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- Author or Editor: Stephan R. de Roode x

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## Abstract

Results from simulations of the stratocumulus-topped boundary layer with one-dimensional versions of general simulation models typically exhibit a wide range of spread in the modeled liquid water path (LWP). These discrepancies are often attributed to differences in the modeled entrainment rate. Results from a large eddy simulation of the First International Satellite Cloud Climatology Project Regional Experiment I stratocumulus case are analyzed. The diagnosed eddy diffusivities for heat and moisture are found to differ by about a factor of 3. Moreover, both have a much larger magnitude than the ones typically applied in boundary layer parameterization schemes. Motivated by these results mean state solutions are analyzed for the specific case in which the vertical fluxes of heat and moisture are prescribed, whereas eddy diffusivity profiles are systematically varied by multiplication with a constant factor. The solutions demonstrate that any value, ranging from zero to a maximum adiabatic value, can be obtained for the LWP. In the subtropical parts over the ocean where horizontally extended stratocumulus fields persist, the surface sensible heat flux is typically small, whereas surface evaporation and entrainment of relatively dry air from above the surface can result in significant moisture fluxes. If the eddy diffusivity values are small, then the mean specific humidity will tend to decrease quite rapidly with height in order to support the humidity flux. This results in erroneous low humidity values in the upper part of the boundary layers causing low LWP values.

## Abstract

Results from simulations of the stratocumulus-topped boundary layer with one-dimensional versions of general simulation models typically exhibit a wide range of spread in the modeled liquid water path (LWP). These discrepancies are often attributed to differences in the modeled entrainment rate. Results from a large eddy simulation of the First International Satellite Cloud Climatology Project Regional Experiment I stratocumulus case are analyzed. The diagnosed eddy diffusivities for heat and moisture are found to differ by about a factor of 3. Moreover, both have a much larger magnitude than the ones typically applied in boundary layer parameterization schemes. Motivated by these results mean state solutions are analyzed for the specific case in which the vertical fluxes of heat and moisture are prescribed, whereas eddy diffusivity profiles are systematically varied by multiplication with a constant factor. The solutions demonstrate that any value, ranging from zero to a maximum adiabatic value, can be obtained for the LWP. In the subtropical parts over the ocean where horizontally extended stratocumulus fields persist, the surface sensible heat flux is typically small, whereas surface evaporation and entrainment of relatively dry air from above the surface can result in significant moisture fluxes. If the eddy diffusivity values are small, then the mean specific humidity will tend to decrease quite rapidly with height in order to support the humidity flux. This results in erroneous low humidity values in the upper part of the boundary layers causing low LWP values.

## Abstract

The vertical transport by shallow nonprecipitating cumulus clouds of conserved variables, such as the total specific humidity or the liquid water potential temperature, can be well modeled by the mass-flux approach, in which the cloud field is represented by a top-hat distribution of clouds and its environment. The mass-flux budget is computed by conditionally sampling the prognostic vertical velocity equation by means of a large eddy simulation of shallow cumulus clouds. The model initialization is based on observations made during the Barbados Oceanographic and Meteorological Experiment (BOMEX). Several different sampling criteria are applied. The presence of liquid water is used to select clouds, whereas additional criteria are applied to sample cloud updraft, downdraft, and core properties. A comparison between the budgets of the mass flux and the vertical velocity variance show that they appear to be qualitatively similar. The mass flux is driven by buoyancy in the lower part of the cloud layer, whereas turbulent transport is important in generating mass flux in the upper part of the cloud layer. Pressure and subgrid-scale effects typically act to dissipate mass flux. Entrainment and detrainment rates for the vertical velocity equation are presented. They are found to be smaller in comparison to the ones for conserved variables. It is explained that the top-hat structure for the virtual potential temperature is degraded by mixing at the cloud boundaries leading to a subsequent evaporative cooling of cloud droplets that supports the formation of negatively buoyant cloud parcels.

## Abstract

The vertical transport by shallow nonprecipitating cumulus clouds of conserved variables, such as the total specific humidity or the liquid water potential temperature, can be well modeled by the mass-flux approach, in which the cloud field is represented by a top-hat distribution of clouds and its environment. The mass-flux budget is computed by conditionally sampling the prognostic vertical velocity equation by means of a large eddy simulation of shallow cumulus clouds. The model initialization is based on observations made during the Barbados Oceanographic and Meteorological Experiment (BOMEX). Several different sampling criteria are applied. The presence of liquid water is used to select clouds, whereas additional criteria are applied to sample cloud updraft, downdraft, and core properties. A comparison between the budgets of the mass flux and the vertical velocity variance show that they appear to be qualitatively similar. The mass flux is driven by buoyancy in the lower part of the cloud layer, whereas turbulent transport is important in generating mass flux in the upper part of the cloud layer. Pressure and subgrid-scale effects typically act to dissipate mass flux. Entrainment and detrainment rates for the vertical velocity equation are presented. They are found to be smaller in comparison to the ones for conserved variables. It is explained that the top-hat structure for the virtual potential temperature is degraded by mixing at the cloud boundaries leading to a subsequent evaporative cooling of cloud droplets that supports the formation of negatively buoyant cloud parcels.

## Abstract

Aircraft measurements made during the “First Lagrangian” of the Atlantic Stratocumulus Transition Experiment (ASTEX) between 12 and 14 June 1992 are presented. During this Lagrangian experiment an air mass was followed that was advected southward by the mean wind. Five aircraft flights were undertaken to observe the transition of a stratocumulus cloud deck to thin and broken stratocumulus clouds penetrated by cumulus from below. From the horizontal aircraft legs the boundary layer mean structure, microphysics, turbulence structure, and entrainment were analyzed. The vertical profiles of the vertical velocity skewness are shown to illustrate the transition of a cloudy boundary layer predominantly driven by longwave radiative cooling at the cloud top to one driven mainly by convection due to an unstable surface stratification and cumulus clouds. During the last flight before the stratocumulus deck was observed to be broken and replaced by cumuli, the total water flux, the virtual potential temperature flux, and the vertical velocity variance in the stratocumulus cloud layer were found significantly larger compared with the previous flights. To analyze the cloud-top stability the mean jumps of conserved variables across the inversion were determined from porpoising runs through the cloud top. These jumps were compared with cloud-top entrainment instability criteria discussed in the literature. It is suggested that enhanced entrainment of dry air is a key mechanism in the stratocumulus–cumulus transition.

## Abstract

Aircraft measurements made during the “First Lagrangian” of the Atlantic Stratocumulus Transition Experiment (ASTEX) between 12 and 14 June 1992 are presented. During this Lagrangian experiment an air mass was followed that was advected southward by the mean wind. Five aircraft flights were undertaken to observe the transition of a stratocumulus cloud deck to thin and broken stratocumulus clouds penetrated by cumulus from below. From the horizontal aircraft legs the boundary layer mean structure, microphysics, turbulence structure, and entrainment were analyzed. The vertical profiles of the vertical velocity skewness are shown to illustrate the transition of a cloudy boundary layer predominantly driven by longwave radiative cooling at the cloud top to one driven mainly by convection due to an unstable surface stratification and cumulus clouds. During the last flight before the stratocumulus deck was observed to be broken and replaced by cumuli, the total water flux, the virtual potential temperature flux, and the vertical velocity variance in the stratocumulus cloud layer were found significantly larger compared with the previous flights. To analyze the cloud-top stability the mean jumps of conserved variables across the inversion were determined from porpoising runs through the cloud top. These jumps were compared with cloud-top entrainment instability criteria discussed in the literature. It is suggested that enhanced entrainment of dry air is a key mechanism in the stratocumulus–cumulus transition.

## Abstract

In many large-scale models mass-flux parameterizations are applied to prognose the effect of cumulus cloud convection on the large-scale environment. Key parameters in the mass-flux equations are the lateral entrainment and detrainment rates. The physical meaning of these parameters is that they quantify the mixing rate of mass across the thermal boundaries between the cloud and its environment.

The prognostic equations for the updraft and downdraft value of a conserved variable are used to derive a prognostic variance equation in the mass-flux approach. The analogy between this equation and the Reynolds-averaged variance equation is discussed. It is demonstrated that the prognostic variance equation formulated in mass-flux variables contains a gradient-production, transport, and dissipative term. In the latter term, the sum of the lateral entrainment and detrainment rates represents an inverse timescale of the dissipation.

Steady-state solutions of the variance equations are discussed. Expressions for the fractional entrainment and detrainment coefficients are derived. Also, solutions for the vertical flux of an arbitrary conserved variable are presented. For convection in which the updraft fraction equals the downdraft fraction, the vertical flux of the scalar flows down the local mean gradient. The turbulent mixing coefficient is given by the ratio of the vertical mass flux and the sum of the fractional entrainment and detrainment coefficients. For an arbitrary updraft fraction, however, flux correction terms are part of the solution. It is shown that for a convective boundary layer these correction terms can account for countergradient transport, which is illustrated from large eddy simulation results. In the cumulus convection limit the vertical flux flows down the “cloud” gradient. It is concluded that in the mass-flux approach the turbulent mixing coefficients, and the correction terms that arise from the transport term, are very similar to closures applied to the Reynolds-averaged equations.

## Abstract

In many large-scale models mass-flux parameterizations are applied to prognose the effect of cumulus cloud convection on the large-scale environment. Key parameters in the mass-flux equations are the lateral entrainment and detrainment rates. The physical meaning of these parameters is that they quantify the mixing rate of mass across the thermal boundaries between the cloud and its environment.

The prognostic equations for the updraft and downdraft value of a conserved variable are used to derive a prognostic variance equation in the mass-flux approach. The analogy between this equation and the Reynolds-averaged variance equation is discussed. It is demonstrated that the prognostic variance equation formulated in mass-flux variables contains a gradient-production, transport, and dissipative term. In the latter term, the sum of the lateral entrainment and detrainment rates represents an inverse timescale of the dissipation.

Steady-state solutions of the variance equations are discussed. Expressions for the fractional entrainment and detrainment coefficients are derived. Also, solutions for the vertical flux of an arbitrary conserved variable are presented. For convection in which the updraft fraction equals the downdraft fraction, the vertical flux of the scalar flows down the local mean gradient. The turbulent mixing coefficient is given by the ratio of the vertical mass flux and the sum of the fractional entrainment and detrainment coefficients. For an arbitrary updraft fraction, however, flux correction terms are part of the solution. It is shown that for a convective boundary layer these correction terms can account for countergradient transport, which is illustrated from large eddy simulation results. In the cumulus convection limit the vertical flux flows down the “cloud” gradient. It is concluded that in the mass-flux approach the turbulent mixing coefficients, and the correction terms that arise from the transport term, are very similar to closures applied to the Reynolds-averaged equations.

## 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.

## 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.

## Abstract

The length scale evolution of various quantities in a clear convective boundary layer (CBL), a stratocumulus-topped boundary layer, and three radiatively cooled (“smoke cloud”) convective boundary layers are studied by means of large-eddy simulations on a large horizontal domain (25.6 × 25.6 km^{2}). In the CBL the virtual potential temperature and the vertical velocity fields are dominated by horizontal scales on the order of the boundary layer depth. In contrast, the potential temperature and the specific humidity fields become gradually dominated by mesoscale fluctuations. However, at the mesoscales their effects on the virtual potential temperature fluctuations nearly compensate. It is found that mesoscale fluctuations are negligibly small only for conserved variables that have an entrainment to surface flux ratio close to −0.25, which is about the flux ratio for the buoyancy. In the CBL the moisture and potential temperature flux ratios can have values that significantly deviate from this number.

The geometry of the buoyancy flux was manipulated by cooling the clear convective boundary layer from the top, in addition to a positive buoyancy flux at the surface. For these radiatively cooled cases it is found that both the vertical velocity as well as the virtual potential temperature spectra tend to broaden. The role of the buoyancy flux in their respective prognostic variance equations is discussed. It is argued that in the upper part of the clear CBL, where the mean vertical stratification is stable, vertical velocity variance and virtual potential temperature variance cannot be produced simultaneously. For the stratocumulus case, in which latent heat release effects in the cloud layer play an important role in its dynamics, the field of any quantity, except for the vertical velocity, becomes dominated by mesoscale fluctuations.

In general, the location of the spectral peak of any quantity becoming constrained by the domain size should be avoided. The answer to the question of how large the LES horizontal domain size should be in order to include mesoscale fluctuations will, on the one hand, depend on the type of convection to be simulated and the kind of physical question one aims to address, and, on the other hand, the time duration of the simulation. Only if one aims to study the dynamics of a dry CBL that excludes moisture, a rather small domain size suffices. In case one aims to examine either the spatial evolution of the fields of any arbitrary conserved scalar in the CBL, or any quantity in stratocumulus clouds except for the vertical velocity, a larger domain size that allows the development of mesoscale fluctuations will be necessary.

## Abstract

The length scale evolution of various quantities in a clear convective boundary layer (CBL), a stratocumulus-topped boundary layer, and three radiatively cooled (“smoke cloud”) convective boundary layers are studied by means of large-eddy simulations on a large horizontal domain (25.6 × 25.6 km^{2}). In the CBL the virtual potential temperature and the vertical velocity fields are dominated by horizontal scales on the order of the boundary layer depth. In contrast, the potential temperature and the specific humidity fields become gradually dominated by mesoscale fluctuations. However, at the mesoscales their effects on the virtual potential temperature fluctuations nearly compensate. It is found that mesoscale fluctuations are negligibly small only for conserved variables that have an entrainment to surface flux ratio close to −0.25, which is about the flux ratio for the buoyancy. In the CBL the moisture and potential temperature flux ratios can have values that significantly deviate from this number.

The geometry of the buoyancy flux was manipulated by cooling the clear convective boundary layer from the top, in addition to a positive buoyancy flux at the surface. For these radiatively cooled cases it is found that both the vertical velocity as well as the virtual potential temperature spectra tend to broaden. The role of the buoyancy flux in their respective prognostic variance equations is discussed. It is argued that in the upper part of the clear CBL, where the mean vertical stratification is stable, vertical velocity variance and virtual potential temperature variance cannot be produced simultaneously. For the stratocumulus case, in which latent heat release effects in the cloud layer play an important role in its dynamics, the field of any quantity, except for the vertical velocity, becomes dominated by mesoscale fluctuations.

In general, the location of the spectral peak of any quantity becoming constrained by the domain size should be avoided. The answer to the question of how large the LES horizontal domain size should be in order to include mesoscale fluctuations will, on the one hand, depend on the type of convection to be simulated and the kind of physical question one aims to address, and, on the other hand, the time duration of the simulation. Only if one aims to study the dynamics of a dry CBL that excludes moisture, a rather small domain size suffices. In case one aims to examine either the spatial evolution of the fields of any arbitrary conserved scalar in the CBL, or any quantity in stratocumulus clouds except for the vertical velocity, a larger domain size that allows the development of mesoscale fluctuations will be necessary.

## Abstract

Previously observed twice-Clausius–Clapeyron (2CC) scaling for extreme precipitation at hourly time scales has led to discussions about its origin. The robustness of this scaling is assessed by analyzing a subhourly dataset of 10-min resolution over the Netherlands. The results confirm the validity of the previously found 2CC scaling for extreme convective precipitation.

Using a simple entraining plume model, an idealized deep convective environmental temperature profile is perturbed to analyze extreme precipitation scaling from a frequently used relation based on the column condensation rate. The plume model simulates a steady precipitation increase that is greater than Clausius–Clapeyron scaling (super-CC scaling). Precipitation intensity increase is shown to be controlled by a flux of moisture through the cloud base and in-cloud lateral moisture convergence. Decomposition of this scaling relation into a dominant thermodynamic and additional dynamic component allows for better understanding of the scaling and demonstrates the importance of vertical velocity in both dynamic and thermodynamic scaling. Furthermore, systematically increasing the environmental stability by adjusting the temperature perturbations from constant to moist adiabatic increase reveals a dependence of the scaling on the change in environmental stability. As the perturbations become increasingly close to moist adiabatic, the scaling found by the entraining plume model decreases to CC scaling. Thus, atmospheric stability changes, which are expected to be dependent on the latitude, may well play a key role in the behavior of precipitation extremes in the future climate.

## Abstract

Previously observed twice-Clausius–Clapeyron (2CC) scaling for extreme precipitation at hourly time scales has led to discussions about its origin. The robustness of this scaling is assessed by analyzing a subhourly dataset of 10-min resolution over the Netherlands. The results confirm the validity of the previously found 2CC scaling for extreme convective precipitation.

Using a simple entraining plume model, an idealized deep convective environmental temperature profile is perturbed to analyze extreme precipitation scaling from a frequently used relation based on the column condensation rate. The plume model simulates a steady precipitation increase that is greater than Clausius–Clapeyron scaling (super-CC scaling). Precipitation intensity increase is shown to be controlled by a flux of moisture through the cloud base and in-cloud lateral moisture convergence. Decomposition of this scaling relation into a dominant thermodynamic and additional dynamic component allows for better understanding of the scaling and demonstrates the importance of vertical velocity in both dynamic and thermodynamic scaling. Furthermore, systematically increasing the environmental stability by adjusting the temperature perturbations from constant to moist adiabatic increase reveals a dependence of the scaling on the change in environmental stability. As the perturbations become increasingly close to moist adiabatic, the scaling found by the entraining plume model decreases to CC scaling. Thus, atmospheric stability changes, which are expected to be dependent on the latitude, may well play a key role in the behavior of precipitation extremes in the future climate.

## Abstract

Large-eddy simulation (LES) models are widely used to study atmospheric turbulence. The effects of small-scale motions that cannot be resolved need to be modeled by a subfilter-scale (SFS) model. The SFS contribution to the turbulent fluxes is typically significant in the surface layer. This study presents analytical solutions of the classical Smagorinsky SFS turbulent kinetic energy (TKE) model including a buoyancy flux contribution. Both a constant length scale and a stability-dependent one as proposed by Deardorff are considered. Analytical expressions for the mixing functions are derived and Monin–Obukhov similarity relations that are implicitly imposed by the SFS TKE model are diagnosed. For neutral and weakly stable conditions, observations indicate that the turbulent Prandtl number (Pr_{T}) is close to unity. However, based on observations in the convective boundary layer, a lower value for Pr_{T} is often applied in LES models. As a lower Prandtl number promotes a stronger mixing of heat, this may cause excessive mixing, which is quantified from a direct comparison of the mixing function as imposed by the SFS TKE model with empirical fits from field observations. For a strong stability, the diagnosed mixing functions for both momentum and heat are larger than observed. The problem of excessive mixing will be enhanced for anisotropic grids. The findings are also relevant for high-resolution numerical weather prediction models that use a Smagorinsky-type TKE closure.

## Abstract

Large-eddy simulation (LES) models are widely used to study atmospheric turbulence. The effects of small-scale motions that cannot be resolved need to be modeled by a subfilter-scale (SFS) model. The SFS contribution to the turbulent fluxes is typically significant in the surface layer. This study presents analytical solutions of the classical Smagorinsky SFS turbulent kinetic energy (TKE) model including a buoyancy flux contribution. Both a constant length scale and a stability-dependent one as proposed by Deardorff are considered. Analytical expressions for the mixing functions are derived and Monin–Obukhov similarity relations that are implicitly imposed by the SFS TKE model are diagnosed. For neutral and weakly stable conditions, observations indicate that the turbulent Prandtl number (Pr_{T}) is close to unity. However, based on observations in the convective boundary layer, a lower value for Pr_{T} is often applied in LES models. As a lower Prandtl number promotes a stronger mixing of heat, this may cause excessive mixing, which is quantified from a direct comparison of the mixing function as imposed by the SFS TKE model with empirical fits from field observations. For a strong stability, the diagnosed mixing functions for both momentum and heat are larger than observed. The problem of excessive mixing will be enhanced for anisotropic grids. The findings are also relevant for high-resolution numerical weather prediction models that use a Smagorinsky-type TKE closure.

## Abstract

Results of four Lagrangian stratocumulus-to-shallow-cumulus transition cases as obtained from six different large-eddy simulation models are presented. The model output is remarkably consistent in terms of the representation of the evolution of the mean state, which is characterized by a stratocumulus cloud layer that rises with time and that warms and dries relative to the subcloud layer. Also, the effect of the diurnal insolation on cloud-top entrainment and the moisture flux at the top of the subcloud layer are consistently captured by the models. For some cases, the models diverge in terms of the liquid water path (LWP) during nighttime, which can be explained from the difference in the sign of the buoyancy flux at cloud base. If the subcloud buoyancy fluxes are positive, turbulence sustains a vertically well-mixed layer, causing a cloud layer that is relatively cold and moist and consequently has a high LWP. After some simulation time, all cases exhibit subcloud-layer dynamics that appear to be similar to those of the dry convective boundary layer. The humidity flux from the subcloud layer toward the stratocumulus cloud layer, which is one of the major sources of stratocumulus cloud liquid water, is larger during the night than during the day. The sensible heat flux becomes constant in time, whereas the latent heat flux tends to increase during the transition. These findings are explained from a budget analysis of the subcloud layer.

## Abstract

Results of four Lagrangian stratocumulus-to-shallow-cumulus transition cases as obtained from six different large-eddy simulation models are presented. The model output is remarkably consistent in terms of the representation of the evolution of the mean state, which is characterized by a stratocumulus cloud layer that rises with time and that warms and dries relative to the subcloud layer. Also, the effect of the diurnal insolation on cloud-top entrainment and the moisture flux at the top of the subcloud layer are consistently captured by the models. For some cases, the models diverge in terms of the liquid water path (LWP) during nighttime, which can be explained from the difference in the sign of the buoyancy flux at cloud base. If the subcloud buoyancy fluxes are positive, turbulence sustains a vertically well-mixed layer, causing a cloud layer that is relatively cold and moist and consequently has a high LWP. After some simulation time, all cases exhibit subcloud-layer dynamics that appear to be similar to those of the dry convective boundary layer. The humidity flux from the subcloud layer toward the stratocumulus cloud layer, which is one of the major sources of stratocumulus cloud liquid water, is larger during the night than during the day. The sensible heat flux becomes constant in time, whereas the latent heat flux tends to increase during the transition. These findings are explained from a budget analysis of the subcloud layer.