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## 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

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

No Abstract Available.

## Abstract

No Abstract Available.

## 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

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

The results of numerical integrations of Sundqvist's (1970) symmetric model for hurricane development modified to include parameterized large-scale eddy fluxes of momentum are presented. The initial wind and moisture distributions, and the prescribed eddy fluxes of momentum, were taken from atmospheric observations of Atlantic developing (prehurricane) and non-developing tropical disturbances as composited by McBride (1981a,b) and McBride and Zehr (1981). For the purposes of the present study, the data for individual stages in the evolution of developing and non-developing disturbances were combined to form a single composite developing hurricane and a single composite non-developing disturbance.

The data reveal the presence of intense, well organized inward eddy fluxes of momentum in developing Atlantic hurricanes and weak, poorly organized fluxes in non-developing disturbances. In the developing disturbances, the eddy fluxes of momentum are organized such that they act as a forcing function for driving the radial circulation, drawing moist air in toward the center of the vortex in the lower troposphere and pumping drier air outward aloft, thereby providing fuel for the explosive growth of the hurricane. In order to test the efficacy of this mechanism, and of Ekman suction and cooperative instability, numerical integrations were performed using the data for the composite developing hurricane, with and without the observed eddy fluxes of momentum, and for the composite non-developing disturbance with the observed eddy fluxes corresponding to this disturbance.

Without eddy flux forcing, the prehurricane developing vortex fails to intensify into a hurricane, even after 20 days of integration. With the observed eddy fluxes of momentum, the same initial vortex intensifies rapidly, reaching hurricane strength within 4 days. Moreover, because of the weak and diffuse pattern of the eddy fluxes of momentum in non-developing tropical disturbances, the initial vortex characterizing these disturbances also fails to develop into a hurricane.

The kinetic energy budgets corresponding to the integrations with the composite developing and non-developing disturbances are presented as a function of time. The calculations reveal that, during the early stages of development of the model hurricane, the conversion (*E _{k}
*) from eddy kinetic energy to the kinetic energy of the mean hurricane circulation is larger than the conversion (

*C*) from potential to kinetic energy. The eddy process is, therefore, directly responsible for the early growth of the model hurricane. This is followed by an explosive increase in the rate of conversion from potential to kinetic energy and in the rate of kinetic energy dissipation (

_{A}*F*). During the latter period,

*C*and

_{A}*F*become almost an order of magnitude greater than the peak attained earlier by

*E*, and the kinetic energy tendency reaches its peak. Without the eddy momentum flux forcing, no such explosive growth takes place.

_{k}The results of these integrations provide evidence that properly organized large-scale eddy fluxes of momentum may be an essential ingredient id the development of Atlantic hurricanes.

## Abstract

The results of numerical integrations of Sundqvist's (1970) symmetric model for hurricane development modified to include parameterized large-scale eddy fluxes of momentum are presented. The initial wind and moisture distributions, and the prescribed eddy fluxes of momentum, were taken from atmospheric observations of Atlantic developing (prehurricane) and non-developing tropical disturbances as composited by McBride (1981a,b) and McBride and Zehr (1981). For the purposes of the present study, the data for individual stages in the evolution of developing and non-developing disturbances were combined to form a single composite developing hurricane and a single composite non-developing disturbance.

The data reveal the presence of intense, well organized inward eddy fluxes of momentum in developing Atlantic hurricanes and weak, poorly organized fluxes in non-developing disturbances. In the developing disturbances, the eddy fluxes of momentum are organized such that they act as a forcing function for driving the radial circulation, drawing moist air in toward the center of the vortex in the lower troposphere and pumping drier air outward aloft, thereby providing fuel for the explosive growth of the hurricane. In order to test the efficacy of this mechanism, and of Ekman suction and cooperative instability, numerical integrations were performed using the data for the composite developing hurricane, with and without the observed eddy fluxes of momentum, and for the composite non-developing disturbance with the observed eddy fluxes corresponding to this disturbance.

Without eddy flux forcing, the prehurricane developing vortex fails to intensify into a hurricane, even after 20 days of integration. With the observed eddy fluxes of momentum, the same initial vortex intensifies rapidly, reaching hurricane strength within 4 days. Moreover, because of the weak and diffuse pattern of the eddy fluxes of momentum in non-developing tropical disturbances, the initial vortex characterizing these disturbances also fails to develop into a hurricane.

The kinetic energy budgets corresponding to the integrations with the composite developing and non-developing disturbances are presented as a function of time. The calculations reveal that, during the early stages of development of the model hurricane, the conversion (*E _{k}
*) from eddy kinetic energy to the kinetic energy of the mean hurricane circulation is larger than the conversion (

*C*) from potential to kinetic energy. The eddy process is, therefore, directly responsible for the early growth of the model hurricane. This is followed by an explosive increase in the rate of conversion from potential to kinetic energy and in the rate of kinetic energy dissipation (

_{A}*F*). During the latter period,

*C*and

_{A}*F*become almost an order of magnitude greater than the peak attained earlier by

*E*, and the kinetic energy tendency reaches its peak. Without the eddy momentum flux forcing, no such explosive growth takes place.

_{k}The results of these integrations provide evidence that properly organized large-scale eddy fluxes of momentum may be an essential ingredient id the development of Atlantic hurricanes.

## Abstract

The nonlinear effects of asymmetries associated with synoptic-scale waves in which hurricanes usually form are simulated by introducing steady-state and time varying eddy fluxes of angular momentum in a parameterized way into Sundqvist's (1970) symmetric model for hurricane development. The equations are integrated numerically using different initial conditions and different distributions of the parameterized eddy fluxes.

It is found that angular momentum flux convergences, with magnitudes comparable to those measured from atmospheric data, markedly accelerate hurricane development, and can initiate a model hurricane when the sea surface temperature is slightly subcritical such that the purely symmetric model fails to produce a vortex of hurricane intensity. Different distributions of the eddy flux of momentum produce different rates of growth, different final intensifies and different vortex sizes. The most effective distributions are those in which the vertical derivative of the angular momentum flux convergence is large near sea level, where it acts as a forcing function for the symmetric radial circulation, drawing moist boundary-layer air into the hurricane from the surroundings. In this way, it enhances the Ekman layer inflow, particularly at the early stages when the sea level vortex is weak. On the other hand, an angular momentum flux *divergence* produced by the eddies is found to suppress model hurricane development, even when the sea surface temperature is supercritical such that the purely symmetric model yields explosive hurricane growth. This is because it produces a radial circulation which opposes the Ekman layer inflow.

The contributions of the different terms in the kinetic energy equation in the purely symmetric integration are compared with those in one of the integrations with an eddy flux convergence of angular momentum. The calculations reveal that the kinetic energy production and dissipation are both larger in the latter case than in the former, and that the production exceeds the dissipation by a greater amount in the latter case, leading to a larger kinetic energy tendency and thereby more explosive hurricane growth.

## Abstract

The nonlinear effects of asymmetries associated with synoptic-scale waves in which hurricanes usually form are simulated by introducing steady-state and time varying eddy fluxes of angular momentum in a parameterized way into Sundqvist's (1970) symmetric model for hurricane development. The equations are integrated numerically using different initial conditions and different distributions of the parameterized eddy fluxes.

It is found that angular momentum flux convergences, with magnitudes comparable to those measured from atmospheric data, markedly accelerate hurricane development, and can initiate a model hurricane when the sea surface temperature is slightly subcritical such that the purely symmetric model fails to produce a vortex of hurricane intensity. Different distributions of the eddy flux of momentum produce different rates of growth, different final intensifies and different vortex sizes. The most effective distributions are those in which the vertical derivative of the angular momentum flux convergence is large near sea level, where it acts as a forcing function for the symmetric radial circulation, drawing moist boundary-layer air into the hurricane from the surroundings. In this way, it enhances the Ekman layer inflow, particularly at the early stages when the sea level vortex is weak. On the other hand, an angular momentum flux *divergence* produced by the eddies is found to suppress model hurricane development, even when the sea surface temperature is supercritical such that the purely symmetric model yields explosive hurricane growth. This is because it produces a radial circulation which opposes the Ekman layer inflow.

The contributions of the different terms in the kinetic energy equation in the purely symmetric integration are compared with those in one of the integrations with an eddy flux convergence of angular momentum. The calculations reveal that the kinetic energy production and dissipation are both larger in the latter case than in the former, and that the production exceeds the dissipation by a greater amount in the latter case, leading to a larger kinetic energy tendency and thereby more explosive hurricane growth.

## Abstract

During a traverse of the regular wave regime in a rotating, differentially heated annulus of viscous liquid (Prandtl number = 86.0), an asymmetrical wave pattern was observed in the transition region between 4 and 5 regular waves. The shape and time development of this pattern is interpreted as being primarily the result of the simultaneous presence of a 4-wave and a 5-wave pattern, and of the dispersion resulting from the relative motion of these two waves.

## Abstract

During a traverse of the regular wave regime in a rotating, differentially heated annulus of viscous liquid (Prandtl number = 86.0), an asymmetrical wave pattern was observed in the transition region between 4 and 5 regular waves. The shape and time development of this pattern is interpreted as being primarily the result of the simultaneous presence of a 4-wave and a 5-wave pattern, and of the dispersion resulting from the relative motion of these two waves.

## Abstract

The radial distribution of the azimuthally averaged temperature at one depth in a rotating, differentially heated annulus of fluid undergoing “amplitude vacillation” is measured with an array of 48 thermocouples suspended on fine wires in the fluid. The properties of amplitude vacillation and the feasibility of making measurements with multi-probe arrays (properly staggered in the fluid to avoid disrupting the organized flow) are discussed.

## Abstract

The radial distribution of the azimuthally averaged temperature at one depth in a rotating, differentially heated annulus of fluid undergoing “amplitude vacillation” is measured with an array of 48 thermocouples suspended on fine wires in the fluid. The properties of amplitude vacillation and the feasibility of making measurements with multi-probe arrays (properly staggered in the fluid to avoid disrupting the organized flow) are discussed.