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- Author or Editor: Frank B. Lipps x
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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.
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 sech2 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 sec1 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 sech2 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 sec1 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 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.
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.
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.
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.
Abstract
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Abstract
The study considers deep moist convection involving only a liquid-vapor phase change. An alternative form of the classical thermodynamic equation for reversible saturated flow is derived. Four approximate forms of this equation are obtained and their relative errors compared to the full equation are evaluated by using parcel theory. The best approximation is found to be an adequate representation of the full equation throughout the total depth of the convection.
The two best approximations are compared with some forms of the thermodynamic equation used by other investigators.
Abstract
The study considers deep moist convection involving only a liquid-vapor phase change. An alternative form of the classical thermodynamic equation for reversible saturated flow is derived. Four approximate forms of this equation are obtained and their relative errors compared to the full equation are evaluated by using parcel theory. The best approximation is found to be an adequate representation of the full equation throughout the total depth of the convection.
The two best approximations are compared with some forms of the thermodynamic equation used by other investigators.
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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