A contour-advective semi-lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields

This paper describes a novel numerical algorithm for simulating the evolution of fine-scale conservative fields in layer-wise two-dimensional flows, the most important examples of which are the earth's atmosphere and oceans. the algorithm combines two radically different algorithms, one Lagrangian and the other Eulerian, to achieve an unexpected gain in computational efficiency. The algorithm is demonstrated for multi-layer quasi-geostrophic flow, and results are presented for a simulation of a tilted stratospheric polar vortex and of nearly-inviscid quasi-geostrophic turbulence. the turbulence results contradict previous arguments and simulation results that have suggested an ultimate two-dimensional, vertically-coherent character of the flow. Ongoing extensions of the algorithm to the generally ageostrophic flows characteristic of planetary fluid dynamics are outlined.

Perturbed Rankine vortices in surface quasi-geostrophic dynamics

An analytical dispersion relation is derived for linear perturbations to a Rankine vortex governed by surface quasi-geostrophic dynamics. Such a Rankine vortex is a circular region of uniform anomalous surface temperature evolving under quasi-geostrophic dynamics with uniform interior potential vorticity. The dispersion relation is analysed in detail and compared to the more familiar dispersion relation for a perturbed Rankine vortex governed by the Euler equations. The results are successfully verified against numerical simulations of the full equations. The dispersion relation is relevant to problems including wave propagation on surface temperature fronts and the stability of vortices in quasi-geostrophic turbulence.

Viscous coupling of shear-free turbulence across nearly flat fluid interfaces

The interactions between shear-free turbulence in two regions (denoted as + and − on either side of a nearly flat horizontal interface are shown here to be controlled by several mechanisms, which depend on the magnitudes of the ratios of the densities, ρ+/ρ−, and kinematic viscosities of the fluids, μ+/μ−, and the root mean square (r.m.s.) velocities of the turbulence, u0+/u0−, above and below the interface. This study focuses on gas–liquid interfaces so that ρ+/ρ− ≪ 1 and also on where turbulence is generated either above or below the interface so that u0+/u0− is either very large or very small. It is assumed that vertical buoyancy forces across the interface are much larger than internal forces so that the interface is nearly flat, and coupling between turbulence on either side of the interface is determined by viscous stresses. A formal linearized rapid-distortion analysis with viscous effects is developed by extending the previous study by Hunt & Graham (J. Fluid Mech., vol. 84, 1978, pp. 209–235) of shear-free turbulence near rigid plane boundaries. The physical processes accounted for in our model include both the blocking effect of the interface on normal components of the turbulence and the viscous coupling of the horizontal field across thin interfacial viscous boundary layers. The horizontal divergence in the perturbation velocity field in the viscous layer drives weak inviscid irrotational velocity fluctuations outside the viscous boundary layers in a mechanism analogous to Ekman pumping. The analysis shows the following. (i) The blocking effects are similar to those near rigid boundaries on each side of the interface, but through the action of the thin viscous layers above and below the interface, the horizontal and vertical velocity components differ from those near a rigid surface and are correlated or anti-correlated respectively. (ii) Because of the growth of the viscous layers on either side of the interface, the ratio uI/u0, where uI is the r.m.s. of the interfacial velocity fluctuations and u0 the r.m.s. of the homogeneous turbulence far from the interface, does not vary with time. If the turbulence is driven in the lower layer with ρ+/ρ− ≪ 1 and u0+/u0− ≪ 1, then uI/u0− ~ 1 when Re (=u0−L−/ν−) ≫ 1 and R = (ρ−/ρ+)(v−/v+)1/2 ≫ 1. If the turbulence is driven in the upper layer with ρ+/ρ− ≪ 1 and u0+/u0− ≫ 1, then uI/u0+ ~ 1/(1 + R). (iii) Nonlinear effects become significant over periods greater than Lagrangian time scales. When turbulence is generated in the lower layer, and the Reynolds number is high enough, motions in the upper viscous layer are turbulent. The horizontal vorticity tends to decrease, and the vertical vorticity of the eddies dominates their asymptotic structure. When turbulence is generated in the upper layer, and the Reynolds number is less than about 106–107, the fluctuations in the viscous layer do not become turbulent. Nonlinear processes at the interface increase the ratio uI/u0+ for sheared or shear-free turbulence in the gas above its linear value of uI/u0+ ~ 1/(1 + R) to (ρ+/ρ−)1/2 ~ 1/30 for air–water interfaces. This estimate agrees with the direct numerical simulation results from Lombardi, De Angelis & Bannerjee (Phys. Fluids, vol. 8, no. 6, 1996, pp. 1643–1665). Because the linear viscous–inertial coupling mechanism is still significant, the eddy motions on either side of the interface have a similar horizontal structure, although their vertical structure differs.

Infrared dynamics of decaying two-dimensional turbulence governed by the Charney-Hasegawa-Mima equation

The low wave number range of decaying turbulence governed by the Charney-Hasegawa-Mima (CHM) equation is examined theoretically and by direct numerical simulation. Here, the low wave number range is defined as values of the wave number k below the wave number kE corresponding to the peak of the energy spectrum, or alternatively the centroid wave number of the energy spectrum. The energy spectrum in the low wave number range in the infrared regime (k →0) is theoretically derived to be E(k) ∼k5, using a quasinormal Markovianized model of the CHM equation. This result is verified by direct numerical simulation of the CHM equation. The wave number triads (k,p,q) responsible for the formation of the low wave number spectrum are also examined. It is found that the energy flux Π(k) for k< kE can be entirely expressed by Π(-)(k), which is the total net input of energy to wave numbers k. Furthermore, the contribution of nonlocal triad interactions to the energy flux is found to be predominant in the range log (k/kE)<-0.5, where the nonlocal interactions are defined to be those triad interactions for which the ratio of the largest leg of the triad to the smallest leg is larger than four. ©2001 The Physical Society of Japan