Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 915793, 14 pages

http://dx.doi.org/10.1155/2015/915793

## Inversion Study of Vertical Eddy Viscosity Coefficient Based on an Internal Tidal Model with the Adjoint Method

^{1}Laboratory of Physical Oceanography, Ocean University of China, Qingdao 266100, China^{2}College of Engineering, Ocean University of China, Qingdao 266100, China

Received 28 March 2014; Revised 17 August 2014; Accepted 18 August 2014

Academic Editor: Fatih Yaman

Copyright © 2015 Guangzhen Jin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on an isopycnic-coordinate internal tidal model with the adjoint method, the inversion of spatially varying vertical eddy viscosity coefficient (VEVC) is studied in two groups of numerical experiments. In Group One, the influences of independent point schemes (IPSs) exerting on parameter inversion are discussed. Results demonstrate that the VEVCs can be inverted successfully with IPSs and the model has the best performance with the optimal IPSs. Using the optimal IPSs obtained in Group One, the inversions of VEVCs on two different Gaussian bottom topographies are carried out in Group Two. In addition, performances of two optimization methods of which one is the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method and the other is a simplified gradient descent method (GDM-S) are also investigated. Results of the experiments indicate that this adjoint model is capable to invert the VEVC with spatially distribution, no matter which optimization method is taken. The L-BFGS method has a better performance in terms of the convergence rate and the inversion results. In general, the L-BFGS method is a more effective and efficient optimization method than the GDM-S.

#### 1. Introduction

Internal tide, which is the internal wave of tidal frequency, is a ubiquitous phenomenon in the oceans. Rattray [1], Baines [2], Bell [3], Baines [4], Craig [5], Gerkema [6], and Llewellyn Smith and Young [7] have developed theoretical models and obtained some analytical solutions of internal tide on ideal topographies. These theoretical models helped them investigate the generation and propagation of internal tide. Although great progress has been made on the internal tide theory and some analytical works were carried out, only a small amount of solutions can be provided, due to the complexity of the problems. For this reason, quantitative analysis with practical significance still has to rely on the combination of numerical simulation, theoretical analysis, experiment, and observation. Numerical simulation is an effective method in marine research and has been widely used in the internal tide research. Kang et al. [8] investigated the internal tide near Hawaii with a two-dimensional, two-layered numerical model and confirmed that the internal tide was generated by barotropic forcing at the Hawaiian Ridge and propagated in north-northeast and south-southwest directions. Based on a high-precision three-dimensional Princeton Ocean Model (POM), Niwa and Hibiya [9] obtained the distribution of the internal tide in the Pacific Ocean using the TOPEX/Poseidon (T/P) satellite data. Cummins et al. [10] simulated the generation and propagation of internal tides near the Aleutian Ridge using T/P altimeter data. The comparison between the altimeter data and their model results showed good agreement for the phase, which also provided evidence for wave fraction near the Aleutian Ridge. With a three-dimensional POM, Niwa and Hibiya [11] investigated the distribution and the energy of the internal tide around the continental shelf edge in the East China Sea. Their numerical experiment results indicated that internal tides are effectively generated over prominent topographies such as sea ridges, island chains, and straits. Jan et al. [12] modified the POM to study the generation of the internal tide and its influence on surface tide in the South China Sea. The conversion from the barotropic energy to the baroclinic energy over topographic ridges in the Luzon Strait was also estimated.

Determination of the vertical eddy viscosity coefficient (VEVC), which describes the vertical mixing in the ocean, plays an important role in the study of energy exchange and material transportation. The VEVC is regularly regarded as a constant in numerical models. Schemes to determine the VEVC mainly include the Prandtl mixing-length hypothesis model, the model, the Pacanowski-Philander mixing model [13], and some turbulent closure models that are more complicated. Many studies have been carried out to investigate the variation of the VEVC [14–18]. All these mentioned studies indicate that due to different intensions of the vertical mixing in sea water, the VEVC should not be treated as a constant but a parameter with spatial distribution.

Satellite remote sensing technology and other related technologies provide us with a large number of data. Thus, it is one of the most important missions in physical oceanography to make use of the data efficiently and precisely as well as to combine the observation data with present numerical models. Indeed, data assimilation with the adjoint method provides an effective access to these missions. The use of the adjoint method in marine science can be traced back to 1980s. The adjoint model is capable of optimizing control parameters in numerical simulation. Bennett and McIntosh [19] applied the weak constraint thought to solve the tidal problem and the geostrophic-flow problem. Yu and O’Brien [20] assimilated both meterological and oceanographic data into an oceanic Ekman layer model and deduced the unknown boundary condition, the unknown vertical eddy viscosity, and the current field. Based on a tidal model with a two-level leapfrog method, Lardner [21] inverted the open boundary conditions in three-test problems. Seiler [22] used the adjoint method to assimilate observations into a quasi-geostrophic ocean model and estimated the lateral boundary values in ideal experiments. Navon [23] wrote a summary of the parameter estimation in meteorology and oceanography in the view of applications with four- dimensional variational data assimilation techniques. Using an automatic differentiation compiler, Ayoub [24] constructed the adjoint model of the Massachusetts Institute of Technology Ocean General Circulation Model and inverted the open boundary conditions in the North Atlantic. Zhang and Lu [25] developed a three-dimensional nonlinear numerical tidal model with the adjoint method and designed several numerical experiments to estimate three kinds of parameters including the open boundary conditions, the bottom friction coefficients, and the vertical eddy viscosity coefficients. Zhang and Lu [26] employed a two-dimensional tidal model to study the inversion of the bottom friction coefficients in the Bohai Sea and the Yellow Sea with the adjoint method. Chen et al. [27] constructed a three-dimensional internal tidal model with the adjoint method and estimated six different kinds of open boundary conditions on fourteen types of topography. Based on a tidal model, Zhang and Chen [28] carried out several semiidealized experiments to estimate the partly and fully spatial varying open boundary conditions. Cao et al. [29] investigated the inversion of open boundary conditions with a three-dimensional internal tidal model and simulated the internal tide around Hawaii by assimilating T/P data.

There are two main objectives of this paper. One is to study the inversion of the VEVC with an internal tidal model and the adjoint method. According to the introductions above, a lot of studies have been carried out to investigate the inversion of the control parameters of internal tide such as the open boundary condition [29, 30] and the bottom friction condition [31, 32]. However, few works are found to study the inversion of VEVC. Since VEVC is a decisive factor to describe the vertical mixing in the ocean, it is necessary to pay attention to the inversion of the VEVC. The other objective is to make a computational investigation on the performance of the gradient descent method and the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method for the inversion of VEVCs based on the model constructed by Chen et al. [27]. Both of the methods do not require any evaluations of the Hessian matrices but gradient vectors and, thus, are computationally feasible. Chen et al. [30] have made a comparative study on several optimization methods but it is on the inversion of the open boundary conditions which is a one-dimension case. The feasibility of these optimization methods for two-dimensional case such as the inversion of the VEVC needs further studies.

Two groups of numerical experiments are carried out to study the inversion of spatially varying VEVCs based on an isopycnic-coordinate internal tidal model with the adjoint method. In Group One, the influences of independent point schemes (IPSs) exerting on parameter inversion are discussed. Group Two investigates the inversions of VEVCs on two different Gaussian bottom topographies and the performances of two optimization methods which are the GDM-S and the L-BFGS methods.

This paper is organized as follows. Section 2 briefly introduces the adjoint tidal model and the methodology. Two optimization methods, including the GDM-S and the L-BFGS methods, are described in Section 3. Section 4 presents design and process of the experiments in detail. Results of the experiments are discussed in Section 5. Finally, we make a summary and draw some conclusions in Section 6.

#### 2. Numerical Model Introduction

An isopycnic-coordinate internal tidal model with adjoint assimilation method is employed in this paper. There are two parts in the internal tide model. One is forward model with the governing equations and the other is adjoint model with the adjoint equations. The two models are used to simulate the internal tide and to optimize the control variables, respectively. Chen et al. [27] had introduced the two parts in great detail and tested the reasonability and feasibility of the model. The formulation will not be presented in this paper. The derivation of VEVC adjustment, introduction of the two optimization method, test of the adjoint method, and the independent point scheme (IPS) are described in details in this part.

##### 2.1. Test of the Adjoint Method

According to the equations and derivations of Chen et al. [27], the formula to invert the VEVC can be derived. The first derivative of Lagrangian function with respect to VEVC is obtained as follows:where is the value of VEVC at grid in the th layer. The gradient of cost functions with respect to the VEVC in the grid can be deduced as follows:where is the potential density in the th layer, and are horizontal velocities at the th time step, and are the adjoint variables of and , respectively, and is the initial thickness of the th layer. The detailed derivation of (2) is presented in the appendix.

Accurately programming the adjoint in such problems as the present one is quite tricky and experience has shown that it is essential to check the accuracy of the adjoint computation before proceeding with the minimization runs [33]. The correctness of the adjoint method is verified in this section. Take the first-order term of a Taylor expansion for the cost function and we obtain the following equation:

Here, is a general point of the control variable, is the computed gradient, and is an arbitrary unit vector in the parameter space. Based on (3) a function of can be written as follows:where is a small real number that is not equal to zero. If the adjoint methodology is correct, it is supposed that according to (4). In this paper, the VEVC variable is treated as and test of the adjoint method is based on (4).

In order to test the accuracy of the adjoint method, two experiments are carried out in which two different types of are used. The different vector directions are and , respectively.

Figure 1 indicates the trends of as approaches to 0. It is clear that in both experiments when is less than 10^{−3}, values of (solid lines) are both very close to 1 (dashed lines). Equation is proved and the correctness of gradient computed in the adjoint model is verified.