Difference between revisions of "Non orographic gravity waves drag"
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== Dedicated flags to call in the ''callphys.def'' == | == Dedicated flags to call in the ''callphys.def'' == | ||
| − | To activate this parametrization: | + | To activate this parametrization: <syntaxhighlight lang="fortran"> calllott_nonoro=True </syntaxhighlight> |
You have to set the maximum value of the Eliassen-Palm flux that can be transported by the wave package: | You have to set the maximum value of the Eliassen-Palm flux that can be transported by the wave package: | ||
Revision as of 11:50, 5 March 2025
Contents
Description of the physical process
Parametrization of the momentum flux deposition due to a discrete number of gravity waves randomly generated by setting their waves characteristics (set as Gaussian distribution).
In the code, the FORTRAN file corresponding to this parametrization isnonoro_gwd_ran_mod.F90
Inherited and adapted from Earth's model (F. Lott)\citep{lott_2012,lott_2013}, Venus' model (F. LOTT, and S. LEBONNOIS)\citep{gilli_2017} and Mars' model (G.GILLI, F. FORGET and D.BARDET)\citep{gilli_2020}. Parametrization implemented in the Generic PCM by D.BARDET is the case of Giant Planets and updated J.LIU
Dedicated flags to call in the callphys.def
To activate this parametrization: calllott_nonoro=True
You have to set the maximum value of the Eliassen-Palm flux that can be transported by the wave package:
epflux_max
Additional parameters can be also change in the callphys.def (do not worry if you do not want to change them, they have default values in the code)
sat ! default gravity waves saturation value = 1. !!
cmax ! default gravity waves phase velocity value = 30. !!
rdiss ! default coefficient of dissipation = 1 !!
kmax ! default Max horizontal wavenumber = 1.e-4 !!
Underlying hypotheses and limitations
- input flux compensated in the deepest layers
- Variable EP-flux according to PBL variation (max velocity thermals)
- reproductibility of the launching altitude calculation
- wave characteristic calculation using MOD
- adding east_gwstress and west_gwstress variables
- The rho (density) at the specific locations is introduced. The equation of EP-flux is corrected by adding the term of density at launch (source) altitude(level)
Equations that are being solved, whenever possible
In this numerical scheme, at each time step and on each point of the horizontal grid, a finite number of waves, with randomly chosen characteristics, are launched upwards. This multi-wave formalism generalises Eckermann's stochastic method and enables a very large number of waves to be generated at low cost. The wave packet sent into the atmosphere is composed of M=NK$\times$NO$\times$NP types of waves, with NK=2 values for the wave number, NO=2 absolute values for the phase velocity and NP=2 propagation directions (eastward and westward) for the non-orbiting gravity waves. This approach enables the GCM to process a large set of waves at a given time $t$ by adding the effect of these M=8 waves to that of the waves launched at previous time steps, in order to calculate trends.
The $\Delta t$ life cycle of gravity waves, i.e. from their generation to their break-up, is significantly longer than the $\delta t$ time step of the GCM.
On Earth, gravity wave theory indicates that atmospheric disturbances induced by convection have life cycles of about 1 day. \citep{lott_2013}.
The spectrum, made up of triple discrete Fourier series, is discretised into 770 stochastic harmonics ($\approx$ M $\times \Delta t/\delta t$) which contribute to the wave field each day and at a given horizontal grid point. At each time $t$, the $\omega'$ vertical velocity field of upward propagating gravity waves can be represented by this sum:
\begin{equation} \omega' = \sum_{n = 1}^{\infty} C_{n} \omega'_{n} \label{eq:GWD_sum_vertical_velocity} \end{equation} \noindent where $C_n$ are the normalisation coefficients, such that $\sum_{n=1}^{\infty} C_{n}^{2}=1$. This is therefore a simple multi-wave representation, based on the principle of wave superposition, which is particularly suitable when linear dynamics is sufficient to describe the flow, where critical levels of wave absorption come into play.
Here each of the vertical velocities $\omega‘_{n}$ can be treated independently of the others, and each coefficient $C_{n}^{2}$ is taken as the probability that the wave field is given by the vertical velocity of the gravity wave $\omega’_{n}$ :
\begin{equation} \omega'_{n} = \Re \{ \hat{\omega}_{n}(z) e^{z/2H} e^{i(k_{n}x+l_{n}y-w_{n}t)} \} \label{eq:GWD_vertical_velocity_one} \end{equation} \noindent where the wave numbers $k_n$, $l_n$ and frequency $w_n$ are chosen at random.
In this formulation, $H$ is a vertical scale characteristic of the mean atmosphere under consideration and $z$ is the logarithmic pressure altitude $z = H \ln(P_r /P)$, with $P_r$ a reference pressure, taken here at source height (i.e. above typical convective cells).
To evaluate the amplitude of $\omega'_{n}$, the vertical velocity is calculated randomly at a given launch altitude $z_0$, then iterated from one model level $z_1$ to the next $z_2$ by a non-rotating Wentzel-Kramers-Brillouin (WKB) approximation :}. Method developed by Léon Brillouin, Hendrik Anthony Kramers and Gregor Wentzel in 1926 to study the semi-classical regime of a quantum system. Based on quantum mechanics and therefore the wave nature of particles, the wave function is developed asymptotically to first order of the power of the $\hbar$ action quantum. The idea is that the Schrödinger equation is derived from the wave propagation equation, where it should therefore be possible to recover classical mechanics in the limit where $\hbar \rightarrow 0$, just as it is possible to recover geometrical optics when $\lambda \rightarrow 0$ in the theory of wave optics}. Using this expression and the polarisation relation between the amplitudes of the large-scale horizontal wind $\hat{u}$ and the vertical wind $\hat{omega}$ (not shown here), it is possible to define the Eliassen-Palm flux of the wave packet sent into the atmosphere:
\begin{equation} \overrightarrow{F^{z}}(k, l, \omega) = \Re \{ \rho_{r}\overrightarrow{\hat{u}\hat{\omega}^{*}} \} = \rho_{r}\frac{\overrightarrow{k}}{\lvert \overrightarrow{k} \rvert^{2}}m(z) \lvert \lvert \hat{\omega}(z) \rvert \rvert^{2} \label{eq:GWD_EPflux} \end{equation} \noindent with $k$, $l$ the horizontal wavenumbers and $w$ the frequency of the vertical velocity field. This is included in the vertical wavenumber $m = \frac{N \lvert \overrightarrow{k} \rvert}{\Omega} $, according to the non-rotating WKB approximation, in the limit $H \rightarrow \infty$, with $\Omega = v - \overrightarrow{k}\overrightarrow{u}$ and $N$ the Brünt-Väisälä frequency \citep{lott_2012}. In equation \ref{eq:GWD_EPflux}, $\rho_{r}$ is the density of the fluid at the reference pressure level $P_r$.
In this scheme, $\hat{\omega}_{n}$ and the Eliassen Palm flow are randomly imposed at a given launch altitude $z_0$.
A constant vertical viscosity $\mu$ (equation 4 in \cite{lott_2012}) controls the vertical distribution of gravity wave drag near the top of the model.
To move from one vertical level to the one just above, the Eliassen Palm flux is essentially conserved, but may undergo a small diffusivity, $\nu = \mu/\rho_0$, which can be included by replacing $\Omega$ by $\Omega + i\nu m^{2}$.
This small diffusivity guarantees that the waves are finally dissipated on the last few levels of the model, if they have not been dissipated before (hence the division by the density $\rho_0$).
Moreover, this new amplitude of the Eliassen Palm flux is limited to that produced by a saturated monochromatic $\hat{omega}_{s}$ wave following \citep{lindzen_1981} :
\begin{equation} \hat{\omega}_{s} = S_{c}\frac{\Omega^{2}}{\lvert \overrightarrow{k}\rvert N}e^{-z/2H}\frac{k^{*}}{\lvert \overrightarrow{k}\rvert} \label{eq:GWD_breaking_vertical_velocity} \end{equation}
\noindent or $\hat{\omega}$ = 0 when $\Omega$ changes sign, to deal with critical levels. In equation \ref{eq:GWD_breaking_vertical_velocity}, $S_{c}$ is a setting parameter and $k^{*}$ a characteristic horizontal wave number corresponding to the largest parameterized wave.
Finally, the $\rho^{-1}\delta_z \overrightarrow{F}^{z}_{n‘}$ ($n’$=1, M) trends produced by the drag of the M generated gravity waves are calculated through the wind trends to be applied to the horizontal zonal wind. Since the $\omega'_n$ speeds are independent realisations, the mean trend produced is the average of these M trends. Thus, the average trend is first redistributed over the longest time scale $\delta t$ by rescaling it by $\delta t / \Delta t$ and then, the first-order autoregressive relationship (AR-1)\footnote{\textbf{First-order autoregressive relationship:} In statistics or signal processing, an AR model is a representation of a time-varying random process (such as nature or the economy). The auto-regressive model specifies that the final variable depends linearly on its previous values and a stochastic term (imperfect predictive term), the specificity of order 1 being that the previous values are those obtained at the time step just prior to the one under consideration. In this case the values of the zonal wind trends at the previous time step $frac{\delta t - \delta t}{\delta t} \left( \frac{\delta \overrightarrow{u}}{\delta t} \right)^{t-. \frac{\delta t}_{GW}$ are added to the drag of the stochastic gravity wave packet of the current time step $\frac{\delta t}{\Delta t}\frac{1}{M}\sum^{M}_{n‘ = 1}\frac{1}{\rho_{0}}\frac{\delta \overrightarrow{F}^{z}_{n’}}{\delta z}$. } described in \cite{lott_2012} is used as follows:
\begin{equation} \left( \frac{\delta \overrightarrow{u}}{\delta t} \right)^{t}_{GW} = \frac{\delta t}{\Delta t}\frac{1}{M}\sum^{M}_{n' = 1}\frac{1}{\rho_{0}}\frac{\delta \overrightarrow{F}^{z}_{n'}}{\delta z} + \frac{\Delta t - \delta t}{\Delta t} \left( \frac{\delta \overrightarrow{u}}{\delta t} \right)^{t-\delta t}_{GW} \label{eq:GWD_zonal_wind_forcing} \end{equation} \noindent to highlight the formalised superposition of stochastic waves from the cumulative sum of zonal wind tendencies due to gravity waves, taking as a normalisation coefficient :
\begin{equation}
C^{2}_{n} = \left( \frac{\Delta t - \delta t}{\Delta t} \right)^{q}\frac{\delta t}{M\Delta t}
\label{eq:normalize_coeff}
\end{equation}
\noindent où $q$ est le nombre entier le plus proche qui arrondit (n-1)/M (c'est-à-dire vers des valeurs plus faibles).
\newpage \subsection{Paramètres de réglage} \label{GWD_parametres}
The characteristics of each wave launched into the GCM are chosen at random with a prescribed probability distribution, the limits of which are the key parameters of the model. These are chosen on the basis of observational constraints (where available) and theoretical considerations.
