Y3

At 7 = 0.25 and observed values of the resistance coefficient C„« 10"3 the index of the exponent in (3.9.19) is equal to 0.4. Therefore, presenting the exponent as a series and discarding all the terms of the expansion, with the

Surface fluxes of momentum, heat and humidity exception of the first two, we obtain the expression

which conforms qualitatively to the expression obtained on the basis of experimental data.

3.10 Methods for estimating surface fluxes of momentum, heat and humidity

At the present time, four methods of estimating momentum, heat and humidity surface fluxes have gained wide acceptance; they are eddy correlation, dissipation, gradient and bulk methods. The first reduces to calculation of the covariances u'w', w'T' and w'q' from measured values of fluctuations of the wind velocity u!, w', temperature T' and specific humidity q', and to subsequent averaging of these covariances in time, as we have already discussed in Section 1.4. We now consider the other three methods.

Dissipation method. We start with the derivation of an equation for velocity, temperature and specific humidity dispersions. For this purpose we represent the pressure p and density p as the sum of their values complying with the hydrostatic equation and depending only on height, and on departures from them. We ascribe the subscript S for the former, and subscript D for the latter. Then taking into account that pD « ps to within the terms of the second order in (pD/ps) we have Vp/p ss Vps/ps + VpD/ps - {Pd/PsWPs/Ps, where V is the gradient operator. For ease of subsequent presentation we turn to tensor designations, setting x1 = x, x2 = y, x3 = z, and assume as usual that all subscripts which are encountered hereinafter take on values 1, 2 and 3, and an subscript which is repeated twice in any term is the equivalent of summation over all values of this subscript. In tensor designations the expression for Vp/p with an allowance for the hydrostatic equation is dp/p cbc; = — gS3i, and the equation of state pD/ps ~ — TD/TS — 0.61gD (see Section 3.8) transforms into the form dp/p dx{ « <9ps/ps ¿bc; -(- dpD/ps dx,- — g(TD/Ts + 0.61gD)(53i, where 5jj is the Kronecker symbol equal to zero at i ± j, and to 1 at i = j. Substitution of the expression for dp/p dxt into the Navier-Stokes equation, previously rewritten in divergence form after rejection of small (within the surface atmospheric layer) terms describing the Coriolis acceleration and changing ps and with height yields dujdt + dutuk/dxk = -dpD/p0 dxt + (/3TD + 0.61gqD)ô3i + v d2ujdxl, (3.10.1)

where ut and uk are instantaneous values of the ith and fcth velocity components; v is the coefficient of kinematic viscosity; ft = g/T0 is the buoyancy parameter; p0 and T0 are reference values of density and absolute temperature.

We decompose uh uk, pD, TD and qD into mean and fluctuating components. The first will be designated by overbars, the second by primes. Then for w; we come to the Reynolds equation:

dujdt + d(UiUk + u[u'k)/dxk = —dp/p0 dxt + (/IT + 0.61 gq)ô3i + v d ujdxk,

and, for m-, to the equation du'Jdt + d(u\uk + utu'k + u'iu'k — u[u'k)/dxk

= -dp'/po ^ + (PT' + 0.6igq')S3i + v dht'Jdxl, (3.10.3)

is obtained by the term-wise subtraction of (3.10.2) from (3.10.3). Hereinafter the subscript D is omitted.

We multiply Equation (3.10.3) by u'j, and the same equation written for u'j by u •; then we sum them, take advantage of the identities

(u'j dp'/dXi — u[ dp'/dxj) = (du'jp'/dxi + du[p'/dXj) — p'(du'j/dxi + du'i/dxj), ou'j d2u'Jdxl + u[ d2u'j/dxi) = Id^u'j/dxl - 2(8u'i/dxk)(du'j/dxk)']

and the continuity equation (3.10.3) for the mean (duk/dxk = 0) and fluctuating ('du'k/dxk = 0) motions, and then we average the expressions obtained. As a result we have d —¡—, - d —r~i —r~i ^ , , , — U(U: + Uk-U{U: + I UjUk--h UtUk-- I H--UjUjUk dt dxk \ dxk dxkJ dxk

0 0

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