Strong and weak constraint variational assimilations for reduced order fluid flow modeling

International audienceIn this work we propose and evaluate two variational data assimilation techniques for the estimation of low order surrogate experimental dynamical models for fluid flows. Both methods are built from optimal control recipes and rely on proper orthogonal decomposition and a Galerkin projection of the Navier Stokes equation. The techniques proposed di er in the control variables they involve. The first one introduces a weak dynamical model defined only up to an additional uncertainty time-dependent function whereas the second one, handles a strong dynamical constraint in which the dynamical system's coe cients constitute the control variables. Both choices correspond to di erent approximations of the relation between the reduced basis on which is expressed the motion field and the basis components that have been neglected in the reduced order model construction. The techniques have been assessed on numerical data and for real experimental conditions with noisy Image Velocimetry data

Fluid flow dynamics under location uncertainty

We present a derivation of a stochastic model of Navier Stokes equations that relies on a decomposition of the velocity fields into a differentiable drift component and a time uncorrelated uncertainty random term. This type of decomposition is reminiscent in spirit to the classical Reynolds decomposition. However, the random velocity fluctuations considered here are not differentiable with respect to time, and they must be handled through stochastic calculus. The dynamics associated with the differentiable drift component is derived from a stochastic version of the Reynolds transport theorem. It includes in its general form an uncertainty dependent "subgrid" bulk formula that cannot be immediately related to the usual Boussinesq eddy viscosity assumption constructed from thermal molecular agitation analogy. This formulation, emerging from uncertainties on the fluid parcels location, explains with another viewpoint some subgrid eddy diffusion models currently used in computational fluid dynamics or in geophysical sciences and paves the way for new large-scales flow modelling. We finally describe an applications of our formalism to the derivation of stochastic versions of the Shallow water equations or to the definition of reduced order dynamical systems

Optimal Nonlinear Eddy Viscosity in Galerkin Models of Turbulent Flows

We propose a variational approach to identification of an optimal nonlinear eddy viscosity as a subscale turbulence representation for POD models. The ansatz for the eddy viscosity is given in terms of an arbitrary function of the resolved fluctuation energy. This function is found as a minimizer of a cost functional measuring the difference between the target data coming from a resolved direct or large-eddy simulation of the flow and its reconstruction based on the POD model. The optimization is performed with a data-assimilation approach generalizing the 4D-VAR method. POD models with optimal eddy viscosities are presented for a 2D incompressible mixing layer at $Re=500$ (based on the initial vorticity thickness and the velocity of the high-speed stream) and a 3D Ahmed body wake at $Re=300,000$ (based on the body height and the free-stream velocity). The variational optimization formulation elucidates a number of interesting physical insights concerning the eddy-viscosity ansatz used. The 20-dimensional model of the mixing-layer reveals a negative eddy-viscosity regime at low fluctuation levels which improves the transient times towards the attractor. The 100-dimensional wake model yields more accurate energy distributions as compared to the nonlinear modal eddy-viscosity benchmark {proposed recently} by \"Osth et al. (2014). Our methodology can be applied to construct quite arbitrary closure relations and, more generally, constitutive relations optimizing statistical properties of a broad class of reduced-order models.Comment: 41 pages, 16 figures; accepted for publication in Journal of Fluid Mechanic

Latent Space Data Assimilation by using Deep Learning

Performing Data Assimilation (DA) at a low cost is of prime concern in Earth system modeling, particularly at the time of big data where huge quantities of observations are available. Capitalizing on the ability of Neural Networks techniques for approximating the solution of PDE's, we incorporate Deep Learning (DL) methods into a DA framework. More precisely, we exploit the latent structure provided by autoencoders (AEs) to design an Ensemble Transform Kalman Filter with model error (ETKF-Q) in the latent space. Model dynamics are also propagated within the latent space via a surrogate neural network. This novel ETKF-Q-Latent (thereafter referred to as ETKF-Q-L) algorithm is tested on a tailored instructional version of Lorenz 96 equations, named the augmented Lorenz 96 system: it possesses a latent structure that accurately represents the observed dynamics. Numerical experiments based on this particular system evidence that the ETKF-Q-L approach both reduces the computational cost and provides better accuracy than state of the art algorithms, such as the ETKF-Q.Comment: 15 pages, 7 figures and 3 table

Controlling overestimation of error covariance in ensemble Kalman filters with sparse observations: A variance limiting Kalman filter

We consider the problem of an ensemble Kalman filter when only partial observations are available. In particular we consider the situation where the observational space consists of variables which are directly observable with known observational error, and of variables of which only their climatic variance and mean are given. To limit the variance of the latter poorly resolved variables we derive a variance limiting Kalman filter (VLKF) in a variational setting. We analyze the variance limiting Kalman filter for a simple linear toy model and determine its range of optimal performance. We explore the variance limiting Kalman filter in an ensemble transform setting for the Lorenz-96 system, and show that incorporating the information of the variance of some un-observable variables can improve the skill and also increase the stability of the data assimilation procedure.Comment: 32 pages, 11 figure

Applications of regional ocean Ensemble Kalman Filter data assimilation

Data assimilation has been widely used in the forecast of oceanic states and tropical cyclones. In this thesis, the Ensemble Kalman Filter (EnKF) based data assimilation algorithm is applied to two applications, a regional ocean data assimilation system for the South Australian Sea and a coupled ocean-atmosphere tropical cyclone (TC) forecast system. The regional ocean data assimilation system consists of the data assimilation algorithm provided by the NCAR Data Assimilation Research Testbed (DART) and the Regional Ocean Modelling System (ROMS). We describe the first implementation of a physical balance operator (temperature-salinity, hydrostatic and geostrophic balance) to DART, to reduce the spurious waves which may be introduced during the data assimilation process. The effect of the balance operator is validated in both an idealised shallow water model and the ROMS model real case study. In the shallow water model, the geostrophic balance operator eliminates spurious ageostrophic waves and produces a better sea surface height (SSH) and velocity analysis and forecast. Its impact increases as the sea surface height and wind stress increase. In the real case, satellite-observed sea surface temperature (SST) and SSH are assimilated in the South Australian Sea with 50 ensembles using the Ensemble Adjustment Kalman Filter. Assimilating SSH and SST enhances the estimation of SSH and SST in the entire domain, respectively. Assimilation with the balance operator produces a more realistic simulation of surface currents and subsurface temperature profile. The best improvement is obtained when only SSH is assimilated with the balance operator. A case study with a storm suggests that the benefit of the balance operator is of particular importance under high wind stress conditions. Implementing the balance operator could be a general bene t to ocean data assimilation systems. The TC forecast system consists of DART and coupled ROMS - WRF (the Weather Research Forecast model). High-frequency (HF) radars can provide high-resolution and frequent ocean surface currents observations during the TC landfall. We describe the first assimilation of such potential observations using idealised Observing System Simulation Experiments. In this system, synthetic HF radar observed coastal currents are assimilated and the forecast performances for weak (Category 2) and strong (Category 4) TCs are examined. Assimilating coastal surface currents improves the 24-hour forecasts of both intensity and track. For the strong case, the errors of the maximum wind speed (Vmax) and the integrated power dissipation (IPD) forecast reduce up to 50%. For the weak case, the improvements in Vmax and IPD forecast are lower (20%), but the track forecast improves 30%. These improvements are similar to the magnitude of the current operational TC forecast error, so that assimilating HF radar observations could be a substantial benefit.Open Acces

Mesoscale ensemble-based data assimilation and parameter estimation

The performance of the ensemble Kalman filter (EnKF) in forced, dissipative flow under imperfect model conditions is investigated through simultaneous state and parameter estimation where the source of model error is the uncertainty in the model parameters. Two numerical models with increasing complexity are used with simulated observations. For lower complexity, a two-dimensional, nonlinear, hydrostatic, non-rotating, and incompressible sea breeze model is developed with buoyancy and vorticity as the prognostic variables. Model resolution is 4 km horizontally and 50 m vertically. The ensemble size is set at 40. Forcing is maintained through an explicit heating function with additive stochastic noise. Simulated buoyancy observations on land surface with 40-km spacing are assimilated every 3 hours. Up to six model parameters are successfully subjected to estimation attempts in various experiments. The overall EnKF performance in terms of the error statistics is found to be superior to the worst-case scenario (when there is parameter error but no parameter estimation is performed) with an average error reduction in buoyancy and vorticity of 40% and 46%, respectively, for the simultaneous estimation of six parameters. The model chosen to represent the complexity of operational weather forecasting is the Pennsylvania State University-National Center for Atmospheric Research MM5 model with a 36-km horizontal resolution and 43 vertical layers. The ensemble size for all experiments is chosen as 40 and a 41st member is generated as the truth with the same ensemble statistics. Assimilations are performed with a 12-hour interval with simulated sounding and surface observations of horizontal winds and temperature. Only single-parameter experiments are performed focusing on a constant inserted into the code as the multiplier of the vertical eddy mixing coefficient. Estimation experiments produce very encouraging results and the mean estimated parameter value nicely converges to the true value exhibiting a satisfactory level of variability

A review of operational methods of variational and ensemble-variational data assimilation

Variational and ensemble methods have been developed separately by various research and development groups and each brings its own benefits to data assimilation. In the last decade or so various ways have been developed to combine these methods, especially with the aims of improving the background error covariance matrices and of improving efficiency. The field has become confusing, even to many specialists, and so there is now a need to summarise the methods in order to show how they work, how they are related, what benefits they bring, why they have been developed, how they perform, and what improvements are pending. This paper starts with a reminder of basic variational and ensemble techniques and shows how they can be combined to give the emerging ensemble-variational (EnVar) and hybrid methods. A key part of the paper includes details of how localisation is commonly represented. There has been a particular push to develop four-dimensional methods that are free of linearised forecast models. This paper attempts to provide derivations of the formulations of most popular schemes. These are otherwise scattered throughout the literature or absent. It builds on the nomenclature used to distinguish between methods, and discusses further possible developments to the methods, including the representation of model error

Method of continuous data assimilation using short-term 4D-VAR analysis

June 30, 1998.Includes bibliographical references.A continuous data assimilation method based on short-term four-dimensional variational data assimilation (4D-Var) is proposed. This method consists of forecast and analysis steps. The analysis increment (analyzed value minus forecasted value) is assumed to be proportional to the gradient of a cost function, which measures the misfit between model prediction and observations over a period of time. The gradient of the cost function is calculated with the adjoint method and is updated cyclically. This technique is a kind of retrospective analysis and can continuously assimilate data in an infinite time period. Different forecast model versions (or models) can be used in the forecast and analysis steps. A two-dimensional shallow-water system with horizontal diffusion, Rayleigh friction and external forcing is used to test the proposed method through identical-twin numerical experiments. The control run represents a typical mesoscale case with energy cascaded in two ways (upscale and downscale). The influence of model error and resolution of the analysis grid on the assimilated results is examined. Results show that when model error is small or moderate, the assimilated wind and geopotential fields correlate well with the true fields. When model error is large, the proposed method can still recover a large portion of small-scale motions which are not resolved by observations. Model error can lead to the generation of spurious small-scale gravity waves because of the inconsistency between model and observations. Numerical experiments show that bounding wind divergence and its time tendency can considerably suppress high-frequency spurious gravity waves and improve the assimilated results.Sponsored by the National Oceanic & Atmospheric Administration under grants NA37RJ0202 and NA67RJ0152; and the Air Force Office of Scientific Research under grant F49620-95-1-0132