2. The model and background theory
where t is the time, is the plasma density, the velocity, p the pressure, the magnetic field and . All quantities are normalised as , , , , and , where all the symbols have their usual meanings. To simplify the notation, the tilde has been dropped in Eqs. (1) to (4) as well as in the rest of this paper.
2.1. The equilibrium
We consider these equations in Cartesian coordinates and assume that there are no variations in the y direction (), thereby reducing the analysis to two dimensions. The equilibrium state is described by a homogeneous magnetic field in the z direction and a density gradient in the x direction:
are allowed in the plasma. The equilibrium conditions (5) and (6) allow the unperturbed total pressure to be constant across the magnetic field.
2.2. Analytical description
Following the standard method (Nakariakov et al. 1995 , 1997), the MHD equations are written in component form using equilibrium (5) and (6) and perturbations (7) and (8). These equations are correct to an arbitrary order of the perturbations.
where the definitions
have been used. Here is the Alfvén speed and the sound speed. Eq. (25) describes the evolution of a linearly polarized Alfvén wave, and the pair of linearly coupled Eqs. (26) and (27) describes the evolution of slow and fast magnetosonic waves. In some limiting cases, e.g. , or , the equations for slow and fast modes are decoupled. The wave equations are correct to an arbitrary order of nonlinearity.
In the geometry considered, a linear Alfvén wave perturbs and , while the magnetosonic waves perturb the remaining physical quantities. However, in the nonlinear regime, the linearly polarized Alfvén wave perturbs these quantities too, and this leads to nonlinear excitation of the magnetosonic waves and to consequent coupling of the Alfvén wave to the magnetosonic perturbations. In other words, this process is self-interaction of the Alfvén wave.
The initial stage of nonlinear generation of magnetosonic waves by Alfvén waves in an inhomogeneous medium has been considered by Nakariakov et al. (1997). To estimate the efficiency of this process, we consider a zero- plasma of the geometry described above. This case is described by Eqs. (25)-(27) with . Initially, there is an Alfvén wave only, perturbing and , and all remaining physical quantities are not perturbed. So, initially, all the nonlinear terms on the right hand side of Eq. (25) are zero, and the Alfvén wave evolves linearly, according to the left hand side of (25). (Note that the same equation has been derived by Ofman & Davila 1997). As is non-zero, the terms and are non-zero too, which give source terms on the right hand sides of (26) and (27). Consequently, the magnetosonic perturbations are generated as
Both motions in the x and z directions perturb the density. The longitudinal motions () are always generated by an Alfvén wave, while the transverse perturbations () are connected with the inhomogeneity of the medium in the transverse direction, . In the initial stage, the transverse perturbations are generated secularly () because of phase mixing of Alfvén waves. In the cold case, the transverse perturbations correspond to obliquely propagating fast magnetosonic waves, and the longitudinal perturbations to degenerated slow waves.
In the case of warm plasma (), slow and fast magnetosonic waves are coupled with each other by the pair of Eqs. (27) and (26). In other words, each wave, slow or fast, perturbs both and components of the velocity. However, the longitudinal and transversal components of the velocity are excited by the same mechanisms (34) and (35) as in the cold case.
The developed stage of the nonlinear interaction of Alfvén waves with compressive perturbations is characterized by the back-reaction of the induced compressive perturbations on the Alfvén waves. Technically, this phenomenon is connected with the growth of the nonlinear terms on the right hand side of Eq. (25). In the case of a homogeneous medium, the self-interaction of Alfvén waves is described by the Cohen-Kulsrud equation (Cohen & Kulsrud 1974, see also discussions in Ofman & Davila 1995, Verwichte et al. 1999and Nakariakov et al. 2000). However, this effect is proportional to the square of the Alfvén wave amplitude and can significantly influence the wave evolution only if the wave amplitude is sufficiently high or the wave propagates over sufficient long distances to allow the nonlinear effects to build up.
Thus, in numerical simulations, we expect to observe weakly nonlinear coupling of Alfvén modes with fast and slow magnetosonic modes during the initial stage of the wave interaction described by Eqs. (34) and (35).
© European Southern Observatory (ESO) 2000
Online publication: December 5, 2000