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Smoothers are increasingly used in geophysics. Several linear Gaussian algorithms exist, and the general picture may appear somewhat confusing. This paper attempts to stand back a little, in order to clarify this picture by providing a concise overview of what the different smoothers really solve, and how. The authors begin addressing this issue from a Bayesian viewpoint. The filtering problem consists in finding the probability of a system state at a given time, conditioned to some past and present observations (if the present observations are not included, it is a forecast problem). This formulation is unique: any different formulation is a smoothing problem. The two main formulations of smoothing are tackled here: the joint estimation problem (fixed lag or fixed interval), where the probability of a series of system states conditioned to observations is to be found, and the marginal estimation problem, which deals with the probability of only one system state, conditioned to past, present, and future observations. The various strategies to solve these problems in the Bayesian framework are introduced, along with their deriving linear Gaussian, Kalman filter-based algorithms. Their ensemble formulations are also presented. This results in a classification and a possible comparison of the most common smoothers used in geophysics. It should provide a good basis to help the reader find the most appropriate algorithm for his/her own smoothing problem.

The real-time reconstruction of the plasma magnetic equilibrium in a tokamak is a key point to access high-performance regimes. Indeed, the shape of the plasma current density profile is a direct o...

Magnetic confinement fusion devices are very integrated systems, difficult to access for measurement, and therefore they pose particular challenges to identification and to the prediction of undesired events. With regard to system identification, Bayesian statistics is a very promising methodology, which provides for the first time a sound way to include physical information about the diagnostics in the evaluation of the error bars. In a prototypical application, a Bayesian statistical approach determines the magnetic topology without any assumption on the equilibrium and provides clear confidence intervals on all the derived quantities. This technique, which complements another more traditional approach based on the Grad–Shafranov equation, can be implemented to provide results about every few milliseconds and therefore it can be envisaged to exploit for feedback purposes. Some phenomena in the evolution of Tokamak plasmas, like disruptions, are too dangerous and prohibitively difficult to control. For these cases avoidance is the best alternative and therefore specific classifiers have been trained and optimized to predict the occurrence of disruptions. The success rates of these predictors, mainly based on Support Vector Machines, are very often of the order or higher than 90%. The generalisation capability of the method has been confirmed by applying the same predictor to new campaigns without retraining. The success rate remains very high (above 80%) even 12 campaigns after the last one used for training.

Data assimilation is the domain at the interface between observations and models, which makes it possible to identify the global structure of a geophysical system from a set of discrete space-time data. After recalling state-of-the-art data assimilation methods, the variational 4DVAR algorithm and sequential Kalman filters, but also the Back and Forth Nudging (BFN) algorithm, we present the Diffusive Back and Forth Nudging (DBFN) algorithm, which is a natural extension of the BFN to some particular diffusive models. We numerically test the DBFN and compare it with other methods on both a 2D shallow water model, and a 3D full primitive ocean model. These numerical results show that the DBFN converges very quickly. Its main advantage is to filter the observation noise, unlike the BFN algorithm, particularly when the model diffusion is high.

The diffusive back-and-forth nudging (DBFN) is an easy-to-implement iterative data assimilation method based on the well-known nudging method. It consists of a sequence of forward and backward model integrations, within a given time window, both of them using a feedback term to the observations. Therefore, in the DBFN, the nudging asymptotic behaviour is translated into an infinite number of iterations within a bounded time domain. In this method, the backward integration is carried out thanks to what is called backward model, which is basically the forward model with reversed time step sign. To maintain numeral stability, the diffusion terms also have their sign reversed, giving a diffusive character to the algorithm. In this article the DBFN performance to control a primitive equation ocean model is investigated. In this kind of model non-resolved scales are modelled by diffusion operators which dissipate energy that cascade from large to small scales. Thus, in this article, the DBFN approximations and their consequences for the data assimilation system set-up are analysed. Our main result is that the DBFN may provide results which are comparable to those produced by a 4Dvar implementation with a much simpler implementation and a shorter CPU time for convergence. The conducted sensitivity tests show that the 4Dvar profits of long assimilation windows to propagate surface information downwards, and that for the DBFN, it is worth using short assimilation windows to reduce the impact of diffusion-induced errors. Moreover, the DBFN is less sensitive to the first guess than the 4Dvar.

Over the past few years, the preparation of ITER-relevant plasma scenarios has been the main focus of experimental activity on tokamaks. The development of integrated, simultaneous, real-time controls of plasma shape, current, pressure, temperature, radiation, neutron profiles and also impurities, ELMs and MHD are now seen to be essential for further development of quasi-steady state conditions with feedback, or the stabilisation of transient phenomena with event-driven actions. For this thrust, the EFDA JET Real-Time Project has developed a set of real-time plasma measurements, experiment control and communication facilities. The plasma diagnostics used for real-time experiments are Far Infra Red interferometry, polarimetry, visible, UV and X-ray spectroscopy, LIDAR, bolometry, neutron and magnetics. Further analysis systems produce integrated results, such as temperature profiles on geometry derived from MHD equilibrium solutions. The actuators include toroidal, poloidal and divertor coils, gas and pellet fuelling, neutral beam injection, radio frequency (ICRH) waves and microwaves (LH). The heating/fuelling operators can either define a power or gas request waveform or select the real-time instantaneous power/gas request from the real-time experiment central control (RTCC) system. The real-time experiment control system provides both a high-level, control-programming environment and interlocks with the actuators. A MATLAB facility is being developed for the development of more complex controllers. The plasma measurement, controller and plant control systems communicate in ATM network. The EFDA Real-Time Project is essential groundwork for future reactors such as ITER. It involves many staff from several institutions. The facility is now frequently used in experiments. This work has been conducted under the European Fusion Development Agreement and is partly funded by Euratom and the UK Engineering and Physical Sciences Research Council.

International audience; This paper presents a reduced-order approach for four-dimensional variational data assimilation, based on a prior EO F analysis of a model trajectory. This method implies two main advantages: a natural model-based definition of a mul tivariate background error covariance matrix $\textbf{B}_r$, and an important decrease of the computational burden o f the method, due to the drastic reduction of the dimension of the control space. % An illustration of the feasibility and the effectiveness of this method is given in the academic framework of twin experiments for a model of the equatorial Pacific ocean. It is shown that the multivariate aspect of $\textbf{B}_r$ brings additional information which substantially improves the identification procedure. Moreover the computational cost can be decreased by one order of magnitude with regard to the full-space 4D-Var method.

Several studies have demonstrated the effectiveness of the singular evolutive extended Kalman (SEEK) filter and its interpolated variant called singular evolutive interpolated Kalman (SEIK) filter in their capacity to assimilate altimetric data into ocean models. However, these filters remain expensive for real operational assimilation. The purpose of this paper is to develop degraded forms of the SEIK filter which are less costly and yet perform reasonably well. Our approach essentially consists in simplifying the evolution of the correction basis of the SEIK filter, which is the most expensive part of this filter. To deal with model instabilities, we also introduce two adaptive tuning schemes to control the correction basis evolution and adjust the variable forgetting factor. Our filters have been implemented in a realistic setting of the OPA model over the tropical pacific zone and their performance studied through twin experiments in which the observations are taken to be synthetic altimeter data sampled on the sea surface. The SEIK filter is used as a reference for comparison. Our new filters perform nearly as well as the SEIK, but can be 2–30 times faster.

The singular evolutive extended Kalman (SEEK) filter has been proposed recently by Pham et al. (1997) for data assimilation into numerical oceanic models. This filter has been applied in different realistic ocean frameworks and has provided satisfactory results (Pham et al., 1997; Verron et al., 1998). However, the SEEK filter remains expensive in real operational assimilation. To reduce cost and obtain a better representativity, we introduce the idea 'local correction basis'. Such basis however cannot be made to evolve according to the model without destroying its locality property. Therefore we shall keep this basis fixed and we augment it by a few global basis vectors which evolve. The resulting semi-evolutive partially local filter is much less costly to implement than the SEEK filter and yet can yield better results. In the first application, validation twin experiments are conducted in a realistic setting of the OPA model over the tropical Pacific Ocean.

We consider the assimilation of satellite altimetric data into a general circulation model of the ocean at basin scale. The satellite observes only the sea-surface height of the ocean. With the assimilation of these data, we aim at reconstructing the four-dimensional space-time circulation of the ocean including the vertical. This problem is solved using the variational technique and the adjoint method. In the present case, a strong constraint approach is assumed, i.e. the quasi-geostrophic ocean circulation model used is assumed to be exact. The control vector is chosen as being the initial state of the dynamical system and it should minimize the mean-square difference between the model solution and the observed data. The assimilation procedure has been implemented and has the ability to transfer the surface data information downward to the deep flows, and hence to reconstruct the oceanic circulation in the various layers used to describe the vertical stratification of the ocean. The paper points out more specifically the crucial influence of the choice of the norm in the vector control space on the convergence speed of the optimization algorithm. Furthermore, various temporal strategies to perform the assimilation are presented and discussed with regard to their ability to properly control the initial state (which is the actual control variable) and the final state.

In this paper, we explore the use of the adjoint method in the oceanographic context for assimilating satellite altimeter data. Experiments with simulated altimeter data are performed in a multi-layer quasigeostrophic ocean model. The interest is mainly with controlling the mesoscale eddy active ocean circulation observed in the mid-latitudes. Due to the surface nature of the observations, one key aspect in the success of assimilation is its ability to transfer the surface data information downwards to the deep flows. Test experiments are performed first, in a coarse resolution regional basin within which several eddies are interacting on the f-plane and second, in a high resolution basin size domain on the s-plane which mimics the Gulfstream-like behavior of mid-latitude jets and western boundary currents and the associated eddy system. As a matter of fact, it is found that the length of the assimilation cycle is crucial to the success of this assimilation. Short assimilation cycles may be efficient in the control of surface flows but rather ineffective with respect to the downward penetration of information. Conversely, long assimilation cycles lead to rather coherent results in terms of efficiency on the vertical but this global efficiency is poor, especially identifying the initial control state. It is suggested that an efficient assimilation strategy can be constructed by dividing the global time sequence in several time sub-periods the individual duration of which must be less than the typical predictability time scale of the flow. However, the whole assimilation cycle must be long enough and larger than the vertical penetration time scale. A "sliced assimilation" strategy which satisfies these conditions is studied in which the assimilation period is divided into equal-length time-intervals. With this approach, an acceptable accuracy can be reached for the recovery of the final flow state. This meets the requirements of the "filtering" objectives of data assimilation and therefore the "forecast" purposes. A so-called "progressive assimilation" strategy is also studied in which successive iterations of increasing durations are performed in order to progressively improve the control of the initial state control variable. This more expensive strategy is more adequate for "smoothing" objectives.

We present a method based on the use of toroidal harmonics and on a modelization of the poloidal field coils and divertor coils for the 2D interpolation and extrapolation of discrete magnetic measurements in a tokamak. The method is generic and can be used to provide the Cauchy boundary conditions needed as input by a fixed domain equilibrium reconstruction code like Equinox (Blum et al 2012 J. Comput. Phys. 231 960–80). It can also be used to extrapolate the magnetic measurements in order to compute the plasma boundary itself. The proposed method and algorithm are detailed in this paper and results from numerous numerical experiments are presented. The method is foreseen to be used in the real-time plasma control loop on the WEST tokamak (Bucalossi et al 2011 Fusion Eng. Des. 86 684–8).

International audience; We study in this article an improvement to the back and forth nudging (BFN) method for geophysical data assimilation. In meteorology or oceanography, the theoretical equations are usually diffusive free, but diffusion is added into the model equations in order to stabilize the numerical integrations and to take into consideration some subscale phenomena. We propose to change the sign of the diffusion in the backward nudging model, which is physically consistent and stabilizes the backward integration. We apply this method to a Burgers equation, study the convergence properties and report the results of numerical experiments. We compare the quality of the estimated initial condition with two other data assimilation techniques that are close from the algorithmic point of view: a variational method, and the quasi-inverse linear method.

A numerical method for the computation of the magnetic flux in the vacuum surrounding the plasma in a Tokamak is investigated. It is based on the formulation of a Cauchy problem which is solved through the minimization of an energy error functional. Several numerical experiments are conducted which show the efficiency of the method.

International audience; In this Note, we propose an improvement to the Back and Forth Nudging algorithm for handling diffusion in the context of geophysical data assimilation. We introduce the Diffusive Back and Forth Nudging, in which the sign of the diffusion term is changed in the backward integrations. We study the convergence of this algorithm, in particular for linear transport equations.; Dans cette Note, nous proposons une amélioration de l'algorithme du Nudging Direct et Rétrograde pour gérer la diffusion dans le contexte de l'assimilation de données en géophysique. Nous introduisons la version Diffusive du Nudging Direct et Rétrograde, dans laquelle le signe du terme de diffusion est inversé dans les équations rétrogrades. Nous étudions la convergence de cet algorithme, en particulier pour l'équation de transport linéaire.

International audience; Free-boundary equilibrium codes are an essential tool for tokamak design studies and scenario development. In this contribution, the CEDRES++ code is presented and applications to ITER, JT-60SA and Tore Supra are reported on. A satisfactory benchmark of CEDRES++ with the DINA code was performed on an ITER Scenario 2 case at 32 different times across the discharge, both in direct and inverse mode. Ongoing design studies for the poloidal field coils system of JT-60SA, performed in the frame of the Broader Approach,werecross checkedwithCEDRES++.Finally,adimensioningstudywascarriedoutforthecoils and power supplies for plasma vertical control in the frame of a feasibility study for the implementation of a divertor in Tore Supra

International audience; The real-time reconstruction of the plasma magnetic equilibrium in a Tokamak is a key point to access high performance regimes. Indeed, the shape of the plasma current density profile is a direct output of the reconstruction and has a leading effect for reaching a steady-state high performance regime of operation. In this paper we present the methodology followed to identify numerically the plasma current density in a Tokamak and its equilibrium. In order to meet the real-time requirements a C++ software has been developed using the combination of a finite element method, a nonlinear fixed point algorithm associated to a least square optimization procedure. The experimental measurements that enable the identification are the magnetics on the vacuum vessel, the interferometric and polarimetric measurements on several chords and the motional Stark effect. Details are given about the validation of the reconstruction on the JET tokamak, either by comparison with 'off-line' equilibrium codes or real time software computing global quantities.

International audience

The main results of the Tore Supra experimental programme in the years 2007–2008 are reported. They document significant progress achieved in the domain of steady-state tokamak research, as well as in more general issues relevant for ITER and for fusion physics research. Three areas are covered: ITER relevant technology developments and tests in a real machine environment, tokamak operational issues for high power and long pulses, and fusion plasma physics. Results presented in this paper include test and validation of a new, load-resilient concept of ion cycotron resonance heating antenna and of an inspection robot operated under ultra-high vacuum and high temperature conditions; an extensive experimental campaign (5 h of plasma) aiming at deuterium inventory and carbon migration studies; real-time control of sawteeth by electron cyclotron current drive in the presence of fast ion tails; ECRH-assisted plasma start-up studies; dimensionless scalings of transport and turbulence; transport experiments using active perturbation methods; resistive and fast-particle driven MHD studies. The potential role of Tore Supra in the worldwide fusion programme before the start of ITER operation is also discussed.

International audience; Magnetic confinement fusion devices are very integrated systems, difficult to access for measurement, and therefore they pose particular challenges to identification and to the prediction of undesired events. With regard to system identification, Bayesian statistics is a very promising methodology, which provides for the first time a sound way to include physical information about the diagnostics in the evaluation of the error bars. In a prototypical application, a Bayesian statistical approach determines the magnetic topology without any assumption on the equilibrium and provides clear confidence intervals on all the derived quantities. This technique, which complements another more traditional approach based on the Grad-Shafranov equation, can be implemented to provide results about every few milliseconds and therefore it can be envisaged to exploit for feedback purposes not only the topology of the ideal flux surfaces but also the confidence intervals. Some phenomena in the evolution of Tokamak plasmas, like disruptions, are too dangerous and prohibitively difficult to control. For these cases avoidance is the best alternative and therefore specific classifiers have been trained and optimised to predict the occurrence of disruptions. The success rates of these predictors, mainly based on Support Vector Machines, are very often of the order or higher than 90%. The generalisation capability of the method has been confirmed by applying the same predictor to new campaigns without retraining. The success rate remains very high (above 80%) even 10 campaigns after the last one used for training.

International audience; The reconstruction of the equilibrium of a plasma in a Tokamak is a free boundary problem described by the Grad-Shafranov equation in axisymmetric configuration. The right-hand side of this equation is a nonlinear source, which represents the toroidal component of the plasma current density. This paper deals with the identification of this nonlinearity source from experimental measurements in real time. The proposed method is based on a fixed point algorithm, a finite element resolution, a reduced basis method and a least-square optimization formulation. This is implemented in a software called Equinox with which several numerical experiments are conducted to explore the identification problem. It is shown that the identification of the profile of the averaged current density and of the safety factor as a function of the poloidal flux is very robust.

International audience; The ultimate purpose of environmental studies is the forecast of its natural evolution. A prerequisite before a prediction is to retrieve at best the state of the environment. Data assimilation is the ensemble of techniques which, starting from heterogeneous information, permit to retrieve the initial state of a flow. In the first part, the mathematical models governing geophysical flows are presented together with the networks of observations of the atmosphere and of the ocean. In variational methods, we seek for the minimum of a functional estimating the discrepancy between the solution of the model and the observation. The derivation of the optimality system, using the adjoint state, permits to compute a gradient which is used in the optimization. The definition of the cost function permits to take into account the available statistical information through the choice of metrics in the space of observation and in the space of the initial condition. Some examples are presented on simplified models, especially an application in oceanography. Among the tools of optimal control, the adjoint model permits to carry out sensitivity studies, but if we look for the sensitivity of the prediction with respect to the observations, then a second-order analysis should be considered. One of the first methods used for assimilating data in oceanography is the nudging method, adding a forcing term in the equations. A variational variant of nudging method is described and also a so-called "back and forth" nudging method. The proper orthogonal decomposition method is introduced in order to reduce the cost of the variational method. For assimilating data, stochastic methods can be considered, being based on the Kalman filter extended to nonlinear problems, but the inconvenience of this method consists in the difficulty of handling huge covariance matrices. The dimension of the systems used for operational purposes (several hundred of millions of variables) requires to work with reduced variable techniques. The ensemble Kalman filter method, which is a Monte-Carlo implementation of the Bayesian update problem, is described. A considerable amount of information on geophysical flows is provided by satellites displaying images of their evolution, the assimilation of images into numerical models is a challenge for the future: variational methods are successfully considered in this perspective.

International audience; This paper deals with a new data assimilation algorithm called the Back and Forth Nudging. The standard nudging technique consists in adding to the model equations a relaxation term, which is supposed to force the model to the observations. The BFN algorithm consists of repeating forward and backward resolutions of the model with relaxation (or nudging) terms, that have opposite signs in the direct and inverse resolutions, so as to make the backward evolution numerically stable. We then applied the Back and Forth Nudging algorithm to a simple non-linear model: the 1D viscous Burgers' equations. The tests were carried out through several cases relative to the precision and density of the observations. These simulations were then compared with both the variational assimilation (VAR) and quasi-inverse (QIL) algorithms. The comparisons deal with the programming, the convergence, and time computing for each of these three algorithms.

This paper deals with a new data assimilation algorithm, called Back and Forth Nudging. The standard nudging technique consists in adding to the equations of the model a relaxation term that is supposed to force the observations to the model. The BFN algorithm consists in repeatedly performing forward and backward integrations of the model with relaxation (or nudging) terms, using opposite signs in the direct and inverse integrations, so as to make the backward evolution numerically stable. This algorithm has first been tested on the standard Lorenz model with discrete observations (perfect or noisy) and compared with the variational assimilation method. The same type of study has then been performed on the viscous Burgers equation, comparing again with the variational method and focusing on the time evolution of the reconstruction error, i.e. the difference between the reference trajectory and the identified one over a time period composed of an assimilation period followed by a prediction period. The possible use of the BFN algorithm as an initialization for the variational method has also been investigated. Finally the algorithm has been tested on a layered quasi-geostrophic model with sea-surface height observations. The behaviours of the two algorithms have been compared in the presence of perfect or noisy observations, and also for imperfect models. This has allowed us to reach a conclusion concerning the relative performances of the two algorithms.

International audience; This paper deals with the numerical reconstruction of the plasma current density in a Tokamak and of its equilibrium. The problem consists in the identification of a non-linear source in the 2D Grad-Shafranov equation, which governs the axisymmetric equilibrium of a plasma in a Tokamak. The experimental measurements that enable this identification are the magnetics on the vacuum vessel, but also polarimetric and interferometric measures on several chords, as well as motional Stark effect or pressure measurements. The reconstruction can be obtained in real-time using a finite element method, a non-linear fixed-point algorithm and a least-square optimization procedure.

Recent advances in experimental measurements of magneto-optic properties of tokamak plasmas and progress in formulation of numerical algorithms for the analysis of magnetic data have allowed the self-consistent determination of the current density in the JET tokamak, in Ohmic and additionally heated discharges. An investigation of the numerical response of a model with finite parameterization to the uncertainties of the available discrete data is carried out. The error propagation is analysed for various types of discharges, and results on the safety factor profile are presented.

This work has been performed in the frame of a collaboration with Association EURATOM-CEA D.R.F.C. (Departement de Recherche sur la Fusion Controlee), Centre d'Etudes Nucleaires de Cadarache, and with the JET team (Joint European Torus) at Culham (G.B.).

International audience; In this Note, we introduce a new algorithm for data assimilation problems, called the back and forth nudging (BFN) algorithm. The standard forward nudging algorithm is first studied for a linear ODE model. The backward nudging algorithm is then introduced in order to reconstruct the initial state of the system. These two algorithms are combined in the new BFN algorithm. The mathematical proof of its convergence is given for a linear ODE system.

International audience; In this paper, we first focus our interest on the use of the variational adjoint method in a relatively simple ocean model in order to try to reconstruct the four-dimensional ocean system from altimetric surface observations of the ocean.

International audience; One of the principal objectives of studying biogeochemical cycles is to obtain precise estimates of the main fluxes, such as total, new and export oceanic productions. Since models can incorporate the a priori knowledge of the most important processes, they are increasingly used for this purpose. However, biogeochemical models are characterized by a large number of poorly known parameters. Moreover, the available data are rather sparse in both time and space, and represent concentrations, not fluxes. Therefore, the major challenge is to constrain the relevant fluxes using information from a limited number of observations and from models incorporating poorly known internal parameters. The present study attempts to meet this challenge. In a 1D framework at the DYFAMED station (NW Mediterranean Sea), near-monthly nitrate and chlorophyll profiles and daily surface chlorophyll concentrations are assimilated in a coupled dynamical–biological model using the tangent linear and adjoint models. Following sensitivity analyses that show that some parameters cannot be recovered from the data set used, assimilation of observed 1997 data is performed. The first inversion considered clearly shows that, in agreement with previous studies, (1) the data impose a C/Chl ratio that varies with depth (i.e. light) and (2) the “initial” conditions (e.g. winter nitrate profile) strongly constrain the annual biogeochemical fluxes. After assimilation of the 1997 data, the agreement between the data and the model is quantitatively improved in 1995 and 1996, which can be considered a good validation of the methodology. However, the order of magnitude of the biogeochemical fluxes, and especially of the particulate export and regenerated production, are not correctly recovered. An analysis of the simulations shows that this result is associated with a strong decrease in zooplankton concentrations. An additional constraint of maintaining acceptable levels of zooplankton is therefore added. The results are improved, but remain unsatisfactory. A final inversion, which takes into account the a priori estimates of the major annual fluxes, is then performed. This shows that there is no inconsistency between the NO3 and chlorophyll data, the order of magnitude of the fluxes and the model. The work therefore demonstrates that recovering biogeochemical fluxes from available data of concentrations and stocks is not a straightforward exercise: the coverage and type of observations, and the nonlinearities of the biogeochemical model all contribute to this difficulty.

International audience; Advanced data assimilation methods are being developed for operational ocean monitoring and prediction systems, with a view to assimilating satellite observations such as altimetry, sea-surface temperature and ocean color into high-resolution models of the ocean circulations and marine ecosystems. The prototype of an operational North Atlantic forecasting system is currently in development. This paper provides an overview of the most recent results obtained with the TOPEX/POSEIDON Science Working Team, and further develops our plans for the Jason-1 mission.

A new data assimilation scheme derived from the singular evolutive extended Kalman (Seek) filter is introduced. The novel feature of the new filter is its correction basis which is partially local in the sense that it consists of "global" and "local" vectors, the later obtained from a local empirical orthogonal functions (Eof) analysis. Such an analysis was introduced in order to better represent the local variability of the ocean. This not only significantly reduces the implementation cost but may also improve the representativeness of the correction basis of the filter. The performance of this scheme is evaluated through twin experiments conducted in a realistic setting of the OPA model over the tropical Pacific zone. The results are compared against those of the Seek filter. The new filter is shown to perform better while it is up to six times faster. Adaptive tuning of the forgetting factor was also used, which enhances performance and improves the stability of the filter during model unstable periods., Un nouveau schéma d’assimilation de données dérivé du filtre de Kalman singulier étendu et évolutif (Seek) est présenté. La nouveauté de ce filtre est sa base de correction partiellement localisée car elle consiste en une partie “globale” et une partie “locale”. Ce dernier provient d’une analyse locale empirique de fonctions orthogonales. Une telle analyse permet une meilleure représentation de la variabilité locale de l’océan. Elle réduit non seulement le coût d’implémentation mais elle accentue également la représentativité de la base de correction du filtre. Sa performance est évaluée par deux expériences jumelles conduites avec une configuration réaliste du modèle OPA sur le Pacifique tropical. Les résultats sont comparés à ceux qui sont issus du filtre Seek. Le nouveau filtre est plus performant tout en étant six fois plus rapide. Un réglage adapté du facteur oublié est également utilisé, ce qui rehausse la performance et accentue la stabilité du filtre durant les périodes d’instabilité du modèle.

This paper deals with the numerical identification of the plasma current density in a Tokamak, from experimental measurements. This problem consists in the identification of a non-linearity in a semi-linear 2D elliptic equation from Cauchy boundary measurements and from integrals of the magnetic field over several chords. A simplified problem, corresponding to the cylindrical approximation, is first solved by a SQP algorithm, where the non-linearity is decomposed in a basis of B-splines and where Tikhonov regularization is used, the regularizing parameter being determined by a cross-validation procedure. An alternative approach based on the decomposition of the non-linearity in a basis of compactly supported wavelets and the use of a new scalar product in the HH2 space has enabled to solve the identification problem in a robust way, without using Tikhonov regularization Finally several test-cases are presented for the determination of the current density in the real toroidal configuration from magnetic, interferometric and polarimetric measurements.

This paper deals with the calculation of the eddy currents in the structures of a Tokamak, which can be assimilated to thin conductors, so that the three-dimensional problem can be reduced mathematically to a two-dimensional one, the variables being two orthogonal coordinates of the considered surface. A variational formulation of the problem in terms of the electric vector potential is then given and a finite element method has been used, which enables to treat the complicated geometry of the toroidal field magnet, the mechanical structures and the vacuum vessels of Tore Supra.

The SCED code solves the equilibrium equations for a Tokamak axisymmetric configuration and the plasma diffusion equations. The equilibrium system takes into account the presence of a magnetic circuit made of iron, the currents in the poloidal coils and the plasma pressure and toroidal field profiles. The plasma boundary is a free boundary defined by its contact with a limiter. A variational formulation of the problem is written and the equations of the poloidal flux ψ are solved by a finite element method; the Picard and the Newton algorithms are tested for the treatment of the non-linearities and compared on TFR and JET equilibrium configurations. The variation of the plasma position with respect to the currents in the coils is then studied by a continuation method and the link between the convergence of the Picard method and the stability of the plasma motions is proved. The evolution problem takes into account the induction of the primary coils and the penetration of the eddy currents in the vacuum vessel. The 2D diffusion equations for the electronic and ionic densities and temperatures and for the poloidal flux are written in terms of the toroidal flux by averaging over the magnetic surfaces. These equations are solved by an implicit scheme in time and by a finite difference scheme in space; the non-linearities are treated by the Newton method. The coupling between the equilibrium system and the diffusion equations enables one to follow the evolution of the plasma boundary. The study of the plasma motions in TFR, in the presence or absence of a feedback system, has been performed with this code and compared with the experimental observations.

This paper reviews plasma equilibrium evolution codes at the resistive diffusion timescale. The two-dimensional axisymmetric equilibrium equations are given, both for air-transformer and iron-transformer tokamaks, for fixed plasma boundary and for free-boundary cases. A model is written which describes the poloidal field circuits, the penetration of the eddy currents in the vacuum vessel and the diffusion of the pressure and toroidal field profiles inside the plasma. The two-dimensional Grad-Shafranov equilibrium equation is coupled with the set of one-dimensional diffusion equations obtained by averaging the transport equations over the magnetic surfaces. Various numerical methods are presented both for the equilibrium equations (flux coordinates or Eulerian grids, finite differences or finite elements, methods of linearization) and for the diffusion system (finite differences in space, implicit scheme in time). The various methods used to couple these two systems are presented and a general algorithm for such an evolution code is given. Finally detailed formulae are written for a particular discretization of these equilibrium and diffusion equations.

A case of male pseudohermaphroditism aged 48 years with systemic hypertension and hypokalaemic alkalosis is described. Results of metabolic studies point to a 17alpha-hydroxylase deficiency demonstrated by low cortisol (0-56 mg/24 h), high corticosterone (270 mg/24 h) and 11-deoxycorticosterone (5 mg/24 h) secretion rates. Adrenocorticotrophin and gonadotrophin levels were markedly raised but plasma androstenedione (3 ng/dl), testosterone (17 ng/dl), oestrone (3 ng/dl) and oestradiol(1-8 ng/dl) were all low. Plasma aldosterone levels and secretion rates in urine were low and were surprisingly unaffected by dexamethasone therapy although low renin levels rose with a marked return of the erect posture effect. Therapeutic levels of dexamethasone were, however, followed by incipient renal failure.