3 Meso-Nh fundamental equations

3.1 General characteristics

The Meso-Nh Atmospheric Simulation System (Lafore et al. 1998) can simulate the time evolution of the atmospheric three dimensional motions ranging from the large meso-alpha scale (100 km) down to the typical microscale of the Large Eddy Simulation (LES) models (50 m). It can forecast several meteorological variables on its 3D computation grid: the three components of the wind, the temperature, the moisture, the pressure and the turbulence kinetic energy. An optical turbulence scheme has been added later and is sketched in Sect. 3.3 (seeing $\varepsilon$, $C_{\rm N}^2$ profiles, isoplanetic angle $\theta_{\rm AO}$, scintillation rate $\sigma ^{2}_{\rm I}$, coherence wavefront time $\tau_{\rm AO}$ and spatial coherence outer scale ${\cal L}_0$). Meso-Nh is a non-hydrostatic numerical model (Lafore et al. 1998) based on the anelastic approximation. The principal characteristic of this approximation is that all acoustic waves are filtered but nevertheless, the model can represent accurately the gravity waves that enhance the shear production of turbulence. The lee waves occurring downstream of the mountain ranges and resulting in flow deceleration are particularly well simulated by this model. These waves can perturb the flow and increase the level of the turbulence activity. Using a high model resolution, we need a non-hydrostatic model to resolve the orographic effects in the turbulence developement and obtain a more accurate estimation of the turbulent kinetic energy because such a model describes this physical phenomenon.

The input data required by the model are:

• a numerical terrain model with high spatial resolution;
• the fields of atmospheric pressure, temperature, humidity and wind known at an initial time t₀. One can use either analyses provided by Meteorologic Centers or radiosoundings provided by Meteorological Stations.

3.2 Dynamical turbulence parameterization

The basic equations of the model that are used to compute the time evolution of the physical system are the conservation of momentum (equations of motion), the first law of thermodynamics combined with equation of state and the equations of dry-air-mass and moisture conservation. The turbulence is among the physical processes that occur at scales too small to be resolved by the model and so it must be parameterized. Although the model offers a choice of 3D turbulent schemes, we used a 1D mode which takes into account only the vertical turbulent fluxes. The method used to describe the turbulence production is detailed in previous papers (Bougeault et al. 1995, 1989). It relies on the turbulence kinetic energy equation which takes the following form in Meso-Nh (Cuxart et al. 1995):

$$\frac{D e}{D t}=-\overline{w^{\prime }u^{\prime }}\frac{\par... ...\overline{w^{\prime }v^{\prime }}\frac{\partial V}{\partial z}$$

$$+\frac{1}{\rho} \frac{\partial}{\partial z}(0.2 \rho L\sqrt{... ...}{2}}}{L}+\beta \overline{w^{\prime }{\theta_{v}} ^{\prime }}$$

where the first and second term on the right hand side represent the shear production, the third term the diffusion and the fourth term the dissipation. The last term is the buoyancy term: $w^{^{\prime }}$ is the vertical wind fluctuation, $\theta_{v} ^{^{\prime }}$ the virtual potential temperature fluctuation and $\beta =g/{\theta_{v}}$ (g is the gravity acceleration). The vertical fluxes of wind and temperature are parameterized following the eddy diffusivity approach:

 

$$\overline{w^{\prime }\theta_{v} ^{\prime }}=-K\frac{\partial \overline{{\theta_{v}}} }{\partial z}\cdot$$ (3)

The physical parameterizations are somewhat different from those of the previous hydrostatic model Peridot (Bougeault 1995) developed at the CNRM in Toulouse (Fr). In the Peridot model, the eddy diffusivity is given by:

 

$$K(x,y,z,t)=0.4L(x,y,z,t)\sqrt{e(x,y,z,t)}$$ (4)

where L is a mixing length. In Meso-Nh the eddy diffusivity is a function given by:

 

$$K(x,y,z,t)=0.16L(x,y,z,t)\sqrt{e(x,y,z,t)}\phi _{3}(x,y,z,t)$$ (5)

where $\phi _{3}(x,y,z,t)$ is an inverse turbulent Prandtl number.

• L is the so called Bougeault-Lacarrere (BL89) (Bougeault 1989) mixing length defined as follows: at any level z in the atmosphere a parcel of air of given turbulence kinetic energy e(z), can move upwards $(l_{\rm up})$ and downwards $(l_{\rm down})$ before being stopped by buoyancy forces. These distances are defined by:

  
  

  $$\int\limits_{z}^{z+l_{\rm up}}\frac{g}{\theta_{v} }\left( \t... ...prime }}\right) \right) {\rm d}z^{^{\prime }}=e\left( z\right)$$ (6)

  and

  
  

  $$\int\limits_{z-l_{\rm down}}^{z}\frac{g}{\theta_{v} }\left( ... ...left( z\right) \right) {\rm d}z^{^{\prime }}=e\left( z\right)$$ (7)

  L is defined as:

  
  

  $$L=\left( l_{\rm up}l_{\rm down}\right) ^{1/2}$$ (8)

• $\phi _{3}$ is a dimensionless function Redelsper & Sommeria (1981). It characterizes the thermal and dynamic stability of the atmosphere and takes into account the spatial variations of temperature, humidity and velocity components. This is an element that permits a parameterization of the model at different scales of motion. Its 3D analytical expression is given in Redelsperger & Sommeria (1981). It is a complex equation linked to the "Redelsperger numbers'' (dimensionless numbers characterizing the thermal and dynamic stability). In a dry 1D option we have a simpler analytical expression:

   
  

  $$\phi _{3}(x,y,z,t)=\frac{1}{1+C_{1} \frac{L^{2}}{e}\frac{g}{\theta_{v} }\frac{\partial \theta_{v} }{\partial z}}$$ (9)

  where C₁=0.139, and L is the above defined mixing length.

L²/e behaves in a different ways in stable and unstable layers. In a very stable layer the mixing length is nearly equivalent to the Deardoff length

 

$$L=\sqrt{\frac{2e}{\frac{g}{\theta_{v} }\frac{\partial \theta_{v} }{\partial z}}}\cdot$$ (10)

Replacing Eq. (10) in Eq. (9) leads to $\phi _{3}=0.78$. In a very unstable layer $\phi _{3}$ takes larger values and an upper bound is set as $\phi _ {3}=2.2$, based on theoretical and experimental results.

Finally, if we substitute Eqs. (5) in (3), we obtain the microscopic quantities related to the macroscopic variables L, e and $\phi _{3}$.

 

$$\overline{w^{\prime }\theta_{v} ^{\prime }}=-0.16L\sqrt{e}\phi _{3}\frac{\partial \overline{{\theta_{v}}} }{\partial z}\cdot$$ (11)

3.3 Optical turbulence parameterization

The Meso-Nh model has been adapted to simulate the optical atmospheric turbulence which is estimated by measuring the structure constant of the temperature fluctuations (Wyngaard et al. 1971; André et al. 1978; Coulman et al. 1986):

 

$$C_{\rm T}^{2}=1.6\varepsilon _{\theta }\varepsilon ^{-1/3}$$ (12)

$\varepsilon _{\theta }$ is the rate of temperature variance destruction by viscous processes and $\varepsilon$ is the rate of energy dissipation related to the turbulence characteristic length L and the energy e by the Kolmogorov law

 

$$\varepsilon =0.7\frac{e^{3/2}}{L}\cdot$$ (13)

Equation (12) assumes that $\theta$ is a passive additive constituent. This is not true in general since buoyancy forces are associated with temperature inhomogeneities and those buoyancy forces are taken into account in other aspects of our computation by the $\phi$ term. This theory is based on the analogy between the velocity fluctuations in a turbulent flow and the concentration fluctuations of a conservative passive additive $\theta$ in a turbulent flow. It is based on a law that links the microscopic and the macroscopic physical parameters. The prognostic equation of the potential temperature variance $\overline{\theta^{\prime 2}}$in the turbulent energy budget is (André et al. 1978):

 

$$\frac{\partial \overline{\theta^{\prime 2}}}{\partial t}=-\f... ...heta}}{\partial z} -\varepsilon _{\theta }-\varepsilon _{\rm R}$$ (14)

where $\varepsilon _{\theta }$ is the molecular dissipation and $\varepsilon _{\rm R}$ the radiative dissipation. Assuming that we can neglect the contributions from the triple correlations $\overline{w^{\prime}\theta^{\prime 2}}$and the radiative dissipation we have

 

$$\frac{\partial\overline{\theta^{2 \prime}}}{\partial t}= -2\... ...partial \overline{\theta}}{\partial z} -\varepsilon _{\theta }.$$ (15)

The steady state balance equation for the rate of destruction of the variance leads to:

 

$$\varepsilon _{\theta }=-2\overline{w^{\prime}\theta^{\prime}}\frac{\partial \overline{\theta}}{\partial z}\cdot$$ (16)

If we substitute (16) and (13) into (12) using Eq. (11) presented in the last paragraph we obtain $C_{\rm T}^{2}$ expressed as a function of macroscopic variables only

 

$$C_{\rm T}^{2}=0.58\phi _{3}L^{4/3}\left( \frac{\partial \overline{\theta} }{\partial z}\right) ^{2}.$$ (17)

Finally, the structure constant of the refraction index is obtained using the Gladstone's law:

 

$$C_{\rm N}^{2}=\left( \frac{80\ 10^{-6}P}{T^{2}}\right) ^{2}C_{\rm T}^{2}.$$ (18)

3.4 Optical turbulence code for Astronomy

Today we need to measure seeing and other parameters such as the coherence wavefront time ($\tau_{\rm AO}$), the isoplanatic angle ($\theta_{\rm AO}$) and the spatial coherence outer scale (${\cal L}_0$). All these parameters are related to the refractive index fluctuations which appear inside the atmospheric turbulent layers and to the wind velocity profile. We summarize, in the following, the parameters which have been coded in the model. All the following parameters are referred to the zenith direction. Starting from the Fried parameter r₀ given by:

 

$$r_{0}=\left[ 0.423\left( \frac{2\pi }{\lambda }\right) ^{2}... ...}^{\infty }C_{\rm N}^{2}\left( h\right) {\rm d}h\right] ^{-3/5}$$ (19)

the seeing $\varepsilon$, defined as the width at the half height of a star image at the focus of a large diameter telescope was expressed as follows (Roddier 1981):

 

$$\varepsilon =0.98\frac{\lambda }{r_{0}}$$ (20)

where $\lambda$ is the optical wavelength. As others authors we choose $\lambda = 0.5\ 10^{-6}$ m for this study.

The isoplanatic angle $\theta_{\rm AO}$ (Roddier et al. 1982; Fried 1979) defined as the maximum angular separation of two stellar objects producing similar wavefronts at the telescope entrance pupil, has the following analytical expression:

 

$$\theta_{\rm AO}=0.31\frac{r_{0}}{h_{\rm AO}}.$$ (21)

Knowing that the seeing layer $h_{\rm AO}$ is a sort of average distance weighted by a 5/3 power law

 

$$h_{\rm AO}=\left[ \frac{\int\limits_{0}^{\infty }h^{5/3}C_{\... ...}^{\infty }C_{\rm N}^{2}\left( h\right) {\rm d}h}\right] ^{3/5}$$ (22)

we obtain  

$$\theta_{\rm AO}=0.057\cdot\lambda^{6/5}\left({\int\limits_{0... ...h^{5/3}C_{\rm N}^{2}\left( h\right) {\rm d}h}\right) ^{-3/5} .$$ (23)

The scintillation rate (Roddier 1981) was expressed as follows

 

$$\sigma ^{2}_{\rm I}=19.12 \lambda^{-7/6}\int\limits_{0}^{\infty }h^{5/6}C_{\rm N}^{2}\left( h\right). {\rm d}h$$ (24)

The coherence wavefront time $\tau_{\rm AO}$ (Roddier et al. 1982) is:

 

$$\tau _{\rm AO}=0.31\frac{r_{0}}{\upsilon _{\rm AO}}$$ (25)

where  

$$\upsilon _{\rm AO}=\left[ \frac{\int\limits_{0}^{\infty}\le... ...{\infty }C_{\rm N}^{2} \left( h\right) {\rm d}h }\right] ^{3/5}$$ (26)

and $\vec{V}$(h) is the horizontal wind velocity vector. Thus, $\tau_{\rm AO}$ can be expressed as:  

$$\tau _{\rm AO} = 0.057 \cdot\lambda^{6/5} \left[\int\limits_... ...ert ^{5/3}C_{\rm N}^{2}\left(h\right) {\rm d}h\right]^{-3/5} .$$ (27)

Finally, the spatial coherence outer scale (Borgnino 1990)

 

$${\cal L}_{0}=\left[\frac{\int\limits_{0}^{\infty } {L(h)}^{-... ...0}^{\infty }C_{\rm N}^{2}\left( h\right) {\rm d}h} \right]^{-3}$$ (28)

where L is the Bougeault-Lacarrere mixing length defined in Sect. 3.2.