Magnetospheric Response to the Solar Wind Dynamic Pressure Pulses inferred from Polar Cap Index

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. A12, PAGES 21,297-21,311, DECEMBER 1, 1993

Magnetospheric Response to Solar Wind Dynamic Pressure Variations' Interaction of Interplanetary Tangential Discontinuities

With the Bow Shock

B. H. Wu

Institute of Space Science, National Central University, Taiwan

M. E. MANDT 1

NASA Goddard Space Flight Center, Greenbelt

L. C. Lr•r• • AND J. K. CHAO

Institute of Space Science, National Central University, Taiwan

Some magnetic impulse events observed in the polar region are related to vortices associated with plasma convection in the ionosphere. Recent analyses of satellite and ground data suggest that the interaction of solar wind dynamic pressure pulses and the magnetosphere may lead to the formation of velocity vortices in the magnetopause boundary layer region. This can in turn lead to the presence of vortices in the polar ionosphere. However, before reaching the Earth's magnetopause, these interplanetary pressure pulses must interact with and pass through the bow shock. A variation of the solar wind dynamic pressure (ApV 2) may be associated with shocks, magnetic holes, or tangential discontinuities (TDs) in the interplanetary medium. We study the interaction of interplanetary TDs with the Earth's bow shock (BS) using both theoretical analysis and MHD computer simulations. It is found that as a result of the collision between a TD and the BS, the jump in the solar wind dynamic pressure associated with the TD is significantly modified, the bow shock moves, and a new fast shock or fast rarefaction wave, which propagates in the downstream direction, is excited. Our theoretical analysis shows that the change in the plasma density across the interplanetary TD plays the most important role in the collision process. In the case with an enhanced dynamic pressure behind the interplanetary TD, the bow shock is intensified in strength and moves in the earthward direction. The dynamic pressure jump associated with the transmitted TD is generally reduced from the value before the interaction. A fast compressional shock is excited ahead of the transmitted TD and propagates toward the Earth's magnetosphere. For the case in which the dynamic pressure is reduced behind the interplanetary TD, the pressure jump across the transmitted TD is substantially weakened, the bow shock moves in the sunward direction, and a rarefaction wave which propagates downstream is excited. We also simulate and discuss the interaction of a pair of tangential discontinuities, which may correspond to a magnetic hole, with the BS.

1. INTRODUCTION

In an attempt to identify the ground magnetic signature of flux transfer events (FTEs), Lanzerotti et al. [1986] studied magnetic impulse events based on observations at South Pole Station. A unipolar or bipolar vertical component of the geomagnetic field disturbance, Bz, is observed during each event. The duration of observed magnetic pulses is typically 1~5 min, and the magnitude ranges from about 20 to 100 nT. Recent observations and theoretical studies seem

• Now at Laboratory for Plasma Research, University of Maryland, College Park.

2 Also at Geophysical Institute, University of Alaska, Fair- banks.

Copyright 1993 by the American Geophysical Union.

Paper number 93JA01013. 0148-0227/93/93JA-01013 $05.00

to infer that solar wind dynamic pressure variations (ApV 2) may lead to the presence of velocity vortices near the magnetopause [Friis-Christensen et al., 1988; Glassmeier et al., 1989; Southwood and Kivelson, 1990; Kivelson and South- wood, 1991]. Vortices in the low latitude-boundary layer (LLBL) map into the ionosphere via a field-aligned current system and can generate the convecting Hall current loops of a single vortex or twin vortices. Therefore one expects that a traveling ionospheric vortex or an induced unipolar (bipolar) magnetic pulse will be observed during such an event [Sibeck, 1990]. This suggestion is supported by the work of Lee [1991] in which MHD simulations of the interaction between the solar wind dynamic pressure pulses and the magnetopause boundary region were studied. The simulation results always show that flow vortices are formed in the LLBL when a shear pressure pulse impinges upon the magnetopause.

In fact, the expected consequences of interplanetary dynamic pressure variations were investigated from another point of view in 1950s. In the beginning stages of a geomagnetic storm, there is often a sudden commencement

21,297

21,298 Wu ET AL.: TANGENTIAL DISCONTINUITY-BOW SHOCK INTERACTION

(but not always associated with substorm events, for example, Nishida [1978]) whose onset is recorded simultane- ously (within about I min) all over the world. These positive/negative sudden impulses ($I + and $I-) were suggested as a consequence of solar wind dynamic pressure variations impinging upon the geomagnetic field [Chapman and Ferraro, 1931]. Following this original suggestion, Gold [1955] proposed that an SI + is caused by an interplanetary shock wave compressing the magnetosphere. Sonerr and Colburn [1965] indicated that the less frequent SI- might be attributed to a reverse shock, and proposed that the forward-reverse shock ensemble to relate an $I + - Si-

pair. This problem (which may be identified with the clas- sical Riemann problem) has been theoretically studied by Sturrock and $preiter [1965] based on the strong-shock approximation. Colburn and Sonerr [1966] also considered tangential discontinuities (TDs) in the solar wind which should be able to provide a large change in the dynamic pressure incident upon the magnetosphere. The effect of a TD on the magnetosphere is evident in satellite observations [e.g., Gosling et al., 1967; Ogilvie et al., 1968].

However, there is a significant aspect of these events that the above authors did not consider, namely the influence of the Earth's bow shock. Because of the supersonic solar wind flow, a standing bow shock wave (BS) is formed a few Earth radii (RE) upstream of the magnetopause. Any interplanetary discontinuity must interact with the BS before impact- ing on the magnetosphere. The average geocentric distance of the subsolar point of the magnetopause and the BS near the ecliptic plane are 11 and 14.6 RE, respectively. However, their actual location at a given time is highly variable due to the fluctuating interplanetary medium. Anderson et al. [1968] showed that the subsolar part of the magnetopause oscillates with periods of 3~15 min and amplitudes in the range of 0.25 ~ 2 RE. They also found that the subsolar region of the BS behaved similarly. This relatively low velocity motion (about 10 km/s) is described as a coupled oscillatory motion of the combined shock-magnetosphere system [Holzer et al., 1966; Greenstadt, 1968; Smit, 1968]. Large- scale displacement and high-speed motion of the BS are also observed. Fairfield [1971] found the shock position about 5~10 RE outside the average position. Greenstadt et al. [1972] measured shock speeds in excess of 100 km/s under low Alfvdn Mach number (M•) conditions. Guha et al. [1972] also observed similar high shock speeds but under more typical conditions of high M•(~ 5.5). Formisano et al. [1973] inferred an average speed of 85 km/s.

Obviously, the dynamic pressure in the solar wind plays a significant role in MHD phenomena of the Earth's shock- magnetosphere system. An intense dynamic pressure in the solar wind implies a BS which is stronger and closer to the magnetopause [Spreiter et al., 1966; Binsack and Vasyliunas, 1968]. There are three common solar wind features that have large amplitude, impulsive dynamic pressure variations associated with them: shocks, magnetic holes, and tangential discontinuities. An interplanetary forward-reverse shock pair can induce a high velocity motion of the BS and a SI + - SI- pair on the ground as a consequence of the compression process. This has been studied by others [e.g., $hen and Dryer, 1972; Dryer, 1973; Chao and Lepping, 1974]. For magnetic holes, the solar wind velocity and temperature do not vary significantly, the density increases but rarely by more than 10%. With

regard to tangential discontinuities, Solodyna et al. [1977] found that only a few solar wind discontinuities provide density (and thus dynamic pressure) variations greater than 35%. Sibeck et al. [1989] has pointed out that the dynamic pressure can vary across an interplanetary TD by as much as a factor of 4, but only on rare occasions. However, TDs are the most prevalent type of interplanetary fluctuation and they can arrive as frequently as several minutes to several hours apart [Burlaga and Ness, 1969]. Because of their abundance in the solar wind, in this study we chose to concentrate on the interaction of TDs with the BS.

In this paper we consider from the viewpoint of magne- tohydrodynamics, a solar wind dynamic pressure variation across a TD which is aligned with the spiral interplanetary magnetic field (IMF), impinging upon the Earth's bow shock. This type of interaction has been discussed by VSlk and Auer [1974], who estimated the average BSs velocity based on the gas-dynamic approximation. Here we concentrate our attention on the plasma variation downstream of the BS (i.e., in the magnetosheath). To remove some of the complicating factors in this problem, we reduce it to a simplified one dimensional interaction of a tangential discontinuity with a perpendicular shock. We theoretically solve this system by numerical analysis. We then use computer simulations to compare with our theoretical results. The effects of varying the parameters which determine the TD-BS system are examined in this context. The influence of magnetic holes and/or diamagnetic cavities is also briefly considered.

The remainder of this paper is organized as follows: In section 2 we present the theoretical framework for the analysis. In section 3 the results of the numerical analysis are described. In section 4 the results of MHD simulations

of the interactions are presented with four cases described in detail. In section 5 we discuss some potential consequences resulting from our theoretical results and explore questions requiring further study. Finally, we summarize our principal results in section 6.

2. THEORETICAL FORMULATION

2.1. Physical Model

As the supersonic solar wind impinges upon the Earth's magnetospheric cavity, a standing fast-mode shock wave or bow shock (BS) is set up in the interplanetary medium a few Earth radii upstream from the magnetopause [e.g., $preiter and Jones, 1963; Fairfield, 1971]. We consider the interaction of a tangential discontinuity (TD) in the solar wind with the Earth's shock-magnetosphere system as illustrated in Figure 1. The TD must be aligned with the interplanetary magnetic field (IMF) which we assume follows the Parker spiral orientation on the average. Associated with this TD, there exists a jump in the solar wind dynamic pressure. The tangential discontinuity will be tangent to the bow shock at the first-encounter point which will generally be in the postnoon region. We focus our attention to the instant before the global variation becoming important.

In the region surrounding the first-encounter point, but over a distance smaller than the radius of curvature of the

bow shock, we can locally consider the interplanetary TD and the Earth's BS as planer surfaces. This problem is then reduced to a one-dimensional problem: A tangential discontinuity convecting with the background solar wind

Wu ET AL.: TANGENTIAL DISCONTINUITY-BOW SHOCK INTEl{ACTION 21,299

Bow S

Solar Wind

• E•arth

Fig. 1. The geometry of an interplanetary TD impinging upon the Earth's bow shock. The TD moves with the radial solar wind, and is aligned with the spiral IMF which encounters the bow shock. The x axis lies on the shock normal toward the Earth at the

first-encounter point. 0 and Ovn represent the garden hose angle of the IMF and the angle between solar wind and shock normal, respectively.

collides with a standing perpendicular fast shock. Figure 2a illustrates the simplified problem: the solid line denotes the total hydromagnetic pressure, Pt = P + B2/2t•0; the jump in Pt denotes the bow shock location and the dashed line is the TD [e.g., Landau and Lifshitz, 1959, Figure 62]. Region 1 is the steady solar wind, region 2 denotes the region downstream of the BS (magnetosheath), and region 3 denotes the interplanetary medium with an enhanced or reduced plasma density behind the TD.

There are only two types of MHD waves which can propagate in the direction perpendicular to magnetic field: the fast-magnetosonic wave and the entropy wave. Correspond- ing to these waves, there are fast shocks or fast rarefaction waves and TDs in the large amplitude limit. In the reference frame of the moving solar wind the collision of TD and BS can be described symbolically by TS._ (following Courant and Friedrichs [1948]), where T denotes the TD, S denotes

the shock and the arrow in the subscript of S denotes the propagation direction of the shock relative to the fluid (the TD is static in the fluid frame).

For a TD with a higher plasma density in region 3 (relative to region 1), the dynamic pressure in this region is enhanced and the following situation may result. When the TD collides with shock, the shock is intensified and compressed (toward the Earth) due to the enhanced dynamic pressure. After the TD penetrates through shock, the intensified shock creates a zone of enhanced compression (compared with region 2) on the downstream and a fast shock is then excited. This new shock must propagate in the opposite direction of BS, as shown in Figure 2b, since only one fast-magnetosonic wave can propagate in each direction [Landau and Lifshitz, 1959]. On the other hand, for a TD with a lower density in region 3, the BS weakens and expands sunward. The new fast wave (FW) becomes a fast rarefaction wave, again propagating toward the downstream. Therefore the result of head-on collision between plane surfaces of TD and BS can be divided into two cases: (1) TS._ =• S._ TS_ if Pa/P• > 1 and (2) TS._ =• S...TR_ if Pa/P• < 1, where R denotes the rarefaction wave, Pa/P• denotes the density ratio across the TD, and the symbol =• denotes the transformation process. Figure 2b illustrates the physical features for the above two results, the details of which are further explained below.

After the TD penetrates the BS, a modified zone appears downstream of the shock from which a fast-mode nonlinear

wave is excited due to the pressure jump between the new downstream region and the original (region 2). The fast wave (FW) is distinguished by this pressure jump and can be either a fast shock (shown by the solid line) or a rarefaction wave (dotted line). There are two distinct regions separated by the transmitted TD which we designate as region 4 and region 5 in the modified zone. VBx and VFw denote respectively the velocity of the moving BS and the transmitted FW after the collision.

In our present study we ignore the effect of the Earth's magnetosphere, which will eventually influence the FW and the transmitted TD. Some aspects of this problem will be discussed in section 5. The problem will be studied in more detail in a later paper. In the following analysis we will obtain an analytic formula from the Rankine-Hugoniot relations to determine the strength and velocity of the BS, the TD and the excited FW after the TD-BS interaction.

5 I 4 I I I ,

VBs ' !

BS BS TD FW

(a) before interaction Co) after interaction

Fig. 2. The one-dimensional model of the TD-BS interaction. The line presents the distribution of the total pressure as function of the normal coordinate x. The jump represents the shock and the dashed line represents a TD. (a) The structure before collision and (b) the structure after collision. A new pressure jump appears ahead of the TD, either as fast shock (upper solid line) or as a rarefaction wave (lower dashed line). VBs and VFW are the velocities of the modified bow shock and excited fast wave, respectively.

2.2. Basic MHD Equations

All calculations for the theoretic analysis and the simulation used in this study are based on the ideal MHD approximation. The ideal MHD equations, which describe the conservation of mass, momentum, magnetic flux, and energy, respectively, are given by

Op O--• =-V'(PV) (1)

O(pV) --V.(pVV +(P+ •-•0)i 1 ) (2) Ot I•o

OB _ -v. ( - (3) ot

O(E ---V. S (4) Ot

where (3)combines Faraday's law (V x • = and

21,300 Wu ET AL.: TANGENTIAL DISGONTINUITY-Bow SHOGK INTEI{AGTION

frozen-in condition ( E + V x B = 0). [ is the unit tensor; 0•, $ are the total energy density and energy flux given by

1 V 2 P B 2 OS= •p + + (S) 7- 1 2tto B 2

tto

where 7 is the specific heat ratio chosen to be the adia- batic constant (5/3) in the energy conservation equation. Although 7 might be larger than 2 upstream from the bow shock during quiet solar activity and slightly less than 5/3 downstream [Shen, 1971] due to the heat flow in the vicinity of the shock [Ogilvie et al., 1971], we ignore its influence in this study.

In one dimensional MHD theory the asymptotic plasma state on one side of a discontinuity can be completely determined in terms of the quantities on the other side by using the Rankine-Hugoniot (R-H) relations. They relate the upstream and downstream plasma states of any type discontinuity under the assumption of steady state. In this study we are only interested in discontinuities whose normal is perpendicular to the magnetic field (i.e., B• = 0) in which case the reduced R-H equations may be expressed as

[p•]=o (7)

[,04 + P + •j•o ] :o (8)

[p•V•] =o

[,,,.,B,] = o (:•o)

[( •-pv •' + *•' • • • +--)•]=0 (•) -- •to

where the subscripts n and t refer to components parallel and perpendicular to the normM of the discontinuity surface and the notation [Q] = Q'-Q indicates the change in the quantity across the surface from downstream (primed) to upstream. Note that the plasma state described in (7)-(11) is with respect to the frame of reference of the discontinuity (i.e., the discontinuity surface is stationary, consistent with the steady state assumption). To apply the R-H equations to a moving discontinuity we must first transform to this reference frame.

For the particular case of field aligned surfaces (i.e., B• = 0) there are only two types of MHD discontinuities distinguished by whether or not a net normal flow exists across the surface. They are the fast shock (v• y• 0) and the tangential discontinuity (v• = 0).

With the definition of the shock strength index, e (the downstream to upstream density ratio), we can rewrite the jump conditions for the fast wave as follows:

I

p =ep (•2) B• =sBt (13)

, 1 • = -• (•)

v,' = v,

1 e2 Bt 2 P'= [2:u• (1-7)+ •- + •1 2•o (]6)

where we use the notation Q and Q' denote the unperturbed and perturbed physical quantity Q. The shock intensity e is determined by an equation of second order originating from (•):

, • -[JUL,+ , (•+•)](• 2-, = 0 for Mz,• y• 0 (17a)

for Ma,• = 0 (17b)

where MAn and/• are the background hydromagnetic fluid parameters defined on upstream of the shock. They are, respectively, the Alfvdn Much number (based on the normal flow component) and the ratio of plasma to magnetic field pressure defined by

2 1 2

:u•_ v•. - •/2•o • = a•/2•o (•9)

The R-H equations above yield the trivial and rather uninteresting solution of a nonexistent shock. This is expressed by (17b) above which shows that in the limit of no net flow across the shock surface, all quantities on one side of the shock are identical to those on the other. This special case is distinct from a TD in which the physical quantities change across the surface.

On the other hand, the change of the density and tangent velocity across the tangential discontinuity can be arbitrary. There are only two constraints relating the plasma state on either side of a TD. Namely that the normal net flow is zero and the total pressure (plasma and magnetic) remains constant across the TD. That is

•=o (20) const (21)

It should be noted that the R-H relations do not represent the exact solution for the case of a rarefaction wave.

However, we have tested solutions of the R-H conditions for rarefactions using simulations and have found that the results are valid even for strong rarefactions. The results shown in section 4.2.1 also verify this.

2.3. Mathematical Analysis

Jump conditions. Theoretically, the R-H conditions relate the parameters of one side of a nonlinear MHD steady state perturbation to those of the other side. They do not provide information about the structures within the perturbation. The jump conditions (12)-(16) allow the solution of fast shock. In addition, another solution is permissible in which the jump conditions are the same as for a fast shock solution, but the normal flows on both sides of this discontinuity are reversed from those of the fast shock (i.e., the flow is divergent at the discontinuity). However, this discontinuity will not maintain a steady state but will spread out in time with its leading and traveling edges moving at constant (but

Wu ET AL.: TANGENTIAL DISCONTINUITY-BOW SHOCK INTERACTION 21,301

differing) velocities. It turns into an expanding rarefaction wave. Note that a steady state for such a discontinuity is not possible.

Parameters. In our calculations, all physical quantities such as density, magnetic field, velocity and pressure are normalized by the steady state solar wind plasma on the upstream of the BS (region 1). We parameterize the background solar wind with Mz,1, /•1, and the normal component of flow velocity is determined by V,1 = Vsw cos0v,, where 0v• is the angle between the radial solar wind and shock normal. The garden-hose angle of the spiral IMF is used to determine the location of the TD-BS interaction on

the bow shock (i.e., the first-encounter point) and is related to Ov• by Ov• = •r/2 -0 (see Figure 1). The change in the plasma state (keep in mind that the shock normal angle, 0z• = 90 ø, has already been specified) across the standing BS is then determined. We describe the altered plasma state behind the interplanetary TD by three parameters: PalP1, Bta/Btl, and ¬a/¬l. The remaining quantities of region 3 are determined by the assumption B, = 0, and the re- quirement that V, and Pt remain constant across the TD. The solar wind dynamic pressure variation, ApV• Is,1 is then determined by the parameter PalP1.

Self-consistent system. For a given set of solar wind conditions in region 1 (MA,1 and •1) and the parameters describing the TD (Pa/Pl, Bta/Btl, and Vta/Vtl), the equations (12)-(16), (20), and (21) provide us with the state of regions 2 and 3. With the assumption that B, = 0, we now have 12 nnknowns in our model (see Figure 2), namely: P4, P4, Vt4, Vn4, Bt4; Ps, Ps, ¬5, Vns, Bts; and VBS, VFW. The numerical subscript refers to the region, VBs is the velocity of the bow shock after the interaction (prior to the interaction Vz•s = 0), and VFw is the velocity of the fast wave exited as a consequence of the interaction. The R-H conditions also provide us with a set of 12 nontrivial equations for determining these unknowns, namely: the five jump conditions relating region 3 to region 5, the same five conditions relating region 2 to region 4, and two constraints relating region 4 to region 5 across the TD. We can represent the plasma state of region 4 as functions of the plasma state in region 2 and VFw by using (12)-(16) and determining the normal velocity upstream of the FW from v• = IVy2- VFw[. Similarly, the plasma state of region 5 can be represented as functions of the plasma state of region 3 and Vz•s based on v• = V•3 - Vz•s. Finally, we add the two constraints for the TD, (20) and (21), into the above set of ten equations. Therefore the above two sets of equations can be used to construct a self-consistent system which can be solved uniquely. We formulate them by a nonlinear simultaneous system of equations which are functions of Vz•s and V•w only, i.e.

V•4 - V•5 = F(Vz•s, VFw) = 0 (22)

Pt4 - Pt5 = G(Vz•s, VFw) = 0

where equation F = 0 is obtained from (20) which states that the normal velocity remains constant across a TD. Once Vz•s and VFw are obtained from (22), the plasma state in each region can then be determined.

Uniqueness of the physical solution. There are seven intersection points of the curves defined by F = 0 and G = 0 on the (Vz•s, VFw) plane indicating that multiple roots exist. Following Landau's argument that there is just one wave (for each mode) propagating in each direction in the fluid,

the inequality Vss < V,•4 < Vrw must be satisfied to maintain the order of the BS, TD and FW. In addition, the bow shock must be maintained by the supersonic condition, V,3 - Vss > Cra (although it is possible for the bow shock to disappear or even be reduced to a rarefaction wave in the limiting case of a very weak dynamic pressure behind the TD, however, we disregard this case). These criteria then determine the unique physical solution.

3. THEORETICAL RESULT

In this section we examine the self-similar sMutions

obtained from the system of equations presented in Section 2. There are two principal types of interactions which can occur in the TD-BS collision depending on whether the solar wind dynamic pressure is enhanced or reduced at the TD (region 3 relative to region 1, see Figure 2a). We designate these types as TD + and TD-, respectively. An example of each type is presented with Pa/Pl = 2.0 for the first type (Case A) and Pa/Pl = 0.5 for the second (case B). All other parameters for these two cases are the same. The results from these two cases will be compared with results from MHD simulations in the next section as

a check on our analytical scheme. In the remainder of this section we examine the dependence of the interaction on the other parameters: PalP1, Vt3/Vtl, and Bt3/Btl across the TD and MA,/•, and the IMF angle 0 ahead of the bow shock. We also examine the limiting case in which there is no dynamic pressure variation across the TD, only changes in the magnetic and thermal pressures.

3.1. Individual Cases

Case A (TD+). In this case we consider a TD carrying an enhanced dynamic pressure jump (TD +) which collides with the bow shock. We parameterize this situation with PalP1 = 2, Vt3/Vtl = 1, Bta/Btl = 1, Man1 = 5, and j•l = 1. In Table 1 we list the normalized plasma properties and the field intensity in each of the regions which are obtained from the numerical solution of the equations presented in section 2. Also listed below the table are Vz•s, VTlv, and VFw. All quantities are expressed in units of Q1, the the physical quantities in region 1. The columns labeled Q3, Qs, Q4, Q2 represent the physical states corresponding to the regions described in Figure 2b. As expected, we find that the enhanced dynamic pressure pushes the bow shock toward the Earth (Vz•s > 0). The increase in solar wind density (in region 3) corresponds to a decrease in Vz (or CF). This results in a higher Much number in the upstream since the solar wind speed remains the same across the TD.

TABLE 1. States of Case A

Quantity Q1 Q3 Q• Q4 Q2

p/p1 1.0000 2.0000 6.6704 3.7501 3.1662 P/(B•2/2go) 1.0000 1.0000 37.7196 34.7799 26.1662

Bt/B1 1.0000 1.0000 3.3352 3.7501 3.1662 Vn/Vnl 1.0000 1.0000 0.4273 0.4273 0.3158 CF/Vnl 0.2708 0.1915 0.5052 0.6776 0.6342

fl 1.0000 1.0000 3.3909 2.4731 2.6118

p Vn•/(pVn•)l 1.0000 2.0000 1.2179 0.6847 0.3158 Pt /(pVn2)l 0.0400 0.0400 0.9769 0.9769 0.7242

YBsIYnl = 0.1821; VTDIVnl = 0.4273; VFwIVnl = 1.0318.


Even though the bow shock is now moving towards the downstream, this motion is small compared to the solar wind speed so the Mach number of the bow shock has increased. The stronger shock produces more dissipation and the region downstream is compressed more than the plasma of region 2. The enhanced compression acts to excite a FW in the downstream. The TD is transmitted and convects with the

plasma between the BS and the FW. The resultant structure is that depicted in Figure 2b.

In this example, the dynamic pressure jump carried by TD is reduced by nearly one half (53% of the initial jump). On the other hand, the presence of the excited fast shock provides another jump both in the dynamic pressure and total hydromagnetic pressure. The dynamic pressure and total hydromagnetic pressure downstream of the newly excited shock (i.e., in region 4) are about 1.37 and 1.25 in normalized units, respectively. Therefore, the effective pressure jump in the magnetosheath is ApV• 21•-4 0.53, ApV•214_2 = 0.37, and APt14-2 = 0.25, while the initial jump is ApV•21a_• - 1. The sum of the jumps in the magnetosheath is larger than that carried by the interplanetary TD initially. The increase is due to a larger amount of dissipation from the stronger bow shock.

Case B (TD-). For the case in which the TD carries reduced dynamic pressure jump (TD-), we use Ps/P• = 0.5. The other parameters are Vta/Vt• = 1, Bta/Bt• = 1, MA• = 5, and /• = 1, as in case A. The results are summarized in Table 2. In this case the BS is weakened

and expands outward (sunward) with VBs < 0 due to the reduced dynamic pressure in the solar wind. The weaker BS now produces less dissipation and the region downstream of it is compressed less than the plasma in region 2. Consequently, a fast rarefaction wave is excited which propagates earthward. The TD is transmitted through the shock and convects with the plasma flow between the BS and the FW. The resulting structure is as depicted in Figure 2b (the dotted line). Note that Vrw is an average velocity of the entire rarefaction wave region separating the steady regions 4 and 2. The nonlinear fast rarefaction wave broadens in time since each edge of the wave propagates at the local fast-mode speed. We can estimate the average velocity from Vc = [P4(V• +Cr)4 +p•(V, +CF)•]/(p4 to obtain V• = 0.86V• ~ Vrw.

In this case, all pressure jumps in the magnetosheath are negative due to the reduced initial dynamic pressure jump. The dynamic pressure jump carded by the TD is weakened by more than a factor of 10 from the initial -0.5 to -0.05. The dynamic pressure jump and total hydromagnetic pressure jump associated with the excited rarefaction wave are -0.21 and -0.20, respectively. The lower value of the

TABLE 2. States of Case B

Quantity Q• Q3 Q• Q4

p/pz 1.0000 0.5000 1.4756 2.6303 3.1662 P/(B•2/2t•o) 1.0000 1.0000 17.3851 19.1761 26.1662

Bt/Bz 1.0000 1.0000 2.9512 2.6303 3.1662 Vn/Vn• 1.0000 1.0000 0.2017 0.2017 0.3158 CF/Vnl 0.2708 0.3830 0.7930 0.5901 0.6343

13 1.0000 1.0000 1.9961 2.7717 2.6118

p Vna/(pVn2)l 1.0000 0.5000 0.0601 0.1071 0.3158 Pt /(PVn•)I 0.0400 0.0400 0.5219 0.5219 0.7242

VB$/Vn• = -0.2074; VTD/Vn• ---- 0.2017;VFw/Vn• = 0.8758.

total pressure jump in the magnetosheath is due to the weakened bow shock.

3.2. Effects oj' the Interplanetary TD Parameters

There may be changes in the density, magnetic field, and/or tangential velocity associated with the interplanetary TD. Here we examine the effects of these changes on the TD-BS interaction. In this section we set the back-

ground solar wind parameters to MA,• = 5 and /• ---- 1. The results are presented as a function of the parameter being investigated, either Po/P•, Vto/Vt•, or Bto/Bt•, with the other two parameters set to unity.

Density variation. Figure 3 shows the resulting conditions in the magnetosheath as a function of the density ratio (PalPs) at the TD with Vt0/Vt• = 1 and B,o/Bt• = 1. The four panels on the top of Figure 3 are, respectively, the solar wind dynamic pressure variation (ApV•21a_•) associated with the initial TD, the modified dynamic pressure jump (ApV•I•_4) of the transmitted TD, and the dynamic pressure jump (ApV• I•-•)and total hydrodynamic pressure jump (APtl•-•) at the excited FW. All of these quantities

-- I I i I i i i I

<1_ 1 1

-1 1

I

i I I I I I I I I

o -1

4

i i i

I I I I I I I I I

I I

0 1 2 3 4 5

/ Pi Fig. 3. The resulting conditions in the magnetosheath after the TD-BS interaction as a function of the density ratio (PalP1)' Other parameters are: MAnl ---- 5, t31 ---- 1, Vta/Vtl = 1, and Bta/Bt• = 1. The four panels on the top show the jumps of the dynamic pressure and total pressure associated with TD and the excited FW, in units of (pVn2)•. The two panels on the bottom show the velocities (in units of Vnl ) and the strength of the BS, TD, and FW, respectively. The dashed line represents the initial states before the interaction.


are normalized by the upstream dynamic pressure, (pV,2)•. The last two panels on the bottom of Figure 3 are, respectively, the velocities (in units of solar wind speed, V,•), and the strength of the modified BS (ess, defined by Ps/Pa) and the FW (eFw, defined by P4/P2)' We divide our discussion into two parts which separately consider the TD + and TD- type interactions. These two types are separated by the dashed line at pa/p• = 1 in Figure 3.

We first consider the situation of an enhanced dynamic pressure behind the interplanetary TD (the region with Ps/Pl > 1 in Figure 3). Some of the features evident here we have Mready discussed: 0 < Vss < V• < Vr•, ess > •0, and • > 1, where •0 is the originM strength of the bow sho& before the collision. As ps/p• increases graduMly, the velocity and strength of the bow sho& and the new fast shock also increase but their variation is graduM. Because of the sho& e•stence condition, v• > Cr > 0, and a constrent that the strength of MHD sho&s never exceeds (7+ 1)/(7-1), the modification of the BS is limited by Vss < Vn3 and •Bs < 4. We estimate the limiting vMue of Vss, within po/p• < 20 in an environment with Mz• = 5 and •1 = 1, to be Vss • 0.SV•. However, the density variation of interplanetary TDs rarely exhibit such a large vMue (in fact, only few observations report up to a factor of 4). Our result infers that the velocity of observed inward movement of the Earth's bow sho&

should not exceed 200 km/s, for the average solar wind conditions of Vsw • 450 km/s. From the information in the pressure panels, we can find that while the solar wind dynamic pressure variation increases linearly with P3/P•, M1 of the pressure jumps in the magnetosheath Mso increase except for the dynamic pressure jump across the transmitted TD which is generMly weakened. However, apV21,-• : (e.•/p• - eoe•w)• Vfo increases more rapidly than •pV•la-x due to the simultaneous increase of PalPs, s•s, and VT•. There is a criticM vMue for the density ratio (• 4.77) above which the dynamic pressure jump across the transmitted TD is enhanced in the TD-BS interaction.

Now we consider the TD- case in whi& the interplanetary TD carries a reduced dynamic pressure. In the region where Pa/P• < 1 in Figure 3, M1 pressure jumps in the magnetosheath are negative. This indicates that the new downstream region of the bow sho& is a rarefaction region. The transmitted rarefaction wave is indicated by s•w < 1. The magnitude of dynamic pressure variation associated with the TD is Mways weakened after the interaction; in fact, it reduces to nearly zero (compared with the originM). Because the upstream Ma& number which determines the shock intensity is proportionM to the square root of the density (or dynamic pressure), the bow sho& strength is reduced (s•s < s0), and it moves away kom the Earth (V•s < 0). As P3/P• decreases further, the flow can hardly m•nt•n the supersonic condition, so the expanding speed of bow sho& increases rapidly until it disappears (s•s = 1). Therefore, the outward velocity of the bow sho& can easily be much greater than that of the inward motion, as reported by observations [e.g., Formisano et al., 1971]. If P3/P• is very smM1, the downstream regions (4 and 5) may even flow outward (away kom the Earth) due to the rarefaction behind the rapidly expanding BS. Thus the magnetosheath plasma may experience a sunward flow if the solar wind dynamic pressure is greatly reduced.

IMF variation. Figure 4 shows essentially the same quantities as Figure 3 but as a function of the ratio of the field intensity Bta/Bt•. The density jump across the TD is set to be Pa/P•: 1 (i.e., no initial dynamic pressure jump), and hence the panel of ApV•la_• is omitted. Because of the different scales, separate panels for the velocities and jump strengths are used in this case. The other parameters are Vta/¬• = 1, M,a,• = 5, and • = 1. We must keep in mind that the variation of the IMF intensity across the TD always corresponds to a plasma pressure variation. From the pressure balance constraint (21) on the TD, the range of permissable variation in the field intensity across the TD is restricted by the condition that the thermal pressure always be positive, i.e., (Bta/Bt•) • < 1 + •. Negative values of Bta/Bt• represent the situation where the field also reverses direction across the TD.

When the TD carries an increased field intensity (Bta/Bt• > 1), the bow shock moves outward and is weakened. However, a fast shock is excited rather than a rarefaction wave. This collision process is different from the type when the TD carries a density jump. In addition, there is now a dynamic pressure jump across the penetrating TD where initially no dynamic pressure jump existed. This is due to the pressure in region 4, which is greater than that on the two sides (region 5 and 2). But the dynamic pressure in the rarefactive region 5 is only slightly smaller than that in region 2. This difference is nearly just offset by the difference of the total hydromagnetic pressure between them. For a TD carrying a decreased field intensity (0 < Bt3/Bt• < 1), the correlations are the opposite. We also find that all of the resulting effects are.very weak compared with those resulting from a density variation. This suggests that the effect of the IMF variation can be ignored when a TD carrying a considerable dynamic pressure variation collides with the bow shock.

Another effect to be considered is a directional variation of

the IMF. In Figure 4, the part with Bta/Btl < 0 presents a 1800 rotation in field direction. We find that it is symmetric with the Bta/Bt• > 0 part. The dotted line represents the special case of a simple directional variation. It produces a structure which is essentially the same as the initial state (the dashed line). In fact, the R-H relations show that the solutions depend only upon the magnitude of the magnetic field for a TD, not its direction. The directional change in the magnetic field remains with the TD and the coplandpity plane of the BS reorient's itself to fit the new IMF direction when the TD passes through it. Thus the basic results for the TD-BS interaction are independent of directional changes of the IMF across the TD.

Velocit•t variation. Since (O) is decoupled from the other equations in the R-H relations, the TD-BS interaction is not affected by change in the tangential velocity (Vt) across the TD. In other words, all of the plasma states and field are unchanged after the collision except for the tangential velocity (however, the ratio of ¬ across the TD remain• the same).

3.3. Back9round Solar Wind

In this section we examine the effects of the solar wind

parameters on the TD-BS interaction. The parameters considered are the Alfvdn MacIt number, the plasma/•, and the IMF garden hose angle. The parameters describing

21,304 Wu ET AL.: TANGENTIAL DISCONTINUITY-Bow SHOCK INTERACTION

0.02

- 0.02 0.01

-0.01

0.01

- O.Ol

0.96

0.94 0.32

o.31 0.02

- o.02 3.32

i i

3.05 1.01

0.99

1.5 - 1 - 0.5 0

Bt3 / Btl O.5 1 1.5

Fig. 4. The resulting conditions of the TD-BS interaction as a function of the field intensity ratio (Bt3/Btl).

the interplanetary TD are Ps/Pl = 2 or 1/2, Vts/Vta = 1, Bt3/Bt• = 1.

Al/vdn Mach number. Figure 5 illustrates the response of a TD+-BS interaction in the magnetosheath corresponding to variations in Mz•, for • = 1. The TD pressure jump is initially ApVn213_1 = 1 which lies on the upper boundary of the top panel. The bow shock is always intensified (ess > e0), and moves inward (Vss > 0). The excited wave is a fast shock since eFW > 1. While Mz•i increases, e0 also increases. We find that (1) eFW increases; (2) velocities of the BS, TD, and FW all decrease; (3) ApV,21•_4 weakens; (4) ApVn•[4_a and APt[4_2 become larger.

When Mz•i decreases, the initial strength of the bow shock becomes weaker. The resultant structure is more

sensitive to the changes introduced by the TD. However, ess remains close to e0 when the Alfvdn Mach number is less than about 4.

The results for Ps/Pl = 0.5 (not shown) are as follows. When Mz• increases gradually, the initial strength of bow shock increases. However, the bow shock is always weakened after the collision (ess < e0) as described in case B. The outward velocity of the BS is also reduced and the modification of the TD dynamic pressure jump becomes large. The difference in the pressures of regions 4 and 2 is

1

I i i i

•w

I I I I I i i I i

i

2 4 6 8 10

MAn1

Fig. 5. The resulting conditions of TD+-BS interaction as a function of the upstream Alfvdn Mach number (MAn1). The

2 upper panel shows the pressure jumps in unit of (pV•)1. The middle panel shows the velocities in unit of Vn•. The lower panel shows the shock strength.

increased. That is, the excited rarefaction wave is intensified (eFW far from unity).

We conclude that, independent of the type of TD (TD + or TD-), the BS intensity (both original and resulting), the intensity of the excited FW, the magnitude of the pressure jumps of the FW, and the change of the TD dynamic pressure jump are positively correlated with MA•i. The speeds of the modified BS, the penetrating TD and the FW show a negative correlation with MA•.

Plasma •. The conditions resulting from a TD+-BS interaction with different upstrea• plasma fl values but the same Alfvdn Mach nnmber (MA•i = 5) are shown in Figure 6. The initial pressure jump of the TD is set to ApV,213_i = 1. The response to a variation in • is less than that of Mzn•. The negative correlation between e0 and •1 is due to the change in the fast mode speed. In fact, we can express it in terms of the fast-mode Mach number, MFn = Man•/41-]-7fll/2. Therefore eFW, ApV214-2, APt[4_2, and the modification of the TD dynamic pressure jump, are negatively correlated with ]3•. The velocities of the BS, TD and FW are positively correlated with ]3x. All of these features are shown in Figure 6. Similarly, these properties hold for the TD- case.

Garden-hose angle. The position of first-encounter point of the TD impinging upon the BS is determined by the IMF garden-hose angle, 0, as shown in Figure 1. It determines the shock normal at the first-encounter point and thereby the upstream normal velocity. The upstream Alfvdn Mach number in region 1 is given by MAn• -- IVsw sin OI/VAx. Within the context of the model presented here the contribution of the IMF angle can be merged into that of the normal Alfvdn Mach number.

WU ET AL.: TANGENTIAL DISCONTINUITY-Bow SHOCK INTERACTION 21,305

0.8

0.1 1.5

0 2 4 6 8 10

Fig. 6. The resulting conditions of TD + -BS interaction as a function of the upstream plasma/3. The format in each panel is same as in Figure 5.

4. NUMERICAL SIMULATION

There are two potentially serious problems with the theoretical formulation presented in section 2. The first is whether the resultant physical structures are indeed as depicted in Figure 2b or if perhaps some other set of structures might result from the interaction of the TD with the shock. The second question is whether the resultant structures are stationary or evolving. Numerical simulations provide a very useful tool for answering these questions and verifying the theoretical results. It should be pointed out here that for the Earth's bow shock system the resultant structures must evolve as they move towards or away from the magnetospheric obstacle which is responsible for the bow shock. However, including the effects of the Earth's magnetosphere is beyond the scope of the present work and we defer such a study until later.

4.1. 1-D MHD Model

In order to verify our theoretical results, we perform simulations using the complete set of the ideal MHD equations as expressed in (1)-(6). In our MHD code, the x axis is along the shock normal, and directed toward the Earth as illustrated in Figure 1. The origin is located at the initial position of the bow shock and the reference frame is chosen such that the shock is at rest. We assume that the Sun and

the Earth are far enough away from our simulation domain that we can ignore their effects. Initially, the plasma state on each side of the bow shock is connected smoothly by a hyperbolic tangent function; i.e.,

1 1 x

Q(x): •(Q2 q- Q1) q- •(Q2 - Q1) tanh(•) (23)

where 5 determines the width of the transition. The

radiation condition [e.g., Miller, 1981] is used at the right boundary. We consider an interplanetary TD with a density jump and a magnetic field reversal. In addition, the plasma and magnetic field pressures on both sides of the TD are unchanged while the temperature is modified to maintain pressure equilibrium across the TD. We parameterize this situation with Bt3/Btl : --1. The TD-BS interaction is independent of the direction of the IMF as discussed in 3.2. We reverse the direction of magnetic field in region 3 only to better distinguish the TD from the shock (or rarefaction wave). We track the positions of the BS, TD, and FW to calculate their respective propagation velocities. The TD is initiated near the left hand boundary at T=0, where T is a normalized time in units given by t0 = 5/V,•1 where 5 is the normalized unit of length and V,•i is the upstream normal velocity of background solar wind. We fix the left boundary (at x = -5) since no MHD wave can propagate upstream from a fast shock. In our scheme, a 4th order Runge-Kutta method is used. We use a uniform-space grid mesh with a total of 1201 grid points in the simulation domain and the normal coordinate ranges from -5 to 25. The entire domain is not always shown in the figures. The resolution is thus Ax = 5/40 and we set the time step to be AT = to/800.

4.2. Simulation Results

We present four different cases of an interplanetary TD collision with the BS. The first two correspond to cases A and B presented in section 3 and are used for a direct comparison with the theoretical results as summarized in Tables I and 2. The last two represent idealized models of a magnetic hole and a diamagnetic cavity. All of the parameters used in the four simulated cases are identical except the density jump behind the TD which distinguishes the TD + and TD- type. We restrict the background solar wind to the conditions of M•4,•1 = 5 and /31 = 1. We parameterize the TD + and TD- types by P3/Pl = 2 and PalP1 : 0.5. The field intensity, plasma pressure and velocity behind the TD are determined by Bta/Btl : --1 and Vts/Vtl = 1.

Collision by an individual TD. The following two cases represent the interaction of an individual TD with an enhanced dynamic pressure (case A) and a reduced dynamic pressure (case B) with the BS.

Case A (TD+): Figure 7 shows the initial and final states of our simulation with the same parameters used for case A in section 3. What is labeled in Figure 7 as Bt is actually the total field since we have assumed a perpendicular geometry. At the beginning of the simulation, the BS and TD are clearly located at x = 0 and x = -4 in Figure 7a. After the TD-BS interaction (T-4), the TD penetrates into the magnetosheath to x ~ 7, the BS moves earthward to x ~ 3, and an excited fast shock propagates ahead of the TD to x ~ 16.5. Qualitatively this is consistent with the theoretical result, and the profile of the total hydromagnetic pressure is the same as in Figure 2. The actual values of physical quantities (p,P, Bt, V,•) are very close to the theoretical solution shown in Table I with the maximum relative error

under 1%. The density profiles at various times are shown in Figure 8. The speeds of the BS, TD, and the new fast shock are constant. In Figure 9 we display the fronts of the discontinuities. Their velocities are calculated by

21,306 Wu ET AL.: TANGENTIAL DISCONTINUITY-BOW SHOCK INTER. ACTION

T=0 T=4

i i i

i i i

b)

0 5 10 15 20 - 5 0

I i i i

J i , , J 10 15 20

Fig. 7. Simulation result of case A (TD + ). Plotted are the profiles of the density, plasma pressure, normal velocity, tangential component of the IMF, the dynamic pressure, and the total (plasma and magnetic) pressure. The left column (T = 0) shows the initial states before collision, where the TD + is located at x = -4 and the BS at x = 0. The right column (T = 4) shows the structure after collision.

linear regression and can be compared to the theoretical predictions in Table 1.

Case B (TD-). When the interplanetary TD carries a reduced dynamic pressure, the transmitted FW is not a shock but a rarefaction wave. We demonstrate this case

by setting a density jump to Pa/Pi '- 1/2. Figure 10 shows that the BS moves sunward, the transmitted TD convects with the downstream flow, and the FW expands as it propagates ahead of the TD. The leading and trailing edges of the rarefaction region move at constant velocities equal to their local fast mode speed. The results from the simulation match the theoretical predictions in Table 2 to within 0.5%. Thus we infer that our theoretical results are correct.

Collision by a pair of TDs. From the previous two simulation cases, we verify the correctness of the theoretical formulation and the accuracy of mathematical scheme used in obtaining quantitative results from it. Now we turn our attention to two types of structures observed in the solar wind: magnetic holes and hot diamagnetic cavities. Turner et al. [1977] have observed isolated, low-field regions (B _< 17) in the interplanetary medium which are generally referred to as magnetic holes. Following the theoretical model of Burlaga and Lemaire [1978], a hole is a pair of TDs bounding a region of weak field and high density. We symbolize this structure as an ensemble of TD+-TD -. Another class of solar wind discontinuities

which has been observed have been termed hot diamagnetic cavities [Thomsen et al., 1986]. Within these structures the

magnetic field is also reduced but the plasma temperature is very high. The density is generally somewhat reduced. As a consequence of the high temperatures inside these structures, they are not in pressure balance and must be expanding with time. However, the variation of the magnetic field and thermal pressure should have little effect

p(t)

T=4

T=3

T=2

T=I

T=0

II

i i I i i i i i i

- 5 0 5 10 15 20

Fig. 8. Time sequence of the density profile for case A.

WU ET AL.: TANGENTIAL DISCONTINUITY-BOW SHOCK INTERACTION 21,307

T

•

_

_

_

o

-5

Fig. 9. The discontinuity fronts of the BS, TD, and FW for case A are plotted in the (x, t) plane. The velocities are derived from the slopes of the lines presented in this figure.

T=3

T=2

T=I

T=O

- 5 0 5 10 15 20

•/•

Fig. 11o Time sequence of the density profile for case C (TD+-TD-). Initially, the TD + is located at x = 1, a TD- at x = -4, and the BS at x = 2.5.

on the TD-BS interaction as discussed in section 3.2. The

diamagnetic cavity may then be roughly modeled by a TD--TD + pair for the purpose of studying its interaction with the BS.

Case C (TD + - TD-): In Figure 11 we show the interaction of the BS with a magnetic hole. Here the density jumps across the TD + and TD- are 2 and 0.5, respectively. The other parameters are the same as in cases A and B. When the leading edge of the magnetic hole (the TD + part) interacts with the BS, the BS is displaced earthward and a fast shock is excited, as in case A. After the trailing edge of the magnetic hole (the TD- part) interacts with the moving BS, the BS once again becomes standing in the simulation frame and a rarefaction wave is generated which eventu- Mly passes through the leading TD +. The rarefaction wave

T=4

T=3

T=2

T=I

T=0

- 5 0 5 10 15 20

Fig. 10. Time sequence of the density profile for case B (TD-).

eventually overtakes the fast shock, producing a complicated interaction which we are not considering here. Note that only one wave of each mode can propagate in each direction in the time asymptotic solution. On the downstream side of the bow shock, the magnetic hole associated with the TD pair has become narrower, and may be regarded as a pressure pulse. The narrowing is due to the compression of the plasma across the shock, as the TDs are frozen into the plasma. Interestingly, the rarefaction wave reduces the plasma density by the same amount as the fast shock en- hances it. Consequently, the magnetic hole retains its sym- metry downstream of the BS. The dynamic pressure pulses in the magnetosheath may be associated with the interplanetary magnetic holes from this study.

Case D (TD--TD+). Figure 12 shows the interaction of a hot diamagnetic cavity with the BS. The conditions for the background solar wind are identical with case C and the density jumps of the TD pair are reversed. The result is very similar to case C with only the order of the processes reversed. When the leading edge of the TD ensemble interacts with the BS, the BS expands outviard and a fast rarefaction wave is produced. When the trailing TD interacts with the BS, the BS recovers and a fast shock is generated. This fast shock passes through the leading TD and eventually catches up with the rarefaction wave (not shown). The cavity associated with the TD pair is much narrower on the downstream side of the BS and is again symmetric.

,5. DISCUSSION

5.1. Response to Field and Plasma Variations Across the TD

The nature of the TD-BS interaction is determined by the change in the plasma upstream of the Earth's bow shock. There are two important factors in the modification of the upstream plasma across the TD, (1) the changes in


T--4

T-3

T--2

T=I

T-0

- 5 0 5 10 15 20

x/5

Fig. 12. Time sequence of the density profile for case D (TD--TD+).

the solar wind dynamic pressure and (2) the changes in the upstream fast magnetosonic speed. Many properties of the interaction can be examined with regard to how the fast magnetosonic speed is altered across the TD. The fast magnetosonic speed is given by CF = (V• + C•) •/•' for perpendicular propagation. When the fast magnetosonic speed ahead of the shock is altered due to the passage of the TD, the Mach number and shock strength are also modified. We can relate the fast mode speed, CF3, in the new upstream state (region 3) to the original speed Cr• (region 1) by

( 2-, ) C}a = p• C}• + ß [ -- 1]V• (24) •33 2 B t l The quantity CFa is inversely proportional to the density variation PalPs, and positively correlated with the field intensity variation Bto/Bt•, if we make the usual assumption of 7 = 5/3. Therefore the response of the bow shock can be examined using (24).

Response to the density variation. For TD+-BS interaction, CFa decreases since Po/P• > 1. The upstream normal velocity, v,a, decreases in the shock frame due to the bow shock being pushed back by the enhanced solar wind dynamic pressure. However, our analysis shows that the bow shock motion is rather small and the increase of the fast

magnetosonic Mach number due to the decrease of Cro results in a stronger shock. For the TD- case, CFa increases which leads to a decrease in the shock Mach number and

shock strength. Response to the IMF variation. Note that C•a increases

with increased (Bta/Btl)•'. The upstream fast-magnetosonic Mach number is then negatively correlated to the variation of field intensity (Bta/Btl) 2. For the enhanced IMF ((Bta/Btl) 2 > 1), the bow shock is weakened. For the weaker IMF ((Bta/Btl) 2 < 1), the bow shock is intensified. This relation is opposite to that of the density variation discussed above.

Note that the coefficient (2- 7)/2 in the term associated with the field variation in (24) is only 1/6 for 7--- 5/3. Our

calculation shows that the resulting effects from Bt3/Bt! are very weak compared with those resulting from a density variation, for MA,I -- 5 and/71 -- 1. We have also calculated the cases with/71 = 0.1 and/71 = 10 and found that the effect of field intensity variation is still very weak. Therefore the response of the magnetosheath due to the IMF variation is small and can be ignored in comparison with the density jump.

In an earlier report, Burgess and Schwartz [1988] studied the kinetic effect of the interaction of an active current sheet

(reversed field) with perpendicular shock in hybrid simulations. They found that the current sheet and the shock re- turn to their original structure after the interaction but were substantially modified during the interaction and produced a volume of plasma with different kinetic characteristics as a result. Their conclusion agree with our result. The directional variation of the IMF contributes very little to the interaction.

5.2. Response off the Magnetosphere

The purpose of this study is to investigate the interaction of solar wind dynamic pressure variations with the Earth's magnetosphere. When an interplanetary TD impinges upon the Earth's shock-magnetosphere system, two types of dynamic pressure jumps will appear in the magnetosheath. These are separately associated with the penetrating TD and the excited fast-magnetosonic wave (a shock or a rarefaction wave). Moreover, we find that a pressure pulse with finite duration may be associated with magnetic holes and/or diamagnetic cavities. After the solar wind dynamic pressure variation penetrates the bow shock, the next obstacle encountered is the magnetopause.

Of course, the complete interaction of any structure in the solar wind with the Earth's magnetospheric system will be very complicated. The interaction will occur over different parts of the magnetosphere at different times producing responses which are likely to mix as they move further into the magnetosphere. In this initial analysis we cannot treat the entire problem, so we begin with a simplified one- dimensional model. We feel that this may be justified for the problem at hand for a number of reasons. The fast-mode wave propagates with the highest velocity perpendicular to the magnetic field. Thus the signal produced at the point of tangency should be the first and clearest to reach the magnetopause. In addition, interactions at oblique angles are likely to produce intermediate and slow mode waves. One would expect this to reduce the strength of the transmitted pulse and produce a more mixed signal as the various waves and structures convect or propagate to the magnetopause. It is of interest to study how the pressure jumps affect the magnetosphere and how flow vortices might occur. It is suggested that the following process may result.

Consider the magnetopause as a tangential discontinuity where the density decreases toward the Earth. The fast wave which propagates normal to the IMF can directly impact the magnetopause. Similar to the TD-BS interaction, the magnetopause should move inward after the excited fast shock penetrates it, and the intensity of the fast shock will be somewhat weakened as characterized by •Bs in Figure 3. A new rarefaction wave is excited which propagates back into the magnetosheath (perhaps eventually interacting with the BS again). When the fast shock interacts with the magnetopause, then vortices may be produced in the

Wu ET AL.: TANGENTIAL DISCONTINUITY-BOW SHOCK INTERACTION 21,309

LLBL which map to the ionosphere leading to magnetic impulse events. In addition, the plasma in magnetosphere becomes denser and hotter when the shock passes through it. In a related study, Mandt and Lee [1991] suggest that a sudden impulse, which might manifest itself as a weak fast shock, passing through the magnetopause may enhance the temperature anisotropy in the magnetosphere. From their hybrid simulation results, they infer that the perpendicular

shock wave exhibits anisotropic heating with Tx > Tii which may lead to the generation of Pc I waves.

The original interplanetary TD penetrates though the BS and follows the fast wave in the magnetosheath. However, the transmitted TD cannot directly pass through a TD type magnetopause in our one-dimensional model since there is no plasma flow across the TDs. In the two- or three- dimensional case, the plasma ahead of and behind the transmitted TD is diverted by the magnetopause and flows tailward around the magnetopause. If the TD flows with the diverted solar wind, the discontinuity can still interact with the magnetosphere. Lee [1991] presented results from an MHD simulation of the interaction between a pressure pulse and the magnetopause boundary layer. He found that a single vortex is formed for a step function in the dynamic pressure profile and a pair of vortices is formed for a pressure pulse with a finite duration (such can be modeled by a TD + and TD- ensemble as in case C in section 4.2). On the other hand, Kivelson and Southwood [1991] presented a different result from a theoretical treatment of the shear Alfvdn mode. They inferred that the induced shear flow is related to the second derivative of the pressure in the azimuthal (longitudinal) direction. Therefore a single pressure jump will produce a pair of vortices and a pulse will produce two pairs of vortices. In general, vortices can be generated in the LLBL by the presence of pressure pulses.

On the basis of ground and satellite observations, transient magnetic pulsations (TRAMs) rarely occur. When observed, they are primarily found in the pre-noon sec- tor [Friis-Christensen et al., 1988; Glassmeier et al., 1989]. However, continuous, nonimpulsive, traveling convection vortices were most often found in the postnoon region [McHenry et al., 1990]. Our results indicate two means by which vortices may be created in the LLBL, a fast shock propagating through the magnetopause and a sheared TD convecting along it. Observations indicate that there are few instances of solar wind discontinuities with a dynamic pressure jump greater than 35%. Not all the interplanetary TDs would correspond to the presence of vortices. We suggest that a strong enough fast shock (or a narrow rarefaction wave) might be related to TRAMs. Transmitted TDs (which usually carries a stronger dynamic pressure jump than the FW) or TD pairs could be related to the more frequently observed traveling vortices.

The magnetosheath pressure pulse modeled by a TD pair might correspond to holes and/or diamagnetic cavities as discussed in section 4.2. Turner et al. [1977] indicated a time scale for magnetic holes of about 50 s. These were observed by spacecraft to be traveling at about 400 km/s. Thomsen [1986] reported that observations of diamagneitic caveties typically last ~ I - 2 min. After passing through the bow shock, the scale length of a hole is generally reduced by a factor of about 1/4 due to compression at the shock. A diamagnetic cavity gets compressed even more. The spatial scale of the related pair of TDs in

the magnetosheath is then about 1 ~ 2RE if regardless of the global change of the shock-magnetosphere system. However, the traveling twin vortices in the magnetosphere have scale length of _> 10RE estimated by [McHenry et al., 1990]. The details of the continued evolution of these structures and their interaction with the magnetopause after the TD-BS interaction will require a more sophisticated two- dimensional model to study. In this paper we provide a possible mechanism for the generation of pressure pulses in magnetosheath and discuss the likely consequences which the mechanism implies. Verification of these predicitons and a detailed study of these other interactions must await further study.

6. CONCLUSION

In this paper we have examined the interaction of a tangential discontinuity with the Earth's bow shock using both theoretic analysis and MHD simulations. The abrupt modification of the solar wind plasma ahead of the Earth's bow shock due to the conditions at a TD leads to some

interesting consequences. We summarize the principal results as follows.

1. When an interplanetary TD convects into a perpendicular shock, both structures are recovered in a modified form after the interaction. The dynamic pressure jump of the transmitted TD is generally less than the value before the collision. In addition, another nonlinear fast wave is excited due to the interaction. For the TD+-BS interaction

the bow shock is intensified, moves inward (earthward), and a fast shock is excited. For the TD--BS interaction the

bow shock is weakened, moves outward, and a rarefaction wave is excited. For a TD with a change in the field intensity but no dynamic pressure jump, the interaction is somewhat different, but the effects are negligible compared to the above mentioned case. For an enhanced IMF a fast

shock is excited (like in the TD + case), but the bow shock is weakened and moves sunward. For a reduced IMF a rar-

efaction wave is excited while the bow shock is intensified

and moves earthward.

2. The outward velocity of the modified bow shock is much greater than the inward velocity for reciprocal density jumps. The inward velocity of the BS after colliding with a TD + approaches a limit of 0.5V,• for very large density variation.

3. The density jump (or dynamic pressure variation) across the TD is the most important factor in the TD-BS interaction. The effects attributable to jumps in the field intensity or thermal pressure on the interaction are negligible in comparison. In addition, if the magnetic field changes direction across the TD, the TD maintains this directional change after the interaction and the coplanarity plane of the shock changes to the new field orientation. The other physical quantities resulting from the interaction are the same as in the case where the fields are parallel on each side of the TD. Similarly, a change in the tangential velocity across the TD has no significant effect on the TD-BS interaction.

4. The interaction of interplanetary holes (or diamagnetic cavities), which can be modeled by an ensemble of a TD + and TD-, with the bow shock will excite both a fast shock and a rarefaction wave. The bow shock is displaced during the interaction but remains stationary after the collision (if the original pulse was symmetric). The structure is


compressed in the interaction and then continues to convect in the downstream flow.

5. The background solar wind parameters effect the interaction primarily by determining the strength of the BS. The interaction is rather sensitive to the shock strength. The solar wind speed, the plasma •, and the IMF geometry (garden hose angle) all contribute to the determination of the normal Mach number. In addition, the IMF direction will determine the position at which the TD-BS interaction takes place. Our results show that when the Mach number is large (strong shock limit), the modifications to the pressure jumps of the transmitted TD and excited FW are greater and the motion of the BS is less. This is true for both

TD+-BS and TD--BS type interactions. 6. Applying the TD--BS case to the interaction of

a fast shock with a TD type magnetopause, the following may result. The fast shock should weaken and propagate faster as it penetrates into the magnetosphere with its increasing magnetic field. The magnetopause is pushed towards the Earth and the density is enhanced in the boundary layer region. The magnetic field intensity and the thermal pressure will increase from the compression. An excited rarefaction wave propagates anti-earthward into the magnetosheath.

In summary, interplanetary fluctuations cannot interact directly with the Earth's magnetosphere. They must first interact with the Earth's bow shock which, in general, acts to modify them. We find that the solar wind density variation (i.e., the dynamic pressure variation) produces the most significant modification in on our TD-BS model. After the interaction with bow shock, the solar wind dynamic pressure jump carried by the TD is modified and energy is redistributed with an additional fast mode wave. The excited fast wave (shock or rarefaction wave) propagates earthward as the leading part of the new structure. The transmitted TD follows the excited nonlinear wave and

carries a reduced dynamic pressure jump. Both the newly excited fast shock (or rarefaction wave) and the transmitted TD can then interact with and modify the magnetosphere.

Acknowledgments. One of the authors (Bor-Han Wu) would like to dedicate this paper to president Chuan-Tau Yu, for his selfless devotion to the students of the National Central University in Taiwan and the inspiration he has provided in my research. Computations are performed at the Institute of Atmospheric Physics in N.C.U. This work is supported by NSF grant ATM 88-

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J. K. Chao and B. H. Wu, Institute of Space Science, National Central University, Chung-Li, Taiwan, R.O.C.

L. C. Lee, Geophysical Institute, University of Alaska, Fair- banks, AK 99775.

M. E. Mandt, Laboratory for Plasma Research, University of Maryland, College Park, MD 20742.

(Received August 12, 1991; revised November 18, 1992;

accepted April 7, 1993.)

Magnetospheric Response to the Solar Wind Dynamic Pressure Pulses inferred from Polar Cap Index

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