# Search Results

## Abstract

In this paper we develop the analytical theory of two-level quasi-geostrophic baroclinic waves without Î²-effect aimed at understanding the role of latent heat release on the development of baroclinic waves.

When the release of latent heat is introduced with pseudo-adiabatic ascent and dry adiabatic descent the width *a* of the ascending region is different from the width *b* of the descending region and, furthermore, a static stability-vertical velocity correlation results in the mean state thickness increasing with time. However, the basic state shell is defined *a priori*, independent of the perturbations, in the formulation of the stability problem. Integro-differential equations for the perturbations are developed. Due to, the mass continuity constraint, the unstable waves in the dry and moist regions are stationary in a frame of reference which translates with the mean zonal wind at the middle level, and the growth rate in the moist region is equal to that in the dry region, the same as in the dry model. We define the parameter *F* = 2*f*
^{2}/*S _{d}p*

_{2}

^{2}

*k*

_{d}^{2}, where

*f*is the Coriolis parameter,

*S*the static stability in the dry region,

_{d}*p*

_{2}the pressure at the middle level, and

*k*= Ï€/

_{d}*b*. The ratio

*a*/

*b*is a function of

*F*. For

*F*> 1, two unstable modes appear. The first mode has a narrow region of strong ascending motion and a wide region of weak descending motion (

*a*/

*b*< 1), and the second mode has a narrow region of strong descending motion and a wide region of weak ascending motion (

*a*/

*b*> 1). As

*F*â†’ 1, the modes become steady and neutral and are characterized by 1)

*a*/

*b*= (

*S*/

_{m}*S*)Â½ (

_{d}*S*is static stability in the moist region), and 2)

_{m}*a*/

*b*â†’ âˆž. As

*F*â†’ âˆž, the modes are steady and neutral and are characterized by 1)

*a*/

*b*â†’ 0, and 2)

*a*/

*b*â†’ 1. In comparison with the dry model, the structure of the first unstable mode shows that the ridge and trough of the streamlines shift slightly toward the region of sinking motion, and the warm advection occurs at the node of the vertical motion, while the structure of the second unstable mode shows that the ridge and trough of the streamlines shift slightly toward the region of rising motion, and the cold advection occurs at the node of the vertical motion.

An analysis of the energetics shows the presence of a latent heat release term which directly contributes to the generation of eddy available potential energy. Although this term is small compared to the vertical and horizontal heat transports, latent heat release causes a significant change in the structure of the waves such that large departure in the horizontal heat transport from dry atmospheric values can occur.

The multicomponent solution is also discussed. It is stressed that the first harmonic must be present and even harmonics are allowed provided the vertical motion is upward everywhere in the moist region of the width *a* and downward everywhere in the dry region of the width *b*. The solution is not Fourier decomposition in the normal sense because except for the first harmonic odd modes are not allowed.

## Abstract

In this paper we develop the analytical theory of two-level quasi-geostrophic baroclinic waves without Î²-effect aimed at understanding the role of latent heat release on the development of baroclinic waves.

When the release of latent heat is introduced with pseudo-adiabatic ascent and dry adiabatic descent the width *a* of the ascending region is different from the width *b* of the descending region and, furthermore, a static stability-vertical velocity correlation results in the mean state thickness increasing with time. However, the basic state shell is defined *a priori*, independent of the perturbations, in the formulation of the stability problem. Integro-differential equations for the perturbations are developed. Due to, the mass continuity constraint, the unstable waves in the dry and moist regions are stationary in a frame of reference which translates with the mean zonal wind at the middle level, and the growth rate in the moist region is equal to that in the dry region, the same as in the dry model. We define the parameter *F* = 2*f*
^{2}/*S _{d}p*

_{2}

^{2}

*k*

_{d}^{2}, where

*f*is the Coriolis parameter,

*S*the static stability in the dry region,

_{d}*p*

_{2}the pressure at the middle level, and

*k*= Ï€/

_{d}*b*. The ratio

*a*/

*b*is a function of

*F*. For

*F*> 1, two unstable modes appear. The first mode has a narrow region of strong ascending motion and a wide region of weak descending motion (

*a*/

*b*< 1), and the second mode has a narrow region of strong descending motion and a wide region of weak ascending motion (

*a*/

*b*> 1). As

*F*â†’ 1, the modes become steady and neutral and are characterized by 1)

*a*/

*b*= (

*S*/

_{m}*S*)Â½ (

_{d}*S*is static stability in the moist region), and 2)

_{m}*a*/

*b*â†’ âˆž. As

*F*â†’ âˆž, the modes are steady and neutral and are characterized by 1)

*a*/

*b*â†’ 0, and 2)

*a*/

*b*â†’ 1. In comparison with the dry model, the structure of the first unstable mode shows that the ridge and trough of the streamlines shift slightly toward the region of sinking motion, and the warm advection occurs at the node of the vertical motion, while the structure of the second unstable mode shows that the ridge and trough of the streamlines shift slightly toward the region of rising motion, and the cold advection occurs at the node of the vertical motion.

An analysis of the energetics shows the presence of a latent heat release term which directly contributes to the generation of eddy available potential energy. Although this term is small compared to the vertical and horizontal heat transports, latent heat release causes a significant change in the structure of the waves such that large departure in the horizontal heat transport from dry atmospheric values can occur.

The multicomponent solution is also discussed. It is stressed that the first harmonic must be present and even harmonics are allowed provided the vertical motion is upward everywhere in the moist region of the width *a* and downward everywhere in the dry region of the width *b*. The solution is not Fourier decomposition in the normal sense because except for the first harmonic odd modes are not allowed.

## Abstract

A second-order theory of baroclinic waves is developed to investigate non-quasi-geostrophic behavior in disturbances in which latent heat release associated with condensation is permitted to occur in an atmosphere saturated with water vapor. A two-level formulation without Î²-effect is used to analyze these disturbances. The analysis involves an expansion of the flow into a basic state zonal flow with superimposed perturbation which is assumed to be independent of the meridional direction. The superimposed perturbation consists of linear combination of a quasi-geostrophic contribution and a non-quasi-geostrophic departure. The basic state flow with the superimposed quasi-geostrophic perturbation has been investigated by the authors in a previous paper. The governing equations for the non-quasi-geostrophic contribution consist of a nonlinear (thermodynamic) integro-differential equation and a nonhomogenous (vorticity) differential equation. The nonlinearity is a direct result of latent heat release associated with pseudo-adiabatic assent; i.e., saturated ascending air parcels and dry descending air parcels. The nonhomogeneity arises from the non-quasi-geostrophic terms in the vorticity equation. In this theory we use the quasi-geostrophic contribution to calculate the non-quasi-geostrophic terms which generate the second-order solution.

The problem is characterized by two parameters, namely a rotational Froude number *F* = 2*f*^{2}(*S*_{dP2}^{2}*k*_{d}^{2})^{âˆ’l} (where *f* is the Coriolis parameter, *S*_{d} the static stability in descending portion of the wave, *p*_{2} the pressure at the middle level, and *k _{d}* = Ï€/

*b*,

*b*being the horizontal extent of the descending or dry portion of the wave) and a moisture parameter Îµ which is proportional to the midlevel vertical gradient of mean flow specific humidity. For Îµ â‰ 0 the disturbances are characterized by

*a*/

*b*â‰ 1, where

*a*is the horizontal extent of the ascending or wet portion of the wave. The quasi-geostrophic contribution to the disturbance is characterized by two modes for

*F*> 1. The first mode has a narrow region of strong ascending motion and a wide region of weak descending motion (

*a*/

*b*< 1), with the reverse for the second mode (

*a*/

*b*> 1). Thew solutions, developed by the authors in an earlier paper, are used to calculate the non-quasi-geostrophic solution terms mentioned above.

For the first moist mode, due to the non-quasi-geostrophic effects, both the trough and ridge are intensified at the upper level with stronger intensification of the trough and are weakened at the lower level with considerable weakening of the ridge. The formation of the frontal zone on the east side of the descending region is a feature similar to that in the dry model with non-quasi-geostrophic effects. For the second moist mode, due to the non-quasi-geostrophic effects, both trough and ridge are weakened at the upper level, but they am intensified at the lower level. The temperature profile in each region is nearly symmetric. The total vertical motion field is asymmetric in each region for both the first and second moist modes.

The main characteristics of the energetics are described by the transports due to the first-order eddy. The transports due to the second-order eddy have only minor influence except for large *F* such as *F* â‰¥ 10 for the first mode and except for Îµ near unity for the second mode.

## Abstract

A second-order theory of baroclinic waves is developed to investigate non-quasi-geostrophic behavior in disturbances in which latent heat release associated with condensation is permitted to occur in an atmosphere saturated with water vapor. A two-level formulation without Î²-effect is used to analyze these disturbances. The analysis involves an expansion of the flow into a basic state zonal flow with superimposed perturbation which is assumed to be independent of the meridional direction. The superimposed perturbation consists of linear combination of a quasi-geostrophic contribution and a non-quasi-geostrophic departure. The basic state flow with the superimposed quasi-geostrophic perturbation has been investigated by the authors in a previous paper. The governing equations for the non-quasi-geostrophic contribution consist of a nonlinear (thermodynamic) integro-differential equation and a nonhomogenous (vorticity) differential equation. The nonlinearity is a direct result of latent heat release associated with pseudo-adiabatic assent; i.e., saturated ascending air parcels and dry descending air parcels. The nonhomogeneity arises from the non-quasi-geostrophic terms in the vorticity equation. In this theory we use the quasi-geostrophic contribution to calculate the non-quasi-geostrophic terms which generate the second-order solution.

The problem is characterized by two parameters, namely a rotational Froude number *F* = 2*f*^{2}(*S*_{dP2}^{2}*k*_{d}^{2})^{âˆ’l} (where *f* is the Coriolis parameter, *S*_{d} the static stability in descending portion of the wave, *p*_{2} the pressure at the middle level, and *k _{d}* = Ï€/

*b*,

*b*being the horizontal extent of the descending or dry portion of the wave) and a moisture parameter Îµ which is proportional to the midlevel vertical gradient of mean flow specific humidity. For Îµ â‰ 0 the disturbances are characterized by

*a*/

*b*â‰ 1, where

*a*is the horizontal extent of the ascending or wet portion of the wave. The quasi-geostrophic contribution to the disturbance is characterized by two modes for

*F*> 1. The first mode has a narrow region of strong ascending motion and a wide region of weak descending motion (

*a*/

*b*< 1), with the reverse for the second mode (

*a*/

*b*> 1). Thew solutions, developed by the authors in an earlier paper, are used to calculate the non-quasi-geostrophic solution terms mentioned above.

For the first moist mode, due to the non-quasi-geostrophic effects, both the trough and ridge are intensified at the upper level with stronger intensification of the trough and are weakened at the lower level with considerable weakening of the ridge. The formation of the frontal zone on the east side of the descending region is a feature similar to that in the dry model with non-quasi-geostrophic effects. For the second moist mode, due to the non-quasi-geostrophic effects, both trough and ridge are weakened at the upper level, but they am intensified at the lower level. The temperature profile in each region is nearly symmetric. The total vertical motion field is asymmetric in each region for both the first and second moist modes.

The main characteristics of the energetics are described by the transports due to the first-order eddy. The transports due to the second-order eddy have only minor influence except for large *F* such as *F* â‰¥ 10 for the first mode and except for Îµ near unity for the second mode.

## Abstract

A set of conditions which justify the application of the Boussinesq approximation to compressible fluids is developed. Two cases are found and compared. In the first, in which the vertical scale of the motion can be of the same order of magnitude as the scale height of the medium, the perturbation momentum must be nondivergent and the effects of perturbations of pressure appear in several places. In the other case, where the vertical scale of the motion is much less than the scale height, the perturbation velocities are non-divergent and the perturbation pressure appears only in the pressure gradient force.

The approximate equations lead to linearized equations controlling the stability of wave motion which are formally equivalent to those for the same problem in the flow of a stratified medium which is incompressible in the sense that the flow is solenoidal. Thus, a variety of results about such motions are made applicable to the problems of convection and gravity wave motion in the atmosphere.

Various properties of the approximate equations are investigated; it is shown that acoustic modes are not permitted; quadratic forms which can serve as energies in various cases are developed; and integral methods of determining stability criteria are reviewed and applied.

In order to give the results wider applicability than to ideal gases, an ideal liquid is defined (*c _{p}
* and the coefficients of expansion all being constant). The thermodynamic functions of this ideal liquid, including the entropy, internal energy and potential temperature, are determined explicitly.

## Abstract

A set of conditions which justify the application of the Boussinesq approximation to compressible fluids is developed. Two cases are found and compared. In the first, in which the vertical scale of the motion can be of the same order of magnitude as the scale height of the medium, the perturbation momentum must be nondivergent and the effects of perturbations of pressure appear in several places. In the other case, where the vertical scale of the motion is much less than the scale height, the perturbation velocities are non-divergent and the perturbation pressure appears only in the pressure gradient force.

The approximate equations lead to linearized equations controlling the stability of wave motion which are formally equivalent to those for the same problem in the flow of a stratified medium which is incompressible in the sense that the flow is solenoidal. Thus, a variety of results about such motions are made applicable to the problems of convection and gravity wave motion in the atmosphere.

Various properties of the approximate equations are investigated; it is shown that acoustic modes are not permitted; quadratic forms which can serve as energies in various cases are developed; and integral methods of determining stability criteria are reviewed and applied.

In order to give the results wider applicability than to ideal gases, an ideal liquid is defined (*c _{p}
* and the coefficients of expansion all being constant). The thermodynamic functions of this ideal liquid, including the entropy, internal energy and potential temperature, are determined explicitly.