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the one hand, and the universal functions ®u, <J>r and <I>9, on the other. To specify functions <I>U, <hr and <$q, one can apply Equations (3.8.11) with <Dr(0) = <D9(0) = 1 or (at large negative values of x3/L) the KEYPS equation providing proper asymptotics in the regime of strongly unstable stratification. In any case, an application of approximating formulae for functions <DU, <£>r and allows us dispense with without measurement data on vertical profiles of the mean velocity, temperature and specific humidity, but in return it entails the realization of an iteration procedure the necessity of which is caused by the presence of fluxes sought in the expression for the Monin-Obukhov length scale.

The dissipation method of estimation of momentum, heat and humidity eddy fluxes has certain advantages as compared with the eddy correlation method: it does not need sensors to be fixed in a strictly prescribed direction; it excludes the necessity of measuring vertical velocity fluctuations, and thereby allows the use of moveable platforms. But it should be remembered that assumptions about local isotropy, the existence of an inertial interval, and smallness of the divergence of the vertical transfer by velocity fluctuations are the basis of this method. The last assumption may not be valid at large distances from the ocean surface. According to Large and Pond (1981, 1982) and Smith and Anderson (1984), discrepancies between estimates of fluxes resulting from the eddy correlation and dissipation methods do not usually exceed 40%.

Gradient method. The essence of this method is the determination of eddy fluxes of momentum, heat and humidity from measurements of vertical profiles of the mean wind velocity, temperature and humidity. The latter, in accordance with the Monin-Obukhov similarity theory, are described by Equations (3.8.2) and (3.8.10) involving the parameters uT^, q^, z0, T0 and q0 and the universal functions ij/u(z/L), i//T(z/L) and t¡/q(z/L). If these functions are known, they obey say, (3.8.11) or any other expressions with proper asymptotes at small and large positive and negative values z/L, see Section 3.8. Then to compile a closed system of equations providing a single-valued determination of the parameters sought, it is basically enough to take measurements of wind velocity, temperature and humidity at only two heights. The number of measurements required can be reduced by discarding the determination of z0, T0 and q0, that is, by passing from u, T and q at a fixed height z to differences u(z2) — «(zj, T(z2) — r(zL) and q(z2) — q{zx). But since these differences are small, particularly if the heights z2 and zx are close to each other, then even small errors in the measurement of differences can entail marked errors in estimating fluxes. Because of this, measurements at three or more heights are preferred as this guarantees more reliable data on the vertical gradients of the meteorological elements.

It is obvious that in this case we do not eliminate errors caused by the still little known forms of the universal functions tj/v(z/L), \¡/T(z/L) and ij/q(z/L), especially for inversion conditions, and by dependence of the Monin-Obukhov length scale on the fluxes sought, converting Equations (3.8.2) and (3.8.10) into a system of non-linear equations. The linear character of the formulae can be preserved if we turn from z/L to another stratification parameter (for example, the gradient Richardson number)

Ri = (jS dT/dz + 0.610 dq/dz)(du/dz)~2, easily calculated from gradient measurements. It will be recalled that, according to the definition of the modified Monin-Obukhov length scale z/L = (;K2z/ul)(pT; + 0.61 gqt), and thus T+ = z(¡>Tl(z/L) dT/dz, q* = z%\z/L) dq/dz, u2Jk2 = z<S>~2(z/L)(du/dz)2. Hence, at <&T(z/L) = <bq(z/L) the equality z/L = <t>x 1(z/L)Q>l(z/L)Ri occurs. From here, after expanding Or(z/L), <£u(z/L) into series in powers of z/L and linearization, we finally have z/L « Ri.

But still the most serious source of errors in determining fluxes by the gradient method is errors of measurement themselves performed on buoys, ships and other moveable and/or bulky platforms, and accordingly they are subjected to the influence of inevitable distortions of the air flow. According to Blanc (1983) typical errors of such origin amount to from 15 to 35% for the eddy heat flux, from 15 to 105% for the eddy humidity flux, and from 10 to 40% for the momentum flux.

Bulk method. However paradoxical it may seem, the main disadvantage of this method is an excess of available, and often contradictory, possibilities. Indeed, this method is based on the presentation of momentum, heat and humidity fluxes in terms of characteristics included in the complex of standard ship measurements and on the use of the so-called bulk formulae ul = Cuu2, —H/pcp = Ctu(T — T0), -E/p = Cqu(q-q0). (3.10.18)

The bulk method would be quite attractive in itself if the coefficients of resistance (C„), heat exchange (CT) and evaporation (Cq) appearing in (3.10.18) were considered as numerical constants or, otherwise, if their dependence on the determining parameters were reliably established. We have neither (see Table 3.1), so the accuracy of fluxes obtained by this method leaves much to be desired. It is important to determine to which consequences the choice of one or other exchange coefficients will lead. An answer to this question was given by Blanc (1983). Using one and the same annual series of standard ship measurements on the Ocean Weather Ship C in the North Atlantic he made a comparison of fluxes calculated with the help of ten different schemes of assignment of resistance, heat exchange and evaporation coefficients. It turned out that mean deviations amount to +0.1 N/m2 (or, in relative units, ±45%) for the momentum flux; ± 17.5 W/m2 (±70%) for the sensible heat flux; and ±18.0 W/m2 (±45%) for the latent heat flux, while maximum relative deviations reach 80%, 100% and 120% respectively.

Such discrepancies are certainly distressing. But even this is not the main consideration. The worst of this is that we do not know which one of the available schemes of assignment of resistance, heat exchange and evaporation coefficients is valid (there are as many opinions here as authors of schemes). It is clear that while estimating results obtained within the framework of the bulk method, one should be very careful and reasonable, and it will take a considerable amount of effort before these results can be qualified not only as promising but as sufficient.

3.11 Methods for estimating COz flux at the ocean-atmosphere interface

In the state of thermodynamic equilibrium of the ocean and atmosphere the solubility (equilibrium concentration) of C02 in water c°, as in fact of any other gas, is determined by its equilibrium concentration in air as well as by temperature T°, and salinity S° of sea water. In other words, C02 follows Henry's law, according to which 4 = cgG(T°, S°), where G(T°, S°) is a solubility coefficient equal to the relation between equilibrium gas concentrations in air and water.

While passing from one equilibrium state to another the change 5c° in the total concentration of inorganic carbon in water, accompanied by a simultaneous change of partial pressure, is described by the formula 5c°/c° = B5pco2/Pco^ where B is the buffer factor or Revelle factor describing the so-called buffer effect - the dependence of the absorbing capacity of the ocean mainly on the content of the final product of the reaction of C02 with water (carbonate ions). Since carbon content in the form of carbonate ions in the surface ocean layer and carbon content in the form of C02 in the atmosphere are in ratio to each other as 1 to 10 then an increase in the atmospheric C02 concentration, say, by 10%, will result in an increase in the total carbon concentration in the upper ocean layer by only 1%. Available data demonstrate that for present-day conditions the Revelle factor varies within the range 0.090 at temperature 17.7 °C through 0.075 at 3.5 °C.

In actuality the atmosphere and the ocean are not in a state of thermodynamic equilibrium and, hence, at any point on the ocean surface at any moment in time, transfer of C02 molecules from one environment to the other takes place. Let us consider the C02 budget at the water-air interface. If the carbon dioxide flux QC02 in the air layer closest to the surface is represented as fcg(c£Q2 — cf ), and in the water layer closest to the surface as /c°(cf — c£02), where and k* are the rates of gas exchange in water and air, and c£02 and Cq0i are the volume concentrations of C02 in these environments, then the budget condition for C02 at the water-air interface can be written in the form QCOl = — c*) = k°(cf — c°02), or taking into account the definitions of the equilibrium concentration

where = (l/fc° + 1/G/Cg)"1 represents the total resistance and l/fc° and 1/Gfcg represent individual resistances of the water-air interface to the gas exchange between them (Figure 3.4).

It is commonly considered that the gas exchange between two media is performed by molecular diffusion. If one adopts this viewpoint and assumes that the rate of gas transfer is equal to the coefficient of molecular diffusion divided by the thickness of the diffusive sublayer then, for example, for the oxygen whose coefficient of molecular diffusion in air is approximately 103 times more than in water, the total resistance will be stipulated only by the resistance in the liquid phase. This conclusion remains valid for carbon dioxide too, although it dissolves much better in water than does oxygen (the oxygen solubility coefficient G is equal to 30, and that of carbon dioxide is equal to 1). This is the result of the diversity of existent forms of C02 in sea water (see Section 2.7). Thus the intensity of C02 exchange between water and air is controlled by processes in the diffusive ocean sublayer closest to the surface.

But since we started by discussing the diffusive ocean sublayer it is worth noting that at small wind velocities when the ocean surface is hydro-dynamically smooth its thickness is a function of the friction velocity in water and the coefficients of kinematic molecular viscosity v and diffusion X. Then from dimensionality considerations we have <5^ = xu~^f(Sc), where Sc = v/x is Schmidt's molecular number. We determine the form of the function f(Sc). For this purpose we take advantage of the approximation f(Sc) = 12.2Sc213 proposed by Deacon (1977) for Sc ranging from 200 to 5000. Its substitution yields = 12.2xu^Sc2/3 or, taking into account the definition of the thickness <5° = Sv/u^ of the viscous sublayer in water, we obtain = 2.4Sc~1/3<5°. As can be seen, at Sc = 103 the thickness of the diffusion sublayer is a fraction of the thickness of the viscous sublayer.

Next, according to the definition, k? = yjd% so that /c° = 0.082u^Sc'213,

2 3 5 8 10 15 20 30 ua m/s Figure 3.4 Dependence of the gas exchange rate on wind velocity at a height of 10 m, according to Bjutner (1986) (a) and Coantic (1986) (b): (1) the curve drawn over mean weighted values (the latter were obtained by recalculation of laboratory measurements carried out before 1978); (2) mean values complying with data from these laboratory measurements in the range of wind velocity change ± 1 m/s (vertical lines are the root-mean-square deviations); (3) data from laboratory measurements in the presence of a heavy alcohol film on the surface of the water; (4) and (5) data of field measurements by the method of radon deficit; the remaining dots in part (b) are data from laboratory measurements carried out in 1978-1985; the dotted line is the result of a calculation by Equation (3.11.2).

whence, on the basis of the continuity condition = (pa/pw)1/2t<* for the momentum flux at the water-air interface, where u^ is the friction wind velocity and pa and pw are air and water densities, it follows that k° = 0.082(pa/pw)1/2Sc"2/3MH!. (3.11.2)

This expression results in understated (compared with experimental data) values for at u^ > 0.2 m/s or at wind velocities over 6-7 m/s. Several explanations for this marked disagreement have been proposed. The most likely one is the rejection of an allowance for the amplification of gas exchange in the process of wind-wave breaking. This can be confirmed by the results of gas exchange measurements in the presence of the surface-active substances (for example, heavy alcohol) causing weakening or even complete cessation of wind-wave breaking and, accordingly, a decrease in k°.

As is well known, wind wave breaking is accompanied by sporadic injections of turbulent energy and by the formation of intermittent turbulent spots in the ocean layer closest to the surface. This in turn, leads to the complete reorganization of its vertical structure. If this feature is taken into account but the ocean surface is still considered to be hydrodynamically smooth, then within the ocean layer closest to the surface one can detect the diffusion sublayer immediately adjoining the water-air interface where the coefficients of the turbulent vT and diffusion Xt are much less than v and as well as the outer part of the viscous sublayer with vT« v, Xt » L and its underlying turbulent layer with vT » v, Xt » X• As this takes place, the coefficients of eddy viscosity and diffusion in the viscous sublayer, according to Kitaigorodskii (1984), can justifiably be prescribed as squared, and in the turbulent layer as power (with an index of 4/3) functions of the vertical coordinate calculated from the water-air interface. In this case the thickness ¿d of the diffusion sublayer and the rate of gas exchange take the form (see Kitaigorodskii, 1984)

where e(0) is the rate of turbulent energy dissipation in the vicinity of the water-air interface; a « 1 is the turbulent Prandtl number (the ratio of eddy viscosity and diffusivity); ab « 0.1 is the non-dimensional numerical constant in the expression for the eddy viscosity coefficient.

In (3.11.3) the only parameter depending on wind velocity is e(0). The simplest way to determine it is to use the condition of local balance between the turbulent energy production ns0 due to wind wave breaking and dissipation e(0)d in the layer with thickness d (the latter is defined as the thickness of the layer within the limits of which the influence of surface tension becomes apparent), that is, ns0 = e(0)d, where e0 = (pa/pw)mw3; d = }'3/5e"2/5(0); y = c/pw; a is the surface tension coefficient; n k 1 is the numerical constant. Substitution of e(0) into (3.11.3) gives, finally,

Comparison of (3.11.4) with (3.11.2) shows that the wind wave breaking

provides stronger dependence of the gas exchange rate on wind velocity than is the case when this effect is being ignored.

Further stronger dependence of the gas exchange rate on wind velocity can be obtained by assuming that the ocean surface is hydrodynamically rough. Following Kerman (1984) we define the scale of gas concentration by the equality QC02 = u^C^s^, where sB = aB(u*/M*c)3 is the relative area of breaking waves; u*c is the critical friction wind velocity when the breaking of wind waves begins; aB = 0.03 is a numerical constant. On the other hand, as we know, QCo2 = k°Ac, where Ac = (cq0JG — c2Ql )■ Equating both equations we have = m!1!W(Ac/CH!)"1sb. We take advantage of the universal dependence for the mass transfer in the vicinity of the immoveable rough surface according to which Ac/C# = cnRey2Sc2t3, where Re0 = u^wh0/v is the surface Reynolds number; h0 = au^/g is the root-mean-square height of breaking waves; a = 0.55 and a = 0.68 are numerical constants. After substitution of these expressions into /c° and application of the continuity condition M*w = (Pa/Pw)1"2"* for the momentum flux at the water-air interface we obtain the following relation for the non-dimensional gas exchange rate k°/(vgu^):

where m = (aB/aa1/2)(pa/pw)1/4 is a new numerical constant.

Comparison of (3.11.5) with (3.11.2) and (3.11.4) shows that a change in the hydrodynamic properties of the ocean surface is accompanied by an abrupt enhancement of the gas exchange rate, a conclusion confirmed in general by field and laboratory measurement data. In view of this we note that estimates of are usually performed by indirect methods. The most precise of these under field conditions is the so-called radon deficit method, which is based on the assumption that a flux of radioactive gas 222Rn from the ocean into the atmosphere is balanced by an integral (within the limits of the upper ocean layer) difference between influx of radon due to the decay of 226Ra and loss as a result of the natural decay of 222Rn. The duration of the measurements should have an order of a week or more to satisfy the condition of quasi-stationarity (the period of half-life of radon is equal to 3.85 days), and the effects of horizontal diffusion and advection should be assumed to be small. The dependence of k° on the wind velocity obtained in such a way is shown in Figure 3.4. Estimates from laboratory measurements are also presented here. Despite the large spread in data it can be seen that the gas exchange rate increases with the wind velocity, and, in addition, at ua « 7 m/s the regime of gas exchange is changed.

From other indirect methods, a method of determination of from measurements of the concentration of the radioactive isotope 14C is worthy of note. As is well known, 14C is formed in the atmosphere due to the action of cosmic rays. Because of this, in the stationary state the flux of 14C through the water-air interface should be balanced by the radioactive decay of 14C in the ocean. This last condition allows us to estimate the global averaged value of using an undisturbed (that is, not distorted as a result of nuclear weapon tests in the atmosphere) redistribution of 14C between the ocean and the atmosphere. The estimation of /c° obtained by this method is about 6 x 10"4 m/s, though its accuracy is not very high.

The first eddy correlation measurements of the carbon dioxide flux in the surface atmospheric layer were carried out in 1977. The estimates of /c° based on these, and on direct recordings of the partial pressure of C02 in water and air, led to an unexpected result (see Smith and Jones, 1985): it turned out that the mean value of QCOl is close to zero, while the partial pressure of C02 in water is markedly higher than in air.

We will return later to an explanation of this fact, but now let us turn to one more important mechanism which operates on wind wave breaking and which, perhaps, is responsible for the enhancement of gas exchange at high wind velocities. By this we mean gas transfer by air bubbles. An analysis of bubble motion in the ocean layer closest to the surface demonstrated (see Memery and Merlivat, 1985) that a flux of gas transported by bubbles is not proportional, generally speaking, to the difference in concentration at the surface cf and in the liquid phase c°. Nevertheless, for so-called trace gases (small atmospheric admixtures to which C02 also belongs) a total flux Qcoz at the water-air interface can be presented in the form Qcc>2 = (k° + k°)(c° - cgo2), where k°h = (c° - cg02)"1 JJ w(#/dz)Q dz dr is the coefficient of gas transport by bubbles; ij/ = ij/(r, z) is the distribution function of bubbles over sizes depending on the depth z; r and w(r) are radius and velocity of a vertical displacement of bubbles; Q = Q(r, z) is the change in the gas content of bubbles during their lifetimes. It is also assumed that bubbles at the depth z > z', where z' = z'(r) is a certain fixed depth, are completely degenerate while at z < z' their dimensions remain constant.

We note that at c® = Cqq2 the gas exchange between bubbles and water will not stop because at concentration Cco2, which does not depend on z, the internal gas pressure in bubbles exceeds its partial pressure in water due to surface tension. Formally, this case is met by = oo, and the equilibrium state, that is, reduction of QCOl to zero, takes place at /c° + /c° = 0 or k° < 0. Accordingly, the surface layer has to be oversaturated (c£02 > c°), and the gas incoming from bubbles into water has to be balanced by its transfer into the atmosphere. The degree of water oversaturation increases with decreasing gas solubility: for C02 it can reach 6.4% (see Memery and

Merlivat, 1985). It is the oversaturation effect of the surface ocean layer in the presence of air bubbles formed in the process of wind wave breaking which explains the above-mentioned result of eddy correlation measurements of C02 flux in the surface atmospheric layer.

Gas transport by bubbles determines the difference between exchange coefficients for different gases and, hence, points to the fact of the illegitimacy of using radon as a tracer while examining C02 exchange between the ocean and atmosphere. Indeed, on the one hand, radon concentration in the atmosphere is close to zero, so that the presence of bubbles does not affect the strength of radon transfer through the water-air interface; on the other hand, C02 concentration in the atmosphere and ocean are comparable and, because of this, gas transport by bubbles has to be important at high wind velocities.

Thus, there is no method among those mentioned above which is free from restrictions preventing, to some extent, the reliable determination of the C02 flux at the ocean-atmosphere interface. Data of its measurements under natural conditions are quite limited as well. So it is no wonder that in global carbon cycle models the laboratory estimates of the gas exchange rate are frequently used for determination of the C02 flux at the ocean surface.

3.12 Features of small-scale ocean-atmosphere interaction under storm conditions

The following lists the specific features of the small-scale interaction of the ocean and atmosphere under storm conditions: firstly, the formation of an intermediate zone near the water-air interface, with a mixture of finite volumes of water and air; secondly, an abrupt intensification of momentum, heat and humidity exchange as a result of macroscopic transfer. The latter is closely connected with sprays in air and bubbles in water, and they, in turn, with wind-wave breaking.

According to Bortkovskii (1983) there are two types of wind-wave breaking: the plunging type, and the sliding type. In the first case a wave crest passes over a slope and, being overturned, plunges into the water. In the second case, the breaking mass of water gathers a large number of air bubbles while moving along a wave slope and forms 'white horses' - turbulent flows of water and air mixture on the wave slope. The density of the mixture is less than that of water, and this density difference is sustained by air entrainment, due to which white horses do not degenerate immediately but slide, under gravity, along wave slopes, as over an inclined wall.

The main penetration of air into water occurs due to wave breaking but the contribution of entrainment through the free surface of liquid is also important. As for the formation of sprays this occurs in the following sequence. When the top of an emerging bubble is higher than the free surface, the water flows down from it and the bubble envelope in the vicinity of the top becomes thinner. This thinning continues until the moment when a hole appears. As soon as this occurs the imbalance of the surface tension force causes the hole to expand and shifts the torn envelope off centre. As a result, a ring-shaped rise appears at the hole periphery and, consequently, ring waves arise. Their interference in the centre of the hole is accompanied by the formation of a vertical jet from which one or several drops (sprays) are separated and soar upwards.

This explanation, based on data from filming laboratory experiments is very simplified. Under natural conditions the sprays are also formed by the separation of water particles from wave crests and by the interaction of steep gravitational waves with surface drift current. But available empirical information about the vertical flux and concentration of sprays in the surface atmospheric layer is both meagre and inaccurate. Little wonder that our knowledge of the mechanisms of momentum, heat and humidity transfer under storm conditions still does not go beyond the limits of qualitative presentations, which, briefly, reduce to the following.

In storm winds, the sprays rise from the surface of the water into the air, where they are accelerated by the wind and fall back into the water, thereby imparting the momentum they have gained to the upper water layer. Momentum transfer with air bubbles occurs in much the same manner, but perhaps less efficiently. Also, the breaking waves result in intensive mixing of the upper water layer. These processes together lead to a decrease in the vertical velocity gradients in the near-surface layers of water and air and to an increase in the resistance coefficient of the sea surface. Another factor which increases the resistance coefficient is the appearance of short, steep gravitational waves and foam which increase the effective roughness of the sea surface.

Turning to the description of the mechanisms of heat and humidity exchange between the ocean and the atmosphere under storm conditions we note first that the temperature in the lower part of the surface atmospheric layer can be higher or lower than that of the water surface, and the humidity, as a rule, is less than the saturating humidity. Thus, the temperature of sprays which, at the moment of their separation, is equal to the temperature of the sea surface, and the saturating humidity at the surface of the sprays, will differ from the temperature and humidity of the surface atmospheric layer. These differences determine heat and humidity exchange between sprays and air.

The sublayer within the limits of which the immediate influence of sprays manifests itself is limited by their rise in height (15-20 cm), that is, the thickness of this sublayer is much less than the thickness of the marine surface atmospheric layer. On the other hand, the thickness of the water layer saturated with bubbles under storm conditions is much greater than the thickness of the near-surface water layer in which the eddy fluxes remain approximately constant in the vertical direction, so that the eddy fluxes in the near-surface water layer make up only a part of the full range of fluxes determined, among other things, by the bubble transport.

For reasons that are easy to understand, the effect of bubbles on transfer processes in the near-surface water layer is difficult to estimate, and thus one has to limit oneself to the analysis of the effect of sprays on the structure of the surface atmospheric layer. Such an analysis, within the framework of a thermodynamic model of an isolated drop with subsequent integration over a range of drop sizes was carried out by Bortkovskii (1983). Of course, this approach is justified if the volume concentration of drops does not exceed the critical concentration at which the distance between separate drops is comparable with the thickness of the boundary layer forming on them. Otherwise, the laws of resistance, heat and humidity exchange for separate drops become unusable for their set.

It is well-known that the critical value of volume concentration of spherical particles amounts to ~0.02, and the humidity content in the lower part of the surface atmospheric layer at wind velocity 20-25 m/s amounts to 10"3-10~4 g/cm3. Accordingly, the mean volume concentration of sprays (ratio of humidity content to water density) is one or two orders less than critical. The assumption as to the spherical nature of drops is also justified over the whole range of spray sizes that is typical of the surface atmospheric layer. Hence, the initial prerequisites of the model mentioned above are realistic. This sustains the hope that, after turning to a set of drops characterized by their distribution over size and space, the final aim - the estimation of the integral transfer of momentum, heat and humidity by bubbles - will be attained. We now give the results of model calculations of most interest to us.

According to Bortkovskii (1983) at a drop radius (from 0.003 to 0.005 cm) typical of the surface atmospheric layer the vertical heat and humidity transfer by sprays turns out to be comparable with the eddy transfer. Along with this, a decrease in the radius down to 0.0015 cm results in the fact that humidity transfer by sprays becomes several times larger than the eddy transfer, and the heat transfer becomes negative (air is cooled down by sprays). This is easily explained: small drops quickly reach equilibrium temperature for which the diffusive heat exchange, losses of heat due to

Table 3.2. Dependence of C*/Cq and C*/C9 on water-air temperature difference and drop radius (according to Bortkovskii, 1983)

Drop radius (cm)

Wind velocity Water-air temperature w a 1 w . • . ♦ °-003 0

Wind velocity Water-air temperature

Table 3.2. Dependence of C*/Cq and C*/C9 on water-air temperature difference and drop radius (according to Bortkovskii, 1983)

(m/s) |
difference (°C) |
c*jcq |
Cfl/Q |
C*JCq | |

20 |
5 |
0.97 |
1.28 |
0.55 |
1.29 |

1 |
1.32 |
0.43 |
0.76 |
1.55 | |

-2 |
2.01 |
1.57 |
0.94 |
0.40 | |

30 |
5 |
1.18 |
1.08 |
0.71 |
1.12 |

1 |
1.85 |
0.25 |
0.87 |
2.23 | |

-2 |
2.53 |
1.80 |
1.20 |
0.49 |

Note: coefficients of heat exchange and evaporation for the environment are assumed to be the same and equal to 1.43 x 10"3 at wind velocity 20m/s, and 1.58 x 10"3 at wind velocity 30m/s; the air temperature is referred to the height 10 m.

evaporation and radiative sources and sinks of heat counterbalance each other, and further evaporation of drops occurs due to the influx of heat from the surrounding air. On the other hand, when the radius increases, those drops with an initial temperature higher than the temperature of the environment give up their heat by diffusive heat exchange.

As it turns out, the contribution of sprays to the total momentum flux is relatively small: at a drop radius of 0.003-0.005 cm it amounts to approximately 10% of the total flux. With decreasing drop radius the ratio of momentum transport by sprays to the total transport diminishes; with increasing drop radius it approaches a finite limit.

In order to illustrate the results we compare, following Bortkovskii (1983), the coefficients of heat exchange (C*) and evaporation (C*) for drops with the usual coefficients of heat exchange(Q) and evaporation (Cq). In doing so, we determine coefficients Qf, C* from Equations (3.8.1) but replace the fluxes appearing in (3.8.1) by heat and humidity exchanges due to sprays. The results of the comparison are given in Table 3.2.

As can be seen, an increase in drop radius from 0.003 to 0.005 cm leads to weakening of evaporation that is connected with a decrease in the overall area of the drops' surface, and with intensification of sensible heat exchange representing the difference between the rate of change in drop heat content and losses of heat due to evaporation. The ratio between the coefficients of heat exchange and evaporation changes similarly. Let us pay attention to the sensitivity of C*/Cq and C*/C9 to variations in water-air temperature difference and in the humidity difference it controls. Under conditions of thermal inversion when the water-air temperature difference becomes negative, an increase in C*/Cq is explained by a reduction in drop cooling, and by the retention of high values of humidity difference at the drop-environment interface. The opposite situation occurs at unstable stratification: its influence contributes to acceleration of the cooling of drops, to a decrease in humidity difference and, as a result, to a decrease in C*/Cq. But the values of C*/Cq and C*/Ce are the most impressive in themselves. They leave no doubt as to the abrupt intensification of heat and moisture exchange under storm conditions.

The contribution of storms to the formation of mean (over a long time period) values of heat exchange and evaporation is determined not only by the intensity of these processes but also by the frequency of storms, by their duration and by temperature and humidity differences at the water-air interface under storm and background (non-storm) conditions. According to data from Ocean Weather Ships (OWS) an increase in heat transfer into the atmosphere manifests itself in autumn, winter and partly in spring. The effect of storms from May to October is not felt in practice. The ratio between total heat fluxes obtained with and without an allowance for storms (this ratio serves as a quantitative measure of storm effects) varies over time and space from 1.38-1.41 in the winter months (OWS A, C, D) to 1.00 in the summer months (OWS D). On average for the year this ratio amounts to 1.32 for OWS A, 1.31 for OWS C, 1.28 for OWS D and 1.15 for OWS E. In other words, in high latitudes (OWS A), and also in the Gulf Stream region (OWS D) and in the North Atlantic Current (OWS C), storm activity determines approximately one-third of the resulting heat and humidity transfer into the atmosphere.

Mesoscale ocean-atmosphere interaction

### 4.1 The planetary boundary layer

The planetary boundary layer (PBL) is the name given to the domain of flow in gaseous and liquid shells of the rotating planet, formed as a result of the simultaneous action of the forces of pressure gradient and turbulent friction and the Coriolis force. We introduce the components of mean velocity u, v and geostrophic wind velocity

C/g = — (f p) ~1 dp/dy = G cos a, V% = (/p) ~1 dp/dx = G sin a, where G is the modulus of the geostrophic wind velocity; a is the angle between an isobar and the axis x; p and p are the mean pressure and density; x and y are axes in the Cartesian coordinate system; and we define the thickness hE of the planetary boundary layer so that the relative departure of mean velocity from geostrophic velocity at level z = hE does not exceed some preassigned value <5, that is, G_1[((7g — u)2 + (Vg — v)2yjJhE < S. Then taking into account that the vertical scale of PBL has to depend on the friction velocity m* and Coriolis parameter /, from dimensional considerations we obtain the Rossby-Montgomery formula hE = yuJ\f\, (4.1.1)

where y is a non-dimensional factor, varying according to different authors, in the range 0.1-0.4. At y = 0.4 (for the atmosphere this value of y corresponds to d « 20%) the PBL thickness amounts to about 1000 m in mid-latitudes, which is an order of magnitude less than the thickness of the troposphere.

Another way of defining hE was proposed by Charney (1969). He postulated that the PBL remains hydrodynamically stable as long as its effective Reynolds number ReE = GhE/kE, where kE is a certain effective value of the eddy viscosity coefficient, does not exceed the critical Reynolds number Recr for the laminar boundary layer. If by analogy with the laminar boundary layer we assume that hE = (2/cE/|/|)1/2 and solve this equation for kE, then we will obtain the estimate hE > 2G/fRecr (4.1.2)

for the PBL thickness.

It is clear that estimates (4.1.1) and (4.1.2) are very rough since they do not take into account the peculiarities of the PBL structure conditioned by the influence of stratification. It is appropriate to note in this connection that there are two types of turbulence: dynamic and convective, generated by Reynolds stresses and buoyancy forces respectively. In the surface atmospheric layer Reynolds stresses dominate at high wind velocities, and buoyancy forces dominate for strong heating of the underlying surface. Dynamic turbulence decreases more rapidly when moving away from the underlying surface than does convective turbulence. As a result, it is convective turbulence which determines the PBL thickness over land. In this case, the upper boundary of the PBL is identified with the lower boundary of high inversion to which the influence of surface heating extends.

At unstable, as well as at stable, stratification, the PBL structure should be determined by the same parameters u^, H/pcp, and [1 as the structure of the surface atmospheric layer, with only one exception that the Coriolis parameter / is now added. From this parameter set along with the Monin-Obukhov length scale L, one more length scale A = Ku^/f (here the von Karman constant k is introduced for convenience) and the non-dimensional combination p0 = A/L = k2 P( — H/pcp)/\f\ul (i.e. the non-dimensional stratification parameter) may be compiled. Hence, hB = yuJf\~1ilf(ji0), (4.1.3)

where ij/(p0) is a universal function equal to 1 for neutral stratification (A*o = 0).

Let us define the function iK^o) and the expression for hE in the regime of strongly unstable stratification. Within the PBL in this case one can distinguish a surface layer, a free convection layer, a mixed layer and an entrainment layer. In the surface layer where dynamic turbulence dominates, the parameters ut, H/pcp, z and ft are the determining parameters, and u^ and are characteristic scales of velocity and temperature (see Section 3.8). Accordingly, the non-dimensional vertical gradients of the mean velocity and temperature obtained by normalizing their dimensional expressions to u* and represent universal functions of the non-dimensional height z/L. From the results of eddy correlation and gradient measurements (see Zilitinkevich,

1970) the surface layer for strongly unstable stratification is limited by height 2 < 0.07|L|.

The height z = 0.07\L\ serves as the lower boundary of the free convection layer where dynamic turbulence becomes negligible as compared with con-vective turbulence. The structure of this layer is determined by three parameters H/pcp, z and /?, so that the scales of velocity w^ and temperature T* composed of these parameters take the form w^ = (zfiH/pcp)1/3, = (H/pcp)/wt. According to the Monin-Obukhov similarity theory the non-dimensional profiles of mean velocity and temperature formed by normalizing to Wj. and remain constant with height, a condition which is valid up to z « \L\. The mixed layer is situated above this height. Its structure depends neither on u# nor on z, being determined only by H/pcp, ft and f. In this case hE is the characteristic length scale, and w^ = (hE[lH/pcp)1/3, and 7^ = (H/pc^/w^ are velocity and temperature scales. Accordingly, non-dimensional vertical gradients of mean velocity and temperature represent functions of z/hE only. The mixed layer extends up to the height z « 0.8/je.

In the range of heights from 0.8hE to 1.2/je the entrainment layer (an intermediate layer between the PBL and the free atmosphere) is arranged. The structure of this layer is determined by the entrainment rate characterizing the intensity of penetration of turbulence from the free convective layer into the non-turbulent inversion layer, by hydrostatic stability of the latter and by conditions in the free atmosphere.

A rough estimate of the PBL thickness for a strong hydrostatic instability (large negative values of p0) can be obtained from dimensional considerations rejecting the entrainment effects and, thereby, assuming that hE is determined by only three parameters H/pcp, ¡3 and f. In this case hE ~ (PH/pcp)1/2\f\~312 ~ l(-/i0)1/2. (4.1.4)

It should be emphasized that the multilayer structure is inherent mainly in the convective PBL over land with a typical thickness exceeding the Monin-Obukhov length scale by 50-100 times. The ratio hE/L is much lower over the ocean (excluding, perhaps, areas of cold deep water formation in the Norwegian and Greenland Seas, and in the Weddell Sea). In such situations, the possibility of ignoring the dynamic turbulence and, therefore, excluding the friction velocity ut from a number of parameters determining the structure of the free convection layer and mixing layer raises some doubts. These doubts can only be eliminated by turning to a description of the PBL over the ocean within the framework of a more general approach, which includes w* and hE in the set of determining parameters.

Under conditions of strongly stable stratification the thickness of the layer containing by turbulence of dynamic origin amounts to about 100 m. An important role in the formation of the structure of the above-mentioned layer is played by the radiating heat influx created by the long-wave emission of the underlying surface and the lowest atmospheric layer. Therefore, when identifying the PBL thickness with the thickness of the surface inversion layer it may be found to be much thicker than the vertical extent of the layer in which the dynamic turbulence predominates. This circumstance should be borne in mind when estimating the expression, hE ~ ul(-PH/pcpyll2\f\~1/2 ~ W2, (4.1.5)

obtained by Zilitinkevich (1972) from dimensional considerations on the assumption that hE is determined only by H/pcp, /? and f. It is also necessary to remember that, due to the suppression of turbulence and weakening of mixing, the response of the stable stratified PBL to a change in external parameters becomes slower. As a result, the condition of quasi-stationarity of the PBL implicitly used in the derivation of (4.1.5) may not be fulfilled. There is only one way to eliminate this obstacle - to replace the diagnostic relationship (4.1.5) by the evolution equation for the PBL thickness.

### 4.2 Problem of closure

We consider the stationary, horizontally homogeneous PBL for parallel, equidistant isobars. We also assume that the horizontal gradients of pressure and density, excluding those cases where the density is involved in a combination with gravity, are equal to their mean values within the PBL. Taking these assumptions into account and neglecting small terms describing the effect of molecular viscosity, the equations for the mean velocity components u, v are written in the form

p dx dz

p dy dz where —u'w', —v'w' are the components of Reynolds stresses normalized to mean density; other symbols are the same.

We complement (4.2.1) and (4.2.2) with the evolution equation for the mean temperature T

dt dz and for the mean specific humidity q

8q 8

8t dz and also by the continuity equation du dv dw

dx dy dz by the condition of hydrostatic equilibrium di=-gp, (4.2.6)

dz and by the equation of state of moist air p = RpT( 1 + 0.61$). (4.2.7)

It should be noted that in (4.2.3) and (4.2.4) we have neglected the sources and sinks of heat and moisture along with molecular diffusion, but have retained, in contrast with (4.2.1) and (4.2.2), terms describing non-stationary effects because otherwise it is impossible to satisfy the condition of vanishing eddy fluxes of heat w'T' and moisture w'q' vanish at the upper boundary of the PBL.

Let us make one more preliminary remark with regard to the type of averaging used. The most convenient method is averaging over an ensemble (an infinite set of independent realizations) having the following properties: ufij = UjUj, «;u'j = u'jiij = 0. To realize such averaging in practice is difficult, if at all possible. And one thus has to apply the so-called ergodic hypothesis, that is, to assume a flow to be statistically stationary, and the ensemble averaging to be equivalent to averaging over time, because in this case the ensemble-average values do not change with time. Usually, the largest time interval in which the time average approaches its stable stationary value with some tolerated accuracy is chosen as an averaging period. To meet this condition the spectrum of the process to be examined must have a deep minimum separating high-frequency eddy fluctuations from low-frequency oscillations of synoptic origin. Then a selection of any averaging period from the band of the spectral minimum will provide filtration of high-frequency fluctuations and at the same time will not introduce large distortions due to low-frequency oscillations. Such a minimum presents in the spectra of the wind velocity and temperature in the surface atmospheric layers over the horizontally homogeneous underlying surface within the range of periods of the order of an hour; it is absent in the spectra of meteorological elements in the upper part of the PBL and in the surface layer over the horizontally non-homogeneous underlying surface.

Let us turn to the system (4.2.1)-(4.2.7) and note that it contains seven equations and 11 unknowns: u, v, w, p, p, T, q, u'w', v'w', w'T' and w'q'. Thus, the system (4.2.1)-(4.2.7) describing the mean (in the Reynolds' sense) motion in the PBL turns out not to be closed because of the appearance of covariances u'w', v'w', w'T' and w'q'. When compiling additional equations for these, as was done, for example, in Section 3.10, new unknowns (third moments) appear. The procedure may be continued and at any time new unknowns (the moments of higher orders) will appear. In short, the strict (in the mathematical sense) closure of the equation system for the eddy motion, and, particularly, the motion in the PBL is impossible. This is termed the problem of closure. Its solution reduces to the use of some a priori assumptions which allow us to express moments of higher order in terms of moments of lower order or by characteristics of the mean flow. Such closures are called closures of the first and second orders respectively.

### 4.2.1 First-order closure

First-order closure is based on the analogy between the eddy and laminar motions, from which the components —u'w' and — v'w' of the vertical eddy flux of momentum can be presented as the product of the eddy viscosity coefficient kM by the vertical gradient of the corresponding component of mean velocity du/dz, dv/dz, that is, —u'w'M = kM du/dz, — v'w' = kM dvjdz. In exactly the same manner the vertical eddy flux w'c' of any other substance with mean concentration c is defined as — w'c' = kc dcjdz, where kc is the eddy diffusion coefficient. Unlike the coefficients of kinematic molecular viscosity v and diffusion x> the coefficients kM and kc are not proper parameters of fluid and depend directly on the peculiarities of the flow structure. For example, for flows in an open channel the eddy viscosity coefficient changes with depth by the parabolic law, and for a plane jet it changes in proportion to the square root of the distance from the source. Therefore, the above-mentioned expressions for eddy fluxes do not in themselves reflect the peculiarities of the eddy transfer structure and serve only as the embodiment of an idea of the proportionality between the substance transfer in a certain direction and the gradient of the substance transported in the same direction. Thus, the problem should consist of setting a connection between the eddy viscosity and diffusion coefficients and the local characteristics of the mean current.

The first step in this direction was undertaken by Prandtl in 1925. As is well known, the coefficient of the molecular viscosity is proportional to the product of the mean velocity of motion of molecules and the mean length between two subsequent collisions. By analogy with the process of molecular transfer Prandtl postulated that vertical transfer in eddy flow is performed by eddies drifting with velocity w' at distance /. The latter is identified with the mixing length, or, as it is often called, the turbulence scale (the mean spatial scale of energy-containing eddies), where the eddy is completely adapted to the conditions of the environment (is mixed with it). Further, considering that the characteristic scale of velocity fluctuations u' and w' are equal to I du/dz, Prandtl obtained — u'w' = l(du2/dz)2. From here and from the expression — u'w' = kM du/dz, kM = l2\du/dz\ follows.

To find the new unknown / additional considerations need to be involved. The simplest of these is the proportionality of the turbulence scale in the near-wall region of the flow to the distance z from the wall, that is, I = kz, where k is the non-dimensional constant (the von Karman constant). Such an assumption is not the only possible one. For example, in the region away from a wall where the proportionality between / and z is not valid the von Karman assumption seems to be more justified. In accordance with this the turbulence scale is considered to be a function of the local characteristics of the mean flow, particularly of the first and second derivatives of velocity in the vicinity of the point examined, but not of the velocity itself: the turbulence scale cannot depend on the value of the mean velocity due to the principle of invariance. Then from dimensionality considerations I = — k x (du/dz)/(d2u/dz2). Generalization of this expression in the case of variable (along height) direction of the mean velocity vector u was proposed by Rossby (1932). It has the form

du |
ld |
du |

dz |
dz |
dz |

The above-mentioned expressions for the turbulence scale do not take into account the effect of stratification. This restriction can be overcome by including a factor depending on stratification in the right-hand side of the expression. Kazanskii and Monin (1961) used this method having rewritten the Prandtl formula in the form I = kz( 1 - oRf)n, where R f = -(JiH/pcp)/ u^du/dz) is the flux Richardson number; ok 12 and n % 0.35 are empirical constants. On the other hand, Zilitinkevich and Laikhtman (1965) used the von Karman formula as an initial assumption. They reasoned as follows (see

Zilitinkevich and Laikhtman, 1965, and Zilitinkevich et al, 1967). If the von Karman formula or any equivalent formula I = — 2i<(du/dz)2/(d/dz)(du/dz)2 is fulfilled for neutral stratification, then for free convection when all the characteristics of turbulence, including the turbulence scale, depend only on the vertical structure of the field of potential temperature 9 or buoyancy p9, the expression / = — 2K(d/dz)((!9)/(d2/dz2)(P9) pertains. The latter, as with the von Karman formula, is obtained from dimensional considerations assuming that the only determining parameters are the first and second derivatives of the function in question. From (du/dz)2 and ¡59 we compile the combination \j/2 = [(du/dz)2 — a'ap 89/dz] having the necessary asymptotic properties, that is, reducing to (du/dz)2 for neutral stratification and to —ct%fid9ldz for strongly unstable stratification where is the reverse of the turbulent Prandtl number (see Section 3.7). Then the expression for the turbulent scale can be presented in the form / = —2K\l/2l(d\ji2/dz) = —K\j//(d\p/dz). Further, when neglecting the effects of non-stationarity, horizontal inhomogeneity and vertical diffusion, then from the budget equation for the kinetic energy of turbulence (see the first relationship in (3.10.15)) it follows that — w'w'(du/dz) + fiw'9' = e. From here and from definitions —u'w' = kM du/dz, w'9' = aekM d9/dz, and also from Kolmogorov's relations of approximate similarity e = ceb3l2/l, kM = c0b1/2l, where b is the average kinetic energy of turbulence referred to density, ce and c0 are non-dimensional constants, we obtain t¡/ = (cjc0)ll2bll2/l. Thus, the generalized von Karman formula for the turbulence scale proposed by Zilitinkevich and Laikhtman is finally written in the form

The concept of mixing length assumes that the eddy viscosity coefficient is equal to zero whenever the gradient of the mean velocity vanishes. In other words, it is suggested that the turbulence is in a state of local equilibrium: generation and dissipation of eddy energy at any point of the flow counterbalance each other, and eddy transfer from neighbouring points and its change over time are absent, or do not play any important role. The fact that this condition is not always valid is obvious. It is enough to recall that in wind tunnels where the turbulence is generated immediately behind a grid and then transported downward along the flow by the mean current according to the concept of mixing length not taking such transport into account, the eddy viscosity coefficient has to take on zero values. As another example: in channels where turbulence is generated mainly in the vicinity of the walls and transported by eddy diffusion to the channel axis, the concept of mixing length neglecting this transport should lead to zero values of the eddy viscosity coefficient at the channel axis. Finally, in the oscillating boundary layer where the intensification of turbulence at the stage of attenuation and the degeneration of turbulence at the stage of amplification of the mean velocity occur, their reproduction within the framework of the concept of mixing length ignoring the effect of non-stationarity on the eddy energy budget is excluded. Thus, the concept of mixing length becomes invalid every time a marked contribution to the eddy energy budget is made by the advective and diffusive transport of eddy energy, and by the pre-history of the process.

The above-mentioned restrictions are a direct consequence of the general hypothesis forming the basis of the concept of mixing length, the hypothesis that turbulence is described by the unique length scale and the unique velocity scale. But reality offers us richer possibilities, an example of which is the convective boundary atmospheric layer at the upper boundary of which a negative (opposite to the direction in the surface layer) heat flux is formed as the result of entrainment of warmer air from the inversion layer located above. An identical situation takes place on convective mixing near the base of the upper mixed layer of the ocean where entrainment of colder water from the lower seasonal thermocline is responsible for the appearance of the positive heat flux (that is, again opposite in direction to the surface flux). Such situations are difficult to describe within the framework of the concept of mixing length.

In this respect the integral models or the models of the mixed layer, as they are often called, based on the assumption that all variables within the PBL depth are constant, can be useful. In models of this type the eddy viscosity coefficient is not an unknown quantity to be determined and the closure problem reduces to the parametrization of eddy fluxes at the external boundary of PBL and of its depth.

### 4.2.2 Second-order closure

The essence of this is that components of Reynolds stresses and eddy fluxes of scalar substances are not approximated by formulae of gradient type which are the basis of the first-order closure (or K-theory as it is also called, in accordance with common designation of eddy exchange coefficients), but they are considered as new unknown quantities derived from equations for the covariations. In Section 3.10 we discussed the method of derivation of equations for the Reynolds stresses. So to avoid repetitions we write Equation (3.10.4), complementing it with terms describing the effect of the Earth's rotation.

For dry air this equation takes the form dt'

8 —— — 8 —r—r i-l—f 8U: —— 8U:\ 8 , , , — W; Uj + Uk — u,Uj + UjUk — + uiuk-J-) + — UiUjUk at 8xk \ 8xk oxk) 8xk i

+ v u', u'j - 2v —- —J- - fk (sjkl u[ui + emu', u'j), (4.2.8)

OXk CXk OXfo where sjk, is the Levi-Civita symbol defined as £i23 = £23i = £3i2 = l> ei32 = e2i3 — £32i = — 1 and sjki = 0 in all other cases.

Equations for the fluxes of scalar characteristics can be obtained in the same manner as (4.2.8). Namely, we multiply Equation (3.10.3), supplemented with a term describing the effect of the Earth's rotation, by T', and Equation (3.10.11) for dry air, by u[, then sum and average them. As a result, we obtain the following budget equation for the eddy heat flux:

-u\T + uk — u'iT' + u'iu'k— + u'k T' —^ + — U',u'kT' dt 8xk \ 8xk oxkJ 8xk

Po ' J' V dxk ' 8xt where the first term on the left-hand side describes the change in time, the second term describes the advective transport, the third term describes the interaction of mean and fluctuating motions, the fourth term describes the transfer by velocity fluctuations (the eddy diffusion), and terms on the right-hand side describe, respectively, changes in the eddy heat flux due to correlations of pressure and temperature fluctuations, buoyancy forces, molecular viscosity and diffusion and the Coriolis force.

Equations (4.2.8) and (4.2.9), together with Equations (3.10.2) and (3.10.9) for mean values of the velocity and temperature f as well as Equation (3.10.3) for temperature dispersion T'2 could form a closed system if it were not for terms describing the influence of dissipating processes, correlations of pressure fluctuations, and the eddy transfer. These terms are not directly connected with the variables sought, and should therefore be considered as additional unknowns which have to be presented in terms of the functions sought, or disregarded wherever possible. This, in fact, is the essence of the problem of second-order closure. We consider newly appearing unknown terms in the order in which they are mentioned.

Dissipation terms. These are v d2u'iu'j/dxl, 2v(du[/dxk)(du'j/dxk), vT' d2u'Jdxk and Xtu'{ S2T'/dxk. The first term, as we have already mentioned, describes the influence of molecular diffusion on the evolution of Reynolds stresses; the second term describes the influence of the viscous dissipation. To estimate these terms we introduce characteristic scales of velocity fluctuations b112, and length / where, as before, b is the kinetic energy of turbulence normalized to the mean density and / is the turbulence scale (the spatial scale of energy-containing eddies). Then v d2ufij/dxi ~ vb/l2 ~ (v/bl/2l)(b3/2/l) ~ Re-\b3t2/l\ where Re = bi/2l/v is the Reynolds number. From this it follows that at large Reynolds numbers (of order 107 in the planetary boundary layer of the atmosphere) the influence of molecular diffusion can be ignored.

Next, for large Reynolds numbers the characteristic spatial scale of energy-containing eddies is much larger than the Kolmogorov length microscale q = (v3/e)1/4 where the dissipation occurs (for an explanation of the sense of e see below). This means that in the energy spectrum the intervals of energy supply and dissipation ar

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