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## Abstract

Absorption of solar radiation in the dayside Venusian thermosphere forces a circulation cell with vertical motions upward on the dayside and downward on the nightside with maximum amplitude greater than a meter per second and horizontal velocities away from the subsolar point with amplitudes up to several hundred meters per second. The first harmonic in temperature determines a several-hundred-degree temperature decrease from dayside to nightside. These conclusions follow from the numerical integration of a dynamic model which includes realistic stratification and temperature-dependent radiative damping. The large day-to-night temperature contrast is a consequence of the addition of extreme ultraviolet (EUV) heating at sufficiently high levels that it must he conducted downward to lower levels before adiabatic and 15 μ cooling can balance it. The observed exospheric temperature of ∼650K near the subsolar point is reproduced with an EUV heating efficiency of 0.3. The calculated nightside exospheric temperature is below 300K for this efficiency.

## Abstract

Absorption of solar radiation in the dayside Venusian thermosphere forces a circulation cell with vertical motions upward on the dayside and downward on the nightside with maximum amplitude greater than a meter per second and horizontal velocities away from the subsolar point with amplitudes up to several hundred meters per second. The first harmonic in temperature determines a several-hundred-degree temperature decrease from dayside to nightside. These conclusions follow from the numerical integration of a dynamic model which includes realistic stratification and temperature-dependent radiative damping. The large day-to-night temperature contrast is a consequence of the addition of extreme ultraviolet (EUV) heating at sufficiently high levels that it must he conducted downward to lower levels before adiabatic and 15 μ cooling can balance it. The observed exospheric temperature of ∼650K near the subsolar point is reproduced with an EUV heating efficiency of 0.3. The calculated nightside exospheric temperature is below 300K for this efficiency.

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## Abstract

The problem of small-amplitude, steady motions and temperature perturbations in a stratified, ideal gas atmosphere with viscosity and heat conduction is investigated. Horizontally varying sources of heat, mass and momentum force these motions. The model assumes two-dimensionality and no rotation. The solutions illustrate the coupling between motions and the temperature field in a stratified fluid with exponential increase in the vertical of the coefficients of kinematic viscosity and thermometric conductivity. Horizontal variation of temperature sets up pressure forces which drive horizontal winds. By continuity the divergence of these winds gives rise to vertical motions which alter the temperature fields. The solutions oscillate with height. Sources at relatively low levels can produce at high levels temperatures and circulations inverse to the directly driven temperature and wind oscillations.

The mathematical analysis reduces to the solution of a sixth-order ordinary differential equation of the hypergeometric type. Power series solutions are obtained in the pressure variable. These are used to satisfy boundary conditions at the zero pressure boundary. Another fundamental set of solutions is constructed in the form of integral representations in order to satisfy a boundedness condition at large pressure. The two solution bases are matched by an expansion of the integrals in power series. The solutions are used to construct a Green's function for the inhomogeneous problem. A numerical example assuming a heat source at zero pressure is discussed. Tabulated values and asymptotic approximations are used to provide a detailed numerical description of the solutions, their derivatives, and the Green's function coefficients.

## Abstract

The problem of small-amplitude, steady motions and temperature perturbations in a stratified, ideal gas atmosphere with viscosity and heat conduction is investigated. Horizontally varying sources of heat, mass and momentum force these motions. The model assumes two-dimensionality and no rotation. The solutions illustrate the coupling between motions and the temperature field in a stratified fluid with exponential increase in the vertical of the coefficients of kinematic viscosity and thermometric conductivity. Horizontal variation of temperature sets up pressure forces which drive horizontal winds. By continuity the divergence of these winds gives rise to vertical motions which alter the temperature fields. The solutions oscillate with height. Sources at relatively low levels can produce at high levels temperatures and circulations inverse to the directly driven temperature and wind oscillations.

The mathematical analysis reduces to the solution of a sixth-order ordinary differential equation of the hypergeometric type. Power series solutions are obtained in the pressure variable. These are used to satisfy boundary conditions at the zero pressure boundary. Another fundamental set of solutions is constructed in the form of integral representations in order to satisfy a boundedness condition at large pressure. The two solution bases are matched by an expansion of the integrals in power series. The solutions are used to construct a Green's function for the inhomogeneous problem. A numerical example assuming a heat source at zero pressure is discussed. Tabulated values and asymptotic approximations are used to provide a detailed numerical description of the solutions, their derivatives, and the Green's function coefficients.

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## Abstract

Small amplitude planetary waves are superimposed on a mean zonal flow with arbitrary horizontal and vertical shears. An expression is derived for the change of the zonal wind and temperature field forced by statistically stationary eddies satisfying a source-free planetary wave equation. This result depends on the existence of singular lines, where the phase speed of an elementary wave is equal to the mean zonal wind speed, or on the presence of a Newtonian cooling process. Second-order interactions vanish when both of these phenomena are absent. The planetary wave-zonal flow interaction is discussed in terms of the eddy transport of potential vorticity. The theory provides a partial interpretation of the maintenance of atmospheric zonal flows, such as that of the wintertime stratosphere, by planetary waves propagating from some other region of the atmosphere.

## Abstract

Small amplitude planetary waves are superimposed on a mean zonal flow with arbitrary horizontal and vertical shears. An expression is derived for the change of the zonal wind and temperature field forced by statistically stationary eddies satisfying a source-free planetary wave equation. This result depends on the existence of singular lines, where the phase speed of an elementary wave is equal to the mean zonal wind speed, or on the presence of a Newtonian cooling process. Second-order interactions vanish when both of these phenomena are absent. The planetary wave-zonal flow interaction is discussed in terms of the eddy transport of potential vorticity. The theory provides a partial interpretation of the maintenance of atmospheric zonal flows, such as that of the wintertime stratosphere, by planetary waves propagating from some other region of the atmosphere.

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## Abstract

We discuss the time-dependent behavior of a Rossby wave on a latitudinally varying flow, near the point where the steady-state wave equation is singular. The wave is forced by the switch-on of a steady forcing. Analytic solutions are obtained for the latitudinal propagation of nondivergent Rossby waves in a linear shear flow and for a large longitudinal wavelength. It is shown that the north–south eddy velocity *v*′ approaches the steady-state solution everywhere when nondimensional time >1, this time being a few days or less for atmospheric planetary waves. The east–west eddy velocity *u*′ takes much longer to approach a steady state near the singularity. One-half the steady-state amplitude of *u*′ is approached in a time inversely proportional to the square root of the distance from the singularity. The solution for *u*′ near the singularity settles down to the steady solution only after a time large compared to the inverse of the distance from the singularity. The steady-state solution for *u*′ is logarithmically singular at the critical level, violating the assumption of small-amplitude motions. However, the initial-value solution indicates that large amplitudes near a critical level probably will not actually occur in the atmosphere since they require a longer time scale to be set up than the time scale for seasonal changes of the zonal winds. At times ≥O(1), the momentum divergence in the initial value calculation is smeared out over the region where the *u*′ component of velocity has not yet settled down. The width of this region is inversely proportional to time.

## Abstract

We discuss the time-dependent behavior of a Rossby wave on a latitudinally varying flow, near the point where the steady-state wave equation is singular. The wave is forced by the switch-on of a steady forcing. Analytic solutions are obtained for the latitudinal propagation of nondivergent Rossby waves in a linear shear flow and for a large longitudinal wavelength. It is shown that the north–south eddy velocity *v*′ approaches the steady-state solution everywhere when nondimensional time >1, this time being a few days or less for atmospheric planetary waves. The east–west eddy velocity *u*′ takes much longer to approach a steady state near the singularity. One-half the steady-state amplitude of *u*′ is approached in a time inversely proportional to the square root of the distance from the singularity. The solution for *u*′ near the singularity settles down to the steady solution only after a time large compared to the inverse of the distance from the singularity. The steady-state solution for *u*′ is logarithmically singular at the critical level, violating the assumption of small-amplitude motions. However, the initial-value solution indicates that large amplitudes near a critical level probably will not actually occur in the atmosphere since they require a longer time scale to be set up than the time scale for seasonal changes of the zonal winds. At times ≥O(1), the momentum divergence in the initial value calculation is smeared out over the region where the *u*′ component of velocity has not yet settled down. The width of this region is inversely proportional to time.

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## Abstract

The role of horizontal wind shears in the vertical propagation of planetary Rossby waves is investigated using an adiabatic linear model. We discuss wave guides formed by regions of weak westerly wind. If the wave guide is formed by trapping of waves between strong westerlies and/or the geometric poles, the ducting occurs as a wave propagation in discrete normal modes of the internal wave guide. On the other hand, for wave guides formed by one or more lines of zero wind, waves are absorbed rather than reflected at the zero wind line so that there are no normal modes of the wave guide. Disturbances excited in the lower stratosphere in the equatorial zero wind wave guide will terminate somewhere in the equatorial stratosphere, but eddy motions may be maintained in the tropics at higher levels by leakage from the Aleutian high planetary wave propagating vertically in a polar wave guide. The Aleutian high should he significantly attenuated by such leakage. The theory of zero wind line absorption suggests a planetary wave coupling with the biennial oscillation.

## Abstract

The role of horizontal wind shears in the vertical propagation of planetary Rossby waves is investigated using an adiabatic linear model. We discuss wave guides formed by regions of weak westerly wind. If the wave guide is formed by trapping of waves between strong westerlies and/or the geometric poles, the ducting occurs as a wave propagation in discrete normal modes of the internal wave guide. On the other hand, for wave guides formed by one or more lines of zero wind, waves are absorbed rather than reflected at the zero wind line so that there are no normal modes of the wave guide. Disturbances excited in the lower stratosphere in the equatorial zero wind wave guide will terminate somewhere in the equatorial stratosphere, but eddy motions may be maintained in the tropics at higher levels by leakage from the Aleutian high planetary wave propagating vertically in a polar wave guide. The Aleutian high should he significantly attenuated by such leakage. The theory of zero wind line absorption suggests a planetary wave coupling with the biennial oscillation.

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## Abstract

A system of equations, including Newtonian cooling and valid away from the equator, is derived for atmospheric zonal winds and the associated zonal temperature and meridional circulation fields. A time scaling is introduced to distinguish three regimes of motion. For short time scales a variation of the usual nondissipative forced symmetric vortex problem is recovered. For intermediate time scales, a “diffusive model” is defined, while for long time scales a “steady state” model is defined. Two examples of the solution of the diffusive equation are given. From the first example we infer that our model is capable of explaining many of the observed features of the downward progression of the stratospheric biennial oscillation. Momentum is carried downward by Coriolis torques resulting from propagating meridional cells. The solutions of our second example describe transient zonal winds which resemble those observed to propagate downward from the stratopause in Meteorological Rocket Network data time-height sections. In general, dissipation relaxes the constraint of conservation of potential vorticity so that wave-like motions are possible when Newtonian cooling is present. The “steady state” model equation is solved by the method of Green's functions to obtain the forced temperature. For this model, there is a direct response to heating and only momentum fluxes force meridional circulations to give subsidence heating. It is found that the domain of influence of the temperature response to momentum fluxes lies below the source point. Our model suggests that observed long-period departures from radiative equilibrium in the atmosphere below the mesopause can be explained as due either to heating by horizontal eddy heat transports or to subsidence heating forced by eddy momentum fluxes.

## Abstract

A system of equations, including Newtonian cooling and valid away from the equator, is derived for atmospheric zonal winds and the associated zonal temperature and meridional circulation fields. A time scaling is introduced to distinguish three regimes of motion. For short time scales a variation of the usual nondissipative forced symmetric vortex problem is recovered. For intermediate time scales, a “diffusive model” is defined, while for long time scales a “steady state” model is defined. Two examples of the solution of the diffusive equation are given. From the first example we infer that our model is capable of explaining many of the observed features of the downward progression of the stratospheric biennial oscillation. Momentum is carried downward by Coriolis torques resulting from propagating meridional cells. The solutions of our second example describe transient zonal winds which resemble those observed to propagate downward from the stratopause in Meteorological Rocket Network data time-height sections. In general, dissipation relaxes the constraint of conservation of potential vorticity so that wave-like motions are possible when Newtonian cooling is present. The “steady state” model equation is solved by the method of Green's functions to obtain the forced temperature. For this model, there is a direct response to heating and only momentum fluxes force meridional circulations to give subsidence heating. It is found that the domain of influence of the temperature response to momentum fluxes lies below the source point. Our model suggests that observed long-period departures from radiative equilibrium in the atmosphere below the mesopause can be explained as due either to heating by horizontal eddy heat transports or to subsidence heating forced by eddy momentum fluxes.

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## Abstract

Atmospheric general circulation models require efficient approximation procedures for including the vertical diffusion or heat into an underlying soil. The most effective procedure for diurnal heating—the “force-restore” approach—uses a prognostic equation for temperature that reproduces exactly the response to periodic heating and uniform thermal properties. Errors result from the neglect of higher harmonies in time in the forcing and from inhomogeneous thermal properties of the underlying soil. Neglect of higher harmonics is tolerable, in part because the response decreases with an increased forcing frequency. Neglect of vertical variation of thermal conductivity and specific heat can introduce major errors in the force-restore treatment, a possibility illustrated by the response to diurnal periodic surface forcing with a snow layer overlying a soil layer. A generalization of the force-restore procedure is developed for inhomogeneous thermal properties and illustrated for a snow layer overlying the soil. A reasonably accurate formulation for coupling the diurnal and seasonal force-restore formalisms is presented.

## Abstract

Atmospheric general circulation models require efficient approximation procedures for including the vertical diffusion or heat into an underlying soil. The most effective procedure for diurnal heating—the “force-restore” approach—uses a prognostic equation for temperature that reproduces exactly the response to periodic heating and uniform thermal properties. Errors result from the neglect of higher harmonies in time in the forcing and from inhomogeneous thermal properties of the underlying soil. Neglect of higher harmonics is tolerable, in part because the response decreases with an increased forcing frequency. Neglect of vertical variation of thermal conductivity and specific heat can introduce major errors in the force-restore treatment, a possibility illustrated by the response to diurnal periodic surface forcing with a snow layer overlying a soil layer. A generalization of the force-restore procedure is developed for inhomogeneous thermal properties and illustrated for a snow layer overlying the soil. A reasonably accurate formulation for coupling the diurnal and seasonal force-restore formalisms is presented.

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## Abstract

The study assumes adiabatic, inviscid, hydrostatic and quasi-geostrophic motions on a mid-latitude β-plane. The domain is assumed vertically unbounded. The stability of a hyperbolic-tangent shear flow to small-amplitude disturbances is discussed. Negative shear zones (wind becoming stronger westward with increasing elevation) are unstable for weaker shears and the resulting instabilities have larger growth rates than in the case with positive shear zones (wind becoming stronger eastward with increasing elevation) and other conditions the same.

Neutral solutions for the hyperbolic-tangent shear flow problem are found analytically, and growth rates and modal structure of unstable modes are found numerically. The unstable modes for a negative shear flow and for sufficiently small longitudinal wavenumber have the structure of vertically propagating Rossby waves. Thus, the shear zone can act as a source of Rossby waves which couple the zonal wind within the shear zone to the mean zonal wind many, scale heights removed from the shear zone.

## Abstract

The study assumes adiabatic, inviscid, hydrostatic and quasi-geostrophic motions on a mid-latitude β-plane. The domain is assumed vertically unbounded. The stability of a hyperbolic-tangent shear flow to small-amplitude disturbances is discussed. Negative shear zones (wind becoming stronger westward with increasing elevation) are unstable for weaker shears and the resulting instabilities have larger growth rates than in the case with positive shear zones (wind becoming stronger eastward with increasing elevation) and other conditions the same.

Neutral solutions for the hyperbolic-tangent shear flow problem are found analytically, and growth rates and modal structure of unstable modes are found numerically. The unstable modes for a negative shear flow and for sufficiently small longitudinal wavenumber have the structure of vertically propagating Rossby waves. Thus, the shear zone can act as a source of Rossby waves which couple the zonal wind within the shear zone to the mean zonal wind many, scale heights removed from the shear zone.

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## Abstract

Global mean sources and sinks of radiative energy are calculated for the upper atmosphere of Venus. We especially consider the region between 90 and 130 km, where the equilibrium temperature is largely controlled through infrared absorption and emission by vibrational-rotational bands of CO_{2}. Source functions for bands deviating from thermodynamic equilibrium are determined as part of the calculation. Radiative transfer in the region of non-overlapping lines is calculated by summing the contribution of individual Voigt lines. Many isotopic and hot bands contribute amounts to 15μ cooling or near-infrared heating at some levels comparable to the contributions by strong bands. The emission of the fundamental 15μ state of C^{12}0_{2}
^{16} is maintained at near local thermodynamic equilibrium values by the radiation field to pressures three orders of magnitude less than would be expected in considering only relative values of radiative and vibrational relaxation rates. Conversion of near-infrared solar photon energy into thermal energy occurs through collisional relaxation of the 15μ fundamental states. Mean near-infrared heating rates increase from 1K (earth day)^{−1} at 65 km to more than 300K (earth day) ^{−1} at 115 km. The 15μ cooling is dominated at most levels by cooling to space. The time scale for radiative damping in the cooling-to-space approximation varies from 30 earth days at 65 km to 1/20 of an earth day at 120 km. The calculated equilibrium temperature profile decreases from 250K at an altitude of 66 km to a minimum of 158K at 88 km, increases to a peak value of 190K at 113 km, and again decreases to a mesopause minimum of 180K at 122 km (8×10μb). The calculated thickness between the 100-mb level and the level of the ionospheric peak differs by less than 1 km from that observed during the Mariner 5 radio occultation experiment.

## Abstract

Global mean sources and sinks of radiative energy are calculated for the upper atmosphere of Venus. We especially consider the region between 90 and 130 km, where the equilibrium temperature is largely controlled through infrared absorption and emission by vibrational-rotational bands of CO_{2}. Source functions for bands deviating from thermodynamic equilibrium are determined as part of the calculation. Radiative transfer in the region of non-overlapping lines is calculated by summing the contribution of individual Voigt lines. Many isotopic and hot bands contribute amounts to 15μ cooling or near-infrared heating at some levels comparable to the contributions by strong bands. The emission of the fundamental 15μ state of C^{12}0_{2}
^{16} is maintained at near local thermodynamic equilibrium values by the radiation field to pressures three orders of magnitude less than would be expected in considering only relative values of radiative and vibrational relaxation rates. Conversion of near-infrared solar photon energy into thermal energy occurs through collisional relaxation of the 15μ fundamental states. Mean near-infrared heating rates increase from 1K (earth day)^{−1} at 65 km to more than 300K (earth day) ^{−1} at 115 km. The 15μ cooling is dominated at most levels by cooling to space. The time scale for radiative damping in the cooling-to-space approximation varies from 30 earth days at 65 km to 1/20 of an earth day at 120 km. The calculated equilibrium temperature profile decreases from 250K at an altitude of 66 km to a minimum of 158K at 88 km, increases to a peak value of 190K at 113 km, and again decreases to a mesopause minimum of 180K at 122 km (8×10μb). The calculated thickness between the 100-mb level and the level of the ionospheric peak differs by less than 1 km from that observed during the Mariner 5 radio occultation experiment.

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## Abstract

The horizontal variation of infrared heating is calculated for the Venusian mesophere and adjoining layers in the stratosphere and lower thermosphere. The calculation assumes no horizontal temperature variation. Very large heating rates at the subsolar point reaching ∼1000K. (earth day)^{−1} are calculated at pressures between 1 and 0.1 μb. Between 0.1 and 0.0l μb, the 15 μ NLTE source function increases with an increase of near-infrared heating. Consequently, there is a large solar zenith variation of 15 μ hot-band cooling that cancels roughly half the near-infrared heating at these levels. It is suggested that the large horizontal variation of heating at 1 μb cannot he completely balanced by adiabatic cooling in the absence of horizontal temperature variation, so the 15 μ emission might be expected to vary horizontally by as much as a factor of 2. At altitudes below the 100-μb level, the horizontal variation of temperature should be entirely negligible.

## Abstract

The horizontal variation of infrared heating is calculated for the Venusian mesophere and adjoining layers in the stratosphere and lower thermosphere. The calculation assumes no horizontal temperature variation. Very large heating rates at the subsolar point reaching ∼1000K. (earth day)^{−1} are calculated at pressures between 1 and 0.1 μb. Between 0.1 and 0.0l μb, the 15 μ NLTE source function increases with an increase of near-infrared heating. Consequently, there is a large solar zenith variation of 15 μ hot-band cooling that cancels roughly half the near-infrared heating at these levels. It is suggested that the large horizontal variation of heating at 1 μb cannot he completely balanced by adiabatic cooling in the absence of horizontal temperature variation, so the 15 μ emission might be expected to vary horizontally by as much as a factor of 2. At altitudes below the 100-μb level, the horizontal variation of temperature should be entirely negligible.