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Max Yaremchuk
and
Paul Martin

Abstract

The sensitivity of model forecasts to uncertainties in control variables is evaluated using the adjoint technique and the ensemble generated by the reduced-order four-dimensional variational data assimilation (R4DVAR) algorithm within the framework of twin-data experiments with a quasigeostrophic model. To simulate real applications where the true state is unknown, the sensitivities were estimated using model solutions that were obtained after assimilating sparse observations extracted from the true solutions. The numerical experiments were conducted in the linear, weakly nonlinear, and strongly nonlinear (NL) regimes with special emphasis on the NL case characterized by the instability of the tangent linear model. It is shown that the ensemble-based R4DVAR method provides better sensitivity estimates in the NL case, primarily due to the better accuracy of the optimized solutions. The concept of sensitivity in the NL case is also considered within the statistical framework. Using analytical arguments and numerical experimentation, averaging the adjoint sensitivity estimates over an ensemble of model trajectories generated by finite perturbations of the optimal control is shown to provide an estimate similar to that obtained with the adjoint model stabilized by enhanced dissipation. This observation allows for evaluation of the sensitivities of strongly nonlinear optimal solutions by using both the adjoint (4DVAR) and ensemble (R4DVAR) optimization algorithms.

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Max Yaremchuk
and
Matthew Carrier

Abstract

Many background error correlation (BEC) models in data assimilation are formulated in terms of a smoothing operator , which simulates the action of the correlation matrix on a state vector normalized by respective BE variances. Under such formulation, has to have a unit diagonal and requires appropriate renormalization by rescaling. The exact computation of the rescaling factors (diagonal elements of ) is a computationally expensive procedure, which needs an efficient numerical approximation.

In this study approximate renormalization techniques based on the Monte Carlo (MC) and Hadamard matrix (HM) methods and on the analytic approximations derived under the assumption of the local homogeneity (LHA) of are compared using realistic BEC models designed for oceanographic applications. It is shown that although the accuracy of the MC and HM methods can be improved by additional smoothing, their computational cost remains significantly higher than the LHA method, which is shown to be effective even in the zeroth-order approximation. The next approximation improves the accuracy 1.5–2 times at a moderate increase of CPU time. A heuristic relationship for the smoothing scale in two and three dimensions is proposed for the first-order LHA approximation.

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Max Yaremchuk
and
Dmitry Nechaev

Abstract

Improving the performance of ensemble filters applied to models with many state variables requires regularization of the covariance estimates by localizing the impact of observations on state variables. A covariance localization technique based on modeling of the sample covariance with polynomial functions of the diffusion operator (DL method) is presented. Performance of the technique is compared with the nonadaptive (NAL) and adaptive (AL) ensemble localization schemes in the framework of numerical experiments with synthetic covariance matrices in a realistically inhomogeneous setting. It is shown that the DL approach is comparable in accuracy with the AL method when the ensemble size is less than 100. With larger ensembles, the accuracy of the DL approach is limited by the local homogeneity assumption underlying the technique. Computationally, the DL method is comparable with the NAL technique if the ratio of the local decorrelation scale to the grid step is not too large.

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Dmitri Nechaev
and
Max Yaremchuk

Abstract

A new representation of the Coriolis terms on the Arakawa C grid is proposed. The approximation dumps the grid-scale noise that arises because of spatial averaging of the Coriolis terms when the grid spacing is larger than the deformation radius. The proposed approximation can also be applied in C-grid schemes with semi-implicit treatment of the Coriolis terms. The new scheme is analyzed in the context of the linear inertial–gravity waves and its advantageous behavior is demonstrated with respect to the conventional technique.

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Max Yaremchuk
and
Konstantin Lebedev

Abstract

Climatological data on the oceanic and atmospheric variability are inverted to study seasonal variation of the Kuroshio Extension (KE) and the recirculation gyre to the south. The processed datasets include climatological fluxes of heat, salt, and momentum at the ocean surface; Levitus hydrography; TOPEX/Poseidon altimetry; and surface drifter data. A variational data assimilation technique is used to retrieve variability in the open ocean region (18°–42°N, 142°E–160°W) bounded at 1000 m from below. By optimizing the open boundary values of oceanic fields in combination with initial conditions and atmospheric forcing, model solutions that are consistent with various climatological datasets within limits of observational errors were found. Using this technique, the mean geographical positions of the Subtropical Mode Water (STMW) and Central Mode Water (CMW) formation sites were estimated, the structures were analyzed, and estimates of the mode water production rates were obtained. Computations indicate that CMW formation is likely to occur 15°–20° west of the location diagnosed formerly without taking salinity data into the account. The optimized seasonal cycle is characterized by the STMW and CMW production rates of 3.8 ± 0.6 and 3.1 ± 0.5 Sv (1 Sv ≡ 106 m3 s−1), respectively. The KE annual mean transport in the upper 1000 m is diagnosed as 68 ± 7 Sv with a maximum of 79 ± 8 Sv in June–July and a minimum of 56 ± 6 Sv in December–January. Analysis of the heat and salt budgets in the region has shown that atmospheric fluxes are counterbalanced by the horizontal divergence of the advective temperature and salinity transports. In the annual mean, horizontal diffusion plays a minor role in the budgets.

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Max Yaremchuk
and
Tangdong Qu

Abstract

The mean seasonal cycle of the western boundary currents in the tropical North Pacific Ocean is studied diagnostically by combining atmospheric climatologies with drifter, satellite altimetry, and hydrographic data in the framework of a simplified numerical model incorporating geostrophy, hydrostatics, continuity, and tracer conservation. The approach enables the authors to diagnose the absolute 3D velocity field and to assess the seasonal cycle of sea surface height (SSH)/total transports. Errors are estimated by considering multiple datasets and averaging over the results of the corresponding diagnostic computations. Analysis shows that bifurcation of the North Equatorial Current (NEC) occurs at 14.3° ± 0.7°N near the Philippine coast. Meridional migration of the NEC bifurcation latitude is accompanied by quantitative changes in the partitioning of the NEC transport between the Kuroshio and Mindanao Current. In February–July, when the NEC transport is 58 ± 3 Sv (Sv ≡ 106 m3 s−1), the Kuroshio transport is 12%–15% higher than the Mindanao Current (MC) transport. In the second half of the annual cycle the situation is reversed: in October the NEC transport drops to 51 ± 2 Sv with the MC transport exceeding the Kuroshio transport by 25%. The net westward transport through the Luzon Strait is characterized by a minimum of 1.2 ± 1.1 Sv in July–September and a maximum of 4.8 ± 0.8 Sv in January– February. A statistically significant correlation is established between the monthly SSH/streamfunction anomalies north of 10°N and the Ekman pumping rate associated with the northeast monsoon developing in the region in October–December. The result provides an indication of the fact that local monsoon is likely to be an important mechanism governing seasonal variation of the NEC partitioning and water mass distribution between the tropical and subtropical North Pacific.

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Max Yaremchuk
,
Dmitri Nechaev
, and
Gleb Panteleev

Abstract

A version of the reduced control space four-dimensional variational method (R4DVAR) of data assimilation into numerical models is proposed. In contrast to the conventional 4DVAR schemes, the method does not require development of the tangent linear and adjoint codes for implementation. The proposed R4DVAR technique is based on minimization of the cost function in a sequence of low-dimensional subspaces of the control space. Performance of the method is demonstrated in a series of twin-data assimilation experiments into a nonlinear quasigeostrophic model utilized as a strong constraint. When the adjoint code is stable, R4DVAR’s convergence rate is comparable to that of the standard 4DVAR algorithm. In the presence of strong instabilities in the direct model, R4DVAR works better than 4DVAR whose performance is deteriorated because of the breakdown of the tangent linear approximation. Comparison of the 4DVAR and R4DVAR also shows that R4DVAR becomes advantageous when observations are sparse and noisy.

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Gleb Panteleev
,
Max Yaremchuk
, and
Oceana Francis

Abstract

We analyzed the feasibility of the reconstruction of the spatially varying rheological parameters through the four-dimensional variational data assimilation of the sea ice velocity, thickness, and concentration into the viscoplastic (VP) sea ice model. The feasibility is assessed via idealized variational data assimilation experiments with synthetic observations configured for a 1-day data assimilation window in a 50 × 40 rectangular basin forced by the open boundaries, winds, and ocean currents and should be viewed as a first step in the developing the similar algorithms which can be applied for the more advanced sea ice models. It is found that “true” spatial variability (∼5.8 kN m−2) of the internal maximum ice strength parameter P * can be retrieved from observations with reasonable accuracy of 2.3–5.3 kN m−2, when an observation of the sea ice state is available daily in each grid point. Similar relative accuracy was achieved for the reconstruction of the compactness strength parameter α. The yield curve eccentricity e is found to be controlled by the data with less efficiency, but the spatial mean value of e could be still reconstructed with a similar degree of confidence. The accuracy of P * , α, and e retrievals is found to increase in regions of stronger ice velocity convergence, providing prospects for better processing of the observations collected during the recent MOSAiC experiment. The accuracy of the retrievals strongly depends on the number of the control variables characterizing the rheological parameter fields.

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Max I. Yaremchuk
and
Denis A. Krot

Abstract

An approximate formula for the reciprocal speed of sound in seawater is obtained in the form of a polynomial that is cubic in potential temperature, and quadratic in pressure and salinity. The expression provides a reformulation of Del Grosso's empirical formula for the speed of sound in seawater in terms of salinity, pressure, and potential temperature, which are the basic thermodynamic parameters describing oceanic state in numerical models. This makes the proposed approximation convenient for constraining the acoustic tomography data by dynamics.

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Max I. Yaremchuk
and
Dmitri A. Nechaev

Abstract

Acoustic tomography (AT) and satellite altimetry (SA) measure properties of the ocean state with high temporal resolution. That makes these data suitable for long-term monitoring of mesoscale features in the open ocean regions, where the open boundaries are the major sources of model forecast uncertainties on timescales larger than 1 week. In this paper, a finite-difference quasigeostrophic model of an open ocean region is considered as a possible tool for interpolating AT–SA data in space and time. The assimilation algorithm is based upon the 4D variational data assimilation scheme controlled by the initial and boundary conditions of the model. The model configuration used in the simulations corresponds to the AT array deployed by the Japan Marine Science and Technology Center (JAMSTEC) in the region of the Kuroshio Extension in 1997. Twin data experiments show that mesoscale features in an area of 1000 km × 1000 km can be effectively monitored by five acoustic transceivers, measuring reciprocal travel times. The quality of assimilation is studied as a function of the position of the transceivers in the vertical and the effective number of monitored rays. It is shown that reciprocal travel time observations (differential tomography) in combination with SA provide a significant improvement of the quality of assimilation.

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