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Criteria for the stability of a line-tied magnetohydrostatic equilibrium in the solar corona
Adv. Space Res. Vot. 6. Nio. 6. pp. 49—52. 1986 Printed in Great Britain. All rights reserved.
0273—117756 5000 # .50 Copyright © COSPAR
CRITERIA FOR THE STABILITY OF A LINE-TIED MAGNETOHYDROSTATIC EQUILIBRIUM IN THE SOLAR CORONA J. Me1viIIe,~A. Hood** and E. R. Priest** ~Department of Mathematics, Napier College, Colinton Road, Edinburgh, Scotland **Departmentof Applied Mathematics, The University, St. Andrews, Scotland ABSTRACT Arcades of loop structures in the solar corona have been associated with the onset of solar flares. Changes in the plasma and/or magnetic pressure could initiate a flare if the equilibrium structure becomes unstable. It is shown that for a model magnetohydrostatic equilibrium, if the plasma 8 > 6’~ where 8* is the 8—value for which a magnetic island just appears on the photosphere, then the closed field lines and some field lines tied to the photosphere are unstable to localised linear perturbations. If the field remains unsheared by photospheric motions, then the condition B < 8* for stability is necessary and sufficient. INTRODUCTION The sun exhibits magnetised plasma structures which appear to remain in a stable State on time-scales of days, weeks or more until some process gives rise to an explosive event such as a solar flare. Here, an energy metl~odis used to examine a theoretical magnetohydro— static equilibrium for ideal stability. For discussion of feasible mechanisms of flare development and recent work on stability theory, see /1,2,3,4/. The equilibrium studied here, which includes the gravitational force, may model an arcade of loops in the solar corona for which variations along the length of the arcade are assumed weak enough to be ignored. Consequently a two—dimensional geometry is used. Furthermore, the atmosmhere is taken to be isothermal, the component of the magnetic field in the direction of the ignorable coordinate (the axial field component) is taken to be zero, and so the field is unsheared by motions of the line—tied footpoints. The equilibrium is described followed by the stability theory and, finally, the results are given and discussed. EQUILIBRIUM Taking the y—axis as vertical, the z—axis along the length of the arcade, and scaling lengths against the pressure scale height H RT/pg, the aforementioned assumptions imply that the magnetic field may be written as B = (8A/8y, —8A/~x, 0) , where A = A)x,y) is a flux function which satisfies 2A +
V
~6
~(F(A)e~’)
=
0
(1)
,
where p = F)A)e~’ is the plasma pressure and F(A) is a function whose form depends on the photospheric boundary conditions /5/. The variables are all dimensionless and the parameter B = 2pP 2, where ~o and B 0/B0 0 are characteristic values of the pressure and field strength respectively. Increasing B sufficiently slowly to preserve equilibrium corresponds to increasing and/or lowering the magnetic pressure. Thus the stability of the equilibrium to variations may be studied. 2, equation (1) has a solution of the form x sin (2Bt~e y/2) When F)A) = A A=A 0cos— ~ ~ where Ag is a constant (see /6/). The field lines are contours 1(b) show field lines for two values of B. It is evident that, magnetic islands exist above the photosphere.
49
the plasma
S
of constant A. Figs. 1(a), if B is sufficiently large,
50
J. Melville. A. Hood and E. R. Priest
-2
-2
Fig. 1(a). The equilibrium magnetic field for 23~ = 0.5. The field lines are all connected to the photos~here.
Fig. 1(b). The equilibrium magnetic field for 25~ = 2.0. The structure now manifests magnetic islands.
STABILITY THEORY A coronal displacement =
—p
—
~o=3 —
(I
3
may be expressed as
B
+
,
z—z_
3+3 i —
where
B
-~---,andB=—. JvA~ — B
Clearly is the component of 3 in the direction of the field (the s-direction) whereas and 3 are transverse components. Writing A 1 = —B41, the second order change in the poten~ia1 energy may be written/4,7/
1~!’~~ )
_L 20 J
=
+
+
B +
(B ~
+ y~o)
BOs
~
B~ + yuo~83 —~ ~—z B ez B)B
(ID
+ f(z,
[8y/H7
—o
,Ao)
=
(y-l)
~0p[
(7.4-o )
±
7.4
=
+
rB(7.4~0 )
—72A,
U -
÷JA1 }2
B
—
j2
eY/H~ (~ e1~l --p
The criterion for stability is that 3w then the ecuilibrium is unstable /8/.
+ VA.VA1
—o
—m
-
where the axial current density J)A,y)
f)4
,A1)J
(r 5Y/H(
+ y.i~)
f’up[e~~7.)3 e~Y~)
-~
+ B~—~Z2
B
=
~
B
+ CA.V.1 -
2}}dAdd
j + JA1~ —
and
~,
(7A.7A
2
1 + JA1)}/(B ~ 0 for all forms of 4.
If
(2)
+
ypo)
6w
<
0 for just one
(3)
4
To oerftrc the stability analysis it is therefore necessary to seek perturbations which will minimise 3W.
with
To minimise respect to 4 , note that the right hand side of (3) dOes not contain 4 and therefore expressinc the ~omoonent~of 4 as single Fourier components of wave_numberZk by the prescriptions /9/, A
-
~z one can set
94 z /z
A
=
1(A,S)
cos kz,
=
(A,S)
cos kz,
=
4 z (A,S)
cos kz,
=
—f (A0
,
Furthermore, since 3 1/k, then 83 /Ss 0 as k fourth terms mi (2) ~rop Out in. the ~arge k limit. in the form /7/
~.
Consequently both the third and 3w, without these terms may now be cast
Stability Criteria of a Line-Tied Equilibrium
6w
2
+
1 Jl(BA]
=
+
bpB(Y
(Bz+y,1p)~]
+
~
-
+
-
51
-
~‘i~•’~~
-
(4)
and since there are no derivatives with respect to A the stability of the equilibrium due to three—dimensional perturbations localised about any particular field line, identified by a value of A, may be determined. A variational method is now needed to perform the minimisation with respect to and At. These perturbations which correspond to a marginally stable state (6w = b) are the solutions of the two Euler-Lagrange equations: y-~-(CP) By~[~
C
wnere -.
~
and
=
~
+
~-
(y
-
1) C-~-(CQ
-
=
0,
(5)
UA 1
B
-
P
-
+
±
+
(Y~liC~~[~:
1]Q
+
+
(P+Q)U}
=
o~
(6)
2 +yup JOB
B~_[~fl]
-
B
A
~
Qj~y~)
-
2B
~~7X
Equations (5) and (6) were solved along field lines which are tied to the photosphere /7/. The boundary condition at the footpoints was taken to be 4 = 0 (equivalently A1 = = 0 on the photospherel . For each integration B—values were found which gave solutions for both odd and even modes of At. The analysis was performed for various tied field lines for the least stable case (y = 1) and for the case when the hydrogen plasma is fully ionised = 5/3) giving the marginal stability curves in Figs. 2 and 3.
L
1
o d
even
2v~ Fig.
2. Marginal stability curves lsolid) for both the odd and even modes of the transverse perturbation component Ag (case y = 1) . The dotted curve is the island curve
0
d
~V~0
2~ Fig. 3. Marginal stability curves and the island curve in case y = 5/3.
)see text)
DISCUSSION OF RESULTS 2/A 2) and 2B~appear naturally in the mathematics of the problem and accordingly they The dotted curve The parameters ~2 were ~= A used 3 to plot the stability curves (solid lines). gives information about the formation of magnetic islandB. The maximum of this curve defines the maximum value of A2 and also the value of 2B~ for which the first magnetic island appears at the photosphere. Denoting the associated B—value as B~ it is seen that for 5 ~ 5* the field is stable to both even and odd modes of A 0 and 1. This is 2 a between necessary and and its maximum the island curve gives B—values. The lower value is that sufficient condition for stability only two when Bz = 0. For each of Afor which the ‘tip’ of the field line just appears at the photosphere. The upper 8—value determines when the field line just closes to form a magnetic-island. Reconnection of field lines must occur or. a time—scale much shorter than that for the quasi—static variation. Note that points ‘under’ the island curve identify field lines which are connected (tied) to the photosphere, and points to the right of it and below the dashed curve identify closed field lines (islands) . It is clear that for B > B~the closed lines and some tied lines are unstable, the extent of the latter increasing as B increases. For the least stable
J. Melsitte. A. Hood and E. R. Priest
52
case Ky =1) the odd modes of A about 0.22, but below tht~ value the reverse is true. 2—values greater than In 1 are less stable than the even ones. For = 5/3 this is general true for the A results suggest that the line—tying in the dense chotosphere is an important stabilising mechanism and, when closed field lines exist, the equilibrium is unstabe.
REFERENCES I.
J. Birn and K. Schindler, Two—ribbon flares: magr.etOstatic equilibria, in Solar Flare Magnetohydrodynamics, ad. E. Priest, Gordon and Breach, 337 (1981)
2.
A. Hood, An energy method for the stability of solar macnetohydrostatic atmospheres, Geophys. Astrophys. Fluid Dynamics 28, 223 (1984).
3.
A. Hood, The stability of magnetic fields relevant to two-ribbon flares, Adv.Soace Des. 4, 49 (1984).
4.
K. Schindler, .3. Birn and L. Janicke, Stability of two dimensional preflare structures, Solar Phys. 87, 103 (1983) -
5.
J. Dungev, A family of solutions of the magnetohydrostatic problem in a conducting atmosphere in a gravitational field, Mon. Not. Roy. Astron. Soc. 113, 150 (1953).
6.
E. Zweibel and A. Hundhausen, Magnetostatic atmospheres: A family of isothermal solutions, Solar Phys. 76, 261 (1982)
7.
J. Melville, A. Hood and E. priest, The ideal macnetohydrodynamic stability of a line— tied coronal macnetohydrostatic equilibrium, Solar Pho’s. (in press).
8.
I. Bernstein, E. Frieman, H. Kruskal and R. Kulsrud, An energy principle for hydromagnetic stability problems, Proc. Roy. Soc. A244, 17 (1958)
9.
E. Zweibel, MHD instabilities of atmospheres with magnetic fields, Astrcphys.J. 731 (1981)
249,
0273—117756 5000 # .50 Copyright © COSPAR
CRITERIA FOR THE STABILITY OF A LINE-TIED MAGNETOHYDROSTATIC EQUILIBRIUM IN THE SOLAR CORONA J. Me1viIIe,~A. Hood** and E. R. Priest** ~Department of Mathematics, Napier College, Colinton Road, Edinburgh, Scotland **Departmentof Applied Mathematics, The University, St. Andrews, Scotland ABSTRACT Arcades of loop structures in the solar corona have been associated with the onset of solar flares. Changes in the plasma and/or magnetic pressure could initiate a flare if the equilibrium structure becomes unstable. It is shown that for a model magnetohydrostatic equilibrium, if the plasma 8 > 6’~ where 8* is the 8—value for which a magnetic island just appears on the photosphere, then the closed field lines and some field lines tied to the photosphere are unstable to localised linear perturbations. If the field remains unsheared by photospheric motions, then the condition B < 8* for stability is necessary and sufficient. INTRODUCTION The sun exhibits magnetised plasma structures which appear to remain in a stable State on time-scales of days, weeks or more until some process gives rise to an explosive event such as a solar flare. Here, an energy metl~odis used to examine a theoretical magnetohydro— static equilibrium for ideal stability. For discussion of feasible mechanisms of flare development and recent work on stability theory, see /1,2,3,4/. The equilibrium studied here, which includes the gravitational force, may model an arcade of loops in the solar corona for which variations along the length of the arcade are assumed weak enough to be ignored. Consequently a two—dimensional geometry is used. Furthermore, the atmosmhere is taken to be isothermal, the component of the magnetic field in the direction of the ignorable coordinate (the axial field component) is taken to be zero, and so the field is unsheared by motions of the line—tied footpoints. The equilibrium is described followed by the stability theory and, finally, the results are given and discussed. EQUILIBRIUM Taking the y—axis as vertical, the z—axis along the length of the arcade, and scaling lengths against the pressure scale height H RT/pg, the aforementioned assumptions imply that the magnetic field may be written as B = (8A/8y, —8A/~x, 0) , where A = A)x,y) is a flux function which satisfies 2A +
V
~6
~(F(A)e~’)
=
0
(1)
,
where p = F)A)e~’ is the plasma pressure and F(A) is a function whose form depends on the photospheric boundary conditions /5/. The variables are all dimensionless and the parameter B = 2pP 2, where ~o and B 0/B0 0 are characteristic values of the pressure and field strength respectively. Increasing B sufficiently slowly to preserve equilibrium corresponds to increasing and/or lowering the magnetic pressure. Thus the stability of the equilibrium to variations may be studied. 2, equation (1) has a solution of the form x sin (2Bt~e y/2) When F)A) = A A=A 0cos— ~ ~ where Ag is a constant (see /6/). The field lines are contours 1(b) show field lines for two values of B. It is evident that, magnetic islands exist above the photosphere.
49
the plasma
S
of constant A. Figs. 1(a), if B is sufficiently large,
50
J. Melville. A. Hood and E. R. Priest
-2
-2
Fig. 1(a). The equilibrium magnetic field for 23~ = 0.5. The field lines are all connected to the photos~here.
Fig. 1(b). The equilibrium magnetic field for 25~ = 2.0. The structure now manifests magnetic islands.
STABILITY THEORY A coronal displacement =
—p
—
~o=3 —
(I
3
may be expressed as
B
+
,
z—z_
3+3 i —
where
B
-~---,andB=—. JvA~ — B
Clearly is the component of 3 in the direction of the field (the s-direction) whereas and 3 are transverse components. Writing A 1 = —B41, the second order change in the poten~ia1 energy may be written/4,7/
1~!’~~ )
_L 20 J
=
+
+
B +
(B ~
+ y~o)
BOs
~
B~ + yuo~83 —~ ~—z B ez B)B
(ID
+ f(z,
[8y/H7
—o
,Ao)
=
(y-l)
~0p[
(7.4-o )
±
7.4
=
+
rB(7.4~0 )
—72A,
U -
÷JA1 }2
B
—
j2
eY/H~ (~ e1~l --p
The criterion for stability is that 3w then the ecuilibrium is unstable /8/.
+ VA.VA1
—o
—m
-
where the axial current density J)A,y)
f)4
,A1)J
(r 5Y/H(
+ y.i~)
f’up[e~~7.)3 e~Y~)
-~
+ B~—~Z2
B
=
~
B
+ CA.V.1 -
2}}dAdd
j + JA1~ —
and
~,
(7A.7A
2
1 + JA1)}/(B ~ 0 for all forms of 4.
If
(2)
+
ypo)
6w
<
0 for just one
(3)
4
To oerftrc the stability analysis it is therefore necessary to seek perturbations which will minimise 3W.
with
To minimise respect to 4 , note that the right hand side of (3) dOes not contain 4 and therefore expressinc the ~omoonent~of 4 as single Fourier components of wave_numberZk by the prescriptions /9/, A
-
~z one can set
94 z /z
A
=
1(A,S)
cos kz,
=
(A,S)
cos kz,
=
4 z (A,S)
cos kz,
=
—f (A0
,
Furthermore, since 3 1/k, then 83 /Ss 0 as k fourth terms mi (2) ~rop Out in. the ~arge k limit. in the form /7/
~.
Consequently both the third and 3w, without these terms may now be cast
Stability Criteria of a Line-Tied Equilibrium
6w
2
+
1 Jl(BA]
=
+
bpB(Y
(Bz+y,1p)~]
+
~
-
+
-
51
-
~‘i~•’~~
-
(4)
and since there are no derivatives with respect to A the stability of the equilibrium due to three—dimensional perturbations localised about any particular field line, identified by a value of A, may be determined. A variational method is now needed to perform the minimisation with respect to and At. These perturbations which correspond to a marginally stable state (6w = b) are the solutions of the two Euler-Lagrange equations: y-~-(CP) By~[~
C
wnere -.
~
and
=
~
+
~-
(y
-
1) C-~-(CQ
-
=
0,
(5)
UA 1
B
-
P
-
+
±
+
(Y~liC~~[~:
1]Q
+
+
(P+Q)U}
=
o~
(6)
2 +yup JOB
B~_[~fl]
-
B
A
~
Qj~y~)
-
2B
~~7X
Equations (5) and (6) were solved along field lines which are tied to the photosphere /7/. The boundary condition at the footpoints was taken to be 4 = 0 (equivalently A1 = = 0 on the photospherel . For each integration B—values were found which gave solutions for both odd and even modes of At. The analysis was performed for various tied field lines for the least stable case (y = 1) and for the case when the hydrogen plasma is fully ionised = 5/3) giving the marginal stability curves in Figs. 2 and 3.
L
1
o d
even
2v~ Fig.
2. Marginal stability curves lsolid) for both the odd and even modes of the transverse perturbation component Ag (case y = 1) . The dotted curve is the island curve
0
d
~V~0
2~ Fig. 3. Marginal stability curves and the island curve in case y = 5/3.
)see text)
DISCUSSION OF RESULTS 2/A 2) and 2B~appear naturally in the mathematics of the problem and accordingly they The dotted curve The parameters ~2 were ~= A used 3 to plot the stability curves (solid lines). gives information about the formation of magnetic islandB. The maximum of this curve defines the maximum value of A2 and also the value of 2B~ for which the first magnetic island appears at the photosphere. Denoting the associated B—value as B~ it is seen that for 5 ~ 5* the field is stable to both even and odd modes of A 0 and 1. This is 2 a between necessary and and its maximum the island curve gives B—values. The lower value is that sufficient condition for stability only two when Bz = 0. For each of Afor which the ‘tip’ of the field line just appears at the photosphere. The upper 8—value determines when the field line just closes to form a magnetic-island. Reconnection of field lines must occur or. a time—scale much shorter than that for the quasi—static variation. Note that points ‘under’ the island curve identify field lines which are connected (tied) to the photosphere, and points to the right of it and below the dashed curve identify closed field lines (islands) . It is clear that for B > B~the closed lines and some tied lines are unstable, the extent of the latter increasing as B increases. For the least stable
J. Melsitte. A. Hood and E. R. Priest
52
case Ky =1) the odd modes of A about 0.22, but below tht~ value the reverse is true. 2—values greater than In 1 are less stable than the even ones. For = 5/3 this is general true for the A results suggest that the line—tying in the dense chotosphere is an important stabilising mechanism and, when closed field lines exist, the equilibrium is unstabe.
REFERENCES I.
J. Birn and K. Schindler, Two—ribbon flares: magr.etOstatic equilibria, in Solar Flare Magnetohydrodynamics, ad. E. Priest, Gordon and Breach, 337 (1981)
2.
A. Hood, An energy method for the stability of solar macnetohydrostatic atmospheres, Geophys. Astrophys. Fluid Dynamics 28, 223 (1984).
3.
A. Hood, The stability of magnetic fields relevant to two-ribbon flares, Adv.Soace Des. 4, 49 (1984).
4.
K. Schindler, .3. Birn and L. Janicke, Stability of two dimensional preflare structures, Solar Phys. 87, 103 (1983) -
5.
J. Dungev, A family of solutions of the magnetohydrostatic problem in a conducting atmosphere in a gravitational field, Mon. Not. Roy. Astron. Soc. 113, 150 (1953).
6.
E. Zweibel and A. Hundhausen, Magnetostatic atmospheres: A family of isothermal solutions, Solar Phys. 76, 261 (1982)
7.
J. Melville, A. Hood and E. priest, The ideal macnetohydrodynamic stability of a line— tied coronal macnetohydrostatic equilibrium, Solar Pho’s. (in press).
8.
I. Bernstein, E. Frieman, H. Kruskal and R. Kulsrud, An energy principle for hydromagnetic stability problems, Proc. Roy. Soc. A244, 17 (1958)
9.
E. Zweibel, MHD instabilities of atmospheres with magnetic fields, Astrcphys.J. 731 (1981)
249,