The sensitivity and predictability of mesoscale eddies in an idealized model ocean

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution April, 1976Two numerical applications of two-level quasigeostrophic theory are used to investigate the interrelationships of the mean and mesoscale eddy fields in a closed-basin ocean model. The resulting techniques provide a more accurate description of the local dynamics, origins, and parametric dependences of the eddies than that available in previous modelling studies. First, we propose a novel and highly efficient quasigeostrophic closed-domain model which has among its advantages a heightened resolution in the boundary layer regions. The pseudospectral method, employing an orthogonal expansion in Fourier and Chebyshev functions, relies upon a discrete Green's function technique capable of satisfying to spectral accuracy rather arbitrary boundary conditions on the eastern and western (continental) walls. Using this formulation, a series of four primary numerical experiments tests the sensitivity of wind-driven single and double-gyred eddying circulations to a transition from free-slip to no-slip boundary conditions. These comparisons indicate that, in the absence of topography, no-slip boundaries act primarily to diffuse vorticity more efficiently. The interior transport fields are thus reduced by as much as 50%, but left qualitatively unchanged. In effect, once having separated from the western wall, the internal jet has no know1edge, apart from its characteristic flow speed, of the details of the boundary layer structure. Next, we develop a linearized stability theory to analyze the local dynamic processes responsible for the eddy fields observed in these idealized models. Given two-dimensional (x, z) velocity profiles of arbitrary horizontal orientation, the resulting eigenfunction problems are solved to predict a variety of eddy properties: growth rate, length and time scales, spatial distribution, and energy fluxes. This simple methodology accurately reproduces many of the eddy statistics of the fully nonlinear fields; for instance, growth rates of 10-100 days predicted for the growing waves by the stability analysis are consistent with observed model behavior and have been confirmed independently by a perturbation growth test. Local energetic considerations indicate that the eddy motions arise in distinct and recognizable regions of barotropic and baroclinic activity. The baroclinic instabilities deîend sensitively on the vertical shear which must exceed 0(5 cm sec-1) across the thermocline to induce eddy growth. As little as a 10% reduction in |uz|, however, severely suppresses the cascade of mean potential energy to the eddy field. In comparison, the barotropic energy conversion process scales with the horizontal velocity shear, |uy|, whose threshold values for instability, a (2 x 10-6 sec-1), is undoubtedly geophysically realizable. A simple scatter diagram of |uy| versus |uz| for all the unstable modes studied shows a clear separation between the regions of barotropic and baroclinic instability. While the existence of baroclinic modes can be deduced from either time mean or instantaneous flow profiles, barotropic modes cannot be predicted from mean circulation profiles (in which the averaging process reduces the effective horizontal shears). Finally, we conduct a separate set of stability experiments on analytically generated jet profiles. The resulting unstable modes align with the upper level velocity maxima and, although highly sensitive to local shear amplitude, depend much less strongly on jet separation and width. Thus, the spatial and temporal variability of the mesoscale statistics monitored in the nonlinear eddy simulations can be attributed almost entirely to time-dependent variations in local shear strength. While these results have been obtained in the absence of topography and in an idealized system, they yet have strong implications for the importance of the mid-ocean and boundary layer regions as possible eddy generation sites.This research has been made possible by National Science Foundation grant OCE74-03001 A03, formerly DES73-00528, and the National Science Foundation funded National Center for Atmospheric Research

Mathematical Theory and Modelling in Atmosphere-Ocean-Science

Participants from around the world gathered to review application and development of mathematics in relation to problems in the atmospheric, oceanic and climate sciences

Realizing surface driven flows in the primitive equations

© Copyright 2015 American Meteorological Society (AMS). Permission to use figures, tables, and brief excerpts from this work in scientific and educational works is hereby granted provided that the source is acknowledged. Any use of material in this work that is determined to be “fair use” under Section 107 of the U.S. Copyright Act September 2010 Page 2 or that satisfies the conditions specified in Section 108 of the U.S. Copyright Act (17 USC §108, as revised by P.L. 94-553) does not require the AMS’s permission. Republication, systematic reproduction, posting in electronic form, such as on a web site or in a searchable database, or other uses of this material, except as exempted by the above statement, requires written permission or a license from the AMS. Additional details are provided in the AMS Copyright Policy, available on the AMS Web site located at (https://www.ametsoc.org/) or from the AMS at 617-227-2425 or [email protected] surface quasigeostrophic (SQG) model describes flows with surface buoyancy perturbations with no interior quasigeostrophic potential vorticity at small Rossby number Ro and O(1) Burger number, where quasigeostrophic dynamics are expected to hold. Numerical simulations of SQG dynamics have shown that vortices are frequently generated at small scales, which may have O(1) Rossby numbers and therefore may be beyond the limits of SQG. This paper examines the dynamics of an initially geostrophically balanced elliptical surface buoyancy perturbation in both the SQG model and the nonhydrostatic Boussinesq primitive equations (PE). In the case of very small Rossby number, it is confirmed that both models agree, as expected. For larger Ro, non-SQG effects emerge and as a result the solution of the PE deviates significantly from that of SQG. In particular, an increase in the Rossby number has the following effects: (i) the buoyancy filaments at the surface are stabilized in that they generate fewer secondary vortices; (ii) the core of the vortex experiences inertial instability, which results in a uniform buoyancy profile in its interior; (iii) the divergent part of the energy spectrum increases in magnitude; (iv) the PE model has significantly more gravity waves that are radiated from the vortex; (v) the magnitude of the vertical velocity increases; and (vi) in the mature stages of evolution, there are gravitational instabilities that develop because of the complicated dynamics inside the vortex. It is demonstrated that significant non-SQG effects are evident when the large-scale Rossby number of the initial flow is about 0.05 and the local Rossby number is O(1).Natural Sciences and Engineering Research Council || RGPIN/386456-201

High-Order Numerical Methods in Lake Modelling

The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint

DNS of bifurcations in an air-filled rotating baroclinic annulus

Three-dimensional Direct Numerical Simulation (DNS) on the nonlinear dynamics and a route to chaos in a rotating fluid subjected to lateral heating is presented here and discussed in the context of laboratory experiments in the baroclinic annulus. Following two previous preliminary studies by Maubert and Randriamampianina, the fluid used is air rather than a liquid as used in all other previous work. This study investigated a bifurcation sequence from the axisymmetric flow to a number of complex flows. The transition sequence, on increase of the rotation rate, from the axisymmetric solution via a steady, fully-developed baroclinic wave to chaotic flow followed a variant of the classical quasi-periodic bifurcation route, starting with a subcritical Hopf and associated saddle-node bifurcation. This was followed by a sequence of two supercritical Hopf-type bifurcations, first to an amplitude vacillation, then to a three-frequency quasi-periodic modulated amplitude vacillation (MAV), and finally to a chaotic MAV\@. In the context of the baroclinic annulus this sequence is unusual as the vacillation is usually found on decrease of the rotation rate from the steady wave flow. Further transitions of a steady wave with a higher wave number pointed to the possibility that a barotropic instability of the side wall boundary layers and the subsequent breakdown of these barotropic vortices may play a role in the transition to structural vacillation and, ultimately, geostrophic turbulence.Comment: 31 page

General circulation modelling of close-in extrasolar giant planets

PhDA large fraction of the extrasolar planets detected so far are giant planets that have such short orbital periods (a few days) that they are thought to be tidally-synchronised with the host star. Such orbits lead to permanent day/night sides on the planets and provide a forcing condition for atmospheric dynamics that is not present in the Solar System. The main subject of this thesis is to model the atmospheric dynamics of these close-in extrasolar giant planets, using an accurate three-dimensional general circulation model (GCM). Using the GCM, the primitive equations are numerically solved, with idealised forcing represented by Newtonian relaxation. A large number of simulations is performed to thoroughly explore the relevant physical and numerical parameter space. First, it is found that different initial flow states lead to markedly different flow and temperature distributions. This result is in contrast with the results or assumptions of many published studies, and underlines the fact that circulation models are currently unsuitable for quantitative predictions without better constrained, and well-posed, initial conditions. Second, the effects of artificial viscosity – particularly in relation to the thermal relaxation timescale – are studied. It is demonstrated that using a large range of thermal time scales, including very short ones ( 1 h), as is common in the literature, leads to dominant noise and/or excessively dissipated fields. Finally, variations of the strength of thermal forcing are studied. Distinct stationary or oscillatory states are identified for different sets of forcing parameters. In addition, multiple long lasting states are observed for a given forcing. Most of the states are characterised by a low number ( 4) of large-scale vortices and planetary waves, which exhibit a periodic time variability. The spatiotemporal variability can be important for observational studies, and provides a strong argument for making repeated measurements of a given planet.European Union Fellowshi

DYNAMICS, STABILITY, AND MAINTENANCE MECHANISMS OF SUPER LONG-LIVED OCEAN VORTICES

Ocean variability in the form of coherent mesoscale eddies is still poorly understood despite more than a century of persistent interest and undeniable geophysical significance. These eddies propagate for thousands of miles and distribute vast amounts of nutrients and energy throughout the world’s oceans. Most intriguing is how analytical theories and physical arguments suggest that larger vortices, those with radii greater than 75 km, should become unstable and break down in the span of months, yet they are observed to last for years. This research explores two phenomena that contribute to the longevity of these features. First is their ability to adjust to ambient large-scale flows—a feature that determines the eddy’s endurance. The second aspect is the role small-scale, irregular topography plays in vortex dynamics. We demonstrate that rough seafloor stabilizes surface-intensified vortices by restricting the motion in the deep layer, thereby allowing the upper ring to perpetuate unhindered. Using an analytical parameterization, we show how this "sandpaper effect" can be accounted for in quasi-geostrophic and full Navier-Stokes models. This research will ultimately allow the U.S. Navy to improve global ocean forecasts without dramatically increasing the spatial resolution.Civilian, Department of the NavyApproved for public release. Distribution is unlimited

Estimating oceanic turbulence dissipation from seismic images

Author Posting. © American Meteorological Society, 2013. 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 Atmospheric and Oceanic Technology 30 (2013): 1767–1788, doi:10.1175/JTECH-D-12-00140.1.Seismic images of oceanic thermohaline finestructure record vertical displacements from internal waves and turbulence over large sections at unprecedented horizontal resolution. Where reflections follow isopycnals, their displacements can be used to estimate levels of turbulence dissipation, by applying the Klymak–Moum slope spectrum method. However, many issues must be considered when using seismic images for estimating turbulence dissipation, especially sources of random and harmonic noise. This study examines the utility of seismic images for estimating turbulence dissipation in the ocean, using synthetic modeling and data from two field surveys, from the South China Sea and the eastern Pacific Ocean, including the first comparison of turbulence estimates from seismic images and from vertical shear. Realistic synthetic models that mimic the spectral characteristics of internal waves and turbulence show that reflector slope spectra accurately reproduce isopycnal slope spectra out to horizontal wavenumbers of 0.04 cpm, corresponding to horizontal wavelengths of 25 m. Using seismic reflector slope spectra requires recognition and suppression of shot-generated harmonic noise and restriction of data to frequency bands with signal-to-noise ratios greater than about 4. Calculation of slope spectra directly from Fourier transforms of the seismic data is necessary to determine the suitability of a particular dataset to turbulence estimation from reflector slope spectra. Turbulence dissipation estimated from seismic reflector displacements compares well to those from 10-m shear determined by coincident expendable current profiler (XCP) data, demonstrating that seismic images can produce reliable estimates of turbulence dissipation in the ocean, provided that random noise is minimal and harmonic noise is removed.This work was funded by NSF Grants 0452744, 0405654, and 0648620, and ONR/DEPSCoR Grant DODONR40027.2014-02-0

Stokes Drift and Meshless Wave Modeling

This dissertation is loosely organized around efforts to improve vertical ocean mixing in global climate models and includes an in-depth analysis of Stokes drift, optimization of a new global climate model wave component, and development of a meshless spectral wave model. Stokes drift (hereafter SD) is an important vector component that appears often in wave-averaged dynamics. Mathematically, SD is the mean difference between Eulerian and Lagrangian velocities and intuitively can be thought of as the near-surface ocean current induced from wave motion. Increasingly, spectral wave models are being used to calculate SD globally. These models solve a 5D wave action balance equation and typically require large computational resources to make short to medium-range forecasts of the sea state. In the first part, a hierarchy of SD approximations are investigated and new approximations that remove systematic biases are derived. A new 1D spectral approximation is used to study the effects of multidirectional waves and directional wave spreading on SD. It is shown that these effects are largely uncorrelated and affect both the magnitude and direction of SD in a nonlinear fashion that is sensitive with depth. In the second part, efforts to add a wave model component to the NCAR Community Earth System Model are discussed. This coupled component will serve as the backbone to a new Langmuir mixing parameterization and uses a modified version of NOAA WAVEWATCH III (a third-generation spectral wave model). In addition, the governing wave action balance equation is reviewed and several variations are derived and formulated. In the third part, construction of a monochromatic spectral wave model using RBF-generated finite differences is described. Several numerical test cases are conducted to measure performance and guide further development. In kinematic comparisons with WAVEWATCH III, the meshless prototype is approximately 70–210 times more accurate and uses a factor of 12 to 17 less unknowns