5 High magnetic field case

We now consider a situation which is more realistic as far as the solar atmosphere is concerned. We consider an isothermal atmosphere extending over several scale heights for which $v_{\rm A} \gg c_{\rm S}$ over most of the atmosphere. This situation is somewhat similar to the atmosphere in sunspots. We consider the solution for small K. There are three types of wave modes present in this situation, the slow, fast and magneto-gravity-Lamb (MgL) modes. The gL-mode acquires a more pronounced magnetic behavior because of the higher magnetic field strength and so we call them MgL-mode (see Banerjee et al. 1995 for further details). From a study of the energy density variation of these modes we find that the fast and MgL-modes are essentially confined to photospheric regions (i.e. the lower part of the atmosphere), whereas the wave energy density of the slow waves is spread over the entire extension of the cavity. As far as the wave heating is concerned the slow modes appear to be a more promising candidate than the other two type of modes.

The frequencies of the slow magneto-acoustic modes or p-modes can be found from Eq. (26) with K = 0, i.e.

$$\Omega_{p,n}=\sqrt{\tilde\gamma\left({n^2\pi^2\over D^2}+\frac 14\right)} ,$$ (32)

where n denotes the order.

 

 

Table 2: Eigenfrequencies (corresponding to a sunspot with radius 5000 km) of different p-modes for a model atmosphere with D = 20, $B = 2~{\rm kG}$, $\tilde\tau_{\rm R} = 0.5$.

Mode	Re($\Omega$)	Im($\Omega$)	P(S)	$\nu$(mHz)	$\tau_D{\rm (S)}$	$\tau_D$/P
p1	0.520	0.0659	186	5.4	233	1.25
p2	0.605	0.0767	160	6.2	200	1.25
p3	0.706	0.089	137	7.3	173	1.26
p4	0.820	0.104	118	8.5	148	1.25

Following Scheuer & Thomas (1981) let us treat the sunspot umbra as a cylinder of radius a. It can easily be shown that our analysis for a plane can be carried over in a straightforward way to cylindrical geometry, by treating axisymmetric modes and regarding $\xi_x$ and k as the radial displacement and wave number respectively. Assuming that the radial component of the displacement vanishes at r=a, we find that k takes discrete values given by $ka=j_{1,\nu}$, where $j_{1,\nu}$ denotes the zero of the Bessel function J₁ of order $\nu$. We consider the lowest-order mode (where order refers to the horizontal direction) corresponding to $\nu=1$. This provides us with a relation between the horizontal wave number and the radius of the spot. Table 1 represents the eigenfrequencies of different order p-modes from our model atmosphere with D = 10, $\tilde\tau_{\rm R} = 0.5$ and $\epsilon = 0.84$ ($B \sim 2$ KG). Table 1 reveals that radiative cooling shifts the eigenfrequencies away from the real axis. Note that the computed frequencies match very well with the ones calculated from expression (32).

Figure 7: MDI intensity-gram showing the location of the slit of the s19332r00 dataset, relative to the sunspot umbra and penumbra. The over-plotted black rectangles are the locations of the slit at the start (right) and at the end time (left) of the observations. Pixel number 67 is marked with a white box.

Radiative cooling leads to a temporal decay of oscillations of the form exp( $-t/\tau_{\rm D}$). The frequency eigenvalues of the four modes are listed in Table 2 for a typical sunspot with radius of 5000 km (K=0.06) together with the ratio of characteristic decay time $\tau_{\rm D} = {\rm Im}(\omega)^{-1}$ and oscillation period = $2 \pi / {\rm Re}(\omega)$. In the presence of Newtonian cooling all four modes are damped by a factor e^-1 within two oscillation periods.

Figure 8: CDS raster images for the s19331r00 dataset, in different temperature lines (as labeled), and the Kitt peak magnetogram. The over-plotted white rectangles are the locations of the slit for the s19332r00 dataset at the start (right) and the end time (left). Pixel 67 is marked as a black box on the images. The contours indicate the location of the umbra and penumbra.

 

 

Table 3: A log of the datasets used in this paper obtained during April 2000.

Active	Date	Dataset	Type of	Pointing	Starting time	Lines used
region			observation	(X, Y)	UT
AR 8951	14 April 2000	s19331r00	Raster	(116, 277)	04:25	O  III, O  V, He  I, Mg  IX, Ca  X
		s19332r00	Temporal	(122, 277)	04:48	O  III, O  V, He  I
		s19333r00	Raster	(135, 277)	06:14	O  III, O  V, He  I, Mg  IX, Ca  X
		s19334r00	Temporal	(136, 277)	06:37	O  III, O  V, He  I
		s19335r00	Raster	(152, 276)	08:02	O  III, O  V, He  I, Mg  IX, Ca  X
		s19336r00	Temporal	(153, 275)	08:26	O  III, O  V, He  I
AR 8963	19 April 2000	s19377r00	Raster	(-1, 418)	18:21	O  III, O  V, He  I, Mg  IX, Ca  X
		s19378r00	Temporal	(4, 412)	18:45	O  III, O  V, He  I
		s19379r00	Raster	(18, 416)	20:10	O  III, O  V, He  I, Mg  IX, Ca  X
		s19380r00	Temporal	(20, 415)	20:34	O  III, O  V, He  I
		s19381r00	Raster	(33, 416)	21:59	O  III, O  V, He  I, Mg  IX, Ca  X
		s19382r00	Temporal	(32, 414)	22:23	O  III, O  V, He  I
AR 8963	20 April 2000	s19387r00	Raster	(204, 402)	18:00	O  III, O  V, He  I, Mg  IX, Ca  X
		s19388r00	Temporal	(204, 402)	18:24	O  III, O  V, He  I
		s19389r00	Raster	(217, 401)	19:49	O  III, O  V, He  I, Mg  IX, Ca  X
		s19390r00	Temporal	(218, 401)	20:13	O  III, O  V, He  I
		s19391r00	Raster	(232, 401)	21:38	O  III, O  V, He  I, Mg  IX, Ca  X
		s19392r00	Temporal	(233, 401)	22:02	O  III, O  V, He  I