A Method on Estimating Time-Varying Vertical Eddy Viscosity for an Ekman Layer Model with Data Assimilation

Abstract

Temporal vertical eddy viscosity coefficient (VEVC) in an Ekman layer model is estimated using an adjoint method. Twin experiments are carried out to investigate the influences of several factors on inversion results, and the conclusions of twin experiments are 1) the adjoint method is a capable method to estimate different kinds of temporal distributions of VEVCs; 2) the gradient descent algorithm is better than CONMIN and L-BFGS for the present problem, although the posterior two algorithms perform better on convergence efficiency; 3) inversion results are sensitive to initial guesses; 4) the model is applicable to different wind conditions; 5) the inversion result with thick boundary layer depth (BLD) is slightly better than thin BLD; 6) inversion results are more sensitive to observations in upper layers than those in lower layers; 7) inversion results are still acceptable when data noise exists, indicating the method can sustain noise to a certain degree; 8) a regularization method is proved to be useful to improve the results for present problem; and 9) the present method can tolerate the existence of balance errors due to the imperfection of governing equations. The methodology is further validated in practical experiments where Ekman currents are derived from Bermuda Testbed Mooring data and assimilated. Modeled Ekman currents coincide well with observed ones, especially for upper layers. The results demonstrate that the assumptions of depth dependence and time dependence are equally important for VEVCs. The feasibility of the typical Ekman model, the imperfection of Ekman balance equations, and the deficiencies of the present method are discussed. This method provides a potential way to realize the time variations of VEVCs in ocean models.

Estimation of Bottom Friction Coefficient in Multi‐Constituent Tidal Models Using the Adjoint Method: Temporal Variations and Spatial Distributions

“…As indicated by Zhang and Lu (2010) and Zhang et al. (2019), the steepest descent method is as efficient and useful as the limited‐memory conjugate gradient algorithm and the limited‐memory Broyden‐Fletcher‐Goldfarb‐Shanno algorithm that have been widely used (Alekseev et al., 2009; Zou, Navon, et al., 1993), which may be due to the clustering of eigenvalues in the spectrum of the problem being minimized (Alekseev et al., 2009). When the BFC is assumed to be spatially varying or temporally varying, is the vector of the spatially varying BFC or temporally BFC arranged in a sequence, as shown in Zhang et al.…”

“…where γ is the step size; l is the lth iteration step of the parameter estimation;  p is the vector of the spatially and temporally varying BFCs arranged in a sequence;  q is the gradient vector of the cost function with respect to  p; and q max is the  L norm of  q. As indicated by Zhang and Lu (2010) and Zhang et al (2019), the steepest descent method is as efficient and useful as the limited-memory conjugate gradient algorithm and the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm that have been widely used (Alekseev et al, 2009;, which may be due to the clustering of eigenvalues in the spectrum of the problem being minimized (Alekseev et al, 2009). When the BFC is assumed to be spatially varying or temporally varying,  p is the vector of the spatially varying BFC or temporally BFC arranged in a sequence, as shown in Zhang et al (2011).…”

A Scheme for Estimating Time-Varying Wind Stress Drag Coefficient in the Ekman Model with Adjoint Assimilation

“…The main purpose of this is to estimate time-varying WSDCs in this study. As mentioned above, the time-varying eddy viscosity has been studied in previous studies, such as Zhang et al [21]. Therefore, we set the eddy viscosity as a variable that only varies with depth.…”

“…Cao et al [20] proposed an estimation scheme to evaluate the eddy viscosity profile (EVP) at the bottom of the Ekman boundary layer based on the adjoint method. Zhang et al [21] successfully inverted the time-varying vertical eddy viscosity coefficients in the Ekman model and also found that the inversion results were much more sensitive to the observations in the upper layers.…”

Parameter Estimation Based on a Local Ensemble Transform Kalman Filter Applied to El Niño–Southern Oscillation Ensemble Prediction

“…Additionally, the study of parameter estimation in atmospheric, oceanic, or coupled models is evolving rapidly in the development of assimilation methods [15][16][17][18]. Parameter estimation is mainly implemented by two methods: the variational method [10,[18][19][20][21][22][23] and ensemble Kalman filter (EnKF) [11,12,21,24,25]. Deriving the linear tangent and adjoint model, the four-dimensional variational method is difficult, even impractical, when implemented in fully coupled circulation models [19,21].…”