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 Author or Editor: Richard S. Lindzen x
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Abstract
It is suggested that an additional source of semidiurnal forcing due to daily variations in tropical rainfall could correct the discrepancy between the calculated phase (boned on forcing due to insulation absorption by ozone and water vapor) and that observed for the surface pressure oscillation. It is also shown that the 180° phase shift in horizontal wind oscillations at 28 km which current calculations predict, but which is not observed, would he eliminated by the proposed additional forcing. The magnitude and phase of the required rainfall oscillation is calculated and found to be consistent with existing observations. Finally, it is shown that the convergence field due to the tide could not directly account for the rainfall oscillation.
Abstract
It is suggested that an additional source of semidiurnal forcing due to daily variations in tropical rainfall could correct the discrepancy between the calculated phase (boned on forcing due to insulation absorption by ozone and water vapor) and that observed for the surface pressure oscillation. It is also shown that the 180° phase shift in horizontal wind oscillations at 28 km which current calculations predict, but which is not observed, would he eliminated by the proposed additional forcing. The magnitude and phase of the required rainfall oscillation is calculated and found to be consistent with existing observations. Finally, it is shown that the convergence field due to the tide could not directly account for the rainfall oscillation.
Abstract
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Abstract
The classic Eady problem is modified to include β ≠ 0, but with the basic distributions of temperature and zonal flow adjusted to preserve zero meridional gradients of basicstate potential vorticity in the fluid interior. Much of the mathematical simplicity of the classic problem is retained; however, the results differ in important ways. Specifically, the instability now has a longwave cutoff in addition to the traditional shortwave cutoff. The former is associated with the fact that the phase speeds of the edge waves begin to differ so much as wavenumber is reduced that the two edge waves can no longer interact in order to form unstable modes. For the unstable modes, this manifests itself in that the steering level for unstable modes is always below the middle of the fluid and approaches the lower boundary near the longwave cutoff. Relatedly, the amplitude of the unstable geopotential perturbations is larger at the upper boundary than at the lower boundary. Finally, below the longwave cutoff, one of the neutral waves has a phase speed that becomes increasingly easterly as wavenumber decreases. This allows a resonant response to planetaryscale stationary forcing.
Abstract
The classic Eady problem is modified to include β ≠ 0, but with the basic distributions of temperature and zonal flow adjusted to preserve zero meridional gradients of basicstate potential vorticity in the fluid interior. Much of the mathematical simplicity of the classic problem is retained; however, the results differ in important ways. Specifically, the instability now has a longwave cutoff in addition to the traditional shortwave cutoff. The former is associated with the fact that the phase speeds of the edge waves begin to differ so much as wavenumber is reduced that the two edge waves can no longer interact in order to form unstable modes. For the unstable modes, this manifests itself in that the steering level for unstable modes is always below the middle of the fluid and approaches the lower boundary near the longwave cutoff. Relatedly, the amplitude of the unstable geopotential perturbations is larger at the upper boundary than at the lower boundary. Finally, below the longwave cutoff, one of the neutral waves has a phase speed that becomes increasingly easterly as wavenumber decreases. This allows a resonant response to planetaryscale stationary forcing.
Abstract
The author reexamines the Charney–Drazin problem with special attention to the concentration of potential vorticity gradient in the neighborhood of the tropopause. It is found that the degree of concentration has a profound effect on the response to stationary forcing, with greater concentration leading to greater response. Smoothing the concentration either analytically or numerically (by using coarser resolution) both lead to reduced responses, especially at higher wavenumbers. The results suggest a potentially important interaction between baroclinically unstable eddies and stationary waves. Insofar as the former act to mix potential vorticity in the troposphere while concentrating gradients at tropopause levels, they significantly condition the basic state for the latter.
Abstract
The author reexamines the Charney–Drazin problem with special attention to the concentration of potential vorticity gradient in the neighborhood of the tropopause. It is found that the degree of concentration has a profound effect on the response to stationary forcing, with greater concentration leading to greater response. Smoothing the concentration either analytically or numerically (by using coarser resolution) both lead to reduced responses, especially at higher wavenumbers. The results suggest a potentially important interaction between baroclinically unstable eddies and stationary waves. Insofar as the former act to mix potential vorticity in the troposphere while concentrating gradients at tropopause levels, they significantly condition the basic state for the latter.
Abstract
It is noted that gravity waves for which u¯−c (u¯=mean flow speed, c=wave phase speed) has a sharp minimum in the upper troposphere or lower stratosphere will have decaying amplitudes above this level despite exponentially decreasing mean density. Eventually this decay ceases and growth resumes. Thus, if a gravity breaks below the level of u¯−c_{min}, it will cease breaking above this level. Breaking will, however, resume at some higher level. This second breaking level is a lower bound for the level of breaking in the mesosphere since waves too weak to break where u¯−c=u¯−c_{min} will break at still higher levels in the mesosphere. Explicit calculations show the “second” breaking levels to be close to observed levels of mesospheric gravity wave breaking. Evidence is also cited for wave breaking in the lower atmosphere, and for the importance of this breaking in the momentum budget of the lower atmosphere.
Abstract
It is noted that gravity waves for which u¯−c (u¯=mean flow speed, c=wave phase speed) has a sharp minimum in the upper troposphere or lower stratosphere will have decaying amplitudes above this level despite exponentially decreasing mean density. Eventually this decay ceases and growth resumes. Thus, if a gravity breaks below the level of u¯−c_{min}, it will cease breaking above this level. Breaking will, however, resume at some higher level. This second breaking level is a lower bound for the level of breaking in the mesosphere since waves too weak to break where u¯−c=u¯−c_{min} will break at still higher levels in the mesosphere. Explicit calculations show the “second” breaking levels to be close to observed levels of mesospheric gravity wave breaking. Evidence is also cited for wave breaking in the lower atmosphere, and for the importance of this breaking in the momentum budget of the lower atmosphere.
Abstract
CISK (Conditional Instability of the Second Kind) is examined for internal waves where lowlevel convergence is due to the inviscid wave fields rather than to Ekman pumping.
It is found that CISKinduced waves must give rise to mean cumulus activity (since there are no negative clouds), and it is suggested that this mean activity plays an important role in the finiteamplitude equilibration of the system. The most unstable CISK waves will be associated with very short vertical wavelengths [O(3 km)] in order to maximize (in some crude sense) subcloud convergence. Thus, the vertical scale is largely determined by the height of cloud base. The vertical scale, in turn, determines the dispersive relations between horizontal and temporal scales. It is found that there exists a waveCISK mode which is independent of longitude, and hence independent of the mean zonal flow. Because of this independence, the period of this oscillation should form a prominent line in tropical spectra. This period turns out to be about 4.8 days which is indeed a prominent feature of tropical spectra. It is shown, due to longitudinal inhomogeneities in the tropics (such as landsea), that the above oscillation must be accompanied by traveling disturbances whose period with respect to the ground will also be 4.8 days and whose longitudinal scales will typically be from 1000–3000 km depending on the mean zonal flow. It is further shown that the existence of the above oscillatory system has two additional implications:

The above system is, itself, unstable with respect to gravity waves with horizontal scales on the order of 100–200 km. Such waves may be associated with cloud clusters.

The above system leads to maximum lowlevel convergence (and hence, a tendency toward mean cumulus activity) in regions centered about ±6°–7° latitude, thus providing a possible explanation for the position of the ITCZ.
Abstract
CISK (Conditional Instability of the Second Kind) is examined for internal waves where lowlevel convergence is due to the inviscid wave fields rather than to Ekman pumping.
It is found that CISKinduced waves must give rise to mean cumulus activity (since there are no negative clouds), and it is suggested that this mean activity plays an important role in the finiteamplitude equilibration of the system. The most unstable CISK waves will be associated with very short vertical wavelengths [O(3 km)] in order to maximize (in some crude sense) subcloud convergence. Thus, the vertical scale is largely determined by the height of cloud base. The vertical scale, in turn, determines the dispersive relations between horizontal and temporal scales. It is found that there exists a waveCISK mode which is independent of longitude, and hence independent of the mean zonal flow. Because of this independence, the period of this oscillation should form a prominent line in tropical spectra. This period turns out to be about 4.8 days which is indeed a prominent feature of tropical spectra. It is shown, due to longitudinal inhomogeneities in the tropics (such as landsea), that the above oscillation must be accompanied by traveling disturbances whose period with respect to the ground will also be 4.8 days and whose longitudinal scales will typically be from 1000–3000 km depending on the mean zonal flow. It is further shown that the existence of the above oscillatory system has two additional implications:

The above system is, itself, unstable with respect to gravity waves with horizontal scales on the order of 100–200 km. Such waves may be associated with cloud clusters.

The above system leads to maximum lowlevel convergence (and hence, a tendency toward mean cumulus activity) in regions centered about ±6°–7° latitude, thus providing a possible explanation for the position of the ITCZ.
Abstract
An equivalent depth of 10 m for the oscillations of the tropical atmosphere was suggested by waveCISK calculations. It is here shown that the dispersive properties of such an atmosphere are in agreement with observed power spectra for southerly and westerly winds.
Abstract
An equivalent depth of 10 m for the oscillations of the tropical atmosphere was suggested by waveCISK calculations. It is here shown that the dispersive properties of such an atmosphere are in agreement with observed power spectra for southerly and westerly winds.
Abstract
The Kelvin–Helmholtz problem deals with the stability of fluids where both shear and stable stratification are restricted to a layer. In observed shear instability in the atmosphere, stable stratification rarely disappears outside the shear zone. In order to get some idea of the implications of this fact, I have investigated the stability properties of a particularly simple configuration: a Helmholtz velocity profile in a continuously stratified, infinite Boussinesq fluid. For a basic discontinuity 2U and BruntVäisälä frequency N, I find that perturbations with horizontal wavenumbers k, such that k ^{2}>N ^{2}/(2U ^{2}), are unstable and decay away from the shear zone. In addition, the shear zone is capable of supplying energy to neutral internal gravity waves, for which k ^{2}<N ^{2}/U ^{2}, which propagate away from the shear zone. A particular wavenumber, k ^{2} = N ^{2}/(2U ^{2}), is shown to be most efficient at carrying energy away from the shear zone. However, additional calculations suggest that for the configuration considered, instabilities ought to be more effective than waves in smoothing the original shear. Comparison with observations suggests, on the other hand, that the waves dominate observed disturbances. The reasons for this are discussed. It is suggested that the waves are enhanced by reflections from the earth's surface which were ignored in the calculations.
Abstract
The Kelvin–Helmholtz problem deals with the stability of fluids where both shear and stable stratification are restricted to a layer. In observed shear instability in the atmosphere, stable stratification rarely disappears outside the shear zone. In order to get some idea of the implications of this fact, I have investigated the stability properties of a particularly simple configuration: a Helmholtz velocity profile in a continuously stratified, infinite Boussinesq fluid. For a basic discontinuity 2U and BruntVäisälä frequency N, I find that perturbations with horizontal wavenumbers k, such that k ^{2}>N ^{2}/(2U ^{2}), are unstable and decay away from the shear zone. In addition, the shear zone is capable of supplying energy to neutral internal gravity waves, for which k ^{2}<N ^{2}/U ^{2}, which propagate away from the shear zone. A particular wavenumber, k ^{2} = N ^{2}/(2U ^{2}), is shown to be most efficient at carrying energy away from the shear zone. However, additional calculations suggest that for the configuration considered, instabilities ought to be more effective than waves in smoothing the original shear. Comparison with observations suggests, on the other hand, that the waves dominate observed disturbances. The reasons for this are discussed. It is suggested that the waves are enhanced by reflections from the earth's surface which were ignored in the calculations.
Abstract
The theory of internal equatorial Planetaryscale waves as developed by Matsuno and by Lindzen is extended to include the effects of shear in the basic state. By means of a numerical study we find that both the equatorial Kelvin wave (the gravest symmetric westerly mode) and the Yanai wave (the gravest anti symmetric easterly wave), the two most commonly observed internal equatorial waves, are absorbed shortly before reaching critical levels where their frequencies are Doppler shifted to zero. Away from such levels, these waves retain their identity within shear zones. However, they become more closely confined to the equator as their Dopplershifted frequencies are reduced. It is found, moreover, for a given frequency and vertical wavelength, that the vertical group velocity of the Yanai wave is smaller than that of the Kelvin wave. In addition, as the Dopplershifted frequency ω^ is reduced, the vertical group velocity of the Yanai wave diminishes as ω^^{3} while the vertical group velocity of Kelvin wave diminishes only as ω^^{2}. As a result of these two features, Yanai waves ten dto be more significantly affected by dissipation than are Kelvin waves.
Abstract
The theory of internal equatorial Planetaryscale waves as developed by Matsuno and by Lindzen is extended to include the effects of shear in the basic state. By means of a numerical study we find that both the equatorial Kelvin wave (the gravest symmetric westerly mode) and the Yanai wave (the gravest anti symmetric easterly wave), the two most commonly observed internal equatorial waves, are absorbed shortly before reaching critical levels where their frequencies are Doppler shifted to zero. Away from such levels, these waves retain their identity within shear zones. However, they become more closely confined to the equator as their Dopplershifted frequencies are reduced. It is found, moreover, for a given frequency and vertical wavelength, that the vertical group velocity of the Yanai wave is smaller than that of the Kelvin wave. In addition, as the Dopplershifted frequency ω^ is reduced, the vertical group velocity of the Yanai wave diminishes as ω^^{3} while the vertical group velocity of Kelvin wave diminishes only as ω^^{2}. As a result of these two features, Yanai waves ten dto be more significantly affected by dissipation than are Kelvin waves.
Abstract
This paper considers the vertical propagation of a longperiod, smallamplitude perturbation in a medium in which radiative transfer and photochemistry play important roles. The perturbation and the basic field are assumed to be axially symmetric and symmetric about the equator; the basic wind field is geostrophic and the basic temperature field is in radiative equilibrium.
It is found that longperiod perturbations can only propagate by virtue of the physical effects of radiative transfer and photochemistry. The computed wave propagates downwards and, for a period of 2.2 years, the phase speed is close to the observed speed of 1.5 km month^{−1} for the “26month” equatorial oscillation. The observed relative phases of velocity and temperature fields, and the sharp attenuation of the oscillation below 20 to 25 km are also found in the model wave.
There are discrepancies between the model and the observed “26month” oscillation, which are to be expected in view of the nonlinearity of the observed phenomenon. However, it appears that, for complex reasons, the observed wave may satisfy equations similar to those occurring in the linear theory.
Abstract
This paper considers the vertical propagation of a longperiod, smallamplitude perturbation in a medium in which radiative transfer and photochemistry play important roles. The perturbation and the basic field are assumed to be axially symmetric and symmetric about the equator; the basic wind field is geostrophic and the basic temperature field is in radiative equilibrium.
It is found that longperiod perturbations can only propagate by virtue of the physical effects of radiative transfer and photochemistry. The computed wave propagates downwards and, for a period of 2.2 years, the phase speed is close to the observed speed of 1.5 km month^{−1} for the “26month” equatorial oscillation. The observed relative phases of velocity and temperature fields, and the sharp attenuation of the oscillation below 20 to 25 km are also found in the model wave.
There are discrepancies between the model and the observed “26month” oscillation, which are to be expected in view of the nonlinearity of the observed phenomenon. However, it appears that, for complex reasons, the observed wave may satisfy equations similar to those occurring in the linear theory.