(mean state, forcing)
Mean State
In specifying the mean state we desire to maintain several important features of the observed case through the spin up period of the simulation. These include the depth and continuity of the cloud layer, and the mixing line structure and inversion characteristics at cloud top. These basic characteristics are quantified in
Table 1: Basic quantities whose values should be preserved through spin up period.
To generate a state which maintains these values we suggest starting with the following profiles of thermodynamic state variables:
Thetal =
289 K, if z < zi 297.5 + (z-zi)1/3 K otherwise qt =
9.0 gkg-1, if z < zi 1.5 gkg-1, otherwise
In the above Thetal was defined from the surface air temperature using values of the physical constants given in Table 2, and a surface pressure of 1017.8 hPa. Note that cp can vary substantially across the cloud top interface: for dry air cpd = 1005 J kg-1 K-1, while for water vapor cpv = 1870 g kg-1 K-1 which implies that the isobaric specific heat above the inversion is approximately 1008 J kg-1 K-1 and 1022 g kg-1 K-1 below. Hence our use of an intermediate value which excludes a dependence on the amount of ambient water vapor.
Table 2: Suggested values of physical constants
To complete the specification of the basic case (absent the radiative forcing which is discussed in the following subsection) requires a determination of geostrophic winds, divergence, sea surface temperatures, and some indication of density for use in Boussinesq models. For the winds, we specify geostrophic values of Ug = 7 ms-1 and Vg = -5.5 ms-1 which produces winds within the boundary layer near 6 and 4.25 ms-1 respectively, as observed. One could choose to align the domain with the mean wind, however we do not do this to facilitate future investigations which may choose to examine the nature of momentum mixing within the stratocumulus topped boundary layer, and its affect on processes such as entrainment. The large-scale divergence of the winds is taken to be D = 3.75 micro s-1 as this seems most consistent with the observed temperature structure above the boundary layer, and the calculated radiative forcing (see below). For the sea-surface temperature we specify a value of 292.5 K, which is 2.1 K warmer than the surface air temperature and should correspond to surface sensible heat fluxes near 15 W m-2 and surface latent heat fluxes of approximately 115 W m-2 given a bulk aerodynamic drag coefficient, Cd = 0.0011. The surface temperature and pressure correspond to a surface air density, rho0 = 1.22 g kg-1 and an air density just below cloud top of rhoi = 1.13 g kg-1.
Figure 4: Plot of idealized soundings used to drive radiative calculations, and resulting radiative fields: qt (upper left); sl (upper center), F# lw and F" lw from calculations (lines) and measured (diamonds and circles respectively) upper right; net flux (lower left); heating rate (lower right). Note the different vertical scale on the lower plots. Adapted from (Stevens et al., 2003)
Radiative forcing (top)
Table 3: Suggested values of tuning parameters for radiative calculations
To parameterize the radiative forcing we use detailed calculations using the Delta-four stream radiative transfer code developed by Fu and Liou 1993.. The radiative fluxes from this model are computed assuming the above specified state, matched to a free atmospheric sounding as discussed by Stevens et al. (2003). Note that to match the observed radiative fluxes a moist layer was incorporated into the radiative sounding at approximately 1500m. The results from this exercise are shown in Fig. 4. Because a radiative calculation similar to the one used to make Fig. 4 is prohibitively expensive to incorporate into every column of a large-eddy simulation, we investigate the extent to which a simpler parameterization of radiative fluxes can be calibrated to represent the expected variations among columns for this case. In designing this scheme we sought to represent the main features of Fig. 4. Namely the significant flux divergence at cloud top, cloud base and just above cloud top, we also chose to use as a basis the simple form of the fluxes exploit in previous inter-comparison studies, (e.g., Bretherton et al., 1999). To do this we propose a three component model of the net radiative flux:
F(z) = F0 exp(-Q(z,1)) + F1 exp(-Q(0, z)) + rhoicpD( 0.25(z - zi)4/3 + zi(z - zi)1/3), (1)
where
Q(a, b) = Integral from a to b of (kappa rho ql dz) ,
and kappa, F0 and F1 are tuning parameters. Note that the third term in (1) was chosen so that it generates the observed (z-zi)1/3 structure in the thetal profile for z > zi and a large-scale subsidence velocity which varies linearly in z such that W = -Dz. It was intended to be applied based on the local zi identified with the height of the first crossing of the qt = 8 g kg-1 isoline in each particular model column.
Figure 5: Plot of idealized net long-wave radiative flux from the Delta-four stream and from Eq. 3 for the base case (left panel) and a drier boundary layer with a thinner cloud (right panel).
We find that this model gives a good fit to the Delta-four stream approximation for the fluxes if we choose the parameter values specified in Table 3. This is illustrated in Fig. 5 where we show the fit for two cases, one being the control case with the specified cloud layer, the other being with a substantially drier boundary layer (qt = 8.5 g kg-1) and hence a much thinner cloud (ql,max = 0.25 g kg-1). These parameter values were adjusted by eye, so although the fit may not be optimal, it appears to be sufficient. Not only does (1) well represent the radiative fluxes as simulated by the Delta-four stream model, it also appears to capture the basic nature of the sensitivity of these fluxes to significant changes in the state of the cloud layer. Note that this parameterization of the radiative fluxes raises a number of subsidiary questions. For instance, is such a parameterization appropriate for an undulating cloud top? Also, if the radiative flux divergences are so significant just above the cloud top, would they be sensitive to multi-dimensional effects within cloud crevices at the base of down-drafts? These are some additional questions which we may be able to address if we can perform reasonable simulations of the base case.
