C

where q^ = —E/ptcu% is the specific humidity scale; and are inverse values of the turbulent Prandtl number (ke/k) and of the Schmidt number (kjk); k, k0 and kq are coefficients of turbulent viscosity, thermal conductivity and diffusion; other designations are the same.

We emphasize once more that Equations (3.7.2) are valid only in the logarithmic boundary layer within the limits of which the effect of molecular exchange on the vertical transfer of the momentum, heat and moisture can be neglected. Considering this and also the fact that for air the molecular Prandtl number (ratio between coefficients of molecular viscosity and thermal conductivity) and molecular Schmidt number (ratio between coefficients of molecular viscosity and diffusion) are close to 1, it may be assumed that the coefficients ae and aq depend only on the stratification of the surface atmospheric layer. Experimental data systematized by Busch (1977) demonstrate an increase in the coefficient ae from 1 for neutral stratification, through 3.5 for unstable stratification. This fact is interesting in itself but for us it is important that under conditions of a weakly stratified liquid the coefficients a0 and aq do not differ very much from 1 and, hence, in the logarithmic boundary layer the coefficients of turbulent viscosity, thermal conductivity and diffusion are equal. But if this is the case then in all cases for hydrodynamically smooth surfaces (ôd0 = ôq0 = 0) on the basis of (3.7.2), we obtain Cu = Ce = Cq. Equality of the coefficients of resistance, heat transfer and evaporation is a direct result of the Reynolds analogy.

In a preceding section we have already noted that extension of the Reynolds analogy to the case of a hydrodynamically rough surface has no justification because of the influence of molecular transfer in the immediate vicinity of the underlying surface. Therefore, until we have reliable information about the dependence of S0o and dq0 on the determining parameters, one has to be satisfied with different suggestions of the Reynolds analogy kind or with simply rejecting the use of Equations (3.7.2) for Ce and Cq and restricting ourselves to experimental estimates of these coefficients. Each of the alternative variants has its own advantages and disadvantages, but the second, because of the absence of an exact analogy between the immovable rough wall and the wave-covered sea surface, is considered to be preferable. We use what can be obtained from analysis of experimental data.

Experimental estimates of the coefficients of resistance, heat exchange and evaporation are presented in Table 3.1. The analysis attests to the fact that available estimates differ quite perceptibly from each other. There are several causes, mainly the adoption of different (eddy correlation, dissipation and gradient, see below) methods of determining eddy fluxes of momentum, heat and moisture; different periods of averaging (sample lengths); and different frequency responses of measuring equipment. One more source of uncertainty is errors in measuring sea surface temperature, which is sometimes identified with the temperature of a certain finite water volume, or with the temperature of the upper mixed layer. And, certainly, one should not neglect the influence of waves and appropriate changes in the hydrodynamic properties of the ocean surface, and the effects of stratification of the surface atmospheric layer.

To illustrate the role of stratification let us direct our attention to the similarity theory, developed by Monin and Obukhov (1954), for a thermally stratified medium.

3.8 The Monin-Obukhov similarity theory

The vertical structure of a thermally stratified surface atmospheric layer is determined by the tangential wind stress t, heat flux H, density p, molecular viscosity v and thermal conductivity x, air heat capacity cp, buoyancy parameter j? = g/T0, and also by one or other parameter describing the geometrical properties of the underlying surface. The creation of the similarity theory is based on the following hypothesis (see Monin and Yaglom, 1965): in the domain of developed turbulence for every interval of the spectrum, excluding the dissipation interval, the turbulent regime does not depend on molecular constants, and at heights much larger than the mean size of the underlying surface undulations, properties of the surface do not affect the

Table 3.1. Coefficients of resistance, heat exchange, and evaporation according to different authors

Range of change

of wind velocity

Author

(m/s)

Cu 10"3

Co 10"3

C„ 10" 3

from 1 to 20

Sheppard (1958) Deacon and Webb (1962) Deacon and Webb (1962) Zubkovskii and Kravchenko (1967) Weller and Burling (1967) Smith (1967) Makova (1968)

Kurnetsov (1970)

Smith (1970) Miyake et al. (1970) Hicks and Dyer (1970) Pond et al. (1971) Sheppard et al. (1972) Denman and Miyake (1973)

Kitaigorodskii et al. (1973) Pond et al. (1974)

Wieringa (1974) Dunckel et al. (1974) Smith (1974) Smith and Banke (1975) Kondo (1975)

Tsukamoto et al. (1975) Garratt and Hyson (1975) Emmanuel (1975) Kriigermeyer (1976) Friehe and Schmitt (1976) Kriigermeyer et al. (1978) Hasse et al. (1978) Wu (1980) Smith (1980) Donelan (1982) Launiainen (1983) Geernaert et al. (1986)

from 1 to 20

from from from from from from from from from from from from from from from from from from from from from from from from from from from from from from from

3 to 16

4 to 18

1 to 22

3 to 10

7 to 21

3 to 13

2 to 21

6 to 22

5 to 21

0.80 + 0.114wlo 1.10 + 0.04h1o 1.10 + 0.07ulo 0.72 + 0.12«lo 1.31 ± 0.36 1.03 ±0.18 0.13«lo -13.1 + u io 1.21 ± 0.24

1.10 ±0.10 1.44 ± 0.40 0.36 + 0.10«10 1.29 + 0.03» 10

l.2Re°015

1.49 + 0.28

1.36 + 0.40

1.48 + 0.21

1.41 ± 0.18

0.86 + 0.058« 10

-

1.56

1.28

1.20

1.20 ±0.30

0.63 + 0.066« 10

-

0.87 + 0.067«iO

1.08 + 0.03

1.2 +0.025«lo

-

0.073w10

-

1.32

1.28

-

1.2

1.15 + 0.20

1.34 + 0.30

1.34(1 - 0.331S)

1.42(1-0.45

-

1.32 ± 0.007

1.30

-

1.25

1.34

0.80 + 0.063

-

0.61 + 0.063«lo

-

0.37 + O.137«10

-

0.80 + 0.065wlo

-

0.43 + 0.097«10

1.15

Note: dashes point to unavailable information; Re = utzjv is the roughness Reynolds number; S = 3.55(7"10 — T0)/u\0 is the dimensionless stratification parameter (an analogue of the Richardson number); Tl0 and Ujq are the temperature and wind velocity (in m/s) at a height of 10 m; T0 is the sea surface temperature.

vertical changes in statistical characteristics of the hydrodynamic fields directly. The number of determining parameters can be reduced by two when it is considered that the mean velocity and temperature do not include mass dimensions and, hence, p and cp have to be incorporated only into combinations i/p and H/pcp. Thus, the mean velocity and temperature vertical distributions should be determined by the values of parameters t/p, ¡3 and H/pcp, from which the following unique (to within a numerical factor) combination is formed:

which is called the Monin-Obukhov length scale.

According to the above-mentioned similarity principle, the dimensionless vertical gradients of the mean velocity u and potential temperature 9 should serve as universal functions of the dimensionless relation z/L, that is, where ut = (t/p)1/2 and = ( — H/pc^/ku^ are the friction velocity and temperature scale; <bu(z/L) and Q>e(z/L) are the universal functions of the argument z/L. The latter represents a criterion of the hydrostatic stability: at z/L < 0 the stratification is unstable; at z/L > 0 it is stable; and at z/L = 0 it is neutral. We note that an increase in the length scale L in z/L is equivalent to a decrease in the height z, and vice versa. Therefore, at sufficiently small heights the effect of the stratification does not manifest itself.

Integrating (3.8.1) we arrive at the expressions

where

and, also, it is assumed that z/z0 « 1, <&u 0 (0) = 1. Let us discuss some asymptotic regimes.

Neutral stratification. In this case (complying with z/L => 0) the buoyancy force does not participate in the formation of the vertical distribution of mean velocity and potential temperature, and the buoyancy parameter /i is excluded from the number of determining parameters. It is impossible to form a non-dimensional combination from the remaining parameters uH/pcp and z. As applied to (3.8.1) it means that at z/L => 0 the functions ®u(z/L) and <59(z/L) approach 1 asymptotically. Then du u* <30 71

OZ KZ OZ z from which, after integration, we obtain usual formulae of the theory of the logarithmic boundary layer:

Stratification close to neutral (small values of z/L). We represent the functions $u(z/L), ®0(z/L) as power series in the vicinity of the point z/L = 0 and restrict ourselves to the first two terms of the expansion. As a result we obtain

Substitution of (3.8.5) into Equations (3.8.1) and their subsequent integration yields the so-called log-linear law for profiles of the mean velocity and potential temperature mo / z z

The numerical constants /?u, f}0 appearing in Equations (3.8.5) and (3.8.6) must be positive. Indeed, an enhancement of instability should be accompanied by an intensification of the turbulent exchange that, in turn, should lead to a decrease in the vertical gradients of the mean velocity and potential temperature. Hence, corrections to the logarithmic terms on the right-hand sides of Equations (3.8.6) have to be negative for unstable stratification (z/L < 0), and positive for stable stratification (z/L > 0). This is possible when

Strongly unstable stratification (large negative values of z/L). We take advantage of the fact that the condition z/L => — oo conforms to either unlimited growth of —H/pcp at fixed u^ or decrease in u* at fixed —H/pcp.

It is natural to assume that in this case the friction velocity should be excluded from the number of parameters determining the vertical structure of the temperature field, but not the velocity field. This condition is fulfilled if at z/L => — oo the function Oe(z/L) in (3.8.1) approaches (z/L)"1/3 asymptotically. If we assume, in addition, that at z/L => — oo the relation <t>u(z/L)/<S>g(z/L) approaches some certain finite limit, then on the basis of (3.8.1) we have u = -zo1'3), k

G ~ = a9TeLxl3(z~1/3 — Zq 1/3), where a„ and a„ are numerical constants.

Strongly stable stratification (large positive values of z/L). We again take advantage of the fact that large positive values of z/L are equivalent either to large z at fixed positive L, or to fixed z at small positive L. But the large positive values of —H/pcp (an abrupt temperature inversion) comply with small positive L, and under such conditions large eddies degenerate because of expenditure of energy on work against buoyancy forces. This leads to a reduction in the exchange between layers, which causes the turbulence to be local in character. But then the characteristics of the turbulent exchange and the vertical gradients of the mean velocity and potential temperature related to them have to cease depending on the vertical coordinate z. This last condition is equivalent to <5u(z/L) => (z/L), <$e(z/L) => (z/L) at z/L => oo. In this case from (3.8.1) it follows that a'u u = — — z, k L

where it is assumed as before that z0/z « 1.

In the general case, in order to estimate the mean velocity and potential temperature according to Equations (3.8.2) it is necessary to know the universal functions ij/u(z/L) and i¡j6(z/L). The first information about them was obtained by Monin and Obukhov (1954) from measurements of the mean wind velocity and temperature in the surface atmospheric layer over land. The results of data handling confirmed the presence of the asymptotic properties of the function Ou(z/L) predicted by the similarity theory. Later it was found that these results were in good agreement with estimates of the function <5u(z/L) from data of covariance measurements.

Conclusions obtained relating to the thermally stratified medium are easily generalized in the case of an allowance for humidity stratification. Such consideration reduces to replacement of the expression for the Monin-Obukhov length scale with where q^ = ( — E/p)/ku^. is the specific humidity scale, and to the addition to (3.8.2) of a similar formula for the vertical profile of the specific humidity q:

where ij/q{z/L) is a universal function having the same asymptotic properties as il/0(z/L).

We demonstrate that the replacement of the Monin-Obukhov length scale by its modified expression (3.8.9) is equivalent to consideration of humidity stratification. Indeed, the equation of dry air state has the form p = RpT, where p, p and T are pressure, density and temperature; R is the gas constant. In the presence of water vapour when the air transforms into a gas mixture the gas constant R = R^/p (here R^ is the universal gas constant; p. is the relative molecular weight) becomes a variable: the relative molecular weights of dry air (pd) and water vapour (pw) are equal to 28.9 and 18 respectively. When considering water vapour as the ideal gas the equation of the moist air state can be presented in the form where q = p/(pw + pd) is the specific humidity; pd and pw are the densities of dry air and water vapour. We take the logarithm of this equation and vary it, taking into account the fact that the density change depends only on changes in temperature and humidity but not on pressure (the Boussinesq approximation). As a result we have p'/p = — T'/T— 0.61^7(1 + 0.6 ky) % — T'/T — 0.61^', where p', T' and q' are fluctuations of density, temperature and specific humidity. Hence, the expression for the buoyancy flux (g/p)p'w' takes the form (6f/p)pV = -fiTw' - OMg^w' = fi(-H/pcp) + 0.61 g(-E/p). Its substitution into L instead of /?( —H/pcp) and using the definitions of T* and q* yield (3.8.9).

Turning to (3.8.1) we note (see Busch, 1977) that the overwhelming majority

P = (RJii)pT=(RJiii)piT+(RJnw)pwT from which, with R = R^/Pi, it follows that p = RpTl 1 + (pd/pw - 1 )q] = RpT{ 1 + 0.61<z)

of experimental data on the vertical distribution of mean wind velocity, potential temperature and specific humidity are described very well by the expressions

We also note that the linear dependence of functions <J>9 and 5>9 for the stable atmospheric stratification (z/L > 0) is universally accepted. As for the form of the function Ou for the unstable stratification (z/L < 0), there is no common opinion on this score. More often, the KEYPS equation (the abbreviation of the names of five authors - Kazanskii, Ellison, Yamamoto, Panofsky and Sellers) is used. It has the form (<J>4 — y(z/L)<1>„) = 1 and follows from the interpolation formula kM = kmh!z( 1 — yRf)~l/4, after substitution of the relations connecting the coefficient of vertical turbulent viscosity kM, and the flux Richardson number Rf with <1>u(z/L). The indicated interpolation formula yields correct results at large (positive and negative) and close to zero values of z/L. Indeed, at z/L => 0 (that is, under conditions of neutral stratification) /?/=> 0 and kM « ku^z; at |z/L| » 1 (that is, under conditions of strongly unstable stratification) — Rf ~ |z/L|4/3 and kM Ku^.z\z/L\li3; finally, at z/L => oo (that is, under conditions of strongly stable stratification) /?/=> 1/y and kM/z => 0. Here and above, y is an empirical constant varying within the limits from 9 to 18 according to different authors.

Values ®0(O) and <5,(0) also serve as the subject of discussion: generally, they are specified as identical, that is, equivalent to the assumption about the similarity of heat and moisture transfer. There is also another point of view according to which the similarity condition is fulfilled not for heat and moisture transfers, but, rather, for momentum and moisture transfers, that is, fl)^ is equal to i>u rather than to <D0.

Now with data on the universal functions <5U, <I>9 and 09 it is possible to obtain expressions specifying the relationships between the coefficients of resistance, heat exchange and evaporation, on the one hand, and the parameter z/L describing atmospheric stratification, on the other hand. These

kM ( = ul/(8u/dz) = Ku^z/Qyjz/L), Rf = (3(-H/pcp)/ul(du/dz) = (z/L)/<Du(z/L)

expressions take the form

= a0 [ln(z/z0) - >Au(z/L)] e" u[ln(z/z0)-^(z/L)]'

The first equation bears a direct relation to the explanation of the following interesting circumstance. When averaging estimates of Cu by different authors, it turns out that the resistance coefficient can be considered constant within the wind velocity range from 4 to 15 m/s and equal to (1.3 + 0.3) x 10-3. But when grouping preliminary estimates according to stratification conditions and subsequently averaging them it turns out (see Bjutner, 1978) that for neutral stratification, or close to neutral stratification, the resistance coefficient increases from 1.1 x 10 at a wind velocity of 4 m/s, to 1.8 x 10"3 at a wind velocity of 14 m/s. As is well known, the marine surface atmospheric layer is characterized by unstable stratification, which leads to an increase in the resistance coefficient; in this case the weaker the wind, the more does the influence manifest itself. Because of this, neglect of the stratification effect leads to a decrease in the difference between estimates of Cu at low and high wind velocities, thereby masking changes in the resistance coefficient.

3.9 Transformation of the thermal regime of the surface atmospheric layer in the presence of wind-wave interaction1

The problem in the title of this section is interesting in itself. But there is a further motive for its solution, i.e. the non-trivial issue, which has not been properly elucidated as yet, about the variability of heat exchange between the ocean and the atmosphere during the process of wind wave development.

So, let us consider the problem of thermal regime transformation of the surface atmospheric layer in the presence of wind-wave interaction. As a basis we use the model of wind-wave interaction presented in Section 3.5. We assume at the same time that the air temperature beyond the surface atmospheric layer (Ta) and the sea surface temperature (T0) remain constant in time, and the eddy heat flux at the upper boundary of the surface atmospheric layer is equal to zero. Then in the absence of radiative sources and sinks of heat and latent heating due to water vapour phase transitions,

1 This section accords with the results of Benilov's research (personal communication).

the equation for the integral, within the surface layer, temperature defect (T — T0) is written in the form d di

where, as before, H0/pcp is the heat flux at the water-air interface normalized to the volume heat capacity of air.

We will assume the temperature difference (T0 — T.d) to be small, that is, the stratification of the surface atmospheric layer is close to the neutral one. In this case to approximate the vertical temperature profile within the height range from the roughness level z0 to the upper boundary h of the surface atmospheric layer, one can use the logarithmic law

where ¿T0 is the near surface temperature jump, = —(H0/pcp)/Ku^ is the temperature scale.

To avoid unnecessary complications we assume as a first approximation that the vertical distribution of temperature within the height range from the water-air interface (z = 0) to the roughness level (z = z0) obeys the linear relationship

Substitution of (3.9.2) and (3.9.3) into (3.9.1) yields d dt

Ho pc p

We take into account the fact that —H0/pcp = ku^ t^ and that in accordance with (3.9.2) for the equality = (7"a — T0 — ST0)/^ is valid, where £ = In h/z0 = Kuh/u^ (see Section 3.5). After substitution of these expressions into (3.9.4) we have dt e

where 0 = 1 + 8T0/(T0 — 7"a) is a new dependent variable uniquely connected to the non-dimensional (normalized to (T0 — Ta)) surface temperature jump.

We define the non-dimensional thickness D of the surface atmospheric layer, non-dimensional time x and non-dimensional heat content defect Q by the formulae

where the symbols have the same meaning as in Section 3.5. According to the last of the displayed formulae,

Therefore incorporating an allowance for (3.9.6) Equation (3.9.5) reduces to the form

0 0

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