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In the files/model/oro_model:Model with an orography and a temperature profile, the radiative equilibrium temperature field at the middle of the atmosphere is specified by a given profile θ⋆ (~.params.AtmosphericTemperatureParams.thetas) and the system relaxes to this profile due to the Newtonian cooling.
In Li et al. li-LHHBD2018, another scheme for the temperature is proposed, based on the mechanism used for the files/model/maooam_model:Coupled ocean-atmosphere model (MAOOAM) and considering the radiative and heat exchanges between the atmosphere and the ground. As in MAOOAM, this mechanism is the one proposed in li-BB1998 and depicted in the files/model/maooam_model:Temperature equations section of the MAOOAM documentation, with the ocean being replaced by the ground (with an orography).
and the ideal gas relation and the vertical discretization of the hydrostatic relation at 500 hPa allows to write the spatially dependent atmospheric temperature anomaly δTa = 2f0θa/R where R (~.QgParams.rr) is the ideal gas constant.
Ordinary differential equations
All the modes of this model version are expanded on the set of Fourier modes Fi detailed in the section files/model/oro_model:Projecting the equations on a set of basis functions:
and as in MAOOAM, the fields, parameters and variables are non-dimensionalized by dividing time by f0 − 1 (~.params.ScaleParams.f0), distance by the characteristic length scale L (~.params.ScaleParams.L), pressure by the difference Δp (~.params.ScaleParams.deltap), temperature by f02L2/R, and streamfunction by L2f0. As a result of this non-dimensionalization, the fields $\theta_{\rm a}$ and $\delta T_{\rm a}$ can be identified: $2 \theta_{\rm a} \equiv \delta T_{\rm a}$.
The equations of the system of ordinary differential equations for this model thus read:
where the parameters values have been replaced by their non-dimensional ones and we have also defined G = − L2/LR2 (~.params.QgParams.G), $\lambda'_{{\rm a}} = \lambda/(\gamma_{\rm a} f_0)$ (~.params.QgParams.Lpa), $\lambda'_{{\rm g}} = \lambda/(\gamma_{\rm g} f_0)$ (~.params.QgParams.Lpgo), $S_{B,{\rm a}} = 8\,\epsilon_{\rm a}\, \sigma_B \, T_{{\rm a},0}^3 / (\gamma_{\rm a} f_0)$ (~.params.QgParams.LSBpa), $S_{B,{\rm g}} = 2\,\epsilon_{\rm a}\, \sigma_B \, T_{{\rm a},0}^3 / (\gamma_{\rm a} f_0)$ (~.params.QgParams.LSBpgo), $s_{B,{\rm a}} = 8\,\epsilon_{\rm a}\, \sigma_B \, T_{{\rm a},0}^3 / (\gamma_{\rm g} f_0)$ (~.params.QgParams.sbpa), $s_{B,{\rm g}} = 4\,\sigma_B \, T_{{\rm a},0}^3 / (\gamma_{\rm g} f_0)$ (~.params.QgParams.sbpgo), $C'_{{\rm a},i} = R C_{{\rm a},i} / (2 \gamma_{\rm a} L^2 f_0^3)$ (~.params.QgParams.Cpa), $C'_{{\rm g},i} = R C_{{\rm g},i} / (\gamma_{\rm g} L^2 f_0^3)$ (~.params.QgParams.Cpgo).
The coefficients ai, j, gi, j, m, bi, j, m and ci, j are the inner products of the Fourier modes Fi:
$$\begin{aligned}
a_{i,j} & = & \frac{n}{2\pi^2}\int_0^\pi\int_0^{2\pi/n} F_i(x,y)\, \nabla^2 F_j(x,y)\, \mathrm{d} x \, \mathrm{d} y = - \delta_{ij} \, a_i^2 \\\
g_{i, j, m} & = & \frac{n}{2\pi^2}\int_0^\pi\int_0^{2\pi/n} F_i(x,y)\, J\left(F_j(x,y), F_m(x,y)\right) \, \mathrm{d} x \, \mathrm{d} y \\\
b_{i, j, m} & = & \frac{n}{2\pi^2}\int_0^\pi\int_0^{2\pi/n} F_i(x,y)\, J\left(F_j(x,y), \nabla^2 F_m(x,y)\right) \, \mathrm{d} x \, \mathrm{d} y \\\
c_{i, j} & = & \frac{n}{2\pi^2}\int_0^\pi\int_0^{2\pi/n} F_i(x,y)\, \frac{\partial}{\partial x} F_j(x,y) \, \mathrm{d} x \, \mathrm{d} y
\end{aligned}$$
These inner products are computed according to formulas found in om-CT1987 and stored in an object derived from the ~.inner_products.base.AtmosphericInnerProducts class.
The vertical velocity ωi can be eliminated, leading to the final equations
with $\boldsymbol{\eta} = (1, \psi_{{\rm a},1}, \ldots, \psi_{{\rm a},n_\mathrm{a}}, \theta_{{\rm a},1}, \ldots, \theta_{{\rm a},n_\mathrm{a}}, \delta T_{{\rm g},1}, \ldots, \delta T_{{\rm g},n_\mathrm{a}})$, as described in the files/technical_description:Code Description. Note that η0 ≡ 1. The tensor 𝒯, which fully encodes the bilinear system of ODEs above, is computed and stored in the ~.tensors.qgtensor.QgsTensor.
Example
An example about how to setup the model to use this model version is shown in files/examples/Lietal:Atmospheric model with heat exchange - Li et al. model version (2017).