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## Abstract

The instability of two-level and two-layer quasi-geostrophic models containing zonally symmetric, meridionally sloping lower boundaries are discussed and compared. The influence of this topography is to reduce the eastward phase speed of the unstable waves and to lengthen the most unstable wave if the slope is upward toward the pole, and to increase the eastward phase speed of the unstable waves and to shorten the most unstable wave if the slope is upward toward the equator. The instability may be slightly enhanced or reduced for a very small slope depending on the model characteristics. For a sufficient slope, such as on the Antarctic Continent, the instability is curtailed.

During the major ice ages in geological history, the ice field extended equatorward to the middle latitudes. Although the meridional temperature contrast (i.e., baroclinicity) is enhanced along the boundary of the ice field, the poleward slope of the ice field tends to diminish instability and the associated poleward transport of sensible heat is reduced or inhibited.

However when the height of the mountains is a large portion of the total depth of the model, the winds at these places will be stronger. Hence the above conclusion should be viewed with caution.

## Abstract

The instability of two-level and two-layer quasi-geostrophic models containing zonally symmetric, meridionally sloping lower boundaries are discussed and compared. The influence of this topography is to reduce the eastward phase speed of the unstable waves and to lengthen the most unstable wave if the slope is upward toward the pole, and to increase the eastward phase speed of the unstable waves and to shorten the most unstable wave if the slope is upward toward the equator. The instability may be slightly enhanced or reduced for a very small slope depending on the model characteristics. For a sufficient slope, such as on the Antarctic Continent, the instability is curtailed.

During the major ice ages in geological history, the ice field extended equatorward to the middle latitudes. Although the meridional temperature contrast (i.e., baroclinicity) is enhanced along the boundary of the ice field, the poleward slope of the ice field tends to diminish instability and the associated poleward transport of sensible heat is reduced or inhibited.

However when the height of the mountains is a large portion of the total depth of the model, the winds at these places will be stronger. Hence the above conclusion should be viewed with caution.

## Abstract

In this paper we develop the analytical theory of two-level quasi-geostrophic baroclinic waves without β-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*= π/

_{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*→ 1, the modes become steady and neutral and are characterized by 1)

*a*/

*b*= (

*S*/

_{m}*S*)½ (

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

_{m}*a*/

*b*→ ∞. As

*F*→ ∞, the modes are steady and neutral and are characterized by 1)

*a*/

*b*→ 0, and 2)

*a*/

*b*→ 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 β-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*= π/

_{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*→ 1, the modes become steady and neutral and are characterized by 1)

*a*/

*b*= (

*S*/

_{m}*S*)½ (

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

_{m}*a*/

*b*→ ∞. As

*F*→ ∞, the modes are steady and neutral and are characterized by 1)

*a*/

*b*→ 0, and 2)

*a*/

*b*→ 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 β-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})^{−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}
* = π/

*b*,

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

*a*/

*b*≠ 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* ≥ 10 for the first mode and except for ε 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 β-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})^{−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}
* = π/

*b*,

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

*a*/

*b*≠ 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* ≥ 10 for the first mode and except for ε near unity for the second mode.

## Abstract

From the analytical model described in the previous article (Part I) we show how second-order non-geostrophic effects modify the initially sinusoidal pattern of vertical motion accompanying an unstable baroclinic wave, and also show how the zonal mean forced Ferrel circulation local to the wave system is altered. The most general modification of the complete ω field is the introduction of strong north-south variations of which the most significant synoptic feature is a region of strong descending motion south of the surface low center. Similar distributions of ω have been noted in observational and numerical studies.

For this two-layer system the vertical transport of heat associated with the second-order ω field is directed downward at the middle level thereby tending to “brake” the overall wave growth. The second-order ω field also leads to a significant downward flux of westerly momentum especially in middle and lower levels, but if this field were weakened, as might be the case, for example, if the static stability increased strongly, the system could exhibit a strong *upward* flux of momentum as revealed in some recent observational and numerical studies.

## Abstract

From the analytical model described in the previous article (Part I) we show how second-order non-geostrophic effects modify the initially sinusoidal pattern of vertical motion accompanying an unstable baroclinic wave, and also show how the zonal mean forced Ferrel circulation local to the wave system is altered. The most general modification of the complete ω field is the introduction of strong north-south variations of which the most significant synoptic feature is a region of strong descending motion south of the surface low center. Similar distributions of ω have been noted in observational and numerical studies.

For this two-layer system the vertical transport of heat associated with the second-order ω field is directed downward at the middle level thereby tending to “brake” the overall wave growth. The second-order ω field also leads to a significant downward flux of westerly momentum especially in middle and lower levels, but if this field were weakened, as might be the case, for example, if the static stability increased strongly, the system could exhibit a strong *upward* flux of momentum as revealed in some recent observational and numerical studies.

## Abstract

An analytical study is made showing how second-order, non-geostrophic effects can modify a two-layer baroclinic wave system that grows exponentially from a small perturbation in a uniform zonal current. Many of the features observed on weather maps are seen to develop, including the relative meridional displacements and intensifications of the high and low pressure areas, the formation of a cutoff low aloft with a splitting of the jet stream, and the formation of zones of strong temperature gradient representing mid-tropospheric “fronts.” An analysis is also given to show the role of shear of the basic zonal current in the development of horizontal tilts of the amplifying waves.

## Abstract

An analytical study is made showing how second-order, non-geostrophic effects can modify a two-layer baroclinic wave system that grows exponentially from a small perturbation in a uniform zonal current. Many of the features observed on weather maps are seen to develop, including the relative meridional displacements and intensifications of the high and low pressure areas, the formation of a cutoff low aloft with a splitting of the jet stream, and the formation of zones of strong temperature gradient representing mid-tropospheric “fronts.” An analysis is also given to show the role of shear of the basic zonal current in the development of horizontal tilts of the amplifying waves.

## Abstract

Using an analytical model similar to that previously applied by the authors to the atmosphere, calculations are made showing how second-order, nongeostrophic effects can modify a two-layer baroclinic wave system that grows exponentially from a small perturbation in a uniform zonal ocean current. It is shown that many of the asymmetric features characteristic of meandering ocean currents develop, including “fronts” and cutoff cyclonic cold pools to the south and anticyclonic warm pools to the north of the axis of the mean current. The implication is that all of these features can be viewed as being the simultaneous consequence of baroclinic instability (with attendant second-order finite-amplitude effects) of a broader, more uniform current that might tend to be forced externally by the wind stress and thermohaline processes.

## Abstract

Using an analytical model similar to that previously applied by the authors to the atmosphere, calculations are made showing how second-order, nongeostrophic effects can modify a two-layer baroclinic wave system that grows exponentially from a small perturbation in a uniform zonal ocean current. It is shown that many of the asymmetric features characteristic of meandering ocean currents develop, including “fronts” and cutoff cyclonic cold pools to the south and anticyclonic warm pools to the north of the axis of the mean current. The implication is that all of these features can be viewed as being the simultaneous consequence of baroclinic instability (with attendant second-order finite-amplitude effects) of a broader, more uniform current that might tend to be forced externally by the wind stress and thermohaline processes.