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D. O. STALEY

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D. O. Staley

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A simplified analysis of the perturbation equations for nonaxisymmetric perturbations in the swirling axial flow of a tornado vortex is used to study inertial instability in the presence of viscosity. It is found that the principal effect of viscosity is to place the most unstable disturbances at finite azimuthal and axial wavenumbers, while having little effect on growth rate and structure. These results are similar to those of Emanuel for mesoscale inertial instability. The instability is not sharply peaked, but remains substantial over a large range of small azimuthal and axial wavelengths, suggesting a mechanism for turbulence.

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D. O. Staley

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The heat conduction constraint on the growth of a droplet is re-derived taking into account the heat transfer associated with mass transfer, and requiring total energy conservation. The derived constraint is identical with that given by Maxwell, who simply equated the rate of latent heat release to the heat conduction associated with a temperature gradient.

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D. O. Staley

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The evolution of Sr-90 distribution following an instantaneous stratospheric or tropospheric source in an annulus defined by latitude walls is investigated by means of a very simple model that uses 12 stratospheric and 12 tropospheric annular boxes and which assumes diffusional transport between boxes and wet and dry scavenging from tropospheric boxes.

Coefficients for interbox transport are expressed in terms of half-residence times, and these are adjusted to be consistent with reasonable eddy diffusivities and/or variable local stratospheric residence times suggested by previous studies. Precipitation scavenging is proportional to precipitation rate, for which a sinusoidal variation was assumed, with annual mean, amplitude of oscillation and phase all determined by climatology.

The spring peak and fall minimum in tropospheric air are obtained at all latitudes. At middle and high latitudes, these extremes result primarily from annual oscillation of mixing downward from the stratosphere, while at low latitudes they result from the phase of the large annual oscillation of precipitation scavenging. The only reasonable way to obtain, in this model, the progressive delay of the spring maximum with latitude in middle latitudes is by a corresponding delay of the maximum rate of transport across the tropopause, a delay suggested by variations of stratospheric mass.

For reasonable values of vertical and horizontal exchange coefficients and a megaton midlatitude source, half-residence times are obtained for the Northern Hemisphere stratosphere, total stratosphere, and Northern minus Southern Hemisphere burdens that agree with observation. Half-residence times for the northern and total stratosphere increase by about two months over a period of several years, as concentration is depleted in middle and high latitudes where transfer to the troposphere is rapid. Maximum concentration develops over the equator.

Tropospheric debris from a tropospheric source is initially rapidly depleted by precipitation scavenging, and after a few months the tropospheric burden is small compared to the stratospheric debris acquired by initial upward diffusion. Thereafter, the stratosphere becomes the source, and both burdens slowly decrease at the rates given by stratospheric residence times.

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D. O. Staley

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The discrete-level baroclinic eigenproblem developed by Satyamurty et al., based on a Boussinesq primitive equation model, is solved by a shooting method for very high vertical resolution. It is found that the vertical resolution used by Satyamurty et al. (0.5 km) is insufficient to eliminate spurious critical layer instability. Peaks in the instability spectrum at very short wavelengths (< ∼ 1000 km) found by them are spurious and are not the result of ageostrophic terms.

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D. O. Staley

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The linear stabilities of simple baroclinic and barotropic flows are investigated by finite differencing of the quasi-geostrophic perturbation equations and reduction to the standard algebraic eigenproblem. In the baroclinic case, four different arrangements of winds and static stability are studied. In the barotropic case, cosine and hyperbolic secant jets are assumed between latitude walls. In all baroclinic and barotropic cases, only the number of levels is varied. In the baroclinic case, 2≤N≤26. In the barotropic case, 3≤N≤20, with solutions assumed to be symmetric across the wind maximum.

Increase of the number of levels, N, by one or a few does not assure a growth-rate spectrum closer to that for a great many levels. As N is increased from the minimum permissible number, the maximum growth rate, the corresponding phase velocity, and the wavelength of maximum growth rate, all describe irregular damped oscillations.

The results in the baroclinic cases depend greatly on the wind and temperature distributions. The two-level model may miss the existence of instability altogether, and in other cases small N may yield very poor descriptions of phase and amplitude, especially in the stratosphere and lower troposphere. Increase of N extends instability to shorter wavelengths, and secondary maxima varying with N are found at short wavelengths. In the barotropic case, small values of N may yield double maxima of similar strength at long wavelengths. For larger N, instability extends to shorter wavelengths, and a secondary maximum may appear at the shortwave end of the spectrum. The barotropic instability spectrum is very sensitive to subtleties of the wind distribution and shows tame variations as truncation errors vary with changing N.

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D. O. Staley

Abstract

Individual potential-vorticity change, vertical motion, vertical advection of vorticity and flow from stratosphere to troposphere are evaluated at different levels and for different times in an extratropical disturbance.

Vertical motions are obtained from trajectories on three isentropic surfaces for two different times. The isentropes pass through or near a front in the upper troposphere, the lower stratosphere, and the high troposphere on the cold side of the front. Vertical motion is also evaluated at 500 mb from an adiabatic method designed to give instantaneous values to the extent that the system moves without change of shape. Negative extremes of 8 to 10 cm sec−1 occur in the upper-air front, with positive extremes of the same magnitude a short distance to the northeast, coinciding with the eastern exit of the frontal zone.

From theory, it is shown that potential vorticity is not conserved if there is either a gradient of diabatic heating or a component of curl, normal to the isentropic surface, of frictional force. The sum of these two effects is evaluated over the United States for two 12-hr periods on two isentropic surfaces which were common to parts of the lower stratosphere, upper troposphere and a front in middle and upper troposphere. Generally large, positive potential-vorticity changes occur in the lower stratosphere and in the upper troposphere on the cold side of the front. Large negative values occur in the frontal zone and around the entire periphery of the positive area, or around the periphery of the trough in the upper air. The potential-vorticity changes are related to simultaneous stability and vorticity changes of like sign. The potential-vorticity changes are positive in the region of the so-called ‘tropopause funnel’ ; changes everywhere appear attributable to vertical gradient of diabatic heating rather than curl of frictional force.

The terms in the vorticity equation which contain vertical velocity (vertical advection of vorticity and tilting terms) are shown by means of the thermal-wind equation to depend only on the vertical motion and temperature fields in an isobaric surface. For the frequent case where negative motion is centered in the baroclinic or frontal zone and rising motion is centered at the exit of the zone, each of the vertical-motion terms has the same characteristic distribution. Positive values of these vertical-motion terms occur on the cold side of the zone and to the right of the downwind exit of the zone, and negative values occur on the warm side of the zone and to the left of the downwind exit of the zone. The vertical advedtion and the sum of vertical advection and tilting terms are evaluated at 500 mb Magnitudes obtained compare with those of the divergence term, although magnitude depends considerably on the distances over which finite differences are evaluated.

Isentropic trajectories trace air initially in the lower stratosphere downward to within 5000 ft of the surface within 24 hr. Diabatic incorporation into the troposphere is also noted. The total adiabatic mass flow into the troposphere associated with the number of typical upper-air disturbances in existence at any time is estimated and found to be sufficient to give the observed short residence times of a few months for radioactive debris injected into the stratosphere by nuclear detonations.

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D. O. Staley

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Short wavelength linear baroclinic instability is examined numerically in two quasi-geostrophic models and one ageostrophic model. The growth rate spectra for the ageostrophic model differ little from those of the quasi-geostrophic models in three cases with different arrangements of shear and static stability. In all models and cases, and for 26 vertical levels, peaks of instability were found at very short wavelengths of a few hundred kilometers. These apparently represent spurious critical layer instability appearing at discrete levels, as suggested by Arakawa. They disappear when very high vertical resolution (e.g., 150 levels) is used.

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D. O. Staley

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The adiabatic lapse rate is derived for a real gas, and the result applied to the Venus atmosphere, which is assumed to consist of pure CO2. The ratio of the real to the ideal gas adiabatic lapse rate is found to exceed unity at pressures and temperatures currently believed to occur in the lower Venus atmosphere. For a pressure of 100 atm and temperatures of 750, 650 and 550 K, the ratios are 1.03, 1.06 and 1.13, respectively. At 70 atm the ratios are 1.02, 1.05 and 1.10, respectively. The expression g/cp, where g is gravity and cp, the specific heat at constant pressure of a real gas, is inferior to the adiabatic lapse rate of an ideal gas as an approximation to the real gas adiabatic lapse rate. Estimates of Venus surface temperature based on downward adiabatic extrapolation for an ideal gas, or even the quantity g/cp, do not lead to results seriously different from extrapolations based on the real adiabatic lapse rate. However, the real gas adiabatic lapse rate will be necessary for accurate analysis of hydrostatic stability.

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D. O. Staley

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The wavelength of maximum baroclinic instability (WMI), maximum growth rate (MGR), phase velocities, and other quantities, are computed for an atmosphere with an internal 100-mb layer having various shears and lapse rates and positioned at various levels in a baroclinically unstable troposphere. It is found that the WMI, MGR, and other properties are more closely related to slope of the isentropes, S θ, in the layer than to Richardson number, Ri, or to shear and lapse rate individually. The results for the layer positioned in the lower, middle and upper troposphere are similar but show systematic variations. The largest WMIs (3000–6000 km) are found for intermediate isentropic slopes of the order of those observed. Much smaller WMIs are found for small and very large isentropic slopes. The MGRs are of the order of 1 d−1 for small and intermediate slopes, but much larger for large slopes (S θ ≳6 × 10−3 or Ri ≲ 0.5). The results suggest temporal and upward increases of WMI as shear and static stability develop in association with growth of the baroclinic wave. They also suggest typical WMIs somewhat larger than the 3100 km for the standard atmosphere.

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