Does the polar cap area saturate?

Transcription

1 Click Here for Full Article GEOPHYSICAL RESEARCH LETTERS, VOL. 34, L09107, doi: /2007gl029357, 2007 Does the polar cap area saturate? V. G. Merkin 1 and C. C. Goodrich 1 Received 15 January 2007; revised 28 March 2007; accepted 5 April 2007; published 9 May [1] We address the question of how the polar cap area (A PC ) and the open magnetic flux in the Earth s ionospheric polar caps depend on the strength of the interplanetary magnetic field (IMF) under conditions of steady driving. We use the Lyon-Fedder-Mobarry (LFM) global magnetohydrodynamic (MHD) model to analyze the relationship between these quantities and compare their behavior to that of the transpolar potential (F PC ). In a series of idealized simulation runs we find that in the LFM model A PC saturates faster than F PC as the IMF strength increases. The ionospheric conductance and the solar wind ram pressure have similar moderate effects on the saturated A PC, while their influence on F PC is very different. It appears that A PC saturates as a result of bulging of the geomagnetic lobes toward the sun, whereby the inner dipole magnetic field is shielded from further reconnection. The process is accompanied by dramatic changes in the global configuration of the magnetic field. Citation: Merkin, V. G., and C. C. Goodrich (2007), Does the polar cap area saturate?, Geophys. Res. Lett., 34, L09107, doi: /2007gl Introduction [2] The phenomenon of the transpolar potential saturation has received a lot of interest over the past few years from experimentalists, modelers, as well as theorists. There now appears to be enough observational evidence of the saturation effect despite the relative rarity of events driven strongly enough to cause it (see Shepherd [2007] for a review). Global MHD models confirm the non-linear nature of the relation between F PC and the solar wind convective electric field. A number of explanations based on these simulations as well as theoretical considerations have been suggested [Siscoe et al., 2004]. The importance of this effect follows from the notion that F PC is a measure of the fraction of the solar wind potential that is applied to the magnetosphere. [3] Of equal importance for understanding the state of the magnetosphere on a global scale is the open magnetic flux threading the ionospheric polar caps. This quantity is to very good accuracy proportional to A PC, since the dipole magnetic field (the approximation used in global MHD models) does not vary significantly across the polar cap. F PC is sometimes regarded as a measure of the open magnetic flux. However, by virtue of Faraday s law, a change in the total closed magnetic flux content in the magnetosphere is due to the inductive electric field. Such a change, resulting, for instance, from a southward rotation of 1 Center for Space Physics, Boston University, Boston, Massachusetts, USA. Copyright 2007 by the American Geophysical Union /07/2007GL029357$05.00 the IMF B Z component and accompanied by an expansion of the polar cap on the dayside, is observed in the ionosphere on time scales of a few minutes to 15 minutes depending on the magnetic local time (MLT), and the corresponding ionospheric convection signatures are not directly mapped into the magnetosphere [Lockwood et al., 1990]. Not only it is the inductive electric field that is responsible for changing the open (closed) magnetic flux, but during such transient periods magnetic field lines cannot be considered equipotential [Hesse et al., 1997], and therefore the instantaneous value of F PC, defined as the difference between the maximum and the minimum electrostatic potential in the ionosphere, is in no direct relation to the rate of change of the open magnetic flux. However, in a hypothetical steady state magnetosphere-ionosphere configuration the two quantities can be related indirectly. Indeed, both the transpolar potential and the open magnetic flux (or A PC ), being global indicators of the state of the solar windmagnetosphere-ionosphere system, can be functions of the system geometry. For instance, in the oversimplified example of a circular polar cap (certainly a bad approximation as viewed in global MHD simulations), the two are related via the polar cap radius, provided the ionospheric electric field is known. [4] Using the LFM global MHD model, in this letter we investigate the behavior of A PC under the influence of the southward IMF of varying strength and compare it to the behavior of F PC. In particular, we address the question of whether A PC and the open magnetic flux saturate as the solar wind convective electric field increases, and consider how this behavior is affected by changing other parameters in the system, e.g. the solar wind dynamic pressure and the ionospheric conductance. Kabin et al. [2004] used the BATS-R-US global MHD model to investigate the geometry and the position of the open-closed field line boundary (the definition of the polar cap we use here as well) under a wide range of idealized solar wind and IMF conditions. That study, while exploring a broad range of input parameters, in particular the IMF strength and orientation, did not investigate specifically the saturation of A PC, the question we address here. 2. Simulation Model [5] For the present study the low resolution LFM simulation code was used with a grid of , or 40704, cells in total, corresponding to the radial, polar and azimuthal directions in the LFM sense (the polar axis is aligned with the Solar Magnetic x-axis) [Lyon et al., 2004]. The inflow solar wind boundary conditions were fixed during a given run and varied between the different runs as follows: the solar wind velocity V X = 500 km/s, V Y = V Z = 0 km/s; IMF B Z = 2.5, 5, 10, 15, 20, 30, 40, 50 nt, B X = B Y =0nT; and the solar wind number density n = 10 cm 3. We L of5

2 repeated this series of runs with two different values of the ionospheric Pedersen conductance S P, which was set to 5 and 10 S uniformly over the entire polar cap, while the Hall conductance was set to 0. In addition, for S P =5Swe repeated the series of simulation runs with a lower solar wind number density n =5cm 3. Thus we were able to probe the influence of the ionospheric conductance and the solar wind dynamic pressure on A PC and the open magnetic flux. [6] In every simulation, the model was steadily driven for 6 hours after the IMF southward turning to allow the magnetosphere to reach a steady state. The latter term is used rather loosely here, since the magnetosphere is never exactly in a steady state even in a global simulation driven by a fixed solar wind with a constant ionospheric boundary condition. The overall variation in F PC during such a configuration of the simulated magnetosphere is usually less than 10%. In order to take this variation into account, F PC was averaged over the final 2 hours of each simulation run. This averaging is more complicated for the calculation of A PC, since it is a rather time consuming process that requires tracing a large number of magnetic field lines. Still, to reduce transient effects, we calculated A PC for 10 instances approximately equally spaced in time during the last hour of every simulation run and averaged these numbers to obtain the values presented in the next section. 3. Simulation Results [7] Figure 1 demonstrates the simulated dependencies of A PC (Figure 1a) and F PC (Figure 1b) on the strength of the interplanetary electric field (IEF) (Figure 1c is discussed in the next section). Three main results follow from Figure 1: [8] (1) A PC saturates faster than the transpolar potential and the effect is more prominent. Indeed, the saturation of F PC in these simulations is mainly manifested in the nonlinearity of the dependence on the IEF, but the potential keeps growing slowly for a series of simulation runs with a fixed ionospheric conductance. This is consistent with results from a similar series of runs accomplished by Merkine et al. [2003]. However, A PC levels off much more firmly and this occurs at smaller values of the IEF. This can be seen especially well in the series of simulation runs with n =5 and n =10cm 3, and S P = 5 S (solid and dashed traces, Figure 1a). The simulations with n =10cm 3 and S P =10S (dotted trace, Figure 1a) exhibit a more complicated behavior, whereby the polar cap area peaks around IEF = 10 mv/m. Considering error bars in Figure 1a, it is still to be determined whether this peak is physically significant, but the result that the polar cap area saturates as the IEF increases is still confirmed by all three series of simulations. [9] (2) The variation of the ionospheric conductance has a much more prominent effect on F PC than on A PC (note that the difference between the solid and the dotted traces in Figure 1a is much smaller than the difference between the corresponding traces in Figure 1b). Considering that the latter is proportional to the open magnetic flux, this means that, in the saturation regime, almost doubling the magnetic flux merging rates on the day- and the night sides (both equal to F PC in a steady state) has a rather moderate effect on the amount of the open and closed magnetic flux in the system. [10] (3) Similar to the effect of S P, the solar wind dynamic pressure P SW does not influence the steady state A PC significantly, considering that doubling P SW results in the polar cap expansion from 0.35 R e 2 to 0.42 R e 2 (the dashed versus the solid trace in Figure 1a). However, in contrast to the S P case, this effect is still greater than the effect of P SW on F PC (note that the dashed trace follows very closely the solid trace in Figure 1b). The weak dependence of F PC on the solar wind dynamic pressure in the LFM model has been discussed before [Merkin et al., 2005] and will not be addressed here any further. We do note here that once again A PC is shown to develop somewhat independently of F PC, at least in the saturation domain. 4. Discussion [11] As mentioned above, the saturation of A PC with increasing magnitude of the southward IMF is equivalent to the saturation of the open magnetic flux. This, in turn, means that for a set of driving parameters (P SW, S P ) there is a lower limit on the amount of the closed flux in the system, and no more flux can be reconnected however strong IEF is applied to the magnetosphere. According to Figure 1 the minimum closed flux in the magnetosphere (the maximum open flux) depends on both P SW and S P, but the dependence is weak. Therefore the question that we ask is what limits the amount of the closed flux in the magnetosphere, i.e. why cannot more flux be reconnected. [12] A theoretical expression for the polar cap radius can be derived as a result of an estimate of the location of the inner edge of the plasma sheet [Siscoe, 1982, 1991; Siscoe and Crooker, 1983]. According to the estimate, the polar cap radius scales as F 3/16 PC. The LFM simulation results depicted in Figure 1 are in qualitative agreement with this estimate, which tells us, in particular, that if F PC saturates then the polar cap radius saturates even faster. Although the particular form of the estimate above follows from a non- MHD description of the plasma sheet and the ring current particle precipitation boundary [Jaggi and Wolf, 1973; Southwood, 1977; Siscoe, 1982], it is a manifestation of a physical pattern that occurs in an ideal MHD plasma as well. This pattern is the sunward convection of the hot plasma sheet particles into the region of the strong magnetic field in the inner magnetosphere. When there is not enough energy to convect any further, the plasma drifts azimuthally to the dayside [Siscoe, 1991]. Once the magnetic flux is reconnected at the nightside reconnection line, the reconnection-driven earthward convection in the plasma sheet establishes the magnetic field configuration, but the total flux is not changed anymore. It is, therefore, a very tempting argument to make that, phenomenologically, the reason for the magnetosphere to have a finite amount of the closed flux on the night side is that there is plasma, which is accelerated in the process of reconnection and this plasma has to go somewhere, i.e. convect on closed field lines. The lower limit on the amount of closed flux then follows from the location of the inner edge of the plasma sheet (in global MHD context this would be the boundary where the earthward flow is diverted to the flanks). In this limiting situation, the nightside reconnection line coincides with the plasma sheet inner boundary. In the ionosphere this translates into the coincidence of the open-closed boundary (the 2of5

3 Figure 1. The dependence of the polar cap area measured in Earth radius, R e = 6380 km, (a) squared, (b) the transpolar potential, and (c) the location of the dayside polar cap boundary for the series of LFM runs indicated, on the strength of the IEF. The dashed line in Figure 1c represents a statistical relation for the location of the equatorward cusp boundary by Carbary and Meng [1986]. A PC is determined by the number of open field lines, N, traced from vertices of a refined ionospheric mesh with a constant elementary cell area, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds. Then, uncertainties in A PC are given by ± ð2sþ 2 þ Nds 2 error bars, where s is the standard deviation and the second term under the square root estimates the error in the A PC calculation arising from boundary effects. Uncertainties in F PC are given by ±2s error bars. polar cap definition used in this letter) and the flow reversal boundary. In fact, this is the limit in which the estimate by Siscoe [1982] and Siscoe and Crooker [1983] was derived, i.e., the limit in which the auroral oval thickness is 0. [13] Figure 2 (right) demonstrates that in the saturation regime the magnetosphere does reach (almost) this limiting case and enters an unusual state where the nightside x-line has moved toward the Earth and is found very close to the region of the strong magnetic field that prevents the flow from further earthward motion. However, despite the appeal of the argument above and the apparent change in the tail magnetic field geometry, close examination of Figure 2 and the nightside open-closed boundary in the ionosphere (not shown) reveals that, in both situations depicted in Figure 2, the amount of closed flux on the nightside is in fact approximately the same. The inflation of the polar cap occurs predominantly owing to the dayside magnetosphere erosion. Indeed, the geometry of the magnetic field is changed dramatically between the two depicted cases not only on the night side, but also on the day side. This unusual configuration of the dayside magnetic field has been discussed before in regard to the saturation of the transpolar potential [Raeder et al., 2001; Siscoe et al., 2004], but it appears to have an even stronger effect on the saturation of the polar cap size via shielding the inner part of the dayside dipole magnetic field from further reconnection. This picture is to some extent supported by inspecting the openclosed boundary location at local noon in the ionosphere (Figure 1c), which equatorward displacement slows down as the IMF increases. [14] Finally, a useful illustration complementing the discussion above is to view the saturation of the open magnetic flux as a geometrical effect following from the global structure of the magnetic field. This is achieved by taking advantage of the idealized conditions assumed for running the global MHD model, which allows us to visualize the footprint of the polar cap in the solar wind, since the magnetic flux threading both of these surfaces has to be the same. Due to the assumed simplicity of the magnetic field geometry above the bow shock (IMF is strictly southward and uniform in space), the magnetic flux through the footprint of the polar cap is given by the strength of the IMF multiplied by the area of that region. Each panel in Figure 3 is a projection of the 3-dimensional structure of the open magnetic field lines on the horizontal plane located at z =75R E (the LFM grid is a cylinder with the radius of 120 R E around the SM x-axis). The view is from above the northern magnetic pole. The purple-color dots are the ends of the open magnetic field lines threading the horizontal plane, so that the colored region as a whole is the footprint of the open polar cap in the solar wind. The elevation of 75 R E is chosen so that in every case shown, this plane is above the bow shock and the magnetic field is guaranteed to be normal to the plane. The panels from the top to the bottom depict the picture described, corresponding to the values of the southward IMF shown on the right. Figure 3 reveals a dramatic decrease in the length of the region shown, as the southward IMF increases. This picture Figure 2. The configuration of open (red) and closed (blue) magnetic field lines for S P = 10 S and the magnitude of the IMF shown in the top left corner. The pair of black field lines on both plots represents the field lines that have just been reconnected on the dayside. The background shows the SM x-component of the plasma velocity. 3of5

4 Figure 3. The open magnetic flux region in the solar wind at SM Z = 75 R E viewed from above the northern pole for n = 10 cm 3 and S P = 5 S. This picture is based on the conceptual representation by Dungey [1965] and Stern [1973] and recent applications by Milan et al. [2004]. appears to be consistent with Alfvén wing formation reported in similar simulations by Ridley [2007], although it remains to be considered whether Alfvén wings do indeed form in the LFM model. The self-consistent changes in the geometry of the magnetic field seen in Figure 3 are needed to keep the open magnetic flux in the system constant in the saturation regime. Figure 3 demonstrates that it is the length of the open magnetic flux region in the solar wind that changes most dramatically with the growing IMF, while its width is reduced roughly by a factor of 2 between the most weakly and the most strongly driven situations. Considering qualitatively that the width of this region determines F PC, while the area determines the magnetic flux, the picture demonstrates why A PC saturates faster than F PC : The length of the polar cap footprint in the solar wind decreases much faster than its width. 5. Summary and Future Work [15] We have shown in this letter results of idealized global MHD simulations and provided their phenomenological interpretation. Clearly, a lot remains to be done. In particular, a more rigorous analysis confirming our conjecture that it is the change in the geometry of the dayside magnetopause that results in the polar cap area saturation should be possible. Effects of the ram pressure and the ionospheric conductance have to be examined in detail. Finally, it still remains to be considered how the results of this work would be affected by including a more realistic model of the ionospheric conductance, i.e., the day-night gradient, the Hall conductance, and the auroral oval contributions. [16] Acknowledgments. VGM would like to thank Prof. G. Siscoe for suggesting this direction of investigations, along with Profs. J. G. Lyon, H. E. Spence, and W. J. Hughes for a number of fruitful discussions. The computations were accomplished on the CISM IBM Power4 supercomputers located at Boston University and supported by the BU SCV group. This research was supported by the National Science Foundation under agreement ATM , which funds the CISM project of the STC program. References Carbary, J. F., and C. I. Meng (1986), Correlation of cusp latitude with Bz and AE (12) using nearly one year s data, J. Geophys. Res., 91(A9), 10,047 10,054. Dungey, J. W. (1965), The length of the magnetospheric tail, J. Geophys. Res., 70(7), Hesse, M., J. Birn, and R. A. Hoffman (1997), On the mapping of ionospheric convection into the magnetosphere, J. Geophys. Res., 102(A5), Jaggi, R. K., and R. A. Wolf (1973), Self-consistent calculation of the motion of a sheet of ions in the magnetosphere, J. Geophys. Res., 78(16), Kabin, K., R. Rankin, G. Rostoker, R. Marchand, I. J. Rae, A. J. Ridley, T. I. Gombosi, C. R. Clauer, and D. L. DeZeeuw (2004), Open-closed field line boundary position: A parametric study using an MHD model, J. Geophys. Res., 109, A05222, doi: /2003ja Lockwood, M., S. W.H. Cowley, and M. P. Freeman (1990), The excitation of plasma convection in the high-latitude ionosphere, J. Geophys. Res., 95(A6), Lyon, J. G., J. A. Fedder, and C. M. Mobarry (2004), The Lyon-Fedder- Mobarry (LFM) global MHD magnetospheric simulation code, J. Atmos. Sol. Terr. Phys., 66, , doi: /j.jastp of5

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