# 3. MHD EVOLUTION OF MAGNETIC TUBE DISTURBED BY

Urban and Civil

Nov 16, 2013 (4 years and 5 months ago)

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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
-

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

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

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

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

region

within

the

frame

of

the

considered

flaring

scenario
.