Christian Le Provost on his way

This paper is an evocation of the scientific career of Christian Le Provost and of the different research topics in oceanography that have been of interest to him: tides, ocean modeling, altimetry, sea level, tidal dissipation, ...     

Mapping nonlinear shallow-water tides: a look at the past and future

Overtides and compound tides are generated by nonlinear mechanisms operative primarily in shallow waters. Their presence complicates tidal analysis owing to the multitude of new constituents and their possible frequency overlap with astronomical tides. The science of nonlinear tides was greatly advanced by the pioneering researches of Christian Le Provost who employed analytical theory, physical modeling, and numerical modeling in many extensive studies, especially of the tides of the English Channel. Le Provost’s complementary work with satellite altimetry motivates our attempts to merge these two interests. After a brief review, we describe initial steps toward the assimilation of altimetry into models of nonlinear tides via generalized inverse methods. A series of barotropic inverse solutions is computed for the M tide over the northwest European Shelf. Future applications of altimetry to regions with fewer in situ measurements will require improved understanding of error covariance models because these control the tradeoffs between fitting hydrodynamics and data, a delicate issue in coastal regions. While M can now be robustly determined along the Topex/Poseidon satellite ground tracks, many other compound tides face serious aliasing problems. In memory of Christian Le Provost     

The oceanic tides in the South Atlantic Ocean

The finite element ocean tide model of Le Provost and Vincent (1986) has been applied to the simulation of the M₂ and K₁ components over the South Atlantic Ocean. The discretisation of the domain, of the order of 200 km over the deep ocean, is refined down to 15 km along the coasts, such refinement enables wave propagation and damping over the continental shelves to be correctly solved. The marine boundary conditions, from Dakar to Natal, through the Drake passage and from South Africa to Antarctica, are deduced from in situ data and from Schwiderski’s solution and then optimised following a procedure previously developed by the authors. The solutions presented are in very good agreement with in situ data: the root mean square deviations from a standard subset of 13 pelagic stations are 1.4 cm for M₂ and 0.45 cm for K₁, which is significantly better overall than solutions published to date in the literature. Zooms of the M₂ solution are presented for the Falkland Archipelago, the Weddell Sea and the Patagonian Shelf. The first zoom allows detailing of the tidal structure around the Falklands and its interpretation in terms of a stationary trapped Kelvin wave system. The second zoom, over the Weddell Sea, reveals for the first time what must be the tidal signal under the permanent ice shelf and gives a solution over that sea which is generally in agreement with observations. The third zoom is over the complex Patagonian Shelf. This zoom illustrates the ability of the model to simulate the tides, even over this area, with a surprising level of realism, following purely hydrodynamic modelling procedures, within a global ocean tide model. Maps of maximum associated tidal currents are also given, as a first illustration of a by-product of these simulations.     

Sequential assimilation of a year-long time-series of SeaWiFS chlorophyll data into a 3D biogeochemical model on the French Mediterranean coast

The objective of this paper was to explore the potentialities of sequential statistical estimation methods to assimilate ocean color observations in a primary production model coupled to a 3D hydrodynamic model. The study site was the gulf of Fos—Rhone delta region on the French Mediterranean coast. The high rate of primary production generally observed in this area is mainly due to strong nutrient inputs of the Rhone River. The assimilation method is derived from the singular evolutive extended Kalman filter (SEEK), which uses an error subspace represented by multivariate empirical orthogonal functions (EOF). SeaWiFS chlorophyll data were assimilated by the ecosystem model during a simulation performed under realistic meteorological conditions for the year 2001. An ‘adaptive’ computing method of the EOF was applied in order to lower the instabilities of the filter. Data assimilation system permitted to reduce the mean absolute error between model and data from 1.51 to 0.77 mg m⁻³ thanks to the SEEK filter, showing a substantial 49% gain. Efficiency of the SEEK filter was then investigated considering several areas of interest inside the modelled domain. Finally, impact of the assimilation scheme on non-observed variables was illustrated and discussed. Throughout this experimentation the data assimilation system showed its potential regarding operational systems.     

Modelling the global ocean tides: modern insights from FES2004

During the 1990s, a large number of new tidal atlases were developed, primarily to provide accurate tidal corrections for satellite altimetry applications. During this decade, the French tidal group (FTG), led by C. Le Provost, produced a series of finite element solutions (FES) tidal atlases, among which FES2004 is the latest release, computed from the tidal hydrodynamic equations and data assimilation. The aim of this paper is to review the state of the art of tidal modelling and the progress achieved during this past decade. The first sections summarise the general FTG approach to modelling the global tides. In the following sections, we introduce the FES2004 tidal atlas and validate the model against in situ and satellite data. We demonstrate the higher accuracy of the FES2004 release compared to earlier FES tidal atlases, and we recommend its use in tidal applications. The final section focuses on the new dissipation term added to the equations, which aims to account for the conversion of barotropic energy into internal tidal energy. There is a huge improvement in the hydrodynamic tidal solution and energy budget obtained when this term is taken into account.     

On the oceanic impact of a data-assimilation method in a coupled ocean-land-atmosphere model

 Based on the Kalman filter theory, a new data-assimilation method has been used to improve the 3-D oceanic temperature field of the Center for Ocean-Land-Atmosphere Studies (COLA) coupled general circulation model. This method is applied to assimilate surface and subsurface temperature of in situ measurements from the Pilot Research Moored Array in the Tropical Atlantic project (PIRATA). The assimilation of the PIRATA data produces an improved representation of the thermal state of the ocean and allows a better estimation of other oceanographic quantities, like meridional heat fluxes and zonal currents. The present paper focuses on the tropical Atlantic and, in particular, it contains new reconstructed temperature profiles. One-month forecast experiments during 1999 were performed and the impact of the assimilation is discussed. Received: 24 April 2001 / Accepted: 8 March 2002     

Erosion probability model of base-soil particle migration into a granular filter under local flow

To avoid soil erosion, lters often are installed in the downstream cross section of dams. However, the probability of piping failure and the lter effect both are related to the soil pore network. Previous erosion probability models for a base soil- lter system do not consider the effect of local ow. Therefore, in this study, an improved erosion probability model is established, in which the deviation between the main ow direction of the soil and the local ow direction in the lter was considered based on a previous model. The improved model was validated by numerical simulation. The erosion probability was found to affect the reasonable evaluation of lter effects and the optimal selection of lter thickness. The controlling con- striction size in a granular lter was increased to Dc* Dc5 with a lter thickness of 59 Dm.     

Effects of wave-induced forcing on a circulation model of the North Sea

The effect of wind waves on water level and currents during two storms in the North Sea is investigated using a high-resolution Nucleus for European Modelling of the Ocean (NEMO) model forced with fluxes and fields from a high-resolution wave model. The additional terms accounting for wave-current interaction that are considered in this study are the Stokes-Coriolis force, the sea-state-dependent energy and momentum fluxes. The individual and collective role of these processes is quantified and the results are compared with a control run without wave effects as well as against current and water-level measurements from coastal stations. We find a better agreement with observations when the circulation model is forced by sea-state-dependent fluxes, especially in extreme events. The two extreme events, the storm Christian (25–27 October 2013), and about a month later, the storm Xaver (5–7 December 2013), induce different wave and surge conditions over the North Sea. Including the wave effects in the circulation model for the storm Xaver raises the modelled surge by more than 40 cm compared with the control run in the German Bight area. For the storm Christian, a difference of 20–30 cm in the surge level between the wave-forced and the stand-alone ocean model is found over the whole southern part of the North Sea. Moreover, the modelled vertical velocity profile fits the observations very well when the wave forcing is accounted for. The contribution of wave-induced forcing has been quantified indicating that this represents an important mechanism for improving water-level and current predictions.     

Joint altimetric and in-situ data assimilation using the GRACE mean dynamic topography: a 1993–1998 hindcast experiment in the Tropical Pacific Ocean

The altimetric satellite signal is the sum of the geoid and the dynamic topography, but only the latter is relevant to oceanographic applications. Poor knowledge of the geoid has prevented oceanographers from fully exploiting altimetric measurements through its absolute component, and applications have concentrated on ocean variability through analyses of sea level anomalies. Recent geodetic missions like CHAMP, GRACE and the forthcoming GOCE are changing this perspective. In this study, data assimilation is used to reconstruct the Tropical Pacific Ocean circulation during the 1993–1996 period. Multivariate observations are assimilated into a primitive equation ocean model (OPA) using a reduced order Kalman filter (the Singular Evolutive Extended Kalman filter). A 6-year (1993–1998) hindcast experiment is analyzed and validated by comparison with observations. In this experiment, the new capability offered by an observed absolute dynamic topography (built using the GRACE geoid to reference the altimetric data) is used to assimilate, in an efficient way, the in-situ temperature profiles from the TAO/TRITON moorings together with the T/P and ERS1&2 altimetric signal. GRACE data improves compatibility between both observation data sets. The difficulties encountered in this regard in previous studies such as Parent et al. (J Mar Syst 40–41:381–401, 2003) are now circumvented. This improvement helps provide more efficient data assimilation, as evidenced, by assessing the results against independent data. This leads in particular to significantly more realistic currents and vertical thermal structures.     

Multi-scale modeling of Puget Sound using an unstructured-grid coastal ocean model: from tide flats to estuaries and coastal waters

Water circulation in Puget Sound, a large complex estuary system in the Pacific Northwest coastal ocean of the United States, is governed by multiple spatially and temporally varying forcings from tides, atmosphere (wind, heating/cooling, precipitation/evaporation, pressure), and river inflows. In addition, the hydrodynamic response is affected strongly by geomorphic features, such as fjord-like bathymetry and complex shoreline features, resulting in many distinguishing characteristics in its main and sub-basins. To better understand the details of circulation features in Puget Sound and to assist with proposed nearshore restoration actions for improving water quality and the ecological health of Puget Sound, a high-resolution (around 50 m in estuaries and tide flats) hydrodynamic model for the entire Puget Sound was needed. Here, a three-dimensional circulation model of Puget Sound using an unstructured-grid finite volume coastal ocean model is presented. The model was constructed with sufficient resolution in the nearshore region to address the complex coastline, multi-tidal channels, and tide flats. Model open boundaries were extended to the entrance of the Strait of Juan de Fuca and the northern end of the Strait of Georgia to account for the influences of ocean water intrusion from the Strait of Juan de Fuca and the Fraser River plume from the Strait of Georgia, respectively. Comparisons of model results, observed data, and associated error statistics for tidal elevation, velocity, temperature, and salinity indicate that the model is capable of simulating the general circulation patterns on the scale of a large estuarine system as well as detailed hydrodynamics in the nearshore tide flats. Tidal characteristics, temperature/salinity stratification, mean circulation, and river plumes in estuaries with tide flats are discussed.     

Assimilation of geodetic dynamic ocean topography using ensemble based Kalman filter

Estimation of ocean circulation is investigated via assimilation of satellite measurements of the dynamic ocean topography (DOT) into the global finite-element ocean model (FEOM). The DOT was obtained by means of a geodetic approach from carefully cross-calibrated multi-mission altimeter data and GRACE gravity fields. The spectral consistency was achieved by consistently filtering both, the sea surface and the geoid. The filter length is determined by the spatial resolution of the gravity field and corresponds to approximately 241 km half width for the GRACE-based gravity field model ITG-Grace03s.The assimilation of the geodetic DOT was performed by employing a local singular evolutive interpolated Kalman (SEIK) filter in combination with the method of weighting of observations. It is shown that this approach leads to a successful assimilation technique that reduced the RMS difference between the model and the data from 16 cm to 5 cm during one year of assimilation. The ocean model returns an optimized mean dynamic ocean topography. The effects of assimilation on transport estimates across several hydrographic World Ocean Circulation Experiment (WOCE) sections show improvements compared to the FEOM run without data assimilation. As a result of the assimilation, DOT estimates are available in the polar or coastal regions where the geodetic estimates from satellite data alone are not adequate. Furthermore, more realistic features of the ocean can be seen in these areas compared to those obtained using the filtered data fields.     

A Tailored Computation of the Mean Dynamic Topography for a Consistent Integration into Ocean Circulation Models

Geostrophic surface velocities can be derived from the gradients of the mean dynamic topography—the difference between the mean sea surface and the geoid. Therefore, independently observed mean dynamic topography data are valuable input parameters and constraints for ocean circulation models. For a successful fit to observational dynamic topography data, not only the mean dynamic topography on the particular ocean model grid is required, but also information about its inverse covariance matrix. The calculation of the mean dynamic topography from satellite-based gravity field models and altimetric sea surface height measurements, however, is not straightforward. For this purpose, we previously developed an integrated approach to combining these two different observation groups in a consistent way without using the common filter approaches (Becker et al. in J Geodyn 59(60):99–110, 2012; Becker in Konsistente Kombination von Schwerefeld, Altimetrie und hydrographischen Daten zur Modellierung der dynamischen Ozeantopographie 2012). Within this combination method, the full spectral range of the observations is considered. Further, it allows the direct determination of the normal equations (i.e., the inverse of the error covariance matrix) of the mean dynamic topography on arbitrary grids, which is one of the requirements for ocean data assimilation. In this paper, we report progress through selection and improved processing of altimetric data sets. We focus on the preprocessing steps of along-track altimetry data from Jason-1 and Envisat to obtain a mean sea surface profile. During this procedure, a rigorous variance propagation is accomplished, so that, for the first time, the full covariance matrix of the mean sea surface is available. The combination of the mean profile and a combined GRACE/GOCE gravity field model yields a mean dynamic topography model for the North Atlantic Ocean that is characterized by a defined set of assumptions. We show that including the geodetically derived mean dynamic topography with the full error structure in a 3D stationary inverse ocean model improves modeled oceanographic features over previous estimates.     

The SEEK filter method for data assimilation in oceanography: a synthesis

This paper reviews the developments of the singular evolutive extended Kalman (SEEK) filter method used for data assimilation in oceanography, since the original paper by Pham et al. (J Mar Syst 16:3–4, 323–340 1998a). First, a short review is presented of the context of data assimilation in oceanography and of the variety of numerical ocean codes and configurations in which the SEEK filter has been implemented using different data sets for assimilation. Then, the paper provides an exposition of the different versions of the SEEK filter developed during the past 10 years and discusses their relative merits for scientific or operational implementations. A classification of the algorithmic variants is proposed, and several possible improvements of the generic methodology are mentioned in the perspective of new assimilation challenges.     

Numerical simulation of flow and aquaculture organic waste dispersion in a curved channel

The dispersion and deposition of particulate organic matter from a fish cage located in an idealized curved channel with a 90° bend are studied for different horizontal grid resolutions. The model system consists of a three-dimensional, random-walk particle tracking model coupled to a terrain-following ocean model. The particle tracking model is a Lagrangian particle tracking simulator which uses the local flow field, simulated by the ocean model, for advection of the particles and random walk to simulate the turbulent diffusion. The sinking of particles is modeled by imposing an individual particle settling velocity. As the homogeneous water flows through the bend in the channel, the results show that a cross-channel secondary circulation is developed. The motion of this flow is similar to a helical motion where the water in the upper layers moves towards the outer bank and towards the inner bank in the lower layers. The intensity of the secondary circulation will depend on the viscosity scheme and increases as the horizontal grid resolution decreases which significantly affects the distribution of the particles on the seabed. The presence of the secondary circulation leads to that most of the particles that settle, settle close to the inner bank of the channel.     

A three-dimensional surface wave–ocean circulation coupled model and its initial testing

A theoretical framework to include the influences of nonbreaking surface waves in ocean general circulation models is established based on Reynolds stresses and fluxes terms derived from surface wave-induced fluctuation. An expression for the wave-induced viscosity and diffusivity as a function of the wave number spectrum is derived for infinite and finite water depths; this derivation allows the coupling of ocean circulation models with a wave number spectrum numerical model. In the case of monochromatic surface wave, the wave-induced viscosity and diffusivity are functions of the Stokes drift. The influence of the wave-induced mixing scheme on global ocean circulation models was tested with the Princeton Ocean Model, indicating significant improvement in upper ocean thermal structure and mixed layer depth compared with mixing obtained by the Mellor–Yamada scheme without the wave influence. For example, the model–observation correlation coefficient of the upper 100-m temperature along 35° N increases from 0.68 without wave influence to 0.93 with wave influence. The wave-induced Reynolds stress can reach up to about 5% of the wind stress in high latitudes, and drive 2–3 Sv transport in the global ocean in the form of mesoscale eddies with diameter of 500–1,000 km. The surface wave-induced mixing is more pronounced in middle and high latitudes during the summer in the Northern Hemisphere and in middle latitudes in the Southern Hemisphere.     

Reduction of thermohaline circulation during deglaciation: the effect on atmospheric radiocarbon and CO2

A two-? ocean reservoir model is employed to examine the combined effects of vertical ocean circulation, organic matter extraction and cosmic ray production on the¹⁴C/¹²C in the atmosphere, ocean and sediment. In this model, dissolution of deep-sea calcium carbonate sediment is assumed to respond to the supply of particulate carbonate from the surface ocean and to the dissolved carbonate-ion concentration of the deep-sea.If the vertical ocean circulation decreased by 50% during the maximum rate of deglaciation, the atmospheric¹⁴C/¹²C would have increased concurrently by 10% relative to the pre-bomb present. Further, if cosmic ray production of¹⁴C was 50% greater than present at about 7800 years B.P. as suggested by archeomagnetic measurements, a double maximum of 10% occurs. The first is at about 10.5 ka and the second at 6.5 ka. This result is similar to the variation of¹⁴C/¹²C over the last 10,000 years calculated from the¹⁴C dates measured by Stuiver on the varved Lake of the Clouds. The result is not sensibly altered if 10¹⁷ moles of organic carbon is extracted from the ocean during sea-level rise.Reduction of the thermohaline ocean circulation by 50% over a one- to two-thousand-year interval would not be sensibly detected in the dating of deep-sea sediment. If Broecker's organic nutrient extraction occurs in conjunction with reduction of vertical circulation, a decrease in the atmospheric P_CO2 will precede the eventual rise.     

Tidal flow forecasting using reduced rank square root filters

The Kalman filter algorithm can be used for many data assimilation problems. For large systems, that arise from discretizing partial differential equations, the standard algorithm has huge computational and storage requirements. This makes direct use infeasible for many applications. In addition numerical difficulties may arise if due to finite precision computations or approximations of the error covariance the requirement that the error covariance should be positive semi-definite is violated. In this paper an approximation to the Kalman filter algorithm is suggested that solves these problems for many applications. The algorithm is based on a reduced rank approximation of the error covariance using a square root factorization. The use of the factorization ensures that the error covariance matrix remains positive semi-definite at all times, while the smaller rank reduces the number of computations and storage requirements. The number of computations and storage required depend on the problem at hand, but will typically be orders of magnitude smaller than for the full Kalman filter without significant loss of accuracy. The algorithm is applied to a model based on a linearized version of the two-dimensional shallow water equations for the prediction of tides and storm surges. For non-linear models the reduced rank square root algorithm can be extended in a similar way as the extended Kalman filter approach. Moreover, by introducing a finite difference approximation to the Reduced Rank Square Root algorithm it is possible to prevent the use of a tangent linear model for the propagation of the error covariance, which poses a large implementational effort in case an extended kalman filter is used.     

Solar activity and dynamic processes in the atmosphere and world ocean heat content

The scenario of climatic changes in the 20th century has been presented in the scope of the developed model concerning the effect of solar activity on the parameters of the climatic system governing the energy flux, outgoing from the Earth into space in the high-latitude regions. The regularities of changes in the circulation in the atmosphere and ocean are discussed. Specific attention is paid to the causes of a “cold snap” in 1940–1976 in the Northern Hemisphere and the nature of an anomalous increase in the heat content in the Earth climatic system (ocean) in 1969–1980. It has been indicated that these phenomena result from changes in the circulation in the atmosphere and ocean (specifically, a change in the thermohaline circulation in the Northern Atlantic), heat exchange between the ocean and the atmosphere and cryosphere.     

A coarse resolution North Atlantic ocean circulation model: an intercomparison study with a paleoceanographic example

Paleoreconstructions suggest that during the Last Glacial Maximum (LGM) the North Atlantic circulation was noticeably different from its present state. However, the glacial salt conveyor belt is believed to be similar to the present-day’s conveyor, albeit weaker and shallower because of an increased freshwater flux in high-latitudes. We present here the investigation of the conveyor operation based on ocean circulation modelling using two numerical models in parallel. The GFDL primitive equation model and a planetary geostrophic model are employed to address the problem of the paleocirculation modelling in cases of uncertain and sparse data comprising the glacial surface boundary conditions. The role of different simplifications that may be used in the ocean climate studies, including the role of grid resolution, bottom topography, coast-line, etc., versus glacial-interglacial changes of the ocean surface climatology is considered. The LGM reverse conveyor gyre appeared to be the most noticeable feature of the glacial-to-interglacial alteration of the ocean circulation. The reversed upper-ocean conveyor, weaker and subducting ‘normal’ conveyor in the intermediate depths, and the change of the deep-ocean return flow route are robust signatures of the glacial North Atlantic climate. The results are found to be ‘model-independent’ and fairly insensitive to all factors other than the onset of the glacial surface conditions.