M. Hoshino1, Y. Saito1, T. Mukai1, A. Nishida1, S. Kokubun2, and T. Yamamoto1
1ISAS, Sagamihara, Kanagawa, 229 Japan, E-mail: firstname.lastname@example.org
2STE Laboratory, Nagoya University, Toyokawa, Aichi, 464-01 Japan
The origin of hot and high speed plasmas observed by the GEOTAIL spacecraft in the magnetotail is discussed in terms of slow shock acceleration and heating. We find that the bulk flow energy for the hot and high speed plasma in the tail plasma sheet is larger than the thermal energy, and there is an lower limit on the ratio of the thermal to bulk flow energy 2p/v2. The lower boundary of 2p/v2 is about 0.2 - 0.4, and the ratio is independent of the plasma temperature in the range of several 100 eV to 10 keV. It is believed that the magnetic reconnection associated with a slow shock can produce the hot and high speed plasma, though the observed lower limit cannot be explained by a standard slow shock heating and acceleration process, because the lower limit of 2p/v2obtained by the standard slow shock Rankine-Hugoniot relation is 0.4 even for the strong slow shock limit. In order to explain the observed lower boundary of 2p/v2, we study the Rankine-Hugoniot relations by taking into account of non-standard MHD effects such as a temperature anisotropy and a heat flux. The observed lower limit can be explained by a slow shock including the temperature anisotropy and the heat flux effects.
The origin of hot plasmas in the plasma sheet is a long standing problem in magnetospheric physics. It is important to understand how and where an effective energy conversion process takes place in the magnetotail. As one of the efficient energy conversion processes, magnetic reconnection is often discussed as a key process on the dynamical change of the magnetosphere, and the slow shocks associated with the magnetic reconnection are thought to be a main engine of the plasma heating and acceleration.
The first model of the steady state magnetic reconnection associated with slow shocks was proposed by Petschek . It is pointed out that, in addition to the magnetic diffusion process around the X-type neutral point, the standing slow shock waves can provide an efficient energy conversion of magnetic field energy to plasma energy. Non-linear evolution of magnetic reconnection was also conducted by using a magnetohydrodynamic simulations [Ugai and Tsuda, 1977; Hayashi and Sato, 1978], and it was shown that the plasma sheet can evolve into the topology of the Petschek's reconnection. Those theoretical and simulation studies of magnetic reconnection motivated to search a slow shock by observations in the magnetotail. Using the data of ISEE 3 in the distant magnetotail, Feldman et al. [1984, 1995] compared the Rankine-Hugoniot relations with the observed jumps of plasma parameters across the boundary between the plasma sheet and the lobe, and confirmed the existence of slow-mode shocks. Recently, Saito et al.  also identified slow shocks during the course of plasma sheet crossing in the distant magnetotail. They investigated the kinetic structure of slow shocks, and discussed the plasma heating and acceleration mechanisms across the shocks.
Although the identification of the slow mode shock supports a paradigm of the magnetic reconnection model as a main engine of plasma acceleration and heating, the occurrence probability to be identified as a slow shock is low. The most of plasma sheet crossing data did not satisfied the Rankine-Hugoniot relations, even if high speed, hot, and high density plasmas are observed in the plasmas sheet side, i.e., in the downstream side of the transition layer [Saito et al. 1995]. The low probability of the slow shock detection might be attributed to time evolution of the slow shock or the multi-dimensional structure of shock. Other processes not indicated by a slow shock acceleration, however, may play an important role on the hot and high speed plasmas. The electric current sheet acceleration in the central plasma sheet is thought to be another important process. The idea is based upon a single particle motion under prescribed magnetic and electric fields, and Speiser particles [Speiser, 1965] in the central plasma sheet gain their energies from the dawn-dusk electric fields. The understanding of the global response of accelerated particles to the fields is still poor.
In this paper, we study first characteristics of hot and high speed plasmas in the magnetotail by using the GEOTAIL/LEP data [Mukai et al. 1994], and discuss the efficiency of plasma acceleration and heating. We show that there is a lower limit in the ratio of thermal and bulk flow energy in the plasma sheet plasmas. We further show that the observed hot and high speed plasma is associated with a temperature anisotropy and a heat flux. We find that the parallel temperature is larger than the temperature perpendicular to the magnetic field in the magnetotail for the most of observed data. A large temperature anisotropy is often observed near the boundary layer between the lobe and the plasma sheet, and the anisotropy decreases with approaching to the neutral sheet. We also find that a heat flux observed in the magnetotail is not negligible compared with the internal energy flux. It is likely that the effects of both the anisotropic plasma and the heat flux modify the shock jump condition. These observations lead us to reexamine the slow shock model with the effects of a temperature aisotropy and a heat flux on the Rankine-Hugoniot relations. We find that a slow mode shocks with the effects of a temperature anisotropy and of a heat flux can adequately explain the observations of the hot and high speed plasmas.
OBSERVATIONS OF HOT AND HIGH SPEED PLASMAS
We discuss characteristics of the hot plasmas observed by GEOTAIL/LEP in the Earth's distant magnetotail. We use the data taken from September 1993 through October 1994 when the GEOTAIL spacecraft was situated from 50 to 200RE in the magnetotail. We study first the relationship between the thermal energy and the bulk flow energy observed in the plasma sheet. In order to subtract the magnetosheath plasmas data, we selected the data which satisfies the conditions of either the ion temperature Tion>400eV or the plasma density n<1cm-3, and then the contamination of the magnetosheath plasma with a high density and low temperature plasma can be removed. The scatter plot in Figure 1 shows the relationship between the ion temperature and the ion bulk flow velocity using the ion moment data. Each dot represents the plasma observation made in the 12 seconds time resolution. The cold plasmas having their temperatures less than about several hundred eV correspond to the lobe or mantle plasmas, and we find that their bulk velocities are ranging from several ten km/s to about several hundred km/s. Those plasmas are flowing tailward. On the other hand, the hot plasmas with their temperature greater than about several hundred eV are thought to belong to the plasma sheet plasmas. We find that the velocities of the hot and high speed plasmas exceed sometimes about 1000 km/s.
In the middle temperature region around 250 eV, one can find a gap region with faint data points. The region probably corresponds to the boundary region between the lobe and the plasma sheet. We think that the thickness of the boundary layer is geometrically thin, so that the occurrence probability of the boundary region becomes low.
Figure 1: The scatter plot of the ion temperature and the ion bulk velocity using the GEOTAIL ion moment data. The dashed line shows 2p/v2=0.4 as reference.
Let us look at the hot and high speed plasmas, which correspond to the upper-right-hand group in Figure 1. We find that the hotter plasma has the larger bulk velocity, and that there is a positive correlation between the ion temperature and the ion bulk velocity. The dashed line shows the ratio of thermal to bulk flow energy 2p/v2 with 0.4 as reference. We find that the lower limit of the ratio of the thermal to the bulk flow energy is about 0.2 - 0.4, and the bulk flow energy is larger than the thermal energy.
The lower limit found in the energy partition between the bulk flow energy and the thermal energy gives an important constrain to understand the origin of hot plasmas in the magnetotail. We will discuss whether or not the lower limit of the energy partition can be explained by the standard magnetic reconnection model associated with slow shocks.
KINETIC AND THERMAL ENERGY PARTITION IN SLOW SHOCKS
The slow shocks associated with magnetic reconnection are thought to be one of the powerful mechanisms to produce the hot and high speed plasmas, which energies are converted from the lobe magnetic field by reducing the amplitude of the magnetic field tangent to the shock. ISEE 3 and GEOTAIL observations in the magnetotail detected slow shocks across the boundary between the lobe and the plasma sheet, in which transition layers the Rankine-Hugoniot relations of slow mode are satisfied [Feldman et al, 1984, 1985; Saito et al, 1995]. The most of the plasma sheet crossing data, however, did not satisfy the slow shock jump condition [Feldman et al, 1984, 1985; Saito et al, 1995]. We reexamine whether the slow shocks can explain the observation. Specifically, we discuss how the energy ratio found in Figure 1 can be understand from a point of view of the slow shock.
Figure 2 shows the Rankine-Hugoniot relations obtained for the standard slow shock in the shock normal incident frame, i.e., the upstream plasma flow vector is perpendicular to the shock plane. Form the left-hand panel in Figure 2, the density compression, the ratio of the plasma thermal energy to the kinetic energy 2p/v2 in the shock downstream, and the bulk flow velocity in downstream are shown as functions of the Alfven Mach number MAand the shock angle NB defined by the angle between magnetic field direction and the shock normal direction. For simplicity, we have assumed a cold plasma in the shock upstream, i.e., the upstream plasma beta 1=0, because the plasma beta in the lobe region of the earth's magnetotail is less than about 5 x 10-3.
Figure 2: The standard Rankine-Hugoniot relations in the shock normal incident frame where the upstream plasma flow is perpendicular to the shock plane. From the left-hand panel, the density compression, the ratio of thermal to bulk flow energy, and the bulk flow speed normalized by the upstream Alfven speed. The horizontal axis is the shock angle between the shock normal and the magnetic field in shock upstream BN, and the vertical axis show the Alfven Mach number in shock upstream MA. The upstream plasma 1 is assumed to be zero.
From the center panel of Figure 2, one can find that the kinetic energy becomes larger than the thermal energy for the strong shock region with NB ~90° and MA~1, and 2p/v2 becomes 0.4 at the strong shock limit of MA=1 and BN-> 90°. Most of magnetic energy in shock upstream are converted into the plasma bulk energy in the shock downstream for the strong shock region. On the other hand, in the weak slow shock region, most of the magnetic field energy is converted in the thermal energy.
From a topological point of view, the shock angle NB is not small for the slow shocks expected in the earth's magnetotail. According to the slow shock observations by GEOTAIL (e.g., Saito et al., 1995), NB is greater than 70°. The inflow Alfven Mach number observed in the lobe side, i.e., in the shock upstream, is about 0.4 to 0.7. Therefore, the ratio of energy partition expected from the standard slow shock Rankine-Hugoniot relation becomes about 0.6 - 0.8. On the other hand, the observed maximum limit of the ratio of thermal energy to bulk flow energy in Figure 1 is about 0.2 - 0.4. It seems that the standard slow shock acceleration model alone cannot provide a sufficient acceleration to explain the observed hot and high speed plasmas in magnetotail. We might need other acceleration processes in addition to the slow shock. In order to explain the additional acceleration, we study the shock jump conditions including non-standard MHD effects such as a temperature anisotropy and a heat flux to explain the observed high efficiency of plasma bulk acceleration. The plasma temperature observed in magnetotail, in general, is not isotropic, and the heat flux does not seem to be negligible compared with the internal energy flux. Those effects are probably important on the dynamical evolution of the slow shock.
OBSERVATIONS OF TEMPERATURE ANISOTROPY AND HEAT FLUX
Before discussing the shock jump conditions including the non-standard MHD effects such as a temperature anisotropy and a heat flux, we study the magnitude of the temperature anisotropy and the heat flux observed by GEOTAIL in the magnetotail. We use the same data set discussed in Figure 1. We select the data satisfying the ion temperature T>700 eV and 2p/v2 <1, namely we choose the hot and high speed plasmas. The threshold value of 700 eV and 2p/v2 <1 is not sensitive to the following discussion. We then plot T||/Tas a function of Bx in Figure 3. We find that there are almost isotropic plasmas near Bx~0, i.e., in the central plasma sheet. The anisotropy increases with increasing the magnitude of Bx. Large anisotropic plasmas such as T||/T>2 might correspond to the plasma sheet boundary layer (PSBL) ion beams, even though our data set is selected by the condition of T > 700 eV. However, most of data points which are located in Bx< 5 nT probably belong to the plasma sheet plasmas, because the lobe magnetic fields on average are larger than 10 nT in the magnetotail. Therefore, we think that the large temperature anisotropy with T||/T ~ 2 exists in the plasma sheet side. The anisotropic plasmas are quickly isotropised as their entering into the central plasma sheet.
Figure 3: The scatter plot of the temperature anisotropy T||/T as a function of Bx for hot and high speed plasmas with Tion >700eV and 2p/v2<1. Data set is same as Figure 1.
Next we study a heat flux observed by GEOTAIL. Figure 4 shows the scatter plot of the heat flux parallel to the magnetic field. We select Tion < 500 eV the data of to study the heat flux in the lobe side, that is, we study the heat flux escaping from the hot plasma sheet to the cold lobe region. The magnitude of heat flux increases with the magnetic field Bx. We compare the heat flux with the internal energy flux in the central plasma sheet. The internal energy flux is estimated by
We find that the heat flux observed in the lobe is not negligible compared with the internal energy flux.
Figure 4: The scatter plot of the total heat flux as a function of Bxfor cold plasmas of Tion < 500 eV.
MODIFIED SLOW SHOCK JUMP CONDITIONS
In this section, we study the shock jump conditions including the effect of temperature anisotropy as well as the effect of heat flux, and discuss how the ratio of energy partition between bulk flow and thermal energy can be modified. An importance of the presence of temperature anisotropy in the various discontinuities has been discussed theoretically by several authors [Hudson, 1970; Neubauer, 1970; Chao 1970; Karimabadi et al., 1995]. We quickly review the one-dimensional, time stationary conservation law, i.e., the Rankine-Hugoniot relations. We choose the plane defined by the velocity v and the magnetic field B to be the (x, z) plane, and the shock normal z to be parallel to the flow velocity v in the shock upstream.
where A=p||/p shows the pressure anisotropy. is the heat flux vector. The brackets around each equation denote the difference between the upstream and downstream values of the enclosed quantities. In order to obtain closure of the magnetohydrodynamic equations in an anisotropic plasma, two equations of state are required for both the parallel pressure and the perpendicular pressure. One of the closure of two equations of state is the CGL theory [Chew-Goldberg-Low, 1956], though the CGL theory is violated at shocks, because the energy dissipation process which should be determined by microscopic processes is not described. If a heat flux is not negligible, in addition to two equations of state another equation is required to obtain a closure of MHD equations. In this paper, we prescribe the pressure anisotropy as well as the heat flux, and eliminate the parallel pressure and the heat flux. After some algebraic manipulations, one arrives at sixth-order equation for Bz,2 as functions of the upstream parameters.
The solutions are shown in Figure 5 for the fixed downstream temperature ratio of T||,2/T ,2 = 3/2 and the upstream plasma beta 1=0 but the heat flux is neglected. We assume that the ratios of the parallel and perpendicular specific heats with respect to the magnetic field are respectively given as ||=3and =2 in the double adiabatic limit. By comparing with the result of an isotropic Rankine-Hugoniot relation in Figure 2, we find that the efficiency of the plasma acceleration va/va1 downstream is reduced, and the ratio of the thermal to bulk flow energy 2p/v2 becomes 0.491 for the strong slow shock limit of MA = 1and BN ->90°. For p||,>p ,the electric current due to the temperature anisotropy flows in opposite direction to the pressure gradient current. As a result the total electric current in the shock front is reduced, and the Lorentz acceleration is suppressed. For the range of 0.4 < MA < 0.7 and 70 ° < BN < 85° expected in the magnetotail slow shock observation, the difference of the ratio 2p/v2 between the isotropic and the anisotropic cases is little. The temperature anisotropic effect cannot improve the efficiency of plasma acceleration. Moreover, the density compression is strongly reduced, which contradicts to the observations of the high density plasma sheet.
Figure 5: The Rankine-Hugoniot relations including a temperature anisotropy. T||/T = 3/2 is assumed. Same format as in Figure 2.
We next study the effect of the heat flux across the plasma sheet boundary between the lobe and the plasma sheet. When a finite heat flux exists, one cannot necessarily expect that the double adiabatic relation holds. Due to the escape of the heat flux from the hot plasma sheet region to the cold lobe region, the ratio of specific heat will be reduced. If there were a significant heat flux along the magnetic field line, an isothermal state with || =1 will be realized.
As one of the simple models to include the effect of a heat flux into the shock jump equations, we assume that the heat flux in shock upstream is proportional to the internal energy flux in shock downstream, and use the observed ratio of
based on Figure 4. We neglect the downstream heat flux. After substituting the above relation into Eq.(5), we can easily find that we may use Eq.(5) without but with a smaller ||, i.e.,
Figure 6 show the result of the Ranking-Hugoniot relations with || == 2, T||,2/T ,2 = 3/2, and 1 = 0 . We find that the ratio 2p/v2 of is reduced compared with the standard slow shock in Figure 2, and the minimum value of 2p/v2 becomes 0.356 at the strong slow shock limit. The density jump 2/ 1becomes large. The shock downstream velocity normalized by the upstream Alfven velocity v2/va1is almost same as the standard slow shock result. Due to the escape of the thermal energy from the downstream, the plasma temperature will decrease. In order to maintain the pressure balance across the shock, the plasma density should increase. We think that the observation of the high efficient energy conversion into the plasma bulk flow energy is originated from the loss of the thermal energy in the shock downstream due to the heat flux escape. The observed hot and high speed plasmas can be explain by the modified slow shock including the temperature anisotropy and the heat flux.
Figure 6: The Rankine-Hugoniot relations including both a heat flux and a temperature anisotropy. T|| /T= 3/2,
|| = =
2 are assumed. Same format as Figure 2.
DISCUSSIONS AND CONCLUSIONS
Non-Maxwellian velocity distribution functions are often observed in the earth's magnetotail, because the relaxation time scale of the velocity distribution function is longer than the dynamical time scale. In addition to an anisotropic plasma, a heat flux also plays an important role in thermodynamics. In this paper, we discussed the shock jump relations under an anisotropic temperature with a tenser ellipsoid axis-symmetric to the direction to the magnetic field, and we also include phenomenologically the effect on a non-zero heat flux.
The efforts of finding slow shocks in magnetotail were done by Feldman [1984, 1985] and Saito et al , who checked the jump of plasma parameters between the lobe side and the plasma sheet side in terms of the Rankine-Hugoniot relations with an isotropic temperature. The occurrence probability to detect a slow shock is low, and the most observed data have the larger density jump and the colder temperature than that theoretically expected from the Rankine-Hugoniot relations [Saito et al, 1995]. We discussed that the shock jump conditions can be improved, if we include the effects of both an anisotropic plasma and a heat flux.
The observational results presented in Figure 1 were made in the frame of spacecraft, that it, the velocity is mesured as the relative speed between the spacecraft and the plasma medium. On the other hand, we used the shock normal incident frame in the Rankine-Hugoniot relations. One might think that the observed ratio of 2p/v2 could be explained just by a standard slow shock process if we could take into account of the diferrence of the frame. The observed large density jump acrros the boundary between the lobe and the plasma sheet, however, cannot be explained by the difference of the frame.
In summary, the origin of hot and high speed plasmas can be explained by the slow shock acceleration and heating under an temperature anisotropy with T||>T =and a finite heat flux. In the modified slow shock Rankine-Hygoniot relations the shock downstream density increases and the plasma temperature decreases compared with an isotropic Rankine-Hugoniot relations.
Discussions with GEOTAIL team members are gratefully acknowledged. MH also thanks H. Karimabadi and K. Quest for their useful comments.
Chew, G. F., M. L. Goldberger and F. E. Low, Proc. Roy. Soc. A 236, 112, 1956
Chao, J. K., Interplanetary collisionless shock waves, Rep. CSR TR-70-3,
Mass. Inst. of Technolog. Cent. for Space Res., Cambridge, Mass., 1970.
Feldman, W. C., et al., Evidence for slow-mode shocks in the deep geomagnetic tail, Geophys. Res. Lett., 11, 599, 1984.
Feldman, W. C., et al., Slow-mode shocks: A semipermanent feature of the distant geomagnetic tail, J. Geophys. Res., 90, 233, 1985.
Karimabadi, H., Steepening of Alfven waves and its effects on the structure of slow shock, Geophys. Res. Lett., 22, 2693 - 2696, 1995.
Karimabadi, H., D. Krauss-Varban, and N. Omidi, Temperature anisotropy effects and the generation of anomalous slow shocks, Geophys. Res. Lett., 22, 2689 - 2692, 1995.
Hayashi, T. and T. Sato, Magnetic reconnection: Acceleration, heating, and shock formation, J. Geophys. Res., 83, 217, 1978.
Hudson, P. D., Rotational discontinuities in an anisotropic plasma, Planet. Space Sci., 19, 1693 - 1699, 1971.
Mukai, T., M. Fujimoto, M. Hoshino, S. Kokubun, S. Machida, K. Maezawa, A> Nishida, Y. Saito, T. Terasawa, and T. Yamamoto, Structure and kinetic properties of the plasmoid and its boundary region, in press, J. Geomag. Geoelectr., 1996.
Neubauer, F. M., Jump relations for shocks in an anisotropic magnetized plasmas, Z. Physik, 237, 205 - 223, 1970.
Petschek, H. E., Magnetic field annihilation, NASA-SP-50, 425, 1964.
Saito, Y., T. Mukai, T. Terasawa, A. Nishida, S. Machida, M. Hirahara, K. Maezawa, S. Kokubun, T. Yamamoto, J. Geophys. Res., 10, 23567 - 23581, 1995.
Speiser, T. W., Particle trajectories in model current sheet 1, Analytical solution, J. Geophys. Res.,, 70, 4129 1965.
Ugai, M. and T. Tsuda, Magnetic field-line reconnection by localized enhancements of resistivity, 1. Evolution in a compressible MHD fluid, J. Plasma Phys., 17, 337, 1977.