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density profiles
(where 
is the ratio of
the gas pressure to the magnetic pressure), when the plasma pressure force is neglected in comparison with
the Lorentz force, and there is a one-dimensional profile of the plasma density, embedded in
the constant and parallel magnetic field, the eigenvalue problem is similar to the quantum
mechanical problem of the interaction of a particle with a nonuniform potential. In this section we
restrict our attention to the Cartesian geometry only. The governing equations, in this case, allow
for exact analytical solutions. We consider wave propagation along the magnetic field (i.e.,
) to avoid the Alfvén resonance. Nakariakov and Roberts (1995b
) established that the
qualitative dispersive properties depend weakly upon the specific profile of the density. Consider a
coronal loop as a magnetic slab with a smooth density profile, given by the profile function
where 
, 
and 
are constant. Here, the parameter 
is the density at the centre of the
inhomogeneity, 
is the density at 
and 
is a parameter governing the inhomogeneity width.
This inhomogeneity, plotted in Figure 8
, is called the symmetric Epstein profile (see, e.g., Nakariakov
and Roberts, 1995b). With an inhomogeneity of this form exact analytical solutions can be
obtained. The plasma is inhomogeneous across the straight and uniform magnetic field 
.
In the zero plasma-
limit considered, the equilibrium total pressure balance is identically
fulfilled.
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), linear perturbations of the transverse plasma velocity
are described by the equation
where 
is the Alfvén speed as 
and 
is the Alfvén speed based upon the excess
density at 
, i.e., 
. As the corresponding profile of the Alfvén speed has a
minimum at the centre of the slab, the slab is a refractive waveguide for fast magnetoacoustic waves
(see Edwin and Roberts, 1988, for discussion).
The eigenvalue problem stated by Equation (20
) supplemented by the boundary conditions
can be solved analytically (Nakariakov and Roberts, 1995b; Cooper et al., 2003). The
eigenfunctions describing kink and sausage modes are respectively given by
is the amplitude. Here 
is given by
with 
being the phase speed, which is determined by the dispersion relations
and
where 
is the Alfvén velocity at the centre of the profile,
.
) and (24) and solutions (21) are a convenient tool for the study of the
effect of the transverse profile on wave properties. In particular, Figure 9 demonstrates that the
sausage mode group speed is affected by the steepness of the profile quite significantly. One of the
observational manifestations of this effect is the shape of the fast wave trains formed by the dispersive
evolution of initially broad band perturbations. Applications of this theory are discussed in
Section 7.
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http://www.livingreviews.org/lrsp-2005-3 |
© Max Planck Society and the author(s)
Problems/comments to |
![2 p r0 = rmax sech [(x/a) ] + r oo](article175x.gif){kind=small}
with the
solid, dashed, dotted and dash-dotted curves corresponding to values of 
, 
, 
,
and 
(effectively 
). The case of 
corresponds to Equation (![2 [ 2 2 ( )] d-U-+ -w---- k2z + -w--sech2 x- U = 0, (20) dx2 C2Ao o C2Ad a](article183x.gif){kind=small}
, with the step function profile (the dashed
line), and with the symmetric Epstein profile (the solid line). The group speed is normalised to the
minimum value of the Alfvén speed, which is equal to the Alfvén speed inside the slab with the
step function profile, represented by the horizontal solid line.