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Astron. Astrophys. 321, 935-944 (1997)

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2. Physical model

2.1. Configuration

Magnetic loops can be considered as the `building blocks' of the solar corona (Priest 1990). They can be up to [FORMULA] long and have radii that vary from [FORMULA] to [FORMULA], where the longer loops have larger radii (Priest 1984). The aspect ratio [FORMULA], with L the length of the loop and a the small radius, is typically 8 (Beaufume et al.:1992). Hence, the cylindrical plasma column considered in the present paper yields a good approximation for the coronal loops. Nothing much is known about the internal structure of these magnetic loops, which makes it difficult to obtain conclusive simulation results on their heating. For resonant absorption, for instance, the radial structure of the loops is very important for the efficiency and the localisation of the heat deposition in the loops (see e.g. Poedts et al. 1989a, 1989b, 1990a, 1990b). Hence, by lack of observational data, in the next subsection we consider a simple but reasonable coronal loop model.

We consider a straight cylindrical plasma column of length L and radius a. The plasma column is side-ways driven by waves which are incident on it. The initial state considered in the present simulations is given in Sect.  4. The external medium is not included in the model. As a matter of fact, the excitation of the coronal loop by an incident wave is numerically simulated by specifying the amplitude and wavenumbers at the surface of the loop (see Sect.  5). The anchoring of the magnetic field lines in the dense photosphere is not taken into account in the present study but its effect will we discussed in a forthcoming paper.

2.2. Nonlinear MHD equations

The results presented in Sect.  4are based on numerical solution of the resistive MHD equations. The single fluid model is used. In dimensionless form, these equations can be written as:


Here, [FORMULA], p, V , and B denote the plasma density, the plasma pressure, the velocity field, the magnetic field, and the gravitational accelleration, respectively. The ratio of specific heats, [FORMULA], is assumed to be 5/3. The magnetic permeability, µ, has been set equal to one. The divergence equation, [FORMULA], serves as an initial condition on B . In the dimensionless set of Eqs. (1)-(5), lengths are normalized to the characteristic scale of the cylindrical system, i.e. to the plasma radius (a), and magnetic fields, velocities, and times are normalized to the initial longitudinal magnetic field on axis ([FORMULA]), the Alfvén velocity ([FORMULA]), and the Alfvén crossing time ([FORMULA]), respectively.

2.3. Approximations

Gravity has been ignored in the equations above. Gravity breaks down the axial symmetry and induces a stratification of the density along the loop and, hence, linear mode coupling in this direction (see e.g. Poedts & Goossens 1991, Beliën et al.:1996). In the present paper, we want to study nonlinear mode coupling and its effects on resonant absorption.
The continuity equation (4), which describes the evolution of the plasma density [FORMULA], is not included in the present formulation. The plasma density is assumed to be constant in space and time, i.e. [FORMULA]. Electric resistivity [FORMULA] is the only dissipative effect considered. The electric resistivity is defined as a scalar quantity with a dependence on the plasma radius only and not on the time. In the present paper, the (dimensionless) plasma resistivity is defined as


with [FORMULA] a constant and [FORMULA] the longitudinal current density in the initial state (see Sect.  2.4). The constant [FORMULA] denotes the resistivity at the position [FORMULA] which will be specified below as the ideal resonance position. The magnetic Reynolds number at the center [FORMULA] is [FORMULA].

2.4. Initial state

All simulations discussed in Sect.  4 start from the same initial state: a static, axisymmetric cylinder with a constant density, a constant z -component of the magnetic field, and a parabolic current density profile in the longitudinal direction. In dimensionless units, we have:


and, hence,


The safety factor [FORMULA], which measures the pitch of the magnetic field, is 1 on axis ([FORMULA]) and 2 at the plasma surface ([FORMULA]) for the typical choice of parameters made here, viz. [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998