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Abstract
A closed form solution for the collection growth equation as used in bulk microphysical parameterizations is derived. Although the general form is mathematically complex, it can serve as a benchmark for testing a variety of approximations. Two special cases that can immediately be implemented in existing cloud models are also presented. This solution is used to evaluate two commonly used approximations. The effect of the selection of different basis functions is also investigated.
Abstract
A closed form solution for the collection growth equation as used in bulk microphysical parameterizations is derived. Although the general form is mathematically complex, it can serve as a benchmark for testing a variety of approximations. Two special cases that can immediately be implemented in existing cloud models are also presented. This solution is used to evaluate two commonly used approximations. The effect of the selection of different basis functions is also investigated.
Abstract
The transient linear response of a quiescent, two-dimensional, nonrotating atmosphere to prescribed heat sources and sinks is investigated. Analytical solutions of the hydrostatic Boussinesq equations are obtained for a rigid lid and for a semi-infinite region. For the rigid lid solution, vertically trapped gravity waves propagate away from the source with a speed that depends on the Brunt–Väisälä frequency and the vertical wavenumber of the heating. The amplitude of the disturbance field in the region of the forcing approaches a constant value. Two modes are of particular interest: 1) a deep fast-moving mode which is responsible for subsidence warming through the depth of the troposphere; 2) a slower moving mode which corresponds to midlevel inflow and lower- and upper-level outflows. A solution is also obtained for a semi-infinite region. Although gravity wave energy can now propagate upward, the structure of the low-level fields still shows many similarities with the rigid lid solution.
An analytical solution is also obtained for the rigid lid case for a pulse forcing function. This solution shows that when the heating is turned off the disturbance separates into two parts moving in opposite directions. The structure of these propagating disturbances is similar to gravity waves produced in two-dimensional numerical simulations of Florida convection. A term analysis is presented that confirms the predominantly linear character of the numerically simulated gravity waves.
Abstract
The transient linear response of a quiescent, two-dimensional, nonrotating atmosphere to prescribed heat sources and sinks is investigated. Analytical solutions of the hydrostatic Boussinesq equations are obtained for a rigid lid and for a semi-infinite region. For the rigid lid solution, vertically trapped gravity waves propagate away from the source with a speed that depends on the Brunt–Väisälä frequency and the vertical wavenumber of the heating. The amplitude of the disturbance field in the region of the forcing approaches a constant value. Two modes are of particular interest: 1) a deep fast-moving mode which is responsible for subsidence warming through the depth of the troposphere; 2) a slower moving mode which corresponds to midlevel inflow and lower- and upper-level outflows. A solution is also obtained for a semi-infinite region. Although gravity wave energy can now propagate upward, the structure of the low-level fields still shows many similarities with the rigid lid solution.
An analytical solution is also obtained for the rigid lid case for a pulse forcing function. This solution shows that when the heating is turned off the disturbance separates into two parts moving in opposite directions. The structure of these propagating disturbances is similar to gravity waves produced in two-dimensional numerical simulations of Florida convection. A term analysis is presented that confirms the predominantly linear character of the numerically simulated gravity waves.
Abstract
Large-eddy simulations that incorporate a size-resolving representation of cloud water are used to study the effect of heavy drizzle on PBL structure. Simulated surface precipitation rates average about 1 mm day−1. Heavily drizzling simulations are compared to nondrizzling simulations under two nocturnal PBL regimes—one primarily driven by buoyancy and the other driven equally by buoyancy and shear. Drizzle implies a net latent heating in the cloud that leads to sharp reductions in both entrainment and the production of turbulent kinetic energy by buoyancy (particularly in downdrafts). Drizzle, which evaporates below cloud base, promotes a cooler and moister subcloud layer that further inhibits deep mixing. The cooling and moistening is in quantitative agreement with some observations and is shown to favor the formation of cumuli rising out of the subcloud layer. The cumuli, which are local in space and time, are responsible for most of the heat and moisture transport. They also appear to generate a larger-scale circulation that differs dramatically from the regularity typically found in nonprecipitating stratocumulus. Time-averaged turbulent fluxes of heat and moisture increase in the presence of precipitation, suggesting that drizzle (and drizzle-induced stratification) should not necessarily be taken as a sign of decoupling. Because drizzle primarily affects the vertical distribution of buoyancy, shear production of turbulent kinetic energy mitigates some of the effects described above. Based on large-eddy simulation the authors hypothesize that shallow, well-mixed, radiatively driven stratocumulus cannot persist in the presence of heavy drizzle. In accord with some simpler models, the simulated case with heavy precipitation promotes a reduction in both liquid-water path and entrainment. However, the simulations suggest that time-integrated cloud fraction may increase as a result of drizzle because thinner precipitating clouds may persist longer if the boundary layer does not deepen as rapidly. These somewhat more complicated dynamics have important implications for a number of hypotheses suggesting that changes in aerosol concentrations, when metabolized by stratocumulus, have a significant effect on climate.
Abstract
Large-eddy simulations that incorporate a size-resolving representation of cloud water are used to study the effect of heavy drizzle on PBL structure. Simulated surface precipitation rates average about 1 mm day−1. Heavily drizzling simulations are compared to nondrizzling simulations under two nocturnal PBL regimes—one primarily driven by buoyancy and the other driven equally by buoyancy and shear. Drizzle implies a net latent heating in the cloud that leads to sharp reductions in both entrainment and the production of turbulent kinetic energy by buoyancy (particularly in downdrafts). Drizzle, which evaporates below cloud base, promotes a cooler and moister subcloud layer that further inhibits deep mixing. The cooling and moistening is in quantitative agreement with some observations and is shown to favor the formation of cumuli rising out of the subcloud layer. The cumuli, which are local in space and time, are responsible for most of the heat and moisture transport. They also appear to generate a larger-scale circulation that differs dramatically from the regularity typically found in nonprecipitating stratocumulus. Time-averaged turbulent fluxes of heat and moisture increase in the presence of precipitation, suggesting that drizzle (and drizzle-induced stratification) should not necessarily be taken as a sign of decoupling. Because drizzle primarily affects the vertical distribution of buoyancy, shear production of turbulent kinetic energy mitigates some of the effects described above. Based on large-eddy simulation the authors hypothesize that shallow, well-mixed, radiatively driven stratocumulus cannot persist in the presence of heavy drizzle. In accord with some simpler models, the simulated case with heavy precipitation promotes a reduction in both liquid-water path and entrainment. However, the simulations suggest that time-integrated cloud fraction may increase as a result of drizzle because thinner precipitating clouds may persist longer if the boundary layer does not deepen as rapidly. These somewhat more complicated dynamics have important implications for a number of hypotheses suggesting that changes in aerosol concentrations, when metabolized by stratocumulus, have a significant effect on climate.
Abstract
The impact of solar heating and infrared cooling on the growth of a population of drops is studied with two numerical modeling frameworks. An eddy-resolving model (ERM) simulation of Arctic stratus clouds is used to generate a dataset of 500 parcel trajectories that follow the mean dynamic motions of the simulated cloud. The 500-parcel dataset is used to drive a trajectory ensemble model (TEM) coupled to an explicit microphysical model that includes the radiative term in the vapor growth equation. The second framework is that of the ERM itself.
Results from the TEM show that the production of drizzle-sized drops is strongly dependent upon parcel cloud-top residence time for both radiative- and nonradiative-influenced growth. Drizzle-sized drops can be produced between 20 and 50 min earlier through the inclusion of the radiative term, which corroborates earlier results. The radiative effect may also cause drops with r < 10 μm to evaporate, producing a bimodal size spectrum. Parcel cloud-top residence times as short as 12 min can initiate this bimodal spectrum. TEM results show that the radiative effect increases drizzle drop mass predominately in parcels that tend to contribute to drizzle even in the absence of the radiative term. Activation of large cloud condensation nuclei appears to have a larger effect on drizzle production than does the radiative term. ERM simulations show a weak overall influence of the radiative term. Drizzle onset occurs earlier when the radiative term is included (about 20 min), but there is no strong change in the overall structure or evolution of the cloud.
Abstract
The impact of solar heating and infrared cooling on the growth of a population of drops is studied with two numerical modeling frameworks. An eddy-resolving model (ERM) simulation of Arctic stratus clouds is used to generate a dataset of 500 parcel trajectories that follow the mean dynamic motions of the simulated cloud. The 500-parcel dataset is used to drive a trajectory ensemble model (TEM) coupled to an explicit microphysical model that includes the radiative term in the vapor growth equation. The second framework is that of the ERM itself.
Results from the TEM show that the production of drizzle-sized drops is strongly dependent upon parcel cloud-top residence time for both radiative- and nonradiative-influenced growth. Drizzle-sized drops can be produced between 20 and 50 min earlier through the inclusion of the radiative term, which corroborates earlier results. The radiative effect may also cause drops with r < 10 μm to evaporate, producing a bimodal size spectrum. Parcel cloud-top residence times as short as 12 min can initiate this bimodal spectrum. TEM results show that the radiative effect increases drizzle drop mass predominately in parcels that tend to contribute to drizzle even in the absence of the radiative term. Activation of large cloud condensation nuclei appears to have a larger effect on drizzle production than does the radiative term. ERM simulations show a weak overall influence of the radiative term. Drizzle onset occurs earlier when the radiative term is included (about 20 min), but there is no strong change in the overall structure or evolution of the cloud.
Abstract
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Abstract
Two dimensional experiments are carried out to determine the effect of various wind and thermodynamic structure on squall line characteristics. Two ideas concerning the effect of shear are found useful in explaining many of the outcomes of the numerical experiments. First, in two dimensions, shear in the absence of vorticity sources and sinks is detrimental to convection (Kuo, Asai). Second, there is a specific value of low-level slim interacting with a cold pool which produces deep uplift and hence strong forcing of convection (Rotunno et al.). Results suggest that moist midlevel air tends to be favorable for squall lines. Increasing the total buoyancy or altering the distribution of buoyancy with height, such that it is increased at low levels, produces stronger systems with updrafts more tilted from the vertical.
The formula by von Kárm´n for the speed of a gravity current gives qualitative agreement with the speed of most of the simulated systems. However, at least two additional factors need to be considered to accurately determine propagation speed. First, the wind speed ahead of the system can be modified from environmental values. Second, the propagation speed depends on the surface pressure jump across the gust front and this is not always accurately determined by the temperature anomaly in the shallow cold pool. A diagnosis of the contributions to the surface pressure jump shows that the warming external to the cold pool and waterloading can be significant. It is found that for vertically oriented or downshear tilted updrafts, the positive contribution to the pressure jump due to waterloading can sometimes exceed the negative contribution due to warming. This results in the system moving faster than predicted by the gravity current model whereas, for an upshear tilted updraft, the effect of warming can outweigh waterloading, causing the system to move slower.
An examination of the vorticity balance that occurs for a case having a vertically oriented updraft suggests that it cannot be regarded as purely one between the environmental shear and the cold pool.
Abstract
Two dimensional experiments are carried out to determine the effect of various wind and thermodynamic structure on squall line characteristics. Two ideas concerning the effect of shear are found useful in explaining many of the outcomes of the numerical experiments. First, in two dimensions, shear in the absence of vorticity sources and sinks is detrimental to convection (Kuo, Asai). Second, there is a specific value of low-level slim interacting with a cold pool which produces deep uplift and hence strong forcing of convection (Rotunno et al.). Results suggest that moist midlevel air tends to be favorable for squall lines. Increasing the total buoyancy or altering the distribution of buoyancy with height, such that it is increased at low levels, produces stronger systems with updrafts more tilted from the vertical.
The formula by von Kárm´n for the speed of a gravity current gives qualitative agreement with the speed of most of the simulated systems. However, at least two additional factors need to be considered to accurately determine propagation speed. First, the wind speed ahead of the system can be modified from environmental values. Second, the propagation speed depends on the surface pressure jump across the gust front and this is not always accurately determined by the temperature anomaly in the shallow cold pool. A diagnosis of the contributions to the surface pressure jump shows that the warming external to the cold pool and waterloading can be significant. It is found that for vertically oriented or downshear tilted updrafts, the positive contribution to the pressure jump due to waterloading can sometimes exceed the negative contribution due to warming. This results in the system moving faster than predicted by the gravity current model whereas, for an upshear tilted updraft, the effect of warming can outweigh waterloading, causing the system to move slower.
An examination of the vorticity balance that occurs for a case having a vertically oriented updraft suggests that it cannot be regarded as purely one between the environmental shear and the cold pool.
Abstract
The joint probability density function (PDF) of vertical velocity and conserved scalars is important for at least two reasons. First, the shape of the joint PDF determines the buoyancy flux in partly cloudy layers. Second, the PDF provides a wealth of information about subgrid variability and hence can serve as the foundation of a boundary layer cloud and turbulence parameterization.
This paper analyzes PDFs of stratocumulus, cumulus, and clear boundary layers obtained from both aircraft observations and large eddy simulations. The data are used to fit five families of PDFs: a double delta function, a single Gaussian, and three PDF families based on the sum of two Gaussians.
Overall, the double Gaussian, that is binormal, PDFs perform better than the single Gaussian or double delta function PDFs. In cumulus layers with low cloud fraction, the improvement occurs because typical PDFs are highly skewed, and it is crucial to accurately represent the tail of the distribution, which is where cloud occurs. Since the double delta function has been shown in prior work to be the PDF underlying mass-flux schemes, the data analysis herein hints that mass-flux simulations may be improved upon by using a parameterization built upon a more realistic PDF.
Abstract
The joint probability density function (PDF) of vertical velocity and conserved scalars is important for at least two reasons. First, the shape of the joint PDF determines the buoyancy flux in partly cloudy layers. Second, the PDF provides a wealth of information about subgrid variability and hence can serve as the foundation of a boundary layer cloud and turbulence parameterization.
This paper analyzes PDFs of stratocumulus, cumulus, and clear boundary layers obtained from both aircraft observations and large eddy simulations. The data are used to fit five families of PDFs: a double delta function, a single Gaussian, and three PDF families based on the sum of two Gaussians.
Overall, the double Gaussian, that is binormal, PDFs perform better than the single Gaussian or double delta function PDFs. In cumulus layers with low cloud fraction, the improvement occurs because typical PDFs are highly skewed, and it is crucial to accurately represent the tail of the distribution, which is where cloud occurs. Since the double delta function has been shown in prior work to be the PDF underlying mass-flux schemes, the data analysis herein hints that mass-flux simulations may be improved upon by using a parameterization built upon a more realistic PDF.
Abstract
A new cloudy boundary layer single-column model is presented. It is designed to be flexible enough to represent a variety of cloudiness regimes—such as cumulus, stratocumulus, and clear regimes—without the need for case-specific adjustments. The methodology behind the model is the so-called assumed probability density function (PDF) method. The parameterization differs from higher-order closure or mass-flux schemes in that it achieves closure by the use of a relatively sophisticated joint PDF of vertical velocity, temperature, and moisture. A family of PDFs is chosen that is flexible enough to represent various cloudiness regimes. A double Gaussian family proposed by previous works is used. Predictive equations for grid box means and a number of higher-order turbulent moments are advanced in time. These moments are in turn used to select a particular member from the family of PDFs, for each time step and grid box. Once a PDF member has been selected, the scheme integrates over the PDF to close higher-order moments, buoyancy terms, and diagnose cloud fraction and liquid water. Since all the diagnosed moments for a given grid box and time step are derived from the same unique joint PDF, they are guaranteed to be consistent with one another. A companion paper presents simulations produced by the single-column model.
Abstract
A new cloudy boundary layer single-column model is presented. It is designed to be flexible enough to represent a variety of cloudiness regimes—such as cumulus, stratocumulus, and clear regimes—without the need for case-specific adjustments. The methodology behind the model is the so-called assumed probability density function (PDF) method. The parameterization differs from higher-order closure or mass-flux schemes in that it achieves closure by the use of a relatively sophisticated joint PDF of vertical velocity, temperature, and moisture. A family of PDFs is chosen that is flexible enough to represent various cloudiness regimes. A double Gaussian family proposed by previous works is used. Predictive equations for grid box means and a number of higher-order turbulent moments are advanced in time. These moments are in turn used to select a particular member from the family of PDFs, for each time step and grid box. Once a PDF member has been selected, the scheme integrates over the PDF to close higher-order moments, buoyancy terms, and diagnose cloud fraction and liquid water. Since all the diagnosed moments for a given grid box and time step are derived from the same unique joint PDF, they are guaranteed to be consistent with one another. A companion paper presents simulations produced by the single-column model.
Abstract
A new single-column model for the cloudy boundary layer, described in a companion paper, is tested for a variety of regimes. To represent the subgrid-scale variability, the model uses a joint probability density function (PDF) of vertical velocity, temperature, and moisture content. Results from four different cases are presented and contrasted with large eddy simulations (LES). The cases include a clear convective layer based on the Wangara experiment, a trade wind cumulus layer from the Barbados Oceanographic and Meteorological Experiment (BOMEX), a case of cumulus clouds over land, and a nocturnal marine stratocumulus boundary layer.
Results from the Wangara experiment show that the model is capable of realistically predicting the diurnal growth of a dry convective layer. Compared to the LES, the layer produced is slightly less well mixed and entrainment is somewhat slower. The cloud cover in the cloudy cases varied widely, ranging from a few percent cloud cover to nearly overcast. In each of the cloudy cases, the parameterization predicted cloud fractions that agree reasonably well with the LES. Typically, cloud fraction values tended to be somewhat smaller in the parameterization, and cloud bases and tops were slightly underestimated. Liquid water content was generally within 40% of the LES-predicted values for a range of values spanning almost two orders of magnitude. This was accomplished without the use of any case-specific adjustments.
Abstract
A new single-column model for the cloudy boundary layer, described in a companion paper, is tested for a variety of regimes. To represent the subgrid-scale variability, the model uses a joint probability density function (PDF) of vertical velocity, temperature, and moisture content. Results from four different cases are presented and contrasted with large eddy simulations (LES). The cases include a clear convective layer based on the Wangara experiment, a trade wind cumulus layer from the Barbados Oceanographic and Meteorological Experiment (BOMEX), a case of cumulus clouds over land, and a nocturnal marine stratocumulus boundary layer.
Results from the Wangara experiment show that the model is capable of realistically predicting the diurnal growth of a dry convective layer. Compared to the LES, the layer produced is slightly less well mixed and entrainment is somewhat slower. The cloud cover in the cloudy cases varied widely, ranging from a few percent cloud cover to nearly overcast. In each of the cloudy cases, the parameterization predicted cloud fractions that agree reasonably well with the LES. Typically, cloud fraction values tended to be somewhat smaller in the parameterization, and cloud bases and tops were slightly underestimated. Liquid water content was generally within 40% of the LES-predicted values for a range of values spanning almost two orders of magnitude. This was accomplished without the use of any case-specific adjustments.
Abstract
An idealized supercell simulation using the Regional Atmospheric Modeling System (RAMS) produced an elongated low-level mesocyclone that subsequently collapsed into a concentrated vortex. Though vorticity continually increased in the mesocyclone due to horizontal convergence, the collapse phase was additionally characterized by rapidly decreasing pressure, closed streamlines, and the creation of a compact vorticity center isolated from the remaining vorticity. It was shown in Part I of this study that the concentration phase was not initiated by an increase in horizontal convergence, suggesting that the proximate cause resided elsewhere.
In this study, the vortex concentration in Part I is examined from a vorticity dynamics perspective. It is shown that concentration occurs when inward radial velocity and vertical vorticity become more spatially correlated in the region surrounding the nascent vortex. It is also emphasized that the anisotropy of the horizontal convergence, which is nearly plane-convergent and of comparable magnitude to the mesocyclonic vorticity, is critical to an understanding of the process. The resultant evolution is intermediate between a state of purely two-dimensional nondivergent dynamics and one in which plane convergence confines vorticity to its axis of dilatation. This intermediate state produces a concentrated vortex more rapidly than either end state. The unsteady nature of the initial vorticity band also serves to increase the circulation and wind speed amplification of the final vortex. It is shown how conceptual models in the fluid dynamics literature can be applied to predicting the time and length scales of tornadic mesocyclone evolution.
Abstract
An idealized supercell simulation using the Regional Atmospheric Modeling System (RAMS) produced an elongated low-level mesocyclone that subsequently collapsed into a concentrated vortex. Though vorticity continually increased in the mesocyclone due to horizontal convergence, the collapse phase was additionally characterized by rapidly decreasing pressure, closed streamlines, and the creation of a compact vorticity center isolated from the remaining vorticity. It was shown in Part I of this study that the concentration phase was not initiated by an increase in horizontal convergence, suggesting that the proximate cause resided elsewhere.
In this study, the vortex concentration in Part I is examined from a vorticity dynamics perspective. It is shown that concentration occurs when inward radial velocity and vertical vorticity become more spatially correlated in the region surrounding the nascent vortex. It is also emphasized that the anisotropy of the horizontal convergence, which is nearly plane-convergent and of comparable magnitude to the mesocyclonic vorticity, is critical to an understanding of the process. The resultant evolution is intermediate between a state of purely two-dimensional nondivergent dynamics and one in which plane convergence confines vorticity to its axis of dilatation. This intermediate state produces a concentrated vortex more rapidly than either end state. The unsteady nature of the initial vorticity band also serves to increase the circulation and wind speed amplification of the final vortex. It is shown how conceptual models in the fluid dynamics literature can be applied to predicting the time and length scales of tornadic mesocyclone evolution.
