Theoretical Understanding of the Linear Relationship between Convective Updrafts and Cloud-Base Height for Shallow Cumulus Clouds. Part I: Maritime Conditions

Zheng and Rosenfeld found linear relationships between the convective updrafts and cloud-base height zb using ground-based observations over both land and ocean. The empirical relationships allow for a novel satellite remote sensing technique of inferring the cloud-base updrafts and cloud condensation nuclei concentration, both of which are important for understanding aerosol–cloud–climate interactions but have been notoriously difficult to retrieve from space. In Part I of a two-part study, a theoretical framework is established for understanding this empirical relationship over the ocean. Part II deals with continental cumulus clouds. Using the bulk concept of mixed-layer (ML) model for shallow cumulus, I found that this relationship arises from the conservation law of energetics that requires the radiative flux divergence of an ML to balance surface buoyancy flux. Given a certain ML radiative cooling rate per unit mass Q, a deeper ML (higher zb) undergoes more radiative cooling and requires stronger surface buoyancy flux to balance it, leading to stronger updrafts. The rate with which the updrafts vary with zb is modulated by Q. The cooling rate Q manifests strong resilience to external large-scale forcing that spans a wide range of climatology, allowing the slope of the updrafts–zb relationship to remain nearly invariant. This causes the relationship to manifest linearity. The physical mechanism underlying the resilience of Q to large-scale forcing, such as free-tropospheric moisture and sea surface temperature, is investigated through the lens of the radiative transfer theory (two-stream Schwarzschild equations) and an ML model for shallow cumulus.

Reply to “Comments on ‘Langmuir Turbulence and Surface Heating in the Ocean Surface Boundary Layer’”

The differences between the conclusions of Noh and Choi and of Pearson et al., which are largely a result of defining different length scales based on different quantities, are discussed. This study shows that the layer over which Langmuir turbulence mixes (nominally hTKE) under a stabilizing surface buoyancy flux should be scaled by a combination of the Langmuir stability length LL and initial/nocturnal boundary layer depth h0 rather than by the Zilitinkevich length.

Turbulence kinetic energy budget during the afternoon transition – Part 2: A simple TKE model

A simple model for turbulence kinetic energy (TKE) and the TKE budget is presented for sheared convective atmospheric conditions based on observations from the Boundary Layer Late Afternoon and Sunset Turbulence (BLLAST) field campaign. It is based on an idealized mixed-layer approximation and a simplified near-surface TKE budget. In this model, the TKE is dependent on four budget terms (turbulent dissipation rate, buoyancy production, shear production and vertical transport of TKE) and only requires measurements of three available inputs (near-surface buoyancy flux, boundary layer depth and wind speed at one height in the surface layer) to predict vertical profiles of TKE and TKE budget terms.This simple model is shown to reproduce some of the observed variations between the different studied days in terms of near-surface TKE and its decay during the afternoon transition reasonably well. It is subsequently used to systematically study the effects of buoyancy and shear on TKE evolution using idealized constant and time-varying winds during the afternoon transition. From this, we conclude that many different TKE decay rates are possible under time-varying winds and that generalizing the decay with simple scaling laws for near-surface TKE of the form tα may be questionable.The model's errors result from the exclusion of processes such as elevated shear production and horizontal advection. The model also produces an overly rapid decay of shear production with height. However, the most influential budget terms governing near-surface TKE in the observed sheared convective boundary layers are included, while only second-order factors are neglected. Comparison between modeled and averaged observed estimates of dissipation rate illustrates that the overall behavior of the model is often quite reasonable. Therefore, we use the model to discuss the low-turbulence conditions that form first in the upper parts of the boundary layer during the afternoon transition and are only apparent later near the surface. This occurs as a consequence of the continuous decrease in near-surface buoyancy flux during the afternoon transition. This region of weak afternoon turbulence is hypothesized to be a “pre-residual layer”, which is important in determining the onset conditions for the weak sporadic turbulence that occur in the residual layer once near-surface stratification has become stable.

Growth and Decay of a Convective Boundary Layer over a Surface with a Constant Temperature

The growth and decay of a convective boundary layer (CBL) over a surface with a constant surface temperature that develops into a linear stratification is studied, and a mathematical model for this system is derived. The study is based on direct numerical simulations with four different Reynolds numbers; the two simulations with the largest Reynolds numbers display Reynolds number similarity, suggesting that the results can be extrapolated to the atmosphere. Because of the interplay of the growing CBL and the gradually decreasing surface buoyancy flux, the system has a complex time evolution in which integrated kinetic energy, buoyancy flux, and dissipation peak and subsequently decay. The derived model provides characteristic scales for bulk properties of the CBL. Even though the system is unsteady, self-similar vertical profiles of buoyancy, buoyancy flux, and velocity variances are recovered. There are two important implications for atmospheric modeling. First, the magnitude of the surface buoyancy flux sets the time scale of the system; thus, over a rough surface the roughness length is a key variable. Therefore, the performance of the surface model is crucial in large-eddy simulations of convection over water surfaces. Second, during the phase in which kinetic energy decays, the integrated kinetic energy never follows a power law, because the buoyancy flux and dissipation balance until the kinetic energy has almost vanished. Therefore, the applicability of power-law decay models to the afternoon transition in the atmospheric boundary layer is questionable; the presented model provides a physically sound alternative.

Turbulence Kinetic Energy budget during the afternoon transition – Part 2: A simple TKE model

A simple model for turbulence kinetic energy (TKE) and the TKE budget is presented for sheared convective atmospheric conditions based on observations from the Boundary Layer Late Afternoon and Sunset Turbulence (BLLAST) field campaign. It is based on an idealized mixed-layer approximation and a simplified near-surface TKE budget. In this model, the TKE is dependent on four budget terms (turbulent dissipation rate, buoyancy production, shear production and vertical transport of TKE) and only requires measurements of three input available (near-surface buoyancy flux, boundary layer depth and wind speed at one height in the surface layer). This simple model is shown to reproduce some of the observed variations between the different studied days in terms of near-surface TKE and its decay during the afternoon transition reasonably well. It is subsequently used to systematically study the effects of buoyancy and shear on TKE evolution using idealized constant and time-varying winds during the afternoon transition. From this, we conclude that many different TKE decay rates are possible under time-varying winds and that generalizing the decay with simple scaling laws for near-surface TKE of the form tα may be questionable. The model's errors result from the exclusion of processes such as elevated shear production and horizontal advection. The model also produces an overly rapid decay of shear production with height. However, the most influential budget terms governing near-surface TKE in the observed sheared convective boundary layers are included, while only second order factors are neglected. Comparison between modeled and averaged observed estimates of dissipation rate illustrate that the overall behavior of the model is often quite reasonable. Therefore, we use the model to discuss the low turbulence conditions that form first in the upper parts of the boundary layer during the afternoon transition and are only apparent later near the surface. This occurs as a consequence of the continuous decrease of near-surface buoyancy flux during the afternoon transition. This region of weak afternoon turbulence is hypothesized to be a "pre-residual layer", which is important in determining the onset conditions for the weak sporadic turbulence that occur in the residual layer once near-surface stratification has become stable.

Scaling Laws for the Heterogeneously Heated Free Convective Boundary Layer

The heterogeneously heated free convective boundary layer (CBL) is investigated by means of dimensional analysis and results from large-eddy simulations (LES) and direct numerical simulations (DNS). The investigated physical model is a CBL that forms in a linearly stratified atmosphere heated from the surface by square patches with a high surface buoyancy flux. Each simulation has been run long enough to show the formation of a peak in kinetic energy, corresponding to the “optimal” heterogeneity size with strong secondary circulations, and the subsequent transition into a horizontally homogeneous CBL. Scaling laws for the time of the optimal state and transition and for the vertically integrated kinetic energy (KE) have been developed. The laws show that the optimal state and transition do not occur at a fixed ratio of the heterogeneity size to the CBL height. Instead, these occur at a higher ratio for simulations with increasing heterogeneity sizes because of the development of structures in the downward-moving air that grow faster than the CBL thickness. The moment of occurrence of the optimal state and transition are strongly related to the heterogeneity amplitude: stronger amplitudes result in an earlier optimal state and a later transition. Furthermore, a decrease in patch size combined with a compensating increase in patch surface buoyancy flux to maintain the energy input results in decreasing KE and a later transition. The simulations suggest that a CBL with a heterogeneity size smaller than the initial CBL height has less entrainment than a horizontally homogeneous CBL, whereas one with a larger heterogeneity size has more.

Energetics of Semienclosed Basins with Two-Layer Flows at the Strait

Examination of the energy budget for semienclosed seas with two-layer exchange flow at the strait shows that the energy flux at the open portion of the boundary (the strait) is proportional to the surface buoyancy flux integrated over the basin area, with the constant of proportionality given by the interface depth. When the surface buoyancy flux is positive, the energy flux is negative: these types of basins have an estuarine circulation. Antiestuarine basins have a negative surface buoyancy flux, which provides a positive energy flux, augmenting the wind work in powering the circulation. The energy budget for the semienclosed seas with vertically separated flows at the strait is examined using reanalysis products for four major semienclosed basins: the Mediterranean and Red Seas (antiestuarine) and the Black and Baltic Seas (estuarine). Important differences in the relative contribution to the energy budget of the wind work versus the surface buoyancy flux are found within basins of the same type, and these differences help explain some qualitative aspects of the basins’ flow.

Eddy-Driven Exchange between the Open Ocean and a Sub–Ice Shelf Cavity

The exchange between the open ocean and sub–ice shelf cavities is important to both water mass transformations and ice shelf melting. Here, the authors use a high-resolution (500 m) numerical model to investigate to which degree eddies produced by frontal instability at the edge of a polynya are capable of transporting dense high-salinity shelf water (HSSW) underneath an ice shelf. The applied surface buoyancy flux and ice shelf geometry is based on Ronne Ice Shelf in the southern Weddell Sea, an area of intense wintertime sea ice production where a flow of HSSW into the cavity has been observed. Results show that eddies are able to enter the cavity at the southwestern corner of the polynya where an anticyclonic rim current intersects the ice shelf front. The size and time scale of simulated eddies are in agreement with observations close to the Ronne Ice Front. The properties and strength of the inflow are sensitive to the prescribed total ice production, flushing the ice shelf cavity at a rate of 0.2–0.4 × 106 m3 s−1 depending on polynya size and magnitude of surface buoyancy flux. Eddy-driven HSSW transport into the cavity is reduced by about 50% if the model grid resolution is decreased to 2–5 km and eddies are not properly resolved.

The Influence of Wind Speed on Shallow Marine Cumulus Convection

The role of wind speed on shallow marine cumulus convection is explored using large-eddy simulations and concepts from bulk theory. Focusing on cases characteristic of the trades, the equilibrium trade wind layer is found to be deeper at stronger winds, with larger surface moisture fluxes and smaller surface heat fluxes. The opposing behavior of the surface fluxes is caused by more warm and dry air being mixed to the surface as the cloud layer deepens. This leads to little difference in equilibrium surface buoyancy fluxes and cloud-base mass fluxes. Shallow cumuli are deeper, but not more numerous or more energetic. The deepening response is necessary to resolve an inconsistency in the subcloud layer. This argument follows from bulk concepts and assumes that the lapse rate and flux divergence of moist-conserved variables do not change, based on simulation results. With that assumption, stronger winds and a fixed inversion height imply larger surface moisture and buoyancy fluxes (heat fluxes are small initially). The consequent moistening tends to decrease cloud-base height, which is inconsistent with a larger surface buoyancy flux that tends to increase cloud-base height, in order to maintain the buoyancy flux at cloud base at a fixed fraction of its surface value (entrainment closure). Deepening the cloud layer by increasing the inversion height resolves this inconsistency by allowing the surface buoyancy flux to remain constant without further moistening the subcloud layer. Because this explanation follows from simple bulk concepts, it is suggested that the internal dynamics (mixing) of clouds is only secondary to the deepening response.