Acoustic‐gravity waves in a viscous and thermally conducting isothermal atmosphere (part II: For small Prandtl number)

Abstract

In part one of these series we investigated the effect of Newtonian cooling on acoustic-gravity waves in an isothermal atmosphere for large Prandtl number. It was shown that the atmosphere can be divided into two regions connected by an absorbing and reflecting layer, created by the exponential increase of the kinematic viscosity with height, and if Newtonian cooling coefficient goes to infinity the temperature perturbation associated with the wave will be eliminated. In addition all linear relations among the perturbation quantities will be modified. In this paper we will consider the effect of Newtonian cooling on acoustic-gravity waves for small Prandtl number in an isothermal atmosphere. It is shown that if the Newtonian cooling coefficient is small compared to the adiabatic cutoff frequency the atmosphere may be divided into three distinct regions. In the lower region the motion is adiabatic and the effect of the kinematic viscosity and thermal diffusivity are negligible, while the effect of these diffusivities is more pronounced in the upper region. In the middle region the effect of the thermal diffusivity is large, while that of the kinematic viscosity is still negligible. The two lower regions are connected by a semitransparent reflecting layer as a result of the exponential increase of the thermal diffusivity with height. The two upper regions are joined by an absorbing and reflecting barrier created but the exponential increase of the kinematic viscosity. If the Newtonian cooling coefficient is large compared to the adiabatic cutoff frequency, the wavelengths below and above the lower reflecting layer will be equalized. Consequently the reflection produced by the thermal conduction is eliminated completely. This indicates that in the solar photosphere the temperature fluctuations may be smoothed by the transfer of radiation between any two regions with different temperatures. Also the heat transfer by radiation is more dominant than the conduction process

Acoustic‐gravity waves in a viscous and thermally conducting isothermal atmosphere (part III: for arbitrary Prandtl number)

“…Consequently, there are four linearly independent solutions, which in the neighborhood of 0 can be written in the following form TI() E an (el),n+.,, T2 () E aS (e2)"+e' + T1 ()log(), T3() E an(e3)"+e" T4() E a, (e4) '+e + T3()log(), (4 5) where el 2, e2 1, e3 e4 0 The prime denotes differentiation of a, and the sums are taken from n 0 to n oo. The coefficients o(e,) are determined from the following three term recursion formula po(n + 2 + e)an+2 + Pl (n + 1 + e)an+l + p2(n + e)an 0, (4.6) Following the same procedure as in Part II (Alkahby [9]), the solution of the differential equation, which satisfies the prescribed boundary conditions can be written in the following form T(z) clTI(z) + c2T3(z). (4 $) To determine the linear combination of T(z) in equation (4.8), the behavior ofT1 (z) and T3(z) for small z must be found Since small z corresponds to large ](] with arg()= 3r/2 + O,, the asymptotic expansions ofT1 () and T3() about infinity should be found.…”

“…[IV] It follows from [III] and the results, which were obtained in Alkahby [8,9], that the effect of thermal conduction on the reflection and dissipation of the wave will be eliminated if the heat exchange between the hotter and cooler region in the atmosphere is intense and the oscillatory process is transformed from the adiabatic form to the isothermal one.…”

“…The combined effect of Newtonian cooling, viscosity and thermal conduction, for large Prandtl number, is investigated in Alkahby [8]. It was shown that the effect of thermal conduction can be excluded and the solution, above the reflecting layer that is created by the viscosity, decays exponentially with altitude before it is influenced by the effect of thermal conduction.…”

“…Moreover, when the Newtonian cooling coefficient is large compared to the frequency of the wave the lower region will be transformed from the adiabatic form to the isothermal one. The effect of Newtonian cooling, viscosity and thermal conduction, for small Prandtl number, on upward propagating acoustic waves in an isothermal atmosphere is investigated by Alkahby [9]. It was shown that when the Newtonian cooling coefficient is small compared to the frequency of the wave the atmosphere may be divided into three distinct regions connected by two different reflecting layers In the lower region the oscillatory process is approximately adiabatic, it is isothermal in the middle region and in the upper region the solution will decay exponentially with altitude.…”

Atmospheric Layers in Response to the Propagation of Gravity Waves under Nonisothermal, Wind-shear, and Dissipative Conditions

“…At the same time, some similar studies took into consideration a horizontal magnetic field (e.g., [30][31][32]). Particularly, in a series of work on the combined effect of Newtonian cooling, viscosity and thermal conduction, Alkahby [33][34][35] demonstrated that, for an arbitrary value of the Newtonian cooling coefficient, on the one hand, a large Prandtl number divides an atmosphere into two distinct regions between which there is an absorbing and reflecting layer produced by the exponential increase of the kinematic viscosity; on the other hand, a small Prandtl number divides the atmosphere into three distinct regions when the Newtonian cooling coefficient is small, while the two lower regions merge into one if the Newtonian cooling coefficient is large. For an arbitrary Prandtl number, it was found that the effect of the thermal conduction is dominated by that of the viscosity if the Newtonian cooling coefficient is small and becomes eliminated completely if the Newtonian cooling is large.…”