Shin-ichi Takehiro

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.

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.

Long-time asymptotic states of forced two-dimensional barotropic incompressible flows on a rotating sphere

This study re-examines a long-time asymptotic state of a two-dimensional barotropic incompressible flow with a small-scale, Markovian random forcing on a rotating sphere. Numerical simulations with different rotation rates of the sphere and different wavenumbers of the forcing are performed from zero initial condition. The integration time is extended to around 100–500 times that of the previous study by Nozawa and Yoden [Phys. Fluids 9, 2081 (1997)]. At an early stage of the time integration, a multiple zonal-band structure or a structure with westward circumpolar jets emerges. However, in the course of time development, a multiple zonal-band structure is found to appear in all cases. The multiple zonal-band structure then enters quasisteady state, showing little energy increase with nearly steady energy spectrum. This is followed by a sudden merger/disappearance of the jets, accompanying an energy increase, and at the final stage of the time integration, a zonal-band structure with only two or three jet...

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).

Physical interpretation of spiralling-columnar convection in a rapidly rotating annulus with radial propagation properties of Rossby waves

To aid the physical understanding of spiralling-columnar convection emerging in rapidly rotating spheres and spherical shells, two-dimensional thermal convection in a rapidly rotating annulus is investigated through the radial propagation properties of topographic Rossby waves. Two kinds of the boundaries containing the fluid in the axial direction are considered: a convex type modelling a spherical geometry and a concave type for comparison. The linear stability of a basic state with no motion and uniformly unstable stratification is examined and spirally elongated structures of critical convection are obtained for small Prandtl numbers. An analysis of the energy budget shows that a part of the kinetic energy generated in the region with slightly inclined boundaries is dynamically transferred and dissipates through viscosity in the region with strongly inclined boundaries. This indicates that the Rossby waves propagate from the region with slightly inclined boundaries to the region with strongly inclined boundaries. It is presented that the appearance of a spiral structure corresponds to an increase of the local radial wavenumber of the Rossby waves propagating in the radial direction. The flow patterns obtained using the dispersion relation of the Rossby waves coincide with those of the tailing part of the spiral structure obtained numerically. As the Prandtl number increases, the Rossby waves barely propagate because of strong viscous dissipation, and the flow pattern is localized in the region with slightly inclined boundaries. For convex boundaries with unstable stratification concentrating near the outer boundary and concave boundaries with unstable stratification confined near the inner boundary, the flow patterns tilt in the direction inverse to the case of uniform unstable stratification. The tilting direction of the flow pattern is not determined by the curvature of the boundaries considered but instead by the radial propagation direction of the Rossby waves excited by thermal convection.

Stability and bifurcation diagram of Boussinesq thermal convection in a moderately rotating spherical shell

Stability and bifurcation of Boussinesq thermal convection in a moderately rotating spherical shell are investigated by obtaining finite-amplitude solutions with the Newton method instead of the numerical time integration. The ratio of the inner and outer radii of the shell and the Prandtl number are fixed to 0.4 and 1, respectively, while the Taylor number is varied from 522 to 5002 and the Rayleigh number is from about 1500 to 10 000. In this range of the Taylor number, the stable finite-amplitude solutions, which have four-fold symmetry in the longitudinal (azimuthal) direction, bifurcate supercritically at the critical points and become unstable when the Rayleigh number is increased up to about 1.2 to 2 times the critical values. When the Taylor number is larger than 3402, propagating direction of the solutions changes from prograde to retrograde continuously as the Rayleigh number is increased. The associated transition of the convection structure is also continuous.

Rossby Waves and Jets in Two-Dimensional Decaying Turbulence on a Rotating Sphere

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.

Energy Accumulation in Easterly Circumpolar Jets Generated by Two-Dimensional Barotropic Decaying Turbulence on a Rapidly Rotating Sphere

Abstract A series of numerical experiments on two-dimensional decaying turbulence is performed for a barotropic fluid on a rotating sphere. Numerical calculations have confirmed two important asymptotic features: emergence of the banded structure of zonal flows and their extreme latitudinal inhomogeneities in which kinetic energy is accumulated into the easterly circumpolar jets. The banded structure of zonal flows is established relatively early on in the initial stage. Later, after extended periods of time integration, only the circumpolar jets are intensified gradually, while there is no further evolution in the banded structure in the low and midlatitudes. Wave activity flux analysis illustrates that the initial vortices in the low and midlatitudes propagate poleward as Rossby waves and converge to produce easterly circumpolar flows. In association with this convergence, accumulation of the mean zonal component of kinetic energy proceeds. The tendency for the accumulation becomes strong as the rotatio...

Retrograde equatorial surface flows generated by thermal convection confined under a stably stratified layer in a rapidly rotating spherical shell

Finite-amplitude thermal convection in a rapidly rotating spherical shell associated with a stably stratified layer placed near the outer surface is investigated. Systematic numerical experiments are performed with an Ekman number of E = 10−3, a Prandtl number of P = 1 and an inner/outer radius ratio of η = 0.4, and the existence of a strongly stratified upper layer is shown to enhance the generation of equatorial surface retrograde flows when the Rayleigh number is approximately ten times larger than the critical value. The existence of the stable layer causes the bottom of the stable layer to behave as a virtual boundary for the convective motion underneath. Its effective dynamic condition varies from the free-slip condition to the no-slip condition as the Rayleigh number increases. The Reynolds stress of the convective vortices beneath the stable layer is weakened and is dominated by the transport of the planetary angular momentum. As a result, the latitudinal temperature gradient produced at the bottom of the stable layer induces the equatorial retrograde flow through the thermal wind balance. This diffuses through the stable layer by viscosity and produces the equatorial surface retrograde flow.

On the retrograde propagation of critical thermal convection in a slowly rotating spherical shell

The retrograde propagation mechanism of critical thermal convection with a sectorial pattern emerging in a slowly rotating spherical shell is investigated through vorticity budget analysis. In the equatorial region, stretching and shrinking of the fluid columns in the direction of the axis of rotation due to the radial component of velocity causes retrograde propagation, whereas in the mid-latitudes, tilting of the radial component of planetary vorticity by the radial shear of the latitudinal component of velocity is dominant. The switching of the propagating direction from retrograde to prograde according to the increase in the rotation speed of the shell originates from the transition of the morphology of vortices from the ‘banana-shaped’ type due to the constraint of the spherical geometry to the columnar type due to the Taylor–Proudman constraint. The variation of the morphology of vortices reverses the tendency of stretching/shrinking of fluid columns accompanied by their cylindrically radial displacement.