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William R. Holland

Abstract

Results from a two-layer, quasi-geostrophic, general circulation model of the ocean with fine horizontal resolution are presented. As in Holland and Lin (1975a.b), mesoscale eddies spontaneously arise due to instabilities in the oceanic currents, giving rise to transient oceanic circulations that reach a statistical equilibrium. In these final equilibrium states, the interaction of the eddy field with the mean state is examined, and it is shown that the eddies determine the character of the large-scale mean flow. In particular, the eddies act to limit the amplitude of the mean flow in the upper ocean, are responsible for a downward energy propagation that fills the deep sea with eddy energy, and create a downward momentum flux which is responsible for the creation of deep, time-mean, abyssal gyres that are an important component of the vertically averaged mass transport in the ocean.

Three new aspects of the mesoscale eddy problem are discussed. First, the Holland and Lin (1975a,b) results are extended to highly nonlinear free jets, a simple but more realistic treatment of the Gulf Stream as the source for mesoscale eddy energy. Second, bottom friction is examined as the likely mechanism for energy dissipation in a quasi-geostrophic turbulent flow; lateral dissipation enters as an important enstrophy sink but not as an important energy sink. Finally, the usefulness of the quasi-geostrophic nature of the model is demonstrated; only one-tenth of the computer time needed for two-layer primitive equation experiments is required for quasi-geostrophic ones with comparable resolution.

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William R. Holland and William J. Schmitz

Abstract

All available observations indicate that the most energetic time-dependent currents are located in the vicinity of intense large-scale oceanic current systems. This characteristic is also a basic property of eddy-resolving gyre-scale numerical models. An initial detailed intercomparison of two-layer eddy-resolving numerical experiments with observation focused on the largest scales of horizontal structure in patterns of abyssal eddy kinetic energy, and on time scales. The numerical experiments examined generally had relevant temporal and meridional scales, but not necessarily realistic zonal scales. The model eddy field did not penetrate as far from the western boundary as observed distributions, by a factor of 2 to 3.

The present study examines the physical processes that govern the model zonal penetration scale and suggests reasons for the previous discrepancy. It is demonstrated that a subtle balance exists between the complex instability processes that tend to tear the jet apart (restricting its zonal penetration) and the tendency for inertial processes to carry the intense current right across the basin. It would seem that any factor that changes the nature of the instability of the thin Gulf Stream jet will alter the penetration scale. In these models this means not only changing physical parameters and including different physics, but also changing such model dependent factors as vertical resolution. Earlier work suggested the need for enhanced vertical resolution to give realistic zonal penetration, but it is now clear that all stabilizing/destabilizing effects conspire together to give a particular penetration scale.

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Paola Malanotte-Rizzoli and William R. Holland

Abstract

In this work we take a first step in the process of assimilating data into models of the ocean general circulation. The goals is not prediction but rather understanding how the data insertion process affects, and is affected by, the dynamics governing the model. The chosen model ocean is steady, weakly nonlinear and highly frictional Strong vertical friction plays the role of eddy fluxes in driving the circulation in the deep layers.

In the data insertion process we capitalize upon the two principles that (i) the available dynamical models are imperfect; (ii) oceanographic data are measured locally. Three major questions are addressed; 1) what is the influence of local data insertion in terms of improving estimates of the model general circulation? 2) how does the model dynamics affect the spreading of information from the data insertion region? 3) what can we learn about the model physics from the effects of data insertion

Density (or temperature) measurements along long hydrographic or tomographic sections or arrays are chosen as data. We vary the location of the section as well as its orientation. In our highly frictional ocean, the most effective sections are meridional, long and located at a distance from the western boundary. Model estimates are then significantly improved over the broad region extending from the data section to the western boundary itself.

Advective effects are minimal and influence the spreading of information only in the intense western boundary current. Rather, the structure of the gyre interior manifests itself through a quite important steering effect exerted by the motion in the intermediate layer upon the spread of information in the surface layer. Due to this effect the region southwest of the data section is consistently preferred for the improvement of the estimates. Simple analytical computations are carried out to rationalize the numerical results. This effect is likely to persist in more realistic, fully eddy-resolving simulations in which the interfacial eddy stresses would play the role here given to vertical friction.

The dependence of spreading of information upon the internal physics and/or external forcing is used to examine what is imperfect in the model parameterizations. In a simple analytical example we scan the two-dimensional parameter space defined by internal friction and wind stress amplitude. The “correct” values of the above parameters cannot be inferred by this simple scanning due to the non-uniqueness of the solution.

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Dale B. Haidvogel and William R. Holland

Abstract

The stability of currents generated in an oceanic eddy-resolving general circulation model EGCM (Holland, 1978) is investigated by solving the eigenvalue problem associated with the finite-difference quasi-geostrophic vorticity equations which govern the flow. In general, both barotropically and baroclinically unstable waves are shown to exist for instantaneous currents found in the EGCM. Although these simulated flows are not always quasi-steady in the sense required by the theory and are themselves modified by the presence of the finite-amplitude eddies, many characteristics of the eddy field and its interaction with the time-mean circulation can nevertheless be deduced by the linear stability analysis.

In particular, these investigations show that linear stability considerations correctly identify regions of instability in the ocean circulation model and accurately predict the low-order statistical features of the eddy field such as wavelength, period and phase speed. The effects of weakly unstable regions which are masked by global diagnostic techniques can be studied with the local stability model. The linear stability analysis also predicts, with some success, higher order statistics such as the sign and structure of intra-eddy energy fluxes that are important indicators of the dynamics of the unstable regions.

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William R. Holland and Liang B. Lin

Abstract

Numerical experiments on the wind-driven ocean circulation in a closed basin show that mesoscale eddiescan appear spontaneously during the integration of the equations of motion for a baroclinic ocean. For somevalues of the basic parameters governing the flow, the solutions reach a steady state while for other valuesfinite-amplitude eddies remain a part of the final statistically steady state. In the eddying cases the solutionscan be regarded as a mean flow upon which is superimposed a set of eddies which propagate westward at afew kilometers per day. The eddies typically have horizontal wavelengths of a few hundred kilometers.

Analyses of die energetics show the eddies to be generated by the process of baroclinic instability. Thepotential energy of the mean flow is released to supply energy to the eddies. The computed Reynoldsstresses, while small compared to the terms in the geostrophic balance of the mean momentum equations, dohave a strong influence on the mean circulation and, in fact, the deep mean circulation is driven entirelyby the eddies. If the flow were steady, there would be no flow in the deep layer in this model. Finally, thecomputed curl of the Reynolds stresses shows that the vorticity balance of the mean flow is strongly affected by the presence of mesoscale eddies.

In the first part of this report we describe the two-layer model and discuss its numerical formulation.Then the results of a preliminary eddy experiment are discussed in detail, showing the spontaneous growthof baroclinic eddies and describing the final statistical steady state that occurs. Energetic analyses andvorticity balances show the important role played by the eddies in determining the character of the oceanicgeneral circulation.

Part II of this paper will discuss a variety of experiments which explore the dependence of results on thebasic parameters and boundary conditions governing the model. In particular the dependence of resultson wind stress magnitude and distribution, lateral viscosity coefficient, basin size, and boundary conditions(free slip and no slip) will be examined.

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William R. Holland and Paola Malanotte-Rizzoli

Abstract

One of the most important forthcoming synoptic datasets for ocean circulation studies will he the sea-surface height data provided by the TOPEX /POSEIDON satellite. The TOPEX/POSEIDON project is in the planning stage and must still decide upon the particular characteristics of the satellite track. The repeat period will be between 10 and 20 days for a variety of technical and strategic reasons. These choices win give a global coverage with spatial resolution (east-west or north-south separation of crossover points) in midlatitudes of roughly 2.8° of latitude and longitude for a 10-day repeat orbit and 1.4° of latitude and longitude for a 20-day repeat orbit. Thus, the crucial question we address in the present study is: what is the effect of changing space or time resolution or both upon the success of a numerical model in reconstructing a four-dimensional picture a the ocean circulation through the assimilation of altimetric data?

To answer this question we carry out a series of numerical experiments with a three-layer, eddy-resolving quasi-geostrophic model of the ocean circulation in which we systematically vary the space and time resolutions of the data available for assimilation experiments. The experiments are carded out under the “best possible” conditions for the assimilation to be successful, namely: (i) the model is “perfect"; (ii) the data have no errors; and (iii) the data are dynamically compatible with the model since they are simulated by the model itself in a control run.

We reach the following conclusions. In principle, assimilation of altimetric data with a simple relaxation (“nudging”) technique can be very successful in driving the assimilation model to the control run even in the deep layers for which no data are supplied. This is achieved with a “nearly perfect” space-time resolution surface height dataset in which data are supplied at every model grid point and every 0.5 day in time. The residual errors after one year of continuous assimilation amount to less than 10% in all three layers. When the altimetric data are provided along tracks with a given realistic separation (but complete time information), the decrease in space resolution degrades the model estimates somewhat. With data provided at every time step but a track separation of 280 km and making use of the best choice of assimilation procedures we have found that the residual rms errors amount to about 45% after six years of continuous assimilation. While the patterns of the circulation are somewhat different from those of the control run and the flow intensifies are slightly underestimated, the correspondences between the assimilation run and the control run are considerable. When the altimetric data are provided with a realistic time sampling period (but with space resolution at every grid point), the intensity of the flow fields also are somewhat underestimated, especially in the deep layer. The assimilation procedure is again capable, however, of reproducing quite faithfully the flow patterns throughout the water column.

When the altimetric data are assimilated along the actual tracks, that is only at the track grid-points and at the actual time of arrival, the best assimilation results achieved with TOPEX repeat periods of 10 or 20 days are about equally effective for improving the model estimates of the circulation. The residual errors after 6 years of continuous time assimilation are from 60% to 70% for both 10- and 20-day repeats. Apparently, the tradeoff between space and time resolutions just about compensate for each other. The results show that under the best of conditions (small errors, good model) a single satellite makes only minor improvements in the model estimates, and it cannot reconstruct the details of the mesoscale eddy field.

It should be kept in mind that these results depend on the space and time scales of motion in the region to be studied. Moreover, the conclusions reached here depend, to an unknown extent, on the assimilation technique used. Better techniques might allow us to better differentiate between the different space-time choices for TOPEX and to reproduce the actual oceanic circulation more faithfully.

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Michael A. Spall and William R. Holland

Abstract

An interactive, nested primitive equation model for oceanic applications is introduced. The model has two components that interact, which we shall call the coarse and the fine grid regions. The fine grid region is nested entirely within the domain of the coarse grid region. The interaction is achieved by an interpolation of the coarse grid fields to obtain boundary conditions for the fine grid region and by an averaging of the tendencies of the prognostic variables on the fine grid to force the coarse grid model. The nested model is applied to two test problems relevant to oceanic phenomena—a barotropic modon and a baroclinic vortex. In each case, nested calculations with 3:1 and 5:1 grid ratios perform quite well, and even ratios of 7:1 are able to reproduce the solution reasonably well while the features are mostly contained within the fine grid region. These results indicate that the interactive nested model approach introduced here may provide an accurate and cost-effective approach to problems that have multiple spatial scales and/or open boundary condition requirements.

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Paola Malanotte-Rizzoli and William R. Holland

Abstract

In Part I of the present work we performed assimilation experiments with a multilayer, quasi-geostrophic (QG) eddy-resolving model of the ocean general circulation. In Part I we studied the quasi-linear, steady state and the assimilated data were density measured along hydrographic sections. The major result of this study was that the most effective sections are long, meridional ones located at distance from the western boundary. The model estimates are significantly improved over the entire region extending from the data section to the western boundary itself.

In this second part we extend the study to the more realistic time-dependent, fully eddy-resolving ocean. Again we capitalize upon the two assumptions that the available models are imperfect and that data are measured only locally at meridional sections. The location of the sections are chosen according to (i) distance from the western boundary; (ii) energetics of the region. Also, here we compare assimilation of density alone versus density and velocity.

A crucial problem emerges when assimilating data into a fully nonlinear, time-dependent model, that is the problem of model predictability The assimilated data can in fact be viewed as “perturbations” introduced into the model at a specific location. The important question is then: is data insertion performed only locally, i.e., along sections, sufficient to “drive” the model to the reference ocean overcoming the model inherent loss of predictability.

Different data sections are compared and the model performance is quantified monitoring two global rms (root mean square) errors, the rms DIFF1 between the model with inserted data and the reference ocean and the rms DIFF2 between the model with inserted data and without.

Two major results emerge from the present study. First, and differently from the quasi-linear steady case, a single data section is very ineffective in driving the model towards the reference ocean over time scales of ∼100 days, comparable with the time scale of predictability loss. The rms-error DIFF2 is used to quantify the effectiveness of the different section as the “true” rms-error DIFF1 exhibits only random fluctuations around a mean equilibration value. The overall error level depends upon the balance between criteria (i) and (ii) above. Results are rationalized by dynamical considerations showing that the internal boundary forcing provided by the data insertion is equivalent to an additional stress-curl (vorticity source) imposed impulsively along a line in each layer. Also, the assimilation of barotropic and baroclinic information versus baroclinic only (velocity and density versus density only) has no effect on the error levels and error growth rates on the short time scale of mesoscale variability. In general, the error growth rates are not significantly different for any of the considered sections, both for the global rms errors measured over the entire basin and for local rms-errors measured over localized regions. On the short time scale of mesoscale variability, all the considered sections are equally ineffective.

A single section of data is shown instead to be quite effective in driving the model to the reference ocean if the data insertion process is carried out for time durations longer than the model equilibration time. With ten years of data assimilation, the climatological mean of the model becomes extremely similar to the climatological mean of the reference ocean. This result can now be quantified using the “true” rms-error DIFF1, which exhibits an unambiguous decreasing trend during the last years of assimilation, thus improving the estimate of the climatology up to 25%. Thus, single hydrographic sections might still be useful in providing a better model climatology if time series of data were available longer than the model equilibration time.

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William R. Holland and Liang B. Lin

Abstract

In this investigation the wind-driven ocean circulation theories are extended to include mesoscale eddiesas an integral part of the general circulation of the ocean. A two-layer numerical model of ocean circulationin a simple, rectangular basin driven by a steady wind stress is used for this purpose. The equations ofmotion are integrated as an initial value problem until the solutions reach either a steady state or, in thecase of an ocean in which eddies have appeared spontaneously as a result of baroclinic instability, a statistically steady state.

Part I of this study discussed the formulation of the numerical model and presented results from a preliminary numerical experiment. Energetic analyses showed that eddies result from baroclinic instability during the spin-up of the ocean from rest and that, in the final statistically steady state, the eddy momentum and buoyancy fluxes played an important role in establishing the mean circulation. In the particular caseexamined there, the region of eddy generation was in the westward return flow and not in the strong boundary jets.

In this part of the study, results from ten additional experiments are examined to understand, in a limitedway, how eddy generation and the resulting eddy statistics depend upon the basic parameters describing themodel ocean. In particular, the dependence of results on the coefficient of lateral viscosity, the wind stressamplitude, the wind stress distribution (one and two gyres), the basin size, and the boundary conditions(slip and no-slip) are discussed. Results show a wide range of model behavior under the conditions examined,but the common result is that the mean circulation of eddying oceans is importantly altered, one mighteven say largely determined, by the statistical nature of the eddy field.

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William R. Holland and Peter B. Rhines

Abstract

Gyre scale and local vorticity balances are examined for a single numerical experiment designed to elucidate the role of eddies in the oceanic general circulation. Due to the complex nature of the flow, a combination of different analyses is needed. In particular the mean potential vorticity fields are calculated and related to local and global vorticity fluxes. The nature of eddy generation and decay is discussed in terms of eddy enstrophy balances in the fluid. Momentum balances in various parts of the gyre are deduced through the application of the circulation theorem. Fields of eddy diffusivity for the mixing of potential vorticity and heat are determined. The applicability of Sverdrup dynamics in various parts of the fluid and the manner in which the deep abyssal gyres are driven are examined.

The net picture is a complex but consistent one. In the upper layer, eddy generation occurs in the separation region of the eastward jet and in the region of westward return flow. Eddy decay occurs principally at the eastern end of the free jet accompanied by upgradient eddy fluxes of heat and potential vorticity. The lower layer is driven from above by inviscid pressure forcing at the interface., this is accompanied by downgradient potential vorticity flux everywhere in the lower layer. The deep dynamics is essentially a “turbulent” Sverdrup balance, Ū3·∇ Q̄3= ∇·κ ∇Q̄3, driven by eddy rather than wind stresses.

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