Application of the Trigonometric Polynomial Interpolation for the Estimation of the Vertical Eddy Viscosity Coefficient Based on the Ekman Adjoint Assimilation Model

Round 1

Reviewer 1 Report

The paper is dedicated to application of 4-dimensional variational data assimilation (4D-Var) to estimate vertical mixing coefficients in the scope of a simple Ekman circulation model. Undoubtably, 4D-Var has a great property absent in most other assimilation techniques: is is an ability of parametric control, i.e., to obtain an optimal solution best fit to observations, the modeller is capable of controlling not only boundary and initial conditons, source functions e t.c., but also internal parameters of the model, which are usually needed to be defined by some indirect techniques, the so-called parameterizations. The typical examples of such parameters are vertical mixing coefficients (diffusivity or viscosity), usually defined by using a certain turbulence model, than can be simple (Pacanowski&Philander, Monin&Obukhov) or complex (Mellor-Yamada, K-KL, K-Epsilon, K-Omega). Using parametric control with involving realistic observation, one can increase quality of estimating vertical mixing coefficients and, consequently, quality of oceanographic simulations.

There are some local comments to the manuscript.

1)    What does the term “inversion” mean? As far as I understood, it is solution of the inverse problem to find the control parameters. As for me, I have not met the term “inversion” for this. If it is widely used in this sense, point it out in the reply. But as for me, this word looks distracting, moreover, in the title of the article, since the phrase “Inversion of the Vertical Eddy Viscosity Coefficient” instantly interpreted as inversion of its vertical profile in the case of its expected monotonicity in depth. Please check the use of this term throughout the paper.

2)    Line 115. A technical recommendation is to write time derivative in the adjoint equations with negative sign, as widely accepted, since they are solved in the reverse time.

3)    Is the VEVC (A) supposed to be time-dependent or stationary?

4)    Line 128. It is not fully clear, what is the reason to use interpolation schemes for this research. You have forward and inverse models with the correction of the control vector. As far as I see, all these tasks can be solved in the standard spatio-temporal grid space, without involving any special spatial representation of the functions. If it is described in details in your previous papers, it should be mentioned in the present paper at least briefly of placed in the appendix, since it is not easy enough to understand the cause-consequence links. Moreover, trigonometric presentations are well justified for regular grid, but it is not so clear, what properties they possess in the case of irregular vertical grid. The latter is usually used for realistic oceanic simulations.

5)    Line 162. What algorithm war used for correction of control parameters? Is it the simplest descent against gradient or anything improved, e.g. selecting optimal descent parameter of quasi-newton algorithm?

6)    Was the twin experiment designed correctly? Such a test is usually performed to check the system for coarse errors. The solution of the forward model must be implemented as observations in the assimilation system. If everything is fully in agreement, the right hand side in the adjoint equations must be exact 0, which yields the null solution for the whole adjoint system. Did you perform such a test, where the initial guess fits to the parameters used in the forward model, or just started the iterative procedure from the disturbed ones?

7)    What is the reason of choosing the VEVC (line 175-178) in such form? It is rather far from the real distribution in the ocean, which is maximum in the upper active/mixed layer and minimum below it. No sin or cos dependency on depth is observed.

8)    What is computational efficiency of the proposed approach? Assimilation of data on velocity is useful in the case of operational forecasting in the real-time mode. However, these data (in contrast to the temperature and salinity) are quickly “forgotten” by the circulation model. Is it worth using 4Dvar with iterative procedure for definition of vertical viscosity coefficient, which is, possibly does not vary in time and presents some mean state for some days? If we need do estimate the time-mean viscosity with using some observations, we can just estimate the Ekman depth and, taking into account Coriolis parameter, just compute the estimate for viscosity.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

In this paper, the authors propose a new method for determining the vertical eddy viscosity coefficient in the Ekman model based on ocean current and wind velocity measurements. The work is interesting, well written and in my opinion should be published in JMSE after the revision. My comments and suggestions are as follows:

1. It is not clear how independent points (IPs) are chosen in models CI and CSI, why such a number of them is chosen as shown in Table 2? If their number is increased, will these models become better than the TPI method proposed by the authors? Are the conclusions of section 3 the result of a small number of independent points? These issues should be given more attention.

2. It would be nice to add in section 4 a comparison with other numerical methods that allow you to solve similar problems. This would greatly enhance the work.

3. Correct the typo in equation (2.9b) and in Figure 2 (Caee1->Case1).

4. Section 2.1 can be extended by adding explanations under what physical conditions this model is valid and what assumptions are underlain in its basis.

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

I thank the authors for their detailed response and have no more question.