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2.3 Zero plasma-b density profiles

The limiting case of zero plasma-b (where b 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., ky = 0) to avoid the Alfvén resonance. Nakariakov and Roberts (1995bJump To The Next Citation Point) 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
(x ) r0 = rmaxsech2 -- + r oo , (19) a
where rmax, r oo and a are constant. Here, the parameter rmax is the density at the centre of the inhomogeneity, r oo is the density at x = oo and a is a parameter governing the inhomogeneity width. This inhomogeneity, plotted in Figure 8View Image, is called the symmetric Epstein profile (see, e.g., Nakariakov and Roberts, 1995bJump To The Next Citation Point). With an inhomogeneity of this form exact analytical solutions can be obtained. The plasma is inhomogeneous across the straight and uniform magnetic field B0 = B0^z. In the zero plasma-b limit considered, the equilibrium total pressure balance is identically fulfilled.
View Image

Figure 8: Symmetric Epstein density profiles of the form 2 p r0 = rmax sech [(x/a) ] + r oo with the solid, dashed, dotted and dash-dotted curves corresponding to values of p = 1, p = 2, p = 4, and p = 100 (effectively p --> oo). The case of p = 0 corresponds to Equation (19View Equation).
According to Nakariakov and Roberts (1995bJump To The Next Citation Point), linear perturbations of the transverse plasma velocity Vx = U (x)exp (iwt- ikzz) are described by the equation
2 [ 2 2 ( )] d-U-+ -w---- k2z + -w--sech2 x- U = 0, (20) dx2 C2Ao o C2Ad a
where C Ao o is the Alfvén speed as x --> oo and C Ad is the Alfvén speed based upon the excess density at x = 0, i.e., 1/2 CAd = B0/(m0rmax). 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 (20View Equation) supplemented by the boundary conditions U (x --> ± oo ) --> 0 can be solved analytically (Nakariakov and Roberts, 1995bJump To The Next Citation PointCooper et al., 2003Jump To The Next Citation Point). The eigenfunctions describing kink and sausage modes are respectively given by

{ a (x) U = A sech (a) ( ) kink mode , (21) A tanh x secha x sausage mode a a
where A is the amplitude. Here a is given by
|kz|a V~ ----------- a = ----- C2A oo - V2ph, (22) CA oo
with Vph = w/kz being the phase speed, which is determined by the dispersion relations
V~ ----------- ( ) C2 - V 2 = |k |aCAo o V2 - C2 (23) A oo ph z CA0 ph A0
|kz| a ( 2 2 ) 2 3 V~ ----------- --2-- V ph - C A0 - ----- = ----- C2Ao o - Vp2h, (24) C A0 |kz| a CAo o
where CA0 = CAo o CAd/ (C2A oo + C2Ad)1/2 is the Alfvén velocity at the centre of the profile, x = 0.
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Figure 9: Comparison of group speeds of a propagating sausage mode guided by a 1D plasma inhomogeneity with the Alfvén speed contrast ratio of 4, 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.
Dispersion relations (23View Equation) and (24View Equation) and solutions (21View Equation) are a convenient tool for the study of the effect of the transverse profile on wave properties. In particular, Figure 9View Image 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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