2. MATHEMATIC MODEL. SELF

SIMILAR APPROACH
The dynamics of plasma in the magnetic tube disturbed by a beam is described by standard set of
MHD equations, where we take into account the effects of viscosity and thermoconductivity, as well
as the Joule heating and radiative energy losses. In order to provide the initial thermodynamical
equilibrium of the magnetic tube without a beam a stationary uniform background heating Q was
introduced.
The Joule heating term
j
'
2
/
s = (
j

j
b
)
2
/ s
in the energy equation includes as well
the return current caused by the propagating beam of non

thermal electrons.
•
We study various types of the magnetic tube response onto injection of a beam of energetic electrons
using dynamical models of a magnetic tube (Khodachenko, 1996), built on the basis of known self

similar solutions of plasma MHD equations (
Imshennik & Syrovatskii, 1967
). In the cylindrical
coordinate system with
z

axis directed along the magnetic tube the self

similar solution have the
following form:
(
1
)
(2)
(3)
(4)
R
,
L

transverse
and
longitudinal
scales
of
a
studied
fragment
of
a
flaring
magnetic
tube
R
*

external
transverse
scale
of
the
whole
magnetic
structure
(r
d
R
<<
R
*
)
(see
Fig
.
1
)
a(t),
b(t)

dimensionless
functions
of
t
,
characterizing
the
degree
of
plasma
compression
in
the
magnetic
tube
(components
of
the
deformation
tensor)
•
Assuming
small
quantities
:
x
1
=
<<
1
and
x
2
=
<<
1
any
function
of
type
T
a
(r,z,t)
in
the
energy
equation
can
be
presented
as
T
a
(r,z,t)
>
T
0
a
(t)[
1

ax
1

ax
2
]
.
This
causes
an
asymptotic
character
of
our
non

adiabatic
model
(Khodachenko,
1996
)
.
3
.
MHD
EVOLUTION
OF
MAGNETIC
TUBE
DISTURBED
BY
A
BEAM
OF
FAST
ELECTRONS
Modeled
by
the
self

similar
solutions
the
respon

se
of
the
magnetic
tube
onto
a
beam
is
defined
by

initial
equilibrium
plasma
parameters
(T
0
(t=
0
)=
=T
00
,
T
1
(t=
0
)=T
10
,
T
2
(t=
0
)=T
20
,
n
00
=
r
(t=
0
)/m
i
)

c u r r e n t
d e n s i t y
of
the
beam
(d
=
j
b
/
j(t=
0
)
)

characteristic
scales
of
the
model
(R,
L,
R
*
)
Three
general
types
of
dynamics
:
(
1
)
compressional
regime
(
2
)
quasi

periodic,
pulsating
regime
(
3
)
decompressional
regime
The
dynamical
regimes
result
from
the
compe

tition
between
grad
P
and
[j
B]
force,
whereas
the
change
of
temperature
is
determined
by
hea

ting
and
cooling
mechanisms
acting
differently
on
different
dynamical
stages
of
the
m
.
tube
evolution
Consider
1
D
case
with
no
variation
of
quantities
along
the
tube
(b(t)=
1
,
T
2
(t)=
0
)
(see
Fig
.
2
)
Fig
.
2
Dynamics
of
r
(t)
and
T
0
(t)
in
the
tube
with
T
10
/
T
00
=
10

2
,
R
=
3
.
10
7
cm,
during
propagation
of
the
beam
d
=
10
5
for
different
values
of
T
00
:
(a)
3
.
10
5
K
;
(b)
1
.
8
.
10
5
K
;
(c)
10
5
K,
and
n
00
:
(
1
)
10
9
cm

3
;
(
2
)
10
10
cm

3
;
(
3
)
10
11
cm

3
•
Possible
observational
output
:
A)
B)
Pulsating
regimes
can
be
analog
of
flaring
events
with
precursors
4. MHD RELAXATION OF MAGNETIC TUBE AFTER DIS

TURBANCE BY A BEAM OF FAST ELECTRONS
The disturbed values of plasma parameters
and m.field in the tube appear as the initial
conditions for modelling its dynamics with

out the beam.
Fig.3
Oscillatory relaxation of magnetic tube with
n
00
=1.5
.
10
11
cm

3
,
T
10
/
T
00
=
T
20
/
T
00
=10

2
,
R
=
3
.
10
7
cm,
L
=
3
.
R
,
R
*
=
3
.
10
8
cm in dependence on the
T
00
:
(1)
10
7
K; (2)
6
.
10
6
K and
B
j
0
(t=0):
(a)
55
G; (b)
45
G.
Plasma velocities and are normalised here to
.
•
Decreasing
initial
temperature
causes
smoothing
of
pulses
and
decrease
of
the
relaxation
time,
whereas
decrease
of
B
j
0
(t=
0
)
results
in
the
decrease
of
a
number
of
pulses
and
transformation
of
the
pulsating
regime
to
a
monotonous
one
(Fig
.
3
)
.
•
The type of dynamical relaxation of m.tube is not strongly influenced by the initial velocity (Fig.4)
5. OBSERVATIONS AND THE MODEL RESULTS
We
compare
processes
predicted
by
the
model
dynamics
in
a
m
.
tube
with
multichannel
observational
data
from
X

ray
spectrometers
on
the
Solar
Maximum
Mission
(SMM)
satellite
.
•
Soft
X

ray
channels
are
an
indicator
of
the
temporal
behaviour
of
hot
plasma
temperature
•
Hard
X

ray
channels
indicate
an
electron
beam,
interacting
with
low

chromospheric
plasma
Fig.5
Examples of events with various types of
soft X

ray emission dynamics
6. CONCLUSION
Considered
here
MHD
response
of
plasma
in
the
low
coronal
/
upper
chromospheric
part
of
a
flaring
magnetic
loop,
caused
by
propagation
of
a
beam
of
fast
non

thermal
electrons,
can
influence
the
observational
manifestation
of
the
flaring
event
.
Presented
ideas
can
be
used
for
interpretation
of
various
types
of
temporal
behaviour
of
the
flaring
electromagnetic
emission
as
well
as
the
for
explanation
of
possible
changes
of
location
of
the
radiating
source
and
specifics
of
existing
material
flows
.
Preliminary
self

similar
modelling
allows
to
define
the
main
possible
dynamic
regimes
of
the
disturbed
magnetic
tube,
which
depend
strongly
on
the
parameters
of
plasma
and
scales
of
event
.
More
detailed
quantitative
study
of
the
effect
requires
an
extensive
numerical
MHD
simulations
.
REFERENCES
Aschwanden, M.J., Kosugi, T., Hudson, H.S., Wills, M.J., Schwartz, R.A., 1996, ApJ, 470, 1198
Brown, J.C., 1973, Solar Phys., 31, 143
Emslie, A.G., 1996, Eos Trans. AGU, 77(37), 355
Emslie, A.G., 1983, Solar Phys., 86, 133
Imshennik, V.S., Syrovatskii, S.I., 1967, Sov.Phys.JETP, 25, 656
Khodachenko, M.L., 1996, Astronomy reports, 40, No.2, 252
Masuda, S., Kosugi, T., Hara, H., Tsuneta, S., Ogawara, Y., 1994, Nature, 371, 495
Van den Oord, G.H.J., 1990, A & A, 234, 496
Acknowledgements
The authors are thankful to E.Rieger for providing the observational data from X1, X2 spectrometers on SMM
Substitution of self

similar
solution into MHD equations
Grouping the terms
proportional to
0

th, 1

st, and 2

nd
power of r and z
Set of ordinary differential
equations for
a(t), b(t), B
j
0
(t), B
z0
(t),
r
(t), T
0
(t), T
1
(t), T
2
(t)
Compressional cooling
(Figs. 2b(2), 2(a)2, 2(a)3)
Conditions of cold and dense photo

sphere

like plasma are formed on the
higher (chromospheric / low coronal) levels
The source of hard X

ray
bursts shifts towards higher
than photospheric levels
Direct collisional heating of plasma by
the beam becomes possible
Fig
.
4
Relaxation
of
the
tube
with
T
00
=
10
7
K,
B
j
0
(t=
0
)=
30
G,
and
all
other
parameters
as
in
the
case
on
Fig
.
3
for
different
V
r
(r=
0
,t=
0
)
:
(a)
0
cm/s
;
(b)

5
.
10
6
cm/s
;
(c)
10
7
cm/s
•
Initial
flat
phase
in
the
emission
on
Fig
.
5
a
is
similar
to
the
model
regime
on
Fig
.
2
c(
1
)
;
•
Fast
increase
of
emission
on
Fig
.
5
b
is
similar
to
the
regimes
on
Figs
.
2
b(
3
),
2
c(
2
),
2
c(
3
)
.
•
The
fact
that
the
second
beam
on
Fig
.
5
b
has
no
influence
on
the
soft
X

ray
emission
can
be
explained
by
the
decompressional
heating
regime
(Figs
.
2
c(
2
),
2
c(
3
))
triggered
in
the
tu

be
by
the
first
beam
.
T
he
second
beam
here
can
not
disturb
the
system
effectively,
and
it
still
does
not
interact
collisionally
with
the
surrounding
plasma
.
•
Oscillatory
decrease
of
emission
on
Fig
.
5
c
is
similar
to
the
regimes
on
Fig
.
3
.
•
Fast
decrease
of
emission
on
Fig
.
5
d
looks
like
the
regimes
on
Fig
.
4
The
similarity
between
the
registered
radiation
of
flaring
events
and
the
plasma
temperature
dynamics
in
our
model
is
just
an
indication
of
their
possible
mutual
connection
.
Further
analy

sis
supposes
a
detailed
study
of
complex
condi

tions
in
the
radiating
region
within
the
frame
of
the
considered
flaring
scenario
.
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