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

## 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 and temperature , stratified under gravity, g, which is taken to be pointing in the z-direction (Fig. 1). The coordinate z decreases with height.

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

The lower layer () represents the convection zone and the overlaying photosphere. The upper layer represents the chromosphere and solar corona which occupy the half-space . The flow quantities below the interface (for ) are denoted by the subscript 1, while these quantities above the interface (for ) are distinguished by the subscript 2. The interface is taken to be located at . 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, ).

Furthermore, we apply the Cowling approximation (Cowling 1941) according to which perturbations to the gravity field are ignored. As long as this is a valid assumption. The solar curvature is negligible as long as . 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.

together with the boundary conditions at the interface,

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

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

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

© European Southern Observatory (ESO) 2000

Online publication: August 17, 2000