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Greg Holloway

Abstract

When applied to oceanic internal waves of observed amplitudes, a class of weak wave-wave interaction theories predict certain very rapid interactions, contradicting the theoretical premise. It has been suggested that the apparent contradiction may be spurious. I attempt to clarify arguments which show that the contradiction is real, hence that weak interaction theories are largely inapplicable over much of the oceanic internal wave spectrum. However, important points of theoretical ambiguity remain.

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Greg Holloway

Abstract

In a recent note, Weinstock reconsiders an argument advanced twenty years earlier by Phillips concerning the buoyancy subrange theory of Lumley. Phillips pointed out that Lumley's theory ought to predict a certain form for temperature fluctuation spectra. Subsequent observations are inconsistent with the predicted spectral form. Weinstock argues that Phillips' analysis is incorrect and that Lumley's theory, suitably extended to treat temperature spectra, is consistent with observations.

Here I will argue that both Lumley and Weinstock theories are incorrect. An alternative theory is described. Field and laboratory tests should be capable of discriminating between the two accounts. The outcome of such tests will bear upon significant practical matters.

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Greg Holloway

Abstract

It is shown that the oceanic internal wave field is too energetic by roughly two orders of magnitude to be treated theoretically as an assemblage of weakly interacting waves. This may be seen both from recent weak wave theoretical calculations which contradict their premises and also from inspection of magnitudes of advection and wave propagation terms. Thus, much recent discussion of results of implications of weak wave theory should be questioned critically. Scaling arguments based on buoyant turbulence are reviewed briefly. The role of vertical mass flux as distinguishing weak wave interactions from stronger turbulence is discussed. Possible progress by renormalization of wave interaction equations is considered.

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Greg Holloway

Abstract

Theoretical predictability of the atmosphere is limited in part by the rate at which small-scale observational errors in the initiation of a forecast grow both in amplitude and in length scale with increasing time. Processes of error growth are examined for the case of equivalent barotropic motion on a β-plane. From direct numerical simulations and from statistical closure theory, it is seen that larger β implies slower rate of error growth. Thus planetary wave propagation enhances predictability. Inclusion of a finite equivalent depth also enhances predictability. For values of β and of the equivalent depth representative of Earth's atmosphere, a theoretical predictability time scale is increased fourfold over earlier calculations based on uniformly rotating, “rigid lid” models.

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Greg Holloway

Abstract

If a field of randomly distributed barotropic eddies interacts with underlying topographic features in a rotating reference frame, the flow is observed to develop, after a time the order of an eddy turnaround time τeddy = L/U a component of steady flow which is locked to the topography in the sense of anti-cyclonic circulation around bills. Thereafter, spectral energy transfer is inhibited and the forms both of the energy spectrum of the motion and of the spectrum of vorlicity-topography correlation are dependent on the topographic variance spectrum.

If the topographic variation is weak, in that τtopo ≫ τeddy where τtopo = π/fδ is a topographic wave period with f twice the uniform rotation rate and δ a characteristic height of topography relative to the depth of fluid, the flow dynamics approaches that of two-dimensional turbulence. If topographic variation is strong. in that τtopo ≪ τeddy, energy readily scatters into smaller scales. decreasing τeddy until a balance is obtained between topographic effects and nonlinear or advective effects. The correlation of vorticity with topography then develops and further evolution is suppressed.

A theory of the statistical evolution of an ensemble of flow realizations averaged over an ensemble of topographies has been described by Herring (1977). In this paper a rather simpler statistical theory is obtained, after the “test field model” of turbulence (Kraichnan, 1971), in which the time evolution of the energy or vorticity variance spectrum and of the vorticity-topography correlation spectrum is described by a pair of coupled integral equations. These equations are seen to correspond to the statistical evolution of a stochastic variable governed by a modified Langevin equation in which topography provides a steady driving force.

Theoretical predictions are compared with numerical flow simulations for various choices of topographic variance spectra. of dissipation mechanisms and of ratios τeddytoppo, Overall quantitative agreement appears to be satisfactory.

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Greg Holloway

Abstract

Interaction of eddies with seafloor topography can exert enormous, systematic forces on the ocean circulation. This interaction has been considered previously under idealized circumstances. Theoretical results are here simplified and extended toward practical application in large-scale ocean circulation models. Among the suggestions is that coarse resolution models can “correct” a depth-independent part of the velocity field toward a velocity given by −z × ∇fLH, where z is the vertical unit vector, f is Coriolis form, L is a length scale O(10 km), and H is the total depth. Absence of this tendency may be implicated in a number of systematic defects that appear in present ocean models.

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Greg Holloway

Abstract

The relation between thickness diffusion in layer and level models is set out. Parameterizations of thickness diffusion are related to a parameterization of eddy effects on momentum. The author anticipates where these parameterizations for thickness and momenta are likely to fail and how the anticipated failures may be overcome.

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Jieping Zou and Greg Holloway

Abstract

The steady-state fit of Arakawa and Lamb's shallow-water equation model to time-mean sea surface height (SSH) has been examined, seeking better performance of the steady-state fit without requiring longer time integrations. It is shown that the minimization problem with steadiness penalties involving one time step is ill conditioned. The difficulty is due to disparity among components of the cost function gradient, when the cost function surface is characterized by steep slopes in some parameter directions and by flat slopes in others. Preconditioning the gradient of cost function by means of an adjoint model is examined. Results show improvement of condition number of the Hessian of the cost function and quality of the fit while retaining the economy of the one-time-step approach. The method is extended to fitting the time-mean SSH from eddy-active simulations.

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Greg Holloway and Tessa Sou

Abstract

From a global inventory of long-term current meter records, the skill of an ocean model is evaluated by measures of differences between modeled and observed currants. A parameterization of eddy – topographic interaction is assessed.

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Greg Holloway and Jane Eert

Abstract

Possible multiple equilibria of large-scale flows over topography have been subjects of many recent investigations. Early suggestions of Charney and De Vore, Hart and Wiin-Nicisen were based upon idealized flows of very few degrees of freedom. Subsequent studies based upon more elaborate systems failed to show intransitive multiple equilibria. On the contrary, we exhibit well-resolved, fully eddy-active model results which show multiple equilibria to be well-established over a range of parameters. “Blocked” regimes are characterized by transient eddy activity. Under steady external forcing, the flow regimes appear to be intransitive.

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