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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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William R. Holland and Dale B. Haidvogel

Abstract

The vacillation of baroclinically unstable waves in a two-layer eddy-resolving oceanic circulation model is described. The vacillation cycle is distinguished kinematically by the mutual coexistence at equilibrium of short (60-day) period mesoscale eddies and a well-defined long (480-day) period modulation to the larger scale flow, as well as the long-term mean ocean circulation. Global energy budgets and related linear stability analyses reveal underlying systematic energy transfers between the slowly varying mean and transient fields of motion. The vacillation phenomenon is shown to occur over a rather narrow range of the nondimensional model parameters. Since the vacillation occurs in the presence of β, a highly structured mean flow field and meridional boundaries, this is perhaps the most complicated geophysical flow situation in which a vacillation cycle has been clearly observed.

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Albert J. Semtner Jr. and William R. Holland

Abstract

An equatorial ocean experiment has been carried out, using the primitive equation model of Semtner and Mintz (1977) with a highly conservative differencing scheme, with high horizontal resolution (Δx = 0.50°, Δy=0.25°) and with 14 levels in the vertical. A turbulent equilibrium state has been reached for a 3300 km × 2200 km equatorial ocean, driven by constant 0.5 dyn cm−2 wind stress, heated at the surface and cooled at the northern and southern walls.

The predicted surface temperature field shows an upwelling-induced cold region along the equator. The temperatures at the equator near the eastern wall are as much as 6°C colder than in the subequatorial regions. Westward moving waves occur in the temperature field a few degrees north and south of the equator. These waves have periods of 33 days, wavelengths of 800 km, and are symmetric about the equator. Their structure is similar to that of equatorially trapped Rossby waves with n=1 in the vertical and m=1 in the horizontal. Shorter wavelength disturbances are found throughout the thermocline near the equator, and these have periods typical of equatorially trapped inertia-gravity waves. The horizontal temperature field at depth suggests that a number of high baroclinic modes are superposed.

The surface flow in the model is characterized by Ekman drift plus transient geostrophic flow off the equator and by weak and variable flow at the equator. A pressure gradient due to the tilt of the sea surface along the equator largely balances the wind stress on the surface layer. Below the surface, this pressure gradient drives an equatorial undercurrent, which slopes upward to the east and intensifies to a maximum of about 100 cm s−1. The undercurrent meanders, with periods of 100 days or more, by as much as 100 km on either side of the equator. Below the current, westward moving cross-equatorial flows with periods of about 44 days sometimes link up the quasi-geostrophic circulations on opposite sides of the equator. These flows appear to be associated with an antisymmetric (in u and p) Rossby wave of the same period having m = 2.

An analysis of energetics shows that the disturbances on either side of the equator are maintained by baroclinic instability, whereas the equatorial undercurrent exhibits mainly barotropic instability. These instabilities lead to transient circulations whose characteristics are similar to those of equatorially trapped neutral waves. Frictional dissipation is concentrated at the equator, and most of the loss of energy is from the eddy circulations rather than from the mean flow.

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Albert J. Semtner Jr. and William R. Holland

Abstract

The purpose of this paper is to compare two numerical models of vastly different complexity and computational requirements, which have been used recently in a number of midlatitude ocean simulations. Specifically, the two-layer quasi-geostrophic (QG) model of Holland (1978) is compared with the five-level primitive equation (PE) model of Semtner and Mintz (1977) for a wind-driven multi-gyre ocean, with effects of bottom topography and thermal forcing included. The dominant feature of the circulation predicted in the previous PE calculations is a strong free jet, with intense mesoscale transients which are maintained by baroclinic instability.

The configuration of the QG experiment is designed to approximate closely that of the PE experiment, while retaining as much of the simplicity of the Holland (1978) model as possible. The QG model spins up to a state of statistical equilibrium, which is characterized by a meandering jet and by mid-ocean mesoscale eddies with periods and wavelengths much like those in the PE experiment. The time-mean circulations and the distributions of eddy energy in both models are very similar. An energy analysis shows that the free jet in the QG model is more barotropically unstable than in the PE model; however, by reducing the QG upper layer depth to be closer to the thickness of the free jet in the PE model (200 m), this discrepancy disappears. Excellent agreement is also obtained between the volume-integrated energetics of the two models, provided one uses the same lateral diffusion coefficients for momentum and heat in both models.

To gain more insight into physical processes, the computational speed of the QG model is exploited to make additional experiments on the influences of bottom topography, thermal forcing and increased vertical resolution. Bottom topography is found to intensify the upper layer jet and to change substantially the pattern of the deep mean flow. While the presence or absence of topography does not alter the degree of baroclinic versus barotropic instability when the upper layer is 500 m thick, topography does cause a greater proportion of baroclinic instability when the upper layer is thinner. Thermal forcing strengthens the flow in both layers. The use of a three-layer QG model removes the arbitrariness associated with the choice of upper layer thickness: the dominant baroclinic instability of the free jet remains and is concentrated at the interface of the upper two layers.

The results of the present intercomparison suggest that QG simulations will produce the same basic dynamics as PE models in the type of problem considered, using a fraction of the computer time. The saying in computer resources can be profitably applied to understanding the important effects of parameter variations on the oceanic general circulation.

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William R. Holland and Mary L. Batteen

Abstract

The two-layer quasigeostrophic model of Holland is modified to include a parameterization of subgrid-scale heat diffusion. Results from a sequence of simple, eddy-resolved calculations illustrate the effects of increasing heat diffusivities. It is clear that even rather small diffusion coefficients (small compared to the viscosity) cause important modifications of the eddy field and of the eddy generation process. In particular, heat diffusion can be very effective at diminishing the baroclinic signal associated with mesoscale processes, making it less likely that baroclinic instability processes can exceed damping.

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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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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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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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