## Q ep Nqdqdx3y

These relations can be used to estimate the eddy fluxes of momentum u2M, heat H/pcp and humidity E/p if the vertical gradients of the mean velocity, temperature and specific humidity, and also the rate e = v(du'/dxk)2 of turbulent energy dissipation and the rates NT = Xr(dT'/dxk)2, Nq = iq(dq'Icxk)2 of equalization of temperature and specific humidity inhomogeneities are known. To determine the latter, information about fifteen spatial derivations is required, nine of them appearing in s, and six in NT and Nq. But if turbulence is considered to be isotropic and Taylor's hypothesis on frozen turbulence is applied, that is, d/dxj = w_1 d/dt is assumed to be valid, then the expressions for e, NT, and Nq will take the form

. = (srv uf\dt)' T u2\dt)' 4 U2\dt from which it follows that the required estimates of s, NT and Nq can be obtained from measurements of the temporal derivatives of wind velocity, temperature and specific humidity fluctuations.

In practice, a frequent assumption is the existence of an inertial interval where the energy supply and dissipation are in fact absent, and there is only the cascade energy transfer from large eddies to smaller eddies. In this interval the spatial velocity spectrum is isotropic, it does not depend on viscosity and is defined only by the energy flux over the spectrum equal to the rate of viscous dissipation. Then, based on dimensional considerations, the spatial spectrum Fx x(k) of any velocity component has to obey the 'five-thirds power' law .Fii(/i) = aifi2'3^"5'3, where k is the wave number assigned in radians per unit length. Similarly, in the domain of overlapping of inertial intervals for velocity and temperature (humidity) where the viscous dissipation and degeneration of temperature (humidity) inhomogeneities become unimportant, the spatial temperature FT(k) and humidity Fq(k) spectra are determined by two parameters: the viscous dissipation and the respective rate of equalization of inhomogeneities. In this case from dimensional considerations FT(k) = pTNTs-1/3k~5'\ Fq(k) = PqNq£~1/ik~513, where ax = 0.55 ± 0.11, pT = 0.80 ±0.16, and = 0.58 ± 0.20 are universal constants (Kolmogorov's constants).

So, if the spatial spectra F1{(k), FT(k), Fq{k), or the corresponding frequency spectra Fn(f), FT(f), Fq(f) defined in accordance with Taylor's hypothesis by transition from k to 2nfjux (here / is a frequency) are known in the inertial interval then estimation of e, NT and Nq reduces to simple calculation by formulae.

Let us return to (3.10.15) and, just as in Section 3.8, introduce non-dimensional vertical gradients of the mean velocity <1>u = (kx3/mh!) dujdx3 temperature 3>r = (x3/Tsl.) dT/dx3 and specific humidity <t>9 = (x3/q^.) dq/dx3, where = ( — H/pc^/Ku^ and q^ = ( — E/p)/KUit. are temperature and specific humidity scales. Then instead of (3.10.15) we have where L is the modified Monin-Obukhov length scale defined by Equation

As may be seen, under known values e, NT and Nq, there is a single-valued correspondence between eddy fluxes of momentum, heat and humidity, on

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