Pages 973-982 |

M. Hoshino^{1}, Y. Saito^{1}, T. Mukai^{1},
A. Nishida^{1}, S. Kokubun^{2}, and T. Yamamoto^{1 }

^{1}ISAS, Sagamihara, Kanagawa, 229 Japan, E-mail: hoshino@gtl.isas.ac.jp

^{2}STE Laboratory, Nagoya University, Toyokawa, Aichi, 464-01
Japan

ABSTRACT

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/v^{2}.
The lower boundary of 2p/v^{2 }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/v^{2}obtained
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/v^{2},
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.

INTRODUCTION

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 [1964]. 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. [1995] 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 200R_{E }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 T_{ion}>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/v^{2}=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/v^{2 }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/v^{2
}in the shock downstream, and the bulk flow velocity in downstream
are shown as functions of the Alfven Mach number M_{A}and 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 M_{A}.
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
M_{A}~1, and 2p/v^{2} becomes
0.4 at the strong shock limit of M_{A}=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/v^{2
}<1, namely we choose the hot and high speed plasmas. The threshold
value of 700 eV and 2p/v^{2
}<1 is not sensitive to the following discussion. We then plot
T_{||}/Tas
a function of B_{x }in Figure 3. We find that there are almost
isotropic plasmas near B* _{x}*~0, i.e., in the central plasma
sheet. The anisotropy increases with increasing the magnitude of B

Figure 3: The scatter plot of the temperature anisotropy
T_{||/}T as
a function of B* _{x}* for hot and high speed plasmas with
T

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 T_{ion
}< 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 B* _{x}*. 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 B_{x}for cold plasmas of T_{ion} < 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 B_{z,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 *v*_{a}*/v*_{a1}* *downstream
is reduced, and the ratio of the thermal to bulk flow energy 2p/v^{2 }becomes
0.491 for the strong slow shock limit of M_{A} = 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 < M_{A} < 0.7 and 70 ° <
_{BN} < 85° expected
in the magnetotail slow shock observation, the difference of the ratio
2p/v^{2 }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/v^{2} of
is reduced compared with the standard slow shock in Figure
2, and the minimum value of 2p/v^{2 }becomes
0.356 at the strong slow shock limit. The density jump _{2/} _{1}becomes
large. The shock downstream velocity normalized by the upstream Alfven
velocity *v*_{2}*/v _{a}*

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 [1995], 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/v^{2 }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.

ACKNOWLEDGMENTS

Discussions with GEOTAIL team members are gratefully acknowledged. MH also thanks H. Karimabadi and K. Quest for their useful comments.

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