George Veronis

Motions at subcritical values of the Rayleigh number in a rotating fluid

A simple analysis is presented for the finite-amplitude, steady motions in a rotating layer of fluid which is heated uniformly from below and cooled from above. The boundaries are considered to be ‘free’ and a solution is obtained for the two-dimensional problem using the eigenfunctions of the stability problem plus the smallest number of higher modes required to represent non-linear interactions. In his analysis of the stability problem Chandrasekhar (1953) concluded that in mercury overstable motions can occur for a value of the Rayleigh number which is as little as 1/67 of the value required for instability to steady motions. In the present paper it is shown that, for a restricted range of Taylor number, steady finite-amplitude motions can exist for values of the Rayleigh number smaller than the critical value required for overstability. The horizontal scale of these finite-amplitude steady motions is larger than that of the overstable motions. A more exact solution to the finite-amplitude problem confirms the above results. The latter solution together with additional physical results will be presented in a later paper.

Wind-driven ocean circulation—Part 2. Numerical solutions of the non-linear problem

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The use of oxygen as a test for an abyssal circulation model

Stommels abyssal circulation velocities are used in the advective-diffusive-decay equation to calculate the dissolved oxygen distribution in an idealized world ocean basin. The use of a weighted difference scheme allows an extension of results obtained earlier to a broader range of values of the parameters for advection, decay and eddy diffusion. Extreme cases of no diffusion and of no advection are used to interpret the results for intermediate values of the parameters. The optimal set of values found for the parameters are: horizontal eddy diffusion = 6 × 106 cm2 sec−1, uniform upwelling velocity = 1.5 × 10−5 cm sec−1, oxygen consumption rate in abyssal waters = 0.002 ml 1.−1 yr−1, recirculation around the Antarctic Circumpolar Current = 35 × 106 m3 sec−1. The optimal pattern is then compared with (an updated) observed oxygen distribution to determine the merits and deficiencies of the circulation model. The major deficiency appears to be the neglect of topographic effects on the abyssal circulation.

Distribution of tracers in the deep oceans of the world

Abstract With the Stommel-Arons model of abyssal circulation of the world ocean a numerical study of the two-dimensional advective-diffusive-decay equation is made to determine those values of the intensity of circulation and eddy diffusivity which give the best agreement with observed distributions of 14 C and dissolved oxygen. Good quantitative agreement with observed data is obtained with a horizontal eddy diffusivity coefficient of 10 7 cm 2 sec −1 and a uniform upwelling beneath the main thermocline of 1·25 × 10 −5 cm sec −1 . The recirculation through the Drake Passage in the Antarctic Circumpolar Current is taken as 10 7 m 3 sec −1 which corresponds to an eastward velocity somewhat more than half of the estimated observed velocity. Because the procedure incorporates a consistent dynamical treatment of the motion of the waters in the interior abyssal oceans, it provides a model for interpreting the observed tracer distribution which is preferable to the direct-circulation model traditionally used to interpret tracer data.

Obtaining Velocities from Tracer Distributions

Abstract A simple advective-diffusive system with uniform, horizontal flow in a channel and fixed boundary concentrations of two tracers is analyzed for the tracer concentrations in the interior. The deduced concentrations are then treated as given information to invert the analysis to obtain the velocity field. The methodology of the inverse procedure is studied here in order to determine the types of information and the conditions under which the original velocity field can be recovered in limited portions of the channel. When the inverse problem is underdetermined, e.g., when only a single tracer distribution and no other information is given, the inversion leads to a flow that bears no resemblance to the known velocity. An overdetermined inverse problem with data that are free of error and noise recovers the original velocity. Also studied are systems with more realistic features in the given data, such as truncation error, noise, qualitatively correct (or incorrect) auxiliary velocity information and...

Large Scale Ocean Circulation

Publisher Summary Theoretical picture of large-scale ocean circulation has grown mostly out of the development of simple models, which isolate the particular phenomenon to be analyzed. This chapter discusses some of the simple theoretical models together with an attempt to extend a few of them to take into account additional features that are not normally included in the models. First, a simple model for deriving the ellipticity of the earth is presented. It is followed by the derivation of the equations of fluid motion in elliptical coordinates and the approximation involved in the use of a spherical coordinate system to analyze oceanographic motions. The approximations encountered in theoretical studies of large-scale flows are then discussed. Next, simple geostrophic flows and their significance are presented. The chapter presents the study of Ekman layers and the role that they play in large-scale circulation and also discusses turbulent transport. The chapter concludes with discussion on laboratory modeling of ocean circulation.

The source-sink flow in a rotating system and its oceanic analogy

Laboratory analogues of theoretical models of wind-driven ocean circulation are based on ideas presented by Stommel (1957). A particularly simple demonstration of the applicability of these ideas is contained in a paper by Stommel, Arons & Faller (1958). The present work develops the source-sink laboratory analogue of ocean circulation models to a point where chosen parametric values allow one to simulate the theoretical models of Stommel (1948) and Munk (1950) exactly. The investigation of the flow in a rotating cylinder generated by a source of fluid near the outer wall leads to a detailed description of the roles of the various boundary layers which occur. This knowledge is used to analyse the more complex source-sink flow in a pie-shaped basin. The laboratory analogue to the Stommel circulation model is analyzed in detail. It is shown that the change in the flow pattern brought about by a radial variation of the position of the eastern boundary in the pie-shaped basin is confined to the interior flow and the boundary layer is largely unaffected. When the bottom of the pie-shaped container slopes, the circulation pattern is changed significantly. For the particular case treated, the depth of the basin along the western boundary is unchanged and the maximum depth occurs at the southeast corner. The circulation generated by a source introduced at the apex of the pie has a gyre whose centre is shifted more toward the southwest corner than the corresponding centre of the gyre for a flat-bottomed basin. Two experiments are reported showing that the western boundary may separate because of the effect of bottom topography or because of the pressure of a cyclonic and an anti-cyclonic gyre generated by suitably placed sources and sinks.

Comparison of Some Recent Experimental and Numerical Results in Bénard Convection

The results of numerical calculations for two‐dimensional Benard convection between rigid boundaries are reported and compared with recent numerical and experimental values reported by other investigators. Values of the Nusselt number, Nu, in the range of Rayleigh number, R, from 2000 to 20 000 are presented. The degree of resolution of a grid point representation necessary to give an accuracy of Nu of 1% is determined. The implications of this requirement show that some previous numerical investigations must have involved fairly large errors in the heat flux and possibly even spurious time oscillations. It is also shown that recent experimental determinations of Nu agree quite well with calculations when the fluid has a large Prandtl number, σ. However, for water (σ=6.8) the discrepancy between numerical calculations and experimental measurements is quite large (14% at R = 20 000).

On the structure and motions of Jupiter's red spot

Abstract A new hypothesis-called the Cartesian diver hypothesis-is proposed to explain the physical nature and observed variations in longitude, size and intensity of Jupiters Great Red Spot. In the light of recent laboratory studies of phase behavior in light gas mixtures, it is suggested that the Red Spot is a region of contrast in the cloud structure of Jupiters outer layers, caused by the presence of a mass of hydrogen-rich solid floating within a fluid layer of hydrogen and helium at some depth below the visible surface. It is shown that this floating solid would exhibit many of the characteristics of a Cartesian diver. Equations of motion for a Cartesian diver in a rotating system are derived, which suggest that the longitudinal motion of the Red Spot consists of several oscillatory components of different amplitudes and frequencies. These predictions are in agreement with motions determined from recent precise measurements of Red Spot longitude. Mechanisms for the source of the observed oscillations are discussed, and estimates are made of the extent of the underlying vertical motions. Effects of the rapid rotation of the planet upon the dynamic behavior of the fluid in the vicinity of the Cartesian diver are qualitatively included, and the resulting physical model is used to explain not only the observed variations of size and intensity of the Red Spot, but also the manner in which these variations correlate with changes in longitude. Laboratory studies and further observations designed to assist in verifying the Cartesian diver hypothesis are suggested.

Experiments on double-diffusive sugar–salt fingers at high stability ratio

In a series of laboratory experiments the growth of double-diffusive salt fingers from an initial configuration of two homogeneous reservoirs with salt in the lower and sugar in the upper layer was investigated. For most of the experiments the stability ratio was between 2.5 and 3, where the latter value is at the upper limit (the ratio of salt to sugar diffusivities) for which fingers can exist. In these experiments long slender fingers are generated at the interface. Essentially all theories or physical bases for models of salt fingers presuppose such a configuration of long fingers. Our measurements show that the length of fingers at high stability ratio increases with time like t 1/2 , with a coefficient that is consistent with the diffusive spread of the faster diffusing component (salt). When the initial stability ratio is closer to unity, fingers penetrate into the reservoirs very rapidly carrying with them large anomalies of salt and sugar which give rise to convective overturning of the reservoirs. The convection sweeps away the ends of the fingers, and when it is intense enough (as it is when the sugar anomaly is large) it can reduce the finger height to a value less than the width. After this initial phase the finger length grows linearly with time as has been found in previous studies. These results show that salt fingers can evolve in quite different ways depending on the initial stability ratio and must cast doubt on the use of simple similarity arguments to parameterize the heat and salt fluxes produced by fingers.