where S¡ is the potential air temperature 6, or water density p2; hi is the thickness of the í th PBL; s is the average (within the PBL) turbulent energy dissipation rate.
We note that in (4.4.8) the diffusive turbulent energy fluxes at the outer boundaries of the PBLs and at the water-air interface are assumed to be equal to zero. Besides the obvious advantages of using this equation to determine the eddy viscosity coefficient, it has its own disadvantages: the appearance of two new unknown parameters (é¡, h¡) and the necessity of approximating the vertical distribution of the eddy viscosity coefficient by some function of height.
In the simplest closed models two methods of determining é are the most commonly used. In the first, average dissipation is assumed to be in proportion to the integral (over the PBL) turbulent energy production of shear and convective origin; the second reduces to the assumption that í:¡ is a function of the eddy viscosity coefficient, averaged within the PBL, and of the PBL thickness. With this assumption we have, on the basis of dimensionality considerations, st = cEk3m/hi, (4.4.9)
where c£ is a numerical constant estimated from experimental data.
As for hh it can be identified by the height at which the derivative of the mean velocity modulus becomes zero initially:
dzt or by the thickness of the friction layer at the upper boundary of which the Ekman condition is fulfilled:
[(",■ - Ugi)2 + (v, - Vgi)2yzL2h = e-«(U2gi + F2)1'2, (4.4.10b)
In the absence of the geostrophic current the last condition reduces to the form
The system of Equations (4.4.1)-(4.4.3) and (4.4.8)-(4.4.10) given above is closed. Its solution at kMi = const was obtained by Kagan (1971). The formulae for the eddy viscosity coefficients in the neutrally stratified PBLs of the atmosphere and the ocean have the form
It follows from here and (4.4.5) that the wind coefficient (ratio between the modulus c0 of the surface drift velocity and the modulus Gt of the geostrophic wind velocity) is independent of wind velocity and approximately equals (Pi/P2)1/2-
The model examined correctly reproduces the relationships between different characteristics of the PBL system and external parameters. But to obtain, with its help, an acceptable agreement between calculated and observed values of characteristics sought is not possible due to the use of eddy viscosity coefficients which are assumed to be constant with height in the atmosphere and with depth in the ocean. Because of this, subsequent improvement of the model has followed the path of replacing the average eddy viscosity coefficients by some function of the vertical coordinate. The sensitivity of the solution to the choice of different changes in eddy viscosity coefficients along the vertical was examined by Kagan (1971). In this work it was shown that replacement of an eddy viscosity coefficient variable over depth in the ocean by its average value does not have any effect on the wind field structure, and only changes the current field structure. This change manifests itself in a small increase in the current velocity in the upper, and a decrease in the lower, part of the friction layer. At the same time use of the average eddy viscosity coefficient in the atmosphere results in an increase in the drift current velocity by 50-60%.
The angle between the direction of the surface drift current and the surface isobar turns out to be quite sensitive to the approximation of the eddy viscosity coefficient in the atmospheric PBL. If for a constant (over height) eddy viscosity coefficient the surface drift current is strictly directed along the isobar (coincidence with the isobar takes place only at constant eddy viscosity coefficients in the atmosphere and the ocean), then for a variable eddy viscosity coefficient in the atmosphere the surface drift current is directed to the side of high pressure, making an angle of 30° to the isobar.
Kagan (1971) has attempted to find an optimal variant of approximation of the vertical distribution of the eddy viscosity coefficient in the atmospheric and ocean PBLs. This requires observed data obtained under strictly controlled conditions, and in view of their almost complete absence he has restricted himself to comparison of calculated and observed values of the wind coefficient. Unfortunately, the observational data for the wind coefficient vary over a rather wide range - from 0.0092 according to Hela, up to 0.0328 according to Mohn - that is explained by errors in determination of the surface current velocity identified from data on the drift of various objects. In other words, available estimates, as a matter of fact, describe not the surface current but, rather, a current in a layer of finite (and different in every separate case) thickness. The accuracy of wind velocity measurements used on determination of the wind coefficient also leaves much to be desired (see Section 1.4). Therefore, from this, quite rich, set of estimates of the wind coefficient, only two estimates, with most statistics and free from the above-mentioned disadvantages, were chosen. These estimates, from Hughes (1956) and Tomczak (1963), were found from measured current velocity values obtained with the help of drifting postcards several mm thick and geostrophic wind velocity values calculated from data on atmospheric surface pressure. Eventually, it was ascertained that approximation of the vertical profile of the eddy viscosity coefficient by a scheme 'with a knee', that is, its approximation by a linear function of height in the surface layer and constant with height in the rest of the atmospheric PBL, and presetting the eddy viscosity coefficient to be constant with depth in the ocean PBL, guarantees quite acceptable accuracy in the calculation of the wind coefficient: according to observed data it changes from 0.0192 through 0.0226; the calculated results yield 0.025.
An ideologically close model of the atmospheric and oceanic PBL system was developed by Yoshihara (1968). Here (as well as in all the other models listed) the atmospheric and ocean PBLs were considered to be horizontally homogeneous. Meanwhile, there are many proofs that in zones of atmospheric and hydrological fronts the horizontal gradients of characteristics can differ markedly from zero. For example, in the zone of the Sargasso Sea hydrological front the temperature difference amounts to more than 1 °C per 10 km. Some parts of the front are known where the temperature difference may be even as much as 3 °C per 100 m. The first model of the atmospheric and oceanic PBL system, taking into account the horizontal inhomogeneity of fields of meteorological and hydrological characteristics, was proposed by Kagan (1971). In this work a stationary solution to the equations of motion, state, heat transport and turbulent energy budget in both environments, and also the budget equations for water vapour in the atmosphere, and for salinity in the ocean was obtained. In doing so the horizontal temperature and salinity gradients were considered to be constant over the vertical, and fixed. The continuity conditions for the velocity, temperature and momentum were accepted at the ocean surface. The surface temperature was determined from the heat balance condition. It was also considered that at the upper boundary of the atmospheric PBL, and at the lower boundary of the ocean PBL the wind and current become geostrophic, and the air temperature and humidity, as well as the water temperature and salinity, take on fixed values. The problem has been solved for constant eddy viscosity coefficients in both media.
The model described was then generalized for the case when the ocean surface is covered by freely drifting ice (see Kagan, 1971). The solution obtained allowed one to explain a number of peculiarities of the thermal and dynamic ocean-atmosphere interactions in the presence of ice cover, and, particularly, intensification of the inversion in the atmosphere in winter, which, as it turns out, is conditioned by the shielding influence of ice contributing to a decrease in the heat influx to the upper ice surface from below, and by a drop in its temperature. However, due to the approximate description of real processes and the assumption of constant eddy viscosity coefficients this explanation has a rather qualitative character: in winter this assumption leads to an overestimation of the temperature at the upper ice surface, and in summer, when the upper ice surface temperature becomes equal to the water-freezing temperature, to considerable (almost seven times) underestimation of heat expenses for ice melting. Because of this, in an improved version of the model, the vertical distribution of the eddy viscosity coefficient in the atmospheric PBL was approximated by the scheme 'with a knee' (see Kagan, 1971). Test results demonstrated that the model reproduces seasonal changes in the ice surface temperature, drift velocity and intensity of ice growth and melting to good precision. The annual mean values of the ice growth or melting rate, and the ice drift velocity, turned out to be equal to 57 cm/year and lOcm/s; observed data yield 30-50 cm/year and 8 cm/s respectively.
We also mention the non-stationary model of the atmospheric ocean PBL system developed by Pandolfo (1969). Here, as in Kagan (1971), the horizontal gradients of temperature and humidity in the atmosphere and the horizontal gradients of temperature and salinity in the ocean were considered to be prescribed, though arbitrary, functions of the vertical coordinate, and instead of the turbulent energy budget equation the Prandtl formula was applied. The modification reduced to accounting for the effects of stratification and wind waves. A detailed discussion of this model can be found in Kagan (1971).
4.4.3 Semiempirical models not using a priori information on the magnitude and profile of the eddy viscosity coefficient
A general feature of all the models discussed is the fact that one or another approximation of vertical profiles of the eddy viscosity coefficient was used in their construction. It was shown in a review article by Zilitinkevich et al. (1967) that, due to the strong variability of the eddy viscosity coefficient in the atmosphere, even a good approximation does not provide a reliable representation of the vertical structure of the atmospheric PBL under all possible conditions. The same can be said with respect to the ocean PBL, the only difference being that information about the eddy viscosity coefficient in the ocean is more limited. A few attempts at estimating the eddy viscosity coefficient from data of direct determinations of Reynolds fluxes in the ocean are detailed in the literature. Panteleyev (1960) was the first to undertake such an attempt. He demonstrated that the eddy viscosity coefficient reaches a maximum at a depth of 10-15 m. Gongwer and Finkle (1960) showed that it decreases below this depth, and in the range of depths from 100-150 m to 400 m it remains practically constant with depth. A report with published data on the magnitude of the eddy viscosity coefficient is contained in Neumann and Pirson (1966), where it is stated that the eddy viscosity coefficient can change, depending on conditions, by two orders of magnitude - from 10 to 103 cm2/s. Thus, data available at present allow determination of only the order and rough form of its profile, and it is natural that, even in spite of the simplicity and clearness of the models where the eddy viscosity coefficient is considered to be a prescribed function of the vertical coordinate, these models are unsatisfactory.
A theoretical model of the system of the atmospheric and ocean PBLs which does not use a priori assumptions of the magnitude and profile of the eddy viscosity coefficient was developed by Laikhtman (1966). We dwell at length on this not only because it is the most efficient of all the models mentioned above but also to show a very elegant method of reducing a two-layer problem to a single-layer one.
When closing the equations of motion by the budget equation for the turbulent energy and by relations of approximate similarity which are common in semiempirical turbulence theory, the equation set for the atmospheric and ocean PBLs takes the form d , dUj -, d Z: az:
Here the last expression is obtained by transformation of the approximate similarity relationship for the eddy viscosity coefficient and the generalized von Karman formula for the turbulence scale (see Section 4.2.1); ab and c are universal constants with numerical values equal to 0.73 and 0.046 respectively; all another symbols are the same.
The turbulent energy budget equation (the third equation in (4.4.12)) includes the function Sh which, as has been already noted, represents either the potential air temperature 6 (at i = 1), or the sea water density p2 (at i = 2). In order not to complicate the problem by the determination of a new unknown function it can be assumed that km dSf/dz; = const or advantage can be taken of any interpolation formula for dS;/dz¡. In both cases the condition dSj/dz,- ^ 0 means that the vertical structure of the atmospheric and ocean PBLs described by Equation (4.4.12) depends on stratification in both media.
Equations (4.4.12) are supplemented by the following boundary conditions:
^miPi dtVdz! = -kM2p2 du2/dz2, kM1p1 dvjdz^^ = -kM2p2 dvjdz
where the last condition follows from the turbulent energy budget equation on the assumption that in the vicinity of the water-air interface the production of turbulent energy is balanced by dissipation. If we now introduce the non-dimensional variables with subscript n,
(;U,, 17,) = (- 1 )l+1(u^/K)(uni, I7m), Z; = (KU+i/\f\)ZH
and then change from velocities to stresses,1 system (4.4.12)-(4.4.14) will take the form d2f?ni dz2 k, nil +
d zli K
VM ni vMnt
> • b
k ni nMni
0 ni +
0, i duni/dzni, <rni = kMni dvni/dzni are non-dimensional com-
ponents of the eddy friction stress; p0i = — K2(gf/Si)(7r0,/p,ci)/|/|u2i; n is the eddy flux of sensible heat (at i = 1), or of mass (at i = 2) at the ocean surface; p^i is the air or water volume heat capacity; /? is a universal constant composed of ab and c and equal to 0.54.
We note that the set (4.4.16)-(4.4.18) looks very much like the analogous system for the atmospheric PBL for which the numerical solution is given in Laikhtman (1970). The only distinction between the problem here and the one in Laikhtman (1970) is in the formulation of boundary conditions at zni => 0 (in Laikhtman, 1970, it is assumed that rjn = 1, an = 0, b„ = 1 instead of (4.4.18)). To determine the universal functions of the two-layer problem (4.4.16)-(4.4.18), an ancillary coordinate system (x',-y\ zni), is introduced where the direction of axis x' and scales are selected to satisfy conditions rjnl = t]n2 = 1, <rnl = a„2 = 0. In accordance with the first two equalities in (4.4.18) this means that (P2u*2/Piu*i) — 1 and that the x' axis is directed along the surface stress and makes a certain angle a with the
1 The use of this technique, proposed by Monin (1950), has the advantage that it allows, firstly, a reduction by two (kV^/u^, kU^/u^i) in the amount of non-dimensional variables and, secondly, to carry over the boundary conditions (4.4.14) from the roughness level z0, to the level z; = 0. Such a replacement of the lower limit of integration is justified because for small z, the friction stress and the turbulent energy do not in practice change with height.
x axis of the standard coordinate system (see Figure 4.3). The friction velocity u^ and the angle a between the direction of the geostrophic wind and surface stress is determined with the help of the, still unused, condition of velocity continuity at the water-air interface (third and fourth relations in (4.4.18)). As a result we obtain the following expressions for the geostrophic friction coefficient x = u^JkGy and the angle a:
da da nl
,dznl V \p2J dzn2J _ 1 — n (N sin y 4- cos y) N — n(N cos y — sin y) '
at z nl
where y is the angle between the geostrophic wind (isobar) and geostrophic current (see Figure 4.3);
We note also that if the roughness parameter z01 of the ocean surface is fixed then z0nl = (k2xRo)~1; if Charnock's formula, z01 = muh/g, is assigned then z0nl = m(Gl\f\/g)x- Here Ro = Gxl\f\z0l is the Rossby number; m is the numerical factor. Accordingly, for z0n2 we have for both cases z0n2 = (y/iPi/PiWxRo)'1 and zon2 = mJ(p1/p2)(Gl\f\/g)x.
The dependence of x and a on the non-dimensional parameter mG^fl/g
and on the stratification parameter /.i0 is presented by Laikhtman (1970) in tabular form so it is easy to find ujj!l and then, with the help of the relation (P2u*2/Piu*i) = 1) t0 calculate ult.2 and the characteristic scales of all variables sought and defined by Equations (4.4.15). After that the vertical distributions of the wind velocity, current velocity and the eddy viscosity coefficients are restored by using the expressions u1 = G1 cos a + xGt danJdznl, \
vl = Gi sin a - xGy dr]nl/dznl, u2 = G2 cos(a + y) - (p1/p2)1/2xG1 dan2/dzn2, v2 = G2 sin(a + y) + (pilp2)ll2xG\ df/„2/dz„2, kM i = K4(X2Gi/\f\)kMnl, kM 2 = (Pi/P2)K4(x2Gl/\f\)kMn2.
Tests showed quite satisfactory coincidence of the calculated and observed dependencies of the resistance coefficient Cu = (u4.1/u10)2 on wind velocity (here u10 is the wind velocity at a height of 10 m). The solution sensitivity to the choice of numerical constants was examined in detail by Radikevich (1968). He also calculated, on the basis of the model in question, seasonal mean fields of wind velocity, drift current, eddy viscosity coefficients and momentum, heat and water vapour fluxes in the North Atlantic.
The model of the ice drift proposed by Laikhtman (1986) was the natural generalization of the model discussed. Its essence is the following. The system of equations for the atmospheric and ocean PBLs, by means of proper rotation of the coordinate axes, reduces to the system of equations for the atmospheric PBL for which the solution is known. The matching of the equations is performed by the equations of motion for ice and the continuity condition for velocity at the water-ice and ice-air interfaces. As a result the ice drift characteristics are determined by the solution of a system of four transcendental equations. The analysis of this system of equations showed that for neutral stratification in both media the ice drift velocity and direction are functions of three non-dimensional parameters: mI|/|/ic2p1G1, GJ\f\z0l and z01/z02. Control calculations performed for fixed values of listed parameters demonstrate that the ice drift coefficient c^/G^ and the angle between the ice drift direction and isobar are insensitive in practice to variations of z01/z02. From the two remaining parameters the first one (m^/l/h;2/?^) determines the velocity and has almost no effect on the direction of ice drift; the second one (GJ\f\z0l) does the opposite.
The description of the models of the atmospheric and ocean PBL system would be incomplete if we did not note the following two circumstances. As is known, the water-air interface oscillates near its mean position in such a way that the different points of the horizontal plane in the vicinity of the mean sea level will be in air and in water simultaneously. The further the considered plane is located from the free surface the larger the changes in the ratio between areas of water and air. If we now recollect the difference between statistical regimes of turbulence in both media, then it becomes absolutely unclear what meaning we should put on the operation of averaging in the zone of water and air contact. It is clear only that the regularities of the processes in this zone cannot be described in the same way as for the atmospheric and ocean PBLs.
In this connection, in a number of works (see, for example, Laikhtman et ai, 1968) the concept of the transitional layer has been introduced, within the limits of which the interface oscillates. Since within this layer all physical characteristics are subjected to considerable changes it cannot be 'contracted to plane' and we cannot expect the continuity conditions to be met. Obviously, this obstacle can be bypassed by using the moveable coordinates consistent with the water-air interface configuration (this is justified only in the case where wind waves do not break), or by identification of the transitional layer with the two-phase liquid layer in the regime of the developed turbulence. Only the future will show whether these, or some other more radical methods, will be successful or not.
Next, in the models examined of the atmospheric and ocean PBL system it was suggested that the turbulent energy in the upper ocean layer is generated only due to the interaction of Reynolds stresses with shears of mean velocity. Because of this the condition of vanishing of the diffusive turbulent energy flux was taken into account as a boundary condition at the ocean surface. There is another viewpoint, which was presented for the first time by Kitaigorodskii (1970), according to which the main mechanism for the production of turbulent energy in the upper ocean layer is diffusion of turbulent energy from the transitional layer where the energy of wind waves transforms into turbulent energy as a result of their breaking. Thus, there are two alternative ideas about the mechanism of turbulent energy generation in the upper ocean layer. Is it crucial to elucidate where the truth lies? Certainly, it is most convenient to take an intermediate position stipulating immediately that both the above-mentioned ideas are equivalent. But let us try to understand which arguments lead to a revision of the traditional presentations. There are two such arguments: (1) either the energy £w transmitted from wind to waves is equal to the energy Ed used to supply drift currents, or at the developed waves it is much larger than the latter; (2) within the framework of Ekman's model of the PBL (shear flow) it is impossible to obtain an estimate of the turbulent energy dissipation typical of the upper layer, and therefore, if all energy transmitted from wind to waves in the regime of developed waves dissipates, then only turbulence of wave origin is able to provide a proper level of dissipation. We have doubts about the second argument, though the first one also needs verification.
So, we will consider Ekman's model of the PBL. Within the framework of this model drift currents are described by the following equations:
T~ &M2 "T^ + fV2 = 0. T~ kM2 J--/"2 = 0, dz2 dz2 dz2 dz2
kM2 du2/dz2 = -(pi/p2)«#i. kMi dt'2/dz2 = 0 at z2 u2, v2 => 0 at z2 => oo,
where it is assumed that the x axis is directed along the tangential wind stress.
We derive an energy equation and then integrate it over z2 from 0 to hy. As a result we obtain
Here the first term on the left describes the energy supply from wind to current; the second term describes turbulent energy production in the ocean PBL.
We will assume that the ocean is stratified neutrally and that the diffusive turbulent energy flux at the ocean surface is equal to zero. Then from the turbulent energy budget equation (4.4.8) it follows that
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