Basic models of turbulent exchange

Anybody who has watched dry leaves or dust blow around on a windy day has noticed that where the air comes up against the surface there arises a complex mass of turbulent eddies. In comparison, the interior of planetary atmospheres are fairly quiescent places, except in the immediate vicinity of rapidly rising buoyant plumes and active cloud systems. The turbulent fluid motions near the planetary surface exchange energy between the surface and the atmosphere, both in the form of sensible heat (energy corresponding to the change of temperature in a mass) and latent heat (energy associated with the change of phase of a condensible substance, with fixed temperature). Representing the effects of turbulence is not like representing radiation, where we can write down some basic physical principles then proceed through a set of systematic approximations until we arrive at a set of equations we can solve. When it comes to turbulence, the state of physics is not yet up to that challenge, and may never be. Instead, one must take a largely empirical approach from the outset, constrained by some fairly broad principles such as conservation of energy.

In this section we will derive the so-called bulk exchange formulae describing the flux of a quantity from the surface to the overlying atmosphere. The general idea is the same whether the quantity is a chemical tracer, sensible heat (associated with temperature fluctuation) or latent heat, so we will first present the formulae for a general tracer. The calculation will be introduce using simple physically-based scaling arguments, and then will be revisited in a more precise and systematic fahion in Section 6.4.

Let c be the specific concentration of some substance, and c' be the fluctuating or "turbulent" part, usually thought of as a deviation from a time or space mean over some suitable interval. Further, let w' be the fluctuating vertical velocity at the top of the surface layer. Then, the flux of the substance, in kg/m2, is where the overbar represents a time or space average and p is the total density of the gas making up the atmosphere. We assume further that the surface layer is thin enough that the variation in pressure and temperature across it is small enough that the variations in density can be neglected. Thus, the density factor can be replaced by a constant typical surface density, ps, and taken outside the average. The ideal gas law states that p = p/RT. If the surface layer has a thickness of a few tens of meters or less, then the hydrostatic law typically guarantees that the contribution of pressure to the density variations is small. It is not inconceivable, however, that the temperature difference across the surface layer could reach 10% of the mean, leading to corresponding changes in the density. With a little more work, the effect of these fluctuations can be brought into the picture, but we will not pursue this refinement as the effects are probably overwhelmed by the uncertainties in the representation of turbulence itself.

Next, we must estimate the correlation w'c'. We build this estimate from a typical vertical velocity 5w, a typical concentration fluctuation 5c, and a non-dimensional factor 0 < a < 1 describing the degree of correlation. Thus, we write w'c' = a ■ 5w ■ 5c. Next, we assume that 5w is proportional to the mean horizontal wind speed U at the top of the surface layer, so 5w = s ■ U. The constant of proportionality s can be thought of as a typical slope characterizing the turbulent eddies, which is in turn roughly related to the roughness of the surface. Note that U is the wind speed, and is therefore positive. We then assume that the typical concentration fluctuation scales with the concentration difference between the air in contact with the ground and the edge of the surface layer, so 5c = f ■ (cg — csa), where csa is the concentration at the edge of the surface layer, cg is that at the ground, and f is a nondimensional constant of proportionality. Putting it all together and lumping the proportionality constants into the drag coefficient CD = a ■ s ■ f, we write

CD is called the drag coefficient because when c is taken to be the turbulent velocity itself, the flux formula gives the flux of momentum, and hence the drag force on the surface. In writing the flux in the form of Eq. 6.9, we have adopted the convention that a positive flux represents a transfer of substance from the ground to the atmosphere. The turbulent flux acts like a diffusion, transferring substance from regions of higher concentration to regions of lower concentration. It is like a bucket-brigade, with partly empty buckets being handed downstairs from the top of the surface layer to the ground, where they are filled and sent back upstairs again (or with full buckets sent downstairs to be partly dumped out on the ground). The mass of substance in a bucket being carried upstairs is proportional to pscg, while the mass of substance in a bucket going downstairs is proportional to pscsa, while CDU gives the rate at which buckets are being handed up or down the stairs.

6.3.1 Sensible heat flux

To obtain the sensible heat flux, we take cpT to be our tracer. This is essentially the dry static energy (see Eq. 2.23), since the surface layer is thin enough that the height z can be taken to be nearly constant. With this choice of tracer, Eq. 6.9 becomes

If the ground is warmer than the air, heat is carried away from the ground at a rate proportional to the temperature difference. If the ground is cooler than the air, the sensible heat flux instead acts to warm the ground.

If CD is independent of temperature, then Fsens is exactly linear in the difference between the ground temperature and air temperature. Hence the coupling coefficient bsens - analogous to bir - is simply bsens = cppsCDU. When the surface layer becomes stably stratified, however, CD can be driven nearly to zero because the energy of turbulence is expended in mixing dense air upward. This effect will be quantified in Section 6.4. The consequent temperature dependence of CD would alter the linearized coupling coefficent.

Note that the sensible heat flux becomes small when the atmosphere has low density. The "wind-chill" factor on present Mars would be exceedingly weak! Conversely, very dense atmospheres like those of Venus or Titan can very effectively exchange heat between the surface and the atmosphere. With CD = .001,U = 10m/s and Tg — Tsa = 1K the sensible heat flux is .13W/m2 on present Mars, 11W/m2 on Earth, 55W/m2 on Titan, and a whopping 540W/m2 on Venus. It is for similar reasons that immersion in near-freezing water is far more life-threatening than walking about scantily clad in air of the same temperature - water is about 1000 times denser than Earth air. One must take care to distinguish thickness of an atmosphere (in terms of density) from optical thickness. An atmosphere can be thick (i.e. dense) while being optically thin, and conversely a thin (low density) atmosphere can nonetheless by optically thick if the greenhouse gas it is made of is sufficiently effective.

Now let's suppose that the sensible heat flux dominates the surface energy budget. By "dominates," we mean that the sensible heat flux due to a small departure from equilibrium (considering the sensible heat flux alone) overwhelms the other terms in the surface energy balance. This would be true if the wind speed and density were large, provided that the ground and atmosphere are dry enough that evaporation remains small. Sensible heat flux vanishes when Tg = Tsa, so this is the state that the system is driven to when sensible heat flux dominates. Taking the radiative and latent fluxes into account would cause a small deviation from this limit.

6.3.2 Latent heat flux

Whatever the condensed substance making up the surface, some of the condensed substance will transform into the vapor phase in the atmosphere contacting the surface, until it reaches the saturation vapor pressure determined by Clausius-Clapeyron. If the winds then carry away this vapor-laden air and replace it with unsaturated air, more mass will evaporate or sublimate from the surface. Since the phase change involves latent heat, a flux of mass away from the surface cools the surface by carrying away latent heat. Conversely, a flux of mass from vapor into the condensed surface will warm the atmosphere where condensation occurs. All substances will evaporate or sublimate to some extent, and whether the latent heat flux is significant is a matter of how big the saturation vapor pressure is at the typical temperature of the surface. For water ice on Titan at 95K, the vapor pressure is under 10-15Pa, so the latent heat flux of water is utterly negligible.

The situation is the same for basalt at 300K on Earth, or even at 750K on Venus. However, the vapor pressure of CO2 on present Mars, of liquid water or water ice on Earth, and of methane on Titan are all high enough to allow substantial latent heat flux. Whatever the condensible substance in question we will use terms like "humidity" by analogy with the archetypal case of water vapor on Earth. Also, for the sake of verbal economy we will often refer simply to "evaporation" in situations where the actually process might be either evaporation or sublimation.

In dealing with latent heat flux, it is more convenient to deal with the mass mixing ratio of the condensible to dry air, rather than specific humidity. This makes it somewhat easier to treat cases where the condensible makes up a substantial part of the total mass. Thus, we use the mass mixing ratio rw as the tracer in Eq. 6.9. If pa is the density of dry air in the surface layer, then the mass of condensible per unit volume is parw and this mass carries a latent heat Lpwrw. we can write the mixing ratio rsa at the edge of the surface layer as hsarsat(Tsa), where where hsa is the relative humidity at the outer edge of the surface layer and rsat(T) is the saturation mass mixing ratio. In terms of saturation vapor pressure, the saturation mass mixing ratio is (Mw/Ma)(psat(T)/pa, with pa being the partial pressure of dry air in the surface layer. Now suppose that at the ground there is a reservoir of a condensed phase of the substance "w" - an ocean, lake, swamp, snow field, glacier or the surface of an icy moon. In this case, the vapor pressure in the air in contact with the surface must be in equilibrium with the condensed phase, and must therefore follow the Clausius-Clapeyron relation evaluated at the temperature of the ground. Equivalently, we can say that rg = rsat(Tg). Using the two mixing ratios, the latent heat flux becomes

Fl = LpaCD U (rsat(Tg ) — hsa • r sat(Ta )) (6.11)

Alternately, using the definition of the mixing ratios and assuming the partial pressure of dry air to be approximately constant within the boundary layer, Eq. 6.11 can be written

Rw T sa

The latter form of the latent heat flux demonstrates that the flux is in fact unaffected by the presence of the dry air. Assuming temperature and wind to be held constant, the evaporation from the Earth's ocean would remain unchanged even if all the N2 were taken out of the atmosphere. This conclusion would no longer be valid if the gases in question had substantial non-ideal behavior, for then the law of partial pressures would no longer hold.

Exercise 6.3.1 Derive Eq. 6.12. What do you have to assume about the air temperature within the surface layer?

In situations where a major constituent of the atmosphere can condense out onto the surface or sublimate or evaporate from it, a constraint on the temperature change across the surface layer enters the problem in a significant way. The constraint arises from the fact that, since the surface layer is thin, the pressure must be nearly constant within the layer. The implications of this constraint are easiest to see when the atmosphere consists of a single condensible component; a concrete example of this situation is provided by the state of the surface layer over seasonal CO2 frost layers on Mars. Let's suppose that the system has a layer of condensate of the atmospheric substance at the surface - an ocean or glacier. Then, since the atmosphere consists of only the one constituent, the surface pressure is fixed in terms of the ground temperature by Clausius-Clapeyron, namely ps = ps(Tg). The pressure at the upper edge of the surface layer must be very nearly equal to this value, otherwise there would be a large unbalanced pressure gradient which would drive a strong flow that would soon transport enough mass to equalize the situation. It follows that if the atmosphere at the upper edge of the surface layer is saturated, we must have Tsa « Tg. In other words, in saturated conditions, the temperature at the ground and the temperature at the upper edge of the surface layer must adjust nearly instantaneously so as to keep the two equal. Under what circumstances can the upper edge of the surface layer be considered saturated? First note that if Tsa were colder than Tg, then the pressure contininuity condition would require the air to be supersaturated. This situation cannot persist for long, so in a case where the free atmosphere is cooling or the ground is heating up, Tsa would adjust nearly instantaneously to remain equal to Tg. This adjustment does involve a transfer of latent heat, which alters the thermal response time of the system. One could treat this transfer in terms of an strong enhancement of CD in such conditions, but there are more natural ways to deal with essentially instantaneous adjustments. The implications for the seasonal cycle of condensible atmospheres will be considered in Section 7.7.5, where such an alternate approach will be illustrated. On the other hand, a situation with Tg < Tsa is perfectly consistent if the atmosphere aloft is subsaturated. In such situations, the transfer of latent heat flux is governed by Eq. 6.12 as usual. The transfer would act both to cool the surface, and to add mass to the atmosphere bringing it closer to saturation. However, in situtations where the atmosphere remains saturated as the system cool down, the previous temperature continuity constraint applies.

From Eq. 6.12 we observe that latent heat flux carries heat away from the ground when the saturation mixing ratio at the ground is less than the mixing ratio of the surface layer. Since typically hsa < 1, this can happen even if the ground is colder than the overlying air. We also note that the latent heat flux becomes insignificant at sufficiently cold temperatures, since both saturation vapor pressures in the equation become small in that limit.

Sensible and radiative heat transport carry no mass away from the surface, but latent heat transport is of necessity accompanied by mass transfer. The mass flux into or out of the ground is simply Fl/L. The mass flux is needed for calculating the rate of ablation of glaciers by sublimation, the drying out of lakes or soil by evaporation, and the rate of salinity change at the surface of an ocean (since evaporation carries away the condensible but not the solute).

Now let's look at how the fluxes behave when the temperature difference between the ground and the outer edge of the surface layer is small. Carrying out a Taylor series expansion of the flux about Tg = Tsa, as we did for the infrared cooling case, we write

Defining the characteristic flux FL = CDUpsat(Tsa), we find

Rw T sa T sa Rw T sa where Rw is the gas constant for the condensible. The Clausius-Clapeyron relation has been used to substitute for dpsat/dT in the expression for bL. E0 is the heat flux due to evaporation or sublimation that would occur with Tg = Tsa; it vanishes if the surface layer is saturated (hsa = 1), but is positive otherwise. Both E0 and bL are proportional to the characteristic flux FL, which vanishes vanishes as Tsa ^ 0, since the saturation vapor pressure vanishes like exp(-L/RwT) in this limit. As one might expect, latent heat flux becomes negligible at sufficiently low temperatures. How low one must go for this to be the case depends on the gas in question. As temperature increases, the characteristic flux becomes large, and hence E0 and bL become large as well. The increase is abetted by the fact that L/Rw T is a large number at typical planetary temperatures (e.g. 18.06 for water vapor at 300K, or 10.3 for methane at 95K). For temperatures high enough that bL becomes large, a modest ground-air temperature difference leads to a very large increase


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Table 6.1: Latent heat flux coefficients for various gases at selected temperatures Ts, with U = 10m/s, CD = .001 and boundary layer relative humidity hsa = 70%.

Table 6.1: Latent heat flux coefficients for various gases at selected temperatures Ts, with U = 10m/s, CD = .001 and boundary layer relative humidity hsa = 70%.

Computed in latent heat flux. This tends to make it hard for the ground temperature to differ much from the free air temperature in such cases.

Table 6.1 gives some typical values of Eo and bL for water, carbon dioxide and methane. In all three cases, we see that the latent heat flux rises very strongly with temperature. For water, latent heat flux is insignificant at temperatures of 230K or lower. The feeble latent flux of a Watt per square meter or so would be utterly dominated by infrared cooling of the surface, or by the sensible heat flux arising from a ground-air temperature difference of as little as 1K. This corresponds to the situation in the Antarctic night of the present Earth, or to the daily average tropical temperatures on a Snowball Earth. However, even at the freezing point of water, the latent heat flux is quite substantial. With a 5K ground-air temperature difference, the flux would be nearly 100W/m2, which is almost half of the typical midlatitude absorbed solar radiation in the ocean, and roughly equal to the typical absorbed solar radiation in ice. The latent heat flux is also comparable to the typical infrared cooling of the surface at such temperatures (inferred from Figure 6.1 ). As temperature is increased further to values characteristic of the modern tropics, the flux increases dramatically; it would take about 90% of the supply of absorbed solar energy going into the ocean in order to sustain the evaporation arising from just a 2K ground-air temperature difference. At these temperatures, the latent flux is considerably in excess of the surface infrared cooling.

For the other gases in the table, the latent heat flux becomes substantial at much lower temperatures. At temperatures comparable to the Martian polar Spring, the latent heat flux due to CO2 sublimation is comparable to the water vapor values for Earth's midlatitudes or tropics (assuming the same degree of boundary layer saturation). These fluxes are particularly consequential in light of weak supply of solar radiation on Mars, relative to Earth. Alternately one may compare the latent flux to the infrared cooling of the surface in the thin Martian atmosphere (aTg4, or 37W/m2 at 160K). Either way, we conclude that latent heat flux plays a key role in determining surface temperature at places on Mars where seasonal CO2 frost is sublimating or being deposited. At Titan temperatures, latent heat flux due to methane evaporation is enormous; the solar radiation reaching Titan's surface is well under 5W/m2, which is two orders of magnitude less than the Methane evaporation one gets under the conditions of Table 6.1. Somehow or other, conditions near Titan's surface must adjust until the evaporation is reduced to the point where it can be balanced by the supply of energy to the surface, but the numbers in the table tell us that methane latent heat flux is the dominant constraint on the adjusted state. Ironically, Titan, at 95K is like an extreme form of the Earth's tropics, in that evaporation dominates the surface energy budget to an even greater extent than it does in Earth's tropics. If the temperature of the Earth's tropics were raised to 320K, as might happen in the high CO2 world following deglaciation of a Snowball Earth, then Eo on Earth, too would greatly exceed the available solar energy, though not to such an extent as it does on Titan. The way the surface conditions adjust to accomodate this state of affairs will be taken up in the Section 6.5.

When the surface is sufficiently cold relative to the air, vapor from the air can be deposited on the surface in the form of dew or frost. In this case the latent heat flux is negative, and carries energy from the atmosphere to the ground. If the boundary layer is saturated (hsa = 1) then frost or dew deposition occurs whenever Tg < Tsa. If the boundary layer is unsaturated deposition won't occur until the ground temperature is made sufficiently cold that the saturation vapor pressure there falls below the partial pressure of the condensible in the overlying atmosphere (a temperature known as the "dew point" or "frost point"). When latent heat is being carried to the surface - as it is during he seasonal polar CO2 frost formation on Mars - the rate of condensation is limited by the rate at which the surface can get rid of the deposited latent heat. Since the surface is colder than the atmosphere during deposition, sensible heat flux carries heat the wrong way to balance the budget, so it is only infrared cooling of the surface that can sustain frost or dew. Otherwise, the surface will simply warm in response to the deposited latent heat until it is no longer cold enough for frost or dew to form.

Over land, there are two further complications that must be considered. The first is that land, unlike a deep ocean or lake or a thick glacier, can dry out. If the land surface is a mix of condensible and (essentially) noncondensible substance, the latent heat flux can exhaust the supply of condensible, whereafter the boundary condition rg = rsat(Tg) is no longer appropriate. In the absence of further supply of condensible at the ground, the latent heat flux must fall to zero. In such a case, one must keep track of the mass of the condensible reservoir at the ground, and zero out the latent heat flux when the reservoir is exhausted. This would be the case for thin snow cover, scattered puddles, or soil moisture on Earth, for CO2 frost layers on Mars and for liquid methane swamps on Titan. For soil moisture, a common simple model is the bucket model, in which each square meter of soil surface is treated as a bucket whose capacity is determined by its porosity and depth. The bucket is filled by rainfall, and emptied by evaporation. Once the bucket is full, any additional rainfall is assumed to run off into rivers (which may or may not be tracked, according to the level of sophistication of the model). As long as the bucket has some water in it evaporation is sustained, but when the bucket is empty latent heat flux is zeroed out and only radiative and sensible heat transfers at the ground are allowed. The bucket model may serve also as a model of conditions at Titan's surface, which may consist not only of liquid methane puddles but also bogs consisting of beds of granular water ice sand or pebbles whose pores are saturated with liquid methane.

The second complication over land concerns the effect of land plants. At present, Earth's climate provides the only example where this must be taken into account. Plants actively pump water from deep storage, at rates determined by their own physiological requirements. This is known as transpiration, and given that moisture flux over vegetated land is always some mix of transpiration and evaporation, the joint process is called evapotranspiration. In this case, the moisture boundary condition at the ground may be more appropriately represented as a flux condition determined by plant physiology rather than setting the moisture mixing ratio at the ground. The moisture flux may be limited by rate at which trees pump moisture, and not by rate at which turbulence carries it away. The mixing ratio at the ground still cannot exceed saturation, so when the transpiration becomes strong enough to saturate the air in contact with the ground, one can revert to the previous model of conventional evaporation. Yet a further complication in vegetated terrain is the very notion of ground and ground temperature. Is "the ground" the forest surface or the elevated leaf canopy? Is the ground temperature that of the leaf surface or the soil? How do we take into acount the mix of illuminated hot leaves and relatively cool leaves in shade? A proper treatment of these factors requires a detailed model of the microclimate in the vegetation layer, which is beyond what we aspire to in this book. One need not abandon all hope of estimating conditions over vegetated terrain, however. As a rule of thumb, dense forests that get enough rainfall to survive in the long term tend to act more or less like the ocean, save for an eleveated CD caused by greater surface roughness. Grasslands, shrub, tundra and prairie can be crudely modeled using the bucket model.

When evaporation dominates the surface budget, equilibrium requires FL = 0, or equiva-lently psat(Tg) = hsapsat(Tsa). Since psat is monotonically increasing in temperature, this relation requires Tg < Tsa if the boundary layer air is unsaturated hsa < 1. Thus, evaporation or sublimation drives the ground temperature to be colder than the overlying air temperature. However, the ground and surface could also achieve equilibrium by transferring enough moisture to the surface layer that it becomes saturated (hsa = 1), in which case Tg = Tsa in equilibrium, as for the case of sensible heat flux. The extent to which equilibrium is attained by adjusting temperature vs. humidity depends on the competition between the rate at which moisture is supplied to the boundary layer and the rate at which dry air from aloft is entrained into the boundary layer. Observed boundary layers on Earth and Titan are significantly undersaturated, leading to the conclusion that the ground temperature would be considerably less than the air temperature, if other fluxes did not intervene. Using the linearized form of the latent heat, the equilibrium ground-air temperature difference is Tg — Tsa « —Eo/bL. For the conditions of Table 6.1, this is —2.6K for Titan at 95K. For a hot Earth at 320K, the difference is about — 5.7K. There are currently no observations of the state of saturation over the sublimating Martian CO2 frost cap, but given the saturation assumed in the table the equilibrium occurs with Tg — Tsa « —2.4K when the air temperature is 260K. Thus, even when evaporation dominates, the equilibrium ground temperature does not differ greatly from the overlying air temperature. This was also found to be the case when the surface budget is dominated by sensible heat flux. It is only the radiative terms that can drive the ground temperature to be substantially different from the overlying air temperature.

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