Leonid I. Piterbarg

Assimilation of drifter observations for the reconstruction of the Eulerian circulation field

[1] In light of the increasing number of drifting buoys in the ocean and recent advances in the realism of ocean general circulation models toward oceanic forecasting, the problem of assimilation of Lagrangian observations data in Eulerian models is investigated. A new and general rigorous approach is developed based on optimal interpolation (OI) methods, which takes into account directly the Lagrangian nature of the observations. An idealized version of this general formulation is tested in the framework of identical twin experiments using a reduced gravity, quasi-geostrophic model. An extensive study is conducted to quantify the effectiveness of Lagrangian data assimilation as a function of the number of drifters, the frequency of assimilation, and the uncertainties associated with the forcing functions driving the ocean model. The performance of the Lagrangian assimilation technique is also compared to that of conventional methods of assimilating drifters as moving current meters, and assimilation of Eulerian data, such as fixed-point velocities. Overall, the results are very favorable for the assimilation of Lagrangian observations to improve the Eulerian velocity field in ocean models. The results of our assimilation twin experiments imply an optimal sampling frequency for oceanic Lagrangian instruments in the range of 20–50% of the Lagrangian integral timescale of the flow field. INDEX TERMS: 4255 Oceanography: General: Numerical modeling; 4263 Oceanography: General: Ocean prediction; 3337 Meteorology and Atmospheric Dynamics: Numerical modeling and data assimilation; KEYWORDS: Lagrangian assimilation, drifters, Lagrangian velocity, Eulerian velocity, numerical model

On the Predictability of Lagrangian Trajectories in the Ocean

The predictability of particle trajectories in oceanic flows is investigated in the context of a primitive equation, idealized, double-gyre ocean model. This study is motivated not only by the fact that this is an important conceptual problem but also by practical applications, such as searching for objects lost at sea, and ecological problems, such as the spreading of pollutants or fish larvae. The original aspect of this study is the use of Lagrangian drifter data to improve the accuracy of predicted trajectories. The prediction is performed by assimilating velocity data from the surrounding drifters into a Gauss‐Markov model for particle motion. The assimilation is carried out using a simplified Kalman filter. The performance of the prediction scheme is quantified as a function of a number of factors: 1) dynamically different flow regimes, such as interior gyre, western boundary current, and midlatitude jet regions; 2) density of drifter data used in assimilation; and 3) uncertainties in the knowledge of the mean flow field and the initial conditions. The data density is quantified by the number of data per degrees of freedom NR, defined as the number of drifters within the typical Eulerian space scale from the prediction particle. The simulations indicate that the actual World Ocean Circulation Experiment sampling (1 particle/[5 83 58 ]o rNR K 1) does not improve particle prediction, but predictions improve significantly when NR k 1. For instance, a coverage of 1 particle/ [1 83 18 ]o rNR ; O(1) is already able to reduce the errors of about one-third or one-half. If the sampling resolution is increased to 1 particle/[0.5 83 0.58] or 1 particle/[0.25 83 0.258 ]o rNR k 1, reasonably accurate predictions (rms errors of less than 50 km) can be obtained for periods ranging from one week (western boundary current and midlatitude jet regions) to three months (interior gyre region). Even when the mean flow field and initial turbulent velocities are not known accurately, the information derived from the surrounding drifter data is shown to compensate when NR . 1. Theoretical error estimates are derived that are based on the main statistical parameters of the flow field. Theoretical formulas show good agreement with the numerical results, and hence, they may serve as useful a priori estimates of Lagrangian prediction error for practical applications.

Estimates of turbulence parameters from Lagrangian data using a stochastic particle model

A new parametric approach for the study of Lagrangian data is presented. It provides parameter estimates for velocity and transport components and is based on a stochastic model for single particle motion. The main advantage of this approach is that it provides more accurate parameter estimates than existing methods by using the a-priori knowledge of the model. Also, it provides a complete error analysis of the estimates and is valid in presence of observation errors. Unlike nonparametric methods (e.g. Davis, 1991b), our technique depends on a-priori assumptions which require that the model validity be checked in order to obtain reliable estimates. The model used here is the simplest one in a hierarchy of random flight models (e.g. Thomson, 1987), and it describes the turbulent velocity as a linear Markov process, characterized by an exponential autocorrelation. Experimental and numerical estimates show that the model is appropriate for mesoscale turbulent flows in homogeneous regions of the upper ocean. More complex models, valid under more general conditions, are presently under study. Estimates of the mean flow, variance, turbulent time scale and diffusivity are obtained. The properties of the estimates are discussed in terms of biases and sampling errors, both analytically and using numerical experiments. Optimal sampling for the measurements is studied and an example application to drifter data from the Brazil/Malvinas extension is presented.

Predictability of Lagrangian particle trajectories: Effects of smoothing of the underlying Eulerian flow

The increasing realism of ocean circulation models is leading to an increasing use of Eulerian models as a basis to compute transport properties and to predict the fate of Lagrangian quantities. There exists, however, a signie cant gap between the spatial scales of model resolution and that of forces acting on Lagrangian particles. These scales may contain high vorticity coherent structures that are not resolved due to computational issues and/or missing dynamics and are typically suppressed by smoothing operators. In this study, the impact of smoothing of the Eulerian e elds on the predictability of Lagrangian particles is e rst investigated by conducting twin experiments that involve release of clusters of synthetic Lagrangian particles into “ true” (unmodie ed) and “ model” (smoothed) Eulerian e elds, which are generated by a QG model with a e ow e eld consisting of many turbulent coherent structures. The Lagrangian errors induced by Eulerian smoothing errors are quantie ed by using two metrics, the difference between the centers of mass (CM) of particle clusters, r, and the difference between scattering of particles around the center of mass, s. The results show that the smoothing has a strong effect on the CM behavior, while the scatter around it is only partially affected. The QG results are then compared to results obtained from a multi-particle Lagrangian Stochastic Model (LSM) which parameterizes turbulent e ow using main e ow characteristics such as mean e ow, velocity variance and Lagrangian time scale. In addition to numerical results, theoretical results based on the LSM are also considered, providing asymptotics of r, s and predictability time. It is shown that both numerical and theoretical LSM results for the center of mass error ( r) provide a good qualitative description, and a quantitatively satisfactory estimate of results from QG experiments. The scatter error ( s) results, on the other hand, are only qualitatively reproduced by the LSM.

THE TOP LYAPUNOV EXPONENT FOR A STOCHASTIC FLOW MODELING THE UPPER OCEAN TURBULENCE

A stochastic model is proposed for multiparticle Lagrangian motion in the upper ocean. The model is based on hydrodynamics equations with random forcing, includes a few well interpreted and well estimated parameters, and implies a common description of the one-particle motion via a Langevin equation for the particle velocity. The dependence of the top Lyapunov exponent on the model parameters is studied as a part of a Lagrangian predictability problem. In particular, it is found that the Coriolis effect can radically improve the prediction of a Lagrangian particle position based on observations of other particles.

Assimilation of drifter observations in primitive equation models of midlatitude ocean circulation

[1] Motivated by increases in the realism of OGCMs and the number of drifting buoys in the ocean observing system, a new Lagrangian assimilation technique is implemented in an idealized, reduced-gravity configuration of the layered primitive equation model MICOM. Using an extensive set of twin experiments, the effectiveness of the Lagrangian observation operator and of a dynamical balancing technique for corrected model variables, which is based on geostrophy and mass conservation, are explored in comparison to a conventional Pseudo-Lagrangian observation operator and an implementation of the Kalman filter method. The results clearly illustrate that the Lagrangian observation operator is superior to the Pseudo-Lagrangian in the parameter range that is relevant for typical oceanic drifter observations, and that the simple dynamical balancing technique works well for midlatitude ocean circulation. INDEX TERMS: 4263 Oceanography: General: Ocean prediction; 4255 Oceanography: General: Numerical modeling; 4594 Oceanography: Physical: Instruments and techniques; 4572 Oceanography: Physical: Upper ocean processes; KEYWORDS: Lagrangian data assimilation, data assimilation, drifter assimilation, ocean prediction, ocean modeling

Inversion for heat anomaly transport from sea surface temperature time series in the northwest Pacific

We describe a heat anomaly transport in the upper ocean mixed layer in the Kuroshio extension region and the subtropical gyre of the northwest Pacific. Emphasis is on behavior in the cool season December-March) during the Asian Winter Monsoon. The heat anomaly transport is estimated by applying an inversion technique to the stochastic partial differential equation for the heat anomaly balance of advection, diffusion, subilizing feedback, and atmospheric forcing. The inversion consists of (1) derivation of statistical parametric model from the heat anomaly balance equation; (2) fitting the derived statistical model to the sea surface temperature (SST) anomaly covariances; and (3) calculation of the heat anomaly net advection velocity, horizontal diffusion coefficient, feedback factor and atmospheric forcing correlation from the parameters of the evaluated statistical model. The inversion was applied to the Comprehensive Ocean-Atmosphere Data Set Compressed Marine Reports SST data averaged at 1 o latitude ×2 o longitude boxes on a 10-day mean basis from 1965 to 1990. The estimates of the net advection velocity are consistent in magnitude and direction with the general circulation in the surface layer of the Northwest Pacific in winter. SST anomalies are transported to the west at ∼0.15 m s -1 in the northern part of the North Equatorial Current. Between 21 o and 29 o N in the recirculating region, SST anomalies propagate westward with the mean velocity less than 0.1 m s -1 . South and east of Honshu the observed pattern of the SST anomaly transport agrees broadly with the circulations of the Kuroshio current and its extension and the Oyashio current. South of Honshu, the eastward transport is about 200-300 km wide; its absolute velocity is up to 0.2 m s -1 . One branch of the uansport separates from the coast near the large meander path of the Kuroshio current and follows the east-southeast direction. The second separation from the coast occurs south of Hokkaido. Over the analysis domain the estimates of the diffusion coefficient are in the range of 3×10 3 to 6x10 3 m 2 s -1 . The h igher values of the diffusion coefficient confirm the enhancement of the mesoscale eddy processes near the subtropical convergence zone. The analysis supports Hasselmanns (1976) theory in which generation of midlatitude SST anomalies lasting the dominant timescale of atmospheric processes is primarily attributed to the short period stochastic weather forcing. However, the analysis indicates that the inertia of SST anomalies to ther memory of earlier winds can not be neglected in the vicinity of the western boundary and in the tropics

CANONICAL VARIABLES FOR ROSSBY WAVES AND PLASMA DRIFT WAVES

Abstract The Poisson bracket defining a hamiltonian formulation of the Rossby wave equation is transformed to the Gardner bracket via a special functional change. The diagonal form of the bracket enables us to introduce the normal canonical variables in the considered hamiltonian system. The first terms of the hamiltonian expansion in powers of the canonical variables are calculated. The proposed method of the Poisson bracket diagonalization is relevant for other physically significant problems: barotropic waves above an uneven bottom, waves in the presence of a scalar nonlinearity and quasigeostrophic flow of a vertically stratified fluid, including the baroclinic effects of topography as dynamical boundary conditions.

Short-Term Prediction of Lagrangian Trajectories

Abstract Lagrangian particles in a cluster are divided in two groups: observable and unobservable. The problem is to predict the unobservable particle positions given their initial positions and velocities based on observations of the observable particles. A Markov model for Lagrangian motion is formulated. The model implies that the positions and velocities of any number of particles form a multiple diffusion process. A prediction algorithm is proposed based on this model and Kalman filter ideas. The algorithm performance is examined by the Monte Carlo approach in the case of a single predictand. The prediction error is most sensitive to the ratio of the velocity correlation radius and the initial cluster radius. For six predictors, if this parameter equals 5, then the relative error is less than 0.1 for the 15-day prediction, whereas for the ratio close to 1, the error is about 0.9. The relative error does not change significantly as the number of predictors increases from 4–7 to 20.

Inversion of Upper Ocean Temperature Time Series for Entrainment, Advection, and Diffusivity

Abstract An inversion technique for estimating the terms of the oceanic near-surface heat transport is extended to include the vertical heat flux at the bottom of the surface mixed layer. The mixed-layer heat balance equation uses a conventional parameterization of the vertical heat flux via entrainment into the mixed layer of interior fluid during the mixed layer deepening. A heat conservation equation defined here for the sea temperature anomalies, deviations from the annual cycle, is driven by stochastic atmospheric forcing, thereby becoming essentially a stochastic partial differential equation. This equation is reduced to the regression estimator aimed on inversion of the sea temperature time series for the unknowns: vertical entrainment velocity, horizontal velocity and diffusivity, feedback factor, and atmospheric forcing parameter. The inversion scheme also involves the velocity divergence norm. The regression estimator is applied to the time series of vertical profiles of temperature anomalies co...