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- Author or Editor: Shin-ichi Takehiro x

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## Abstract

The concept of wave over-reflection is applied to the unstable normal mode of the Eady problem. In order to consider propagation properties of the Rossby waves trapped at the boundaries, two thin boundary layers are introduced at the top and bottom of the fluid layer of the original model. It is shown that the Rossby waves in the boundary layers are always over-reflected as long as their critical levels exist in the constant-shear evanescent region and the waves are transmitted to the opposite boundary layer. The dispersion relation obtained by using laser formula and quantization qualitatively coincides with that of the normal mode. Although the growth rate is systematically overestimated, the short-wave cutoff is well described by the over-reflection solution.

The mechanism of over-reflection obtained in this study is understood by conservation of momentum. Since the pseudomomentum of the transmitted and the incident waves have opposite signs to each other, the amplitude of the reflected wave should become larger than that of the incident wave to satisfy constant momentum flux constraints. This mechanism corresponds to that of over-reflection in a linear shear flow of the shallow water model shown in a paper by Takehiro and Hayashi.

## Abstract

The concept of wave over-reflection is applied to the unstable normal mode of the Eady problem. In order to consider propagation properties of the Rossby waves trapped at the boundaries, two thin boundary layers are introduced at the top and bottom of the fluid layer of the original model. It is shown that the Rossby waves in the boundary layers are always over-reflected as long as their critical levels exist in the constant-shear evanescent region and the waves are transmitted to the opposite boundary layer. The dispersion relation obtained by using laser formula and quantization qualitatively coincides with that of the normal mode. Although the growth rate is systematically overestimated, the short-wave cutoff is well described by the over-reflection solution.

The mechanism of over-reflection obtained in this study is understood by conservation of momentum. Since the pseudomomentum of the transmitted and the incident waves have opposite signs to each other, the amplitude of the reflected wave should become larger than that of the incident wave to satisfy constant momentum flux constraints. This mechanism corresponds to that of over-reflection in a linear shear flow of the shallow water model shown in a paper by Takehiro and Hayashi.

## Abstract

A numerical study on the runaway greenhouse state is performed by using a general circulation model (GCM) with simplified hydrologic and radiative processes. Except for the inclusion of three-dimensional atmospheric motion, the system utilized is basically equivalent to the one-dimensional radiative–convective equilibrium model of Nakajima et al. in which the runaway greenhouse state is defined.

The results of integrations with various values of solar constant show that there exists an upper limit of the solar constant with which the atmosphere can reach a statistical equilibrium state. When the value of solar constant exceeds the limit, 1600 W m^{−2}, the atmosphere sets in a “thermally runaway” state. It is characterized by continuous increase of the amount of water vapor, continuous decrease of the outgoing longwave radiation, and continuous warming of the atmosphere and the ground surface.

The upper-limit value of the solar constant obtained by the GCM experiments corresponds to the upper limit of outgoing longwave radiation determined by the one-dimensional model of Nakajima et al. with a fixed value of relative humidity, 60%, which is a typical value obtained by the GCM. The thermally runaway states realized in the GCM are caused by the radiation structure found by Nakajima et al. that prohibits the existence of thermal equilibrium states. The calculated values of the upper limit of radiation and water vapor content cannot be directly applied to describing real planetary atmospheres, since the model physical processes are quite simple—gray radiation scheme without clouds. However, because of this simplification, the GCM gives deeper insight into the structure of a runaway atmosphere.

## Abstract

A numerical study on the runaway greenhouse state is performed by using a general circulation model (GCM) with simplified hydrologic and radiative processes. Except for the inclusion of three-dimensional atmospheric motion, the system utilized is basically equivalent to the one-dimensional radiative–convective equilibrium model of Nakajima et al. in which the runaway greenhouse state is defined.

The results of integrations with various values of solar constant show that there exists an upper limit of the solar constant with which the atmosphere can reach a statistical equilibrium state. When the value of solar constant exceeds the limit, 1600 W m^{−2}, the atmosphere sets in a “thermally runaway” state. It is characterized by continuous increase of the amount of water vapor, continuous decrease of the outgoing longwave radiation, and continuous warming of the atmosphere and the ground surface.

The upper-limit value of the solar constant obtained by the GCM experiments corresponds to the upper limit of outgoing longwave radiation determined by the one-dimensional model of Nakajima et al. with a fixed value of relative humidity, 60%, which is a typical value obtained by the GCM. The thermally runaway states realized in the GCM are caused by the radiation structure found by Nakajima et al. that prohibits the existence of thermal equilibrium states. The calculated values of the upper limit of radiation and water vapor content cannot be directly applied to describing real planetary atmospheres, since the model physical processes are quite simple—gray radiation scheme without clouds. However, because of this simplification, the GCM gives deeper insight into the structure of a runaway atmosphere.

## Abstract

Jet formation in decaying two-dimensional turbulence on a rotating sphere is reviewed from the viewpoint of Rossby waves. A series of calculations are performed to confirm the behavior of zonal mean flow generation on the parameter space of the rotation rate Ω and Froude number Fr. When the flow is nondivergent and Ω is large, intense easterly circumpolar jets tend to emerge in addition to the appearance of a banded structure of zonal mean flows with alternating flow directions. When the system allows surface elevation, circumpolar jets disappear and an equatorial easterly jet emerges with increasing Fr. The appearance of the intense easterly jets can be understood by the angular-momentum transport associated with the generation, propagation, and absorption of Rossby waves. When the flow is nondivergent, long Rossby waves tend to be absorbed near the poles. In contrast, when Fr is large, Rossby waves can hardly propagate poleward and tend to be absorbed near the equator.

## Abstract

Jet formation in decaying two-dimensional turbulence on a rotating sphere is reviewed from the viewpoint of Rossby waves. A series of calculations are performed to confirm the behavior of zonal mean flow generation on the parameter space of the rotation rate Ω and Froude number Fr. When the flow is nondivergent and Ω is large, intense easterly circumpolar jets tend to emerge in addition to the appearance of a banded structure of zonal mean flows with alternating flow directions. When the system allows surface elevation, circumpolar jets disappear and an equatorial easterly jet emerges with increasing Fr. The appearance of the intense easterly jets can be understood by the angular-momentum transport associated with the generation, propagation, and absorption of Rossby waves. When the flow is nondivergent, long Rossby waves tend to be absorbed near the poles. In contrast, when Fr is large, Rossby waves can hardly propagate poleward and tend to be absorbed near the equator.