Alfven Soliton and Intermediate Shock in the Solar Wind

We show in this paper that intermediate shocks are formed from ki­ netic Alfven waves through wave form continuous steepening. From the action of nonlinear interaction and dispersion Alfven solitons are formed and then evolve into shocks. We calculated the traveling stationary struc­ tures directly from nonlinear autonomous equations which we had derived. These traveling stationary structures correspond to intermediate shock, switch-on shock or Alfven solitary waves. Another kind of rare discontinu­ ity was discovered in the calculation from the rarefying solitary waves. The solitary wave series remain on the front of the shocks because of no strong dissipation appears in the equation. When we added some anomalous dissi­ pation terms to the equations, intermediate shocks with increased in en­ tropy appeared.


INTRODUCTION
Is there any intermediate shock in the solar wind? How is intermediate shock produced? These are problems discussed by space scientists recently. Usually, magnetohydrodynamic (MHD) shocks are investigated through Rankine-Hugoniot jump relations. MHD shocks which satisfy the principle of discontinuity and entropy increase can be divided into three categories according to MHD wave modes: fast shock, slow shock and intermediate shock. But interme diate shock is usually referred to as non-evolutionary or unstable, and has been thought non physical reality (Jeffrey and Taniuti 1964;Anderson 1965). Wu (Wu 1987(Wu , 1988(Wu , 1990) has presented numerical solutions to dissipative MHD equations showing that MHD-intermediate shocks can be formed from continuous wave form steepening, and changing the parameters of dissipation can make intermediate shock stable through its structure layer variation. The exist ence and formation of intermediate shock has become an important subject for discussion. Some people pointed out, using numerical simulations and analyses, that in the shocks' evolu-352 TAO, Vol. 12, No. 2, June 2001 tion process the existence of intermediate shock is absolutely necessary (Kennel et al. 1989;Steiwolfson and Hundhauson 1990;Hu 1994). Furthermore, the existence of intermediate shocks is a necessary condition for a hydrid shock formed by linking up different types of MHD shock (Hu 1994). _ Up to now, most interplanetary shocks measured have been affirmed to be fast shocks and some to be slow shocks (Chao 1974;Whang 1982, l 986a;Richter 1985Richter , 1987, only one intermediate shock measured in the solar wind was reported (Chao 1993). Though intermedi ate shocks are thought to exist, it is still not clear, in theory how they are formed in a collisionless plasma. Corresponding to MHD wave modes, intermediate shock should be formed through Alfven wave evolution. But the Alfven wave is a kind of transverse wave propagating along the magnetic field and has no nonlinear interaction and dissipation in the theory. Some people offer nonlinear term (V. V) V in the oblique propagation of the Alfven wave. But, group velocity of the Alfven wave, i.e., the wave energy still persists in the propagation along the magnetic field. Below we show that intermediate shock is evolved from the kinetic Alfven wave instead of from the MHD Alfven wave. The kinetic Alfven wave was first introduced in space physics by Hasegawa (Hasegawa 1976). It is weak dispersion, the dispersion relation is expressed approximately as: Where k11 and k J. are the wave vector component parallel and perpendicular to the magnetic field respectively, and k J. >> k11 stand for the kinetic Alfven wave. p1 is the ion gyroradius. In the solar wind it seems reasonable for k1p; 2<<1, although kl. >> k11 for the kinetic Alfven wave energy is propagating along the magnetic field:

am
( 1 2 2) �II = -= V A 1 + -k J.Pj ;;::: VA Jk 11 2 , am k 11 ( 2 2) As there are longitudinal disturbed velocity and electric field £11 in the kinetic alfven wave the governing equations are nonlinear. From the action of the dispersion, the nonlinear kinetic Alfven wave first evolves into Alfven solitons, then into intermediate shock by the dissipation effect. The kinetic Alfven wave is quite different from the MHD Alfven wave . Besides having the dispersion and Landau dissipation, the kinetic Alfven wave is a longitudinal-transverse coupling mode, i.e., the Alfven wave and the ion acoustic wave coupling mode. Its transverse velocity and electric field in the wave yield elliptic polarization, so it easily evolves into kinklike solitons or eddies. In general, a shock must be a traveling stationary structure that can be numerically calculated to be found. We can directly calculate this traveling stationary structure. We will show by numerical analysis that the dispersive, nonlinear kinetic Alfven wave can evolve into intermediate shocks, Alfven solitons and magnetic kink along with the different parameters in the solar wind. The solar wind is low f3 plasma, m e << f3<<1 (where f3 is the ratio of thermal pres m i sure to magnetic pressure in the plasma). The inertial term of electron in the two-fluid equa tions can be omitted (Lysak and Lotko 1996). Suppose the background magnetic field is Here we have used the quasi-neutrality condition n = n1 = n e , where c i ' c e are the ther mal speed of electrons and ions. As m e << m i the electron polarization has been ignored. The displacement current in the Maxwell equations is omitted for OJ << 01• Here we use Gauss units system , c is the speed of light. Canceling vix in the equations above, we can normalize the equations as CJ N +_?_(NU ) + j_(N CJ E x) = 0 ar ;;z, ax ar , au (2) velocity; n0 is the density of background plasma. The Equations (1)-(5) can be transfered to the travelling wave phase: Integrating equation (2) with boundary condition s � -=, N = 1, U = � , yields Substituting E z and U into equations (3) and (4), with tedious derivations we obtain the nor mal nonlinear autonomous equations on s (10) where y = JE x and a= Vi -M A . These equations are similar to equations which describe a; r · self-organization phenomena in nonlinear dynamics. From these start equations we will calcu late the self-organization strucutre, such as shocks and solitons from equations (8)-(11).

INTERMEDIATE SHOCK
In (8) (2) Intermediate shock exists only when Vi -M A /y>O. This means that speed at up stream of shock should be more than M A . The iptensity of shock (by the jump of density) sensitively depends on two parameters, the initial disturbed velocity Vi and the character speed ratio c; IV}. The shock is stronger when Vi is bigger or c ; IV} is smaller. We can roughly estimate the thickness of the shock: ; = (aK + y,l -M A T)/ Mi, a ---+ 0,  We can conclude that the shock is thicker when c;tv}, is smaller. Figure 2 shows another example of the calculated result of switch-on shock. The value of parameters has listed in the figure. This switch-on shock evolves from the kinetic Alfven wave propagating toward the sun in the solar wind frame, but carried out by the solar wind. The speed at downstream of shock is smaller than it is at upstream, but it is still larger than the Alfven speed. This seems like a fast shock in the MHD discontinuity theory. But the tangential component of magnetic field of this shock reverses 90°. This result might not have been thought to be an intermediate shock in previous observations. The calculated result indicates that we can find this kind of stationary solution whose velocity at downstream of the shock is still larger than the Alfven velocity. We do not know if anyone has observed such discontinuity in the solar wind. We sort it into switch-on shock. In the shocks mentioned above, the magnetic field at downstream is larger than it is at upstream (lb type). We did not find the switch-off shock. All such shocks are formed by the compressive solitary wave, with asymmetry £2. It is interesting that the rare density at solitary wave (cavitons) can evolve into a rare discontinuity (Fig. 3). Such rare discontinuity also is a type of the stationary solutions of the equations.

ALFVEN SOLITON AND MAGNETIC FIELD KINK
In various stationary solutions of nonlinear equations (8)-(11), besides the intermediate shocks, there are other important stationary structures, the Alfven solitons and the magnetic field kinks. The existence of Alfven soliton has been proved by Hasegawa (1976) with double electric potential. (Hasegawa and Mirna 1976). Figure 4 shows the Alfven soliton solved from (8)-( 11 ). The density is the series of solitary waves (Fig. 4a ). The speeds at upstream and down stream are the same (Fig. 4b). The tangential component of magnetic and electric field rotate 180° (B2/B11 = -1 ) (Figs. 4e,4f). We have tried to find out Alfven shock in various parameters; but we did not find the shock with B2/B11 = -1, and v; >Ur In the solutions of Alfven solitons, when the initial impact v;. is small, we can find a kind of magnetic kink without the density variation and only the tangential component of magnetic and electric field reverse 180° (B2/B11 = -1). This magnetic kink is different from the MHD rotating discontinuit y, just just an Alfven soliton, it propagates with a velocity higher than the Alfven velocity.  where Y=--x, F = -. E1= --, and k1 =k(--1-2). Cvis the specific heat ofplasma.

INTERMEDIATE SHOCK WITH DISSPATION
J s J s Cvm; C v B o k 1 is the thermal conductivity. Here we have omitted tedious derivation. We add the equation (15) which comes from the energy equation and determines the temperature variation as fluid passes through the shock. The parameter M i disappears in equations ( 1 2)-(16), because it has been drawn in S for simplification. Figure 5 shows the results of calculation. We can see in Fig. Sf that the temperature increases as fluid passes through the shock, other parameters changes like that in the intermediate shock in Fig. 3. This is intermediate shock with entropy increase as usually understood.
Nonlinear dynamics has an intrinsic property that the solutions of a dynamic system de pending on the parameters display complicated behavior: traveling stationary solution appears for some parameters and chaos or turbulence for other parameters. Among so many param eters y = cos e' where e is propagating angle only when y ;::: y c (i.e., e � ec) the stationary solutions or shocks exist. This is just the requirement of the kinetic Alfven wave. Intermediate shock exists for Alfven Mach number MA>l, otherwise the solitary wave or magnetic kink  Yi (a == Yiy). When Yi is greater than a critical value the shocks appear; the solitons or magnetic kink corresponding to smaller initial impact Yi.

CONCLUTION AND DISCUSSION
The shear Alfven wave with the longitudinal component of electric field and velocity is a coupling mode of the Alfven wave with ion acoustic wave. This wave mode, the called kinetic Alfven wave in nature, is different from the MHD Alfven wave. Hasegawa and Mirna (1978) have pointed out that the kinetic Alfven wave is easily exited and develops into the turbulence. The Alfven turbulence in the solar wind perhaps comes from the kinetic Alfven wave, not from MHD turbulence. The kinetic Alfven wave has nonlinear effect, dispersion and even weak dissipation, it easily evolves into a soliton or shock by nonlinear wave form steepening. Transforming independent variable of the dynamic equations into the travelling phase, we derive the nonlinear autonomous equations whose stationary solution is a self-organized structure. This structure can be soliton, shock or something else. If we define shock as a propa gating discontinuity, it can exist in the conservative system, but shock with entropy increase can only exist in the dissipated system. Here we have obtained following shocks or solitons by numerical calculation.
(1) intermediate shock (lb type) (2) Switch-on shock (3) Alfven soliton ( 4) Rare discontinuity The method we used to find the solitons can also be used to find the stationary travelling solution of the nonlinear dynamic equations. These equations include the kinetic Alfven wave mode, and allow us to successfully obtained the intermediate shocks. At the front of the shock there are remains of the solitary wave series. The reason is that the kinetic Alfven wave has dispersion and lacks dissipation. We believe that if we add strong dissipation to the dynamic equaitons, such as ion acoustic anomalous resistance, the solitary wave series in front of the shock will dampen off and an intermediate shock with temperature increase will be produced.
In fact, we have got intermediate shock with entropy increase (Fig. 5) by Joule dissipation of longitudinal electric field.
As the start equations for calculation, in equation (8)-(11), initial impact Yi is a parameter. When Yi is small, the soliton appears, when Yi increases to a critical value, the shock is produced. In the initial equilibrium state N=l, Y=O, G=O, but B (tangential magnetic field) is uncertain; when B=O we get switch-on shock, when B<O we get the intermedite shock or soliton. In collisionless plasma, the formation of shock is related to solitary wave, compres sive solitary wave corresponds to shock, and the rare solitary wave evolves into a rare discon tinuity (Fig. 3).
Once the kinetic Alfven wave is exited it easily evolves into the turbulence. So, the inter mediate shock which is produced by the kinetic Alfven wave should be found where the ki-netic Alfven wave is excited. At Alfven point of the solar wind ( the speed of the solar wind is equal to the Alfven velocity, U0 = V A ) there is kinetic exiting of the kinetic Alfven wave ( Song and Lu 1998 ). The new exited kinetic Alfven wave is propagating in two directions along the magnetic field. A branch of the kinetic Alfven wave propagating toward the sun (M A < -1) is carried by the solar wind outward from the sun. It is mere conjecture that this branch of the kinetic Alfven wave will evolve more easily into the intermediate shock with