MODELLING THE INTERACTION BETWEEN THE ATMOSPHERIC BOUNDARY LAYER AND EVAPORATING SEA SPRAY DROPLETS

Jeff D. Kepert
Bureau of Meteorology Research Centre, Melbourne, Australia

Chris W. Fairall
NOAA Environmental Technology Laboratory, Boulder, Colorado

Based on a paper submitted at The Third Conference on Boundary Layers and Turbulence, Dallas, Texas, January 10-15, 1999.

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1. INTRODUCTION

Attempts to quantify the effects of sea spray evaporation must contend with many uncertainties. Some, such as the evolution of the temperature and mass of a single droplet for as long as it remains suspended in the atmosphere, are now quite well understood (Andreas, 1990). Others, such as the length of time the droplet actually remains suspended, are more difficult. The processes by which droplets are produced are known, but there remain orders of magnitude of variation between estimates of the rate at which these processes operate at various wind speeds. A further important area of investigation is the impact of an elevated sink of heat and source of moisture on the atmospheric boundary layer.

The thermodynamic effect of evaporating droplets is a battle between their evaporation rate and their atmospheric suspension lifetime. The suspension lifetime is a battle between the substantial fall velocity of large droplets and vertical transport by turbulence. However, the full evaporation of droplets is controlled also by the sources of heat available to power the evaporation process. In dynamic equilibrium, there are only two sources: (1) upward turbulent transfer by sensible heat from the ocean, and (2) downward turbulent transfer of heat from the marine boundary layer (MBL) above the droplet layer. Because evaporation of droplets cools and moistens the MBL and the surface layer, these heat sources adjust internally until some equilibrium is approached. This feedback limits the amount of evaporation produced regardless of the number of droplets ejected by the ocean. Thus, this problem cannot be investigated without realistic incorporation of full MBL-scale physics.

Several recent studies of feedback effects have yielded valuable information (Rouault et al. 1991; Edson and Fairall 1994; Makin, 1998), but have all imposed upper boundary conditions at heights on the order of tens of meters. Fixing the mean thermodynamic conditions at such a low height is tantamount to allowing the atmospheric boundary layer to supply essentially infinite fluxes to evaporate droplets.

The two major issues that must be resolved to develop a simple but accurate parameterization of sea spray effects are: (1) a characterization of the oceanic droplet source strength as a function of wind speed and (2) a characterization of the feedback effects in reducing the total droplet contribution. The first issue can be resolved only by direct measurements of droplet spectra and evaporation rates over the open ocean; the second can be attacked with a suitable numerical model. Available measurements are of droplet concentrations (instead of droplet fluxes), so an evaporation/turbulent transport model is required to deduce the surface droplet source strength. Thus, a model is required for both aspects of the sea spray problem.

Here, we describe some recent results using our full boundary layer model with explicit evaporating droplets. We will focus particularly on determining and quantifying factors which limit droplet evaporation, including the adjustment of the boundary layer.

2. MODEL DESCRIPTION

The model is a one-dimensional high order turbulence closure model of the cloud-topped marine boundary layer. This is coupled to an Eulerian model of saline droplet transport and evaporation. The specific elements of the coupling are that the droplet transport depends on the atmospheric turbulence field, that the droplet evaporation sees the prevailing atmospheric conditions, and that the sink of sensible heat and source of latent heat through droplet evaporation are realised by the atmosphere.

We model the droplet evaporation using salinity-modified cloud microphysics equations (Pruppacher and Klett, 1978; Andreas, 1990), including a ventilation factor (Fairall et al., 1994). Andreas (1990) showed that the time scale for a droplet=s temperature to adjust is much less than that for it to evaporate. Accordingly, we may assume that it instantaneously attains this temperature, which we calculate by the method given in Kepert (1996). This allows us to characterise spray droplets by just two variables, radius and salinity, with its temperature a function of these.

Droplet diffusion is modelled by a flux-gradient relation using the diffusivity from the turbulence closure, modified for inertial effects according to Rouault et al. (1991). Their fall velocity is calculated according to Pruppacher and Klett (1978). At the lower boundary, we combine the effects of diffusion and fall velocity into a deposition velocity, calculated as in Slinn and Slinn (1980).

Conditions were prescribed to give, at equilibrium and in the absence of spray, a 10 m wind of 25 ms^-1, potential temperature of 298.9 K and relative humidity of 81%, while the sea surface temperature was 300 K. The friction velocity of 1.13 ms^-1 implies that the standard deviation of vertical velocity fluctuations, _w, in the surface layer is 1.4 ms^-1 and we inject the droplets at the level immediately below the equilibrium wave height, which is 7.0 m. After the initial steady state was attained, the spray source was switched on and integration was then continued for a further 24 hours, or until a steady-state was again attained.

3. RESULTS AND DISCUSSION

There is a large variation - several orders of magnitude - in recent estimates of the production rates of sea spray by different authors (Andreas, 1998). While many of the cases we consider will be within the bounds of current best estimates of what is reasonable for a source function, we prefer at this stage to not tie our results too firmly to any particular source function.

To facilitate this, we consider the efficiency with which the atmosphere converts the liquid water contained in the spray droplets into vapour. Thus, for a given production rate of spray droplets S(n,r), we define the evaporation potential E_p as the droplet-mediated latent heat flux Q_l if all the droplets evaporated to dryness.

3.1 The model response to medium droplets

Here we consider droplets with an initial radius of 128 µm and therefore a fall velocity of 0.96 m.s^-1 which is somewhat less than _w. The evaporation potential E_p is 300 Wm^-2. Figure 1 contains the flux profiles and droplet distribution. Droplets are diffused up to a few tens of metres, and a small proportion of droplets evaporate down to the equilibrium radius.

Although a small proportion of the droplets reach their equilibrium radius, the efficiency factor at 0.057 is still quite small. We may define an atmospheric residence time scale

which is several times the fall velocity time scale defined by Andreas (1992) but much less than his evaporation time scale, here 118 s. We conclude that droplet evaporation in this case is limited by droplet suspension.

To investigate the sensitivity of evaporation efficiency Q_l/E_p to source rate, two further calculations with this initial radius, but with one tenth, and ten times, the source function. Reducing the source rate has little impact on the evaporation efficiency Q_l /E_p, but increasing it reduces it to 0.030. The reason for this largely the decrease in evaporation rate due to the increase in low level humidity. This is thus a negative feedback, whereby spray-mediated moistening and cooling of the air increases the relative humidity and slows droplet evaporation. Further, the more slowly evaporating droplets will spend more time at larger radii and hence with larger fall velocities. Finally, reduced turbulence resulting from the increased stability caused by the droplet-mediated cooling also shortens the residence time.

3.2 The model response to small droplets

Here E_p = 30 Wm^-2 and with an initial radius of 32 mm these droplets are from the spume part of the production spectrum. However, their fall velocity of 0.12 m.s^-1 is much less than s_w and accordingly we expect them to be efficiently supported by the turbulence.

The diagram of concentration relative to radius and height (Figure 2) supports this expectation. We see that the maximum at the equilibrium radius has grown in relative importance and now contains the majority of the droplets. Consistent with this, the residence time scale _res = 10078 s has become effectively infinite. The weak decrease in concentration with height for droplets at the equilibrium radius (17.2 mm) contrasts strongly with the marked height dependence for larger, less evaporated droplets. This is not because the larger droplets are unable to be diffused upward, but rather that they evaporate more rapidly than they are transported.

Consistent with the above is that the efficiency ratio is now Q_l /E_p = 0.78. As the equilibrium radius is 17.2 µm, the maximum realisable efficiency is 1 - (17.2/32)³ = 0.84, so this case is approaching the maximum realisable efficiency.

The upper part of the Figure 2 shows that the maximum of droplet concentration tilts towards the right, or larger radii, at a height of several hundred metres. The axis of this ridge of droplet concentration follows the line along which r/t = 0; that is, where the droplets are at the equilibrium size. The reason for this is that the relative humidity increases as cloud base is approached. As the droplets are diffused upward and encounter this relatively moister air, they begin to grow again and move to larger radii.

3.3 The Effects of Initial Radius and Source Rate

Figure 3 shows evaporation efficiency as a function of initial radius and potential evaporation, and we see there are several distinct regimes. The first of these is in the lower left of the figure, bounded by the 0.85 contour, where efficiency is essentially constant and all droplets reach their equilibrium radius.

To the right of this region, efficiency falls off as approximately r₀^-2 - see also Figure 4 (upper). The significance of an r₀^-2 dependence is that this is what would be expected from a fall velocity parameterisation of residence time, since fall velocity is approximately proportional to r and r/t is approximately inversely proportional to r. A couple of relevant factors are missing from this simple argument. First is the effect of the ventilation factor, which increases evaporation rate and is more significant for the larger droplets with greater fall velocities. Opposing this is that the larger droplets= residence time will be more constrained by their fall velocity and less affected by the turbulence. We thus see that, while turbulent enhancement of the residence time for droplets with r₀ in the range of 50 - 200 µm is insufficient to evaporate them to dryness, it nevertheless produces a marked increase in Q_l beyond that expected from the fall velocity formulation. This is important, as the peak volume production in proposed droplet source functions typically lies in this range (Andreas, 1998). Evaporation efficiency in this lower right portion of the E_p - r₀ phase space is thus determined by atmospheric residence time.

Above the latter of the two regions, the contours curve toward the left, showing that the increased source rate has not resulted in a proportional increase in droplet-mediated flux. This reduction is a consequence of two negative feedbacks: first, the moistening and cooling of the near-surface layer reduces the droplet evaporation rate, and second, the damped turbulence reduces the residence time.

The smaller droplets, whose residence time is sufficient to evaporate almost all of them to equilibrium when E_p is low, also show a decrease in efficiency as the source rate increases. However, in this case it is not the reduction in evaporation rate but rather the increase in equilibrium radius that causes the reduction in efficiency.

The picture is completed by considering the dependence of the depth of the droplet evaporation layer on initial radius and E_p (Figure 4 lower). For low to moderate E_p, the scale height Z_DEL below which 95% of droplet evaporation occurs is independent of E_p and, as discussed before, is limited for large droplets by their very strong resistance to upward diffusion, and for small droplets by the fact that they are fully evaporated before they can get very far from the surface. In between, there is a maximum, reaching 70 m for r₀ = 64 µm, where neither limiting process acts strongly. This is remarkably deep, but not implausibly so, as a crude calculation shows:

_wevap. 1.3 m/s H 60 s 80 m

suggesting some droplets at least reach this height before evaporating to equilibrium.

We note that this tends to suggest that existing measurements of thermodynamic fluxes at high wind speeds have been taken within the DEL, which may help explain the lack of obvious spray effect observed. We note also that this peak in Z_DEL occurs for droplets near the peak volume production radius in various source functions, and suggest that measuring the flux divergence across this layer may be a route to confirming the spray effect.

4. CONCLUSIONS

Several model runs were considered in detail. It was found that for very large droplets, the previously proposed "fall velocity formulation" gave a slight overestimate of droplet residence time and evaporation. However, as the initial radius decreased, turbulence had a much stronger influence on droplet transport and there was a dramatic increase in residence time and evaporation efficiency. The latter reached its maximum possible value, and residence time became essentially infinite, for the smaller droplets where v_fall << _w.

A variety of feedbacks were found to have significant impact on the results. As the droplet evaporation increased, the stabilisation of the boundary layer resulted in a readjustment of the fluxes which reduced the stress and the total thermodynamic flux. In addition, the now well known perturbations in the sensible and latent flux profiles were found, and it was shown that about 70% of the droplet mediated latent flux was realised above the droplet evaporation layer, once the effects of the overall reduction in flux were taken to account.

A further process, the reduction of droplet transport by the damped turbulence, was also shown to be an important negative feedback at high source rates, due to the stabilisation which reduced both the droplet residence time, and the ability of the atmosphere to supply warm dry air to evaporate more droplets.

Acknowledgments. This work is partly supported by the US Office of Naval Research under grant N00014-94-I-0556.

Andreas E.L., 1990: Time constants for the evolution of sea spray droplets. Tellus, 42B, 481-497.

Andreas, E.L., 1998: A new sea spray generation function for wind speeds up to 32 m/s. J. Phys. Oceanogr., in press.

Edson, J.B., and C.W. Fairall, 1994: Spray droplet modeling. I:  Lagrangian model simulation of the turbulent transport of evaporating droplets. J. Geophys. Res., 99, 25229-25311.

Fairall, C.W., J.D. Kepert, and G.J. Holland, 1994: The effect of sea spray on surface energy transports over the ocean. The Global Atmospheric Ocean System, 2, 121-142.

Kepert, J.D., 1996: Comments on "The temperature of evaporating sea spray droplets". J. Atmos. Sci., 53, 1634-1645.

Makin, V.K., 1998:Air-sea exchange of heat in the presence of wind waves and spray. J. Geophys. Res., 103, 1137-1152.

Pruppacher, H.R., and J.D. Klett, 1978: Microphysics of Clouds and Precipitation. D. Reidel Publishing Company, 714 pp.

Rouault, M.P., P.G. Mestayer, and R. Schiestel, 1991: A model of evaporating spray droplet dispersion. J. Geophys. Res., 96, 7181-7200.

Slinn, S.A. and W.G.N. Slinn, 1980: Predictions for particle deposition in natural waters. Atmospheric Environment, 14, 1013-1016.

Fig 1: Results from for model run where the droplet initial radius is 128 mm and the potential evaporation is 300 Wm-2. At top, the droplet number concentration (m-3mm-1) as a function of droplet radius and height, with the star indicating the source radius and height. Below, profiles of sensible and latent heat flux in the presence (solid) and absence (dashed) of spray. The triangles on the height axes indicate the cloud base.

Fig 2: The same as Fig 1, except the initial radius is 32 mm and the potential evaporation is 30 Wm-2.

Fig 3: Contours of evaporation efficiency as a function of potential evaporation and initial radius. The dashed line shows where Ql = 1 W/m2.

Fig 4: Droplet-mediated flux Ql (top) and depth of the droplet evaporation layer, zDEL (below), as functions of initial radius, when Ep = 1 Wm-2. The dotted line in the upper frame shows a r-2 dependence.

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