Yoshi-Yuki Hayashi

Stable and unstable shear modes of rotating parallel flows in shallow water

This article considers the instabilities of rotating, shallow-water, shear flows on an equatorial β-plane. Because of the free surface, the motion is horizontally divergent and the energy density is cubic in the field variables (i.e. in standard notation the kinetic energy density is ½ h ( u 2 + v 2 )). Marinone & Ripa (1984) observed that as a consequence of this the wave energy is no longer positive definite (there is a cross-term U h ′ u ′ ). A wave with negative wave energy can grow by transferring energy to the mean flow. Of course total (mean plus wave) energy is conserved in this process. Further, when the basic state has constant potential vorticity, we show that there are no exchanges of energy and momentum between a growing wave and the mean flow. Consequently when the basic state has no potential vorticity gradients an unstable wave has zero wave energy and the mean flow is modified so that its energy is unchanged. This result strikingly shows that energy and momentum exchanges between a growing wave and the mean flow are not generally characteristic of, or essential to, instability. A useful conceptual tool in understanding these counterintuitive results is that of disturbance energy (or pseudoenergy) of a shear mode. This is the amount of energy in the fluid when the mode is excited minus the amount in the unperturbed medium. Equivalently, the disturbance energy is the sum of the wave energy and that in the modified mean flow. The disturbance momentum (or pseudomomentum) is defined analogously. For an unstable mode, which grows without external sources, the disturbance energy must be zero. On the other hand the wave energy may increase to plus infinity, remain zero, or decrease to minus infinity. Thus there is a tripartite classification of instabilities. We suggest that one common feature in all three cases is that the unstable shear mode is roughly a linear combination of resonating shear modes each of which would be stable if the other were somehow suppressed. The two resonating constituents must have opposite-signed disturbance energies in order that the unstable alliance has zero disturbance energy. The instability is a transfer of disturbance energy from the member with negative disturbance energy to the one with positive disturbance energy.

A Study on the “Runaway Greenhouse Effect” with a One-Dimensional Radiative–Convective Equilibrium Model

Abstract A simple one-dimensional radiative–convective equilibrium model is used to investigate the relationship between the surface temperature and the outgoing infrared radiation at the top of the atmosphere. The model atmosphere has a gray infrared absorption coefficient and is composed of a radiative equilibrium stratosphere and a moist adiabat troposphere. An upper limit of the outgoing infrared radiation is found to exist. The existence of the upper limit is characterized by the radiation limits that appear when the optical depth of the entire atmosphere becomes sufficiently deep and the temperature structure around the levels where the optical depth is about unity approaches a fixed profile. This appearance of an upper limit differs from that found by Komabayashi and Ingersoll, which is obtained from the constraint of the stratospheric radiation balance. As one of those radiation limits, the outgoing infrared radiation has an asymptotic limit as the surface temperature increases. This is caused by ...

OVER-REFLECTION AND SHEAR INSTABILITY IN A SHALLOW-WATER MODEL

The characteristics of shallow-water waves in a linear shear flow are studied, and the relationship between waves and unstable modes is examined. Numerical integration of the linear shallow-water equations shows that over-reflection occurs when a wave packet is incident at the turning surface. This phenomenon can be explained by the conservation of momentum as discussed by Acheson (1976). The unstable modes of linear shear flow in a shallow water found by Satomura (1981) are described in terms of the properties of wave propagation as proposed by Lindzen and others. Ripass (1983) theorem, which is the sufficient condition for stability of flows in shallow water, is also related to the wave geometry. The Orr mechanism, which is proposed by Lindzen (1988) as the primary mechanism of wave amplification, cannot explain the over-reflection of shallow-water waves. The amplification of these waves occurs in the opposite sense to that of Orrs solution.

Numerical modeling of Jupiter's moist convection layer

Moist convection of Jupiters atmosphere is examined using a large-domain two-dimensional fluid dynamical model with simplified cloud microphysics of water. The result shows that the water condensation level acts as a dynamical and compositional boundary. The convection below the condensation level is characterized by a steady regular cellular structure and a homogeneous distribution of water mixing ratio. Above the condensation level, cloud elements accompanied by the upward motion develop and disappear irregularly but successively, and water mixing ratio is highly inhomogeneous. The horizontal average of mixing ratio decreases rapidly with height just above the condensation level, resulting in a distinctive stable layer at 5 bar. The stable layer prevents the air masses above and below it from mixing with each other. As a result, the upper dry air does not reach 20 bar level, where the Galileo probe observed low humidity.

A Numerical Study on Appearance of the Runaway Greenhouse State of a Three-Dimensional Gray 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 ...

Performance benchmarks for a next generation numerical dynamo model

Numerical simulations of the geodynamo have successfully represented many observable characteristics of the geomagnetic field, yielding insight into the fundamental processes that generate magnetic fields in the Earths core. Because of limited spatial resolution, however, the diffusivities in numerical dynamo models are much larger than those in the Earths core, and consequently, questions remain about how realistic these models are. The typical strategy used to address this issue has been to continue to increase the resolution of these quasi-laminar models with increasing computational resources, thus pushing them toward more realistic parameter regimes. We assess which methods are most promising for the next generation of supercomputers, which will offer access to O(106) processor cores for large problems. Here we report performance and accuracy benchmarks from 15 dynamo codes that employ a range of numerical and parallelization methods. Computational performance is assessed on the basis of weak and strong scaling behavior up to 16,384 processor cores. Extrapolations of our weak-scaling results indicate that dynamo codes that employ two-dimensional or three-dimensional domain decompositions can perform efficiently on up to ∼106 processor cores, paving the way for more realistic simulations in the next model generation.

Simple Cumulus Models in One-Dimensional Radiative Convective Equilibrium Problems

Abstract The cumulus model presented by Lindzen et al. for calculating one-dimensional radiative convective equilibria is examined. When only the balance of moist static energy is considered, the value of the convective mass flux Mc is required to be externally specified. Dependency of equilibrium solutions on Mc shows that an upper limit of the value of Mc exists above which the temperature in the region of upward motion is lower than that in the region of downward motion; that is, the buoyancy is negative. Lindzen et al. tried to specify the value of Mc by introducing the surface heat fluxes. However, it is found that the buoyancy of their solution is negative. In order to obtain an appropriate equilibrium solution where the buoyancy is positive, the balance of kinetic energy, especially the dissipative process, should be considered. It is found that the value of Mc, which gives a realistic value of the dissipation rate, is close to the upper limit. In order to have a solution with a more realistic temp...

Linear stability of thermal convection in rotating systems with fixed heat flux boundaries

Linear stability of rotating thermal convection in a horizontal layer of Boussinesq fluid under the fixed heat flux boundary condition is examined by the use of a vertically truncated system up to wavenumber one. When the rotation axis is in the vertical direction, the asymptotic behavior of the critical convection for large rotation rates is almost the same as that under the fixed temperature boundary condition. However, when the rotation axis is horizontal and the lateral boundaries are inclined, the mode with zero horizontal wavenumber remains as the critical mode regardless of the rotation rate. The neutral curve has another local minimum at a nonzero horizontal wavenumber, whose asymptotic behavior coincides with the critical mode under the fixed temperature condition. The difference of the critical horizontal wavenumber between those two geometries is qualitatively understood by the difference of wave characteristics; inertial waves and Rossby waves, respectively.

Emergence of Circumpolar Vortex in Two Dimensional Turbulence on a Rotating Sphere

Characteristics of the decaying non-divergent two dimensional turbulence on a rotating sphere is considered. The spontaneous appearance of circumpolar easterly vortices reported in the previous study (Yo-den and Yamada, 1993) is interpreted by the Rossby wave property from the framework of weak-nonlinear theory. The resolution of the numerical model utilized by Yoden and Yamada (1993) T85 is not high enough to represent the upward energy cascade of two dimensional turbulence. The initial energy spectral peak adapted there is located almost in the range of wave regime especially when the rotation rate is large, which justifies the explanation of angular momentum redistribution by Rossby waves. A series of new experiments with the higher resolution T341 and new sets of initial energy spectral distributions are performed to confirm that the circumpolar vortices appear even after the full nonlinear upward cascade of turbulent energy and that the band structure of angular momentum emerges even when the rotation rate of the system is large.

The effects of thermal conditions on the cell sizes of two-dimensional convection

The effects of thermal conditions on the patterns of two-dimensional Boussinesq convection are studied by numerical integration. The adopted thermal conditions are (i) the heat fluxes through both upper and lower boundaries are fixed, (ii) the same as (i) but with internal cooling, (iii) the temperature on the lower boundary and the heat flux through the upper boundary are fixed, (iv) the same as (iii) but with internal cooling, and (v) the temperatures on both upper and lower boundaries are fixed. The numerical integrations are performed with Ra = 104 and Pr = 1 over the region whose horizontal and vertical lengths are 8 and 1, respectively.The results confirm that convective cells with the larger horizontal sizes tend to form under the conditions where the temperature is not fixed on any boundaries. Regardless of the existence of internal cooling, one pair of cells spreading all over the region forms in the equilibrium states. On the other hand, three pairs of cells form and remain when the temperature on at least one boundary is fixed. The formation of single pairs of cells appearing under the fixed heat flux conditions shows different features with and without internal cooling. The difference emerges as the appearance of a phase change, whose existence can be suggested by the weak nonlinear equation derived by Chapman & Proctor (1980).