Singular Bohr-Sommerfeld Rules for 2D Integrable Systems

In this paper, we describe Bohr-Sommerfeld rules for semi-classical completely integrable systems with 2 degrees of freedom with non degenerate singularities (Morse-Bott singularities) under the assumption that the energy level of the first Hamiltonian is non singular. The more singular case of {\it focus-focus} singularities is studied in [Vu Ngoc San, CPAM 2000] and [Vu Ngoc San, PhD 1998] The case of 1 degree of freedom has been studied in [Colin de Verdiere-Parisse, CMP 1999] Our theory is applied to some famous examples: the geodesics of the ellipsoid, the $1:2$-resonance, and Schroedinger operators on the sphere $S^2$. A numerical test shows that the semiclassical Bohr-Sommerfeld rules match very accurately the ``purely quantum'' computations.Comment: postscript, 61 pages, figures best seen in color. Preprint Institut Fourie

Quasigeostrophic flows and turbulence in a rotating homogeneous fluid

Submitted in partial fulfillment of the requirements for the degree of Doctor of Science at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution August, 1977Stimulated by new evidence from both "in situ" oceanic observations and results from numerical modelling, a laboratory study of quasigeostrophic flow and turbulence in a rotating homogeneous fluid has been undertaken. Two dimensional turbulence driven by a uniform distribution of sources and sinks which oscillate in time, can be fairly well reproduced in this context. Inertial time scales are about ten times smaller than Ekman spinup time, and typical Reynolds numbers read 2000. The observations emphasize the spectral tendency of the energy containing eddies. The case of no topography is first discussed. In steadily forced turbulence, it is observed that the energy containing scale is significantly larger than the forcing scale. In the decaying stage the red cascade is observed and rates of interaction are measured. Theoretical arguments for both behaviors are presented; the former concerning the forced turbulence case is believed to be new. The forcing is next applied over various large scale topographies, modelling the geophysical beta effect. The polar beta plane geometry preserves the above spectral characteristics but at the same time introduces anisotropy into the flow pattern. A broad westward mean flow develops in the north and is surrounded by a belt of cyclones lying on its southward side. The calculated second-order Eulerian mean flows induced by steadily and uniformly forced Rossby waves in a long zonal channel, exhibit much of the same momentum distribution in the inertial regime. In contrast, the "sliced cylinder" geometry which possesses no closed geostrophic contours drastically modifies the above picture. Both mean flow production and a large scale tendency for the eddies are inhibited. The geographical distribution of the eddy intensities and scales is now wildly inhomogeneous. The second aspect of this work is a study of the interaction of Rossby waves with mean flows. A zonally traveling, forced wave is generated near the southern boundary of a polar beta plane. Due to energy radiation in the free interior and (or) potential vorticity mixing by the finite amplitude waves, a westward zonal flow develops. The effect of the mean flow upon the forced steady waves is to weaken the anticyclones and intensify the cyclones. Pressure time series reveal a growth of harmonics and general spectral broadening as the waves travel freely inwards, suggesting active nonlinear interactions. An experimental test of Rhines' (1977) potential vorticity mixing theory is also presented at free latitudes. The decay period when the driving is suppressed shows that a net transfer from the waves to the mean flow kinetic energy occurs. Connection with hydrodynamic stability theory is discussed. Interaction of Rossby waves with an externally generated westward mean flow allows one to make a controlled study of the critical layer problem. For small amplitude waves, the mean flow is accelerated in the entire region between the forcing and the critical latitude which acts as a wall for mean wave momentum. In nonlinear runs the steady profile of the westward flow indicates that an accelerating force is acting everywhere, revealing the increasing transmission of wave momentum through the critical layer. At the same time, pressure measurements near the critical point show considerable fine structure developing over a long time scale. The third part deals with steady isolated source-sink flows in the sliced cylinder geometry. The response of the fluid to a meridionally oriented steady dipole extends exclusively westward of the forcing. The viscously balanced solutions are discussed and relevance to oceanic abyssal circulation is emphasized. With strong driving, the combination of a cyclone to the north and an anticyclone to the south is absolutely stable although the reverse configuration is not. A connection with a certain class of free, steady, isolated, inertial solutions developed recently by Stern (1976) is made.The DGRST . (FRACE) and the Joint Program in Oceanography, Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution offered a fellowship for the first two years. The National Science Foundation under Grant OCE75-2l 674 and the Office of Naval Research under Contract N00014-74-C0262-NR-083-004 supported this study for the final two years

Lagrangian eddy statistics from surface drifters in the eastern North Atlantic

In 1976–1977, 16 surface drifters were seeded regularly in an area about 100 km in diameter, providing a good opportunity to study diffusion induced by the mesoscale eddies present in this part of the eastern North Atlantic. It is found herein that Taylor\u27s formula relating the diffusivity (rate of dispersion) of marked fluid parcels to the integral of the Lagrangian correlation function is accurate up to 30 days. The diffusivity, initially zero, reaches a plateau after about 10 days. The diffusivities associated with the time scale of the energy containing eddies (30 days) are 2.3 107 cm2/s and 1.7 107 cm2/s in the zonal and meridional direction respectively. This is small in view of the surface eddy kinetic energy of the order of 100 cm2/s2. Accordingly the Lagrangian integral time scales are as short as 2 days. After 30 days, the diffusivity decreases and the dispersion saturates. The Lagrangian eddy energy content is therefore weak at low frequencies in contrast with subsurface Eulerian energy content obtained in the same area. The center of gravity of the Lagrangian and Eulerian energy spectrum indicates time scales respectively of 12 and 26 days. At frequencies higher than 0.1 cpd the Lagrangian velocity spectrum falls off with a −2 slope. This may be an indication of a balance between the Lagrangian acceleration of water parcels and the essentially white noise wind stress computed from a nearby weathership. Eulerian spectra at greater depths which had a similar −2 slope in this band but a much reduced energy level suggest an oceanic response strongly trapped at the surface

Binary-swap volumetric rendering on the T3D

Journal ArticleThis paper presents a data distributed parallel raytraced volume rendering algorithm and its implementation on the CRI T3D. This algorithm distributes the data and the computational load to individual processing units to achieve fast and high-quality rendering of high-resolution data. The volume data, once distributed, is left intact. The processing nodes perform local raytracing of their subvolume concurrently. No communication between processing units is needed during this local ray-tracing process. A subimages is generated by each processing unit and the final image is obtained by compositing subimages in the proper order by the Binary-Swap algorithm. Performance of this algorithm on the T3D is presented and compared to an implementation on the CM-5

On the movements of the North Atlantic Subpolar Front in the preinstrumental past

Author Posting. © American Meteorological Society, 2016. This article is posted here by permission of American Meteorological Society for personal use, not for redistribution. The definitive version was published in Journal of Climate 29 (2016): 1545-1571, doi:10.1175/JCLI-D-15-0509.1.Three sediment records of sea surface temperature (SST) are analyzed that originate from distant locations in the North Atlantic, have centennial-to-multicentennial resolution, are based on the same reconstruction method and chronological assumptions, and span the past 15 000 yr. Using recursive least squares techniques, an estimate of the time-dependent North Atlantic SST field over the last 15 kyr is sought that is consistent with both the SST records and a surface ocean circulation model, given estimates of their respective error (co)variances. Under the authors’ assumptions about data and model errors, it is found that the 10°C mixed layer isotherm, which approximately traces the modern Subpolar Front, would have moved by ~15° of latitude southward (northward) in the eastern North Atlantic at the onset (termination) of the Younger Dryas cold interval (YD), a result significant at the level of two standard deviations in the isotherm position. In contrast, meridional movements of the isotherm in the Newfoundland basin are estimated to be small and not significant. Thus, the isotherm would have pivoted twice around a region southeast of the Grand Banks, with a southwest–northeast orientation during the warm intervals of the Bølling–Allerød and the Holocene and a more zonal orientation and southerly position during the cold interval of the YD. This study provides an assessment of the significance of similar previous inferences and illustrates the potential of recursive least squares in paleoceanography.OM acknowledges support from the U.S. National Science Foundation. CW acknowledges support from the European Research Council ERC Grant ACCLIMATE 339108.2016-08-1