Difference between revisions of "Non orographic gravity waves drag"

Latest revision as of 11:22, 30 April 2025

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).

nonoro_gwd_ran_mod.F90

Inherited and adapted from Earth's model (F. Lott), this parametrization has been implemented in the Venus PCM (F. LOTT, and S. LEBONNOIS), in the Mars PCM (G.GILLI, F. FORGET and J.LIU), and in the Generic PCM (D.BARDET and J.LIU). However, we would like to draw the reader's attention to the fact that there may be some subtle differences from one model to another, which may be specific to Mars and/or Venus. But, the formalism in the Generic model is intended to remain very generic.

Dedicated flags to call in the callphys.def

 calllott_nonoro=True

You have to set the maximum value of the Eliassen-Palm flux that can be transported by the wave package:

nonoro_gwd_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)

nonoro_gwd_sat ! default gravity waves saturation value = 1.  !!
nonoro_gwd_cmax ! default gravity waves phase velocity value = 30.  !!
nonoro_gwd_rdiss ! default coefficient of dissipation = 1 !!
nonoro_gwd_kmax ! default Max horizontal wavenumber = 1.e-4 !!

Underlying hypotheses and limitations

• Wave characteristic calculation using MOD
• Variable EP-flux according to PBL variation (max velocity thermals)
• Reproducibility of the launching altitude calculation

More advance description of the stochastic non-orographic gravity waves drag parametrization

$$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$$

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.

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} 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} 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} 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. 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 Lott et al. 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 :

\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}

where $$\hat{\omega}$$ = 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) 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} 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} where $$q$$ is the nearest integer that rounds (n-1)/M (i.e. towards lower values).

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.

Reference papers

Lott et al. (2012) [1] Lott and Guez (2013) [2]