Form 3

Annual Report
LaMer, Ehime University
Date (25, February, 2018 )To Director of LaMer

Principle Investigator:
Affiliation Second Institute of Oceanography, SOA
Position

Ph.D Candidates＿＿＿＿＿＿ ＿＿＿Name in print

Ze-Nan Zhu

Include the report on the result of the project/meeting in a separate sheet.
1. Project / Meeting title
Coastal tomographic mapping of nonlinear tidal currents and residual currents
2. Members of project / meeting
NameAffiliation

PositionContribution part

PIZe-Nan Zhu

Second Institute Ph.Dof Oceanography, Candidates
SOA

Data processing

MembersXiao-HuaZhu

Second Institute Seniorof Oceanography, ResearchSOA
Scientist

Beneficial discussion

Ehime University

Beneficial discussion

LaMerFaculty

member in charge

Xinyu GuoProfessor

1/6＿3. Contents (please write in separate sheet, A4-size, within 5 pages including figures andtables. Itemize “Title, members’ names and affiliations, aim, procedure, result,publication/conference presentation, perspectives in future”).
This report will be opened to the public through LaMer website and printed matter. Please
send additional products, i.e., abstract book, reprint, if available toLaMer Office of Research Support Division,Ehime University, Bunkyo 2-5, Matsuyama, 790-8577 Japan
E-mail: lamer@stu.ehime-u.ac.jp
Tel and Fax: +81-89-927-8187

Title
Coastal tomographic mapping of nonlinear tidal currents and residual currents

Members’ names and affiliations
NameInstitutionand Employment position
Department
Zhu
Second Institute Senior
Xiao-Hua of Oceanography, Research Scientist
SOA
Xinyu
Faculty member Professor
Guoof LaMer

E-mailxhzhu@sio.org.cnguoxinyu@sci.ehimeu.ac.jp

Aim
Compared with traditional techniques of measuring the current field, Coastal Acoustic
Tomography (CAT) can easily achieve long-term and synchronous measurement over a large area,obtaining the rapidly varying current field in coastal seas. This project aims to carry out a full
discussion on the nonlinear tidal dynamics based on CAT data in coastal regions with Prof. Guo Xinyu
of Ehime University. In addition, a discussion with Prof. Guo about how to assimilate the CAT data
into the ocean model with triangular mesh by using the ensemble Kalman Filter scheme is also
necessary.

Procedure
1. The principal investigator (PI) made an oral presentation “A study on assimilation andapplication of Coastal Acoustic Tomography” and made a briefly introduction of Coastal
Acoustic Tomography and its application in coastal regions.
2. The PI carried out further research with Prof. Guo Xinyu on nonlinear tidal currents in
Chinese coastal regions. The PI discussed with Prof. Guo about the generation mechanismsof the nonlinear tidal current (i.e. M4 and M6 tides) observed by Coastal Acoustic
Tomography.
3. The PI discussed with Prof. Guo about the schemes of assimilating the Coastal Acoustic
Tomography data into the ocean model with unstructured grid.

Results
We select a typical coastal regions of China (Zhitouyang Bay) to discuss the characteristics of2/6

nonlinear tidal currents. The 15-min mean depth-averaged current data obtained by Coastal Acoustic
Tomography (CAT), which covers about 27 hours in Zhitouyang Bay on the western side of the East
China Sea, are used to estimate the semidiurnal tidal current (M2) as well as its first two overtidecurrents (M4 and M6).
To obtain the M4 and M6 tidal currents generated by the advection and quadratic bottom friction
terms, we carry out the derivation of these terms in the two-dimensional shallow-water equations.
These equations can be written as follows:𝜕𝑢𝜕𝑡{ 𝜕𝑣
𝜕𝑡+𝑢+𝑢𝜕𝑢
𝜕𝑥𝜕𝑣
𝜕𝑥+𝑣+𝑣𝜕𝑢𝜕𝑦𝜕𝑣
𝜕𝑦− 𝑓𝑣 = −𝑔+ 𝑓𝑢 = −𝑔

𝜕𝜂
𝜕𝑥
𝜕𝜂
𝜕𝑦

−−𝐶𝑑ℎ+𝜂
𝐶𝑑ℎ+𝜂

𝑢√𝑢2 + 𝑣 2(1)𝑣√𝑢2 + 𝑣 2

Here, x, y are the horizontal spatial coordinates (positive denoting eastward and northward
respectively, unit is m), t is time, u, v are the horizontal velocity components; 𝑔 is gravitational
acceleration (=9.8 m s-2); 𝜂 is surface elevation; Cd is bottom drag coefficient (=0.0025); h is the𝜕𝑢water depth; f is Coriolis parameter (=7.27×10-5 s-1). Thus, the nonlinear advection terms are 𝑢 +𝑣

𝜕𝑢𝜕𝑦and 𝑢

𝜕𝑣
𝜕𝑥+𝑣

𝜕𝑣
𝜕𝑦, the nonlinear friction terms are𝐶𝑑ℎ+𝜂

𝑢√𝑢2 + 𝑣 2 and𝜕𝑥𝐶𝑑ℎ+𝜂

𝑣√𝑢2 + 𝑣 2 .

The tidal velocity components of M2 (i.e., 𝑢𝜎 and 𝑣𝜎 ) can be written as𝑢𝜎 = 𝑈𝑐 (𝑥, 𝑦) cos(𝜎𝑡) + 𝑈𝑠 (𝑥, 𝑦) sin(𝜎𝑡){𝑣𝜎 = 𝑉𝑐 (𝑥, 𝑦) cos(𝜎𝑡) + 𝑉𝑠 (𝑥, 𝑦) sin(𝜎𝑡)(2)Here, 𝑈𝑐 (𝑥, 𝑦)，𝑈𝑠 (𝑥, 𝑦), 𝑉𝑐 (𝑥, 𝑦), 𝑉𝑠 (𝑥, 𝑦) are harmonic constants of the M2 tidal current at eachspatial point, 𝜎 is the M2 angular frequency.
The M4 tide is mainly caused by the nonlinear advection terms of the M2 tidal current (Parker, 1991).
Substituting Eq. (2) into Eq. (1), we find that the nonlinear advection terms can be written as follows:
𝐴 + 𝐵𝑐 cos(2𝜎𝑡) + 𝐵𝑠 sin(2𝜎𝑡){ 0𝐶0 + 𝐷𝑐 cos(2𝜎𝑡) + 𝐷𝑠 sin(2𝜎𝑡)1𝜕𝑈𝑐2𝜕𝑈𝑐

𝜕𝑥

Here, 𝐴0 = (𝑈𝑐1(𝑈𝑐𝜕𝑈𝑐+ 𝑉𝑐+ 𝑈𝑠+ 𝑉𝑐
𝜕𝑈𝑠𝜕𝑈𝑐+ 𝑈𝑠

𝜕𝑦+ 𝑉𝑠

𝜕𝑈𝑠𝜕𝑈𝑠𝜕𝑥+ 𝑉𝑠𝜕𝑈𝑠𝜕𝑦11𝜕𝑉𝑐2𝜕𝑥) ; 𝐶0 = (𝑈𝑐) ; 𝐵𝑠 = (𝑈𝑐𝜕𝑈𝑠+ 𝑈𝑠𝜕𝑈𝑐(3)+ 𝑉𝑐+𝑉𝑑𝑈𝑠

𝑐 𝜕𝑦𝜕𝑥
𝜕𝑉𝑐
𝜕𝑉𝑠
𝜕𝑉𝑠
𝜕𝑉𝑠
𝜕𝑉𝑠
𝜕𝑉𝑐
𝑉𝑐+ 𝑈𝑠+ 𝑉𝑠 ) ; 𝐷𝑠 = (𝑈𝑐+ 𝑈𝑠+ 𝑉𝑐+ 𝑉𝑠 ).
𝜕𝑦𝜕𝑥
𝜕𝑦2𝜕𝑥
𝜕𝑥
𝜕𝑦𝜕𝑦2𝜕𝑥𝜕𝑦𝜕𝑥

𝜕𝑦12𝜕𝑉𝑐𝜕𝑥𝜕𝑉𝑐
𝜕𝑦+ 𝑈𝑠+ 𝑉𝑠

𝜕𝑈𝑐𝜕𝑦𝜕𝑉𝑠
𝜕𝑥+ 𝑉𝑠𝜕𝑉𝑠𝜕𝑦1) ; 𝐵𝑐 =) ; 𝐷𝑐 = (𝑈𝑐2𝜕𝑉𝑐
𝜕𝑥+Then we can obtain the currents from the advection terms (i.e., M4_cal) by integrating with respect
to time the M4 angular frequency terms in Eq. (3), as follows:
𝐵𝑢2𝜎 = 0 sin(2𝜎𝑡 − 𝛼𝑢 )2𝜎{(4)𝐷𝑣2𝜎 = 0 sin(2𝜎𝑡 − 𝛼𝑣 )2𝜎𝐵𝐷𝐵𝑐𝐷𝑐Here, 𝐵0 = √𝐵𝑐 2 + 𝐵𝑠 2 ; 𝐷0 = √𝐷𝑐 2 + 𝐷𝑠 2 ; 𝛼𝑢 = arctan ( 𝑠 ) ; 𝛼𝑣 = arctan ( 𝑠 ). The velocities inEq. (4), having the M4 angular frequency, are considered to be the predicted M4 tidal currents. Then,because the predicted M4 currents are derived from the advection terms, if the predicted M4 are similar
to the M4 currents measured by CAT, we can confirm that the advection terms make a primarycontribution to the generation of M4 tidal currents, otherwise the M4 tidal currents may also begenerated by other factors.
In the deep area, where water depths are larger than 60 m, M4 velocity amplitudes measured by CAT
agree well with those predicted by the advection terms in the shallow water equations, indicating thatM4 in the deep area is predominantly generated by the advection terms (Figure 1).

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Figure 1. Tidal current ellipses of (a) M4 and (b) M4_cal. Red lines indicate similar ellipses for M4 andM4_cal. The area enclosed by the dashed lines indicates the CAT observational region.
On the other hand, the overtide that has three times the M2 angular frequency (i.e., 3σ) is mainly
caused by the friction terms (Fang, 1987; Parker, 1991). The quadratic bottom friction terms of thelinear tide can be written in the following form:𝜏maj = 𝐶𝑑 |√𝑢maj 2 + 𝑢min 2 | 𝑢maj /ℎ(5)𝜏min = 𝐶𝑑 |√𝑢maj 2 + 𝑢min 2 | umin /ℎ{here, 𝜏𝑚𝑎𝑗 and 𝜏min are the bottom friction along the major and minor axis directions of the lineartide, respectively; 𝑢𝑚𝑎𝑗 and 𝑢𝑚𝑖𝑛 are the velocities along the major and minor axis directions of thelinear tide, respectively; h is the water depth. Considering 𝑢𝑚𝑎𝑗 ≫ 𝑢𝑚𝑖𝑛 (Fig.1), Eq. (5) can bewritten as𝜏maj = 𝐶𝑑 |𝑢maj |𝑢maj /ℎ{(6)𝜏min = 𝐶𝑑 |𝑢maj |𝑢min /ℎEq. (2) also can be written in the following form:𝑢maj = 𝑈maj (𝑥, 𝑦) cos 𝜎𝑡{(7)𝑢min = 𝑈min (𝑥, 𝑦) sin 𝜎𝑡Here, 𝑈maj and 𝑈min are the amplitudes along the major and minor axis directions of the linear
tide, respectively.
The Fourier expansion of Eq. (6) can be written as follows when we consider only one linear
tidal constituent:8𝜏maj = 𝑈maj (𝑥, 𝑦)2 ∑𝑚=0,1,2,3,…(−1)𝑚+1 (2𝑚−1)(2𝑚+1)(2𝑚+3)𝜋 cos(2𝑚 + 1)𝜎𝑡{8𝜏min = 𝑈maj (𝑥, 𝑦)𝑈min (𝑥, 𝑦) ∑𝑚=0,1,2,3,…(−1)𝑚 (2𝑚−1)(2𝑚+3)𝜋 sin(2𝑚 + 1)𝜎𝑡(8)The predicted M6 tidal currents (i.e., M6_cal) are obtained by setting m=1 and integrating Eq. (8) withrespect to time as follows:𝑢3𝜎 = 𝑈maj (𝑥, 𝑦)28𝐶𝑑cos 3𝜎𝑡{4𝐶𝑣3𝜎 = −𝑈maj (𝑥, 𝑦)𝑈min (𝑥, 𝑦) 𝑑 sin 3𝜎𝑡45ℎ𝜋(9)15ℎ𝜋The velocities in Eq. (9) having the M6 angular frequency are considered be the predicted M6 tide.
From the similarity between the measured M6 and predicted M6 currents, we can confirm thecontribution of quadratic bottom friction to the generation of the M6 tide.
M6 measured by CAT and predicted by the nonlinear quadratic bottom friction terms agrees well inareas where water depths are less than 20 m, indicating that friction mechanisms are predominant for
generating M6 in the shallow area (Figure 2).

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Figure 2 Tidal current ellipses of (a) M6 and (b) M6_cal. Red lines indicate similar ellipses for M6 andM6_cal. The area enclosed by the dashed lines indicates the CAT observational region.
Then, we also made the first challenge to assimilate the CAT data into an ocean model with an
unstructured triangular grid by using the ensemble Kalman filter scheme (Evensen, 1994).
In order to establish a relationship between CAT data and model states, the relationship between thesound transmission lines and the triangular mesh is described in Figure 3. The original CAT data aretravel time differences (∆𝜏) between two acoustic CAT stations. At first, we transfer ∆𝜏 to average̅̅̅̅̅̅velocity in the direction along the transmission lines (𝑉𝐶𝐴𝑇 ) between the two stations using the equation
𝐶0 2 ∆𝜏̅̅̅̅̅̅𝑉
𝐶𝐴𝑇 = − 2𝐿 , where 𝐶0 is the reference sound speed and 𝐿 is the station-to-station distance.
̅̅̅̅̅̅̅̅Meanwhile, the average velocity along the transmission line in the ocean model (𝑉𝑚𝑜𝑑𝑒𝑙 ) can be𝑛 𝑙𝑖̅̅̅̅̅̅̅̅
calculated by 𝑉𝑚𝑜𝑑𝑒𝑙 = ∑𝑖=1 (𝑢𝑖 𝑐𝑜𝑠𝜃 + 𝑣𝑖 𝑠𝑖𝑛𝜃), where 𝑢 and 𝑣 are the eastward and northward𝐿ocean model velocity components, respectively; n is the number of triangular grids that are crossed bythe sound transmission line; the subscript i indicates the ith triangular grid in the sound transmission
line and the corresponding distance is li; 𝜃 is the direction of the sound transmission line, rotating
anticlockwise from due east.
In theory, ̅̅̅̅̅̅𝑉𝐶𝐴𝑇 equals ̅̅̅̅̅̅̅̅
𝑉𝑚𝑜𝑑𝑒𝑙 , so the relationship between CAT data and model states can bewritten as:
𝐶 2 ∆𝜏

− 02𝐿

𝑙= ∑𝑛𝑖=1 𝑖 (𝑢𝑖 𝑐𝑜𝑠𝜃 + 𝑣𝑖 𝑠𝑖𝑛𝜃)𝐿(10)Thus, the relationship between CAT measurements and state vectors of the simulation can be writtenas:
𝑢1
𝑣1
𝐿1𝐿1
𝐿1
𝐿1𝐿1⋮𝑙𝑗_𝑛𝑗𝑙𝑗_𝑛𝑗
𝑙𝑗_1
𝑙𝑗_1
𝐶 2 ∆𝜏𝑢𝑁𝑖
𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝜃
− 0 𝑗
⋯𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝜃
⋯⋯ 𝐿𝑗𝑗𝑗𝑗
𝐿𝑗
𝐿𝑗=(11)
𝐿𝑗
𝐿𝑗𝑗𝑣𝑁𝑖
⋮⋮
⋮
⋮
𝑙𝑚_1
𝑙𝑚_1
𝑙𝑙
𝐶0 2 ∆𝜏𝑚𝑢𝑁⋯ 𝐿 𝑐𝑜𝑠𝜃𝑀 𝐿 𝑠𝑖𝑛𝜃𝑚 ⋯ 𝑚_𝑛𝑚 𝑐𝑜𝑠𝜃𝑚 𝑚_𝑛𝑚 𝑠𝑖𝑛𝜃𝑚 ⋯−()(𝑣 )𝐿𝑚(𝑚𝑚𝐿𝑚𝐿𝑚 )𝑁
where N is the element number in the model mesh; the subscript m denotes the sound transmission
𝐶 2 ∆𝜏1− 0⋯

𝑙1_1

𝑐𝑜𝑠𝜃1

𝑙1_1

𝑠𝑖𝑛𝜃1

⋯

𝑙1_𝑛1

𝑐𝑜𝑠𝜃1

𝑙1_𝑛1

𝑠𝑖𝑛𝜃1

⋯line number and j denotes the jth sound transmission line. Thus, the first matrix in the right hand ofequation 11 is H. Here, we just show some coefficients that are crossed by the transmission lines forexample.

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Figure 3. An example of the sound transmission line between C1 and C2 crossing the triangularmesh (gray grids).
Reference
Evensen G., 1994. Sequential data assimilation with a nonlinear quasi-geostrophic model using
Monte Carlo methods to forecast error statistics, J. Geophys. Res., 99, C5, 10143-10162.
Fang G. H., 1987. Nonlinear effects of tidal friction. Acta Oceanologica Sinica, 6(supp I), 105-122.
Parker B. B., 1991. The relative importance of the various nonlinear mechanisms in a wide range oftidal interactions (review). In: Parker, B. B (Ed.), Tidal Hydrodynamics. Wiley New York, USA, pp.
237-268.

Publication/conference presentation
Oral presentation:
Title: A study on assimilation and application of Coastal Acoustic Tomography
Lecturer: Ze-Nan Zhu
Time: August 1, 2017
Location: Ehime University.

Perspectives in futureThis project help me a lot to improve the research on nonlinear tidal currents and assimilation
scheme based on Coastal Acoustic Tomography data. For the further study, I hope to discuss morewith Prof. Guo to improve the assimilation scheme. Meanwhile, I hope to popularize the applicationsof Coastal Acoustic Tomography in coastal regions.

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