Atmospheric primitive equations

The atmospheric primitive equations are a simplified version of the Navier-Stokes equations. They are applicable in the case of a fluid to the surface of a sphere while posing as assumptions that the vertical component of the movement is much weaker than the horizontal component and than the layer of fluid is very thin relative with the ray of the sphere. These assumptions correspond in general to flow with large scales, known as synoptic scale, of the terrestrial atmosphere and these equations are thus applied in Météorologie and Océanographie. The digital model of forecast of time solve these equations or an alternative of those in order to simulate the future behavior of the atmosphere.

In addition, the primitive equations applied to the Océanographie make it possible to simulate the behavior of the seas. Reduced to only one dimension, they solve the equations of Laplace of the tide, a problem of eigenvalues which one analytically obtains the athwartship structure of oceanic circulation.

Definitions

In general, all the forms of primitive equations connect five variables and their evolution in time:

• $u$ is speed Zonale (direction is/western tangent with the sphere).

• $v$ is speed Méridional E (NORTH-SOUTH).
• $\backslash \; omega$ is the vertical movement
• $T$ is the Température
• $\backslash \; phi$ is the Géopotentiel

They also use known variables:

• $f$ is the factor of Coriolis which is equal to $2\; \backslash \; Omega\; sin\; (\backslash \; phi)$ with $\backslash \; Omega$ the rate of rotation of the Ground ($2\; \backslash \; pi/24$ radians/hour) and $\backslash \; phi$ the latitude.

• $R$ is the constant of the Perfect gas
• $p$ is the Pression
• $c\_p$ is the Specific heat to constant pressure
• $J$ is the flow of Chaleur per unit of time and of mass
• $\backslash \; pi$ of Exner
• $\backslash \; theta$ is the potential Température

is the function

Various forms of the primitive equations

The representation of the primitive equations depends on the vertical coordinates used. One can use the Pression, the Logarithme of the pressure or the coordinates known as “sigma” which are a relationship between the pressure on a level and the pressure of surface. Moreover, the Speed, the Temperature and the Géopotentiel can be broken up into their median value and their value of disturbance according to the decomposition of Reynolds.

In coordinates of pressure and Cartesian born

If one uses the pressure as coordinates vertical and (X, there) like coordinates horizontal tangential with the Sphère, by neglecting the curve of the Ground, one obtains a simple representation of the physical processes concerned:

• the movement of the air or the seas in a system in rotation is a balance between various forces: the Force of Coriolis, the gradient of pressure, the Revolved, the centripetal Force and the Friction. According to the second principle of Newton, one adds these forces to know the total force which is exerted on the fluid:

$\backslash \; frac\; \{D\; \backslash \; vec\; \{V\}\}\; \{dt\}\; =\; F\; \backslash \; vec\; \{V\}\; -\; (\backslash \; nabla\; p\; \backslash \; rho)\; -\; \backslash \; vec\; g^*\; \backslash \; +\; \backslash \; frac\; \{\backslash \; vec\; V^2\}\; \{R\_c\}\; +\; F\_\; \{R\}\; \backslash \; qquad\; \backslash \; begin\; \{boxes\}\; F\_r\; =\; Friction\; \backslash \; \backslash \; \backslash \; vec\; g^*=constante\; \backslash \; of\; \backslash \; climbs\; \backslash \; acute\; \{E\}\; \backslash \; verticale=\; 9,8\; m/s^2\; \backslash \; \backslash \; R\_c\; =\; Rayon\; \backslash \; of\; \backslash \; curve\; \backslash \; of\; the\; \backslash \; flow\; \backslash \; end\; \{boxes\}$

the geostrophic equations of the movement (the pressure and forces of being opposed Coriolis) are obtained by neglecting the Friction and the centripetal Force. As the movement is horizontal $\backslash \; vec\; g^*$ is null being vertical. Thus by changing for the coordinates into Geopotential $\backslash \; phi$, one obtains:

$\backslash \; frac\; \{of\; the\}\; \{dt\}\; -\; F\; v\; =\; -\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; phi\}\; \{\backslash \; partial\; X\}$

$\backslash \; frac\; \{FD\}\; \{dt\}\; +\; F\; U\; =\; -\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; partial\; phi\}\; \{\backslash \; there\}$

• the equation of the hydrostatic balance for the special case of the vertical movement in which there is no acceleration:

$\backslash \; frac\; \{\backslash \; partial\; \backslash \; phi\}\; \{\backslash \; partial\; p\}\; =\; -\; \backslash \; frac\; \{R\; T\}\; \{p\}$

• the equation of continuity which connects the convergence/divergence of horizontal mass with the vertical movement (no creation/loss of mass right change of level):

$\backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; X\}\; +\; \backslash \; frac\; \{\backslash \; partial\; v\}\; \{\backslash \; partial\; there\}\; +\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; Omega\}\; \{\backslash \; partial\; p\}\; =\; 0$

• the Thermodynamic equation of energy coming from the First law of thermodynamics

$\backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; T\}\; +\; U\; \backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; X\}\; +\; v\; \backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; there\}\; +\; \backslash \; Omega\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; p\}\; +\; \backslash \; frac\; \{R\; T\}\; \{PC.\; \_p\}\; \backslash \; right)\; =\; \backslash \; frac\; \{J\}\; \{c\_p\}$

By adding an equation of composition which connects the water contents of the air (or of salt in the sea) and its variation in space one a whole of variables obtains which describe the behavior of the atmosphere (sea).

In coordinates sigma and stereographic polar projection

If one not divides the atmosphere into absolute pressure but into levels having the same relationship with the pressure of surface, one speaks about coordinates sigma. For example, if one divides the atmospheric layer into three levels: surface, level or the pressure is half of that on the ground and top of the atmosphere (null pressure); there would be sigma = 1, sigma = 0,5 and sigma = 0.

In addition, stereographic polar projection can be represented using a plan which one poses on a pole and on which one projects the contour of the continents as if a light illuminated the sphere since the center of the Earth. One obtains like result a plane cartographic projection. This projection can be regarded as a sight of the Earth above the north pole. This projection is appropriate for the fields located in the vicinity of the northern latitude 60 degrees. It is valid to the north pole but it is not recommended for areas close to the equator because the deformations increase in an important way as soon as one approaches some.

By using these two types of coordinates, one can simplify the primitive equations as follows:

• Temperature: $\backslash \; frac\; \{\backslash \; delta\; T\}\; \{\backslash \; delta\; T\}\; =\; U\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; T\}\; \{\backslash \; delta\; X\}\; \backslash \; right)\; +\; v\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; T\}\; \{\backslash \; delta\; there\}\; \backslash \; right)\; +\; W\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; T\}\; \{\backslash \; delta\; Z\}\; \backslash \; right)$

• Wind U: $\backslash \; frac\; \{\backslash \; delta\; U\}\; \{\backslash \; delta\; T\}\; =\; \backslash \; eta\; v\; -\; \backslash \; frac\; \{\backslash \; delta\; \backslash \; Phi\}\; \{\backslash \; delta\; X\}\; -\; C\_p\; \backslash \; Theta\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; \backslash \; pi\}\; \{\backslash \; delta\; X\}\; \backslash \; right)\; -\; Z\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; U\}\; \{\backslash \; delta\; \backslash \; sigma\}\; \backslash \; right)\; -\; \backslash \; frac\; \{\backslash \; delta\; \backslash \; frac\; \{\backslash \; left\; (u^2\; +y\; \backslash \; right)\}\{2\}\}\; \{\backslash \; delta\; there\}$

• Wind v: $\backslash \; frac\; \{\backslash \; delta\; v\}\; \{\backslash \; delta\; T\}\; =\; \backslash \; eta\; \backslash \; left\; (\backslash \; frac\; \{U\}\; \{v\}\; \backslash \; right)\; -\; \backslash \; frac\; \{\backslash \; delta\; \backslash \; Phi\}\; \{\backslash \; delta\; there\}\; -\; C\_p\; \backslash \; Theta\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; \backslash \; pi\}\; \{\backslash \; delta\; there\}\; \backslash \; right)\; -\; Z\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; v\}\; \{\backslash \; delta\; \backslash \; sigma\}\; \backslash \; right)\; -\; \backslash \; frac\; \{\backslash \; delta\; \backslash \; frac\; \{\backslash \; left\; (u^2\; +y\; \backslash \; right)\}\{2\}\}\; \{\backslash \; delta\; there\}$

• Contained out of water: $\backslash \; frac\; \{\backslash \; delta\; W\}\; \{\backslash \; delta\; T\}\; =\; U\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; W\_x\}\; \{\backslash \; delta\; X\}\; \backslash \; right)\; +\; v\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; W\_y\}\; \{\backslash \; delta\; there\}\; \backslash \; right)\; +\; Z\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; W\_z\}\; \{\backslash \; delta\; Z\}\; \backslash \; right)$

• Thickness of pressure: $\backslash \; frac\; \{\backslash \; delta\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; p\}\; \{\backslash \; delta\; sigma\}\; \backslash \; right)\}\{\backslash \; delta\; T\}\; =\; U\; \backslash \; left\; \backslash \; frac\; \{\backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; p\}\; \{\backslash \; delta\; sigma\}\; \backslash \; right)\; \_x\}\; \{\backslash \; delta\; X\}\; \backslash \; right\; +\; v\; \backslash \; left\; \backslash \; frac\; \{\backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; p\}\; \{\backslash \; delta\; sigma\}\; \backslash \; right)\; \_y\}\; \{\backslash \; delta\; there\}\; \backslash \; right\; +\; Z\; \backslash \; left\; \backslash \; frac\; \{\backslash \; left\; (\backslash \; frac\; \{\backslash \; delta\; p\}\; \{\backslash \; delta\; sigma\}\; \backslash \; right)\; \_z\}\; \{\backslash \; delta\; Z\}\; \backslash \; right$

Where $\backslash \; begin\; \{boxes\}\; \backslash \; Theta:\; Temp\; \backslash \; acute\; \{E\}\; potential\; erasure\; \backslash \; \backslash \; \backslash \; \backslash \; \backslash \; Phi:\; G\; \backslash \; acute\; \{E\}\; opotentiel\; \backslash \; \backslash \; \backslash \; sigma:\; level\; \backslash \; sigma\; \backslash \; end\; \{boxes\}$

In this frame of reference, several variables (the such temperature, the potential Temperature and water contents) remain on the same level sigma and move with the wind on this level. This last is calculated by using the height of geopotential, the specific heat, the function of Exner ($\backslash \; pi$) and the change of level sigma.

Solution of the primitive equations

The analytical solution of the primitive equations give waves sine oïdales which vary in time and space. One can thus break up it into Harmonique S whose coefficients are connected to the Latitude and altitude, which gives atmospheric waves and tides.

$\backslash \; begin\; \{Bmatrix\}\; U,\; v,\; \backslash \; phi\; \backslash \; end\; \{Bmatrix\}\; =\; \backslash \; begin\; \{Bmatrix\}\; \backslash \; hat\; U,\; \backslash \; hat\; v,\; \backslash \; hat\; \backslash \; phi\; \backslash \; end\; \{Bmatrix\}\; e^\; \{I\; (S\; \backslash \; lambda\; +\; \backslash \; sigma\; T)\}$

$S\; \backslash \; and\; \backslash \; \backslash \; sigma$ is respectively the zonal number of wave and the angular frequency.

To obtain this solution, it is necessary to linearize the equations and in general to simplify them by using often unrealistic assumptions (not dissipation of the waves, isothermic atmosphere, etc). In the practical applications of these equations, the forecast of time for example, one uses methods of numerical Analyze which call upon the division of the waves in discrete values in order to take account of all the variables.

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