The second equation of our system is the conservation of mass, (12)\begin{equation} \dfrac{\partial \rho}{\partial t} + \boldsymbol{\nabla}. \left( \rho {\vec V} \right) = 0, \end{equation}∂ρ∂t+∇.ρV=0,which, in spherical coordinates, writes (13)\begin{equation} \dfrac{\partial \delta \rho}{\partial t } + \frac{1}{r^2} \dfrac{\partial }{\partial r} \left( r^2 \rho_0 V_r \right) + \frac{\rho_0}{r \sin \theta} \left[ \dfrac{\partial }{\partial \theta} \left( \sin \theta V_\theta \right) + \dfrac{\partial V_\varphi}{\partial \varphi} \right] = 0. \label{conservation_masse} \end{equation}∂δρ∂t+1r2∂∂r(r2ρ0Vr)+ρ0rsinθ∂∂θsinθVθ+∂Vϕ∂ϕ=0.The thermal forcing (J) appears on the right-hand side of the linearized heat transport equation (see CL70, Gerkema & Zimmerman 2008) given by (14)\begin{equation} \frac{1}{\Gamma_1 p_0} \dfrac{\partial \delta p}{\partial t} - \frac{1}{\rho_0} \dfrac{\partial \delta \rho}{\delta t} + \frac{N^2}{g} V_r = \kappa \frac{\rho_0}{p_0} \left[ J - J_{\rm rad} \right], \label{transport_chaleur_1} \end{equation}1Γ1p0∂δp∂t−1ρ0∂δρδt+N2gVr=κρ0p0J−Jrad,where \hbox{$ \kappa = \left( \Gamma_1 - 1 \right)/\Gamma_1 $}κ=Γ1(−1)/Γ1 and Jrad is the power per mass unit radiated by the atmosphere, supposed to behave as a grey body. We consider that Jrad∝δT. This hypothesis is known as “Newtonian cooling” and was used by Lindzen & McKenzie (1967) to introduce radiation analytically in the classical theory of atmospheric tides (see also Dickinson & Geller 1968). Physically, it corresponds to the case of an optically thin atmosphere in which the flux emitted by a layer propagates upwards or downwards without being absorbed by the other layers. In optically thick atmospheres, such as on Venus (Lacis 1975), this physical condition is not verified. Indeed, because of a stronger absorption, the power emitted by a layer is almost totally transmitted to the neighbourhood. Therefore, this significant thermal coupling within the atmospheric shell should ideally be taken into account in a rigorous way, which would lead to great mathematical difficulties (e.g. complex radiative transfers, Laplacian operators) in our analytical approach. However, recent numerical simulations of thermal tides in optically thick atmospheres by Leconte et al. (2015) show behaviour of the flow that is in good agreement with a model that uses radiative cooling. Therefore, in this work we assumeNewtonian cooling as a first model of the action of radiation on atmospheric tides. Newtonian cooling brings a new characteristic frequency, denoted σ0, which we call radiative frequency and which depends on the thermal capacity of the atmosphere. The radiative power per unit mass is thus written (15)\begin{equation} J_{\rm rad} = \frac{p_0 \sigma_0}{\kappa \rho_0 T_0} \delta T. \label{phirad} \end{equation}Jrad=p0σ0κρ0T0δT.Like the basic fields p0, ρ0, and T0, the radiative frequency varies with r and defines the transition between the dynamical regime, where the radiative losses can be ignored, and the radiative regime, where they predominate in the heat transport equation. Assuming that the radiative emission of the gas is proportional to the local molar concentration C0 = ρ0/M, it is possible to express Jrad and σ0 as functions of the physical parameters of the fluid (Appendix D), (16)\begin{equation} J_{\rm rad} = \frac{8 \epsilon_a}{M} \mathscr{S} T_0^3 \delta T \label{phirad2} \end{equation}Jrad=8ϵaMST03δTand (17)\begin{equation} \sigma_0 \left( r \right) = \frac{8 \kappa \epsilon_a \mathscr{S} }{\mathscr{R}_{\rm GP} } T_0^3, \label{sigma0} \end{equation}σ0(r)=8κϵaSRGPT03,the parameter ϵa being an effective molar emissivity coefficient of the gas and S= 5.670373 × 10-8 W m-2 K-4 the Stefan-Boltzmann constant (Mohr et al. 2012). The substitution of Eq. (15) in Eq. (14) yields (18)\begin{equation} \frac{1}{\Gamma_1 P_0} \dfrac{\partial \delta p}{\partial t} - \frac{1}{\rho_0} \dfrac{\partial \delta \rho}{\delta t} + \frac{N^2}{g} V_r = \kappa \frac{\rho_0}{p_0} J - \sigma_0 \frac{\delta T}{T_0}\cdot  \label{transport_chaleur_2} \end{equation}1Γ1P0∂δp∂t−1ρ0∂δρδt+N2gVr=κρ0p0J−σ0δTT0·Finally, the system is closed by the perfect gas law (19)\begin{equation} \frac{\delta p}{p_0} = \frac{\delta T}{T_0} + \frac{\delta \rho}{\rho_0}\cdot \label{GPlaw} \end{equation}δpp0=δTT0+δρρ0·Substituting Eq. (19) in Eq. (18), we eliminate the unknown δT, and obtain (20)\begin{equation} \frac{1}{\Gamma_1 p_0} \left( \dfrac{\partial \delta p}{\partial t} \! +\! \Gamma_1 \sigma_0 \delta p \right) + \frac{N^2}{g} \dfrac{\partial \xi_r}{\partial t} = \frac{ \kappa \rho_0}{p_0} J + \frac{1}{\rho_0} \left( \dfrac{\partial \delta \rho}{\partial t}\! +\! \sigma_0 \delta \rho \right). \label{transport_chaleur_3} \end{equation}1Γ1p0∂δp∂t+Γ1σ0δp+N2g∂ξr∂t=κρ0p0J+1ρ0∂δρ∂t+σ0δρ.Because of the rotating motion of the perturber in the equatorial frame (RE;T), a tidal perturbation is supposed to be periodic in time (t) and longitude (ϕ). So, any perturbed quantity f of our model can be expanded in Fourier series of t and ϕ(21)\begin{equation} f = \sum_{m,\sigma} f^{m,\sigma} \left( \theta , r \right) e^{i \left( \sigma t + m \varphi \right)}, \label{perturbation} \end{equation}f=∑m,σfm,σ(θ,r)eiσt+mϕ,the parameter σ being the tidal frequency of a Fourier component and m its longitudinal degree1. We also introduce the spin parameter (22)\begin{equation} \nu \left( \sigma \right) = \frac{2 \Omega}{\sigma}, \label{nu} \end{equation}ν(σ)=2Ωσ,which defines the possible regimes of tidal gravito-inertial waves:
Citation: Leconte et al. (2015)
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