= -/[ps]M + [ps]{MM + ["*«*] + ["'«']} —


Here O = gz; co = dpjdt is the isobaric vertical velocity; a = p/ps, ps is the surface pressure; R is the gas constant for air; Fx, F(fn FT and Fq are sources and sinks of momentum, heat and moisture, respectively; other symbols are the same.

System (5.7.1) contains 20 unknown second moments (additional unknowns appear in zonal averaging of the terms Fx, Fv, FT and Fq if they have non-linear structure and when formulating boundary conditions at the underlying surface if its topography is taken into consideration). Among the 20 unknown second moments some (with primes) owe their origins to transient synoptic disturbances, others (with asterisks) owe their origins to steady motions. The common method of description of second moments with the help of the hypothesis of the semiempirical theory of turbulence (second moments are considered to be proportional to gradients of appropriate averages with proportionality factors having the sense of coefficients of virtual diffusion of momentum, heat and moisture) does not work here because these factors turn out to be complicated functions of spatial coordinates and some of them (say, those appearing by the parametrization of [Vu']), even become negative.

A more acceptable method of parametrization of the meridional flux [u V] of zonal momentum was proposed by Williams and Davis (1965). They proceeded from the fact that synoptic disturbances draw their energy from the available potential energy of zonal motions, a measure of which is the meridional temperature gradient. Subsequently, the momentum flux

[mV] is presented in the form

providing the possibility of the existence of negative viscosity in jet streams. Here L0 = c/Q is the horizontal scale of barotropic synoptic disturbances; c is the sound velocity; il is the angular velocity of the Earth's rotation; k is the non-dimensional factor assumed to be proportional to (z/H0)2; z is the vertical coordinate; H0 = c2/rcg is the height of the homogeneous atmosphere; k = cp/vv is the ratio of heat capacities at constant pressure and constant volume.

The remaining second moments created by transient synoptic disturbances were described by ordinary diffusion formulae with positive coefficients of virtual viscosity, heat conductivity and diffusion. Such a parametrization of second moments of the [a'fc'] type was used by Dymnikov et al. (1979), and by Williams and Davis (1965).

A different method of parametrization of the second moments [a'fe'] was developed by Green (1970). The essence of this is to connect zonal average fluxes of conservative characteristics created by transient synoptic disturbances with the average gradients of these characteristics using the matrix of transport coefficients, the dependence of which on spatial coordinates can be established from an analysis of zonal flow instability. Considering this, the intensity of the potential temperature transport is expressed in terms of horizontal and vertical gradients of potential temperature (anisotropic diffusion), and the intensity of quasi-geostrophic potential vorticity transport is expressed only in terms of the horizontal gradient of potential vorticity.

Following Green (1970) we obtain the expression for second moments. We start with the definition of the quasi-geostrophic potential vorticity. In Cartesian coordinates it has the form where n and £ are the potential and relative vorticities; rj is the logarithm of potential temperature; B = dlnri/dz is the parameter of static stability in the environment; p is the density; j? is the change in the Coriolis parameter / with latitude; y is the meridional coordinate; the symbol 3 signifies the departure from an undisturbed state which is a function of the vertical coordinate only.

On the basis of (5.7.3) the meridional potential vorticity flux will be

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