Proton pitch-angle diffusion rate and polytropic index values in the magnetosheath: Model and experiment

M. I. Pudovkin and S. A. Zaitseva

Physics Research Institute, St. Petersburg State University, St. Petersburg, Russia

C.-V. Meister

Astrophysikalisches Institut Potsdam, Potsdam, Germany

B. P. Besser

Institut für Weltraumforschung, Österreichische Akademie der Wissenschaften, Graz, Austria

Contents

Abstract

1. Introduction

In the thermodynamics of gases and isotropic plasmas the relationship between variations of the gas pressure p (or temperature T ) and density for adiabatic processes is determined by the expressions

(1)

where r and n are plasma mass and number densities, g=c_p/c_v is the adiabatic index, and g=5/3 for ideal one-atomic gases. Besides thermodynamic quantities the value of g determines also some characteristics of the gas flow, such as the standoff distances of the bow shock in front of obstacles, as well as the jump of plasma parameters across the shock. This has provoked physicists to try to estimate the value of g in the solar wind plasma from corresponding experimental data [Fairfield, 1971; Farris et al., 1991; Peredo et al., 1995; Slavin et al., 1983; Zhuang and Russell, 1981]. However, results of these estimates proved to be rather contradictory and sometimes even absurd (see a short review of the problem presented by Pudovkin et al. [1997]). On the other hand, it is known that the magnetized solar wind plasma is essentially anisotropic, and the value of g obtained for it from (1) being valid for an isotropic plasma, in an anisotropic plasma, represents only some empirical relationship between the total proton (or electron) temperature and the plasma number density variations, and so it may be considered as a formal or effective adiabatic index g_eff.

It seems evident that in a general case, g_eff is not equal to the ratio of the specific heats c_p/c_v, and its relationship to plasma temperature ( T, T ) component variations and to flow characteristics is not clear.

Using the double-adiabatic laws, Pudovkin et al. [1997] obtained recently a general expression for the effective adiabatic index g_eff in a convecting anisotropic collisionless plasma. In their analysis, g_eff was defined as an index which characterizes the relationship between the total proton temperature ( T=(T + 2 T)/3 ) and plasma number density ( n ) variations expressed in the form of (1). In particular, they showed that the anomalous value of g_eff<1, which seems to occur in the vicinity of the magnetopause [Pudovkin et al., 1995a, 1995b] may be explained by peculiarities of a magnetized plasma compression. However, in their analysis the authors assumed that the transfer between the perpendicular and the parallel (with respect to the magnetic field B ) components of the temperature is negligible.

At the same time, a detailed analysis of the experimental data carried out by Denton et al. [1994] and Hill et al. [1995] has shown that such a transfer does exist and seems to be rather effective, at least in the magnetosheath regions adjacent to the magnetopause. Then there arises a problem on the applicability of the above mentioned model to studies of the real magnetosheath plasma.

Besides, an effective transfer between the perpendicular and parallel proton temperatures has to change significantly the relationship of the proton temperature ( T_p ) and its component T and T variations to variations of the plasma density and hence the values of g, g, and g_eff. In this connection the results of Hau [1996] and Hau et al. [1993] are of a great interest. They used the data from the AMPTE/IRM spacecraft for 29 magnetosheath crossings and obtained the values of g=0.94 0.1 and g = 1.14 0.13 for the entire magnetosheath. These relatively low values of g and g were explained by them in terms of nonadiabatic effects. On the other hand, the same experimental profiles of T and T were explained by Denton et al. [1994] and Hill et al. [1995] by an effective energy transfer from T to T.

In this paper we consider in detail characteristics of the magnetosheath plasma flow in the vicinity of the subsolar streamline predicted by the model [Pudovkin and Lebedeva, 1987; Pudovkin et al., 1995a, 1995b]. The aim of the consideration is to estimate expected values of the polytropic indices g, g, and g_eff and their variations across the magnetosheath for an adiabatic flow. Special attention is paid to the influence of the proton pitch-angle diffusion on the expected values of g, g, and g_eff. The results obtained will be used for the interpretation of experimental data, and some conclusions will be drawn on to what degree the magnetosheath plasma flow is adiabatic. Also, the characteristic time of relaxation of the proton temperature anisotropy ( t ) will be estimated.

2. Basic Equations

In the analysis below we shall suppose that the temperature anisotropy in the convecting collisionless plasma arises due to the splitting of the parallel and perpendicular (with respect to the magnetic field) degrees of freedom and, as a result, to independent heating or cooling of the plasma in these two directions. Variations of the perpendicular and parallel pressures ( p and p, respectively) and of corresponding components of the plasma temperature ( T and T ) are given by three sets of equations. The double-adiabatic equations which may be obtained from the conservation of two adiabatic invariants: m=/B, and I= l², where l is the length of the "fluid'' segment of a field line [Chew et al., 1956]:

(2)   (3)

The continuity equation

(4)

where v is the plasma velocity. And the frozen-in magnetic field equation

(5)

Using (2)-(5), we can obtain the values of the adiabatic indices as [Belmont and Mazelle, 1992; Pudovkin et al., 1995a, 1995b]:

(6)   (7)

(8)

Here v= v_z/ z and v = v_x / x + v_y / y are calculated in a local Cartesian coordinate system with the Z axis directed along the magnetic field line at the point of observation (or calculation), and X and Y axes lying in the plane perpendicular to the local B. C= v / v is a modification of the well-known parameter of the magnetic compressibility ( C=C_p-1, where C_p is the magnetic compression index [Belmont and Mazelle, 1992]) introduced to distinguish more obviously between the flows in which the plasma compression along the magnetic field is associated with the plasma compression or decompression in the perpendicular direction. In particular, while analyzing the plasma flow within the magnetosheath, we suppose that the plasma moving earthward along the subsolar stream line is compressed in the direction perpendicular to the magnetic field (  v_x/ x < 0 ). Thus C<0 corresponds to v_z/ z>0, i.e., to plasma spreading from the subsolar streamline, and C>0 corresponds to the flow with streamlines converging to the subsolar line.

It should be noted that contrary to the generally accepted point of view, formulae (6) and (7) result in g=2 only for pure perpendicular compression of the plasma ( C=0 ), and g differs from 2 for any oblique compression. Respectively, one has g=3 only for a pure parallel compression, and in any other case, g 3. Thus a deviation of g and g indices from their "canonical'' values does not necessarily mean that the plasma flow is nonadiabatic.

The value and gradients of the plasma velocity are determined by the equation of motion. However, to make the results of the model profile analysis comparable with those of experimental profiles (see below), we shall express (  v ), (  v ), and (  v ) in terms of gradients of the magnetic field intensity and plasma density:

(9)   (10)   (11)

As for the temperature profiles, they are easily obtained from (2) and (3) as

(12)   (13)

The proton pitch-angle diffusion will be taken into account by introducing into the right-hand sides of (11) and (12) the relaxation terms (the so-called t approximation) in the form [see Denton et al., 1994; Hesse and Birn, 1992; Pudovkin and Smolin, 1988; Shishkina et al., 1995]

where T_eq is an equilibrium temperature, and t is the temperature anisotropy relaxation time. Then equations (12) and (13) may be rewritten in the form

(14)   (15)

Provided the plasma flow is constant in time, so that we can replace d/dt by v d/dx, we have

(16)   (17)

Having inverted (16) and (17), one may obtain expressions which allow one to estimate the values of t from observed profiles of the magnetosheath plasma parameters:

(18)   (19)

where DB and DT are the differences of B and T over the spatial interval Dx. In the case of an adiabatic flow the values of t, determined from the variations of T and T, which we refer to as t and t, respectively, have to coincide, since in this case there is no input or output of energy to (from) the plasma, and the energy lost by the transversal motion of protons has to be transferred to the energy of their longitudinal motion (the thermal energy of the magnetosheath electrons is small [Phan et al., 1994] and may be neglected).

Using this model, we shall analyze the variations of the plasma parameters across the magnetosheath to investigate possible influence of the pitch-angle diffusion on the proton temperature anisotropy and values of the adiabatic indices and to estimate the temperature anisotropy relaxation time  t.

3. Variations of Plasma Parameters Across the Magnetosheath for a Stationary Profile: A Model

Analyzing the observed variations of the plasma parameters across the magnetosheath, one has to bear in mind that in the presence of the magnetic field reconnection at the magnetopause resulting in a nonzero normal component of the velocity v_n=(0.1-0.2)v_a [Feldman, 1986] ( v_a is the Alfvén velocity), it takes about 10-20 min for the solar wind plasma to cross the magnetosheath along the subsolar streamline and, on average, about 2 hours for a spacecraft. Correspondingly, in order to consider a crossing as a "stationary'' one, solar wind parameters should preserve their values for at least 2 hours. However, we have no solar wind data during the crossings discussed by Hau et al. [1993] and by Hill et al. [1995] and therefore do not know whether this condition was fulfilled during these crossings. Even more, there are reasons to believe, as will be shown below, that the solar wind was highly fluctuating during the first of these crossings. Nevertheless, we shall consider at first a stationary model profile, for which we shall use one of the magnetosheath parameter profiles calculated using the model by Pudovkin and Lebedeva [1987] and Pudovkin et al. [1995a, 1995b].

3.1. Magnetosheath Parameter Profiles in a Flow Without Proton Pitch-Angle Diffusion

In this model, variations of the plasma and magnetic field parameters along the subsolar streamline are calculated under the assumption that the plasma is spreading from the symmetry plane (which contains the subsolar streamline in the magnetosheath and the reconnection line at the magnetopause [see Pudovkin and Lebedeva, 1987] in the direction perpendicular to that plane.

At this point we have to note that this model was developed for an isotropic plasma with g=5/3, which apparently restricts the applicability of the model while dealing with an anisotropic plasma. As the results of numerical modeling show, n(x) and B(x) profiles calculated for the isotropic and anisotropic plasmas differ insignificantly [Pudovkin et al., 1999], which proves applicability of the isotropic model.

With this in mind we present in Figure 1 variations of the plasma density ( n ), velocity ( v ), and magnetic field intensity ( B ) across the magnetosheath calculated along the subsolar streamline for Q=150^o ( Q is the angle between the solar wind magnetic field vector projection on the YZ plane of the GSM coordinate system and the Z axis of that system), Mach number M=8, and Alfvénic Mach number M_a=10. B and n values in the figure are normalized using their values in the solar wind ( B_sw=4.65 nT and n_sw=5 cm ^-3 ), and values of v are normalized using the velocity just after the bow-shock crossing ( v_bs=125 km s ^-1 ).

One can see that on the whole, the model profiles of the magnetosheath parameters are similar to the experimental ones, and the model reproduces the gross features of the magnetosheath plasma flow.

In addition, we show in Figure 1 the variations of T and T (normalized by their values at the bow shock, taken as T₀=6 10⁶ K and T ₀=4.8 10⁶ K) calculated (again, in a nonself-consistent manner) from the B(x) and n(x) profiles in accordance with the adiabatic equations (2) and (3).

These profiles show that the drift of the magnetosheath plasma in a magnetic field increasing on approach to the magnetopause results in an increase of T and simultaneous decrease of T, which contradicts the experimental data. As was said above, this contradiction was explained by Hau et al. [1993] by some nonadiabatic effects and by Denton et al. [1994] and Hill et al. [1995] by the pitch-angle diffusion of protons. In the next sections we shall try to distinguish between these two effects.

For the given profiles of B(x) and n(x) the values of (  v ), (  v ), and their ratio C(x)=(  v )/(  v ) may be calculated using (10) and (11), and the results are presented in Figure 2.

As is seen from Figure 2, C(x) is negative throughout the magnetosheath (this corresponds to a plasma compression along the X axis and its expansion in perpendicular directions), gradually decreasing from C(x)=-0.75 near the bow shock to C(x)=-3 in the magnetopause vicinity.

This regular behavior of the C(x) index obtained for a stationary model will be compared below with experimental data to judge on the stationarity of the experimental profiles under consideration.

Values of g and g may be calculated from the profiles of T, T, and n presented in Figure 2. Indeed, from the adiabatic laws one has

(20)   (21)

and therefrom

(22)   (23)

Profiles of g and g obtained in this manner are also given in Figure 2. In spite of rather rapid variations of g and g the sum S_g (g+2g ) is equal to 5 throughout the entire magnetosheath, consistent with an adiabatic process [Belmont and Mazelle, 1992].

3.2. Influence of the Pitch-Angle Diffusion on the Proton Temperature Profile

Until now, we have neglected the possible energy transfer between parallel and perpendicular degrees of freedom in the magnetosheath plasma. To illustrate the influence of such a transfer, we have integrated (using the Runge-Kutta method) the system of equations (16) and (17) for different values of t, and the results are presented in Figure 3. The curves in Figure 3 show that as should be expected, the pitch-angle diffusion significantly alters the temperature profiles across the magnetosheath, and when t Dt (where Dt  300-1000 s is the time taken by the solar wind plasma to cross the magnetosheath), T and T rapidly approach an equilibrium state, and then vary together. In the region adjacent to the magnetopause, both components of the proton temperature decrease, which agrees with the experimental data by Denton et al. [1994], Hau [1996], Hau et al. [1993], and Hill et al. [1995].

The change of T (x) and T (x) profiles immediately results in a change of the polytropic indices g and g (see Figure 4). In particular, in the case t Dt (e.g., for t=5 s and t=10 s), T T and the magnetosheath plasma behaves like an isotropic plasma with g g 5/3.

Thus the values of g and g can differ strongly from their adiabatic values; however, this does not imply the existence of sources or sinks of energy; all the calculations were carried out for an adiabatic process.

It seems interesting to consider the variations of g_eff for given n(x), T (x), and T (x) profiles across the magnetosheath for various values of t. In this connection we present in Figure 5 profiles of g_eff, calculated as

(24)

for t = 5 s, 100 s, and t . One can see from Figure 5 that in the case t= 5 s, g_eff 5/3 through the entire magnetosheath. When t=100 s, g_eff varies from about 2 at the beginning of the profile to approximately 1.5 on approach to the magnetopause. When t , g_eff, being greater than 2 in the outer magnetosheath, proves to be less than unity in the inner regions of the sheath.

3.3. Determination of the Characteristic Time of Relaxation of the Proton Temperature Anisotropy

Finally, we shall consider the possibility of determining the value of the temperature anisotropy relaxation time t from the T (x), T (x), B(x), n(x), and v(x) profiles.

A solution of that problem (in connection with the proton temperature anisotropy within the plasma sheet in the magnetotail) was proposed by Hesse and Birn [1992], Pudovkin and Smolin [1988], and Shishkina et al. [1995] on the basis of equations similar to (16) and (17); an estimate of t was obtained from a comparison of calculated profiles of the temperature anisotropy with the observed ones. Using the same equations, Denton et al. [1994] obtained the value of t in the magnetosheath in the close vicinity of the magnetopause. In their analysis, Denton et al. [1994], as well as Pudovkin and Smolin [1988], assumed the variations of T and T to be determined by the same values of t, which is correct only in the case of adiabatic processes [see Hau, 1996; Hau et al., 1993]. In this connection it seems useful to determine t and t independently and thus to judge to what degree the process under consideration is adiabatic. These estimates may be obtained using equations (18) and (19). To test the method, we used T (x) and T (x) profiles calculated for t=50 s (see Figure 1). The results of recalculating t (x) and t (x) from these profiles are presented in Figure 6. One can see that t and t vary around the value t=t= 49.9 0.1, which is rather close to the true value of t.

A more detailed analysis of the problem shows that the values of t may be obtained with reasonable accuracy when t varies in the range 0.1 Dt < t< 10 Dt, where Dt is, as earlier, the time taken by the solar wind plasma to cross the magnetosheath.

4. Analysis of Experimental Data

Now we shall use the model proposed above for the analysis of experimental data, in particular the magnetosheath parameter profiles presented by Hau et al. [1993] and by Hill et al. [1995] obtained onboard the AMPTE/IRM satellite. Both profiles took place in the magnetosheath in the vicinity of the subsolar streamline: the latitude of the satellite and LT varied from -4^o to -3^o and from 1202 to 1229, respectively, for the Hill et al. [1995] crossing and from -5.9^o to -4^o and 1311 to 1350, respectively, for the Hau et al. [1993] crossing. Of course, a deflection of the satellite's latitude from zero, and of its local time from 12 hours, means that the satellite trajectory does not coincide with the subsolar streamline, and this may essentially embarrass the comparison of the model and experimental data. However, we have no better data. To our regret, we had no original data for these crossings and therefore had to digitize the curves published in the journal articles. Practically, we could only obtain values of magnetosheath parameters at a few characteristic points of the profiles, and all other values were found by interpolation between these points. This is why the "experimental'' data are represented by smooth lines and do not exhibit any scatter. The values taken from the profiles are listed in Tables 1 and 2.

4.1. The Magnetosheath Crossing, November 17, 1985 [Hill et al., 1995]

First of all, we consider the character of the plasma flow in the magnetosheath during the crossing on November 17, 1985 [Hill et al., 1995]. Unfortunately, we could not get hold of any data on the solar wind for this period, and to judge to what degree the conditions in the solar wind were (or were not) stationary, the profile of the plasma compression parameter C(x) was calculated using the data given in Table 1 and formulae (9)-(11). The results are presented by the curves at the bottom of Figures 7a and 7b. The curves show that C(x) is negative over the entire magnetosheath and gradually varies from C(x) -0.8 near the bow shock to C(x) -1.5 in the magnetopause vicinity (a relatively rapid jump of the C(x) value at x 5500 km is most probably associated with an inaccuracy in digitizing the experimental curves). This variation is similar to the C(x) variation shown in Figure 2, and the crossing on November 17, 1985 may be considered to be in a quasi steady state regime. This allows one to use the results of the model presented above for the interpretation of data of this crossing. Curves in the top part of Figure 7a show the expected variations of T calculated from the B(x), v(x), and n(x) profiles according to equations (16) and (17) for various values of t. Again, we have to remind here that a rapid bending of all the curves at X=5500 km is associated with inaccuracy in digitizing experimental profiles B(x) and n(x) and by substitution of the real smooth curves by a sequence of segments with linearly changing variables. One can see that experimental values (also given in Figure 7a) are in a close agreement with the model curves with 5 < t < 50 s, which suggests a rather intense output of energy from transversal degrees of freedom, supposingly, to the field-aligned ones.

However, the variations of the parallel temperature T given in the top part of Figure 7b seem to be more complicated. Indeed, in the first half of the profile the experimental curve of T almost coincides with the model curve with t 500 s, which means that the decrease of T proceeds almost adiabatically, and only in the inner part of the magnetosheath the slope of the experimental curve corresponds to the model curves with t varying in the range 5-100 s.

This estimate may be confirmed by the direct calculation of t and t from the T (x) and T (x) profiles with the use of formulae (18) and (19). Corresponding values of relaxation times are presented in Figure 8, in which one can see that in the first half of the profile, t is equal to 50-150 s, while t varies from 300 to 1300 s, and only in the inner part of the magnetosheath it drops to 100-200 s. This estimate is close to (though somewhat larger than) the value of t obtained by Denton et al. [1994] for the flow near the magnetopause.

Thus the plasma flow in the magnetosheath in the case under consideration seems to pass two distinctly different stages: in the outer regions of the magnetosheath the energy transfer between the perpendicular and the parallel temperatures is relatively small and does not play any significant role, at least in the behavior of T. Whereas in the inner magnetosheath this energy transfer increases to such a degree that it affects the variations of both T and T and causes a significant deflection of the T(x) profile from the adiabatic one.

Besides, it has to be noted that even in the outer magnetosheath, the rate of the T increase is somewhat less than it is predicted by the adiabatic law, which means that the transverse thermal energy is dissipated. At the same time, as we have seen, this energy is not transferred to the parallel temperature energy, which suggests the existence of some other sinks of perpendicular energy (e.g., the energy may be transmitted from protons to electrons, where the latter, due to their very high heat conduction, move almost isothermically).

In the inner magnetosheath, t is also 1.5-2 times greater than t ; however, taking into account the low accuracy of the experimental data, we cannot be sure of the reality of this difference.

What may be the cause of the change of the proton pitch-angle diffusion? It seems reasonable to suppose that it is associated with the change of the intensity of the plasma wave turbulence responsible for the proton pitch-angle scattering. Hill et al. [1995] suppose that the mirror-wave instability plays the main role in that process. This conclusion seems to be convincingly confirmed by observations of plasma density and magnetic field intensity variations identified by the authors with mirror waves and by fulfillment of the mirror-wave-instability criterion in most of the cases analyzed by them.

In this connection we present, in the bottom panels of Figures 7a and 7b, curves showing the variations of the value of K_ph=T /T -(1+ 0.85 (b )^1/2 ) with b = 8 pn k T /B², which determines the threshold of the development of the mirror-wave instability [Phan et al., 1994]. Figures 7a and 7b show that during the entire crossing the value of T /T is close to the instability threshold, and an intensive proton pitch-angle diffusion starts when the temperature anisotropy T /T exceeds the threshold value, which seems to confirm the idea of the leading role of the mirror-wave instability in the process under consideration. On the other hand, the variations of the K_ph value are relatively small, so some doubts arise whether they really can cause such significant changes of the rate of the pitch-angle diffusion. In this connection we should like to draw attention to the fact that the change of the pitch-angle diffusion regime coincides in time with a rather rapid increase of the plasma expansion along the magnetic field lines, manifested by the decrease of the C(x) value (see Figures 7a and 7b), which allows one to suppose that this change in the plasma flow character, and consequently in the rate of the T decrease, may be responsible for the observed increase of the energy transfer rate. The next question we should like to discuss here is the variation of the polytropic index values across the magnetosheath.

Expected variations of g and g for a flow with T /T 1.2-1.3 and t 50 s have been discussed earlier and are shown in Figure 4. It shows that in the bulk of the magnetosheath the g and g values are expected to vary in the range 1.5-2.

The observed values of g and g are given in Figure 9. One can see in Figure 9 that the behavior of g generally agrees with the model prediction. At the same time, the observed variation of g drastically differs from the expected one and is much closer to that of an adiabatic flow (see Figure 2). This result confirms once more the conclusion that the energy transfer from the perpendicular component of the proton temperature to the parallel one is small in the outer regions of the magnetosheath.

Concerning the S_g index, one can see from Figure 9 that at the first part of the profile, it ranges between -2 and -4, which greatly differs from its adiabatic value S_g=5. This also confirms the assumption that the proton temperature in that part of the magnetosheath changes nonadiabatically.

On the other hand, within the inner layers of the magnetosheath, S_g varies around the value S_g=5, which agrees with the adiabatic model.

Beside the g and g profiles, Figure 9 shows also variations of the effective adiabatic index g_eff calculated using (24). The figure shows that at the outer part of the crossing the observed variations of g_eff significantly differ from the expected ones for both t=50 s and t (see Figure 5), which may be explained by the adiabatic decrease of T along with the nonadiabatic change of T with t 100 s. Within the inner magnetosheath, g_eff is seen to be close to the value predicted by the model with t=100 s.

At the same time, it is interesting to note that the "mean" value of g_eff characterizing the variation of the proton temperature across the entire magnetosheath and calculated substituting into (24) the values of T and n measured at the magnetopause edge of the profile and in the vicinity of the bow shock is

g_eff = 1.7

This agrees with the results of Fairfield [1971], Farris et al. [1991], Peredo et al. [1995], Slavin et al. [1983], and Zhuang and Russell [1981], where g was found using such integral characteristics as the magnetosheath thickness.

4.2. Crossing on October 24, 1985 [Hau et al., 1993]

Values of the magnetosheath parameters obtained for this crossing were analyzed in the same manner as above. The bottom curve in Figures 10a and 10b shows the variation of the plasma compression index C(x). One can see from Figure 10a that contrary to the previous case, the value of C(x) at the beginning of the profile is positive. For a stationary flow this would mean that the magnetosheath plasma, instead of diverging from the subsolar streamline, converges toward it, which is impossible. This suggests that the flow is highly variable during this period, and this is not surprising. Indeed, judging on the plasma velocity near the bow shock ( v 250 km s ^-1 ), the solar wind velocity at that time has a value of about 1000 km s ^-1, which corresponds to a sporadic high-velocity stream which as a rule is greatly irregular. Thus we may expect some contradictory results for this part of the profile.

At x>10,000 km, C(x) is negative, and its variation corresponds to the stationary flow in the remaining part of the magnetosheath with the exception in the close vicinity of the magnetopause, where C(x) rapidly decreases. The variation of C(x) may be explained by the fact that plasma streamlines in the vicinity of the magnetopause are tangential to the magnetopause, so the satellite is moving there not along a streamline but perpendicularly to the line.

The observed profile of T along with profiles of T calculated for different values of t from the observed profiles of B(x), n(x), and v(x) are also shown in Figure 10a. From a comparison of the model curves with the experimental one, one can see that the latter is close to the model profile with t =5 s.

As for the longitudinal temperature T (Figure 10b), the experimental curve is initially below all the model curves, which means that T decreases at that time more rapidly than adiabatically. This may be explained either by the existence of rather effective sinks of the parallel energy or, and this seems to be more likely, by the nonstationarity of the flow during this time. For x>15,000 km (with exception of a close vicinity to the magnetopause), the variation of T corresponds, judging on the slope of the curves, to a flow with t=50-100 s.

Results of calculation of t and t from the B(x), n(x), T (x), and T (x) profiles using the formulae (18) and (19) are presented in Figure 11. The value of t for x<15,000 km could not be obtained for the reasons mentioned above. At x>15,000 km, where the plasma flow seems to be relatively stationary, one can see that t and t are rather close to each other and equal to 10-50 s, which is by 0.5-1 order of magnitude less than in the previous case. This may be explained by different values of the magnetosheath parameters and plasma velocity during the two crossings.

The mirror-wave instability criterion K_ph is also shown in Figures 10a and 10b, and as in the previous case, the rapid increase of the pitch-angle diffusion observed at x=10,000-15,000 km seems to be associated with the increase of the rate of the plasma expansion along the magnetic field rather than with any variation of the K_ph value.

Variations of the adiabatic indices are presented in Figure 12. As one can see from Figure 12, variations of g, g, and S_g indices along the first (outer) part of the crossing significantly differ from the model ones, which seems to be explainable in this case mainly by the nonstationary character of the flow at that time. As in the previous case, the values of the three polytropic indices within the inner layers of the magnetosheath are close to those predicted by the adiabatic model.

The profile of g_eff is seen to agree, on the whole, with the model; g_eff= 1.6 for the entire profile under consideration, which is close to the g_eff value obtained for the previous case.

Thus in spite of the difference in the flow characteristics during corresponding periods the analysis of the experimental variations of the plasma and magnetic field parameters during the two magnetosheath crossings shows that one can distinguish two regions in the magnetosheath: an outer one, from the bow shock to approximately the point where the plasma density approaches its maximum, and an inner one, leading up to the magnetopause. In the first region the energy transfer between the perpendicular and the parallel temperatures is relatively slow, which taking into account the observational noise makes it difficult to obtain there values of the adiabatic indices and of the characteristic relaxation time t. Nevertheless, the results obtained allow us to suppose that the T variations in this region proceed almost adiabatically. At the same time, the perpendicular temperature T increases in this region significantly slower than is predicted by the double-adiabatic law, which suggests the existence of some dissipative process [Hau, 1996; Hau et al., 1993].

In the second region the estimates of t, obtained from T and T, are close to each other, which suggests an effective energy transfer between the perpendicular and the parallel temperatures. The model of the plasma turbulence responsible for the proton pitch-angle diffusion is not quite clear. However, on the basis of the fact that the mirror-wave instability criterion is not fulfilled in the second part of the profile on October 24, 1985, this turbulence may be associated with the development of the ion-cyclotron instability [Gary, 1992].

5. Conclusions

Results of the analysis presented above may be summarized as follows:

1. Variations of the perpendicular ( T ) and parallel ( T ) temperatures along the subsolar streamline in the magnetosheath are calculated for a model adiabatic flow. The profiles of T(x) and T(x) obtained are used to estimate variations of the adiabatic indices g and g across the magnetosheath. It is shown that both indices significantly change along the streamline.

2. The analysis of the experimental profiles of T and T shows that they essentially differ from the adiabatic ones, which suggests the existence of some nonideal processes which may be associated with the pitch-angle diffusion of protons.

3. As the numerical model simulation shows, the energy transfer between the perpendicular and the parallel proton temperatures significantly changes not only the temperature ( T, T, and T ) profiles but also the values of the polytropic indices g, g, and g_eff. Thus the observed deviation of those indices from their adiabatic values does not necessarily mean that the process under consideration is not adiabatic (see the discussion by Denton et al. [1994], Hau [1996], and Hau et al. [1993]), and some additional analysis of experimental data is needed.

4. The relaxation time of the proton temperature anisotropy obtained from experimental T profiles ( t ) within the outer magnetosheath is systematically greater than the time obtained from the T variations ( t ). This means that the rate of the perpendicular energy decrease is greater than the rate of the parallel energy increase, which suggests the existence of perpendicular energy sinks, the nature of those sinks being yet unclear.

5. In spite of a rather complicated behavior of g and g indices, the effective polytropic index g_eff calculated for the entire magnetosheath is equal to 1.7 for the crossing presented by Hill et al. [1995], and g_eff = 1.6 for the crossing discussed by Hau et al. [1993]. This may explain why the estimates of the polytropic index obtained from the characteristics of the magnetosheath as a whole [Fairfield, 1971; Farris et al., 1991; Peredo et al., 1995; Zhuang and Russell, 1981] give a value of g approximately equal to 5/3.

Acknowledgments

References

Belmont, G., and C. Mazelle, Polytropic indices in collisionless plasmas: Theory and experiment, J. Geophys. Res., 97, 8327, 1992.

Chew, G. F., M. L. Goldberger, and F. E. Low, The Boltzmann equation and the one fluid hydromagnetic equations in the absence of particle collisions, Proc. R. Soc. London, Ser. A, 236, 112, 1956.

Denton, R. E., B. J. Andersson, S. P. Gary, and S. A. Fuselier, Bounded anisotropy fluid model for ion temperatures, J. Geophys. Res., 99, 11,225, 1994.

Fairfield, D. M., Average and unusual location of the Earth's magnetopause and bow shock, J. Geophys. Res., 76, 6700, 1971.

Farris, M. H., S. M. Petrinec, and C. T. Russell, The thickness of the magnetosheath: Constraints on the polytropic index, Geophys. Res. Lett., 18 (10), 1821, 1991.

Feldman, W. C., Magnetic reconnection in the terrestrial magnetosphere, in Magnetospheric Phenomena, pp. 145-156, Astrophys. Workshop, New York, 1986.

Gary, S. P., The mirror and ion cyclotron anisotropy instabilities, J. Geophys. Res., 97, 8519, 1992.

Hau, L.-N., Nonideal MHD effects in the magnetosheath, J. Geophys. Res., 101, 2655, 1996.

Hau, L.-N., T.-D. Phan, B. U. Ö. Sonnerup, and G. Paschmann, Double-polytropic closure in the magnetosheath, Geophys. Res. Lett., 20 (20), 2255, 1993.

Hesse, M., and J. Birn, MHD modeling of magnetotail instability for anisotropic pressure, J. Geophys. Res., 97, 10,643, 1992.

Hill, P., G. Paschmann, R. G. Treumann, W. Baumjohann, N. Sckopke, and H. Lühr, Plasma and magnetic field behavior across the magnetosheath near local noon, J. Geophys. Res., 100, 9575, 1995.

Peredo, M., J. A. Slavin, E. Mazur, and S. A. Curtis, Three-dimensional position and shape of the bow shock and their variation with Alfvénic, sonic and magnetosonic Mach numbers, and interplanetary magnetic field orientation, J. Geophys. Res., 100, 7907, 1995.

Phan, T.-D., G. Paschmann, W. Baumjohann, N. Sckopke, and H. Lühr, The magnetosheath region adjacent to the dayside magnetopause: AMPTE/IRM observations, J. Geophys. Res., 99, 121, 1994.

Pudovkin, M. I., and V. V. Lebedeva, Solar wind parameters in the magnetosheath in a model with a magnetic barrier, Geomagn. Aeron. (in Russian), 27 (1), 22, 1987.

Pudovkin, M. I., and S. V. Smolin, Pitch-angle particle distribution in the convecting magnetospheric plasma, Geomagn. Aeron. (in Russian), 28 (2), 268, 1988.

Pudovkin, M. I., B. P. Besser, V. V. Lebedeva, S. A. Zaitseva, and C.-V. Meister, Magnetosheath model in the Chew- Goldberger-Low approximation, Physics of Plasma, 6 (7), 2887, 1999.

Pudovkin, M. I., C.-V. Meister, B. P. Besser, and H. K. Biernat, The effective polytropic index in a magnetized plasma, J. Geophys. Res., 102, 27,145, 1997.

Pudovkin, M. I., V. S. Semenov, H. K. Biernat, G. A. Bachmaier, and M. T. Kiendl, Structure of the subsolar magnetopause on November 1, 1978, 1510-1530 UT: An effect of reconnection?, J. Geophys. Res., 100, 19,239, 1995a.

Pudovkin, M. I., S. A. Zaitseva, and B. P. Besser, Magnetopause magnetic barrier parameters in dependence on the solar wind magnetic field orientation, Ann. Geophys., 13, 828, 1995b.

Shishkina, E. M., M. I. Pudovkin, and N. G. Matveeva, Peculiarities of pitch-angle diffusion in auroral plasma at initial and active phases of substorms, Proc. Arc. Antarct. Res. Inst. (in Russian), 436, 35, 1995.

Slavin, J. A., D. S. Chaussee, R. E. Holzer, J. H. Spreiter, and S. S. Stahara, Solar wind flow about the terrestrial planets, 2, Comparison with gas dynamic theory and implications for solar-planetary interactions, J. Geophys. Res., 88, 19, 1983.

Zhuang, H. C., and C. T. Russell, An analytic treatment of the structure of the bow shock and magnetosheath, J. Geophys. Res., 86, 2191, 1981.