Shallow precipitating cumuli within the easterly trades were investigated using shipboard measurements, scanning radar data, and visible satellite imagery from 2 weeks in January 2005 of the Rain in Cumulus over the Ocean (RICO) experiment. Shipboard rainfall rates of up to 2 mm h 21 were recorded almost daily, if only for 10-30 min typically, almost always from clouds within mesoscale arcs. The precipitating cumuli, capable of reaching above 4 km, cooled surface air by 1-2 K, in all cases lowered surface specific humidities by up to 1.5 g kg 21 , reduced surface equivalent potential temperatures by up to 6 K, and were often associated with shortlived increases in wind speed. Upper-level downdrafts were inferred to explain double-lobed moisture and temperature sounding profiles, as well as multiple inversions in wind profiler data. In two cases investigated further, the precipitating convection propagated faster westward than the mean surface wind by about 2-3 m s 21 , consistent with a density current of depth ;200 m. In their cold pool recovery zones, the surface air temperatures equilibrated with time to the sea surface temperatures, but the surface air specific humidities stayed relatively constant after initial quick recoveries. This suggested that entrainment of drier air from above fully compensated the moistening from surface latent heat fluxes. Recovery zone surface wind speeds and latent heat fluxes were not higher than environmental values. Nonprecipitating clouds developed after the surface buoyancy had recovered (barring encroachment of other convection). The mesoscale arcs favored atmospheres with higher water vapor paths. These observations differed from those of stratocumulus and deep tropical cumulus cold pools.

Equations that follow from the Navier-Stokes equation and incompressibility but with no other approximations are "exact.". Exact equations relating second-and third-order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal, and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment, and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: i) the trace of the structure functions, ii) DNS that has periodic boundary conditions, and iii) an average over a sphere in r-space. The last case (iii) introduces the average over orientations of r into the structure function equations. The energy dissipation rate ε appears in the exact trace equation without averaging, whereas in previous formulations ε appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.

Several models are developed for the high-wavenumber portion of the spectral transfer function of scalar quantities advected by high-Reynolds-number, locally isotropic turbulent flow. These models are applicable for arbitrary Prandtl or Schmidt number, v/D, and the resultant scalar spectra are compared with several experiments having different v/D. The ‘bump’ in the temperature spectrum of air observed over land is shown to be due to a tendency toward a viscous-convective range and the presence of this bump is consistent with experiments for large v/D. The wavenumbers defining the transition between the inertial-convective range and viscous-convective range for asymptotically large v/D (denoted k* and k1* for the three- and one-dimensional spectra) are determined by comparison of the models with experiments. A measurement of the transitional wavenumber k1* [denoted (k1*)s] is found to depend on v/D and on any filter cut-off. On the basis of the k* values it is shown that measurements of β1 from temperature spectra in moderate Reynolds number turbulence in air (v/D = 0·72) maybe over-estimates and that the inertial-diffusive range of temperature fluctuations in mercury (v/D ≃ 0·02) is of very limited extent.

Exact equations are given that relate velocity structure functions of arbitrary order with other statistics. "Exact" means that no approximations are used except that the Navier-Stokes equation and incompressibility condition are assumed to be accurate. The exact equations are used to determine the structure function equations of all orders for locally homogeneous but anisotropic turbulence as well as for the locally isotropic case. The uses of these equations for investigating the approach to local homogeneity as well as to local isotropy and the balance of the equations and identification of scaling ranges are discussed. The implications for scaling exponents and investigation of intermittency are briefly discussed.