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Paola Malanotte-Rizzoli and William R. Holland


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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Paola Malanotte-Rizzoli and Paul J. Hancock


In a series of previous papers, a local theory was formulated to model the persistent atmospheric patterns known as blocking events. The adopted model was the fully nonlinear, baroclinic quasi-geostrophic potential vorticity equation with a mean zonal wind having vertical and horizontal shear. Solutions were found consisting of localized dipole structures with an equivalent barotropic vertical structure. The basic “recipe” provided by the theory was that, in order to form a block characterized by a split flow with an embedded vortex pair, the upstream mean zonal wind ū(y, z) must have a structure which allows for local confinement. Specifically, the function V = ¼ − y/ū, with y, the meridional gradient of mean potential vorticity, must have the shape of a potential well. The bound states of this potential well are structures localized in the (y, z) plane and trapped by the well's positive barriers.

The data analysis carried out here and the results presented are designed to establish whether such a trapping structure exists for the positive blocking cases when compared with the winter climatological mean or other patterns such as the negative anomaly cases of Dole. The unambiguous and robust results emerging from the data analysis are: (i) the composite of the positive anomaly cases shows a strong northern barrier centered in the latitude band 62° to 72°N, in agreement with the northern confinement of the block. The southern barrier, if present, is not covered by the available data. The northern, positive barrier is not present in the climatology. Its presence and significance are doubtful and debatable for the negative anomaly composite. (ii) For the individual positive cases of blocking in which the vortex pair is sufficiently north to be fully covered by the analysis and for which a smooth and zonal upstream wind can be defined, the V-function shows both northern and southern positive brriers at the latitudes of block confinement.

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William R. Holland and Paola Malanotte-Rizzoli


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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M. Ross Vennell and Paola Malanotte-Rizzoli


Many coastal regions in the world ocean are characterized by well-mixed conditions to shelf depth in the density field during the winter season. In these situations it is appropriate to construct a model based on the assumption that the shelf is vertically well mixed. Such a model has been constructed assuming that (i) vertical mixing of momentum is stronger than either horizontal mixing or inertial effects; (ii) the density field is also vertically well mixed, i.e., varies only in horizontal at zero order in the expansion in the vertical Peclet number, (iii) the cross-shelf scale is small compared to the alongshelf scale; (iv) depth varies only in cross-shelf direction. The transport streamfunction equation and the advective density equation can then be combined into a single model equation by noting that in the vertically well-mixed flow, density is conserved as it is advected along streamlines.

This model is used to study two different configurations quite common for shelf circulations in the world ocean. The first configuration considers the effect of a deep baroclinic ocean in driving the shelf circulation. Past studies show that a barotropic deep-ocean pressure gradient cannot drive significant shelf flow and that the continental slope effectively “insulates”the shelf from the deep ocean. Thus, the basic question we want to answer is: how does the baroclinic structure of a deep ocean flow affect its ability to penetrate the shelf and determine its circulation? The second configuration to which the model is applied is that of a coastal current driven by an alongshore buoyancy source, such as a river discharge or an alongshore jet.

In all model applications studied, the basic mechanism by which the flow is able to cross topography is bottom friction. In the first situation the flow is driven by prescribing the velocity and density at the outer edge of the shelf. Three types of shelf forcing by the deep ocean are studied: a wide inflow, a narrow inflow and forcing by a Gulf Stream ring. A specific application is made to the northern Adriatic coastal shelf forced by a dense water pool formed in wintertime in the Adriatic interior. In all cases, the main conclusion for the baroclinic deep-ocean inflow onto the well-mixed shelf is that, like in the barotropic case, the tendency for the flow to follow isobaths is much stronger than the degree to which bottom friction allows cross-isobath motion. The deep ocean inflow forces a horizontal boundary layer against the shelf edge. The width of this boundary layer, and therefore the shelf penetration, is larger if the drag coefficient is higher, the latitude lower or the bottom slope weaker. Also, surface-intensified deep ocean flows penetrate the shelf few strongly than bottom intensified flows.

In the two examples studied of flow entering the shelf region near shore, i.e. a coastal river outflow and a coastal jet, it is again shown that the vertical shear of the shore inflow again determines the overall flow pattern. Because of bottom friction, a light alongshore jet or a low density river, i.e., surface intensified flow, expands across the topography more slowly than a bottom intensified flow, such as a high density river or a heavy jet.

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Paola Malanotte-Rizzoli and William R. Holland


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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Paola Malanotte-Rizzoli and Roberta E. Young


The major objective of oceanic data assimilation studies has been thus far to obtain a four-dimensional realization (space plus time) of the oceanic flow simultaneously consistent with the observations and the model dynamics. In these latest years, however, the forecasting of oceanic motions has emerged as a legitimate and important goal per se. In particular, the operational prediction of mesoscale flows and frontal systems has been the objective of recent assimilation applications in various regional systems of the World Ocean. One such effort focused on the short-term prediction of the Gulf Stream system in the DAMEE GSR (Data Assimilation and Model Evaluation Experiments Gulf Stream Region) sponsored by the U.S. Navy. The objective of DAMEE GSR phases I and II was 1–2-week forecast experiments. Phase III extended the suite of case studies by adding a 2-month-long assimilation experiment to assess the impact of long-term assimilations on model performance and forecasting skill.

In this paper the authors report the results of DAMEE GSR phase III but broaden the perspective by addressing two further issues, namely, the model sensitivity to the choice of the initial fields and the frequency of intermittent data assimilation. Two versions of the OTIS-3 (Optimum Thermal Interpolation System) of the U.S. Navy Fleet Numerical Oceanography Center were available, providing slightly different distributions of temperature and salinity over the entire Gulf Stream system. They are referred to as OTIS-3a, available with biweekly frequency from 4 May 1988 through 28 December 1988, in the context of a different assimilation work; and OTIS-3b, provided by the DAMEE GSR phase III effort, for the 2-month period 4 May–4 July 1988, with a slightly irregular frequency, weekly on the average. The main results can be summarized as follows.

The intermittent assimilation of the OTIS-3b datasets with average weekly frequency profoundly improves the model forecasting skill. Without assimilation the model never beats persistence. With the assimilation, the model-predicted Gulf Stream north wall is in excellent agreement with the verification infrared (IR) north wall, remaining always within the error bar of the IR north wall estimate, ±15km.

Two types of sensitivity experiments to the initial conditions were carried out: first, reconstruction of the initial fields with the two different OTIS-3a and OTIS-3b datasets but with the same initialization method; second, reconstruction of the initial fields with the same OTIS-3a dataset but with two different initialization methods. The results show that the initial velocity field is much more crucial in affecting the model evolution and hence its predictive skill as it determines the stability properties of the Gulf Stream jet. Hence, it is very important to use the same dynamical initialization for velocity when starting from different distributions of temperature and salinity, as the jet profiles thus obtained will be very similar in structure and strength. This identical dynamical initialization will allow meaningful comparisons of experiments that start from slightly different density distributions.

Finally, we compare weekly assimilations of OTIS-3b with biweekly and monthly assimilations of OTIS-3a, initialized with the same procedure. The authors conclude that a weekly assimilation of the global OTIS-3 dataset is not necessary and that a biweekly assimilation is equally effective in improving the model predictive skill.

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Paola Malanotte-Rizzoli and Willam R. Holland


Ocean Acoustic Tomography was proposed by Munk and Wunsch as a method for making measurements of ocean variability over large areas. After the successful demonstration of the feasibility of the idea in the 1981 three-dimensional Mesoscale Experiment the tomography group has proposed a new experiment to be carried out in the Eastern Pacific Ocean, on ranges as long as the subtropical gyre scale.

This paper address the question of which average quantities of importance for the ocean general circulation and ocean climate can be measured by tomography and with what accuracy. The paper focuses upon the following quantities i) measurement of the heat content vertical profile horizontally averaged over a tomographic section; ii) time variability of the average heat content, or average pycnocline displacement, at different depths; iii) measurement of the average pycnocline slope at different depths.

To answer the above question the tomographic experiment is simulated in a given model ocean, using Holland's eddy-resolving general circulation quasi-geostrophic model. The results of the modeling simulations can be summarized as follows.

  • The tomographic technique bars upon the use of inverse methods to reconstruct the interior sound speed perturbation field, or, equivalently, the heat content field. Over ranges as long as the gyre scale, the typical result of a single inversion is to provide an ocean with warm or cold biases. A simple iterative procedure allows the removal of these biases. The final estimates of the mean heat content (averaged over the tomographic section) at different depths is very good.

  • Through a time-evolution experiment carried out for the duration of a full year, the time evolution of the average pycnocline displacement can be monitored at various depths. Thus tomography can measure the frequency spectrum of the average pycnocline displacement in layers below the surface mixed layer in which the circulation is basically wind-driven.

  • The initial estimate of the average heat content can be significantly improved through a better specification of the statistics of the region, like the inclusion of a spatial mean in the horizontal covariance function for the sound speed perturbation. In particular, the inclusion of an inhomogeneous covariance modeling long-scale pycnocline trends allow us to estimate the average pycnocline slope at various depths. The obtained slope estimates are very good. Measurement of isopycnal slopes averaged in time could be used for β-spiral calculations. Thus, simple “density” tomography would provide a tool to evaluate the absolute velocity field and not only the geostrophic velocity shear.

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Paola Malanotte-Rizzoli, Dale,B. Haidvogel, and Roberta E. Young


Increasing evidence suggests that the Gulf Stream system is the origin for much of the eddy variability observed in the northwestern Atlantic Ocean. However, the dynamical mechanisms by which eddy energy, once generated in the neighborhood of the Gulf Stream, penetrates into the midocean regions are not fully understood. Here we explore the proposition that radiation away from the Gulf Stream can be interpreted as an oscillatory response of the ocean gyre interior to the forcing associated with translation and temporal evolution of the Gulf Stream current and its meanders. Specifically, we propose as a possible mechanism responsible for pules of Rossby wave radiation the sudden growth to finite amplitude and successive amplitude pulsations of quasi-stationary or eastward-moving finite-amplitude meanders. To examine this mechanism in detail, we construct a model for radiation in which the quasi-geostrophic equivalent barotropic potential vorticity equation is forced to its northern boundary by assigned distributions of streamfunction and vorticity corresponding to 1) a stationary pulsating meander, 2) a slowly propagating meander, and 3) a combination of 1 and 2.

We use both analytical and numerical techniques to solve the linear boundary-forced initial value problem with and without lateral friction and with and without bottom topography. We focus not only on the steady forced response but also on the transient component of the interior Rossby radiation field. On the β-plane without topography, it is shown that radiation into the far field is only possible for meander pulsation below a critical frequency. Below this frequency, the transient far-field response is large and force basin modes are necessary, together with the forced equilibrium solution, to accommodate the prescribed initial state of the system. With the addition of viscosity to damp the transient basin modes, and for frequencies higher than the critical value, the response is trapped at the northern wall and no radiation can be supported in the far field.

The inclusion of a simple parabolic relief along the full channel does not modify the qualitative behavior of the interior field. The only effect of the parabolic relief is to modify the effective value of planetary β and the cross-channel structure of the westward-propagating Rossby modes. The above results are, however, profoundly altered with the addition of nonlinearity. In particular, new mechanisms are possible for the development of far-field radiation and for the appearance of nonlinear, coherent solutions, as will he shown in Part II of this paper.

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Antonietta Capotondi, William R. Holland, and Paola Malanotte-Rizzoli


The improvement in the climatological behavior of a numerical model as a consequence of the assimilation of surface data is investigated. The model used for this study is a quasigeostrophic (QG) model of the Gulf Stream region. The data that have been assimilated are maps of sea surface height that have been obtained as the superposition of sea surface height variability deduced from the Geosat altimeter measurements and a mean field constructed from historical hydrographic data. The method used for assimilating the data is the nudging technique. Nudging has been implemented in such a way as to achieve a high degree of convergence of the surface model fields toward the observations.

Comparisons of the assimilation results with available in situ observations show a significant improvement in the degree of realism of the climatological model behavior, with respect to the model in which no data are assimilated. The remaining discrepancies in the model mean circulation seem to be mainly associated with deficiencies in the mean component of the surface data that are assimilated. On the other hand, the possibility of building into the model more realistic eddy characteristics through the assimilation of the surface eddy field proves very successful in driving components of the mean model circulation that are in relatively good agreement with the available observations. Comparisons with current meter time series during a time period partially overlapping the Geosat mission show that the model is able to “correctly” extrapolate the instantaneous surface eddy signals to depths of approximately 1500 m. The correlation coefficient between current meter and model time series varies from values close to 0.7 in the top 1500 m to values as low as 0.1–0.2 in the deep ocean.

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Paola Malanotte-Rizzoli, Roberta E. Young, and Dale B. Haidvogel


In Part I of the present work we studied the transient Rossby wave radiation excited in the far field by a northern boundary forcing. We proposed as a possible mechanism responsible for transient pulses of Rossby waves the sudden growth to finite amplitude and successive amplitude pulsations of a stationary or eastward moving meander of an intense current like the Gulf Stream. In Part I the linear regime was thoroughly analyzed and the major findings were: 1) the transient part of the response initially excited by “switching on” the meander is the one responsible for the Rossby waves radiating and being clearly detectable in the far field; 2) the Rossby radiation is excited only when the meander pulsation frequency is below a critical value.

In this paper, we extend the results of Part I to the highly nonlinear regime addressing the problem of the production of nonlinear, coherent structures through the same boundary forcing mechanisms. The model used is the quasi-geostrophic, potential vorticity equation on a β-plane with or without topography, in a periodic channel forced by a northern boundary function. The forcing functions are designed to simulate (i) a stationary pulsating meander, (ii) a slowly propagating meander, (iii) a combination of both. This may constitute the simplest possible idealization of meander growth to finite amplitude, its successive steepening and bending with pinching off of eddies as frequently observed in the Gulf Stream system.

Using a boundary forcing idealizing a steady meander with a pulsating amplitude, we excite nonlinear vortices on the β-plane without topography. The related regime is called the “modon” regime as, at every pulsation cycle, two circulation patterns are produced, one cyclonic, the other anticyclonic. These vortices are part of a dipole pair and self-advect eastward or westward depending on their polarity (low-above-high or inverse). Through diagnostic tools we show that these vortices are ‘reasonable’ modons, that is highly nonlinear, exact solutions of the equivalent barotropic model on the β-plane.

The second type of boundary forcing capitalizes upon a resonance mechanism which was previously studied analytically in the weakly nonlinear regime. In it, an interior vortex is resonantly excited by a meander with constant amplitude which propagates at one of the eigenspeeds of the free linear modes of the channel. This mechanism needs topography to be effective when using quasi-geostrophic dynamics. With parabolic relief all along the channel, a resonantly forced nonlinear eddy is excited in the interior which propagates eastward leaving the forcing wave trailing behind it. In the highly nonlinear limit, the excited vortex remains coherent against the effects of dispersion even when the northern boundary forcing is turned off. Through a process of Rossby-wave shedding, it adjusts itself to the free-modon solution of the potential vorticity equation with a parabolic topography, as shown by using modon diagnostics.

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