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The above relationships are in fact the laws of resistance and heat and humidity exchange for the PBL. With their help and knowing the form of the functions A, B, C and D, one can set a connection between the geostrophical friction coefficient u^/G, the angle a, the coefficients of heat exchange T^/(6h — 60) and evaporation q*/(qh — q0\ as well as the stratification parameter /i0 = K3(u!|!/G)~1(j8r!|! + 0.61^^.)/|/|G on the one hand, and external parameters G, (6h — 0o), (qh — q0), f and z0 of the planetary boundary layer on the other.

The law of resistance for the neutrally stratified PBL was first obtained by Kazanskii and Monin (1961). The relations (4.3.4) for the stratified PBL were recommended by Zilitinkevich (1970). Later they were repeatedly verified. As a result it was shown that experimental data generally confirm the universal character of dependencies A(p0), B(p0), C(p0) and D(p0) despite the inevitable spread.

Following Zilitinkevich (1970) we consider the asymptotic properties of the functions A, B, C and D for stable (p0 < 0) and unstable (p0 > 0) stratification. According to (4.2.1), (4.2.2) and considerations presented in Section 4.1 the divergence of the vertical eddy momentum flux in the outer part of the PBL is of the order of u^/\f\hE, where hE is the PBL thickness defined depending on conditions of hydrostatic stability by Equation (4.1.4) or Equation (4.1.5). Taking this circumstance into account as well as the fact that hE ~ (fiH/pcp)l/2\f\~312 for the strongly unstable stratification and that pH/pcp and p0 are connected to each other by the relationship ¡}H/pcp = (|/|uI/k2)( — p0), we have the following expressions for defects in the velocity components in the outer part of the PBL:

where \J/U, i¡/y are universal functions which differ from those in (4.3.1).

We match (4.3.5) with respective asymptotic formulae u(z) = (ujK)[_au + cu(z/-L)~1/3 - ln(z0/-L)], v(z) = 0, complying with the surface atmospheric layer for the free convection regime (see (3.8.7)), where au and cu are non-dimensional constants. As a result we arrive at the same law of resistance as in (4.3.4) but now

A(p0) = (-po)" 1/2Ml5 B(ji0) = (-p0yll2M2 + ln{-p0)/K - au.

For strongly stable stratification when hE ~ — fiH/pcf)~l,2\f\ ~1/2 the expressions for velocity defects in the outer part of the PBL take the form u(z) - G cos a = (u*/K)pll2il/u(z/hE), j v(z) - G sin a = (uJk^^Vv^/M- J

Combining them with the asymptotic formulae u(z) = (ujK)lba(z/L) - In(z0/L)], v(z) = 0, (4.3.8)

complying with the surface atmospheric layer for the strong stability regime (see (3.8.8)), we arrive again at the first two relations in (4.3.4) but now

Similarly, the expressions for defects of temperature and specific humidity in the outer part of the PBL in the regimes of strong instability and strong

stability are written in the form

where, depending on conditions of stratification, hE is given either in the form of (4.1.4) or in the form of (4.1.5). Matching these expressions with their respective expressions

0(z) -90= T£ae + ce(z/-L)"1/3 - (l/a°) ln(z0/-L)], q(z) ~q0 = q*la9 + cq(z/-L)~^ - (l/a°) ln(z0/-L)], for the surface atmospheric layer in the regime of free convection and

0{z) -0O = T,[be(z/L) - (l/a°) ln(z0/L)], q(z) -Go = q*lbq(z/L) - (l/a°) ln(z0/L)],

in the regime of strong stability we obtain the last two relations in (4.3.4). As this takes place it turns out that C(n0) = ln( — ¡Xq/k) + <xqM3 — u%ae, D(n0) = In( — h0/k) + a°M4 — aqaq in the first case and C(/i0) = ln(/i0/fc) + ocg N3, D(fi0) = \n(/x0/K) + aqN4 in the second case. Here M3, M4, N3, iV4, as well as Mu M2, N2, are the numerical constants derived from experimental data. We note that experimental estimates of the universal functions A, B, C and D are subjected to large scattering (see Figure 4.1), caused, firstly, by the approximate character of the theory ignoring the influence of non-stationarity and horizontal inhomogeneity of real fields of wind velocity, air temperature and humidity, and, secondly, by errors of handling of experimental data arising from indeterminacy in selection of the PBL thickness. We pay attention also to the strong sensitivity of the universal functions A, B, C and D to variations in the parameter /i0, which dictates the necessity of considering the effects of stratification when calculating the characteristics of the interaction between the PBL and the underlying surface.

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