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- Author or Editor: RICHARD A. ANTHES x
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Abstract
A method for parameterizing the effects of deep cumulus clouds on the larger scale thermodynamic and moisture fields in numerical models is proposed. Rigorous derivations of the effect of cumulus clouds on their environment are derived for two definitions of the large-scale averaged variables. In the first, the classical Reynolds averaging method is used and the averaged variables vary continuously over the domain. In the second method, which has been popular in the derivation of cumulus parameterization schemes, the averages are defined by dividing an incremental area of the domain (usually the mesh aim) into a region occupied by convection and the remainder of the region which is free of convection. In this method, the large-scale averages assume discrete values over each incremental area. The differences between the large-scale equations that result from these two methods and some possible difficulties that may be encountered when the averaging interval approaches the aim of the convective clouds are discussed.
The process that determine the effect of deep cumulus convection on the larger scale variables are discussed. The vertical distribution of the net heating of the large scale by the cumulus clouds is determined primarily by the vertical distribution of beating on the cloud scale. A secondary effect is the vertical eddy flux of heat by warm updrafts, which shifts the large-scale heating maximum to slightly higher levels than the level of maximum cloud-scale beating. The major effect of convection on the large-scale moisture equation is to dry the lower troposphere and moisten the upper troposphere.
A method for determining the fractional area covered by deep cumulus updrafts is proposed. This method requires large-scale moisture convergence and estimates of the thermodynamic properties of the typical updraft.
The parameterization scheme conserves total energy in the large-scale equations. It requires representative values of temperature and moisture in the deep convection, as well as an estimate of the vertical distribution of cloud-scale heating. Any cloud model that provides these parameters may be used to complete the scheme; here a one-dimensional cloud model is utilized.
Vertical profiles of the net convective heating rate and the convective effects on the large-scale moisture field are computed for three clouds of different radii using a tropical and an extratropical sounding. The vertical partitioning of the net convective heating as determined by this method is compared to the partitioning given by Kuo's scheme.
Abstract
A method for parameterizing the effects of deep cumulus clouds on the larger scale thermodynamic and moisture fields in numerical models is proposed. Rigorous derivations of the effect of cumulus clouds on their environment are derived for two definitions of the large-scale averaged variables. In the first, the classical Reynolds averaging method is used and the averaged variables vary continuously over the domain. In the second method, which has been popular in the derivation of cumulus parameterization schemes, the averages are defined by dividing an incremental area of the domain (usually the mesh aim) into a region occupied by convection and the remainder of the region which is free of convection. In this method, the large-scale averages assume discrete values over each incremental area. The differences between the large-scale equations that result from these two methods and some possible difficulties that may be encountered when the averaging interval approaches the aim of the convective clouds are discussed.
The process that determine the effect of deep cumulus convection on the larger scale variables are discussed. The vertical distribution of the net heating of the large scale by the cumulus clouds is determined primarily by the vertical distribution of beating on the cloud scale. A secondary effect is the vertical eddy flux of heat by warm updrafts, which shifts the large-scale heating maximum to slightly higher levels than the level of maximum cloud-scale beating. The major effect of convection on the large-scale moisture equation is to dry the lower troposphere and moisten the upper troposphere.
A method for determining the fractional area covered by deep cumulus updrafts is proposed. This method requires large-scale moisture convergence and estimates of the thermodynamic properties of the typical updraft.
The parameterization scheme conserves total energy in the large-scale equations. It requires representative values of temperature and moisture in the deep convection, as well as an estimate of the vertical distribution of cloud-scale heating. Any cloud model that provides these parameters may be used to complete the scheme; here a one-dimensional cloud model is utilized.
Vertical profiles of the net convective heating rate and the convective effects on the large-scale moisture field are computed for three clouds of different radii using a tropical and an extratropical sounding. The vertical partitioning of the net convective heating as determined by this method is compared to the partitioning given by Kuo's scheme.
Abstract
A cumulus parameterization scheme that utilizes a one-dimensional cloud model is tested in a revised, axisymmetric hurricane model. The details of how the parameterization scheme may he incorporated into a larger wale model are presented. With a mean tropical sounding, the cumulus parameterization scheme yields a vertical distribution of heating that is appropriate for tropical cyclone development. The structure of the model hurricane and the properties of the convective clouds at various stages of the model storm development are described. The vertical distribution of the cloud-scale heating and the vertical eddy fluxes of heat, moisture and momentum are given as a function of radius from the storm center during the mature stage. The vertical fluxes of heat and moisture cool and dry the lower troposphere while they warm and moisten the upper troposphere. The vertical transport of radial momentum by the cumulus convection is relatively unimportant, however, the transport of tangential momentum is significant in reducing the vertical shear of the tangential wind.
Abstract
A cumulus parameterization scheme that utilizes a one-dimensional cloud model is tested in a revised, axisymmetric hurricane model. The details of how the parameterization scheme may he incorporated into a larger wale model are presented. With a mean tropical sounding, the cumulus parameterization scheme yields a vertical distribution of heating that is appropriate for tropical cyclone development. The structure of the model hurricane and the properties of the convective clouds at various stages of the model storm development are described. The vertical distribution of the cloud-scale heating and the vertical eddy fluxes of heat, moisture and momentum are given as a function of radius from the storm center during the mature stage. The vertical fluxes of heat and moisture cool and dry the lower troposphere while they warm and moisten the upper troposphere. The vertical transport of radial momentum by the cumulus convection is relatively unimportant, however, the transport of tangential momentum is significant in reducing the vertical shear of the tangential wind.
Abstract
Notable asymmetric features of an early experiment with a three-dimensional hurricane model were spiral bands of convection and large-scale asymmetries (eddies) in the outflow layer. Using an improved version of the model, we describe the formation and maintenance of these features in greater detail in this paper. The spiral bands in the model propagate cyclonically outward in agreement with bands in nature. The breakdown of symmetry into a chaotic pattern of eddies in the outflow region is shown to be the result of dynamic (inertial) instability, with the eddy kinetic energy derived from the kinetic energy of the azimuthal flow. This instability does not contribute to the overall intensification of the model storm, however.
We observe a curious anticyclonic looping of the vortex center in these experiments. This looping appears to be associated with asymmetries in the divergence pattern associated with the eddies in the outflow layer.
This paper also summarizes improvements made in the original version of the model. In contrast to the earlier model, the current version contains an explicit water vapor cycle. A staggered horizontal grid is used to provide a higher resolution in evaluating the pressure gradient forces. Some of the pragmatic assumptions made in the earlier model, notably those involving horizontal diffusion of heat and momentum, have been eliminated in the current version.
Abstract
Notable asymmetric features of an early experiment with a three-dimensional hurricane model were spiral bands of convection and large-scale asymmetries (eddies) in the outflow layer. Using an improved version of the model, we describe the formation and maintenance of these features in greater detail in this paper. The spiral bands in the model propagate cyclonically outward in agreement with bands in nature. The breakdown of symmetry into a chaotic pattern of eddies in the outflow region is shown to be the result of dynamic (inertial) instability, with the eddy kinetic energy derived from the kinetic energy of the azimuthal flow. This instability does not contribute to the overall intensification of the model storm, however.
We observe a curious anticyclonic looping of the vortex center in these experiments. This looping appears to be associated with asymmetries in the divergence pattern associated with the eddies in the outflow layer.
This paper also summarizes improvements made in the original version of the model. In contrast to the earlier model, the current version contains an explicit water vapor cycle. A staggered horizontal grid is used to provide a higher resolution in evaluating the pressure gradient forces. Some of the pragmatic assumptions made in the earlier model, notably those involving horizontal diffusion of heat and momentum, have been eliminated in the current version.
Abstract
Steady-state solutions to the complete axisymmetric Navier-Stokes equations are obtained by an iterative technique for an intense pressure-gradient force representative of the tropical cyclone. Solutions for the horizontal and vertical components of motion are compared for various horizontal and vertical mixing coefficients, drag coefficients, and Coriolis parameters. A multilevel model is used, and the results are compared with those from a simple one-level model.
Abstract
Steady-state solutions to the complete axisymmetric Navier-Stokes equations are obtained by an iterative technique for an intense pressure-gradient force representative of the tropical cyclone. Solutions for the horizontal and vertical components of motion are compared for various horizontal and vertical mixing coefficients, drag coefficients, and Coriolis parameters. A multilevel model is used, and the results are compared with those from a simple one-level model.
Abstract
The role of asymmetries (large-scale horizontal eddies) in satisfying the mean angular momentum budget for the steady-state hurricane is studied by computing transverse circulations for a prescribed tangential vortex on the scale of 1000 km. For realistic diabatic heating rates at large distances from the hurricane center, the correlation between radial velocity and absolute vorticity must be negative in the upper troposphere and positive in the lower troposphere.
The transverse circulations show small-scale oscillations that increase as internal mixing is increased. This paradox results from balancing “noise” in the prescribed tangential wind profiles by oscillations in the radial and vertical advection terms.
Abstract
The role of asymmetries (large-scale horizontal eddies) in satisfying the mean angular momentum budget for the steady-state hurricane is studied by computing transverse circulations for a prescribed tangential vortex on the scale of 1000 km. For realistic diabatic heating rates at large distances from the hurricane center, the correlation between radial velocity and absolute vorticity must be negative in the upper troposphere and positive in the lower troposphere.
The transverse circulations show small-scale oscillations that increase as internal mixing is increased. This paradox results from balancing “noise” in the prescribed tangential wind profiles by oscillations in the radial and vertical advection terms.
Abstract
A two-dimensional horizontal variable grid is derived that has maximum resolution at the center and minimum resolution near the boundaries of the grid. By using the analytic transformation that defines the variable grid, the equations of motion for a free-surface model are transformed in terms of new independent space variables in a computational domain with constant resolution. Numerical experiments utilize the variable grid to (1) increase the domain size with a fixed resolution at the center and (2) increase the resolution at the center with a fixed domain size.
Several finite-difference analogs and three time-integration schemes are tested. For a given domain size and number of grid points, several variable grid experiments show superior results in the mass and momentum fields compared to constant grid results. Most variable grid experiments, however, show a small (less than 1 percent) increase in total energy after 2000 time steps due apparently to the presence of additional nonlinear terms in the forecast equations.
The results show that, although care must be taken with the nonlinear terms, the variable grid may be effectively used in certain physical problems to economically gain resolution at the center of the domain.
Abstract
A two-dimensional horizontal variable grid is derived that has maximum resolution at the center and minimum resolution near the boundaries of the grid. By using the analytic transformation that defines the variable grid, the equations of motion for a free-surface model are transformed in terms of new independent space variables in a computational domain with constant resolution. Numerical experiments utilize the variable grid to (1) increase the domain size with a fixed resolution at the center and (2) increase the resolution at the center with a fixed domain size.
Several finite-difference analogs and three time-integration schemes are tested. For a given domain size and number of grid points, several variable grid experiments show superior results in the mass and momentum fields compared to constant grid results. Most variable grid experiments, however, show a small (less than 1 percent) increase in total energy after 2000 time steps due apparently to the presence of additional nonlinear terms in the forecast equations.
The results show that, although care must be taken with the nonlinear terms, the variable grid may be effectively used in certain physical problems to economically gain resolution at the center of the domain.
Abstract
A diagnostic axisymmetric model in isentropic coordinates is developed to study the effect of differential heating on the dynamics and energetics of the steady-state tropical cyclone. From the thermal forcing specified by various heating distributions, slowly varying solutions for the mass and momentum fields are obtained by an iterative technique.
The theory of available potential energy for open systems is utilized to study the energy budget for the hurricane. In the slowly varying state, the gain of available potential energy by diabatic heating and lateral boundary processes balances the conversion of potential to kinetic energy that, in turn, offsets frictional dissipation. For a domain of radius 500 km, the boundary flux of available potential energy is about 40 percent of the generation by diabatic heating. For a domain of radius 1000 km, however, the boundary flux is about 15 percent of the generation.
Horizontal and vertical mixing are studied through the use of constant exchange coefficients. As the horizontal mixing decreases, the maximum surface wind increases and moves closer to the center.
Several horizontal and two vertical distributions of latent heating are investigated. The maximum surface wind is dependent primarily on heating within 100 km. The transverse (radial) circulation is closely related to the heat release beyond 100 km. In experiments in which the vertical variation of heating is pseudoadiabatic, the temperature and outflow structures are unrealistic. A vertical distribution that releases a higher proportion of heat in the upper troposphere yields results that are more representative of the hurricane.
Abstract
A diagnostic axisymmetric model in isentropic coordinates is developed to study the effect of differential heating on the dynamics and energetics of the steady-state tropical cyclone. From the thermal forcing specified by various heating distributions, slowly varying solutions for the mass and momentum fields are obtained by an iterative technique.
The theory of available potential energy for open systems is utilized to study the energy budget for the hurricane. In the slowly varying state, the gain of available potential energy by diabatic heating and lateral boundary processes balances the conversion of potential to kinetic energy that, in turn, offsets frictional dissipation. For a domain of radius 500 km, the boundary flux of available potential energy is about 40 percent of the generation by diabatic heating. For a domain of radius 1000 km, however, the boundary flux is about 15 percent of the generation.
Horizontal and vertical mixing are studied through the use of constant exchange coefficients. As the horizontal mixing decreases, the maximum surface wind increases and moves closer to the center.
Several horizontal and two vertical distributions of latent heating are investigated. The maximum surface wind is dependent primarily on heating within 100 km. The transverse (radial) circulation is closely related to the heat release beyond 100 km. In experiments in which the vertical variation of heating is pseudoadiabatic, the temperature and outflow structures are unrealistic. A vertical distribution that releases a higher proportion of heat in the upper troposphere yields results that are more representative of the hurricane.
Abstract
Additional experiments with the slowly varying axisymmetric hurricane model described by Anthes are presented. Variable horizontal eddy viscosity coefficients are utilized to study circulations that result from an increased heating function and a heating function with a double maximum along the radial direction. Infrared radiative cooling is modeled for the clear tropical atmosphere following Sasamori. A representative cooling rate is used to study the effect of cooling in the hurricane environment on the dynamics and energetics of the slowly varying tropical cyclone.
Abstract
Additional experiments with the slowly varying axisymmetric hurricane model described by Anthes are presented. Variable horizontal eddy viscosity coefficients are utilized to study circulations that result from an increased heating function and a heating function with a double maximum along the radial direction. Infrared radiative cooling is modeled for the clear tropical atmosphere following Sasamori. A representative cooling rate is used to study the effect of cooling in the hurricane environment on the dynamics and energetics of the slowly varying tropical cyclone.
Abstract
This paper investigates the problem of initializing operational hurricane models with several types of real data. Imbalances in real data generate inertia-gravity waves with periods that vary strongly in different regions of the hurricane domain. The energy of these waves is removed by propagation out of the domain, by the horizontal diffusion process, and by the truncation errors associated with the Matsuno time-differencing scheme. Several initialization schemes are tested with a symmetric hurricane model. Random and bias errors superimposed on perfect data produce imbalances that lend to significant errors in short-range forecasts. A general dynamic initialization scheme that is suitable for diabatic, viscous models yields very promising results.
The dynamic initialization technique is utilized in an effort to determine the types of data that will be most useful in initializing operational hurricane models. In general, observations are most useful near the center of the storm at low levels. Temperature and wind observations are about equally effective in reducing initial analysis errors. Specific humidity observations, on the other hand, seem less important. Finally, the sensitivity of the initialization method is tested with observations that include rather large bias errors.
Abstract
This paper investigates the problem of initializing operational hurricane models with several types of real data. Imbalances in real data generate inertia-gravity waves with periods that vary strongly in different regions of the hurricane domain. The energy of these waves is removed by propagation out of the domain, by the horizontal diffusion process, and by the truncation errors associated with the Matsuno time-differencing scheme. Several initialization schemes are tested with a symmetric hurricane model. Random and bias errors superimposed on perfect data produce imbalances that lend to significant errors in short-range forecasts. A general dynamic initialization scheme that is suitable for diabatic, viscous models yields very promising results.
The dynamic initialization technique is utilized in an effort to determine the types of data that will be most useful in initializing operational hurricane models. In general, observations are most useful near the center of the storm at low levels. Temperature and wind observations are about equally effective in reducing initial analysis errors. Specific humidity observations, on the other hand, seem less important. Finally, the sensitivity of the initialization method is tested with observations that include rather large bias errors.
Abstract
A two-dimensional mesoscale model is applied to study the evolution of a strong sea breeze on a stagnant base state. In contrast to previous studies, this paper considers the relationship of the planetary boundary layer (PBL), the thermodynamic structure and the vertical circulation associated with the sea breeze in detail.
The development of the sea breeze circulation is studied quantitatively using the circulation theorem. The circulation in the vertical plane normal to the coast develops as a result of the solenoid term. The vertical diffusion of momentum acts as the most important brake on the developing circulation in agreement with previous theoretical results. The Coriolis term is small until 6 h after the beating cycle. Late in the cycle, however, it reaches a value of 45% that of the solenoid term. Horizontal and vertical advective effects are small.
Under zero geostrophic wind conditions, the return flow occurs entirely above the PBL. Therefore, neutrally buoyant pollutants emitted at the surface can only enter the return flow through the narrow zone of upward motion at the sea breeze front. Trajectories indicate that considerable recirculation toward the shore of these pollutants as well as pollutants left over in the previous day's mixed layer may occur. For the time and space scale of the sea breeze considered here, Coriolis forces are important in causing significant transports along the coast.
The depth of the circulation and the trajectories are sensitive to the rate of heating over land and the initial static stability. For strong heating in a relatively unstable environment, a significant component to the return circulation exists up to 5 km. For moderate heating in a more stable environment, there is very little return circulation above 3 km.
Abstract
A two-dimensional mesoscale model is applied to study the evolution of a strong sea breeze on a stagnant base state. In contrast to previous studies, this paper considers the relationship of the planetary boundary layer (PBL), the thermodynamic structure and the vertical circulation associated with the sea breeze in detail.
The development of the sea breeze circulation is studied quantitatively using the circulation theorem. The circulation in the vertical plane normal to the coast develops as a result of the solenoid term. The vertical diffusion of momentum acts as the most important brake on the developing circulation in agreement with previous theoretical results. The Coriolis term is small until 6 h after the beating cycle. Late in the cycle, however, it reaches a value of 45% that of the solenoid term. Horizontal and vertical advective effects are small.
Under zero geostrophic wind conditions, the return flow occurs entirely above the PBL. Therefore, neutrally buoyant pollutants emitted at the surface can only enter the return flow through the narrow zone of upward motion at the sea breeze front. Trajectories indicate that considerable recirculation toward the shore of these pollutants as well as pollutants left over in the previous day's mixed layer may occur. For the time and space scale of the sea breeze considered here, Coriolis forces are important in causing significant transports along the coast.
The depth of the circulation and the trajectories are sensitive to the rate of heating over land and the initial static stability. For strong heating in a relatively unstable environment, a significant component to the return circulation exists up to 5 km. For moderate heating in a more stable environment, there is very little return circulation above 3 km.
