Warwick PX420 Solar MHD 2007-2008: Oscillations of Quiescent Prominences
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Oscillations of Quiescent Prominences
1. Equilibrium
A quiescent prominence is a huge, almost vertical sheet of dense, cool plasma surrounded
by a hotter and rare coronal environment.
The basic modes of oscillation of a prominence sheet are closely analogous to the modes
of oscillation of a stretched elastic string of non-uniform density.
The string analogy is most simply illustrated for the case of two uniform strings of differing
densities joint together:
ρp
ρc
−L
x
−a
0
a
L
Consider a section of string, extending from x = −a to x = +a, of density ρp and
Alfvén speed CAp (A PROMINENCE) joined to string of density ρc and speed CA0 (THE
CORONA); the ends are tied at x = ±L, where the field line reaches the base of the
corona.
The total pressure balance along the string prescribes
2
2
CA0
ρc = CAp
ρp .
(1)
(Derive this condition from the total pressure balance.)
2. Equations for Perturbations
Transverse perturbations of the magnetic string (say, in the y-direction) are described by
the Alfvén wave equation, which contains information about the medium.
Outside the prominence, in the coronal part of the magnetic string:
d2 Vy(c)
ω 2 (c)
+
Vy = 0,
2
dx2
CA0
(2)
Warwick PX420 Solar MHD 2007-2008: Oscillations of Quiescent Prominences
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where the index (c) means “in the corona”.
Inside the prominence,
d2 Vy(p)
ω 2 (p)
+ 2 Vy = 0,
dx2
CAp
(3)
where the index (p) means “in the prominence”.
At x = ±a, solutions of these equations should be matched with each other by boundary
conditions. In the case of the purely transverse waves, we require
and
Vy(c) (x = ±a) = Vy(p) (x = ±a)
(4)
dVy(c)
dV (p)
(x = ±a) = y (x = ±a)
dx
dx
(5)
At the base of the corona, the line-tying boundary conditions are
Vy(c) (x = ±L) = 0.
(6)
3. Solutions of Governing Equations
Solutions of wave equation (2) outside the prominence, in the corona, satisfying line-tying
boundary conditions (6) are
¸
·
Vy(c)
ω
(x + L) ,
= A sin
CA0
·
in
− L < x < −a,
(7)
a < x < L,
(8)
¸
ω
(x − L) ,
CA0
where A and B are arbitrary constants.
Vy(c) = B sin
in
Inside the prominence, in −a < x < a, the solution of wave equation (3) is
Ã
Vy(p)
!
Ã
!
ω
ω
= C cos
x + D sin
x ,
CAp
CAp
(9)
where C and D are arbitrary constants. This solution is a sum of an even function and
an odd function.
In the following, we restrict our attention to the even (symmetric) solutions only, with
D = 0,
Ã
!
ω
(p)
x .
(10)
Vy = C cos
CAp
This mode of oscillations is called a kink mode. The kink mode is characterized by the
displacement of the centre of the prominence.
If we choose the odd solution, it would be a sausage mode. This mode does not perturb
the centre of the prominence.
Warwick PX420 Solar MHD 2007-2008: Oscillations of Quiescent Prominences
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4. Derivation of Resonant Conditions
The arbitrary constants A, B and C have to be chosen to satisfy boundary conditions (4)
and (5).
First of all, we need the derivatives of the solutions:
·
¸
dVy(c)
ω
ω
=A
cos
(x + L) ,
dx
CA0
CA0
in
·
¸
dVy(c)
ω
ω
cos
(x − L) ,
=B
dx
CA0
CA0
Ã
dVy(p)
ω
ω
= −C
sin
x
dx
CAp
CAp
− L < x < −a,
(11)
a < x < L,
(12)
in
!
in
− a < x < a.
(13)
Applying boundary conditions (4) and (5), we obtain
at x = −a:
·
"
¸
#
ω
ω
A sin
(−a + L) = C cos
(−a) ,
CA0
CAp
·
(14)
"
¸
#
ω
ω
ω
ω
A
cos
(−a + L) = −C
sin
(−a) ,
CA0
CA0
CAp
CAp
at x = a:
"
¸
·
#
ω
ω
(a − L) = C cos
a ,
B sin
CA0
CAp
·
"
¸
(15)
(16)
#
ω
ω
ω
ω
B
cos
(a − L) = −C
sin
a .
CA0
CA0
CAp
CAp
(17)
From equations (14) and (16), it follows that
A = −B,
(18)
(indeed, the mode is symmetric!). With this relation, equations (15) and (17) become
identical.
Finally, we express A from equation (14),
Ã
!
·
¸
ω
ω
A = C cos
a / sin
(L − a) .
CAp
CA0
(19)
Substituting A to equation (15), we obtain, with the use of (1), the resonant condition
Ã
ωa
tan
CAp
!
s
=
·
¸
ρc
ω
cot
(L − a) ,
ρp
CA0
(20)
which expresses the frequency ω through the parameters of the prominence and the corona.
Warwick PX420 Solar MHD 2007-2008: Oscillations of Quiescent Prominences
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4. The Global Mode Frequency
It is convenient to rewrite resonant condition (20) in the dimensionless form,
tan Ω − R cot [RΩ(d − 1)] ,
(21)
with the dimensionless parameters
ωa
Ω=
,
CAp
s
R=
ρc
,
ρp
d=
L
.
a
(22)
The equation has a number of solutions, corresponding to different kink modes. The
lowest frequency mode is called the global mode.
The figure shows dependence of the q
frequency of the global mode on the ratio of densities
outside and inside the prominence, ρc /ρp , for three different ratios L/a:
The solid curve corresponds to L/a = 30, dotted to L/a = 60 and dashed to L/a = 90.
Now, let us estimate the period of global mode oscillations. A typical parameters of a
prominence are
half-width a
shoulder L
Alfvén speed CAp
density ρp
magnetic field
3 Mm
100 Mm
75 km/s
2 × 10−10 kg m−3
12 G
In the corona, the Alfvén speed can be taken as 1000 km/s, which gives us
q
ρc /ρp ≈ 0.27
and
L/a ≈ 33.3.
Warwick PX420 Solar MHD 2007-2008: Oscillations of Quiescent Prominences
Thus, determining
Ω=
ωa
≈ 0.13,
CAp
we estimate the period of the global mode oscillations as
P =
2πa
2π × 3000
≈
≈ 1933 s ≈ 32 min.
ΩCAp
0.13 × 75
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