The surface layer theory based on dimensional analysis tells us most of what we need to know, but it doesn't tell us how the drag coefficient depends on the height at which the top-of-layer conditions are applied, nor does it say precisely how the coefficient depends on stratification or the surface roughness. We will now re-do the surface layer theory using a more precise form of the similarity assumption. The most important thing we will get out of this is a quantification of the suppression of turbulent mixing in stable conditions. This exerts a very important control on the fluxes at night-time and over ice or snow, where surface layer conditions are often stable. In particular, when ice or snow is melting the temperature is pinned at the freezing point, so if the atmospheric temperature is significantly above freezing, the surface layer is very stable, and this limits the delivery of heat available for melting.

Let c be any quantity whose flux we wish to determine in the surface layer. It might be temperature, water vapor, methane or some other chemical tracer. We will also consider the flux of horizontal momentum (proportional to horizontal wind u) using the similarity theory. Though we are not attempting to do much dynamics in this book, we will nevertheless need to talk a bit about momentum flux since this is what will tell us how the mean wind varies with height within the surface layer.

Within the surface layer, the fluxes of tracer and momentum are constant, by definition. This allows us to define the following velocity and tracer scales:

As a consequence of the second definition the tracer flux is just u*c*. The velocity scale ut is called the friction velocity, and is taken to be positive by convention. When the flux of tracer is upward,then the tracer fluctuation scale c* is positive.

Next we derive equations for the vertical gradient of mean tracer and mean wind. We'll first consider the neutrally stratified case, in which buoyancy forces are negligible. Strictly speaking, this case only applies when the density within the surface layer is constant. Any situation with heat transport would involve some temperature fluctuations, and hence some density fluctuations. In practice, though, when the temperature difference across the surface layer is sufficiently weak, the neutrally stratified calculation yields accurate results. The definition of "sufficiently weak" will be made precise later, when we come to incorporate buoyancy forces.

When buoyancy is insignificant in the surface layer, the only length scale appearing in the problem is the height z above the ground. Since the only tracer scale is c* and the only velocity scale is u*, dimensional analysis then tells us that the equations for the vertical gradients must be

dz z where Kvk is a nondimensional constant called the Von Karman constant. In principle, the nondi-mensional constant appearing in the tracer equation could be different from that in the momentum equation, but laboratory experiments indicate that in fact the same constant applies to both. The Von Karman constant has been measured in a wide range of turbulent laboratory experiments, which indicate that Kvk « .4. The sign choice in the tracer equation is dictated by the physical requirement that the tracer flux be upward when the concentration is greater at the surface than it is aloft.

The similarity equations allow us to relate the flux of tracer to the difference in tracer concentration between the ground and the upper edge of the surface layer, and similarly for momentum. Let zi be the upper edge of the surface layer. Integrating from a smaller height z* to zi and assuming that the winds vanish at the height z*, we find

The height z* at which we set the lower limit of integration cannot be sent to zero because of the logarithmic divergence in that limit. In fact, it has a physical meaning, and is called the roughness height. It corresponds to the height at which the airflow is so perturbed by the irregularities in the boundary that the mean flow is essentially zero. The roughness height corresponds loosely to the typical height of the bumps on the surface, but is generally smaller than one would intuit from the physical height of the bumps. In practice, it is determined by fitting the observed mean wind profile with the logarithmic form. Over open water, the roughness length is on the order of 0.0002 m, though at strong wind speeds the wind-driven waves increase the roughness significantly. Over ice or smooth land, the roughness length is more like 0.005 m, increasing to 0.03m if there is grass or low vegetation, 0.5 m for low forest and 2 m for large forests or urban areas.

The logarithmic profile of wind and concentration is called the law of the wall, and has been verified in a great variety of turbulent flows, ranging from wind and water tunnel experiments to atmospheric measurements to velocity profiles in tidal surges in the Bay of Fundy. Next, the tracer and velocity equations can be combined to give the tracer flux

w'c' = u*c* = (ln( Vk))2 u(zi)(c(z*) - c(zi)) (6.18)

from which we identify the drag coefficient

This formula allows us to explicitly compute the drag coefficient given the roughness height and the height at which one chooses to apply the boundary condition at the upper edge of the surface layer; one is free to choose zi as a matter of convenience, so long as it is low enough that the fluxes are constant within the surface layer. Though different roughness lengths are sometimes applied for moisture and momentum, it is generally adequate to use the same drag coefficent for all mixed quantities. Using the previous values for roughness length and assuming zi = 10m, we get CD = 0.0014,0.0028,0.0047, 0.018, 0.062 for open water, ice or smooth land, grassland, low forest and large forest, respectively.

Now let's introduce the effects of buoyancy. To do this, we must first define buoyancy quantitatively. Let pg be the mean density at the ground. Then the net force (per unit volume) on an air parcel with density p will be g ■ (pg — p) while the parcel is near the ground. The acceleration of the air parcel is obtained by dividing by the force by the mass, and is thus g ■ (pg/p — 1). The buoyancy acceleration is a form of reduced gravity, reflecting the fact that buoyancy forces cancel part of the gravitational forces, leading to a reduction in acceleration. We will refer to the buoyancy acceleration as simply "buoyancy" for short, and denote it by the symbol p. The buoyancy is affected both by the temperature and the composition of the atmosphere. For uniform composition, warm air will be positively buoyant when surrounded by colder air. However, air that is rich in a low molecular weight substance will be buoyant when surrounded by air that has lower concentration of that substance, even if the temperature is uniform. For example, since water vapor has lower molecular weight than dry Earth air, moistening an air parcel adds to its upward buoyancy and drying it tends to make it sink. The same can be said for adding methane to N2 in Titan's atmosphere. Similarly, the Martian atmosphere contains a few percent of Ar on average, which has a higher molecular weight than CO2. Thus, when pure CO2 sublimates from the Martian CO2 seasonal frost cap, it will be positively buoyant in the background mixture of CO2 and Ar. One can imagine a variety of situations in which an atmospheric constituent is released from or absorbed into the surface, but the most common situation involves a condensible substance which condenses onto or sublimates/evaporates from a reservoir at the surface. That could be N2 ice on Triton, liquid CH4 on Titan, CO2 ice on Mars, or solid or liquid H2O on Earth. Using the ideal gas law, the density is p = RpT(1 — *> + RT"' = -¿t (1 + ( M — ^ (620)

where Ma and Mc are the molecular weights of the noncondensible background gas and the con-densible component, respectively. Since the surface layer is thin, p can be assumed nearly constant within the layer. The buoyancy is then

Tg 1 + (e — 1)nc where e = Mc/Ma and nc,g is the molar concentration of the condensible substance at the ground. When nc = V',g = 0, or when e = 1, buoyancy is simply g ■ (Tg — T)/Tg. In the general case, if e < 1 then increasing nc makes the parcel more positively buoyant, while if e > 1 increasing nc makes the parcel more negatively buoyant. In general, the buoyancy is a nonlinear function of temperature and concentration, but in the special case where both (T — Tg )/Tg and nc are small, the buoyancy takes on the simple form

Tg in which the buoyancy is the sum of a temperature contribution and a composition contribution. When using either form of the buoyancy, it is assumed that the the buoyancy-generating tracer is saturated at the ground, in the typical case where its flux is maintained by a condensible reservoir there. Thus, nc,g = Psat(Tg)/p, where psat is given by Clausius-Clapeyron.

We can treat the buoyancy flux and mean buoyancy profile much as we did the tracer flux and tracer profile in the neutral case, defining w'3' = u*3*. However, the buoyancy scale 3* is now a dynamically signficant quantity with dimensions of acceleration, which can affect the profile. This has the important consequence that z is no longer the only length scale that enters into the problem - in addition we can define the Monin-Obukhov length

1 u2

Kvk 13*1

The inclusion of the Von-Karman constant in the definition of the Monin-Obukhov length is purely a matter of convention, and has no particular physical significance. The Monin-Obukhov length is on the order of the height to which a negatively buoyant plume with velocity u* would rise before exhausting its kinetic energy, or the height to which a positively buoyant plume initially at rest would rise before attaining velocity u*. For distances much closer to the boundary than i, the turbulence is dominated by the kinetic energy of the wind shear, as in the neutral case. For heights much greater than i buoyancy suppresses or enhances the turbulence. Using the Monin-Obukhov length, we define the non-dimensional depth Z = z/i. In contrast to the neutral case, the equations for the gradient of wind, tracer or buoyancy can each depend on some function of Z; in principle, because the function has a dimensionless argument, one can take data in the field or laboratory, and determine the function once and for all, as if it were a sine or cosine. The test of the validity of this bold assumption is to evaluate the functions from a wide variety of different field and laboratory datasets, and see if one gets essentially the same result from each. This is the assumption upon which the similarity theory rests, and it seems to work out well in practice. In terms of the similarity functions, the equations for buoyancy and wind gradient become

In general, one should allow for different scaling functions for the buoyancy and momentum equations, and indeed this is sometimes necessary to provide a good fit to data. For the stably stratified (i.e. negatively buoyant) case, it has been found that using the same scaling function for both equations is adequate. Some remarks on the positively buoyant case will be given later. In any event, let's assume Fu = . In nondimensional form, the equations become

dZ Z

which integrate out to yield the relations

where G(Zi) = J^1 (F(Z)/Z)dZ. To eliminate the buoyancy and velocity scales, we divide the first equation by the square of the second and multiply by zi — z*, which results in

in which we have rewritten the mean buoyancies as a function of the dimensional height. The left hand side is a nondimensional number called the bulk Richardson number, denoted henceforth by Ri. The Richardson number can be computed in terms of known quantities at the upper and lower edges of the surface layer, and gives the relative importance of potential and kinetic energy; when |Ri| ^ 1, buoyancy forces are negligible, and the surface layer can be treated as if it were neutral.

Given Ri, Eq. 6.27 can be solved either explicitly or iteratively for Zi, which allows ft* and u* to be determined from Eq. 6.26. The buoyancy flux is then

w'ft' = u*ft* = u(ft(Z*) — ft(Zi)) (6.28) from which we identify the drag coefficient

This gives the drag coefficient for the buoyancy flux, but what we really want is the drag coefficient for sensible and latent heat flux. When the atmosphere has uniform composition, the buoyancy flux is proportional to the sensible heat flux, and so the above-derived drag coefficient can be unambigously used for sensible heat flux. When the atmosphere has nonuniform composition contributing to buoyancy, however, the drag coefficients for sensible and latent flux could in principle be different from that for buoyancy. This is sometimes handled by introducing separate empirically determined scaling functions for moisture and heat flux. Refinements of the theory are quite straightforward, and can be found in the references given in the Further Readings section for this chapter. There is a fair amount of data pertinent to similarity functions for moisture flux on Earth, where the concentration of moisture reaches a few percent of the total atmosphere. These can almost certainly be applied to other buoyancy-generating substances at similar concentrations. However, behavior of the similarity functions when the buoyancy generating component accounts for a substantial fraction of the atmosphere, as is the case for methane on Titan, is essentially unexplored. In our calculations, we will be content to use the same drag coefficient for all fluxes. In the stably stratified case the resulting errors are probably not too consequential, since we'll see shortly that the main effect of the stratification is to choke off essentially all turbulent fluxes when the Richardson number exceeds a critical value; the refinements to the theory only modify the fluxes in the rather narrow window between neutral conditions and nearly complete suppression.

To proceed further, we need to specify an explicit similarity function F(Z). In the stably stratified case (Ri > 0) field and laboratory experiments can be adequately fit by functions of the form F(Z) = 1 + Z/Ric, with Ric « 0.2. With this definition,

G(Zi) = Ri-(Zi — Z*) + ln Z1 = R-(Zi — Z*) + ln - (6.30)

With this simple form of the similarity function, Eq. 6.27 can be analytically solved for Zi in terms of Ri, though for more complicated functions commonly in use a numerical iteration is generally required. With this form of G it is a straightforward matter to solve Eq. 6.27 for Zi — Z* in terms of Ri, evaluate G(Zi), and then compute CD from Eq. 6.29. Note that with the assumed form of G the right hand side of Eq. 6.27 has a maximum value of Ric when the argument approaches infinity. Thus, there is no consistent solution when Ri > Ric. It is assumed that the turbulence is completely suppressed, and that the turbulent fluxes vanish, for more stable values of Ri. Complete suppression of turbulence is somewhat unrealistic, and some formulations use alternate forms of G so as to allow a bit of flux to persist into the very stable case. However, the applicability of Monin-Obukhov theory to very stable conditions is a matter of considerable dispute.

Carrying out the above procedure, we find that the drag coefficient is

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