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- Author or Editor: Richard L. Pfeffer x

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## Abstract

The dynamics of eddy-induced fluctuations of the zonal-mean current in the troposphere and lower stratosphere is examined from the perspective of conventional and transformed Eulerian diagnostics using ECMWF FGGE IIIb data. Spatial correlations among different terms in the transformed Eulerian angular momentum equation for the entire FGGE year reveal a significant positive relationship between the divergence of the horizontal component of the Eliassen-Palm (E-P) flux (which is associated with meridional eddy fluxes of momentum) and day-to-day changes in the shape and intensity of the zonal current. It is found also that a very high *negative* correlation exists between the Coriolis torque associated with the eddy-induced residual circulation and the divergence of the vertical component of the E-P flux (which is associated with poleward eddy fluxes of heat and baroclinic wave development).

An explanation is given as to why the Coriolis term in the transformed Eulerian momentum equation so effectively cancels the vertical E-P flux divergence and why the zonal current responds with such great sensitivity to the horizontal divergence. It is shown on theoretical grounds that in regions in which the stratification parameter *S* (which is proportional to the ratio of the square of the Brunt-Väisälä frequency to the square of the Coriolis frequency) is sufficiently small, the response of the zonal mean flow to the E-P flux divergence must be in the form of a residual circulation, with little change in the zonal angular momentum distribution. This is found to be the situation in regions of baroclinic wave development and intense poleward eddy heat fluxes. In regions of sufficiently large *S* the response to the E-P flux divergence must be in the form of an acceleration or deceleration of the zonal flow in the direction of the eddy torque, with very little effect on the residual circulation. This is found to be the situation in regions of horizontal wave propagation outside the source region where poleward eddy momentum fluxes dominate.

A case study is selected from the FGGE dataset to illustrate the processes involved in a characteristic fluctuation of the zonal-mean current using both conventional and transformed Eulerian diagnostics. In this context the often neglected role of the poleward eddy heat flux at the lower boundary in accelerating the zonal current throughout the depth of the troposphere and lower stratosphere is shown to be important.

## Abstract

The dynamics of eddy-induced fluctuations of the zonal-mean current in the troposphere and lower stratosphere is examined from the perspective of conventional and transformed Eulerian diagnostics using ECMWF FGGE IIIb data. Spatial correlations among different terms in the transformed Eulerian angular momentum equation for the entire FGGE year reveal a significant positive relationship between the divergence of the horizontal component of the Eliassen-Palm (E-P) flux (which is associated with meridional eddy fluxes of momentum) and day-to-day changes in the shape and intensity of the zonal current. It is found also that a very high *negative* correlation exists between the Coriolis torque associated with the eddy-induced residual circulation and the divergence of the vertical component of the E-P flux (which is associated with poleward eddy fluxes of heat and baroclinic wave development).

An explanation is given as to why the Coriolis term in the transformed Eulerian momentum equation so effectively cancels the vertical E-P flux divergence and why the zonal current responds with such great sensitivity to the horizontal divergence. It is shown on theoretical grounds that in regions in which the stratification parameter *S* (which is proportional to the ratio of the square of the Brunt-Väisälä frequency to the square of the Coriolis frequency) is sufficiently small, the response of the zonal mean flow to the E-P flux divergence must be in the form of a residual circulation, with little change in the zonal angular momentum distribution. This is found to be the situation in regions of baroclinic wave development and intense poleward eddy heat fluxes. In regions of sufficiently large *S* the response to the E-P flux divergence must be in the form of an acceleration or deceleration of the zonal flow in the direction of the eddy torque, with very little effect on the residual circulation. This is found to be the situation in regions of horizontal wave propagation outside the source region where poleward eddy momentum fluxes dominate.

A case study is selected from the FGGE dataset to illustrate the processes involved in a characteristic fluctuation of the zonal-mean current using both conventional and transformed Eulerian diagnostics. In this context the often neglected role of the poleward eddy heat flux at the lower boundary in accelerating the zonal current throughout the depth of the troposphere and lower stratosphere is shown to be important.

## Abstract

An iterative numerical procedure, suitable for use on electronic computers, is presented for calculating the horizontal propagation speeds of acoustic-gravity waves in a stratified atmosphere. The atmosphere is represented by an arbitrarily large series of isothermal layers, so that propagation can he readily studied in an atmosphere with a complex stratification. The basic approach is similar to that used in seismology. A sample calculation shows reasonable agreement with the exact two-layer solution of Pekeris.

## Abstract

An iterative numerical procedure, suitable for use on electronic computers, is presented for calculating the horizontal propagation speeds of acoustic-gravity waves in a stratified atmosphere. The atmosphere is represented by an arbitrarily large series of isothermal layers, so that propagation can he readily studied in an atmosphere with a complex stratification. The basic approach is similar to that used in seismology. A sample calculation shows reasonable agreement with the exact two-layer solution of Pekeris.

## Abstract

The nature and extent of the interactions between the zonally averaged current (*ū*) and synoptic-scale eddies, including the contribution of the eddy-induced men meridional circulation, are assessed. Eddy fluxes of heat and momentum determined by Oort and Rasmussen are used as forcing functions in the diagnostic equation for the streamfunction corresponding to the mean meridional circulation. This equation is solved by relaxation and the solution is used to evaluate the contribution of the eddy-induced meridional circulation to the acceleration of the zonal current (∂*ū*/∂*t*). The *net* acceleration (i.e., the sum of the accelerations due to this meridional circulation and to the convergence of the eddy flux of momentum) is evaluated and is presented as a function of latitude and pressure height for each of the four seasons. Also presented is the meridional distribution of the net kinetic energy generation (*ū*∂*ū*/∂*t*) by the eddies, and the hemispheric integral of this quantity. The net annual generation of zonal kinetic energy by the eddies, including the contribution of the eddy-induced meridional circulation, is found to be ∼70% of the net annual generation neglecting the contribution of this meridional circulation. The distribution of ∂*ū*/∂*t* and of *ū*∂*ū*/∂*t* due to the eddies reveals, however, that the maxima of these quantities are centered poleward of the zonally-averaged jet core.

In an effort to assess the extent to which diabatic beating is responsible for the location and seasonal changes in the position of the jet core, a similar calculation is made using only the diabatic heating distribution given by Newell *et al.* as a forcing function for the mean meridional circulation. The acceleration (∂*ū*/∂*t*) and the net kinetic energy generation (*ū*∂*ū*/∂*t*) due to diabatic heating are found to be centered in the tropics, equatorward of the zonally averaged jet core. Hence, diabatic heating alone does not maintain the zonally averaged jet in the position we find it. Apparently, the combined contributions of the eddies and of diabatic heating, where neither has its maximum, create the positive acceleration and kinetic energy generation required to maintain the jet core where we find it.

In middle latitudes, poleward of the jet core, internal dissipation and the meridional Ekman circulation induced by surface friction must combine to decelerate the air being accelerated by the eddies. In the tropics, equatorward of the jet core, the acceleration of the westerlies by diabatic heating is largely counterbalanced by the deceleration by the eddies, leaving a small residual acceleration to be taken care of by dissipative processes.

## Abstract

The nature and extent of the interactions between the zonally averaged current (*ū*) and synoptic-scale eddies, including the contribution of the eddy-induced men meridional circulation, are assessed. Eddy fluxes of heat and momentum determined by Oort and Rasmussen are used as forcing functions in the diagnostic equation for the streamfunction corresponding to the mean meridional circulation. This equation is solved by relaxation and the solution is used to evaluate the contribution of the eddy-induced meridional circulation to the acceleration of the zonal current (∂*ū*/∂*t*). The *net* acceleration (i.e., the sum of the accelerations due to this meridional circulation and to the convergence of the eddy flux of momentum) is evaluated and is presented as a function of latitude and pressure height for each of the four seasons. Also presented is the meridional distribution of the net kinetic energy generation (*ū*∂*ū*/∂*t*) by the eddies, and the hemispheric integral of this quantity. The net annual generation of zonal kinetic energy by the eddies, including the contribution of the eddy-induced meridional circulation, is found to be ∼70% of the net annual generation neglecting the contribution of this meridional circulation. The distribution of ∂*ū*/∂*t* and of *ū*∂*ū*/∂*t* due to the eddies reveals, however, that the maxima of these quantities are centered poleward of the zonally-averaged jet core.

In an effort to assess the extent to which diabatic beating is responsible for the location and seasonal changes in the position of the jet core, a similar calculation is made using only the diabatic heating distribution given by Newell *et al.* as a forcing function for the mean meridional circulation. The acceleration (∂*ū*/∂*t*) and the net kinetic energy generation (*ū*∂*ū*/∂*t*) due to diabatic heating are found to be centered in the tropics, equatorward of the zonally averaged jet core. Hence, diabatic heating alone does not maintain the zonally averaged jet in the position we find it. Apparently, the combined contributions of the eddies and of diabatic heating, where neither has its maximum, create the positive acceleration and kinetic energy generation required to maintain the jet core where we find it.

In middle latitudes, poleward of the jet core, internal dissipation and the meridional Ekman circulation induced by surface friction must combine to decelerate the air being accelerated by the eddies. In the tropics, equatorward of the jet core, the acceleration of the westerlies by diabatic heating is largely counterbalanced by the deceleration by the eddies, leaving a small residual acceleration to be taken care of by dissipative processes.

## Abstract

Angular momentum relative to a fixed vertical axis is used as a measure of rotational flow on the scale of cyclones and anticyclones. An exact equation derived by Starr for the time rate of change of this momentum is expressed in an approximate form, and a long-period observational study is made in two regions to determine the validity and prognostic implications of the resulting relationship.

It is found that the fluctuations of angular momentum can be accounted for primarily by a single effect—the horizontal transport of momentum across the boundaries of the regions. Moreover, a sufficient lag relationship is found to exist so that the transport serves as a predictor of the 12-hour change of momentum.

## Abstract

Angular momentum relative to a fixed vertical axis is used as a measure of rotational flow on the scale of cyclones and anticyclones. An exact equation derived by Starr for the time rate of change of this momentum is expressed in an approximate form, and a long-period observational study is made in two regions to determine the validity and prognostic implications of the resulting relationship.

It is found that the fluctuations of angular momentum can be accounted for primarily by a single effect—the horizontal transport of momentum across the boundaries of the regions. Moreover, a sufficient lag relationship is found to exist so that the transport serves as a predictor of the 12-hour change of momentum.

## Abstract

Two types of vacillation are distinguished in a rotating differentially heated annulus of fluid: (1) Tilted-trough vacillation, which has been discussed in a number of previous papers in the literature, is characterized by periodic changes in the wave shapes, in which the waves first tilt “north-west” to “south-east”, and later “north-east” to “south-west”, relative to the rotating annulus. (2) Amplitude vacillation, which was first observed by the authors in a series of student demonstrations conducted at the Lamont Geological Observatory, is characterized by periodic expansions and contractions of the wave pattern with no noticeable change in the tilt of the disturbances. It is suggested that the details of the energy conversions are different in the two cases. Some speculations are also presented concerning the role of viscosity in limiting the number of degrees of freedom available for vacillation.

## Abstract

Two types of vacillation are distinguished in a rotating differentially heated annulus of fluid: (1) Tilted-trough vacillation, which has been discussed in a number of previous papers in the literature, is characterized by periodic changes in the wave shapes, in which the waves first tilt “north-west” to “south-east”, and later “north-east” to “south-west”, relative to the rotating annulus. (2) Amplitude vacillation, which was first observed by the authors in a series of student demonstrations conducted at the Lamont Geological Observatory, is characterized by periodic expansions and contractions of the wave pattern with no noticeable change in the tilt of the disturbances. It is suggested that the details of the energy conversions are different in the two cases. Some speculations are also presented concerning the role of viscosity in limiting the number of degrees of freedom available for vacillation.

## Abstract

The effect of the vertical transport of horizontal momentum by cumulus clouds on the development of a symmetric model hurricane is investigated. This is accomplished by using Sundqvist's symmetric hurricane model with parameterized cumulus friction. The scheme used to include cumulus friction in the model is essentially the same as that given by Stevens and Lindzen in 1978 and Lindzen in 1981. The results of two sets of numerical integrations are presented. In one, the initial wind and moisture distributions were derived from atmospheric observations in Atlantic intensifying cyclones as composited by McBride. In the other, the initial vortex was specified as that which corresponds to the linearly most unstable mode in Mak's 1980 linear analysis of the effect of cumulus friction on hurricane formation. Given each initial wind, temperature and moisture distribution, numerical integrations were performed with and without cumulus friction present in the model.

With cumulus friction included, the growth rates of the initial disturbances and their final intensities are smaller than those obtained in the absence of cumulus friction. The Atlantic intensifying cyclone with cumulus friction reaches storm strength, whereas without cumulus friction it develops into a hurricane. In the second pair of numerical integrations with the initial vortex specified as described above, the model develops hurricanes with and without cumulus frictions, but the rate of intensification and final strength of the vortex are significantly smaller when cumulus friction is included. The damping effect of cumulus friction is attributed to the fact that the angular momentum transported from the lower into the upper troposphere by cumulus mixing is not fully replenished in the lower troposphere by the cumulus induced secondary (radial) circulation. This contrasts with the effect of the inward eddy flux of momentum, reported on previously, which was found to enhance the intensification of hurricanes. The crucial difference between the two mechanisms, both of which induce secondary radial circulations due to a vertical differential in cyclonic torque, appears to be the net increase of momentum in the vortex due to inward eddy flux of momentum, which is not present in the case of cumulus friction. The latter mechanism simply redistributes the momentum vertically, actually reducing the strength of both the low-level cyclone and the upper-level anticyclone.

## Abstract

The effect of the vertical transport of horizontal momentum by cumulus clouds on the development of a symmetric model hurricane is investigated. This is accomplished by using Sundqvist's symmetric hurricane model with parameterized cumulus friction. The scheme used to include cumulus friction in the model is essentially the same as that given by Stevens and Lindzen in 1978 and Lindzen in 1981. The results of two sets of numerical integrations are presented. In one, the initial wind and moisture distributions were derived from atmospheric observations in Atlantic intensifying cyclones as composited by McBride. In the other, the initial vortex was specified as that which corresponds to the linearly most unstable mode in Mak's 1980 linear analysis of the effect of cumulus friction on hurricane formation. Given each initial wind, temperature and moisture distribution, numerical integrations were performed with and without cumulus friction present in the model.

With cumulus friction included, the growth rates of the initial disturbances and their final intensities are smaller than those obtained in the absence of cumulus friction. The Atlantic intensifying cyclone with cumulus friction reaches storm strength, whereas without cumulus friction it develops into a hurricane. In the second pair of numerical integrations with the initial vortex specified as described above, the model develops hurricanes with and without cumulus frictions, but the rate of intensification and final strength of the vortex are significantly smaller when cumulus friction is included. The damping effect of cumulus friction is attributed to the fact that the angular momentum transported from the lower into the upper troposphere by cumulus mixing is not fully replenished in the lower troposphere by the cumulus induced secondary (radial) circulation. This contrasts with the effect of the inward eddy flux of momentum, reported on previously, which was found to enhance the intensification of hurricanes. The crucial difference between the two mechanisms, both of which induce secondary radial circulations due to a vertical differential in cyclonic torque, appears to be the net increase of momentum in the vortex due to inward eddy flux of momentum, which is not present in the case of cumulus friction. The latter mechanism simply redistributes the momentum vertically, actually reducing the strength of both the low-level cyclone and the upper-level anticyclone.

## Abstract

Drazin's (1972) weakly nonlinear theory of the self interaction of a single, slightly unstable, normal mode in a viscous, baroclinic fluid with a continuous density distribution is used to determine eddy fluxes of heat and eddy temperature variances as a function of rotation and internal thermal gradients. The calculations are applied to annulus experiments in that portion of the regular wave regime in which the observed flow is dominated by a single mode. V. Barcilon's (1964) linear theory for a viscous fluid is used for the purpose of mode selection at each point in dimensionless-parameter space. The geostrophic eddy heat flux and temperature variance are then determined from Drazin's lowest order nonlinear theory (which is a direct extension of Barcilon's theory).

It is found that the eddy beat flux and temperature variance depend upon internal thermal gradients at marginal stability and upon the distance of the point in dimensionless-parameter space from marginal stability for each mode. This result is different from those obtained from linear theory (e.g., by Green, 1970; Saltzman and Vernekar, 1971; Stone, 1972) but agrees in essence with that obtained by Hart (1974) from a nonlinear analysis of a two-layer model.

Numerical calculations from the theoretically derived formulas show that the eddy beat flux and temperature variance corresponding to each mode increase with increasing rotation rate when the imposed temperature contrast is held constant, but that there is an abrupt drop in the magnitude of both quantities when the wavenumber changes to the next higher integral value. Experimental evidence obtained by Pfeffer *et al.* (1978) verifies that real fluids behave in this way. Another feature observed in laboratory experiments and predicted by the theory is that at fixed rotation rate (or Taylor number) the eddy heat flux and the eddy temperature variance may be either larger or smaller at larger values of the imposed temperature contrast, depending on the location of the experiments in dimensionless-parameter space. If we think of seasons being brought about by changes in imposed temperature contrast at constant rotation rate, this result implies that only in certain regions of dimensionless-parameter space is winter (defined as the season with the highest imposed temperature contrast) the season with the largest eddy heat flux or eddy temperature variance.

## Abstract

Drazin's (1972) weakly nonlinear theory of the self interaction of a single, slightly unstable, normal mode in a viscous, baroclinic fluid with a continuous density distribution is used to determine eddy fluxes of heat and eddy temperature variances as a function of rotation and internal thermal gradients. The calculations are applied to annulus experiments in that portion of the regular wave regime in which the observed flow is dominated by a single mode. V. Barcilon's (1964) linear theory for a viscous fluid is used for the purpose of mode selection at each point in dimensionless-parameter space. The geostrophic eddy heat flux and temperature variance are then determined from Drazin's lowest order nonlinear theory (which is a direct extension of Barcilon's theory).

It is found that the eddy beat flux and temperature variance depend upon internal thermal gradients at marginal stability and upon the distance of the point in dimensionless-parameter space from marginal stability for each mode. This result is different from those obtained from linear theory (e.g., by Green, 1970; Saltzman and Vernekar, 1971; Stone, 1972) but agrees in essence with that obtained by Hart (1974) from a nonlinear analysis of a two-layer model.

Numerical calculations from the theoretically derived formulas show that the eddy beat flux and temperature variance corresponding to each mode increase with increasing rotation rate when the imposed temperature contrast is held constant, but that there is an abrupt drop in the magnitude of both quantities when the wavenumber changes to the next higher integral value. Experimental evidence obtained by Pfeffer *et al.* (1978) verifies that real fluids behave in this way. Another feature observed in laboratory experiments and predicted by the theory is that at fixed rotation rate (or Taylor number) the eddy heat flux and the eddy temperature variance may be either larger or smaller at larger values of the imposed temperature contrast, depending on the location of the experiments in dimensionless-parameter space. If we think of seasons being brought about by changes in imposed temperature contrast at constant rotation rate, this result implies that only in certain regions of dimensionless-parameter space is winter (defined as the season with the highest imposed temperature contrast) the season with the largest eddy heat flux or eddy temperature variance.

## Abstract

The role of large scale eddy processes in the transformation of cloud clusters and depressions into hurricanes is investigated by using different initial conditions in numerical integrations of the Naval Research Laboratory limited-area hurricane model. With initial conditions specified from the Colorado State University composite datasets of Professor William Gray for Atlantic *nondeveloping* cloud clusters, wave trough clusters and depressions, no hurricane formation takes place in any of the model integrations. With initial conditions specified from the datasets for Atlantic *developing* cloud clusters and depressions, a hurricane develops in the course of each model integration. One characteristic difference between the developing and nondeveloping disturbances is that the former exhibit large, well-organized eddy flux convergences of angular momentum associated with wavelike disturbances in the upper troposphere and lower stratosphere, whereas the latter exhibit weak, poorly organized eddy momentum fluxes.

In order to assess the role played in hurricane formation by the large-scale eddy processes in the developing cases, we performed additional integrations with initial conditions in which the eddies were removed from the datasets for the developing cloud cluster and depression. This was accomplished by using only the symmetric components of the wind and moisture fields in these datasets. In these integrations, the initial disturbances failed to develop. We take this as evidence to support the view that wavelike asymmetries in the upper troposphere and lower stratosphere may be necessary for hurricane development from Atlantic cloud clusters and depressions. Such asymmetries may act through the agency of eddy fluxes of heat and/or eddy fluxes of momentum. In this paper, we concentrate mainly on the role of eddy fluxes of momentum. Mechanistically, these fluxes exert an upper-level cyclonic torque on the initially weak vortex. Such a process induces upper level divergence and lower level convergence. The air converging in the lower boundary layer over a broad stretch of warm ocean brings moisture inward, organizing and concentrating the convection which fuels the development of the hurricane.

## Abstract

The role of large scale eddy processes in the transformation of cloud clusters and depressions into hurricanes is investigated by using different initial conditions in numerical integrations of the Naval Research Laboratory limited-area hurricane model. With initial conditions specified from the Colorado State University composite datasets of Professor William Gray for Atlantic *nondeveloping* cloud clusters, wave trough clusters and depressions, no hurricane formation takes place in any of the model integrations. With initial conditions specified from the datasets for Atlantic *developing* cloud clusters and depressions, a hurricane develops in the course of each model integration. One characteristic difference between the developing and nondeveloping disturbances is that the former exhibit large, well-organized eddy flux convergences of angular momentum associated with wavelike disturbances in the upper troposphere and lower stratosphere, whereas the latter exhibit weak, poorly organized eddy momentum fluxes.

In order to assess the role played in hurricane formation by the large-scale eddy processes in the developing cases, we performed additional integrations with initial conditions in which the eddies were removed from the datasets for the developing cloud cluster and depression. This was accomplished by using only the symmetric components of the wind and moisture fields in these datasets. In these integrations, the initial disturbances failed to develop. We take this as evidence to support the view that wavelike asymmetries in the upper troposphere and lower stratosphere may be necessary for hurricane development from Atlantic cloud clusters and depressions. Such asymmetries may act through the agency of eddy fluxes of heat and/or eddy fluxes of momentum. In this paper, we concentrate mainly on the role of eddy fluxes of momentum. Mechanistically, these fluxes exert an upper-level cyclonic torque on the initially weak vortex. Such a process induces upper level divergence and lower level convergence. The air converging in the lower boundary layer over a broad stretch of warm ocean brings moisture inward, organizing and concentrating the convection which fuels the development of the hurricane.