Simulating the Atmosphere : the Navier-Stokes Equations


The progress accomplished in the last years in the numerical forecasting of the weather has been considerable. The quality of the forecasts produced by Meteo-France for three days hence is the same as that of forecasts for 24 hours into the future from 20 years ago. In part, that's due to mathematics.

The atmosphere is a complex machine. It is a fluid in perpetual motion, composed of different gases, of dust and of liquids in suspension. It moves around, it deforms, it mixes, transforms itself ceaselessly although governed by its own physical and chemical constraints and its interactions with its external surroundings. It receives solar radiation, which it re-emits in part into interplanetary space, and exchanges energy and water vapor with the oceans, glaciers, the ground and with vegetation. It induces a greenhouse effect without which the earth would be an icy planet. It redistributes a part of the solar energy received by the tropical regions towards the polar and temperate regions . By virtue of friction, it drags along the ocean surface and creates surface currents. Its flows are perturbed by mountains, valleys, changes of surface roughness and contrasts between land and ocean. Man-made pollution affects its chemical composition and its aptitude to form clouds or precipitation.

Simplifying the Equations

As it respects the laws of Fluid Mechanics and of Thermodynamics, the evolution of the atmosphere could be simulated with the aid of mathematical models. So as to forecast the weather consisting of simulating the behavior which the atmosphere will adopt in the hours or the days to come, in response to its internal constraints (behavior of gas, conservation of mass, change of phase of water,....) and its exchanges with the surroundings (friction, conduction, radiation, evaporation, ....).

The principal equations involved in the simulation of the atmosphere are the following

  • The equations of motion allowing us to describe the movement of the air, in three dimensions (in an ($x,y,z$) basis - $u$ and $v$ are the horizontal components of the speed of displacement of the particles, $w$ is the vertical component) on taking account of the terrestial attraction, of the pressure p, of the viscosity and of the interial forces related to the rotation of the Earth (centripetal acceleration and Coriolis acceleration).
  • The equation of state of the air represents the behavior of molecules in a volume of fluid (the pressure exerted by the air molecules is a function of the density and their speed of displacement).
  • The equation of continuity describes the conservation of mass of the atmospheric air.
  • The equation of thermodynamics allows the changes in thermodynamic energy to be represented as a function of radiation, of the conduction of air, of evaporation and of the condensation of water.
  • The equations of the change of phase of water describing the transformations of atmospheric water in its forms as vapor, as cloud (droplets, crystals of ice) or as precipitation (rain, snow, sleet, hail,...)
  • The equations describing exchanges with the surroundings allow the representation of exchanges which occur in the form of radiation, of heat, of friction, of precipitation - evaporation. The models can in particular take into account various parameters like the type of surface (land,water, ice sheet,...), the type of surface (sand, clay, rock, urban zone, ...) or the type of vegetation (farming, forest,...).

You can see that the parameters which come into play are extremely numerous and complex. The number of arithmetic operations that need to be performed in order to produce a forecast is colossal, so that a forecast covering 24 hours could take several days, or even several months, with current computers, even the most powerful. In order to bypass this difficulty and simplify the operations, the different parameters undergo an analysis of their order of magnitude which allows the elimination of those which have the least effect on the forecast, or at least to represent them in a fashion more simple and more approximate in order to minimise their effect on the time of calculation and the allocation of computer resources. Certainly the results will be less precise, but it enables us to obtain them within an appropriate time, sufficiently early for the forecast to be used. In the models, the atmosphere is generally represented by its principal parameters (pressure, wind, temperature, humidity). After numerous simplifications, the principal equations could be written as follows.

Let us commence with the equations of motion, in which the motions are accelerated or slowed by the forces of pressure with components \[ -\frac{1}{\rho} \frac{\partial p}{\partial x}, \ \ \ - \frac{1}{\rho} \frac{\partial p}{\partial y},\ \ \ - \frac{1}{\rho} \frac{\partial p}{\partial z} \] ($\rho$ is the density of the atmosphere)

the inertial forces related to the rotation of the Earth \[ -2\Omega (w\cos\phi - v\sin \phi) \] and \[ -2\Omega u \sin \phi\] ($\Omega$ indicates the speed of rotation of Earth and $\phi$ is the latitude)

and the horizontal component $F_{fx}$ and $F_{fy}$ of the frictional forces.

The equations resulting from the above considerations are \[ \left\{ \begin{array}{l} \frac{du}{dt} = -\frac{1}{\rho} \frac{\partial p}{\partial x} - 2 \Omega(w\cos\phi - v\sin \phi) + F_{fx}\\ \frac{dv}{dt} = -\frac{1}{\rho} \frac{\partial p}{\partial y} - 2 \Omega u\sin\phi + F_{fy}\\ \frac{dw}{dt} = -\frac{1}{\rho} \frac{\partial p}{\partial z} - g\end{array}\right.\]

In the equation of thermodynamics, the variations in temperature T as a function of time t are governed by the variations of pressure \[ \frac{R}{C_p} \frac{T}{p} \frac{dp}{dt} \] (where $R= 8.3145 \ J \ mol^{-1} \ K^{-1}$ is the ideal gas contant and $C_p$ is the specific heat of humid air at constant pressure in $J \ kg^{-1}\ K^{-1}$)

and by the rate Q of heat contributed by radiation, conduction, mixing and changes of phase of the water. It is written \[ \frac{dT}{dt} = \frac{R}{C_p} \frac{T}{p} \frac{dp}{dt} + Q \] The equation of state connects the pressure, temperature and density. If the air is considered to be an ideal gas, it can be written in the form \[ p= \rho RT \] The equation of continuity describes the conservation of mass of the atmospheric air, as well as the increase in density due to compression and the reduction due to expansion. It can be formulated thus \[ \frac{1}{\rho}\frac{\partial\rho}{\partial t} = - \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} - \frac{\partial w}{\partial z} \]

As one can see in the preceding articles, one can still simplify the problem by supposing the atmosphere is isolated, without any exchanges with the exterior, and on neglecting the components $F_{fx}$ and $F_{fy}$ of the frictional force as well as the contributions of heat by radiation and by conduction. Another simplification, a classic in the forecasting on a grand scale, is the hydrostatic approximation, which involves considering that the vertical acceleration of speeds are negligible, or \[ \frac{\partial w}{\partial t} = 0 \] Finally, one can re-formulate the third equation of motion as follows \[ \frac{\partial p}{\partial z} = -g\rho \] In practice, the forecast cannot be realised via the small packets of fluids which possess a great variability difficult to measure and which would be too numerous to permit a reasonable time for the calculations. In the models, as the preceding articles have shown, the principal parameters are therefore averaged over the small "virtual boxes" and only processes of dimensions noticeably greater to that of the box are represented in an explicit manner.

Finally the system of equations thus obtained does not have exact solitions, directly calculable. The numerical methods employed yield only aproximate solutions. The space is divided into small juxtaposed geometrical volumes, forming a network in the space. The dimension of the volumes of this network defines the power of resolution of the model. The equations are thus written to represent the variation of different parameters in the volume of interest (Eulerian derivatives of the form $\frac{\partial A}{\partial t}$) and not within the parcels of air in motion themselves (Lagrangian derivatives of the form $\frac{dA}{dt}$). For all parameters A, the local variation is thus written \[ \frac{\partial A}{\partial t} = -u\frac{\partial A}{\partial x} -v \frac{\partial A}{\partial y} - w \frac{\partial A}{\partial z} + \frac{dA}{dt} \] The equations of motion can thus now be written in the form :

\[ \left\{ \begin{array}{l} \frac{du}{dt} = -u\frac{\partial u}{\partial x} -v \frac{\partial v}{\partial y} - w \frac{\partial w}{\partial z} -\frac{1}{\rho} \frac{\partial p}{\partial x} - 2 \Omega(w\cos\phi - v\sin \phi) + F_{fx}\\ \frac{dv}{dt} = -u\frac{\partial u}{\partial x} -v \frac{\partial v}{\partial y} - w \frac{\partial w}{\partial z} -\frac{1}{\rho} \frac{\partial p}{\partial y} - 2 \Omega u\sin\phi + F_{fy}\\ \frac{dw}{dt} = -u\frac{\partial u}{\partial x} -v \frac{\partial v}{\partial y} - w \frac{\partial w}{\partial z} -\frac{1}{\rho} \frac{\partial p}{\partial z} - g\end{array}\right.\]

The mathematical treatment of such a system of equations, attributed to Henri Navier and George Gabriel Stokes, is horribly complex, so much so that even today no solutions can be found except for approximate solutions. Its complete resolution constitutes one of the problems of the Prix de Millenium, a collection of seven maths challenges posed in 2000 by the Clay Institute of Mathematics. Each of the problems is endowed with a prize of a million dollars offered by the institute. Up to now, only one of the problems has been resolved : the Poincaré Conjecture. Which will be the next on the list?

Jean-Pierre Chalon, Tangente 160

  • Henri Navier

    Henri Navier (1785 - 1836)

    George Stokes

    George Stokes (1819-1903)