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## Abstract

Obserations of longitudinally-averaged zonal flows in the atmosphere and ocean tend to display north–south symmetry about the equator, with a characteristic wind maximum or minimum, and therefore little horizontal wind shear locally near the equator. It is shown that this configuration is required for balanced flow on a sphere to be inertially stable. If dissipation can be neglected, any horizontal wind shear at the equator will cause inertial instability to develop. effectively eliminating the horizontal shear. It follows that the potential vorticity (*q*) must vanish at the equator for the symmetric circulation to be stable; and that it must increase to the north and decrease to the south. Balanced cross-equatorial flow can occur only if there is a north–south gradient in the torque or the diabatic heating at the equator. These conclusions are obtained under the assumption of a balanced zonal flow; i.e., acceleration and dissipation are explicitly neglected in the meridional momentum equation.

The characteristics of the equatorial symmetric instability that develops if the mean flow is horizontally sheared at the equator are investigated. The analysis with Rayleigh friction and Newtonian cooling (i.e., scale-independent dissipation) extends We treatment of inviscid instability on the equatorial beta-plane by Dunkerton (1981). Quadratic as well as linear shear is treated, thereby enabling application to tropical jets.

The instability is confined to the region in which the vertical component of absolute vorticity is of opposite sign to the local Coriolis parameter, i.e., where the square of the inertial frequency *f*(*gz;*+) is negative. The mode of greatest instability is a moridional overturning with a single cell in the horizontal dimension which tends to mix angular momentum, thereby eliminating the horizontal gradient of angular momentum at the equator. With the scale-independent parameterization of mechanical and thermal dissipation, the mixing occurs most readily at the smallest vertical scales and the gravest (*n*=0) meridional mode. When the low curvature is much less than β, the maximum growth rate for symmetric instability is approximately one-half the magnitude of the relative vorticity of the mean flow at the equator minus the mechanical dissipation rate. Hence the horizontal shear at the equator must exceed twice the Rayleigh friction coefficient for instability. Thermal dissipation does not affect the instability criterion.

Many recent studies have been undertaken which investigate the effect of mean zonal flow on tropical waves and instabilities. A consequence of the present analysis is that a stationary. non-dissipative basic mate flow with horizontal shear at the equator is not an appropriate basic state for neutrally propagating waves because it is not a stable solution of the symmetric governing equations. The instability tends to eliminate the latitudinal shear of the zonal flow at the equator if mechanical dissipation is not 100 great. While Dunkerton (1981) focused on the ramifications for the middle atmosphere, this study applies the results primarily to tropospheric and oceanic circulations.

## Abstract

Obserations of longitudinally-averaged zonal flows in the atmosphere and ocean tend to display north–south symmetry about the equator, with a characteristic wind maximum or minimum, and therefore little horizontal wind shear locally near the equator. It is shown that this configuration is required for balanced flow on a sphere to be inertially stable. If dissipation can be neglected, any horizontal wind shear at the equator will cause inertial instability to develop. effectively eliminating the horizontal shear. It follows that the potential vorticity (*q*) must vanish at the equator for the symmetric circulation to be stable; and that it must increase to the north and decrease to the south. Balanced cross-equatorial flow can occur only if there is a north–south gradient in the torque or the diabatic heating at the equator. These conclusions are obtained under the assumption of a balanced zonal flow; i.e., acceleration and dissipation are explicitly neglected in the meridional momentum equation.

The characteristics of the equatorial symmetric instability that develops if the mean flow is horizontally sheared at the equator are investigated. The analysis with Rayleigh friction and Newtonian cooling (i.e., scale-independent dissipation) extends We treatment of inviscid instability on the equatorial beta-plane by Dunkerton (1981). Quadratic as well as linear shear is treated, thereby enabling application to tropical jets.

The instability is confined to the region in which the vertical component of absolute vorticity is of opposite sign to the local Coriolis parameter, i.e., where the square of the inertial frequency *f*(*gz;*+) is negative. The mode of greatest instability is a moridional overturning with a single cell in the horizontal dimension which tends to mix angular momentum, thereby eliminating the horizontal gradient of angular momentum at the equator. With the scale-independent parameterization of mechanical and thermal dissipation, the mixing occurs most readily at the smallest vertical scales and the gravest (*n*=0) meridional mode. When the low curvature is much less than β, the maximum growth rate for symmetric instability is approximately one-half the magnitude of the relative vorticity of the mean flow at the equator minus the mechanical dissipation rate. Hence the horizontal shear at the equator must exceed twice the Rayleigh friction coefficient for instability. Thermal dissipation does not affect the instability criterion.

Many recent studies have been undertaken which investigate the effect of mean zonal flow on tropical waves and instabilities. A consequence of the present analysis is that a stationary. non-dissipative basic mate flow with horizontal shear at the equator is not an appropriate basic state for neutrally propagating waves because it is not a stable solution of the symmetric governing equations. The instability tends to eliminate the latitudinal shear of the zonal flow at the equator if mechanical dissipation is not 100 great. While Dunkerton (1981) focused on the ramifications for the middle atmosphere, this study applies the results primarily to tropospheric and oceanic circulations.

## Abstract

The budgets of vorticity, momentum and divergence are evaluated for the average synoptic-scale wave composited by Thompson *et al*. (1979) from Phase III data of the GARP Atlantic Tropical Experiment (GATE). The data are analyzed into a Phase III mean component and a time-varying wave component. The geopotential field which is required for the momentum and divergence budgets is obtained from vertical integration of the hydrostatic equation.

It is found that nonlinear terms in all wave budgets tend to be small, so that the disturbances are governed by linear dynamics. In contrast with the composite wave analyzed by Shapiro (1978), the waves are not approximately advected by the horizontal wind. Accelerations have the same amplitude as the Coriolis force in the momentum balances of the disturbances, indicating that the disturbances are not quasi-geostrophic. However, thermal wind balance appears to be a good approximation for the mean zonal wind. None of the usual simplifications to the divergence equation is satisfactory for the traveling waves.

Each of the large-scale budgets has a significant residual imbalance, suggesting that subsynoptic-scale circulations strongly affect the wave dynamics. Cumulus transports must be parameterized in the dynamic budgets for a proper treatment of the dynamics and energetics of the wave disturbances. Neither Rayleigh friction nor a simple vertical momentum exchange is an adequate parameterization of the apparent momentum sources.

## Abstract

The budgets of vorticity, momentum and divergence are evaluated for the average synoptic-scale wave composited by Thompson *et al*. (1979) from Phase III data of the GARP Atlantic Tropical Experiment (GATE). The data are analyzed into a Phase III mean component and a time-varying wave component. The geopotential field which is required for the momentum and divergence budgets is obtained from vertical integration of the hydrostatic equation.

It is found that nonlinear terms in all wave budgets tend to be small, so that the disturbances are governed by linear dynamics. In contrast with the composite wave analyzed by Shapiro (1978), the waves are not approximately advected by the horizontal wind. Accelerations have the same amplitude as the Coriolis force in the momentum balances of the disturbances, indicating that the disturbances are not quasi-geostrophic. However, thermal wind balance appears to be a good approximation for the mean zonal wind. None of the usual simplifications to the divergence equation is satisfactory for the traveling waves.

Each of the large-scale budgets has a significant residual imbalance, suggesting that subsynoptic-scale circulations strongly affect the wave dynamics. Cumulus transports must be parameterized in the dynamic budgets for a proper treatment of the dynamics and energetics of the wave disturbances. Neither Rayleigh friction nor a simple vertical momentum exchange is an adequate parameterization of the apparent momentum sources.

## Abstract

The movement of a set of Lagrangian parcels in the two-dimensional pressure field associated with a tropical convective line is considered. The results are compared with observational characteristics of GATE quasi-two-dimensional convective lines. It is shown that for the composite, slow-moving GATE convective line it is necessary to account for horizontal pressure gradients in order to obtain realistic momentum flux. The role of vertical pressure gradients and initial conditions for an air parcel's movement is studied. Based on the results, some suggestions concerning proper formulation of parameterization of the convective momentum flux are made.

## Abstract

The movement of a set of Lagrangian parcels in the two-dimensional pressure field associated with a tropical convective line is considered. The results are compared with observational characteristics of GATE quasi-two-dimensional convective lines. It is shown that for the composite, slow-moving GATE convective line it is necessary to account for horizontal pressure gradients in order to obtain realistic momentum flux. The role of vertical pressure gradients and initial conditions for an air parcel's movement is studied. Based on the results, some suggestions concerning proper formulation of parameterization of the convective momentum flux are made.

## Abstract

The paper examines the role of the development of outflow-layer instabilities on the motion of tropical cyclones. The influence of barotropic instability is examined by comparing the time changes in the storm tracks with the frequencies of free, unstable barotropic modes. For intense vortices barotropic instability is shown to contribute to the slow (periods of a few days) trochoidal motion of a cyclone. The development of instability depends on the horizontal distribution and frequency of environmental forcing. The strongest response occurs when the frequency of the forcing matches the frequency of an unstable mode.

## Abstract

The paper examines the role of the development of outflow-layer instabilities on the motion of tropical cyclones. The influence of barotropic instability is examined by comparing the time changes in the storm tracks with the frequencies of free, unstable barotropic modes. For intense vortices barotropic instability is shown to contribute to the slow (periods of a few days) trochoidal motion of a cyclone. The development of instability depends on the horizontal distribution and frequency of environmental forcing. The strongest response occurs when the frequency of the forcing matches the frequency of an unstable mode.

## Abstract

Measuremenits of the momentum transport in tropical convective lines suggest that horizontal momentum can be generated by the pressure low located near the center of the convective part of the line.

A simple convective parameterization is used to evaluate this effect. The parameterization is a version of the Fritsch and Chappell scheme, modified in order to evaluate the influence of the horizontal pressure gradients on momentum transport. The results suggest that in modeling of convective (particularly slow-moving) lines with 20 km resolution, subgrid horizontal pressure gradients should be taken into account. Sensitivity studies show that the magnitude of the calculated momentum flux strongly depends on the average vertical velocity and the vertical velocity in clouds.

## Abstract

Measuremenits of the momentum transport in tropical convective lines suggest that horizontal momentum can be generated by the pressure low located near the center of the convective part of the line.

A simple convective parameterization is used to evaluate this effect. The parameterization is a version of the Fritsch and Chappell scheme, modified in order to evaluate the influence of the horizontal pressure gradients on momentum transport. The results suggest that in modeling of convective (particularly slow-moving) lines with 20 km resolution, subgrid horizontal pressure gradients should be taken into account. Sensitivity studies show that the magnitude of the calculated momentum flux strongly depends on the average vertical velocity and the vertical velocity in clouds.

## Abstract

We investigate the temporal and spatial characteristics of unstable normal modes in a horizontally sheared flow on a sphere using the shallow water equations. Both inertial and barotropic instabilities are identified in cases where the appropriate necessary conditions are satisfied.

A primary focus is determining what conditions favor asymmetric modes of inertial instability rather than symmetric modes. With the Bickley jet profile, the region of instability [*f*(*f* + &xi) ≤ 0] is confined to the anticyclonic side of the jet in a limited region. We find that symmetric instability is preferred only for modes of very small vertical wide, for which the pressure gradient force is secondary. Relatively small dissipation is needed to stabilize these modes. With deeper vertical scales, asymmetric instabilities are preferred in which the zonal scale of the instability is comparable to the width of the unstable region.

This study extends previous results for linear shear on an equatorial beta plane to the midlatitude jet case. Our results suggest that deep atmospheric circulations in spatially confined regions of negative potential vorticity may develop as asymmetric rather than symmetric instabilities.

## Abstract

We investigate the temporal and spatial characteristics of unstable normal modes in a horizontally sheared flow on a sphere using the shallow water equations. Both inertial and barotropic instabilities are identified in cases where the appropriate necessary conditions are satisfied.

A primary focus is determining what conditions favor asymmetric modes of inertial instability rather than symmetric modes. With the Bickley jet profile, the region of instability [*f*(*f* + &xi) ≤ 0] is confined to the anticyclonic side of the jet in a limited region. We find that symmetric instability is preferred only for modes of very small vertical wide, for which the pressure gradient force is secondary. Relatively small dissipation is needed to stabilize these modes. With deeper vertical scales, asymmetric instabilities are preferred in which the zonal scale of the instability is comparable to the width of the unstable region.

This study extends previous results for linear shear on an equatorial beta plane to the midlatitude jet case. Our results suggest that deep atmospheric circulations in spatially confined regions of negative potential vorticity may develop as asymmetric rather than symmetric instabilities.

## Abstract

The linear, zonally symmetric modes of the basic state of a Hadley cell are examined. We find that the inclusion of the divergent basic state leads to the formation of a new class of slowly oscillating modes, some of which have periods in the range of 40–50 days. The modes have many features in common with the observed tropical 40–50-day oscillation; however, an explanation for the observed fluctuations in convective cloudiness remains a topic for future work.

## Abstract

The linear, zonally symmetric modes of the basic state of a Hadley cell are examined. We find that the inclusion of the divergent basic state leads to the formation of a new class of slowly oscillating modes, some of which have periods in the range of 40–50 days. The modes have many features in common with the observed tropical 40–50-day oscillation; however, an explanation for the observed fluctuations in convective cloudiness remains a topic for future work.

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## Abstract

The barotropic instability of basic states with downstream and asymmetric cross-stream variations is investigated using the linearized, non-divergent barotropic vorticity equation in a periodic beta channel. A single expression for streamfunction defines the basic state; the variations are introduced by parameter changes. Eigenvalue and time integration methods are employed to determine the instabilities.

When downstream variation is present, the maximum amplitude of the unstable streamfunction is always located downstream from the location of maximum latitudinal shear. This occurs because the disturbance propagates through a range of longitudes where the flow is locally barotropically unstable. The instability is sensitive to the degree of downstream variation. When there is strong variation there is more concentration of streamfunction amplitude in the regions downstream of the jet maximum, smaller disturbance scales and smaller growth rates than when the flow is more parallel. The smaller growth rate for the downstream variation case results from the propagating disturbance being subjected to strong shear for only a finite time before moving into regions of lesser shear where it cannot grow as fast. Asymmetric cross-stream variation has little effect on the growth rate and frequency of the most unstable mode but significantly affects the structure of the instability. Larger amplitude, and more pronounced tilt opposite to the shear occur on the side of the jet with the strongest shear. This can be expected from a theoretical consideration of the energetics. Instabilities in the presence of both downstream and asymmetric cross-stream variations combine the effects of each of those individual kinds of variations.

The time integration and eigenvalue methods compare very well; in fact, they complement one another. Both may be needed for a thorough understanding of these types of instabilities.

Preliminary nonlinear calculations show that as the disturbances grow to finite amplitude, they split, with high centers moving to the north and low centers to the south. The size of the disturbances increases as well. Downstream variation causes split centers of significant amplitude to be concentrated in the eastern part of the channel while cross-stream asymmetry introduces an asymmetry in the strength of the highs and lows. Additional studies are proposed to investigate the hypothesis that the development of blocking patterns may depend crucially on whether the large-scale flow is conducive to instabilities with blocking characteristics.

## Abstract

The barotropic instability of basic states with downstream and asymmetric cross-stream variations is investigated using the linearized, non-divergent barotropic vorticity equation in a periodic beta channel. A single expression for streamfunction defines the basic state; the variations are introduced by parameter changes. Eigenvalue and time integration methods are employed to determine the instabilities.

When downstream variation is present, the maximum amplitude of the unstable streamfunction is always located downstream from the location of maximum latitudinal shear. This occurs because the disturbance propagates through a range of longitudes where the flow is locally barotropically unstable. The instability is sensitive to the degree of downstream variation. When there is strong variation there is more concentration of streamfunction amplitude in the regions downstream of the jet maximum, smaller disturbance scales and smaller growth rates than when the flow is more parallel. The smaller growth rate for the downstream variation case results from the propagating disturbance being subjected to strong shear for only a finite time before moving into regions of lesser shear where it cannot grow as fast. Asymmetric cross-stream variation has little effect on the growth rate and frequency of the most unstable mode but significantly affects the structure of the instability. Larger amplitude, and more pronounced tilt opposite to the shear occur on the side of the jet with the strongest shear. This can be expected from a theoretical consideration of the energetics. Instabilities in the presence of both downstream and asymmetric cross-stream variations combine the effects of each of those individual kinds of variations.

The time integration and eigenvalue methods compare very well; in fact, they complement one another. Both may be needed for a thorough understanding of these types of instabilities.

Preliminary nonlinear calculations show that as the disturbances grow to finite amplitude, they split, with high centers moving to the north and low centers to the south. The size of the disturbances increases as well. Downstream variation causes split centers of significant amplitude to be concentrated in the eastern part of the channel while cross-stream asymmetry introduces an asymmetry in the strength of the highs and lows. Additional studies are proposed to investigate the hypothesis that the development of blocking patterns may depend crucially on whether the large-scale flow is conducive to instabilities with blocking characteristics.

## Abstract

Dynamic budgets of an average synoptic-scale wave have been made by Stevens (1979) from GARP Atlantic Tropical Experiment Phase III B- and A/B-scale data composited by Thompson *et al*. (1979). In the present study the apparent sources of momentum and vorticity, computed from the large-scale budgets, are compared with parameterized sources from independently derived cumulus mass fluxes and one-dimensional steady-state cloud models. The cloud models include spectral and bulk, as well as single-cloud models. The cumulus mass fluxes are determined from a thermodynamic budget analysis of Johnson (1 978).

A simple single-cloud model is found to adequately account for the net effect of the cumulus transport and production of vorticity. The one-dimensional cloud models, however, do not account for the apparent momentum source in the upper troposphere. An evaluation is made of the sensitivity of the results to the assumed cloud-base vorticity and radiative heating rate. The limitations of the simple cloud models for the parameterization of convective effects in both the momentum and vorticity budgets are discussed.

## Abstract

Dynamic budgets of an average synoptic-scale wave have been made by Stevens (1979) from GARP Atlantic Tropical Experiment Phase III B- and A/B-scale data composited by Thompson *et al*. (1979). In the present study the apparent sources of momentum and vorticity, computed from the large-scale budgets, are compared with parameterized sources from independently derived cumulus mass fluxes and one-dimensional steady-state cloud models. The cloud models include spectral and bulk, as well as single-cloud models. The cumulus mass fluxes are determined from a thermodynamic budget analysis of Johnson (1 978).

A simple single-cloud model is found to adequately account for the net effect of the cumulus transport and production of vorticity. The one-dimensional cloud models, however, do not account for the apparent momentum source in the upper troposphere. An evaluation is made of the sensitivity of the results to the assumed cloud-base vorticity and radiative heating rate. The limitations of the simple cloud models for the parameterization of convective effects in both the momentum and vorticity budgets are discussed.