Electrodynamic Coupling in the Solar Wind-Magnetosphere-Ionosphere System ∗

This paper presents a brief summary of our recent work based on global MHD simulations of the Solar wind-Magnetosphere-Ionosphere (SMI) system with emphasis on the electrodynamic coupling in the system. The main conclusions obtained are summarized as follows. (1) As a main dynamo of the SMI system, the bow shock contributes to both region 1 Field-Aligned Current (FAC) and cross-tail current. Under strong interplanetary driving conditions and moderate Alfvén Mach numbers, the bow shock’s contribution may exceed more than fifty percent of the total of either region 1 or cross-tail currents. (2) In terms of more than 100 simulation runs with due southward Interplanetary Magnetic Field (IMF), we have found a combined parameter f = EswPswM −1/2 A (Esw, Psw, and MA are the solar wind electric field, ram pressure, and Alfvén Mach number, respectively): both the ionospheric transpolar potential and the magnetopause reconnection voltage vary linearly with f for small f , but saturate for large f . (3) The reconnection voltage is approximately fitted by sin(θIMF/2), where θIMF is the IMF clock angle. The ionospheric transpolar potential, the voltage along the polar cap boundary, and the electric fields along the merging line however defined they may be, respond differently to θIMF, so it is not justified to take them as substitutes for the reconnection voltage.


Introduction
In the Solar Wind-Magnetosphere-Ionosphere (SMI) system, one question of most interest is the energy transfer and the transformation between mechanical and magnetic energies.The former involves and the latter entirely belongs to electrodynamic processes.In the framework of the MHD theory, the solar wind is considered to be the main energy source, and it deforms the Earth's magnetic field and causes plasma convection in the system, leading to various magnetospheric and ionospheric activities.Electric field and current, determined by (1) appear everywhere in the system as a result, and in turn they serve as important manifestations and provide quantitative descriptions of relevant electrodynamic coupling.For steady cases we have ∇ × E = 0 from Faraday's law, so an electrostatic potential ϕ may be introduced to express E via E = −∇ϕ.
In situ measurements of electric field and current in space are local measurements, and global configurations of electric field and current have been constructed mainly by inference.Then a quantitative investigation of the electric field and current heavily relies on global MHD simulations of the SMI system.A large number of papers appeared in the lit-erature on the electric field and current in the SMI system.This paper does not intend a comprehensive review of them, but is limited to a brief summary of our recent work on the issue, quoting some references that are closely related to our work.Our review contains 3 aspects: the bow shock contributions to the region 1 Field-Aligned Current (FAC) and the crosstail current, the saturation of the ionospheric transpolar potential and the magnetopause reconnection voltage, and the reconnection voltage versus Interplanetary Magnetic Field (IMF) clock angle (the angle between the IMF and the z-axis), covered by the following 3 successive sections in proper order.We conclude this paper with brief remarks in Section 5.

Bow Shock Contribution to Region 1 and Cross-tail Currents
There exist various currents in the SMI system, such as the bow shock current, the magnetopause current, the ring current, the plasma sheet current, the FAC, and the ionospheric current.However, it is difficult to find observational evidence for the interconnection between these currents, and presently we can only examine such interconnection by drawing current streamlines based on data obtained by global MHD simulations of the SMI system.For instance, one may locate the source regions of the region 1 current by tracing the associated current streamlines rooted at the grid points of the ionosphere.Some of these current streamlines may pass through the bow shock layer, and then we conclude that the bow shock contributes to the region 1 current.The contribution of each of the current streamlines may be measured by the product of the normal current density at the grid point in the ionosphere and the corresponding mesh area.The sum of the contributions from relevant current streamlines denotes the total contribution to the region 1 current from the bow shock.
Similarly, we may evaluate the contributions made by the bow shock and the magnetopause to the cross-tail current, and so on.
Since the current density is derived from the Ampere's law, i.e., Equation (2), the current streamlines must be closed curves.Consider a closed thin current tube, of which the volume is V and the central line is a closed loop C. The current strength I is constant along the tube, so we have for steady cases where dl is the the arc element vector of the loop.Here we have used the condition of ∇ × E = 0. From Equation (3), we conclude that if j • E > 0 in one segment of the loop, there must exist another segment, in which j • E < 0. Or in terms of the language of electric circuit theory, we may say that if there is a load (j •E > 0) somewhere on the loop, there must be a dynamo elsewhere on the same loop.Such a simple analysis establishes a physical link between currents in load and dynamo areas.
By drawing current streamlines based on MHD simulation data, Fedder et al. (1997) and Siscoe and Siebert (2006) noted the bow shock contribution to the region 1 current.Recently, Guo et al. (2008) put such a conclusion on a more quantitative basis, and pointed out that a large fraction of the region 1 current originates from the bow shock under realistic solar wind conditions.In their numerical runs, the solar wind is along the Sun-Earth line, the IMF is due southward, the ionosphere has a uniform Pedersen conductance and a zero Hall conductance, and the simulation solution is in a quasi-steady state.Each solution is characterized by three adjustable parameters, the solar wind speed v sw , the southward IMF B z , and the Pedersen conductance Σ P , where all other solar wind parameters are fixed, including the number density n = 5 cm −3 and the plasma pressure p sw = 0.0126 nPa.The contribution of the bow shock to the region 1 current was found to change with solar wind and IMF conditions and the Pedersen conductance of the ionosphere.Table 1 lists part of the results: the total region 1 FAC strength, I 1 , the contribution from the bow shock, I 1bs , and the ratio between the two, I 1bs /I 1 , versus v sw and Σ P for a fixed value of B z = −20 nT.I 1 increases monotonically with increasing v sw and Σ P as expected.It is interesting that I 1bs also increases monotonically with both v sw and Σ P , not only in magnitude, but also in percentage of the total region 1 current.A larger solar wind speed or a higher ionospheric Pedersen conductance leads to a greater contribution from the bow shock to the region 1 current.Similar results are listed in Table 2, where the IMF B z replaces Σ P as one of the adjustable parameters, whereas Σ P is fixed to be 10 S. The contribution of the bow shock to the region 1 current increases monotonically with increasing v sw and |B z | in both magnitude and percentage.A stronger southward IMF strength also leads to a greater contribution from the bow shock to the region 1 current.Under the extreme case of v sw = 800 km•s −1 , IMF B z = −20 nT, and Σ P = 10 S, about 62% of the total region 1 current originates in the bow shock (see either Table 1 or Table 2).On the other hand, when these three parameters are small in magnitude, the bow shock makes little contribution to the region 1 current.This was probably the reason why region 1 current streamlines that originate from the bow shock were not identified under standard interplanetary and ionospheric conditions.Incidentally, calculations also demonstrated that no contribution is made by the bow shock to the region 1 current if the IMF is due northward, the same conclusion as reached by Tanaka (1995) and Siscoe et al. (2000).
The Alfvén Mach number of the solar wind directly affects the bow shock strength and current, so it must affect the contribution of the bow shock to the region 1 current.As a result, the bow shock contribution should exhibit a non-monotonic variation with increasing southward IMF strength.Peng and Hu (2009) found that I 1bs increases with increasing southward IMF strength for M A > 2, a similar result  Guo et al. (2008).However, I 1bs has a nonmonotonic variation with B z in both magnitude and percentage, when the solar wind number density is 5 and 10 cm −3 .The turning points are B z = −20 nT for n sw = 5 cm −3 , and −30 nT for n sw = 10 cm −3 .It happens that the turning points are located at nearly the same Alfvén Mach number, M A = 2, as seen from Table 3.As far as the case with n sw = 20 cm −3 is concerned, M A is always larger than 2, I 1bs exhibits a monotonic variation with B z as expected.
Cross-tail current in the magnetotail was conventionally believed to be entirely closed on the magnetopause, forming a θ-like structure while viewing from the Sun.It is the case if the IMF vanishes.The situation changes for non-vanishing IMF cases.By drawing current streamlines starting from the crosstail current region, Tang et al. (2009) found that the bow shock also contributes to the cross-tail current.Therefore, the cross-tail current closes on both the nightside magnetopause and the bow shock.Figure 1 shows a global picture of the closure of the crosstail current.The distribution of j • E is plotted in color in a plane perpendicular to the Sun-Earth line at x = −15R e (R e is the radius of the Earth), and 4 typical streamlines of the cross-tail current starting in this plane are projected to the plane in order to show the closure patterns.The two dynamos with j • E < 0, colored in white, are clearly identified as the magnetopause and the bow shock, respectively.The cross-tail current region serves as a load, colored in red, in which magnetic reconnection may occur from time to time.The current streamlines in blue, which are closer to the equatorial plane than those in red, close on the bow shock, whereas the red current streamlines close on the magnetopause.The whole picture looks like two overlapped Greek letters of θ,  et al., 2009).
in contrast with a traditional single θ structure for the cross-tail current.As one of the reconnection currents in the magnetosphere, the cross-tail current has two dynamos, the bow shock and the magnetopause, in comparison with the magnetopause reconnection current, of which the only dynamo is the bow shock, as concluded by Siebert and Siscoe (2002).Tang et al. (2009) also studied the dependence of the total cross-tail current and the contribution made by the bow shock upon solar wind and IMF conditions.They concluded that an increase of southward IMF strength leads to an increase of the crosstail current and the contribution from the bow shock, but a decrease of the contribution from the magnetopause.For the case of B z = −10 nT with normal solar wind conditions (v sw = 400 km•s −1 , n sw = 5 cm −3 , p sw = 0.0126 nPa, Σ P = 5 S), the bow shock contribution was found to account for more than 80 percent of the cross-tail current.Besides, an increase of the solar wind speed results in an increase of the total cross-tail current and the contribution from the magnetopause, and a slight decrease of the contribution from the bow shock.As far as the Pederson conductance of the ionosphere is concerned, it does not affect the total cross-tail current significantly, but changes the share of the bow shock and the magnetopause in the cross-tail current.A higher Pedersen conductance leads to a larger share of the bow shock and a smaller share of the magnetopause.

Saturation of the Transpolar Potential and Reconnection Voltage
Transpolar potential V pc , defined as the difference of the positive and negative electric potential peaks in the polar cap ionosphere, serves as a key parameter of the SMI coupling.Observations show that V pc saturates under strong solar wind electric field (Reiff and Luhmann, 1986;Russell et al., 2001;Hairston et al., 2003;Shepherd et al., 2007).Several parameterized models were developed based on either observational records (Ridley, 2005) or physical arguments Siscoe et al., 2002Siscoe et al., , 2004;;Kivelson and Ridley, 2008) in order to give a quantitative description of the saturation behavior of V pc .In these models, the solar wind electric field E sw was taken to be the only driving source, whereas other parameters, such as the ionospheric Pedersen conductance Σ P , the solar wind ram pressure P sw and Alfvén Mach number M A , play roles in affecting the saturation point at which the transpolar potential starts to saturate and the saturation level (Fedder and Lyon, 1987;Siscoe et al., 2002;Merkine et al., 2003;Ridley, 2005).The transpolar potential is intimately related to the reconnection voltage V r , which is defined as the total electric potential drop along the dayside merging line on the magnetopause and characterizes the global reconnection rate.The reconnection voltage is presently unobservable, so it has been inferred from either V pc by assuming that magnetic field lines are equipotentials or E sw multiplied by an effective length of the merging line.However, the presence of a parallel electric field (Siscoe et al., 2001;Merkine et al., 2005;Hu et al., 2007) and the uncertainty involved in the estimation of the effective length of the dayside merging line have made these inferences somewhat ambiguous.Based on MHD simulation data for steady states of the SMI system, a direct integration of electric field was used to calculate the electric potential along the merging line so as to determine V r (Merkine et al., 2003(Merkine et al., , 2005;;Siscoe et al., 2001).Hu et al. (2007) proposed three methods for evaluating the potential along the merging line, referred to as reconnection potential, which are all based on the curvelinear integration of electric field, but differ in the choice of the reference potential and integration path.For due southward IMF cases, they found the differences among the answers via different methods to be within 20% when an equivalent numerical resistivity is carefully selected.In particular, one of these methods takes radial rays that connect the inner boundary of the magnetosphere (r = 3 R e ) with the merging line as the integration path, and the potential at the inner boundary as the reference potential, which is set to be the potential at the footpoints in the ionosphere of the Earth's dipole field lines that pass through the inner boundary.It happens that the obtained reconnection potential is insensitive to the magnitude of the numerical resistivity, and approximately determined by where Φ R is the reconnection potential at point P on the merging line, r p is the geocentric distance of point P , and Φ c is the potential at the intersection point between the radial ray and the inner boundary, which equals the potential at the footpoint in the ionosphere of the Earth's dipole field line through the intersection point.Physically, this reflects the fact that the plasma convection in the magnetosphere is driven by magnetic reconnection, and that we have neglected the effect of resistivity and viscosity in the magnetosphere, which should be negligible in both numerical and physical senses except in regions with steep gradients or MHD discontinuities.As pointed out by Hu et al. (2009), this method embodies the magnetic flux balance in the magnetosphere: the magnetic flux in the dayside magnetosphere extracted by magnetic reconnection is balanced by the inflow flux created by plasma convection.In order to reveal the complex relations of V pc and V r to solar wind and IMF conditions, Peng et al. (2009) made more than 100 MHD simulation runs and obtained various steady solutions of the SMI system for due southward IMF cases.Each solution was characterized by four parameters, Σ P , E sw , P sw , and M A , and the associated V pc and V r were calculated.
Figure 2 shows all data points obtained in E sw − V pc and E sw − V r planes.The data points are scattered, except that an upper and lower envelop could be seen dimly in each panel, which means an upper and a lower limit cut off of the two potentials for extremely strong and weak solar wind driving conditions, respectively.While the variation of V pc and V r with E sw is clear, compelling, and systematic in Figure 2, the wide range of the values of the two potentials at a given value of E sw (different values of P sw and M A ) indicates an equally systematic variation of V pc and V r with P sw and M A .Therefore, it is not justified to attribute the saturation of V pc or V r solely to the increase of E sw regardless of the variation of P sw and M A .
Based on the fact that all three parameters, E sw , P sw and M A , affect V pc and V r , Peng et al. (2009) introduced a combined parameter as powers of E sw (mV/m), P sw (nPa) and M A given by and reset the data points in the f − V pc and f − V r planes, as shown in Figure 3, where the cross signs, solid circles, and plus signs correspond to data points with Σ P = 1, 5, and 10 S, respectively.The simulation data points divided in groups according to the value of Σ P are concentrated nearby several separate curves in Figure 3(a) and 3(b), which may be approximately fitted by where f is given by Equation ( 5) and Σ P is in the unit of S. For the available simulation data points in  corresponding fitting curves are 16, 24, and 12 kV for V pc with Σ P = 1, 5, and 10 S, respectively, and their counterparts for V r are 23, 26, and 15 kV, respectively.The relative RMS deviations are 8%, 17%, and 13% for V pc and 9%, 15% and 7% for V r , respectively.
Based on these fitting formulas, Peng et al. (2009) made the following three conclusions: (1) Both V pc and V r saturate with respect to the increase of f instead of E sw as often assumed; any variation of the interplanetary conditions in favor of the increase of f may cause the saturation of V pc and V r .
(2) The saturation point, defined from Equation ( 6) and ( 7) by setting the potential values to be one half of the saturation levels, is found to be f c = 6.6 for V pc and f c = 14.4 − 0.9Σ 1/2 p for V r , when the units are mV/m for E sw and nPa for P sw in the expression of f (see Equation ( 5)).The ionospheric Pedersen conductance controls the saturation level, reached as f → ∞, which does not depend on the solar wind driving conditions.
(3) The two potentials, V pc and V r , stem from the SMI coupling and exhibit similar saturation behaviors in response to the coupling.They are positively correlated because of sharing the same driving source and the close coupling between the ionosphere and the magnetosphere.

Reconnection Voltage Versus IMF Clock Angle
In order to achieve a quantitative description of the Recently, multi-spacecraft in-situ observations were used to infer the global geometry of the magnetic merging line, or X-line as often termed in the literature (see a brief review by Paschmann (2008)).For instance, Phan et al. (2006) used the Geotail and Wind data during stable dawnward dominated IMF to infer the presence of a tilted X-line hinged near the subsolar point and spanning the entire dayside magnetopause.Based on a statistical study of 290 fast flow events measured by Double Star/TC-1 in low latitudes and Cluster in high latitudes, Pu et al. (2007) suggested that a possible S-shaped X-line exists for generic dawnward IMF cases.The configuration of the merging line inferred from these observations is consistent with the prediction from the component reconnection hypothesis.Hu et al. (2009) proposed a method to quantitatively trace the whole merging line, which is applicable for magnetospheric magnetic fields associated with arbitrary IMF clock angles.The method is still based on the relation between the merging line and last closed field lines.First, last closed field lines are drawn starting from the the inner boundary of the magnetosphere (r = R e ) in both northern and southern hemispheres, and the starting points are uniformly disposed in longitude at a constant interval of 4 • .Each field line is traced by the fourthorder Runge-Kutta method with a step-arc-length 0.1 (R e ).For a given longitude of the starting point, a last closed field line is obtained by iteration with an accuracy of 10 −4 degree in the latitude of the starting point.A last closed field line thus obtained must touch the merging line at a certain point, where the magnetic field strength is assumed to be a minimum.On this basis, the point with minimum magnetic field strength is found for each last closed field line, being regarded as a point right on the merging line, referred to as a "touch point".The touch points thus obtained tend to concentrate in groups on the magnetic nulls, and there exist several wide spaces between adjacent groups, in which no touch points appear.Each space is then filled up by one of the two last closed field lines, which pass through the two touch points right next to the space, respectively.The touch points and the filled spaces form a closed curve around the Earth, that is regarded as the merging line wanted.This method was tested by tracing the merging line of the compound field superposed by the Earth's dipole field and a uniform field, for which analytic solution is available (Yeh, 1976), and was successfully applied to trace the whole merging line of magnetospheric magnetic fields associated with different IMF clock angles.
Once the merging line is obtained for a specific magnetospheric magnetic field, the radial-rayintegration method (Hu et al., 2007) mentioned above may be used to calculate the reconnection potential and the associated tangential electric field along it so as to determine the distribution of the reconnection rate and its integration along the merging line, i.e., the reconnection voltage.
Using the above-mentioned diagnosis methods, Hu et al. (2009) studied the IMF clock angle dependence of the reconnection voltage.The simulation runs were made with all interplanetary conditions to be fixed (v sw = 400 km•s −1 , n sw = 5 cm −3 , p sw = 0.0126 nPa, and B IMF = 10 nT) except for the IMF clock angle, which is taken to be θ IMF = 22.5 • , 45 • , 90 • , 135 • , and 180 • , respectively.Once a quasisteady solution is obtained, the diagnosis methods described above were used to determine its merging line and the reconnection potential Φ R along the merging line.Moreover, the tangential electric field E R and the magnetic field strength B R were also evaluated along the merging line.The results are shown in Figure 4, where each column corresponds to a fixed IMF clock angle θ IMF , labeled on the top of the top panels, whereas rows 1 to 4 denote the projections of the merging line in the r − ϕ and λ − ϕ (λ is the latitude and ϕ is the longitude that is zero at the subsolar point and positive duskside) planes and Φ R and E R versus ϕ along the merging line, respectively.
The two magnetic nulls closest to the subsolar point, as marked out by two vertical dashed lines in each of the right 4 panels, are correctly captured by the obtained merging line: the northern null (λ > 0 or ϕ > 0) is type A or A S , and the southern null (λ < 0 or ϕ < 0), type B or B S , as they should be for the magnetospheric magnetic field (Lau and Finn, 1990).
The sunward part of the diagnosed merging line looks smooth and steady with time, whereas the the tailward part is somewhat irregular especially in the r − ϕ profile and changes with time.This comes from oscillations of the simulation solutions, which become more serious in the magnetotail (Hu et al., 2005).The merging line is not equidistant from but closer to the Earth on the sunward side due to the action of the solar wind.The λ − ϕ profiles of the merging line in A horizontal dot-dashed line displays the zero line of either ΦR or ER in each panel of the bottom two rows (Hu et al., 2009).
solid look similar to those in thick dashed of the compound field superposed by the Earth's dipole field and the IMF, especially on the sunward side.To some extent, this justifies the analogy in topology between the magnetospheric magnetic field and the compound field.
When θ IMF decreases, both Φ R and E R decrease in magnitude and thus V r is reduced.A dip appears in the E R − ϕ profile across the subsolar point, a similar result to that obtained by Siscoe et al. (2001) for the due duskward IMF case.The subsolar dip increases in relative amplitude with decreasing θ IMF , and it reaches almost the zero line of E R for the θ IMF = 22.5 • case, implying that the reconnection rate becomes negligible nearby the subsolar point when the IMF tilts northward.In the presence of a subsolar dip, the peaks of E R shift to high latitudes, but the associated reconnection still belongs to component reconnection (Sonnerup et al., 1981), since an appreciable component of magnetic field parallel to the merging line exists at the reconnection site.Nearby the nulls closest to the subsolar point, E R almost vanishes.This means that antiparallel reconnection (Crooker, 1979), which is supposed to take place nearby the nulls, does not actually occur for dayside reconnection of quasi-steady states of the SMI system.
Figure 5(a) shows the θ IMF dependence of the reconnection voltage V r , the ionospheric transpolar potential V pc , and the reconnection voltage V RI , defined by Fedder et al. (1995) based on projections of ionospheric potential contours on the open-closed field boundary.The three voltages are all normalized to the values for θ IMF = 180 • .The curves of sin 3/2 (θ IMF /2), sin 2 (θ IMF /2), and sin 3 (θ IMF /2) are shown for reference.V pc and V RI show different depe-  (Hu et al., 2009).
ndence from that of V r , so it is not justified to take them as substitutes for the reconnection voltage.For generic southward IMF cases (θ IMF > 90 • ), V pc and V RI are very close to each other, better fitted by the sin 3 (θ IMF /2) curve, being essentially consistent with the observational conclusion made by Reiff and Luhmann (1986).For the generic northward IMF cases (θ IMF < 90 • ), on the other hand, the decrease of the two voltages with decreasing θ IMF slows down.V pc approaches a finite value as θ IMF goes to zero, which may arise from other coupling processes between the solar wind and the magnetosphere (see Fedder et al., 1991).V r is better fitted by the sin 3/2 (θ IMF /2) curve.Based on 10 observed magnetospheric state variables, Newell et al. (2007) inferred a solar windmagnetosphere coupling function, which is proportional to sin 8/3 (θ IMF /2).If the coupling function is assumed to be proportional to the square of the reconnection voltage (Kan and Lee, 1979), then the index in the coupling function is 3, which is slightly larger than but close to 8/3 inferred by Newell et al. (2007).
Figure 5(b) shows three electric fields associated with the merging line as functions of θ IMF , E M , the maximum electric field along the merging line, E A , the mean electric field over the sunward merging line between the two peaks of reconnection potential, and E S , the electric field at the subsolar point of the merging line that was used by Borovsky et al. (2008) to characterize the dayside reconnection rate.The three electric fields are all normalized to the values for θ IMF = 180 • .Interestingly, the three electric fields show similar patterns of variation with θ IMF except that E S decreases more rapidly with decreasing θ IMF .However, such patterns differ from that for V r shown in Figure 5(a).Therefore, electric fields on the merging line, however defined they may be, are not justified to replace the reconnection voltage in characterizing the coupling between the solar wind and the magnetosphere.The reason is partly associated with another factor which controls the coupling, i.e., the length L of the dayside merging line between the two peaks of reconnection potential.As Figure 5(c) shows, L varies non-monotonically with θ IMF , peaked at about θ IMF = 90 • .

Concluding Remarks
Quantitative analyses of the electric current and electric field based on global MHD simulation data will greatly promote our understanding of the electrodynamic coupling in the Solar Wind-Magnetosphere-Ionosphere (SMI) system.Among these analyses, diagnosis of the current closure helps us to set up links between load and dynamo areas in the system.For instance, it is known from such diagnoses that the bow shock as a dynamo supplies reconnection current on the magnetopause, the region 1 current into the ionosphere, and the cross-tail current in the central current sheet, and becomes a predominant contributor under strong interplanetary driving conditions.Thus we have realized a direct influence of the bow shock on the magnetospheric and ionospheric activities.The present studies of electric current were limited to steady cases, but similar studies may be extended to time-dependent cases, since the current is always closed according to the Ampere's law.Then we may reveal the temporal variation of various electric currets and their closure, which would be an interesting subject for future efforts.
We have found a single parameter, which essentially determines the behavior of the transpolar potential and reconnection voltage in response to the solar wind and IMF conditions.According to Equation (5), this parameter is Here we have avoided the outstanding issue of the physical mechanism that leads to such a single parameter f .Although some suggestions were made on the saturation mechanism (Siscoe et al., 2004), we still have a long way to go in seeking a single, unified mechanism for the saturation of the ionospheric transpolar potential and the magnetopause reconnection voltage as well.
Through calculating the reconnection potential along the merging line, we have acquired a quantitative knowledge of the magnetic reconnection on the magnetopause and the relation between the reconnection voltage and the transpolar potential.Since magnetic reconnection is believed to be the main means of transformation of magnetic energy to mechanical energy, a more quantitative study of magnetic reconnection and its relation to solar wind and IMF conditions is obviously important.Nevertheless, present calculations were limited to steady cases, in which an electric potential is well defined and may be evaluated by an integration of the convectional electric field along a properly selected path, where the ideal MHD condition approximately holds.For time-dependent cases, which are most important, one has to seek other ways to calculate the reconnection electric field or the reconnection rate.

Figure 1
Figure 1 The colored contour of j • E in the plane of x = −15 Re and projections of 4 typical streamlines of cross-tail current, which close on the magnetopause (red lines) and the bow shock (blue lines), respectively (Tang

Figure 2
Figure 2 Distribution of available simulation data points in (a) Esw − Vpc and (b) Esw − Vr planes.Cross signs, solid circles and plus signs correspond to different ionospheric Pedersen conductances, which are 1, 5, and 10 S, respectively(Peng et al., 2009).

Figure 3
Figure 3 Reorganization of the simulation data points in (a) f − Vpc and (b) f -Vr planes.The fitting curves for Vpc come from Equation (6), and those for Vr from Equation (7).Cross signs, solid circles and plus signs correspond to different ionospheric Pedersen conductances, which are 1, 5, and 10 S, respectively.
reconnection by global MHD simulations in general cases, one needs to determine the merging line and calculate the electric potential distribution along it.For magnetospheric magnetic fields, the merging line should be a closed space curve which encircles the Earth.Attempts were made to allocate the merging line of quasi-steady magnetospheric magnetic fields obtained by global MHD simulations.Fedder et al. (1995) identified the merging line with the intersection between the magnetopause and the topological separatrix between the IMF lines and the inmost open field lines that connect to the ionosphere.They showed the IMF lines for different IMF clock angles in their plates 1 and 2, from which one can see the merging line by the naked eye.Nevertheless, they did not display the merging line explicitly, and only the angle between the topological separatrix and the IMF was measured.For a precisely duskward IMF case,Siscoe et al. (2001) did find the sunward part of the merging line explicitly, which was identified with the middle part of the last closed field line that comes closest to the two magnetic nulls.Using a similar method,Dorelli et al. (2007) located the merging line for a generic northward IMF case with an IMF clock angle of 45 • .

Figure 4
Figure 4 Magnetic merging line and its distributions of magnetic field strength (BR), reconnection potential (ΦR), and reconnection electric field (ER) for several separate IMF clock angles (θIMF).Rows 1 to 4 denote the r − ϕ and λ − ϕ profiles of the merging line, and the distributions of ΦR and ER, respectively.The analytic solutions for the merging line of the corresponding compound field are shown by thick dashed curves in each panel of the top two rows.The magnetic field strength distribution is shown in each panel of the second row by thick dotted curves.The vertical dot-dashed and dashed lines mark out the peaks of ΦR and the magnetic nulls closest to the subsolar point respectively.

Figure 5
Figure 5 (a) Normalized reconnection voltage (Vr), transpolar potential (Vpc), and projected reconnection voltage in the ionosphere (VRI) versus the IMF clock angle θIMF.Also shown for reference are the curves of sin 3/2 (θIMF/2), sin 2 (θIMF/2), and sin 3 (θIMF/2).(b) Normalized electric fields versus θIMF, including the maximum field (EM), the mean field (EA), and the field at the subsolar point (ES).(c) Normalized length of the merging line (L) between the positive and negative peaks of reconnection potential, geocentric distance (rS) of the subsolar point of the merging line, and radius (rN) of the merging line for the compound field superposed by the Earth's dipole field and the IMF versus θIMF.Solid and open circles in each panel denote the simulation results ≡ |B z |).

Table 3 The total region 1 current I 1 (MA) and the contribution of the bow shock I 1bs (MA) to I 1 versus the solar wind number density nsw (cm −3 ) and the southward IMF Bz (nT) (Peng and Hu, 2009).
Guo et al. (2008) et al. (2008).However, if M A becomes close to or falls below 2, I 1bs will decrease with increasing |B z | in both magnitude and percentage (i.e., I 1bs /I 1 ) because of the resultant reduction of the bow shock strength.Table 3 lists their simulation results.It can be seen that I 1 increases monotonically with increasing |B z |, a similar conclusion as reached by