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6.2 Theoretical modelling

Theoretical models of the propagation of longitudinal waves in stratified coronal structures, such as polar plumes (Ofman et al., 1999Jump To The Next Citation Point2000aJump To The Next Citation Point) and coronal loops (Nakariakov et al., 2000), describe the evolution of the wave shape and amplitude with the distance along the structure s in terms of the extended Burgers equation
@A @2A @A @s- - a1A - a2@q2- + a3A @q- = 0, (52)
where the coefficients a1, a2, and a3 are, in general, functions of s and describe: a1 - the effects of stratification, radiative losses, and heating; a2 - dissipation by thermal conductivity and viscosity; a3 - nonlinearity. The q = s- C t s is the running coordinate. When a = 0 1, Equation (52View Equation) reduces to the Burgers equation. Specific expressions for the coefficients depend upon the geometry of the problem and are different in the plume and loop cases. Let us illustrate the derivation of Equation (52View Equation) in a specific situation. Following Tsiklauri and Nakariakov (2001Jump To The Next Citation Point), we consider a semi-circular loop of the curvature radius RL, with the inclination angle a (measured from the normal to the solar surface) and a non-zero offset of circular loop centre from the coronal base line, Z0 (i.e., distance from the circle’s centre to the solar surface). The loop cross-section is taken to be constant. The loop is filled with a gravitationally stratified magnetised plasma of constant temperature. The sketch of the model is shown in Figure 16View Image.
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Figure 16: The sketch of the coronal loop model suggested by Tsiklauri and Nakariakov (2001Jump To The Next Citation Point). A coronal loop is considered as a magnetic field line with density and gravitational acceleration varying along the axis of the cylinder.
The gravitational acceleration along the loop is
[X0- Z0- ] g(s) = GMo. -[------(RL-x---RL-h-cos-a)------]-, (53) R2 o. 1 + RL- X0h - Z0(1 - x) cos a 2 Ro . RL RL
where h = sin(s/RL), x = cos(s/RL), and X0/RL = (1- Z20/R2L)1/2. The use of the hydrostatic equilibrium equations and the isothermal equation of state, allows us to write the stationary density profile along the loop as
[ gg(0) ] ---C2s--[X[0h----Z0(1---x)]co]s-a-- r0(s) = r0(0) exp - 1 + -RL X0h - -Z0(1- x) cos a . (54) Ro . RL RL
Assuming that the wavelength is much less than the scale height, the dissipation and the nonlinearity lengths, and applying the multi-scale expansion method, we obtain the evolutionary equation for weakly nonlinear, weakly dissipative slow magnetoacoustic waves in stratified coronal loops,
@V 1 g + 1 @V 1 [4j0 k ||(g - 1)2 ]@2V ----- ------ V + ------V ----- --------- ----+ ----------- ---2-= 0, (55) @s 2H(s) 2Cs @q 2r0(s)Cs 3 Rg @q
where an effective scale height is introduced as H(s) = C2s/gg(s), k|| is the coefficient of thermal conduction, and j0 denotes the volume viscosity or thermal conductivity.

Similarly, in the radially stratified vertical polar plume, the evolutionary equation is

( ) @r 1 g(R) 1 @r 2j @2r ---+ -- + ----2 r + --r ---- -----------2-= 0, (56) @R R 2Cs r0 @q 3Csr0(R) @q
where R is the radial (or vertical) coordinate (see Ofman et al., 2000aJump To The Next Citation Point, for details; both the viscosity and thermal conduction are included in the dissipative coefficient j).

Solutions of Equation (52View Equation) are in a satisfactory agreement with the observed evolution of the wave amplitude: For example, Figure 17View Image demonstrates the comparison of the observed and theoretically predicted growth of the longitudinal wave amplitude with height. Also, full MHD 2D numerical modelling of these waves gives similar results (see Ofman et al., 1999Jump To The Next Citation Point2000a). The comparison of the observed and modelled evolution scenarios give the possibility of estimating, in particular, the efficiency of non-adiabatic processes, such as the competition of heating and radiative losses.

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Figure 17: The comparison of the theoretically predicted (the continuous curve) and observed amplitudes of the upwardly propagating EUV disturbances observed in polar plumes (from Ofman et al., 1999).
The energy carried and deposited by the observed waves is certainly insufficient for heating of the loop. However, Tsiklauri and Nakariakov (2001) have shown that sufficiently broadband slow magnetoacoustic waves, consistent with currently available observations in the low frequency part of the spectrum, can provide the rate of heat deposition sufficient to heat the loop. In this scenario, the heat would be deposited near the loop footpoints which agrees with observationally determined positioning of the heat source.
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