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Astron. Astrophys. 360, 707-714 (2000)

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2. Setup of the problem

In this section, we consider the f -mode that propagates along an interface between two semi-infinite layers of perfect gas of equilibrium density [FORMULA] and temperature [FORMULA], stratified under gravity, g, which is taken to be pointing in the z-direction (Fig. 1). The coordinate z decreases with height.

[FIGURE] Fig. 1. The geometry of the problem. The chromosphere and solar corona occupy the upper half-plane. The photosphere and convection zone occur at [FORMULA]. The convective random flow is located at the lower half-plane and the f -mode is represented by the sinusoidal line.

The lower layer ([FORMULA]) represents the convection zone and the overlaying photosphere. The upper layer represents the chromosphere and solar corona which occupy the half-space [FORMULA]. The flow quantities below the interface (for [FORMULA]) are denoted by the subscript 1, while these quantities above the interface (for [FORMULA]) are distinguished by the subscript 2. The interface is taken to be located at [FORMULA]. This model is a special case of the model which was considered by Vanlommel & Cadez (1998) and Vanlommel & Goossens (1999) (valid for the width of the chromosphere, [FORMULA]).

Furthermore, we apply the Cowling approximation (Cowling 1941) according to which perturbations to the gravity field are ignored. As long as [FORMULA] this is a valid assumption. The solar curvature is negligible as long as [FORMULA]. Under these assumptions the Sun can be modeled as plane-parallel with constant gravity g.

It is assumed that the f -mode is incompressible and the plasma is magnetic field-free, i. e. the motions in the solar atmosphere are described by hydrodynamic equations, viz.

[EQUATION]

[EQUATION]

[EQUATION]

together with the boundary conditions at the interface,

[EQUATION]

Here, [FORMULA] is the mass density, [FORMULA] is the velocity, p is the pressure, g is the gravitational acceleration and [FORMULA] describes the wavy interface. Henceforth, indices with the comma denote the partial derivatives, e.g. [FORMULA].

In what follows we assume two-dimensional motions with [FORMULA] and [FORMULA] and consider the case when the transitional layer [FORMULA] becomes a sharp discontinuity of density and temperature. In particular, we take into account the equilibrium state in which the flow [FORMULA] occurs in the lower medium only and it depends both on x, [FORMULA] and t. The mass density [FORMULA] and the pressure [FORMULA] are functions of z only, viz.

[EQUATION]

Assuming that perturbations are small, we expand the fluid variables around this equilibrium and introduce the flux function [FORMULA] such that

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: August 17, 2000
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