8 dt 8z pcp where 6 and T are potential and absolute temperatures and cp is the heat capacity of air at constant pressure.
If the left-hand side of Equation (3.1.2) has order 0.1 x 10" 3 °C/s and the characteristic value of \H\/pcp is equal to 2.0 x 10_2oCm/s then for a thickness of the layer within the limits of which the relative variation of the vertical heat flux H is less than 20% the estimate h2 > 40 m is valid.
Similarly, if the variation of specific humidity q in time has order 1.5 x 10" 7 g/kg s, and the vertical flux E of the water vapour is 0.5 x 101 g/m2s, then on the basis of the averaged budget equation for specific humidity at dz p and of the assumption that water vapour phase transitions are absent, the thickness h3 of the layer where the vertical flux of water vapour varies only up to 20% is h3 > 50 m.
Thus, conditions of approximate constancy with height for the vertical fluxes of momentum, heat and moisture are fulfilled simultaneously if the thickness h of the surface atmospheric layer is defined as h = min^, h2, h3).
Let us now find the characteristic relaxation time ia for the surface atmospheric layer. On the basis (3.1.1) the time has to be equal to ia « uoh/0.2(t/p), which, after substitution of the typical values u0, h and |x|/p yields ia«2 hours. The characteristic time tw of the wind wave development has the same order. And, indeed, judging by the measurement data the limiting root mean square height of wind waves is achieved at the fetch X (a distance counted in the direction of the wave propagation), which is X « 104u2/#, where ua is the mean wind velocity at the standard measurement level in the surface atmospheric layer, g is gravity. Assuming as a rough estimate that X « c0iw, where c0 is a phase velocity complying with the frequency of maximum in the spectrum of wind waves, we find that ia « 10*u2/gc0. Next, assuming that the entire wind momentum ss puah in the surface atmospheric layer is spent only on wave development, and for the wave momentum the estimate Jiw « pwga2/c0 is valid (here a is the characteristic wave height; pw and p are water and air densities), we have that c0 ~ (pw/p)(gra2/uaft). Substitution of this expression into tw yields iw « iOA(p/pw)(ulh/g2a2). From this it follows at (p/pw) ~ 10~3, wa « 10 m/s, h x 100
m, a tt 1 m, g « 10 m/s2 that tw « 3 h. Thus, the statistical character istics of the quasi-stationary surface atmospheric layer have to depend not only on height over the underlying surface and other determining parameters, but also on the stage of wind wave development.
3.2 Vertical distribution of the mean velocity over an immovable smooth surface; viscous sublayer; logarithmic boundary layer
We orient the x axis along the tangential wind stress. Then the condition of approximate constancy with height of the momentum flux takes the form where u is a component of mean wind velocity along the axis x; u' and w' are turbulent fluctuations of velocity along axes x and z; v is the kinematic viscosity; the overbar signifies averaging.
Equation (3.2.1) does not permit determination of the unique vertical distribution of mean velocity because, apart from u, it contains one more unknown function: the Reynolds stress —u'w'. But some conclusions about the possible form of the function u(z) can be derived with the help of dimensional analysis. Indeed, the mean velocity u near a wall depends only on the friction stress x, distance z from the underlying surface and also on the kinematic viscosity v and density p of the medium. In addition, t and p can only be in the form of the combination x/p not containing the dimension of mass. Usually, instead of x/p it is customary to use the value u^ = (x/p)112 with dimensions of velocity. This value, called friction velocity (or dynamic velocity), defines the velocity scale for the flow near a wall. On the basis of the 7r-theorem of dimensional analysis the dependence of u on u^, z and v can be presented as where /„(zu^/v) is a universal function of the argument zu^/v.
Breaking the established order of discussion somewhat we recall the formulation and formal proof of the 7c-theorem which forms the central statement of dimensional analysis and which will be repeatedly applied subsequently.
The n-theorem. Any dependence among n + 1 dimensional variables of which the k <n variables have independent dimensions can be represented in the form of a dependence among the n + 1 — k non-dimensional combinations.
Let there be a functional dependence a = f(a1,..., ak,ak + a„),
where the arguments a1,...,ak have independent dimensions (i.e. their dimensions cannot be expressed by means of the dimensions of the remaining variables), and the dimensions of the variable a being determined and the arguments ak+l,..., a„ are expressed by means of dimensions of the first k arguments as here, as is common practice, the symbol [ ] denotes a dimension.
Let us pass from one system of units of measurement to another inside a given class so that the first k arguments al,...,ak change (i = 1,..., k) times. Then, a[ = ...,a'k = pkak.
Accordingly, the variable a is determined and all of the remaining arguments ak +!,..., a„ change too, and relationship (3.2.3) is rewritten as
Since Pi is an arbitrary change in scales of the units of measurement, we can choose it such that piai = 1. Then
or, which is equivalent,
If we introduce the designations
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