Formation of Current Sheets in Magnetohydrostatic .../67531/metadc674232/m2/1/high... · Formation...

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AGU 1996 Fall Meeting

Formation of Current Sheets in

Magnetohydrostatic Atmospheres

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G. S. Choe and C. Z. Cheng

Plasma Physics Laboratory, Princeton University

Princeton, NJ 08543-0451

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DISCL A~MER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liabili- ty or mponsibility for the accuracy, completeness, or usefulness of any information, appa- ratus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessar- ily state or reflect those of the United States Government or any agency thereof.

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ABSTRACT

It is demonstrated that a 2-D magnetic field configuration in a magnetohy-

drostatic equilibrium without any nullpoint can be deformed into a configuration

with current sheets, ie., tangential discontinuities, either by temperature change

or by footpoint displacement.

The magnetohydrostatic solutions by Low (A & Ap, 1992), which have a

quadrupolar field geometry, are chosen as our initial configurations.

When the whole atmosphere is uniformly heated, the expansion of plasma is more effective in the outer flux tubes than in the inner ones. The expanding

plasma pushes out the field lines in each bipolar regon so that a current sheet of a finite length is formed where the field lines from each region come into

contact. The resulting pressure profile at the base has pressure maxima st the

center of each bipolar regions. The smooth equilibrium solution with the same pressure distribution contains an X-point.

If the pressure is initially higher in the outer tubes than in the inner ones,

cooling of the atmosphere can also lead to current sheet formation. A s the

pressure scale height decreases by cooling, the magnetic field pressure dominates

the plasma pressure in the upper part of the flux tubes. The subsequent expamion

of field lines creates a tangential discontinuity. If resistivity is considered jn this weak equilibrium state, magnetic reconnection results in a new Kippenhahn-

Schliiter type field configuration with a magnetic island. It is expected that a

prominence can stably reside within the magnetic island. When the field footpoints undergo a shearing motion with a continuous

shearing profile, a current sheet can be formed beyond a critical amount of

shear.

Our results suggest that the formation of a Current sheet and the subse:quent magnetic reconnection can be ubiquitous in the solar atmosphere. The resulting field configurations are quite favorable for prominence formation.

1

MOTIVATION OF THE STUDY

0 A current sheet is a pre-condition for magnetic reconnection, which.is considered to play a central role in solar flares and

coronal heating.

0 Solar prominences are likely to form in current sheets; be-

cause they are mostly observed above the polarity inversion lines, especially between two bipolar regions.

0 Considering the commonness of the above phenomena, cur-

rent sheet formation must be quite a general process, but is

not well understood.

GOAL OF THE STUDY

We construct MHS equilibria with current sheets of different ge- ometry by various physical processes. Since a field configuration

with a null-point tends to be deformed to a current sheet, we start

from a smooth equilibrium without any null-points. Specifically,

we consider two different types of change in the environment of

the system.

1.

2.

Thermodynamic Changes How does the equilibrium change when the system is either

heated or cooled? H a current sheet is to form, what is the

critical temperature at which the system enter the weak equi- librium state?

Motion of Field Line Footpoints at the Boundary Can a current sheet be formed if the boundary motion is

continuous in space so that no discontinuity in field line con-

nectivity is allowed? What is the critical amount of footpoint displacement?

I

be in

If f i e ffie

a.

, 4

Case 1 A

T) T O

b e. dd ‘7 A

A-NALYTIC AL SOLUTION OF lVlHS ~ I L I B R I U M

Linear Grad-Shafranov Equation for MHS Equilib- rium.

dP z

dA H V2A + -exp(--)= 0

P ( A ) = ~ ( ~ 4 : z = 0) = aA2 + p i *

p ( A , Z ) = P(A)exp( -L) H

- 0 Solutions With a Periodic Quadrupolar Field Geom-

1.5

I .o 2

etry (Low, 1992) - (i) Potential Field ( a = 0 )

0.5

0.0

A ( x , z ) = - Bo [exp(-kz)coskx-a~exp(-3kz)cor33kx] k

-1.0 -0.5 0.0 0.5 I .o -0.5 0.0 0.5 1.0

x / * X / r

Fig. la and b. The potential held5 generated by Eq. i 12) fora (I, = r) 2.' 2nd b 0 = 9.-0. 1 ~ 1 t h 4 = 0 5 The mayef ir lincsofforce shown ;Ue @m!Wn of constant sm3m funcrion A k I D , , drawn 31 a constant increment in 4 s I S evident from the contour values given. In a. the shaded regions b i p d x fields

bounded b! thc separatnx line o i force Ak;B,, = 0.75. in b. the & q h d lines are lines of force additional to those provided P the consat k - e m e n t in AklBu. which Serve to bring out the presence of an X-type magnetic ncuini point

(ii) a < 0; Pressure Depletion in the Lower Arcades

where s = 5 2 k H and q2 = 8aH2

1.5

I .o z

0.5

0.0 -1.0 -0.5 0.0 0.5 1.0 -0.5 0.0 0.5 1.0

?/IT WT

Fig . 2a and b. The case of u = 0.25. u = - i j 8 l r showing a the non-potential, equilibrium magnetic field comprcsscd downward relative 10 he initial potential held. 3nd b [he contours of constant. negative depyrure L p of the pressurr from the initial plane-parallel dismtution. In this ,3nd subsequent figures. =e set f = H = 0.5. In a. the magnetic lines of force shown as thick m w e d lines are superposed upon the lines of forrr (thin. unarmtwed lines) Of the initial held. The same set of lines of force are drawn so that each corrcsponding pair of lines of force from chc initial and fid smts M shown roofed to the same point at the txmdary. as an aid to visualization of the mapciic tieid svoiucion.fhe contours of A p in b are ploned at a consant inm:ment in some unit

2

1.5

I .3

C. 5

nn

(iii) a > 0; Higher Pressure in the Lower Arcades

coskx A ( x , i ) = - Bo ( J s ( q e x p ( - & ) )

k Js (Q)

cos3 kx J3s (Q exp ( - &- ) ) J3s ( 4 )

- a3

"." -1.0 - 0.5 0.0 0.5 I .o -0 5 0.0 0.5 1.0

Y/x X / r

NUMERICAL METHODS

1. Cases 1A and 1B

To obtain an MHS equilibrium for a given temperature, a

magnetofrictional method is employed as follows.

= -v (p.;) 0 at -

4 - = -c V A

T = const.

dA at

where Q is a friction coefficient.

2. Case2

r3

To follow the quasi-static evolution of the magnetic field, a time-dependent MHD simulation is performed with a ffoot-

point velocity as a boundary condition.

Field Lines

I

= 2

T = 1.0 To

- 2 - 1 0 1 2

X

0

N 1

-2 -1 0 1

X 2

Field Lines T = 1.5 T,, 2 "

t

N I

0

-2 - 1 0 i 2 X

Field Lines T = 2.0 'L 2

0 -2 - 1 0 I 2

Field Lines

X

T = 2.5 "&

- 2 - 1 0

X 1 2

Field Lines T = 3.0 To 2

3 -2 - 1 0 1 2

X

Field Lines T = 5.0 To 2

N l

3 -2 - 1 0

X 1 2

a

4

2 2

2 P

a 0

4 0

rt;

I

1 2 3 4 c

3 6 7 8 9 10

1 J

A

0 0

Fieid Lines T = 1 0 0 T, I ? 12

S 6 -

- 2 - 1 0 1 x/ 7l

- 2 - 1 i 2 x s n

P / P * T = 1.00 To 12

(i

N 6 -

1 - , -2 -1 0 1 2

x / n

Field Lines T = 0.50 To

I / 12

-2 - 1 0 1 2

x / n

Field Lines T = 0 2 5 TO

- , - 2 3 2 x z

0

*l

Field Lines T = 0.iO To

I - 0- I '

,,<;

i \* P ". .-..

0

0

- 2 - 1 0 1 2

x/n

1.0

0.0 0 1

X/7T

2

Field Lines T = 0.16 To

0 - 2 -1 0 1

-1 0 X / k

2

0 . 3

Field Lines

T = 0.13 To

-0 .4 0.0 x

0.4

4 0

4 0

- 1 0 X

Field Lines i = 1.0 I

2)

I -I 0 I

x

Y r-- (D fb cs-

d-

X 5

Z C

i!

0 ...

L

-JY -4 0 4 0

x

L

-50 0 -Jy 3000

‘ I

SUMMARY

We show that a magnetohydrostatic equilibrium can evolve into a

confi,ouration with a tangential discontinuity in various ways even though there exists no null-point in the initial field confi,ouration.

1. Thermodynamic Variations

0 M e n an atmosphere embedding a quadrupolar magnetic

field is heated, the plasma pressure increases most iin the center of the lowerlying arcades. This plasma pressure

pushes the arcades towards each other to form a current sheet.

0 When the magnetic field in equilibrium is confined by the

outside plasma pressure, drainage of material by plasma

cooling can cause expansion of the magnetic field i a the

upper atmosphere, where a current sheet c m be fonmed.

2. Footpoint Motions

0 Even when the footpoint motion is spatially continuous,

expansion of the lowerlying arcades by shearing can ex- pel the field of the overlying arcade and form a tangential

discontinuity.