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Fast Magnetohydrodynamic
Waves in Line-tide Solar
Coronal Loops
Antonio Jesús Díaz Medina
Ramón Oliver Herrero
Jose Luís Ballester Mortes
Departament de Física
Universitat de les Illes Balears
SPAIN
Coronal loop oscillations
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• Solar Corona: outer layer of the solar atmosphere.
• Regions with closed magnetic field: plasma confined in loops.
• Loop oscillations currently reported.
• Line-tied (footpoints remain
undisturbed).
• Periods of the order of 5 min.
Aschwanden et al. (2000, 2002)
Loop models
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Loop models
• Straightened loop
• Hydrostatic equilibrium.
• Homogeneous, with chromospheric
layers or static heating profiles.
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MHD equations
• Linearized ideal MHD equations and low-beta plasma
approximation. Two partial differential equations are derived,
⎡ ∂2
2
2⎤
⎢ 2 − c A ( z )∇ ⎥ p T = 0
⎣ ∂t
⎦
⎡ ∂2
⎤
∂p
ρ 0 ⎢ 2 − c A2 ( z )∇ 2 ⎥ v ⊥ = −∇ ⊥ T .
∂t
⎣ ∂t
⎦
• Line-tying boundary condition on the photosphere: v(z=±L)=0.
• Separation of variables leads to the dispersion relations after
applying the suitable boundary conditions at the plasma interfaces:
⎡ 1
⎤
1
(l)
(c)
(l)
(c)
H
B
I
(
λ
b
)
K
'
(
λ
b
)
I
'
(
λ
b
)
K
(
λ
b
)
−
∑
sn s ⎢ ( c ) m
s
m
n
s
m
n
⎥ = 0.
(l) m
λs
s =1
⎣ λn
⎦
∞
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Homogeneous loop
• Dispersion diagrams, frequencies vs. loop radius.
(l)
• Straightforward dependences: hn ( z ) ≈ cos(k n z );
Φ m (ϕ ) = e imϕ .
• Modification from unbounded tube: more than one n family.
sausage modes (m=0)
kink modes (m=1)
Very few modes are trapped!
Fluting modes (m=2)
Cut-off frequencies:
ωcutL (n +1)π
cAc
=
2
• The previous results can be extended to non trapped modes, but
the spatial structure grows exponentially far away from the tube!
.
Effect of chromospheric layers
• Structure along the loop: chromospheric layers of uniform density
(ρch/ρc=200) and given height (Hch/L=0.1).
sausage (m=0)
kink (m=1)
fluting (m≥2)
• Necessary to combine all the modes with
different structure in the z-direction to fulfill
the boundary conditions.
• Only one cut-off frequency for all the modes.
Really few
modes are
trapped!
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Effect of heating
• Dependence on the heating to conduction ratio and the footpoint
density.
ωL/cAc
ωL/cAc
fundamental
kink mode
first kink
harmonic
ρl/ρc
H
ρl/ρc
H
• However, the frequency depends slightly on the footpoint density
for a given value of the density in the summit.
• Reason: the amplitude in the dense part is very small.
• Important result: the density should be measured in the summit.
• Very small differences between the heating models used.
Conclusions
• Few modes are trapped under coronal loops conditions.
• The addition of a very dense chromospheric layer only introduces
slightly corrections because of the restriction of the amplitude
near the footpoints.
• The inclusion of a heating term in the equilibrium varies
significantly the resulting frequencies. It is very difficult to
estimate which is the heating profile from the wave analysis.
• The resulting frequencies are very similar to the ones of a
homogeneous tube with the apex density.
Related publications:
•Díaz, Oliver, Ballester & Roberts 2004, A&A, in press.
•Díaz, Oliver & Ballester, 2004, in preparation
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