Current Sheet Thinning During the Substorm Growth Phase.
Antonius Otto and Fred Hall IV
Abstract: 
A typical property during the growth phase of geomagnetic substorms is the thinning of the nearEarth current sheet, most pronounced in the region from about 6 to 15 R_{E}. We propose that the cause for the current sheet thinning is convection from the midnight tail region to the dayside to replenish magnetospheric magnetic flux which is eroded at the dayside as a result of dayside reconnection. The convection from the nearEarth tail region toward the dayside has to conserve the entropy on magnetic field lines. This constraint prohibits the replacement of flux from a region deeper in the magnetotail such that the nearEarth tail region is largely evacuated of magnetic flux [the Erickson and Wolf (1980) problem]. The absence of magnetic flux crossing the equatorial plane forces the formation of thin current layers. 

Introduction 

The substorm growth phase and the thinning of the nearEarth current sheet set the stage for the onset of the expansion phase. Whatever physics plays a role in the expansion onset is determined during the prior slow evolution in the growth phase. Substorms typically occur during periods of southward IMF, i.e. times with enhanced dayside reconnection. The growth phase lasts about 30 minutes to 1 hour. Typical growth phase properties are an increase in dayside convection, the expansion of the auroral oval, often an equatorward motion of quiet auroral arcs, a stretching of the magnetotail magnetic field, and the thinning of the nearEarth current sheet. The current sheet thinning appears to be particularly interesting (a) because it is not easily explained and (b) because one of the most significant changes at substorm onset is the dipolarization of the nearEarth magnetic field, which diverts the nearEarth crosstail current (previously confined to the thin current sheet) into fieldaligned currents. The current sheet thinning during the growth phase cannot be explained by simple compression of the current sheet. Considering a onedimensional Harris sheet, an increase of the external pressure by a factor of factor of 2 (lobe magnetic field by a factor of sqrt(2)) implies a number density increase by 2^{1/gamma}=1.52, or a compression of the current sheet by this factor  far short of the observed factor of 10 reduction in current sheet thickness. Prior attempts to explain the current layer thinning are the work by Lee et. al. (1995, 1998), who proposed a local diffusion of thermal plasma energy (`entropy antidiffusion'); and the work of Schindler and Birn (1993) and Birn, Hesse, and Schindler (1998), who suggested the evolution of a singular current layer if boundary conditions change in a way which does not permit a continuous (smooth) solution. These mechanisms do not provide a straightforward explanation for either the location of the thinning or the time scale of the growth phase. 
State of the Magnetosphere After Southward Turning of the IMF 


After southward turning of the IMF, dayside reconnection removes closed magnetic flux from the dayside magnetosphere. This flux is replenished by threedimensional plasma flow from the morning and evening sides as sketched in Figure 1. The cause of this flow is the removal of magnetic flux and an imbalance between increasing nightside pressure (in part due to an increase of the flaring and lobe field pressure on the nightside). Without this convection the location of the dayside magnetopause would continue to move inward as long as the IMF stays southward. There is various evidence for the azimuthal convection: (a) Observations show that the dayside magnetopause erodes by about 0.5 to 1 R_{E} during a time period of about 15 minutes after the southward turning (Berchem and Russell) and then remains at this distance. (b) Ionospheric convection increases after the southward turning of the IMF on a time scale of 8 to 15 minutes including the return flow toward the dayside. This time scale is consistent with 2 to 4 Alfven bounce times of about 2 minutes to generate the dayside convection. On the nightside, magnetospheric convection is established in part due to a fast expansion wave [See also (c).] (traveling at about 1000 km/s this takes another 2 minutes) and in part due to increased pressure due to the increase of the magnetic flaring angle which takes at least the time for the solar wind to travel from the dayside magnetopause to about 10 R_{E} behind the terminator, i.e., some additional 7 minutes for a solar wind speed of 300 km/s. These numbers are consistent with corresponding observations. (c) Global simulations show this flow immediately after dayside reconnection starts (Fedder, private communication, GEM meeting, 1999). The flow channels propagate as fast expansion waves from the dayside to the nightside.
Since flow is converging close to the subsolar point on the dayside (in the equatorial plane), there must be a region of diverging flow on the nightside. Kan (1990) has suggested this as a possible cause for current sheet thinning. Why, then, isn't flux replenished from the deeper tail region? 

Conservation of Specific Entropy on Magnetic Flux Tubes 


Frequently it is assumed that the magnetosphere is in a steady state, for instance by mapping the solar wind electric field (or a certain fraction thereof) into the magnetosphere. However, there are two major constraints which prevent steady convection. (a) Magnetic reconnection at the magnetopause is changing the amount of open/closed magnetospheric magnetic flux. Thus in order to maintain a steady state, dayside reconnection (creating open flux) has to be matched by nightside reconnection such that the total closed flux remains constant. (b) The second constraint to steady convection is the entropy on magnetic flux tubes. Locally, the quantity p /n^{gamma}=constant (with p=pressure, n=number density, gamma=ratio of specific heats) is conserved for ideal convection (assuming ideal Ohm's law). Integrating this along magnetic field lines leads to pV^{gamma}=constant where V is the differential flux tube volume =Int(dl/B). Background on the entropy and entropy on field lines can be found here (link to come). The flux tube volume scales approximately as V~L_{cs}/B_{cs} where L_{cs }is the length of the field line path through the current sheet and B_{cs }is the magnetic field in the current sheet. In other words, if a field line contracts strongly in length without loss of particles, the pressure has to increase strongly during the process. Now, convection from the midtail region toward the nearEarth region implies a drastic reduction of the length of the field line and an increase of the typical magnitude of B such that the flux tube volume decreases by several orders of magnitude. Entropy conservation during this convection implies that the pressure should increase according to p=p_{0}(V_{0}/V)^{gamma} where p_{0 }and V_{0 }are pressure and flux tube volume in the midtail. Since the ratio V_{0}/V for convection from, let's say, 30 R_{E} to 10 R_{E }changes by several orders of magnitude, the pressure has to increase by a corresponding amount. Figure 2 illustrates the field geometry, pressure along the xaxis, flux tube volume, and specific entropy computed for a Tsyganenko (1996) magnetic field. It demonstrates that the flux tube volume change by about 2 orders of magnitude between 30 R_{E} and 10 R_{E }. Thus the pressure must increase about 1000 fold to allow steady convection. We have computed the pressure from the field tension yielding the specific entropy as shown in the plot. As one can see in Figure 1, there is a two order of magnitude change in the entropy such that steady convection is impossible without loss of particles and energy from the flux tube. The required loss is inconsistent with observations (link to come) such that we have to conclude that steady convection from 30 R_{E} to 10 R_{E }does not occur! This is basically the result of the Erickson and Wolf (1980) analysis. This is consistent with corresponding observations (Borovsky et. al., 1998). It should be noted that similar conservation laws also apply to the total number of particles on a flux tube. 

Synthesis and Conclusions 


We have constructed an equatorial map of specific entropy in Figure 3. Stars indicate the location of the magnetopause. Lines are contours of constant specific entropy, i.e., paths along which convection is preferable. Assuming that the contours of 0.1 (dashdotted) and the next one out (solid) mark approximately the boundary of magnetic flux on which dayside erosion takes place, these contours also mark the approximate boundaries of the region from which magnetic flux can be transported to the dayside, i.e., the region which may serve as the reservoir of (replacement) magnetic flux. The reservoir is expected to be in the nearEarth tail close to midnight because of the higher compression of this region due to the largest flaring at the noonmidnight meridian. The most pronounced effects would be found about 9 R_{E} and 10 R_{E}. Thus magnetic flux crossing the tail region within these boundaries is convected toward the dayside (approximately along lines of constant specific entropy), depleting the tail region between about 9 R_{E} and 10 R_{E} of magnetic flux because no convection replenishes this flux from the tail. The result of this divergent flow is a decrease of B_{z} in the nearEarth tail. We expect that the evacuation of this region, together with the decrease of B_{z}, leads to the formation of a thin current sheet with decreasing thickness. [Note that the lobe field does not decrease. (link to come)] Considering a region of size 6 R_{E} by 10 R_{E,} an average B_{z} of about 50 nT in the nearEarth region and a potential of about 50 keV yields a time of 2000 seconds to deplete this tail region of its magnetic flux  which is consistent with the typical duration of the growth phase. Thus the proposed mechanism can explain the location of the strongest current sheet thinning and the approximate duration of the growth phase. It should also be noted that the formation of the current layer implies the transition from dipolar to taillike magnetic field in the region around 10 R_{E}, implying the equatorward expansion of the trapping boundary. 

Speculation and Future Work 


References: 

Birn, J., M. Hesse, and K. Schindler, Formation of thin current sheets in space plasmas, J. Geophys. Res., 103, 6843, 1998 Borovsky, J.E., M.F. Thomsen, R.C. Elphic, T.E. Cayton, and D.J. McComas, The transport of plasma sheet material from the distant tail to geosynchronous orbit, J. Geophys. Res., 103, 20297, 1998 Erickson, G.M. and R.A. Wolf, Is steady convection possible in the Earth's magnetotail?, Geophys. Res. Lett., 7, 897, 1980 Kan, J. R., Taillike reconfiguration of the plasma sheet during the substorm growth phase, Geophys. Res. Lett., 17, 2309, 1990 Lee, L.C., L. Zhang, G.S. Choe, and H.J. Cai, Formation of a very thin current sheet in the nearearth magnetotail and the explosive growth phase of substorms, Geophys. Res. Lett., 22, 1137, 1995 Lee, L.C., L. Zhang, A. Otto, G. Choe, and H.J. Cai, Entropy antidiffusion instability and formation of a thin current sheet, J. Geophys. Res., 103, 29419, 1998 Schindler, K. and J. Birn, On the Cause of Thin Current Sheets in the NearEarth Magnetotail and Their Possible Significance for Magnetospheric Substorms, J. Geophys. Res., 98, 15477, 1993 Tsyganenko, N.A., Effects of the solar wind conditions on the global magnetospheric configuration as deduced from databased field models, ESA SP389, 181, 1996. 