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## Abstract

The dissipation parameterizations developed for higher-order closure are used to parameterize lateral entrainment and detrainment in a mass-flux model. In addition, a subplume-scale turbulence scheme is included to represent fluxes not captured in the conventional mass-flux framework. These new parameterizations are tested by simulating trade wind cumulus from the Barbados Oceanographic and Meteorological Experiment (BOMEX).

## Abstract

The dissipation parameterizations developed for higher-order closure are used to parameterize lateral entrainment and detrainment in a mass-flux model. In addition, a subplume-scale turbulence scheme is included to represent fluxes not captured in the conventional mass-flux framework. These new parameterizations are tested by simulating trade wind cumulus from the Barbados Oceanographic and Meteorological Experiment (BOMEX).

## Abstract

Shallow-water equations discretized on a perfect hexagonal grid are analyzed using both a momentum formulation and a vorticity-divergence formulation. The vorticity-divergence formulation uses the unstaggered Z grid that places mass, vorticity, and divergence at the centers of the hexagons. The momentum formulation uses the staggered ZM grid that places mass at the centers of the hexagons and velocity at the corners of the hexagons. It is found that the Z grid and the ZM grid are identical in their simulation of the physical modes relevant to geostrophic adjustment. Consistent with the continuous system, the simulated inertia–gravity wave phase speeds increase monotonically with increasing total wavenumber and, thus, all waves have nonzero group velocities.

Since a grid of hexagons has twice as many corners as it has centers, the ZM grid has twice as many velocity points as it has mass points. As a result, the ZM-grid velocity field is discretized at a higher resolution than the mass field and, therefore, resolves a larger region of wavenumber space than the mass field. We solve the ∇^{2}
*f* = *λf* eigenvalue problem with periodic boundary conditions on both the Z grid and ZM grid to determine the modes that can exist on each grid. The mismatch between mass and momentum leads to computational modes in the velocity field. Two techniques that can be used to control these computational modes are discussed. One technique is to use a dissipation operator that captures or “sees” the smallest-scale variations in the velocity field. The other technique is to invert elliptic equations in order to filter the high wavenumber part of the momentum field.

Results presented here lead to the conclusion that the ZM grid is an attractive alternative to the Z grid, and might be particularly useful for ocean modeling.

## Abstract

Shallow-water equations discretized on a perfect hexagonal grid are analyzed using both a momentum formulation and a vorticity-divergence formulation. The vorticity-divergence formulation uses the unstaggered Z grid that places mass, vorticity, and divergence at the centers of the hexagons. The momentum formulation uses the staggered ZM grid that places mass at the centers of the hexagons and velocity at the corners of the hexagons. It is found that the Z grid and the ZM grid are identical in their simulation of the physical modes relevant to geostrophic adjustment. Consistent with the continuous system, the simulated inertia–gravity wave phase speeds increase monotonically with increasing total wavenumber and, thus, all waves have nonzero group velocities.

Since a grid of hexagons has twice as many corners as it has centers, the ZM grid has twice as many velocity points as it has mass points. As a result, the ZM-grid velocity field is discretized at a higher resolution than the mass field and, therefore, resolves a larger region of wavenumber space than the mass field. We solve the ∇^{2}
*f* = *λf* eigenvalue problem with periodic boundary conditions on both the Z grid and ZM grid to determine the modes that can exist on each grid. The mismatch between mass and momentum leads to computational modes in the velocity field. Two techniques that can be used to control these computational modes are discussed. One technique is to use a dissipation operator that captures or “sees” the smallest-scale variations in the velocity field. The other technique is to invert elliptic equations in order to filter the high wavenumber part of the momentum field.

Results presented here lead to the conclusion that the ZM grid is an attractive alternative to the Z grid, and might be particularly useful for ocean modeling.

## Abstract

Higher-order closure (HOC) models have been proposed for parameterization of the turbulent planetary boundary layer (PBL). HOC models must include closures for higher-order moments (e.g., fourth moments in third-order closure models), for pressure terms, and for dissipation terms. Mass-flux closure (MFC) models have been proposed for parameterization of cumulus convection and, more recently, the convective PBL. MFC models include closures for lateral mass exchanges and for pressure terms (which are usually ignored). The authors developed a new kind of model that combines HOC and MFC, which they hope will be useful for the parameterization of *both* the PBL and cumulus convection, in a unified framework. Such a model is particularly well suited to regimes in which the PBL turbulence and the cumulus convection are not well separated, for example, the broken stratocumulus and shallow cumulus regimes.

The model makes use of an assumed joint probability distribution for the variables of interest, and the equations typically used in HOC models can be derived by integrating over the distribution. Accordingly, the model is called Assumed-Distribution Higher-Order Closure (ADHOC). The prognostic variables of ADHOC are the mean state, the second and third moments of the vertical velocity, and the vertical fluxes of other quantities of interest. All of the parameters of the distribution can be determined from the predicted moments; thereafter the joint distribution is effectively known, and so any and all moments can be constructed as needed. In this way, the usual closure problem of “higher moments” is avoided. The pressure-term parameterizations previously developed for HOC models are used to predict the convective fluxes and the moments of the vertical velocity.

In companion papers, parameterizations of lateral mass exchanges and subplume-scale fluxes are presented, and then ADHOC is applied to several observationally based tropical, subtropical, and dry convective boundary layers.

## Abstract

Higher-order closure (HOC) models have been proposed for parameterization of the turbulent planetary boundary layer (PBL). HOC models must include closures for higher-order moments (e.g., fourth moments in third-order closure models), for pressure terms, and for dissipation terms. Mass-flux closure (MFC) models have been proposed for parameterization of cumulus convection and, more recently, the convective PBL. MFC models include closures for lateral mass exchanges and for pressure terms (which are usually ignored). The authors developed a new kind of model that combines HOC and MFC, which they hope will be useful for the parameterization of *both* the PBL and cumulus convection, in a unified framework. Such a model is particularly well suited to regimes in which the PBL turbulence and the cumulus convection are not well separated, for example, the broken stratocumulus and shallow cumulus regimes.

The model makes use of an assumed joint probability distribution for the variables of interest, and the equations typically used in HOC models can be derived by integrating over the distribution. Accordingly, the model is called Assumed-Distribution Higher-Order Closure (ADHOC). The prognostic variables of ADHOC are the mean state, the second and third moments of the vertical velocity, and the vertical fluxes of other quantities of interest. All of the parameters of the distribution can be determined from the predicted moments; thereafter the joint distribution is effectively known, and so any and all moments can be constructed as needed. In this way, the usual closure problem of “higher moments” is avoided. The pressure-term parameterizations previously developed for HOC models are used to predict the convective fluxes and the moments of the vertical velocity.

In companion papers, parameterizations of lateral mass exchanges and subplume-scale fluxes are presented, and then ADHOC is applied to several observationally based tropical, subtropical, and dry convective boundary layers.

## Abstract

Assumed-PDF methods for the parameterization of subgrid-scale processes in atmospheric models provide many benefits. Many currently used assumed-PDF schemes reconcile the high number of required PDF parameters with the relative paucity of input moments by employing simplifying assumptions that are difficult to test. This paper explores the possibility of constructing a trivariate double-Gaussian PDF from the first three orders of moments without simplifying assumptions and proves that no unique solution exists. In an effort to provide a path for future improvement of current assumed-PDF schemes, the expectation maximization (EM) algorithm for Gaussian mixture models is used with LES output of shallow cumulus, stratocumulus, and deep convection cases to determine “best fit” PDFs using from one through four Gaussian clusters. The EM PDFs are evaluated using PDF-diagnosed higher-order moments, PDF-diagnosed cloud statistics, and the Akaike information criterion. It was found that two Gaussian clusters were almost always adequate to represent both higher-order moments and cloud statistics like cloud fraction, water content, and vertical fluxes of cloud water and buoyancy in layered clouds such as stratocumulus and deep convective anvils. However, higher-order moments and higher-order cloud statistics were only properly represented when three or four Gaussians were used in the upper regions of shallow cumulus layers and throughout the active portion of deep convection. Evidence is also provided that several common assumptions employed to diagnose trivariate double-Gaussian PDFs from a minimum number of input moments are weak.

## Abstract

Assumed-PDF methods for the parameterization of subgrid-scale processes in atmospheric models provide many benefits. Many currently used assumed-PDF schemes reconcile the high number of required PDF parameters with the relative paucity of input moments by employing simplifying assumptions that are difficult to test. This paper explores the possibility of constructing a trivariate double-Gaussian PDF from the first three orders of moments without simplifying assumptions and proves that no unique solution exists. In an effort to provide a path for future improvement of current assumed-PDF schemes, the expectation maximization (EM) algorithm for Gaussian mixture models is used with LES output of shallow cumulus, stratocumulus, and deep convection cases to determine “best fit” PDFs using from one through four Gaussian clusters. The EM PDFs are evaluated using PDF-diagnosed higher-order moments, PDF-diagnosed cloud statistics, and the Akaike information criterion. It was found that two Gaussian clusters were almost always adequate to represent both higher-order moments and cloud statistics like cloud fraction, water content, and vertical fluxes of cloud water and buoyancy in layered clouds such as stratocumulus and deep convective anvils. However, higher-order moments and higher-order cloud statistics were only properly represented when three or four Gaussians were used in the upper regions of shallow cumulus layers and throughout the active portion of deep convection. Evidence is also provided that several common assumptions employed to diagnose trivariate double-Gaussian PDFs from a minimum number of input moments are weak.

## Abstract

A model that employs a new form of mass-flux closure (described in Part I of this paper) is applied to a variety of clear and cloudy planetary boundary layers (PBLs) including dry convection from the Wangara Experiment, trade wind cumulus from the Barbados Oceanographic and Meteorological Experiment (BOMEX), and marine stratocumulus from the Atlantic Stratocumulus Experiment (ASTEX). For Wangara, the simulated variances and fluxes match that expected from similarity arguments, while the mean state is a little less mixed than the observations. In the BOMEX simulation, the shape and magnitude of the fluxes and the turbulence kinetic energy budget agree with LES results and observations. However, the liquid water mixing ratio is too large. This is attributed to an underprediction of the skewness. In agreement with observations from the ASTEX experiment, many of the model-simulated fields distinctly reflect a regime in transition between the trade wind cumulus and the classic stratocumulus-topped boundary layers.

In general, the simulated entrainment rate tends to be a little underpredicted in regimes where there is little cloud-top radiative cooling (Wangara and BOMEX), while it is overpredicted in regimes where this process is more critical (e.g., ASTEX). Prior work suggests that this may be related to the manner in which the pressure terms are parameterized in the model. Overall, the model is able to capture some key physical features of these PBL regimes, and appears to have the potential to represent both cloud and boundary layer processes. Thus, this approach is a first step toward unifying these processes in large-scale models.

## Abstract

A model that employs a new form of mass-flux closure (described in Part I of this paper) is applied to a variety of clear and cloudy planetary boundary layers (PBLs) including dry convection from the Wangara Experiment, trade wind cumulus from the Barbados Oceanographic and Meteorological Experiment (BOMEX), and marine stratocumulus from the Atlantic Stratocumulus Experiment (ASTEX). For Wangara, the simulated variances and fluxes match that expected from similarity arguments, while the mean state is a little less mixed than the observations. In the BOMEX simulation, the shape and magnitude of the fluxes and the turbulence kinetic energy budget agree with LES results and observations. However, the liquid water mixing ratio is too large. This is attributed to an underprediction of the skewness. In agreement with observations from the ASTEX experiment, many of the model-simulated fields distinctly reflect a regime in transition between the trade wind cumulus and the classic stratocumulus-topped boundary layers.

In general, the simulated entrainment rate tends to be a little underpredicted in regimes where there is little cloud-top radiative cooling (Wangara and BOMEX), while it is overpredicted in regimes where this process is more critical (e.g., ASTEX). Prior work suggests that this may be related to the manner in which the pressure terms are parameterized in the model. Overall, the model is able to capture some key physical features of these PBL regimes, and appears to have the potential to represent both cloud and boundary layer processes. Thus, this approach is a first step toward unifying these processes in large-scale models.

## Abstract

In 2001, the authors presented a higher-order mass-flux model called assumed distributions with higher-order closure (ADHOC), which represents the large eddies of the planetary boundary layer (PBL) in terms of an assumed joint distribution of the vertical velocity and scalars such as potential temperature or water vapor mixing ratio. ADHOC is intended for application as a PBL parameterization. It uses the equations of higher-order closure to predict selected moments of the assumed distribution, and diagnoses the parameters of the distribution from the predicted moments. Once the parameters of the distribution are known, all moments of interest can be computed.

The first version of ADHOC was incomplete in that the horizontal momentum equations, the vertical fluxes of horizontal momentum, the contributions to the turbulence kinetic energy from the horizontal wind, and the various pressure terms involving covariances between pressure and other variables were not incorporated into the assumed distribution framework. Instead, these were parameterized using standard methods.

This paper describes an updated version of ADHOC. The new version includes representations of the horizontal winds and momentum fluxes that are consistent with the mass-flux framework of the model. The assumed joint probability distribution is replaced by an assumed joint spatial distribution based on an idealized coherent structure, such as a plume or roll. The horizontal velocity can then be determined using the continuity equation, and the momentum fluxes and variances are computed directly by spatial integration. These expressions contain unknowns that involve the parameters of the assumed coherent structures. Methods are presented to determine these parameters, which include the radius of convective updrafts and downdrafts and the wavelength, tilt, and orientation angle of the convective rolls. The parameterization is tested by comparison with statistics computed from large-eddy simulations. In a companion paper, the results of this paper are built on to determine the perturbation pressure terms needed by the model.

## Abstract

In 2001, the authors presented a higher-order mass-flux model called assumed distributions with higher-order closure (ADHOC), which represents the large eddies of the planetary boundary layer (PBL) in terms of an assumed joint distribution of the vertical velocity and scalars such as potential temperature or water vapor mixing ratio. ADHOC is intended for application as a PBL parameterization. It uses the equations of higher-order closure to predict selected moments of the assumed distribution, and diagnoses the parameters of the distribution from the predicted moments. Once the parameters of the distribution are known, all moments of interest can be computed.

The first version of ADHOC was incomplete in that the horizontal momentum equations, the vertical fluxes of horizontal momentum, the contributions to the turbulence kinetic energy from the horizontal wind, and the various pressure terms involving covariances between pressure and other variables were not incorporated into the assumed distribution framework. Instead, these were parameterized using standard methods.

This paper describes an updated version of ADHOC. The new version includes representations of the horizontal winds and momentum fluxes that are consistent with the mass-flux framework of the model. The assumed joint probability distribution is replaced by an assumed joint spatial distribution based on an idealized coherent structure, such as a plume or roll. The horizontal velocity can then be determined using the continuity equation, and the momentum fluxes and variances are computed directly by spatial integration. These expressions contain unknowns that involve the parameters of the assumed coherent structures. Methods are presented to determine these parameters, which include the radius of convective updrafts and downdrafts and the wavelength, tilt, and orientation angle of the convective rolls. The parameterization is tested by comparison with statistics computed from large-eddy simulations. In a companion paper, the results of this paper are built on to determine the perturbation pressure terms needed by the model.

## Abstract

Result obtained with a mixed layer model are used to study the dynamics of stratomulus formation and dissipation in subtropical marine stratocumulus cloud regimes. The model used allows entrainment to be driven by shear as well as buoyancy, and includes a very crude parameterization of the partial blackness of thin cloud layers. Model results show that for some values of the large-scale divergence there are three equilibrium mixed layer structures, two of which are stable. One of the stable equilibria is cloudy, deep, and buoyancy-driven, while the other is clear, shallow, and shear-driven. It is found that as a result of hysteresis effects a transient increase in the large-scale divergence can produce a long-lasting break in the clouds.

## Abstract

Result obtained with a mixed layer model are used to study the dynamics of stratomulus formation and dissipation in subtropical marine stratocumulus cloud regimes. The model used allows entrainment to be driven by shear as well as buoyancy, and includes a very crude parameterization of the partial blackness of thin cloud layers. Model results show that for some values of the large-scale divergence there are three equilibrium mixed layer structures, two of which are stable. One of the stable equilibria is cloudy, deep, and buoyancy-driven, while the other is clear, shallow, and shear-driven. It is found that as a result of hysteresis effects a transient increase in the large-scale divergence can produce a long-lasting break in the clouds.

## Abstract

Statistical-equilibrium (SE) states of radiative–convective systems in tropical oceanic conditions are simulated with a cloud ensemble model (CEM) in this study. Typical large-scale conditions from the Marshall Islands and the eastern tropical Atlantic regions are used to drive the CEM.

The simulated SE precipitable water, column temperature, and relative humidity are only slightly higher than those of the observed mean states in both regions when time-invariant large-scale total advective cooling and moistening effects are imposed from observations. They are much higher than the observed if time-invariant observed large-scale ascent is imposed for the Marshall Islands region (i.e., ignoring horizontal advective effects). Compared with results from two similar studies, this SE state is somewhere between the cold/dry regime by Sui et al. and the warm/humid regime by Grabowski et al. Temporal variations of the imposed large-scale vertical motion that allows for subsidence make the SE state colder and drier. It remains about the same, however, if the magnitude of the imposed large-scale vertical motion is halved. The SE state is also colder and drier if solar radiation is absent. In general, all the SE states show that wet columns are thermally more stable (unstable) and dry columns are thermally more unstable (stable) in the lower (upper) troposphere.

Column budget analyses are performed to explore the differences among the simulations performed in this study and among the different studies.

## Abstract

Statistical-equilibrium (SE) states of radiative–convective systems in tropical oceanic conditions are simulated with a cloud ensemble model (CEM) in this study. Typical large-scale conditions from the Marshall Islands and the eastern tropical Atlantic regions are used to drive the CEM.

The simulated SE precipitable water, column temperature, and relative humidity are only slightly higher than those of the observed mean states in both regions when time-invariant large-scale total advective cooling and moistening effects are imposed from observations. They are much higher than the observed if time-invariant observed large-scale ascent is imposed for the Marshall Islands region (i.e., ignoring horizontal advective effects). Compared with results from two similar studies, this SE state is somewhere between the cold/dry regime by Sui et al. and the warm/humid regime by Grabowski et al. Temporal variations of the imposed large-scale vertical motion that allows for subsidence make the SE state colder and drier. It remains about the same, however, if the magnitude of the imposed large-scale vertical motion is halved. The SE state is also colder and drier if solar radiation is absent. In general, all the SE states show that wet columns are thermally more stable (unstable) and dry columns are thermally more unstable (stable) in the lower (upper) troposphere.

Column budget analyses are performed to explore the differences among the simulations performed in this study and among the different studies.

## Abstract

The influence of large-scale advective cooling and/or moistening on the quasi-equilibrium behavior of simulated, tropical oceanic cumulus ensembles is examined in this study. Two sensitivity simulations are performed by imposing time varying/invariant large-scale advective cooling effects and time invariant/varying large-scale advective moistening effects. The results are compared with a control simulation performed with both large-scale advective cooling and moistening effects that are time varying.

It is found that the generalized convective available potential energy (GCAPE) tendency is almost one order of magnitude smaller than the GCAPE production in all simulations. This indicates that the quasi-equilibrium assumption of Arakawa and Schubert is well justified. The higher-order behavior of quasi-equilibrium cumulus ensemble is then examined. It is found that the GCAPE variations are nearly equally contributed by temperature and water vapor variations in the control simulation. In the sensitivity simulations, they are mainly contributed by the temperature (water vapor) variations even though the imposed large-scale advective cooling (moistening) is time invariant. A significant finding of this study is that there is a negative lag correlation between GCAPE and the intensity of cumulus convection. The lag corresponding to the largest negative correlation ranges from 1 to 5 h in various simulations. The existence of a negative correlation and the maximum lag of a few hours is independent of the character and period of the imposed large-scale advective forcing. The maximum lag can be interpreted as the adjustment timescale from disequilibrium to quasi-equilibrium states in the presence of time-varying large-scale forcing.

## Abstract

The influence of large-scale advective cooling and/or moistening on the quasi-equilibrium behavior of simulated, tropical oceanic cumulus ensembles is examined in this study. Two sensitivity simulations are performed by imposing time varying/invariant large-scale advective cooling effects and time invariant/varying large-scale advective moistening effects. The results are compared with a control simulation performed with both large-scale advective cooling and moistening effects that are time varying.

It is found that the generalized convective available potential energy (GCAPE) tendency is almost one order of magnitude smaller than the GCAPE production in all simulations. This indicates that the quasi-equilibrium assumption of Arakawa and Schubert is well justified. The higher-order behavior of quasi-equilibrium cumulus ensemble is then examined. It is found that the GCAPE variations are nearly equally contributed by temperature and water vapor variations in the control simulation. In the sensitivity simulations, they are mainly contributed by the temperature (water vapor) variations even though the imposed large-scale advective cooling (moistening) is time invariant. A significant finding of this study is that there is a negative lag correlation between GCAPE and the intensity of cumulus convection. The lag corresponding to the largest negative correlation ranges from 1 to 5 h in various simulations. The existence of a negative correlation and the maximum lag of a few hours is independent of the character and period of the imposed large-scale advective forcing. The maximum lag can be interpreted as the adjustment timescale from disequilibrium to quasi-equilibrium states in the presence of time-varying large-scale forcing.

## Abstract

Observations show that cumulus clouds often occur in long-lived mesoscale groups, or clumps. Five possible explanations of clumping are surveyed. The “mutual protection hypothesis,” that clumps occur because cumulus clouds create and maintain, in their near environments, relatively favorable conditions for the development of succeeding clouds, is examined at length. This idea is tested through the use of a simple time-dependent model in which clouds, triggered at randomly selected locations, tend to stabilize their environment in the face of a prescribed constant forcing. Results show that clumping occurs when the cloud-induced stabilization rate is strongest at an intermediate distance from a cloud, and that it does not occur when the stabilization rate decreases monotonically away from a cloud.

## Abstract

Observations show that cumulus clouds often occur in long-lived mesoscale groups, or clumps. Five possible explanations of clumping are surveyed. The “mutual protection hypothesis,” that clumps occur because cumulus clouds create and maintain, in their near environments, relatively favorable conditions for the development of succeeding clouds, is examined at length. This idea is tested through the use of a simple time-dependent model in which clouds, triggered at randomly selected locations, tend to stabilize their environment in the face of a prescribed constant forcing. Results show that clumping occurs when the cloud-induced stabilization rate is strongest at an intermediate distance from a cloud, and that it does not occur when the stabilization rate decreases monotonically away from a cloud.