Abstract
We derive the dispersion equation for gravito-magnetohydrodynamical (MHD) waves in an isothermal, gravitationally stratified plasma with a horizontal inhomogeneous magnetic field. Sound and Alfvén speeds are constant. Under these conditions, it is possible to derive analytically the equations for gravito-MHD waves. The high values of the viscous and magnetic Reynolds numbers in the solar atmosphere imply that the dissipative terms in the MHD equations are negligible, except in layers around the positions where the frequency of the MHD wave equals the local Alfvén or slow wave frequency. Outside these layers the MHD waves are accurately described by the equations of ideal MHD.
We consider waves that propagate energy upward in the atmosphere. For the plane boundary, z=0, between two isothermal plasma regions with horizontal but different magnetic fields, we discuss the boundary conditions and derive the equations for the reflection and transmission coefficients.
In the simpler case of a gravitationally stratified plasma without magnetic field, these coefficients describe the reflection and transmission properties of gravito-acoustic waves.
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Notes
Equations (11) and (12) show that the amplitudes of the fluid displacement ξ 1z and the total pressure perturbation P 1 are exponential functions of z. However, the phase-averaged energy density 〈ϵ〉 of these waves remains z invariant. Namely, 〈ϵ〉 is the sum of the corresponding thermal, kinetic and magnetic mean energy densities
$$\langle\epsilon\rangle=\langle\epsilon_{T} \rangle+\langle \epsilon_{K} \rangle+\langle\epsilon_{M} \rangle, $$i.e.
$$\langle\epsilon\rangle=\frac{v^2_{\mathrm{s}}}{2\rho_{0}(z)}\rho_{1}\rho ^{*}_{1}+\frac{\rho_{0}(z)}{2}\boldsymbol{v}_{1} \cdot\boldsymbol{v}_{1}^{*}+\frac{1}{2\mu_{0}} \boldsymbol{B}_{1}\cdot\boldsymbol{B}_{1}^{*}, $$where ∗ denotes the complex conjugate quantities, and ρ 1 and v 1=(v 1x ,v 1y ,v 1z ) are the perturbed density and plasma velocity. From the definitions of the fluid displacement ξ 1z =iv 1z /ω and the total pressure perturbation P 1=p 1+B 0⋅B 1, one can easily write the following proportionalities: |v 1|∼ξ 1z , for the perturbed plasma velocity, ρ 1∼P 1 for the perturbed plasma density, and |B 1|∼P 1/B 0 for the perturbed magnetic field. According to Equations (6), (11) and (12)
$$\boldsymbol{v}_{1}\cdot\boldsymbol{v}_{1}^{*} \equiv|\boldsymbol{v}_{1}|^2\sim \operatorname{exp}(z/H), \qquad \rho_{1}\rho^{*}_{1}\equiv|\rho _{1}|^2\sim \operatorname{exp}(-z/H), \qquad\boldsymbol{B}_{1} \cdot\boldsymbol{B}_{1}^{*}\equiv|\boldsymbol{B}_{1}|^2 \sim \mbox{const.}, $$which finally gives that the averaged wave density 〈ϵ〉 is z invariant.
The second equation in Equation (9) can be rewritten in the form
$$\frac{\mathrm{d}}{\mathrm{d}z}\bigl(P_{1}-g\rho_{0}(z) \xi_{1z}\bigr)=C_{3}\xi_{1z}-C_{4}P_{1}-g \rho_{0}(z)\,\frac{\mathrm{d}\xi_{1z}}{\mathrm{d}z}. $$Integrating this equation in the interval z=±ζ yields boundary conditions for pressure perturbation.
For the horizontal phase velocities \(V_{\mathrm{h}}<1/\sqrt{s}\) the modified acoustic waves are evanescent.
For the pure acoustic case the reflection coefficient is \(R=\frac{ (s\sqrt{V^{2}_{\mathrm{h}}-1}-\sqrt {sV^{2}_{\mathrm{h}}-1} )^{2}}{ (s\sqrt{V^{2}_{\mathrm{h}}-1}+\sqrt {sV^{2}_{\mathrm{h}}-1} )^{2}}\). It is easy to see that for \(V_{\mathrm{h}}=\sqrt {(s+1)/s}\), R=0.
For the horizontal phase velocities V h>ΩBV/Ωco≈0.97 gravity waves are evanescent.
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This work was performed in the framework of the Montenegrin National Project “Physics of Ionized Gases and Ionized Radiation”.
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Jovanović, G. Reflection Properties of Gravito-MHD Waves in an Inhomogeneous Horizontal Magnetic Field. Sol Phys 289, 4085–4104 (2014). https://doi.org/10.1007/s11207-014-0579-6
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DOI: https://doi.org/10.1007/s11207-014-0579-6