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- Author or Editor: Frank B. Lipps x

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## Abstract

The dynamics of thermal convection through a shallow layer with vertical shear is examined using an idealized numerical model. The convection is assumed to take the form of two-dimensional rolls. The mean shear flow and the unstable temperature gradient are maintained by no-slip, conducting boundary conditions applied at the upper and lower boundaries of the model. When the convective rolls are transverse to the mean current, the flow approaches a steady state with time for the cases of primary interest. In agreement with previous numerical studies the shear has a stabilizing influence on the convection: the transformation of potential energy into disturbance kinetic energy is reduced, and disturbance kinetic energy is transformed into basic kinetic energy. A new result, in agreement with linear stability theory, is that shear can significantly increase the horizontal distance between disturbances over that expected with no shear.

Steady-state results were also obtained when the rolls are parallel to the mean current. In this case basic kinetic energy is transformed into disturbance kinetic energy. Results for momentum transfer and heat transfer obtained from the present numerical model are compared with the experimental results of Ingersoll. This comparison suggests that for low values of the Rayleigh number his convection is primarily in the form of rolls parallel to the shear flow. However, for Rayleigh numbers >20,000 the experimental and numerical results start to diverge, suggesting that three-dimensional effects are becoming important in this range.

## Abstract

The dynamics of thermal convection through a shallow layer with vertical shear is examined using an idealized numerical model. The convection is assumed to take the form of two-dimensional rolls. The mean shear flow and the unstable temperature gradient are maintained by no-slip, conducting boundary conditions applied at the upper and lower boundaries of the model. When the convective rolls are transverse to the mean current, the flow approaches a steady state with time for the cases of primary interest. In agreement with previous numerical studies the shear has a stabilizing influence on the convection: the transformation of potential energy into disturbance kinetic energy is reduced, and disturbance kinetic energy is transformed into basic kinetic energy. A new result, in agreement with linear stability theory, is that shear can significantly increase the horizontal distance between disturbances over that expected with no shear.

Steady-state results were also obtained when the rolls are parallel to the mean current. In this case basic kinetic energy is transformed into disturbance kinetic energy. Results for momentum transfer and heat transfer obtained from the present numerical model are compared with the experimental results of Ingersoll. This comparison suggests that for low values of the Rayleigh number his convection is primarily in the form of rolls parallel to the shear flow. However, for Rayleigh numbers >20,000 the experimental and numerical results start to diverge, suggesting that three-dimensional effects are becoming important in this range.

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## Abstract

A brief review of the scale analysis of Lipps and Hemler is given without any reference to the parameters *G* and *B*. The resulting anelastic equations conserve energy, in contrast to the modified anelastic set of equations analyzed by Durran. In addition, the present equations give an accurate solution for the frequency of gravity waves in an isothermal atmosphere. The present anelastic equations have these characteristics in common with the pseudo-incompressible equations introduced by Durran.

The equations obtained from the scale analysis are appropriate for numerical integration of deep convection. The associated Poisson equation can be solved using standard procedures. For the pseudo-incompressible set of equations, the Poisson equation is more difficult to solve.

## Abstract

A brief review of the scale analysis of Lipps and Hemler is given without any reference to the parameters *G* and *B*. The resulting anelastic equations conserve energy, in contrast to the modified anelastic set of equations analyzed by Durran. In addition, the present equations give an accurate solution for the frequency of gravity waves in an isothermal atmosphere. The present anelastic equations have these characteristics in common with the pseudo-incompressible equations introduced by Durran.

The equations obtained from the scale analysis are appropriate for numerical integration of deep convection. The associated Poisson equation can be solved using standard procedures. For the pseudo-incompressible set of equations, the Poisson equation is more difficult to solve.

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## Abstract

A diagnostic second-order turbulence parameterization has been incorporated into a shallow anelastic three-dimensional numerical cloud model. The turbulence closure scheme for the subgrid-scale motions includes the effects of buoyancy, condensation and liquid water drag. This model has been used to study trade wind cumuli which are roughly 1200 m thick. The simulated cloud has many features in common with observed clouds (Malkus, 1954); however, the observed clouds are made up of several thermal elements instead of one as in the numerical simulation, and they persist over a much longer time period.

When comparing the present model with another using deformation eddy viscosity, the following results are obtained: 1) The deformation model has a larger smoothing effect on the horizontally averaged potential temperature and water vapor mixing ratio. 2) Early in the cloud's development, the subgrid-scale kinetic energy is larger than the computed-scale kinetic energy. At the mature stage, the subgrid-scale energy is about one-half to three-quarters the magnitude of the computed-scale kinetic energy. In the deformation model the subgrid-scale turbulence is less, especially in the early stages of the cloud's history. 3) It is found that buoyancy effects can be dropped from the Reynold's stress equation without significant loss of accuracy.

The results of both models are highly sensitive to changes of external parameters. This type of sensitivity is either a characteristic of clouds in general, or is a special property of the present models.

## Abstract

A diagnostic second-order turbulence parameterization has been incorporated into a shallow anelastic three-dimensional numerical cloud model. The turbulence closure scheme for the subgrid-scale motions includes the effects of buoyancy, condensation and liquid water drag. This model has been used to study trade wind cumuli which are roughly 1200 m thick. The simulated cloud has many features in common with observed clouds (Malkus, 1954); however, the observed clouds are made up of several thermal elements instead of one as in the numerical simulation, and they persist over a much longer time period.

When comparing the present model with another using deformation eddy viscosity, the following results are obtained: 1) The deformation model has a larger smoothing effect on the horizontally averaged potential temperature and water vapor mixing ratio. 2) Early in the cloud's development, the subgrid-scale kinetic energy is larger than the computed-scale kinetic energy. At the mature stage, the subgrid-scale energy is about one-half to three-quarters the magnitude of the computed-scale kinetic energy. In the deformation model the subgrid-scale turbulence is less, especially in the early stages of the cloud's history. 3) It is found that buoyancy effects can be dropped from the Reynold's stress equation without significant loss of accuracy.

The results of both models are highly sensitive to changes of external parameters. This type of sensitivity is either a characteristic of clouds in general, or is a special property of the present models.

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## Abstract

The northward momentum transfer across an asymmetric jet in a three-dimensional atmosphere is examined by means of an initial-value problem. The flow is contained between two latitude circles in the horizontal and between the ground and the tropopause in the vertical. The motion is governed by the potential vorticity equation. The initial horizontal flow consists of a disturbance superimposed upon a slightly asymmetric west-cast zonal current. The initial momentum transfer vanishes identically.

A solution is obtained for the tendency of the momentum transfer. It is found that the small asymmetry in the zonal current is responsible for the strong northward momentum transport in the region of the strongest zonal flow. The variation of the momentum transfer both in the horizontal and in the vertical agrees satisfactorily with the observed transfer. It is found that an initial maximum meridional velocity of 10 m sec^{−1} is required in order that after one day the predicted transfer becomes equal to the observed transfer at 200 mb and 3ON.

The transfer of energy between the zonal flow and the disturbance is discussed. The transformation from perturbation kinetic energy to basic-flow kinetic energy has a small positive value and the transformation from basic-flow potential energy to perturbation potential energy has a larger positive value. The disturbance will increase its energy by about 10 per cent after one day.

## Abstract

The northward momentum transfer across an asymmetric jet in a three-dimensional atmosphere is examined by means of an initial-value problem. The flow is contained between two latitude circles in the horizontal and between the ground and the tropopause in the vertical. The motion is governed by the potential vorticity equation. The initial horizontal flow consists of a disturbance superimposed upon a slightly asymmetric west-cast zonal current. The initial momentum transfer vanishes identically.

A solution is obtained for the tendency of the momentum transfer. It is found that the small asymmetry in the zonal current is responsible for the strong northward momentum transport in the region of the strongest zonal flow. The variation of the momentum transfer both in the horizontal and in the vertical agrees satisfactorily with the observed transfer. It is found that an initial maximum meridional velocity of 10 m sec^{−1} is required in order that after one day the predicted transfer becomes equal to the observed transfer at 200 mb and 3ON.

The transfer of energy between the zonal flow and the disturbance is discussed. The transformation from perturbation kinetic energy to basic-flow kinetic energy has a small positive value and the transformation from basic-flow potential energy to perturbation potential energy has a larger positive value. The disturbance will increase its energy by about 10 per cent after one day.

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## Abstract

The stability of a two-layer incompressible fluid system on a rotating earth is investigated. The upper layer has infinite depth and is inert; the lower layer has finite depth and a basic west to east zonal velocity of form sech^{2}
*y*. The linearized potential vorticity equation is used for the stability investigation. It is found that both the *beta* effect due to the curvature of the earth and the divergence tend to stabilize the jet if the winds are from west to cast everywhere. However, if there are easterly winds away from the center of the jet, the divergence may not be stabilizing.

This stability theory is applied to a jet at 45 deg latitude in the atmosphere. The maximum wind is 60 m sec^{1} and the half-width of the jet is 1000 km. For the case of no divergence the most unstable wavelength is 5500 km and this disturbance has an *e*-fold amplification in 1.8 days. If we include divergence, the most unstable wavelength is again 5500 km but the *e*-fold amplification time is 14 days.

The theory can also be applied to the Gulf Stream. For a current with a maximum velocity of 1.5 m sec^{−1}, a half-width of 31 km and a depth of 550 m, the most unstable wavelength is 180 km and the *e*-fold amplification time is 4 days.

## Abstract

The stability of a two-layer incompressible fluid system on a rotating earth is investigated. The upper layer has infinite depth and is inert; the lower layer has finite depth and a basic west to east zonal velocity of form sech^{2}
*y*. The linearized potential vorticity equation is used for the stability investigation. It is found that both the *beta* effect due to the curvature of the earth and the divergence tend to stabilize the jet if the winds are from west to cast everywhere. However, if there are easterly winds away from the center of the jet, the divergence may not be stabilizing.

This stability theory is applied to a jet at 45 deg latitude in the atmosphere. The maximum wind is 60 m sec^{1} and the half-width of the jet is 1000 km. For the case of no divergence the most unstable wavelength is 5500 km and this disturbance has an *e*-fold amplification in 1.8 days. If we include divergence, the most unstable wavelength is again 5500 km but the *e*-fold amplification time is 14 days.

The theory can also be applied to the Gulf Stream. For a current with a maximum velocity of 1.5 m sec^{−1}, a half-width of 31 km and a depth of 550 m, the most unstable wavelength is 180 km and the *e*-fold amplification time is 4 days.

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## Abstract

This paper attempts to determine under what conditions horizontal shear in the mean zonal flow can provide the initial source of energy for the traveling disturbances of low latitudes. A three-zone barotropic model is constructed in order to examine the stability of an idealized mean zonal current. The width and total wind shear associated with this mean current are varied. The form of growing disturbances and their amplification rates are found.

A stability analysis is also carried out for a basic flow which has a hyperbolic tangent variation with latitude. Results obtained by numerical integration for this basic flow are similar to those found previously with the three-zone model. In discussing his easterly wave model, Yanai indicates a basic flow which has a total wind shear of about 8 m sec^{–1} occurring over approximately 6° of latitude. Results obtained for a basic flow with these characteristics show that the fastest growing wave has a wavelength near 2500 km and an *e*-folding time of about 7 days.

## Abstract

This paper attempts to determine under what conditions horizontal shear in the mean zonal flow can provide the initial source of energy for the traveling disturbances of low latitudes. A three-zone barotropic model is constructed in order to examine the stability of an idealized mean zonal current. The width and total wind shear associated with this mean current are varied. The form of growing disturbances and their amplification rates are found.

A stability analysis is also carried out for a basic flow which has a hyperbolic tangent variation with latitude. Results obtained by numerical integration for this basic flow are similar to those found previously with the three-zone model. In discussing his easterly wave model, Yanai indicates a basic flow which has a total wind shear of about 8 m sec^{–1} occurring over approximately 6° of latitude. Results obtained for a basic flow with these characteristics show that the fastest growing wave has a wavelength near 2500 km and an *e*-folding time of about 7 days.

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## Abstract

A three-dimensional numerical model with warm rain bulk cloud physics is used to investigate the shallow convection observed on day 226 of GATE. This convection had cloud tops at 3.0 km, cloud bases at 0.4 km and approximately 0.1 cm of rain at the surface. The simulated convection shows a strong sensitivity to the criterion for the onset of autoconversion of cloud water into rain water. The strongest convection occurs for the case in which no rain water forms. This case, however, does not conform to the observed convection, lacking the downdraft below cloud base and the observed strong surface outflow.

The primary simulation produces a “finger” of convection propagating to the northeast, perpendicular to the northwest–southeast orientation of the larger-scale line of convection. The orientation and propagation speed of the calculated convection are in excellent agreement with observed radar data. This simulation also has a well-defined leading edge and strong surface outflow as observed. In poorer agreement, the cloud base was too high and the rainfall at the surface was less than observed.

Present calculations indicate that the boundary layer air is flowing through the line from southwest to northeast below cloud base. The primary moisture source for the cloud is the upper half of the subcloud layer, with nearly horizontal flow entering the cloud.

## Abstract

A three-dimensional numerical model with warm rain bulk cloud physics is used to investigate the shallow convection observed on day 226 of GATE. This convection had cloud tops at 3.0 km, cloud bases at 0.4 km and approximately 0.1 cm of rain at the surface. The simulated convection shows a strong sensitivity to the criterion for the onset of autoconversion of cloud water into rain water. The strongest convection occurs for the case in which no rain water forms. This case, however, does not conform to the observed convection, lacking the downdraft below cloud base and the observed strong surface outflow.

The primary simulation produces a “finger” of convection propagating to the northeast, perpendicular to the northwest–southeast orientation of the larger-scale line of convection. The orientation and propagation speed of the calculated convection are in excellent agreement with observed radar data. This simulation also has a well-defined leading edge and strong surface outflow as observed. In poorer agreement, the cloud base was too high and the rainfall at the surface was less than observed.

Present calculations indicate that the boundary layer air is flowing through the line from southwest to northeast below cloud base. The primary moisture source for the cloud is the upper half of the subcloud layer, with nearly horizontal flow entering the cloud.

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## Abstract

A set of four-hour simulations has been carried out to study deep moist convection characteristic of the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE). The present model includes warm rain bulk cloud physics and effects associated with a large-scale, time-invariant convergence. The convection took approximately two hours to develop from a random moisture disturbance. The cloud efficiency, in terms of the total water vapor condensed, was near 40%.

The heat and moisture budgets and the time–mean vertical fluxes of mass, heat, and moisture were calculated for the last 80 minutes of the simulations. In this study the primary emphasis was placed upon run A, the three-dimensional calculation. For this calculation, the layer centered near 4.0 km was a region of low mean cloudiness but of strong convection. The upward mass flux was strong and upward heat and moisture fluxes had maximum values in this layer. The strongest downward mass flux was due to weak downward velocities in the rainy area below cloud base.

Time-mean data were also calculated for vertical velocity cores and compared with observed data. In run A, virtually all updraft cores are in-cloud and for a deep layer between 2.5 and 8.0 km the in-cloud up-ward mass flux is nearly all associated with cores. In this layer the upward mass flux due to cores is approximately twice the mass flux associated with the large-scale convergence. The fractional area of updraft cores is small, varying between 2.5% and 4.0% for vertical levels between 1 and 11 km. Calculated values of core diameter D̄ are in relatively good agreement with the observed data. For values of mean vertical velocity ¯ω, however, the agreement is not nearly as good. For downdraft cores, values of ¯ω, are significantly smaller than the observations. For updraft cores, values of ω at lower levels are small, whereas values in the upper levels are in reasonable agreement with observations. The weak updraft cores at lower levels may be related to the absence of strong gust fronts in the present simulations.

## Abstract

A set of four-hour simulations has been carried out to study deep moist convection characteristic of the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE). The present model includes warm rain bulk cloud physics and effects associated with a large-scale, time-invariant convergence. The convection took approximately two hours to develop from a random moisture disturbance. The cloud efficiency, in terms of the total water vapor condensed, was near 40%.

The heat and moisture budgets and the time–mean vertical fluxes of mass, heat, and moisture were calculated for the last 80 minutes of the simulations. In this study the primary emphasis was placed upon run A, the three-dimensional calculation. For this calculation, the layer centered near 4.0 km was a region of low mean cloudiness but of strong convection. The upward mass flux was strong and upward heat and moisture fluxes had maximum values in this layer. The strongest downward mass flux was due to weak downward velocities in the rainy area below cloud base.

Time-mean data were also calculated for vertical velocity cores and compared with observed data. In run A, virtually all updraft cores are in-cloud and for a deep layer between 2.5 and 8.0 km the in-cloud up-ward mass flux is nearly all associated with cores. In this layer the upward mass flux due to cores is approximately twice the mass flux associated with the large-scale convergence. The fractional area of updraft cores is small, varying between 2.5% and 4.0% for vertical levels between 1 and 11 km. Calculated values of core diameter D̄ are in relatively good agreement with the observed data. For values of mean vertical velocity ¯ω, however, the agreement is not nearly as good. For downdraft cores, values of ¯ω, are significantly smaller than the observations. For updraft cores, values of ω at lower levels are small, whereas values in the upper levels are in reasonable agreement with observations. The weak updraft cores at lower levels may be related to the absence of strong gust fronts in the present simulations.

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## Abstract

In this note, a more rational approach is given to specify the parameters *G* and *B* in the scale analysis of Lipps and Hemier. The thermodynamic equation is written in a different form so that a closed expression for *B* can be derived. The present values of *G* and *B* are very similar to those in the previous scale analysis. A new result is that the time scale is expressed in terms of the moist convective instability rather than the inverse of the Brunt-Vsisälä frequency.

The ratio of volume integrated kinetic energy to volume integrated first-order sensible heat is also discussed in more detail. It is found that for an accurate estimate of sensible beat the region of compensating downward motion between the active clouds must be taken into account. As indicated by earlier authors, the amount of sensible heat produced inside the clouds is relatively small.

## Abstract

In this note, a more rational approach is given to specify the parameters *G* and *B* in the scale analysis of Lipps and Hemier. The thermodynamic equation is written in a different form so that a closed expression for *B* can be derived. The present values of *G* and *B* are very similar to those in the previous scale analysis. A new result is that the time scale is expressed in terms of the moist convective instability rather than the inverse of the Brunt-Vsisälä frequency.

The ratio of volume integrated kinetic energy to volume integrated first-order sensible heat is also discussed in more detail. It is found that for an accurate estimate of sensible beat the region of compensating downward motion between the active clouds must be taken into account. As indicated by earlier authors, the amount of sensible heat produced inside the clouds is relatively small.

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## Abstract

A 4-h simulation is carried out for the 22 May 1976 squall line that passed through the mesonetwork of the National Severe Storm Laboratory in central Oklahoma. This squall line was more than 100 km wide, oriented north-south and traveled eastward at approximately 14 m s^{−1}. It produced rainfall of 2-h duration at surface stations.

The simulation was obtained from a three-dimensional convective cloud model with open lateral boundary conditions on the east and west, and periodic conditions on the north and south boundaries. The model domain is 96 km long (east–west) and 32 km wide (north-south) with a horizontal grid resolution of 1.0 km and a vertical resolution of 0.5 km. A squall line develops and moves eastward at 13.7 m s^{−1} during the last two hours of the simulation. The present mesoγ-scale model, however, can only simulate the leading edge of the squall line, with rain at specific surface locations lasting only 30 min. Realistic features of the modeled flow include the surface westerlies moving faster than the line behind the gust front, the strong easterlies in the lower cloud levels, and the cold boundary layer behind the gust front.

Two-hour time means of the vertical momentum flux are calculated in a 60-km-wide domain (east–west) following the squall line. The vertical disturbance momentum flux for momentum normal to the line agrees with observations and is primarily confined to this region adjacent to the squall line. Horizontal-averaged time-mean momentum budgets are also calculated in this domain. For the normal component of momentum, this budget is in a quasi-steady state. It cannot be in a fully steady state as the gust front moves 1.2 m s^{−1} faster than the area of rain behind the line for the 2-h time mean.

The parameterization of Schneider and Lindzen for the vertical momentum flux associated with active clouds is compared with mean data from the simulation. Their parameterization accounts for the in-cloud vertical momentum flux reasonably well, but ignores the remaining flux associated with convective-scale downdrafts, which is significant in lower levels.

## Abstract

A 4-h simulation is carried out for the 22 May 1976 squall line that passed through the mesonetwork of the National Severe Storm Laboratory in central Oklahoma. This squall line was more than 100 km wide, oriented north-south and traveled eastward at approximately 14 m s^{−1}. It produced rainfall of 2-h duration at surface stations.

The simulation was obtained from a three-dimensional convective cloud model with open lateral boundary conditions on the east and west, and periodic conditions on the north and south boundaries. The model domain is 96 km long (east–west) and 32 km wide (north-south) with a horizontal grid resolution of 1.0 km and a vertical resolution of 0.5 km. A squall line develops and moves eastward at 13.7 m s^{−1} during the last two hours of the simulation. The present mesoγ-scale model, however, can only simulate the leading edge of the squall line, with rain at specific surface locations lasting only 30 min. Realistic features of the modeled flow include the surface westerlies moving faster than the line behind the gust front, the strong easterlies in the lower cloud levels, and the cold boundary layer behind the gust front.

Two-hour time means of the vertical momentum flux are calculated in a 60-km-wide domain (east–west) following the squall line. The vertical disturbance momentum flux for momentum normal to the line agrees with observations and is primarily confined to this region adjacent to the squall line. Horizontal-averaged time-mean momentum budgets are also calculated in this domain. For the normal component of momentum, this budget is in a quasi-steady state. It cannot be in a fully steady state as the gust front moves 1.2 m s^{−1} faster than the area of rain behind the line for the 2-h time mean.

The parameterization of Schneider and Lindzen for the vertical momentum flux associated with active clouds is compared with mean data from the simulation. Their parameterization accounts for the in-cloud vertical momentum flux reasonably well, but ignores the remaining flux associated with convective-scale downdrafts, which is significant in lower levels.