A&A 420, 11071115 (2004)
DOI: 10.1051/00046361:20040119
Slow magnetoacousticlike waves in postflare loops
A. N. Kryshtal

S. V. Gerasimenko
Department of Cosmic Plasma Physics, Main Astronomical
Observatory of the National Academy of Sciences of Ukraine, 03680
Zabolotnogo Str., 27, Kiev 127, Ukraine
Received 12 February 2003 / Accepted 16 February 2004
Abstract
We investigate the stability to the development of
plasma waves in the preflare situation of a loop structure at the
chromospheric part of a current circuit of a loop. We investigate
the conditions under which lowfrequency plasma instabilities can
develop, assuming the absence of beam instabilities.The
largescale quasistatic electric field in the loop circuit is
assumed to be "subdreicer" and weak. Thus the percentage of
"runaway" electrons is very small and their influence on the
process of instability development is negligible. The pair Coulomb
collisions are described by a BGKmodel integral. We consider the
situation when the plasma at the surface layer of a loop has a
spatial gradient of density. In accordance with
HeyvaertsPriestRust theory, such a preflare situation would
typically exist when the amplitude of the weak electric field
in the circuit of an "old"
loop in an active region begins to increase when "new" magnetic
flux emerges from under the photosphere. We have found that two
types of waves are generated in such a plasma due to the growth
of instabilities: the "kinetic Alfvenlike" waves and new type of
waves, in the range of magnetoacoustic ones. The instability of
these latter waves has a clear threshold and it can be considered
as an "indicator" of the development of a preflare situation in an
active region.
Key words: plasmas  Sun: flares  Sun: chromosphere
Plasma instabilities are traditionally assumed to be one
of the main sources of waves in the flare atmosphere (Priest 1982;
Somov 1994; Zaitsev et al. 1994). Their hierarchy is usually
considered as an important element in the theory of cyclotron
maser emission (CME) (Melrose 1989; Zaitsev et al. 1994; Mel'nikov
et al. 2002) as well as in the dynamics of current sheets which
form in flaring arcades (Podgorny & Podgorny 2001), namely where
flares most frequently occur (De Jager 1959; Priest 1982). Thus
the loops of arcades are "postflare" loops of a previous flare
and the preflare of the next one. The investigation of the
preflare plasma state on the basis of HeyvaertsPriestRust (HPR)
theory (Heyvaerts et al. 1977; Podgorny & Podgorny 2001) is
current practice. In this theory a flare is considered to be the
result of the interaction of a "new" magnetic flux, which emerges
from under the photosphere and an "old" one, which passes through
a current circuit of a loop in an arcade. Such interaction results
in an adiabatic slow increase of the amplitude of a weak
quasistatic electric ("DC") field in the circuit of the "old"
loop. From general physical considerations (De Jager 1959;
Kadomtsev & Pogutse 1967) it is clear that, in such circumstances,
different lowenergetic wave instabilities of the "stream" type
will grow in plasma. If the growth time of the largescale
electric field amplitude
is much larger than the time of instability development,  this
situation occurs most frequently (Poletto & Kopp 1986),  then
the mechanism of "direct initiation" of instability by the
electric field will take place. The corresponding instabilities
will have a clearly expressed threshold character (Kryshtal &
Kucherenko 1995). The value of the amplitude
,
expressed in units of the local Dreicer field ,
will be the threshold value in this case. We
assume that in preflare situation there are only very few
"superenergetic" charged particles (i.e. plasma "without beams").
When the electric field is weak, i.e. when

(1) 
the percentage of "runaway" electrons is so small that
their influence on the instability development can be regarded as
negligible (Alexandrov et al. 1988). Of course,this is only one of
the possible scenarios for the preflare situation development
(Zaitsev et al. 1994).
In connection with the "perpetual" problem of solar physics, the
problem of coronal heating, Aschwanden (2001) has formulated
interesting conclusions based on the recent soft Xray and EUV data from space observations with the YOHKOH, SOHO and TRACE satellites. We consider that the inclusion of the chromosphere is
very important not only "in conventional AC and DC models"
(Aschwanden 2001) but in the investigation of the dynamics of the
preflare plasma state in a loop in general. Preferential footpoint
heating (Aschwanden 2001) can be explained from our point of view
in the framework of the wellknown Ionson model (Ionson 1978). The
most important necessary condition is the generation of the
kinetic Alfven waves (KAW) on the "chromospheric floor" of the
loop current circuit. These waves propagate almost perpendicular
to the direction of the magnetic field
of the
loop in its surface layer (Ionson 1978). Second, the observed
overdensity of order
(Aschwanden 2001),
which, of course, is not less in the chromosphere, seems to be a
good reason to assume that strong gradients of the plasma density
can have a larger effect than the similar values of temperature and
magnetic field amplitude
.
Here
we denote by
the inverse of the gradient scale
,
is the reduced spatial gradient of
density. The value of
is proportional to the inverse
value of the mean scale of the density inhomogeneity of the
electrons and ions (
). Analogous values for the
temperature and the magnetic field can be introduced in a similar
way. Since the end of the 1950s (De Jager 1959), it is well known
that overdensity is the most distinguishing property of postflare
loops. When the effect of the reduced spatial gradient of density
dominates, it is possible to neglect the shear influence on the
instability development. This needs some special condition
regarding the plasma state and the characteristics of the
perturbation. The increasing number of detections of flarelike
events at temperatures of
MK in EUV (with
SOHO/ETT and TRACE) (Aschwanden 2001) allows us to assume that
the early stage of the preflare process can correspond to a
temperature of 0, 5 MK. The plasma parameters should be
obtained from the wellknown semiempirical model of chromospheric
flare regions (Machado et al. 1980).
We investigate the physical conditions for the growth of
lowfrequency plasma wave instabilities in the case of the
longwave perturbations propagating almost perpendicular to the
direction of the loop magnetic field
.
We use a
local rectangular Cartesian coordinate system with the Zaxis
directed along the field vectors
and
(we assume that
). Taking into account that we consider a plasma in the "chromospheric floor" near the footpoint of a loop, this means that the XYplane is actually parallel to
the surface of the photosphere. For intermediate calculations we
have used a cylindrical coordinate system in velocity space
(
)
(Alexandrov et al. 1988).
We assume that the plasma has a onedimensional spatial gradient
of its density along the Xaxis. The thickness of the surface
layer of the loop is the mean spatial scale of inhomogeneity of
the plasma density. For the local solutions of the dispersion
relation for the lowfrequency plasma waves (Michailovsky 1963) we
consider that the origin of the Cartesian coordinates is placed
near the inner border of the surface layer. We assume
(Michailovsky 1963), that the wave frequency
satisfies the inequality

(2) 
where
is the local ion gyrofrequency.
In the present investigation, we use the same mechanism of "direct initiation" of instability and the same plasma model as in our previous work (Kryshtal & Kucherenko 1995; Kryshtal 2000), the
most important properties of which are the following:
 1.
 The weak largescale electric field
is quasistatic, which implies that

(3) 
This can be considered as typical for flares processes and for
different types of plasma instabilities (Zaitsev et al. 1994).
On the other hand there is observational evidence for a
correlation between the time of maximum E_{0} (t) and the
moment at which the first energy release occurs in a flare
(Poletto & Kopp 1986; Zaitsev et al. 1994). The lefthand side
of inequality (3) approximately equals the value
,
where
is the time for the amplitude E_{0} (t) to grow to its maximum value.
We assume that the electric field under consideration is extremely
weak and that the threshold value
,
at which the instability, with growth rate
and growth time
,
starts developing, does not much exceed the equilibrium value
.
This value corresponds to the
origin of the preflare process when interaction between the "old"
and "new" magnetic fluxes is absent. In this situation
equals the time of the shortperiod prediction of a flare and the instability appearance can be considered in a sense as a
"forerunner" of a flare process.
 2.
 As earlier (Kryshtal 2000) the plasma under consideration is
assumed to be fully ionized and collisions are described by a BGK model integral (Alexandrov et al. 1988). The equilibrium velocity distribution function for ions is assumed to be pure Maxwellian,
but the same function for the electrons is described by a shifted
Maxwellian distribution with electron shift velocity:

(4) 
here,
is the ionelectron collision frequency. When
a weak ("external") electric field exists and there is an enough
strong magnetic field in the plasma, the ionelectron collisions
dominate (Alexandrov et al. 1988).The contribution of other mutual
collisions of charged particles can be taken into account in a
phenomenological way with the help of the factor
(Kryshtal & Kucherenko 1995).
 3.
 Taking into account the real scales of inhomogeneity of the
plasma (electron or ion) densities in the loops (Zaitsev et al. 1994) we consider the longwave approximation for the perturbations:

(5) 
where
and
are the electron and ion
kinetic parameters respectively,
is the transverse
component of perturbation wavevector
(
;
),
and
are respectively
the electron and ion thermal velocities and
is
the electron gyrofrequency.
 4.
 The spatial density inhomogeneity is supposedly "weak", which
means, according to Michailovsky (1963) that the conditions

(6) 
where

(7) 
have to be satisfied by the drift frequencies of electrons and ions, respectively.
 5.
 We have used the condition of quasineutrality of plasma

(8) 
for the equilibrium densities of the charged particles. When Eq. (8) is valid for an arbitrary x, the analogous equation for the spatial gradients of densities is also
valid:

(9) 
Equation (9) allows us to make use of the wellknown simple connection between the drift frequencies of electrons and ions (Michailovsky 1963; Alexandrov et al. 1988)

(10) 
where

(11) 
In our calculations we have assumed that the density profiles in the surface layer have the form

(12) 
and that

(13) 
is satisfied. In this case the condition (6) is practically equivalent to the wellknown inequality (Michailovsky 1963; Kadomtsev & Pogutse 1967) for plasma with weak spatial
inhomogeneity of density

(14) 
where
is the ion gyroradius. Equation (14), as a rule, is easily satisfied in the loops (Zaitsev et al. 1994).
 6.
 Under the mentioned above conditions, it is natural to make
in the calculation the local approximation to the dispersion
relation (DR), which is justified if (Michailovsky 1963)

(15) 
where
(for onecharged ions) and
is the wellknown "plasma ":

(16) 
The local approximation to the DR allows us to neglect the influence of the boundaries on the process of instability development. Since we consider a "low" plasma (Michailovsky 1963), then the following inequality has to be satisfied (Krall & Trivelpiece 1973)

(17) 
We assume everywhere that

(18) 
Then the condition (15) for the solutions to be local turns out to be even more stringent than the condition

(19) 
for the transverse wavelength of the perturbation. Inequality (19) is the condition of the approximation of geometrical optics.
 7.
 Taking into account the fact that in a surface layer the main spatial inhomogeneity of density is in the direction perpendicular to the magnetic field
,
as well as the fact that
the lowfrequency waves under consideration are of a quasielectrostatic type, we assume that the condition

(20) 
is satisfied for the components of the perturbation wavevector (Ionson 1978).
 8.
 We take the "longitudinal" phase velocity of the perturbation to vary in the range

(21) 
This range is typical of the "Alfvenlike" and "driftlike" waves in a plasma (Krall & Trivelpiece 1973). We have also made use of some additional conditions:
 9.
 In the case of "weak inhomogeneity" we consider a "moderately
nonisothermal" plasma with

(22) 
On the one hand this allows one to neglect the possible influence of
ionacoustic turbulence on the process of instability development.
Actually it means that

(23) 
On the other hand, the relatively low values of t in (22) imply that threshold value
,
at which the instability arises, may be low too. This may be important for the problem of preheating of the loop footpoints (Aschwanden 2001).
 10.
 We noted that we neglect in our calculations the reduced spatial gradients of temperature and
magnetic field compared to that of density. According to Kadomtsev & Pogutse, this is equivalent to the neglect of the influence of "shear" (Kadomtsev & Pogutse 1967). For a
characteristic scale of the perturbation of order
,
i.e. for
,
this assumption is
correct if the condition for the ion "plasma "

(24) 
is satisfied. In Eq. (24)
,
where
is
the Alfven velocity and
is the ion plasma
frequency. The analogous condition for the electron "plasma " has the form

(25) 
Taking the conditions (5) and (20) into account, these last two inequalities, (24) and (25), point to the importance of the "right choice" of plasma parameters, i.e. appropriate values of the densities and temperatures of the electrons and ions. With the mentioned above restrictions and conditions, this is not so simple. Semiempirical models of the
flare chromosphere (Machado et al. 1980) fortunately provide us with necessary parameters. In our calculations we used the following values:

(26) 
We supposed that in the local thermodynamical equilibrium
before the interaction of the "old" and "new" magnetic
fluxes, and
can exceed
according to Eq. (22) in the case of weak inhomogeneity in the early stage of the preflare process. In the paper of Machado et al. (1980) the
values of
and
correspond to the height h=1459 km above the photosphere. Strictly speaking the magnetic field amplitude in our model cannot be arbitrary. This is
clear in view of the conditions and approximations of the employed
model. There is still some freedom in the choice of this parameter
(Gopasyuk 1987). We assumed
.
This value is
very closed to the value
from Aschwanden (1987).
For such magnetic field amplitudes and with the parameters given in Eq. (26) and extremely small values of
,
the conditions (24) and (25) can be easily satisfied in the case of a weak inhomogeneity. In the opposite case of a strong inhomogeneity, for
extremely small values of ,
when instability can develop on the background of turbulence and the value of
can sharply increase, the conditions (24) and (25) practically determine an upper limit for
.
In the case of weak inhomogeneity, when the ionelectron collisions dominate, we can estimate for the parameters (26) the equilibrium value
,
which corresponds to the steady state in the loop circuit (without interaction of the "new" and "old" magnetic fluxes). If we assume that the loop under consideration is a "semitorus" with small radius R_{0}, it can be shown that

(27) 
For a current in this loop I = 1
(Gopasyuk 1987) and
(Zaitsev et al. 1994), we then obtain

(28) 
Of course, it seems very problematic to obtain the exact value
of R_{0}, so the estimate (28) can give only an order
of magnetude
.
The starting point for obtaining the dispersion relation (DR) for this plasma model is the expression for the scalar dielectric permeability of a hot magnetoactive plasma with
spatial inhomogeneity. (Alexandrov et al. 1988).
When the electric field in the lowfrequency wave
in the inhomogeneous medium can be considered as quasipotential,
the equation for this permeability in the inhomogeneous plasma,

(29) 
plays the same role as the standard DR in homogeneous plasma. In this case, Eq. (29) can be considered as the eikonal equation for electrostatic waves in the zero order of
geometrical optics. For the lowfrequency waves, which satisfy condition (2), Eq. (29) can be simplified in a standard way (Kryshtal 2000) and reduced to the form

(30) 
where
and
are the longitudinal and transverse
parts of the dielectric permeability respectively, c is the
speed of light in vacuum. This form of DR was first investigated
by Michailovsky (1963) in the local approximation at
,
and t = 1. In the present plasma
model with
,
and t>
1 the form of the DR remains the same (Kryshtal 2000), but, if we
take into account the existence of subdreicer field and the
influence of the collisions, additional terms appear in the
expressions of
and
.
These expressions become much more
complicated, with terms that describe the ion thermal motion. But
the DR becomes suitable for analysis when
takes extremely small values. This form of the DR, which is
modified by the presence of the external electric field, pair
Coulomb collisions and ion thermal motion, is refered to as the
modified dispersion relation (MDR). Each root of the equation
(Krall & Trivelpiece 1973)

(31) 
corresponds to a certain kind of plasma wave, with dispersion law
,
where m is a number labeling the root. We consider this plasma wave as the solution of MDR. In the present plasma model,
actually depends not on x, but on L(see Eqs. (12)(13)). We determine the growth rate (positive or negative) of the instability from the equation

(32) 
In the framework of our approximations, Eq. (31) can be reduced to the polynomial form

(33) 
with



(34) 

(37) 

(38) 
where

(39) 
In the Eq. (33)

(40) 
is the reduced frequency (or reduced "longitudinal" phase velocity). At the same time in the Eqs. (34)(38)

(41) 
where

(42) 
are the reduced drift frequencies of the charged particles. In this paper we only investigated the real roots of the algebraic equation of the fourth order (33), because
we want to exclude the possible cases of "aperiodic" instability
or "aperiodic" damping (Alexandrov et al. 1988). In numerical
simulations, we have found that complex roots of Eq. (33) appear at
.
This is why we have taken
as the upper limit of the
interval in (39).
Equation (33) can be solved by the standard Euler method, but for extremely small values of
(actually as
)
it becomes impossible to
obtain the solution in an analytical form. So, we had to make use
of the numerical calculations based on exact formulae (Mishina & Proskuryakov 1962). We have used the roots of the resolvent equation. For all roots to be real and positive, and thus to
obtain all four roots of the MDR (33) as real, the condition

(43) 
should be satisfied for the discriminant D of the resolvent equation (Mishina & Proskuryakov 1962). Here the following notations have been used:
where P, q and r are the coefficients of the fourth order equation in the reduced form
y^{4}  Py^{2} + qy + r = 0.
This equation can be obtained from the initial equation
through the standard transformation
The resolvent equation of the third order has the form
Equation (43) imposes the most stringent restrictions on the main
plasma characteristics of the employed model. However, these
restrictions allow us to consider separately the cases of the
"weak" and "strong" inhomogeneities.
The obtained real roots of the MDR (33) have to be substituted into formula (32) to analyse the expressions for the growth rates. Only if

(44) 
the corresponding waves grow during the linear stage of the process of instability development. The reduced growth rates for the all four roots of the MDR (33) have the following form:

(45) 
with

(46) 

(47) 
where

(48) 
C_{1} 
= 





(49) 



(52) 

(53) 
S_{2} 
= 





(57) 

(58) 

(59) 
Numerical simulation has shown that the requirement of the absence of an imaginary part in the roots of MDR (33) has to be supplemented by the condition

(60) 
Practically, this means that the use of the linear approximation of perturbation theory is valid. These two conditions (43) and (60) allow us to consider separately the cases of "weak" and "strong" inhomogeneity. The case of weak inhomogeneity corresponds to extremely small values of
and extremely large possible values of
and .
At the same time the conditions (5) and (20) hold for
and
respectively, as well as condition (22) for t. Specifically, we assumed that the parameters
,
t,
and
vary in the following ranges:

(61) 

(62) 

(63) 

(64) 
By analogy with
we designated the value of t at which the growth rate becomes positive by
.
For the set of parameters (26) and
we have

(65) 
and

(66) 
In the framework of our model, the need for a numerical solution evidently increases. For very narrow intervals of variations of
and
and constant values
and
,
corresponding areas on the surfaces of the
reduced longitudinal phase velocities
are practically flat. Their local
"topology" (i.e. existence of the local extrema) does not play any
significant role. In this situation the specific ranges for
and ,
where condition (43) is satisfied, as well as specific orientation in "parameter space" of
these "locally flat" surfaces, seem much more important, because
they allow us to determine, in principle, the dispersion law, i.e.
the type of plasma wave. Strictly speaking, this is just an
estimate, but this estimate turns out to be reasonably good.
The calculations show that four roots of the MDR (33) can
be formally split into two pairs. The roots, which we call
"
" and "
waves", have in the ranges
(61)(64) very close (but not the same)
values of
and
,
but opposite signs (
is positive
and
is negative). The
wave can be
easily interpreted as the "right" (i.e. with k_{z} > 0) kinetic
Alfven wave, modified by the pair collisions and largescale
subdreicer electric field. Unfortunately, we cannot interpret the
wave in the same way as the inverse kinetic
Alfven wave. In the framework of the present model we have used in calculations
only
and
as variable parameters to
obtain the maximum effect. We do not know the details of the
polarization of perturbation, so, we cannot say whether the
negative values of
are the result of
incompleteness of our description of perturbation, or whether they
demonstrate the wellknown "parasitic" effect, when negative
frequencies appear in DR (Michailovsky 1963; Alexandrov et al.
1988). For the same reason in the other pair of roots we consider
only the
wave (with positive values) and suppose
that only this root corresponds to a "real" wave. It is hard
enough to determine exactly the dispersion law for this wave. We
can only approximately describe it as a wave from the range of the
slow magnetoacoustic ones, because in our model the dispersion
law for these SMAwaves has the approximate form
.
This wave is definitely modified by the weak
drift motions, pair Coulomb collisions and subdreicer electric
field. It practically does not depend on
(more exactly,
this dependence is very weak) and depends on
in an unusual way in comparison to KAW. This last fact
does not permit us to consider this
wave as the
"exact" SMAwave, even taking into account the corrections to the
dispersion law due to the collisions, subdreicer field and drift
motions. Here we meet an interesting phenomenon: at
,
t = 1 and
("Michailovsky's"case, 1963), without the collisions and electric field, the DR (33) becomes a polynomial of third order. The roots of
this DR are the two Alfven (in our case  the kinetic Alfven)
waves and slow driftAlfven wave (which contains in its dispersion
law the drift frequency (41)(42) as the
factor, and because of this is very slow) (Michailovsky 1963).
When
,
t > 1 and
,
the KAWs are, of course, modified, but remain KAWs. At the same
time the slow drift Alfven wave vanishes, and instead of this the
waves appear, which contain the drift frequency (41)(42) in their dispersion laws not as the small factor, but as the small correction. Figures 1 and 2
show the behaviour of the functions
and
at

(67) 

(68) 
Figures 3 and 4 show the behaviour of the corresponding reduced
growth rates
and
at the same values of
and
as in (67)(68). It can
be easily seen that
,
thus the
wave (modified KAW) damps. At the same time
becomes positive for some values of
and ,
when
and
.
This means that the instability of the
wave has a clearly expressed threshold and can be considered as being an indicator of the
dynamics of preflare process. In a sense the generation of the
wave during the linear stage of the process of instability development can be considered as a forerunner of a
flare in a loop structure.

Figure 7:
The growth rate of the inverse KAWlike instability
at
and
. 
Open with DEXTER 

Figure 8:
The growth rate of inverse SMAWlike instability
at
and
. 
Open with DEXTER 
An interesting situation occurs for k_{z} < 0. The
 and
waves become the physical meaningful ones, and
waves in a sense lose their physical meaning. Figures 5 and 6 show the behaviour of the inverse KAWlike
wave and
the inverse SMAWlike
wave.Their growth rates
and
are shown in Figs. 7 and 8. In this case the
wave plays the role of a forerunner
of a flare. But it has one very important defect: at the same value of
it has too high value of
.
In the framework of our model this is important.
The role of magnetoacoustic waves in the flare models (Priest 1982; Somov
1994; Zaitsev et al. 1994; Zaitsev & Stepanov 1982), especially in the laboratory modelling of this phenomenon (Somov 1994; Zaitsev et al. 1994), is wellknown. There is a considerable
number of investigations (Rosenraukh & Stepanov 1988; Terekhov et al. 2002; Zaitsev & Stepanov 1989) of the pulsations of the flare emission as well as investigations of the transverse waves
in the loop structures (Podgorny & Podgorny 2001; Mel'nikov et al. 2002; Schrijver et al. 2002). From our point of view, the most interesting peculiarities of the present plasma model and of the
results that we have obtained are the following:
 1.
 The instability is of a "nonbeam" type. The only analogue of any kind of a beam in
this plasma model is the "beam" of "runaway" electrons. Due to the extremely small values of
in the case of weak inhomogeneity its influence on the process of instability
development is negligible.
 2.
 The extremely small value of
from (68) points to the fact that we really study a very early stage of the preflare process (see
from (28)).
 3.
 The relatively small value of
from (67) demonstrates that in the framework of this plasma model no considerable preheating near the footpoints is needed
for the generation of an
wave.
 4.
 The small values of the reduced growth rate
definitely
point to the fact that we study a clearly expressed wave process,
with a large number of periods of the generated waves.
Evidently, the generation of an
wave is not enough for the real shorttime prediction of a flare in a loop structure. But it is one of its most important necessary conditions.
Acknowledgements
The authors thank Dr. K. V. Alikajeva and Prof. S. I. Gopasyuk for useful discussions of the present work.

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