.5 [Robinson and Vondrak, 1984] and Σ_p,e = (40 < E_e > ε^0.5)/(16 + <E_e>²), S_a is the radio flux with 10.7 cm wave length, χ is the solar zenith angle where <E_e> is the mean electron energy in keV, ε is the electron energy flux in ergs cm⁻² [Robinson et al., 1987]. This formula for combining the effects of solar illumination and electron precipitation is deemed appropriate when the two ionization sources are well separated in altitude, providing two parallel channels for the currents to flow [Wallis and Budzinski, 1981; Galand and Richmond, 2001]. The peak photoionization rate for solar EUV occurs around 100 km, while the peak ionization rate for dayside boundary layer or open field line electrons with energies up to a few hundred eVs occurs above 200 km [Robinson and Vondrak, 1984; Rees, 1963].

However, when the two sources of ionization occur at identical altitude, then it may be more appropriate to use the formula ${\sum }_p={({{\sum }_{p,s}}^2+{{\sum }_{p,e}}^2)}^{0.5}$ [Wallis and Budzinski, 1981]. This formula is more applicable for the closed field line regions where the precipitating electrons originating from the magnetosphere have energies of a few tens of keVs. Although we do not include such regions in our study, this alternative formula would not give significantly different results because Σ_p,s dominates over Σ_p,e in the region of interest (open field line/boundary layer, near noon/early afternoon).

4. Observational Confirmation

Figure 1 shows the value of J_‖,max predicted by equation (5) using Δ_i obtained from the measured value of Λ and equation (6) versus observed large-scale J_‖ that are located in the upward R1 at the boundary layer and on open field lines from 11 to 17 magnetic local time (MLT). In the present statistical study, for simplicity, our method uses an automated procedure to obtain large-scale FAC based on an infinite current sheet approximation. If the measured J_‖ is linearly proportional to J_‖,max (J_‖,max ~ c J_‖), then on the log-log plot, (1) the slope would be unity and (2) any nonunity proportionality constant would shift the points in Y direction by log(c) [the least squares fit would yield a Y intercept of log(c)]. As can be seen in Figure 1, the points tend to cluster along the dashed line, which has a slope of 1, suggesting that this scenario has some basis. Figure 1 provides an empirical relationship between J_‖,max calculated from equation (5) and the observed J_‖. The figure shows that the calculated J_‖,max is highly correlated with the observed J_‖ with correlation coefficient r = 0.81. The correlation is highly significant, the probability for two uncorrelated variables to give |r| ≥ 0.81 is < 0.01 (P < 0.01, number of points n = 361). The least squares fit returns log(J_{‖,max_cal}) = (0.93 ± 0.04) log(J_{‖_obs}) + (0.03 ± 0.01) where J_{‖,max_cal} = calculated maximum J_‖,max from equation (5) and J_{‖_obs} = large-scale J_‖ derived from the DMSP magnetometer. The least squares fit is plotted as a black solid line, which can be compared with a dashed line.

Figure 1.

Calculated J_‖,max versus observed J_‖ for MLT = 11–17. The calculated J_‖,max is obtained from equation (5). If the calculated J_‖,max is linearly proportional to observed J_‖, then all the points would lie along the dashed line, which has a slope of 1. The least squares fit is plotted as solid line, which nearly matches the dashed line.

Figure 1 shows that the approximation J_{‖,max_cal} ~c J_{‖_obs} is fairly good, as can be seen by comparing the points, the black solid line, and the dashed line. However, the slope of the solid line, 0.93 ± 0.04, is slightly lower than that of the dashed line (slope = 1). Figure 1 includes all points from 11 to 17 MLT. However, J_‖ near noon may respond differently because the magnetic field lines map to the vicinity of the subsolar region where (1) the magnetosheath densities and temperatures are high, which would lead to increased thermal currents and (2) the kinetic pressure gradient may contribute significantly to J_‖ because the velocity shear is minimal [e.g., De Keyser and Echim, 2013]. To check this effect, Figure 2 plots only the points from noon (11:30–12:30 MLT). The least squares fit returns log(J_{‖,max_cal}) = (0.83 ± 0.07) log(J_{‖_obs}) + (0.05 ± 0.03) and the correlation is highly significant r = 0.79 (P < 0.01, n = 89). The least squares figt is plotted as the solid line, which deviates further from the dashed line (slope = 1) than the solid line in Figure 1.

Figure 2.

Calculated J_‖,max versus observed J_‖ for noon (MLT = 12). The calculated J_‖,max is obtained from equation. (5). The least squares fit is plotted as solid line. If the calculated J_‖,max is linearly proportional to observed J_‖, then all the points would lie along the dashed line, which has a slope of 1. The slope of least squares fitted line (0.83 ± 0.07) is smaller than 1, which may be attributed to the higher thermal current near noon and/or kinetic pressure gradient.

Figure 3 plots the points from MLT = 11–17, excluding the points near noon. The least squares fit returns log (J_{‖,max_cal}) = (0.96 ± 0.04) log(J_{‖_obs}) + (0.03 ± 0.01) and the correlation is highly significant r = 0.81 (P < 0.01, n = 272). The least squares fit (solid line) seems to match the dashed line the best among all three figures (Figures 1–3). Two things are worth noting about the fit. First, the slope is nearly unity suggesting J_{‖,max_cal} is linearly proportional to J_{‖_obs}. Second, the small Y intercept of 0.03 suggests that the proportionality constant is nearly unity, J_{‖,max_cal} ~ 1.07 J_{‖_obs}.

Figure 3.

Calculated J_‖,max versus observed J_‖ for MLT = 11–17, excluding data near MLT = 12. The calculated J_‖,max is obtained from equation (5). The least squares fitted line (solid line) has a slope of 0.96 ± 0.04, which agrees very well with that of the dashed line, which has a slope of 1.

The use of log-log plots is particularly useful to identify scaling relationships. Such plots were employed in Johnson and Wing [2015] to isolate the power law dependence of some parameters. Log-log plots have the useful property that unknown constants of proportionality are eliminated from the scaling relationships and just become an offset. Many of the model input parameters are only proportional to measured solar wind parameters, so it makes sense to use a log-log plot, and these parameters would provide an offset. Using the same data points in Figure 3, the correlation of predicted and observed J_‖ in linear scale is r = 0.75, which is still highly significant, but lower than r = 0.81 for the correlation in the log-log scale, as expected.

5. Sources of Uncertainties

The data in Figures 1–3 exhibit some scatter. In our analysis, due to the lack of boundary layer observations, we assume simple scaling relations between the solar wind parameters and those in the boundary layer, assuming V₀ = 0.15 V_sw and boundary layer n_e = n_sw. These simple scaling relations may be adequate to capture power law dependence, but velocity and density obviously vary along the boundary, leading to scatter in the data. A variation by a factor of 2 or 3 in V₀ [Fujimoto et al., 1998; Phan et al., 1997; Vaisberg et al., 2001; Dimmock and Nykyri, 2013] would introduce a shift in the Y direction by 0.3–0.5 in the log-log plots in Figures 1–3. Similar considerations also apply to the magnetosheath density. Moreover, for simplicity, the present study uses an automated procedure to obtain large-scale J_‖ from magnetometer. If J_{‖_obs} is linearly proportional to J_{‖,max_cal} (J_{‖,max_cal} ~ c J_{‖_obs}), then any nonunity proportionality constant would introduce a shift in the Y direction, as discussed in section 4. However, if J_{‖_obs} is not linearly proportional to J_{‖,max_cal}, then it would affect the slope as well, but the result seems to suggest that the slope is nearly 1, except for the points near noon.

Σ_p is estimated from the Robinson and Vondrak [1984] and Robinson et al. [1987] empirical formulas, both of which have uncertainties (see also discussion in section 3).

The κ parameter in equation (2) was calculated from n_e = n_sw (as discussed above), but T_e is assumed to be 1× 10₆ K [Phan et al., 1997]. We have also used B₀ = 30 nT [Phan et al., 1997; Vaisberg et al., 2001]. A variation by a factor of 2 would introduce a shift in the Y direction by 0.3, as the case for V₀. Interestingly, at the boundary layer, from the subsolar region to the dusk flank, V₀ would increase while B₀ would decrease. Hence, the product V₀B₀ ~ E₀ would not vary much, as can be seen in MHD simulations (S. Merkin, private communication, 2014). The value of B_i/B₀ is assumed to be 1000, but the real values can vary along the boundary.

It is difficult to evaluate the uncertainty in Newell and Meng [1988] and Newell et al. [1991a, 1991b, 1991c] algorithm that is used to determine whether the FAC is located on open or closed field lines or at the boundary layer. Although this algorithm has been widely used for over two decades, it has never been quantitatively validated.

6. Discussion and Summary

Johnson and Wing [2015] derives an analytical theory for J_‖ generation from the velocity shear at the magnetopause boundary layer. The theory gives a simple formula for calculating J_‖,max using parameters that can be observed in ionosphere and solar wind, empirical formulas for conductivities, and simple scaling of magnetosheath velocity and density. Our results suggest that the analytical theory can predict observed J_‖ fairly well (our method, which uses an automated procedure, does not allow us to observe J_‖,max). The least squares fit of calculated J_‖,max versus observed J_‖ returns log(J_{‖,max_cal}) = (0.96 ± 0.04) log(J_{‖_obs}) + (0.03 ± 0.01) for data points from 11:00–17:00 MLT, excluding data from noon. Near noon (MLT = 12), the slope of the least squares fitted line is lower, log(J_{‖,max_cal}) = (0.83 ± 0.07) log(J_{‖_obs})+ (0.05± 0.03).

Near the subsolar region, the magnetosheath densities and temperatures are higher, which may increase the thermal current, J_t = n_eeV_t/2π, where J_t = thermal current, n_e = electron density at the boundary layer, and V_t = electron thermal velocity at the boundary layer. The accelerating potential can be overestimated when J_‖ is comparable to or smaller than J_t. To get a rough estimate of the thermal current, if we assume n_e ~ 10 cm⁻³ and T_e ~ 2 × 10⁶ K at the boundary layer near the subsolar region, then J_t ~1 × 10⁻⁶ Am⁻², which is comparable to points in the lower quartile in Figure 2. The kinetic pressure gradient could also modify J_‖ and the accelerating potential [De Keyser and Echim, 2013], particularly near the subsolar region where the velocity shear is minimal. Both thermal currents and kinetic pressure gradient are not taken into account in our model, e.g., equation (5). These factors may contribute to the less than unity slope obtained from the least squares fit and larger Y intercept in Figure 2. However, Figures 1–3 show that despite not including thermal currents and kinetic pressure gradient, the model remarkably agrees with the data fairly well.

Besides the thermal currents and kinetic pressure gradient, there may be other sources for the cross-field electric field in the boundary layer that are not related to velocity shear. All of these may introduce scatter in Figures 1–3. For example, the magnetopause processes such as reconnection, flux transfer events, and Kelvin-Helmholtz Instability may introduce small-scale FACs [Miura and Pritchett, 1982; Johnson et al., 2014; Wing et al., 2014], which would introduce noise or scatter in our figures. The present study only examines large-scale FACs and we will leave smaller-scale FACs for a future study.

The remarkable agreement between theory and observations suggests that Johnson and Wing [2015] theory captures the essential physics governing the solar wind interaction with the magnetosphere and energy transfer to the magnetosphere-ionosphere system via field-aligned currents.

Key Points.

• A shear flow boundary layer model for generating upward FAC distributions is validated

• Model neglects pressure gradient and thermal J_‖, which contributes to J_‖ near the subsolar region