Leonid M. Ivanov

Analysis of Sparse and Noisy Ocean Current Data Using Flow Decomposition. Part I: Theory

A new approach is developed to reconstruct a three-dimensional incompressible flow from noisy data in an open domain using a two-scalar (toroidal and poloidal) spectral representation. The results are presented in two parts: theory (first part) and application (second part). In Part I, this approach includes (a) a boundary extension method to determine normal and tangential velocities at an open boundary, (b) establishment of homogeneous open boundary conditions for the two potentials with a spatially varying coefficient k, (c) spectral expansion of k, (d) calculation of basis functions for each of the scalar potentials, and (e) determination of coefficients in the spectral decomposition of both velocity and k using linear or nonlinear regressions. The basis functions are the eigenfunctions of the Laplacian operator with homogeneous mixed boundary conditions and depend upon the spatially varying parameter k at the open boundary. A cost function used for poor data statistics is introduced to determine the optimal number of basis functions. An optimization scheme with iteration and regularization is proposed to obtain unique and stable solutions. In Part II, the capability of the method is demonstrated through reconstructing a 2D wind-driven circulation in a rotating channel, a baroclinic circulation in the eastern Black Sea, and a large-scale surface circulation in the Southern Ocean.

Reconstruction of oceanic flow characteristics from quasi-Lagrangian data: 1. Approach and mathematical methods

We describe a nontraditional method for processing data derived from quasi-Lagrangian tracers. The approach relies on a representation of the thermohydrodynamic field as a set of a finite number of functions (modes) with the subsequent computation of mode amplitudes from quasi-Lagrangian data. This requires special methods for calculating mode components of the thermohydrodynamic field from Lagrangian observations. Also, because of the quasi-Lagrangian nature of the data there are special problems caused by the need to filter in both space and time and to account for motions unresolved by such observations. The former problem is addressed by a filtering procedure based on a variational criteria and Pontryagins principle. To handle the latter problem, a subgrid parameterization scheme appropriate for Lagrangian data is proposed.

Reconstruction of oceanic flow characteristics from quasi-Lagrangian data: 2. Characteristics of the large-scale circulation in the Black Sea

The circulation in the western portion of the Black Sea in 1987 is assessed from surface drifter trajectories, climatological data, and numerical modeling. These diverse data sources are combined by the paradigm reported in the companion article. The primary emphasis is on the low-frequency and wave number circulation characteristics. The analysis suggests that in the Black Sea there may be a seasonal change in the direction of the large-scale circulation.

Analysis of Sparse and Noisy Ocean Current Data Using Flow Decomposition. Part II: Applications to Eulerian and Lagrangian Data

Abstract The capability of the reconstruction scheme developed in Part I is demonstrated here through three practical applications. First, the nonlinear regression model is used to reproduce the upper-layer three-dimensional circulation of the eastern Black Sea from model data distorted by white and red noises. Second, the quasigeostrophic approximation is used to reconstruct the shallow water circulation pattern in an open domain with various sampling strategies. Third, the large-scale circulation in the Southern Ocean is reproduced from the First Global Atmospheric Research Program (GARP) Global Experiment (FGGE) drifter data with noncontrollable noise statistics. All three cases confirm that the theoretical approach is robust to various noise-to-signal ratios, number of observations, and station disposition. Using the simplified open boundary condition for analyzing long-term observational data is recommended because the nonlinear regression procedure requires considerable computer resources.

Probabilistic Stability of an Atmospheric Model to Various Amplitude Perturbations

Every forecast should include an estimate of its likely accuracy, as a measure of predictability. A new measure, the first passage time (FPT), which is defined as the time period when the model error first exceeds a predetermined criterion (i.e., the tolerance level), is proposed here to estimate model predictability. A theoretical framework is developed to determine the mean and variance of FPT. The low-order Lorenz atmospheric model is taken as an example to show the robustness of using FPT as a quantitative measure for prediction skill. Both linear and nonlinear perspectives of forecast errors are analytically investigated using the self-consistent Nicolis model. The mean and variance of FPT largely depends on the ratio between twice the maximum Lyapunov exponent (s) and the intensity of attractor fluctuations ( q2), l 5 2s/q2. Two types of predictability are found: l . 1 referring to low predictability and l , 1 referring to high predictability. The mean and variance of FPT can be represented by the e-folding timescales in the low-predictability range, but not in the high-predictability range. The transition between the two predictability ranges is caused by the variability of the attractor characteristics along the reference trajectory.

System of quasi‐zonal jets off California revealed from satellite altimetry

[1] A discrete wavelet transform was applied to satellite altimetry data for the period 1992―2007 off California to decompose the SSH signal into inter-annual, annual, semi-annual and shorter period components. For the lowest frequency (inter-annual) component, a system of alternating quasi-zonal jets was detected. The jet system was delineated by a north-south series of quasi-zonal bands of co-rotating eddies; that is, the eddies were embedded in a shearing zonal flow. The direction of eddy rotation alternated between adjacent bands. The temporal behavior of the jet system showed the existence of quasi-stationary states and transitions between them. Observed non-linear effects of the evolution of the jets included southward drift at about 0.2 cm sec ―1 , deviations of the jets from the zonal direction, and re-forming of the jet system through decay and merging of eddy chains.

Fall-winter current reversals on the Texas-Louisiana continental shelf

Abstract Fall–winter recurrence of current reversal from westward to eastward is identified on the Texas–Louisiana continental shelf using the current-meter [Texas–Louisiana Shelf Physical Oceanography Program (LATEX-A)] and near-surface drifting buoy [Surface Current and Lagrangian Drift Program (SCULP-1)] observations in 1993 and 1994. Reversal events roughly satisfy the Poisson distribution with one current reversal nearly every 12 days. Synoptic winds seem responsible for the current reversal events. Other processes such as offshore eddies shed from the Loop Current and river runoff are less important to change alongshore flow direction at synoptic scales. A statistical model is established to predict the synoptic current reversal using the surface wind observations.

ROTATION METHOD FOR RECONSTRUCTING PROCESS AND FIELD FROM IMPERFECT DATA

Reconstruction of processes and fields from noisy data is to solve a set of linear algebraic equations. Three factors affect the accuracy of reconstruction: (a) a large condition number of the coefficient matrix, (b) high noise-to-signal ratio in the source term, and (c) no a priori knowledge of noise statistics. To improve reconstruction accuracy, the set of linear algebraic equations is transformed into a new set with minimum condition number and noise-to-signal ratio using the rotation matrix. The procedure does not require any knowledge of low-order statistics of noises. Several examples including highly distorted Lorenz attractor illustrate the benefit of using this procedure.

Backward Fokker‐Planck equation for determining model valid prediction period

[1] A new concept, valid prediction period (VPP), is presented here to evaluate ocean (or atmospheric) model predictability. VPP is defined as the time period when the prediction error first exceeds a predetermined criterion (i.e., the tolerance level). It depends not only on the instantaneous error growth but also on the noise level, the initial error, and the tolerance level. The model predictability skill is then represented by a single scalar, VPP. The longer the VPP, the higher the model predictability skill is. A theoretical framework on the basis of the backward FokkerPlanck equation is developed to determine the mean and variance of VPP. A one-dimensional stochastic dynamical system [Nicolis, 1992] is taken as an example to illustrate the benefits of using VPP for model evaluation. INDEX TERMS: 4263 Oceanography: General: Ocean prediction; 4255 Oceanography: General: Numerical modeling; 3367 Meteorology and Atmospheric Dynamics: Theoretical modeling; KEYWORDS: backward Fokker-Planck equation, instantaneous error, Lorenz system, predictability, tolerance level, valid prediction period

Filtering noise from oceanographic data with some applications for the Kara and Black Seas

Abstract If a reconstruction process is reduced to the solution of ill-posed algebraic systems, we suggest several procedures to improve the accuracy of reconstruction from noisy and irregular data. These procedures transform ill-posed equations to their well-posed analogies, thereby reducing both the contribution of noise to the equation system and the condition number of the system matrix. One of the techniques, the so-called “regularizing filter”, can be applied to observation samples of limited size when the ratio of the number of estimated field parameters to the number of field observations and noise to signal ratio are under 0.5–0.6 and 4–5, respectively. Furthermore, the filter is constructed without any preliminary knowledge of low-order noise statistics. The regularizing filter combined with a conventional function fitting procedure is illustrated through linear mapping scalar oceanographic fields, such as the surface temperature in the Black Sea observed from the NOAA-11, SiO2 in the Kara Sea, cesium and chlorophyll in the Black Sea. Herein comparing our approach to optimal interpolation, generalized cross-validation and smoothing spline-interpolation is also given.