Further evidence for the role of magnetotail current shape in substorm initiation

Data base

The shape of the magnetospheric current sheet is varying continuously every moment. At close geocentric distances, it is controlled by the orientation of the Earth’s magnetic dipole axis and periodically changes in response to the planet’s diurnal rotation and its orbital motion around the Sun. The distant current sheet extends anti-sunward and follows the randomly changing SW direction in a windsock-like manner. If the SW is purely radial (flows in the anti-solar direction) and the Earth’s dipole is perpendicular to the ecliptic plane (dipole tilt Ψ _GSM—angle between the Z _GSM axis and Earth’s dipole axis—is zero), we expect to have a planar current sheet and north–south symmetric field lines. The deflection of SW from the radial direction would cause the displacement of the distant current sheet from the equatorial plane in accordance with SW flow. The dipole axis rotation (dipole tilt increase) will move the near-Earth portion of the current sheet from the ecliptic plane to the magnetic equator creating a bent and shifted current sheet at closer equatorial distances. These three configurations are schematically shown in Fig. 1 in the top, middle, and bottom panels. The most recent data-based studies of the global shape of the neutral sheet and its position with respect to the magnetospheric equatorial plane were carried out by Tsyganenko and Fairfield (2004), Tsyganenko and Andreeva (2014), and Tsyganenko et al. (2015). In these works, the observed deformation of the neutral sheet was described by empirical equations representing its average shape in the Geocentric Solar Wind (GSW) coordinate system with the X _GSW axis antiparallel to the observed SW flow vector rather than to the Sun–Earth line [as assumed in the standard Geocentric Solar Magnetospheric (GSM) system]. Larger tilt angles result in stronger deformation of the near-Earth current sheet at r≤10−15R _E and in larger shifts from the equatorial plane at greater distances. The effective tilt angle Ψ _GSW (the angle between the Z _GSW axis and Earth’s dipole axis) depends not only on the date/UT (which defines the tilt angle Ψ _GSM in the standard solar-magnetospheric coordinate system) but also on the orientation of the SW velocity vector V _SW. The angle Ψ _GSW can be calculated by means of a standard software package GEOPACK-2008, which is available from http://geo.phys.spbu.ru/~tsyganenko/Geopack-2008.html.

In further analysis, our main interest is focused, however, on the effects of current sheet bending and rotation in the X Z _GSM plane, which may be described with the combined variations of regularly varying standard dipole tilt Ψ _GSM and rapidly changing SW flow direction in the X Z _GSM plane. For that purpose, instead of using the highly variable GSW coordinate system and Ψ _GSW based on the three-component SW velocity vector, we remained in the frame of standard GSM coordinates and introduced the effective tilt Ψ _SW=Ψ _GSM+ arctan(V z _SW/V x _SW). Here, V z _SW and V x _SW are the SW flow velocity components.

The statistical distribution of the dipole tilt angle Ψ _GSM over a year is illustrated by a histogram in the left panel of Fig. 2, and the data show the numbers of 5-min intervals with Ψ _GSM values falling into consecutive 2° intervals. The distribution is nearly symmetric with respect to Ψ _GSM=0, there are two peaks at ±12°–14°, and the values rapidly fall off towards the edges at |Ψ _GSM|≈33°. The slight asymmetry is due to the ellipticity of the Earth’s orbit around the Sun, which results in a somewhat longer (by ∼4 %) total time with positive tilt angles accrued during summer months. The right panel in the same figure presents a similar histogram for the SW flow deflection angle in the XZ-plane, which is calculated as α= arctan(V _zSW/V _xSW), where V _zSW and V _xSW are the SW flow velocity components. One can see that the number of 5-min periods with a given flow direction during 1 year (here, 2007) has a very regular, nearly Gaussian distribution, and in most of the 5-min periods the α angle does not exceed 4°. The 5-min average SW data were downloaded from the OMNIWeb site (http://omniweb.gsfc.nasa.gov/).

To analyze the possible dependence of substorm probability on current sheet bending, we used substorm onsets (beginnings of the expansion phase) identified as abrupt decreases of the magnetic AL index. This method for automatic substorm phase detection has been described in more detail in Juusola et al. (2011) and Partamies et al. (2013). Out of all detected expansion phases, we selected those with AL minima of at least −50 nT. For the Northern Hemisphere, we used THEMIS ground-based observations during 2007–2009, and for the Southern Hemisphere, all available data were employed. The southern AL index data were downloaded from the virtual magnetic observatory at vmi.gsfc.nasa.gov as prepared by the University of California Los Angeles (UCLA). As a result, we detected 6091 substorms in the Northern Hemisphere and 2170 substorms in the Southern Hemisphere. The resulting numbers of substorms during the periods with different dipole tilt values (Ψ _GSM) are given for each hemisphere separately and for both hemispheres in the top row of Fig. 3. All three histograms show distinct minima near the zero tilt, two local maxima around |Ψ _GSM|∼15°, and gradual decreases towards the ends of the tilt angle range. In general, the distributions are very similar to that shown in the left panel of Fig. 2, but with one important difference; there is a relatively larger number of substorms for Ψ _GSM<−20° in the Southern Hemisphere and for Ψ _GSM>20° in the Northern Hemisphere. Additionally, note that the total distributions are not quite symmetric, with a larger number of substorms for positive tilt values in the Northern Hemisphere and for negative tilts in the Southern Hemisphere (i.e., corresponding to the local summer periods in each hemisphere).

It is important to note, however, that in order to explore the current sheet bending effects on substorm triggering, one should take into account the changing direction of the SW flow, namely, instead of the standard dipole tilt Ψ _GSM, one should use the effective tilt Ψ _SW=Ψ _GSM+α=Ψ _GSM+ arctan(V _zSW/V _xSW). To that end, we complemented our substorm onset data with OMNI 5-min interplanetary data, i.e., corresponding values of the IMF and V _SW components were ascribed to each substorm onset. The bottom row of Fig. 3 gives the number of substorms in relation to the effective tilt angle Ψ _SW. Here, we are primarily interested in the bending and latitudinal deflection of the tail, i.e., the effects in the X Z _GSM-plane, such that the V _y component was assumed to be of minor importance. The main features of the distributions in the bottom row remained generally the same as those in the upper row, but the range of values was somewhat larger (up to ± 40°).

Statistical results

To analyze the substorm probability, we combined the data from both hemispheres and then excluded events from the Northern Hemisphere substorm list that had onsets that were simultaneous (within ± 5 min) with those detected in the Southern Hemisphere in order to avoid duplications. After that, we normalized the substorm numbers within each bin of the tilt angle by the corresponding values of its occurrence frequency shown in the left panel of Fig. 1. The normalized values are shown in Fig. 4.

A clear increase of the substorm probability with growing tilt can be readily seen for all tilt values, except in the ±24°–32° interval, but those values are offset by a steep increase in the next interval ±32°–40°, so that by averaging over these two bins (with a relatively small total number of substorms, see Fig. 3), this again results in an increase of the probability. However, for the negative tilts, the probability values are smaller, which is apparently a result of the more than twice lower number of events from the Southern Hemisphere. On the basis of these data, we conclude that during periods with larger dipole tilt angles, substorm onsets occur with a larger probability than under smaller tilt angles.

Another way to reveal the dependence of substorm occurrence on the effective tilt value is to check whether the SW deflection is more effective when it increases the absolute value of the dipole tilt, i.e., when it makes the magnetotail current sheet more curved. To quantitatively evaluate this possibility, we introduced the function Ψ _norm=Ψ _GSM·sign(α), where Ψ _GSM is the standard dipole tilt in the GSM coordinate system and α is the SW velocity inclination in the X Z _GSM-plane (α= arctan(V z _SW/V x _SW)). Then, positive values of Ψ _norm would correspond to the same sign of the dipole tilt and the additional tilt due to the SW deflection, while its negative values would correspond to a decrease of effective tilt and, hence, a smaller current sheet curvature. By normalizing Ψ _GSM with sign(α), we kept the Ψ _norm values within the standard range of ±33°. The resulting histogram of the substorm numbers as a function of Ψ _norm is shown in Fig. 5, where blue (red) columns and blue (red) numbers correspond to the number of substorms for negative (positive) values of Ψ _norm. The effect was rather distinct, and all the red columns were 10–20 % higher than the blue ones. The only bin with two equal numbers was the one for the smallest tilt angle range, which was quite naturally expected. This result can be explained if we agree that the substorm initiating instability depends on the total tilt (or degree of current sheet bending). Thus, the already loaded magnetic flux, which is not large enough for a substorm to start under the currently existing dipole tilt, may become efficient after the tilt is further increased by a sudden deflection of the SW in a favorable direction.

The same arguments can also be applied to periods of increasing or decreasing dipole tilt. Then, substorms would preferably start during periods with steadily increasing dipole tilt, while decreasing tilt would result in delayed substorm onsets, even under constant energy loading. Using our data from the Northern Hemisphere (with more regular observations), we compared the number of substorms that occurred during the periods with decreasing and increasing dipole tilt and found that, indeed, the difference does exist, though it is not large, i.e., a little less than 10 % (2917 events with decreasing tilt versus 3173 during tilt increase). To further investigate our conclusion, we performed an identical analysis for the periods of steady magnetospheric convection (SMC) events, when no substorm instability was initiated and substorm expansion did not take place, in spite of a continuous loading. We used a similar data base, which was produced by Kissinger et al. (2011) based on auroral electrojet indices and visual inspection of data, and selected the SMC periods from the same time interval from 2005 through 2007. The obtained result was opposite to that for the substorm events; specifically, we found that about 20 % more SMC events started during the periods of decreasing tilt (218 versus 175). One of the many possible reasons for this finding may be, for example, the growth of the substorm instability threshold due to the constantly decreasing tilt.

To summarize, we conclude that during the periods with larger current sheet bending and deflection from the Sun–Earth line, the number of substorms is larger than during the periods with a more symmetric plasma sheet. This can be seen best in the events when fast SW deflections increase the existing dipole tilt, thus, leading to premature substorm onset. In such situations, the difference from the opposite case (when the SW deflections decrease the effective dipole tilt) may be quite significant, with as much as a 10–20 % increase in the number of substorm onsets.

Data selection and binning

A critical factor in these kinds of studies is the density and evenness of the data coverage of the modeling region in both geometric and parametric space. Since our ultimate object of interest is the distribution of the total current per unit tail length (i.e., the volume density j integrated in the z-direction), each interval of the tailward distance should contain enough data taken both northward and southward from the current sheet. This study used a large multi-year set of space magnetometer data that were obtained by Geotail (1995–2013), Cluster (2001–2012), and five THEMIS probes (2007–2013), these data were taken tailward from the plane x=−10 R _E. Figure 7 illustrates the data distribution in the eight intervals of x in projection onto the XZ and XY-planes in the GSW coordinate system. In order to keep the models mathematically simple, the data were also restricted to limited ranges of y- and z-coordinates, with the selection window size gradually widening down the tail, as shown in the plots. All the data had 5-min resolution, the number of records in the individual (overlapping) subsets varied from ∼32,000 to ∼82,000, and the total number of records used in this study was 264,552. Each data record was provided with concurrent information on the interplanetary medium conditions, which were obtained from 5-min average yearly files with OMNI data.

Model

which includes the geodipole contribution B _x,dip; the remaining terms represent the field of external sources. Here we used simple functions f ₁(z)=z/(z ²+D ²)^1/2 and f ₂(z)=z/(z ²+D ²) to represent the local north–south profiles of B _x (z). The first factor, f ₁, is an odd smoothed step function which reverses its sign at z=0 and varies over a characteristic thickness 2D between asymptotic limits −1<f ₁<+1. The second factor, f ₂, differs from f ₁ only in that it asymptotically tends to zero at z→±∞, which allows for gradual north–south variation of B _x in the lobes and thus lends more flexibility to the model. The half-thickness D of the current sheet was assumed in each distance bin as a symmetric parabolic function of the normalized distance y _n =y/10 across the tail: \(D=D_{0}+D_{1}{y_{n}^{2}}\).

The bracketed terms in Eq. (1) quantify the response of the model B _x to the SW ram pressure P _n =P/〈P〉, which is normalized by its average value 〈P〉=1.6 nPa, the driving parameter F _n associated with the IMF, and the dipole tilt angle Ψ. The function \(F_{n}=10^{-4}\cdot V_{\text {SW}}^{4/3}B_{t}^{2/3}\sin ^{8/3}(\theta /2)\) is an index by Newell et al. (2007), which is normalized by the value F=10⁴ corresponding to the SW speed 450 km/s and a purely southward IMF with B _z =−5 nT.

The dipole tilt Ψ controls both the magnitude of the tail current and its geometry; accordingly, the tilt effects enter in the model described by Eq. (1) in a twofold manner. The first mode of response is explicitly represented by the terms with the factors Ψ and Ψ ². Note here that both brackets in Eq. (1) include two types of terms that correspond to opposite symmetry properties with respect to z and Ψ. Terms of the first type correspond to the overall decrease or increase of |B _x | in both lobes and, as such, include only even powers of Ψ. The other type represents the north–south asymmetry induced by Ψ≠0, which implies only odd powers of Ψ. Owing to a general symmetry constraint on the model B _x , which requires that B _x (x,y,z,Ψ)=−B _x (x,y,−z,−Ψ) (e.g., Mead and Fairfield 1975; Eq. (4)) terms that are odd with respect to Ψ are even with respect to z and vice versa.

where \(R_{\mathrm {H}_{0}}\phantom {\dot {i}\!}\), \(R_{\mathrm {H}_{1}}\phantom {\dot {i}\!}\), and χ are free model parameters. Since our region of interest lies tailward from the current sheet hinging area, the parameter α was fixed at a constant value of α=3 based on previous estimates in TF04.

In total, the model Eqs. (1) −(3) contain 22 unknown free parameters including ten linear coefficients a _i , i=1,…10 and 12 nonlinear parameters D ₀, D ₁, \(R_{H_{0}}\), \(R_{H_{1}}\), χ, G, and γ ₁−γ ₆. Their values were derived for each distance bin using a search code based on the downhill simplex Nelder–Mead method, combined with a singular value decomposition (SVD) algorithm (Press et al. 1992) to calculate the linear coefficients at each simplex step.

Model results

Figure 8 shows distributions of the absolute value |B _x | in four cross-tail sections, each bounded within its corresponding data selection window in the y- and z-directions. All the plots correspond to the dipole tilt angle Ψ=30°, SW pressure P=3 nPa, and the IMF driver F _n =1. In the midnight meridian plane, the current sheet center is located around Z _s ≈4 R _E, which is consistent with the best-fit value of the hinging distance of R _H≈8 R _E. Both the absolute magnitude |B _x | and its north–south asymmetry are greatest in the nearest distance bin and then gradually decrease tailward (note the different range of the color bars in the successive plots).

This agrees with an earlier finding (Tsyganenko 1998, Fig 4) based on a much smaller dataset, as well as with the result of a magnetohydrodynamic (MHD) simulation run,which is presented next in Fig. 9. To check if similar magnetic field behavior with dipole tilt was produced by global MHD modeling, a set of global MHD simulations with synthetic SW input was performed through the Community Coordinated Modeling Center in the Goddard Space Flight Center (http://ccmc.gsfc.nasa.gov/). Here, we present the results of one of this simulation that was made using Block-Adaptive-Tree-Solarwind-Roe-Upwind-Scheme (BATSRUS) model (Powell et al. 1999). This simulation was performed in a numerical grid with 1 million cells [minimal grid size is 0.25 Earth radii (R _E) in the inner magnetosphere, 0.5 R _E in the dayside regions of the magnetopause and bow-shock, in mid-tail plasma sheet, and 1 R _E in the mid-tail magnetopause and lobes]. The simulation had a duration of 30 min (in addition to 1 h of preconditioning) with fixed SW input in order to obtain the quasi-stationary solution. Namely, the SW speed V _x =−400 km/s, V _y =V _z =0, proton temperature T=105 K, number density N=5 cm ⁻³, and IMF (B _x ,B _y ,B _z ) =(0,0,−4) nT. Ionospheric conductance was also fixed with uniform Hall and Pedersen components. The dipole tilt was set to Ψ=+30 (North Pole tilted towards the Sun) without updating.

Based on the results of fitting the model Eq. (1) to the data subsets for the eight sliding bins of x- coordinates shown in Fig. 7, we calculated for each interval \(\Delta B_{x}=0.5(B_{x}^{(N)}-B_{x}^{(S)})\), where \(B_{x}^{(N)}\) and \(B_{x}^{(S)}\) correspond to the midnight locations at z=Z _s ±3 R _E northward and southward of the current sheet center. The plots of Δ B _x shown in Fig. 10 represent the variation of the average tail lobe field (and thereby the current) along the Sun–Earth line in the interval −30≤x≤−10 R _E for four combinations of the SW pressure P and the IMF parameter F. Each panel shows two profiles, which were calculated for the case of a planar current sheet (Ψ=0, green solid line) and a warped current sheet (Ψ=30°, black solid line). Red and blue dashed lines show, respectively, the southern and northern lobe fields \(-B_{x}^{(S)}\) and \(B_{x}^{(N)}\) entering in the half-sum Δ B _x for the tilted case Ψ=30°. Error bars in the plots correspond to the values of the residual root mean square (rms) difference between the observed and model fields for each distance bin. As expected, in all four cases, the field decreases downtail almost monotonically, except in the farthest distance bins beyond 25 R _E. The principal effects of interplanetary conditions are also clearly seen, which involve a significant increase of the tail field with growing ram pressure P and the IMF index F, in agreement with earlier results (e.g., Fairfield and Jones 1996; Tsyganenko 2000).

In the context of the present study, the most interesting feature is a distinct overall decrease of the tail current in the tilted case; in all four plots, the black line is positioned significantly lower than the green line. The largest relative difference (nearly 25 %) was observed in the three nearest bins of x for the case with P=4 nPa and F _n =0.