Parameterization of the Vertical Velocity Equation for Shallow Cumulus Clouds

Abstract

The application of a steady-state vertical velocity equation for parameterized moist convective updrafts in climate and weather prediction models is currently common practice. This equation usually contains an advection, a buoyancy, and a lateral entrainment term, whereas the effects of pressure gradient and subplume contributions are typically incorporated as proportionality constants a and b for the buoyancy and the entrainment terms, respectively. A summary of proposed values of these proportionality constants a and b in the literature demonstrates that there is a large uncertainty in their most appropriate values. To shed new light on this situation an analysis is presented of the full vertical budget equation for shallow cumulus clouds obtained from large eddy simulations of three different Global Energy and Water Cycle Experiment (GEWEX) Cloud System Study (GCSS) intercomparison cases. It is found that the pressure gradient term is the dominant sink term in the vertical velocity budget, whereas the entrainment term only gives a small contribution. This result is at odds with the parameterized vertical velocity equation in the literature as it employs the entrainment term as the major sink term. As a practical solution the damping effect of the pressure term may be parameterized in terms of the lateral entrainment rates as used for thermodynamic quantities like the total specific humidity. By using a least squares method, case-dependent optimal values are obtained for the proportionality constants a and b, which are linearly related with each other. This relation can be explained from a linear relationship between the lateral entrainment rate and the buoyancy.

Evaluation of the convection‐resolving climate modeling approach on continental scales

“…To this end the hourly vertical wind output fields are filtered with an ideal first‐order high‐pass filter: The filter retains all signals with a period shorter than 2 h, and waves with a period of 5 h are damped by a factor 0.5. To yield the final index of deep convective activity, the absolute seasonal average of the filtered vertical wind on the 500 hPa level is computed: To assess the sensitivity of CI, the following alternative methods have been tested: higher‐order filters, updraft sampling with a threshold W > 1 m/s [ De Roode et al , ], or combinations of the two. The higher‐order filters reduced the magnitude of CI but did not change the spatial distribution.…”

Stereo photogrammetry reveals substantial drag on cloud thermals

“…In other words, an acceleration of the form (3 c d /8 a ) w 2 can, with an appropriate choice of c d , accommodate all types of drag: form drag, wave drag, and entrainment drag. Studies have found, however, that the deceleration from entrainment is small [ Dawe and Austin , ] and may be even slightly negative [ de Roode et al , ; Sherwood et al , ]. Consistent with these findings, we will see below that form drag and wave drag are sufficient to explain the observations.…”

“…Section 2 described the three potential sources of drag that can contribute to an effective c d : form drag, wave drag, and entrainment drag. Previous studies have presented evidence indicating that entrainment is not a significant source of drag [ de Roode et al , ; Sherwood et al , ]. This leaves us with the following question: can form drag and wave drag generate the effective drag coefficient of c d ≈1 that the previous section demonstrated must be present?…”

Understanding Convective Extreme Precipitation Scaling Using Observations and an Entraining Plume Model

“…Here, a 1 5 1 and a 2 5 0.6 are constants (de Roode et al 2012). The buoyancy is computed from the difference between cloud and environmental virtual temperature T y and gravitational constant g. The model is implemented on a 20-m vertical grid.…”