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- Author or Editor: Ka Kit Tung x
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Abstract
A nongeostrophic theory of zonally averaged circulation is formulated using the nonlinear primitive equations on a sphere, taking advantage of the more direct relationship between the mean meridional circulation and diabatic heating rate which is available in isentropic coordinates. Possible differences between results of nongeostrophic theory and the commonly used geostrophic formulation are discussed concerning (i) the role of eddy forcing of the diabatic circulation, and (ii) the “nonlinear nearly inviscid” limit versus the geostrophic limit.
A set of general diagnostic tools comparable in scope to their geostrophic counterparts is given in Part I, including (i) a generalized definition of Eliassen–Palm flux divergence (without restriction to small amplitudes, to steady state or to adiabatic flows), the vanishing of which is a necessary condition for nonacceleration, (ii) a generalized nonlinear Taylor formula that relates the flux of Ertel's potential vorticity to the Eliassen–Palm flux divergence and (iii) a relationship between the Eliassen–Palm flux divergence and isentropic mixing coefficient, Kyy , used in chemical tracer transport equations in isentropic coordinates. From the mean momentum budget, we give in Part II in estimate of the Eliassen–Palm flux divergence using fitted “observed” field of net radiative heating rate. From this an estimate of the magnitude and latitudinal/seasonal variation of Kyy is also provided.
Abstract
A nongeostrophic theory of zonally averaged circulation is formulated using the nonlinear primitive equations on a sphere, taking advantage of the more direct relationship between the mean meridional circulation and diabatic heating rate which is available in isentropic coordinates. Possible differences between results of nongeostrophic theory and the commonly used geostrophic formulation are discussed concerning (i) the role of eddy forcing of the diabatic circulation, and (ii) the “nonlinear nearly inviscid” limit versus the geostrophic limit.
A set of general diagnostic tools comparable in scope to their geostrophic counterparts is given in Part I, including (i) a generalized definition of Eliassen–Palm flux divergence (without restriction to small amplitudes, to steady state or to adiabatic flows), the vanishing of which is a necessary condition for nonacceleration, (ii) a generalized nonlinear Taylor formula that relates the flux of Ertel's potential vorticity to the Eliassen–Palm flux divergence and (iii) a relationship between the Eliassen–Palm flux divergence and isentropic mixing coefficient, Kyy , used in chemical tracer transport equations in isentropic coordinates. From the mean momentum budget, we give in Part II in estimate of the Eliassen–Palm flux divergence using fitted “observed” field of net radiative heating rate. From this an estimate of the magnitude and latitudinal/seasonal variation of Kyy is also provided.
Abstract
A zonally averaged model of stratospheric tracer transport is formulated in isentropic coordinated There are some conceptual and computational advantages, as well as some disadvantages in adopting the potential temperature, instead of pressure, as the vertical coordinate. The main disadvantage is that the “density” (mass per unit coordinate volume) in isentropic coordinates is no longer a constant as in the pressure coordinate system under the hydrostatic approximation. However, it can be shown that this density effect is almost negligible in the calculation of the mean diabatic circulation and the eddy advective transports. What is gained by adopting the new formulation is a conceptually simpler picture of the interplay of diabatic and adiabatic process in the transport of tracers. Mean diabatic heating (cooling) forces a direct rising (descending) mean mass flow. Along the streamlines of this mean mass circulation, tracers are advected in the mean. These surfaces slope downward and poleward in the lower stratosphere. In addition to advection, tracers are also dispersed from their mean path by transient adiabatic processes in a direction parallel to the local isentropic surface. As a result, the lines of mean constant tracer mass mixing ratio slope less steeply than the mean streamlines, but more steeply than the isentropic surfaces. The effect of eddy transport on chemically reacting minor constituent gases is also discussed.
Abstract
A zonally averaged model of stratospheric tracer transport is formulated in isentropic coordinated There are some conceptual and computational advantages, as well as some disadvantages in adopting the potential temperature, instead of pressure, as the vertical coordinate. The main disadvantage is that the “density” (mass per unit coordinate volume) in isentropic coordinates is no longer a constant as in the pressure coordinate system under the hydrostatic approximation. However, it can be shown that this density effect is almost negligible in the calculation of the mean diabatic circulation and the eddy advective transports. What is gained by adopting the new formulation is a conceptually simpler picture of the interplay of diabatic and adiabatic process in the transport of tracers. Mean diabatic heating (cooling) forces a direct rising (descending) mean mass flow. Along the streamlines of this mean mass circulation, tracers are advected in the mean. These surfaces slope downward and poleward in the lower stratosphere. In addition to advection, tracers are also dispersed from their mean path by transient adiabatic processes in a direction parallel to the local isentropic surface. As a result, the lines of mean constant tracer mass mixing ratio slope less steeply than the mean streamlines, but more steeply than the isentropic surfaces. The effect of eddy transport on chemically reacting minor constituent gases is also discussed.
Abstract
The problem of barotropic instability of zonal flows to infinitesimal normal-mode perturbations is considered. The zonal flow is assumed to be continuous. but is allowed to be monotonic or nonmonotonic, and can have one or more inflection-points (which are the zeroes of the mean vorticity gradient., the zeroes are allowed to be of any order). A sufficient condition for instability is derived for this general flow profile. The present result complements the condition for stability found by Arnol'd (1965).
Abstract
The problem of barotropic instability of zonal flows to infinitesimal normal-mode perturbations is considered. The zonal flow is assumed to be continuous. but is allowed to be monotonic or nonmonotonic, and can have one or more inflection-points (which are the zeroes of the mean vorticity gradient., the zeroes are allowed to be of any order). A sufficient condition for instability is derived for this general flow profile. The present result complements the condition for stability found by Arnol'd (1965).
Abstract
Through a critical analysis of the convergence properties of spectral series, it is shown that Clark's method of solution leads to a divergent series; hence all his recent results on quasi-geostrophic wave propagation in distorted background flows are erroneous. A general condition for convergence is derived. The convergent solution (if it exists) to a general second-order recurrence formula is given, which is then applied to Clark's problem, yielding an exact closed form solution. The solution consists of an interacting trio of waves whose wavenumbers add up to zero. With results thus obtained, it is found that the propagation of wavenumber 2 disturbances is not affected by wavenumber 1 finite-amplitude distortions in the background flow, in disagreement with the result of Clark.
Abstract
Through a critical analysis of the convergence properties of spectral series, it is shown that Clark's method of solution leads to a divergent series; hence all his recent results on quasi-geostrophic wave propagation in distorted background flows are erroneous. A general condition for convergence is derived. The convergent solution (if it exists) to a general second-order recurrence formula is given, which is then applied to Clark's problem, yielding an exact closed form solution. The solution consists of an interacting trio of waves whose wavenumbers add up to zero. With results thus obtained, it is found that the propagation of wavenumber 2 disturbances is not affected by wavenumber 1 finite-amplitude distortions in the background flow, in disagreement with the result of Clark.
Abstract
For quasi-geostrophic stationary long waves forced by topography, the nonlinear lower boundary condition is derived in terms of the geopotential height and compared with the linearized version. The common practice of replacing terms describing the flow over and around a mountain by upstream zonal flow over the mountain and evaluating the resulting condition at sea level is found to be a good approximation for the cases considered and does not need to be modified as sometimes suggested. Specifically, it is found that this approximation does not affect, for most cases, the lower boundary condition expressed in terms of the geopotential height provided that the stationary wave is not near resonance. At resonance, the eddy advection terms may become important for large-amplitude waves when dissipation and surface diabatic heating are taken into account
Abstract
For quasi-geostrophic stationary long waves forced by topography, the nonlinear lower boundary condition is derived in terms of the geopotential height and compared with the linearized version. The common practice of replacing terms describing the flow over and around a mountain by upstream zonal flow over the mountain and evaluating the resulting condition at sea level is found to be a good approximation for the cases considered and does not need to be modified as sometimes suggested. Specifically, it is found that this approximation does not affect, for most cases, the lower boundary condition expressed in terms of the geopotential height provided that the stationary wave is not near resonance. At resonance, the eddy advection terms may become important for large-amplitude waves when dissipation and surface diabatic heating are taken into account
Abstract
The time-dependent Hadley circulation is studied numerically in a nonlinear, nearly inviscid, axially symmetric primitive equation model, with the heating varying periodically on an annual cycle. The annual average of the Hadley circulation strength in this model with time-dependent heating is about a factor of 2 stronger than the steady-state response to the annual mean heating and is closer to the observed strength in the real atmosphere. This is caused by the fact that heating centered off-equator tends to produce stronger meridional circulation in the winter hemisphere than in the case when the heating maximum is located at the equator, as pointed out previously by Lindzen and Hou. However, unlike the steady-state solutions, there is no abrupt change as the heating center is moved off the equator.
The temperature response in this time-dependent model is simple to understand. In the tropical region, where there is a variable, but persistent, Hadley circulation, the temperature is homogenized latitudinally. In the high-latitude region, where there is no meridional circulation (in the absence of the eddies), the temperature response goes through an annual cycle with a phase lag relative to the phase of the heating. This response is as predicted by the simple time-dependent temperature equation in the absence of meridional circulation.
Abstract
The time-dependent Hadley circulation is studied numerically in a nonlinear, nearly inviscid, axially symmetric primitive equation model, with the heating varying periodically on an annual cycle. The annual average of the Hadley circulation strength in this model with time-dependent heating is about a factor of 2 stronger than the steady-state response to the annual mean heating and is closer to the observed strength in the real atmosphere. This is caused by the fact that heating centered off-equator tends to produce stronger meridional circulation in the winter hemisphere than in the case when the heating maximum is located at the equator, as pointed out previously by Lindzen and Hou. However, unlike the steady-state solutions, there is no abrupt change as the heating center is moved off the equator.
The temperature response in this time-dependent model is simple to understand. In the tropical region, where there is a variable, but persistent, Hadley circulation, the temperature is homogenized latitudinally. In the high-latitude region, where there is no meridional circulation (in the absence of the eddies), the temperature response goes through an annual cycle with a phase lag relative to the phase of the heating. This response is as predicted by the simple time-dependent temperature equation in the absence of meridional circulation.
Abstract
A two-and-a-half-dimensional interactive stratospheric model (i.e., a zonally averaged dynamical-chemical model combined with a truncated spectral dynamical model), whose equatorial zonal wind was relaxed toward the observed Singapore zonal wind, was able to reproduce much of the observed quasi-biennial oscillation (QBO) variability in the column ozone, in its vertical distribution in the low and middle latitudes, and also in the high southern polar latitudes. To reveal the mechanisms responsible for producing the modeled QBO signal over the globe, several control runs were also performed. The authors find that the ozone variability in the lower stratosphere—and hence also in the column—is determined mainly by two dynamical mechanisms. In the low to midlatitudes it is created by a “direct QBO circulation.” Unlike the classic picture of a nonseasonal two-cell QBO circulation symmetric about the equator, a more correct picture is a direct QBO circulation that is strongly seasonal, driven by the seasonality in diabatic heating, which is very weak in the summer hemisphere and strong in the winter hemisphere at low and midlatitudes. This anomalous circulation is what is responsible for creating the ozone anomaly at low and midlatitudes. Transport by the climatological circulation and diffusion is found to be ineffective. At high latitudes, there is again a circulation anomaly, but here it is induced by the modulation of the planetary wave potential vorticity flux by the QBO. This so-called Holton–Tan mechanism is responsible for most of the QBO ozone signal poleward of 60°. During spring in the modeled northern polar region, chaotic behavior is another important source of interannual variability, in addition to the interannual variability of planetary wave sources in the troposphere previously studied by the authors.
Abstract
A two-and-a-half-dimensional interactive stratospheric model (i.e., a zonally averaged dynamical-chemical model combined with a truncated spectral dynamical model), whose equatorial zonal wind was relaxed toward the observed Singapore zonal wind, was able to reproduce much of the observed quasi-biennial oscillation (QBO) variability in the column ozone, in its vertical distribution in the low and middle latitudes, and also in the high southern polar latitudes. To reveal the mechanisms responsible for producing the modeled QBO signal over the globe, several control runs were also performed. The authors find that the ozone variability in the lower stratosphere—and hence also in the column—is determined mainly by two dynamical mechanisms. In the low to midlatitudes it is created by a “direct QBO circulation.” Unlike the classic picture of a nonseasonal two-cell QBO circulation symmetric about the equator, a more correct picture is a direct QBO circulation that is strongly seasonal, driven by the seasonality in diabatic heating, which is very weak in the summer hemisphere and strong in the winter hemisphere at low and midlatitudes. This anomalous circulation is what is responsible for creating the ozone anomaly at low and midlatitudes. Transport by the climatological circulation and diffusion is found to be ineffective. At high latitudes, there is again a circulation anomaly, but here it is induced by the modulation of the planetary wave potential vorticity flux by the QBO. This so-called Holton–Tan mechanism is responsible for most of the QBO ozone signal poleward of 60°. During spring in the modeled northern polar region, chaotic behavior is another important source of interannual variability, in addition to the interannual variability of planetary wave sources in the troposphere previously studied by the authors.
Abstract
The process of baroclinic equilibration in the atmosphere is investigated using a high-resolution two-layer quasigeostrophic model in a β-plane channel. One simple channel geometry is investigated for which only two zonal waves are initially unstable, with the shorter being linearly more unstable but nonlinearly less effective. It is discovered that the mechanism of nonlinear baroclinic adjustment, formerly proposed by Cehelsky and Tung, including a nonlinear wavenumber selection process, can explain the equilibration at all levels of forcing for this case. At small forcings the most unstable wave dominates the heat flux, consistent with the quasi-linear equilibration of Stone’s simple baroclinic adjustment. At high forcings the longer, less unstable wave dominates, and the equilibration involves both quasi-linear dynamics by this dominant wave and nonlinear transfer from the shorter to the longer wave. For intermediate forcings there is a transition between the low and high regimes; no single wave dominates.
At every forcing except in the intermediate regime there is critical equilibration by the dominant wave. For intermediate forcings, the model equilibrates at a value between the critical shear of the two waves.
The wavenumber selection process involves a threshold of heat transport for each wave. Above this, the amplitude of the wave would be so large as to cause itself to break and saturate. The shorter wave’s threshold occurs at moderate forcings, at which point it relinquishes dominance to the longer wave. A method for calculating these thresholds is proposed, which involves only robust features of the equilibrium.
Abstract
The process of baroclinic equilibration in the atmosphere is investigated using a high-resolution two-layer quasigeostrophic model in a β-plane channel. One simple channel geometry is investigated for which only two zonal waves are initially unstable, with the shorter being linearly more unstable but nonlinearly less effective. It is discovered that the mechanism of nonlinear baroclinic adjustment, formerly proposed by Cehelsky and Tung, including a nonlinear wavenumber selection process, can explain the equilibration at all levels of forcing for this case. At small forcings the most unstable wave dominates the heat flux, consistent with the quasi-linear equilibration of Stone’s simple baroclinic adjustment. At high forcings the longer, less unstable wave dominates, and the equilibration involves both quasi-linear dynamics by this dominant wave and nonlinear transfer from the shorter to the longer wave. For intermediate forcings there is a transition between the low and high regimes; no single wave dominates.
At every forcing except in the intermediate regime there is critical equilibration by the dominant wave. For intermediate forcings, the model equilibrates at a value between the critical shear of the two waves.
The wavenumber selection process involves a threshold of heat transport for each wave. Above this, the amplitude of the wave would be so large as to cause itself to break and saturate. The shorter wave’s threshold occurs at moderate forcings, at which point it relinquishes dominance to the longer wave. A method for calculating these thresholds is proposed, which involves only robust features of the equilibrium.
Abstract
By forcing an interactive chemical–dynamical model of the stratosphere with the observed Singapore zonal winds and with the observed daily varying planetary wave heights just above the tropopause over the period from March 1980 to February 1993, much of the observed interannual variability (IAV) in the monthly mean ozone column from pole to pole was able to be reproduced. The best correlations were obtained equatorward of 50° during winter and autumn in both hemispheres. These correlations were due to the model’s interaction with the specified equatorial zonal wind. In the Southern Hemisphere, the observed high-latitude ozone anomaly in November (the month with the largest anomaly) is well anticorrelated with the 20-mb Singapore wind, and correlations suggest the anomaly propagates poleward from 45°S in August and possibly even from 15°S in June. The model reproduces this behavior well. In the Northern Hemisphere (NH), by contrast, the observed high-latitude ozone anomaly is not well correlated with the equatorial quasi-biennial oscillation nor does it propagate from lower latitudes. Model results demonstrate that the NH high-latitude ozone anomaly is influenced strongly by the IAV in the forcing of the planetary waves. Model results show a large IAV in the ozone column during polar night in the NH (where the Total Ozone Mapping Spectrometer is not able to observe) that is due to the IAV in the forcing of planetary waves.
Abstract
By forcing an interactive chemical–dynamical model of the stratosphere with the observed Singapore zonal winds and with the observed daily varying planetary wave heights just above the tropopause over the period from March 1980 to February 1993, much of the observed interannual variability (IAV) in the monthly mean ozone column from pole to pole was able to be reproduced. The best correlations were obtained equatorward of 50° during winter and autumn in both hemispheres. These correlations were due to the model’s interaction with the specified equatorial zonal wind. In the Southern Hemisphere, the observed high-latitude ozone anomaly in November (the month with the largest anomaly) is well anticorrelated with the 20-mb Singapore wind, and correlations suggest the anomaly propagates poleward from 45°S in August and possibly even from 15°S in June. The model reproduces this behavior well. In the Northern Hemisphere (NH), by contrast, the observed high-latitude ozone anomaly is not well correlated with the equatorial quasi-biennial oscillation nor does it propagate from lower latitudes. Model results demonstrate that the NH high-latitude ozone anomaly is influenced strongly by the IAV in the forcing of the planetary waves. Model results show a large IAV in the ozone column during polar night in the NH (where the Total Ozone Mapping Spectrometer is not able to observe) that is due to the IAV in the forcing of planetary waves.
