GUO Zheng, CAO Anzhou, and WANG Jianfeng

Tide-Induced Mixing in the Bottom Boundary Layer in the Western East China Sea

GUO Zheng1), CAO Anzhou2), and WANG Jianfeng3), 4), *

1) Marine Science and Technology College, Zhejiang Ocean University, Zhoushan 316022, China 2) Ocean College, Zhejiang University, Zhoushan 316021, China 3) Institute of Oceanology, Chinese Academy of Sciences, Qingdao266071, China 4) Center for Ocean Mega-Science, Chinese Academy of Sciences, Qingdao 266071, China

The East China Sea (ECS) boasts a vast continental shelf, where strong tidal motions play an important role in the substance transport and energy budget. In this study, the tide-induced mixing in the bottom boundary layer in the western ECS is analyzed based on records measured by moored acoustic Doppler current profilers from June to October 2014. Results show that the M2tide is strong and shows a barotropic feature, whereas the O1tide is much weaker. Based on the M2tidal currents, the eddy viscosity in the bottom Ekman boundary layer is estimated with three schemes. The estimated eddy viscosity values vary within 10?4–10?2m2s?1, reaching a maximum at approximately 5m height from the bottom and decreasing exponentially with the height at all three stations. Moreover, the shear production of turbulent kinetic energy is calculated to quantify the mixing induced by different tidal constituents. The results show that the shear production of the M2tide is much stronger than that of the O1tide and shows a bottom intensified feature.

tides; East China Sea; bottom boundary layer; mixing; eddy viscosity

1 Introduction

As a basic form of seawater motion, tides play an important role in ocean mixing (Munk and Wunsch, 1998). Studies indicate that one-third of barotropic tidal energy is dissipated in the deep ocean, whereas the remaining energy is lost directly to bottom friction in shallow seas and on the continental shelf (Egbert and Ray, 2001; Carter, 2012). Strong tidal currents are reduced near the bottom, thus creating a vertical shear. Shear-generated turbulence enhances vertical mixing, contributing to upward nutrient transportation and biological activities (Thorpe, 2007). Therefore, the knowledge of turbulent mixing induced by tidal currents near the bottom is important for understanding both physical motions and biochemical processes.

The eddy viscosity is an important parameter for quantifying the mixing structure in the bottom boundary layer. The eddy viscosity is estimated from the observed quantities, such as current velocity. Attempts have been madewhere eddy viscosity profiles are specified as vertically constant or linear (Prandle, 1982; Soulsby, 1983), or investigated relying on the turbulence closure model (Werner, 2003). Yoshikawa and Endoh (2015) compared three methods for estimating turbulent mixing structures. These methods are considered because the mean velocity profiles are determined by the mean eddy viscosity in the bottom boundary layer. Therefore, the mean eddy viscosity can be estimated by solving the Ekman balance equations using the least squares methods. Cao(2017) proposed an alternative scheme consisting of the bottom Ekman boundary layer model and its adjoint model as well as a minimization algorithm. They validated its performance with twin experiments but did not apply it to the practical context for lack of observations.

The East China Sea (ECS) boasts a vast continental shelfand strong tidal motions (Kang, 2002; Niwa and Hibiya, 2004). Many observations have been made to investigate the turbulence in this area (Liu, 2009; Lozovatsky, 2012, 2015; Song, 2019; Wang, 2020); however, these observations are still limited compared to the vastness of the ECS. Therefore, the investigation of tidal motions and turbulent mixing induced by tides in various regions is important and beneficial in deepening our understanding of the dynamics in the ECS. In this study, based on three records of the current velocity measured by acoustic Doppler current profilers (ADCPs) in the western ECS, we estimated the eddy viscosity profiles in the bottom Ekman boundary layer for two purposes. First, we need to assess the performance of the scheme proposed by Cao(2017) in the real sea and compare it with the performance of other schemes. Second, we needto provide a turbulent mixing structure in the western ECS near Yangtze Estuary.

The paper is organized as follows. The observations and methods for estimating eddy viscosity are introduced in Section 2. In Section 3, characteristics of tidal currents andturbulence mixing induced by tides at three stations are presented. Section 4 summarizes the paper.

2 Data and Methods

2.1 Current Data

Current data of Wang(2019) were used in this study. The data were measured using moored ADCPs at three stations (Fig.1): Station DH1 located at 122?56.149?E, 31?59.803?N, where the water depth is 34.3m; Station DH2 at 122?56.882?E, 31?01.450?N, with a depth of 47.0m; and Station DH3 at 122?50.684?E, 28?54.826?N, with a depth of 61.8m. The ADCPs deployed at the three stations operated from June 12 to August 22, June 11 to September 4, and June 12 to October 20, 2014, respectively. All velocity profiles were collected every 30 minutes with a 2.0-m bin size. The first available bin of ADCP is 4.8m above the bottom of the ocean.

Fig.1 Locations of three ADCP stations (DH1, DH2, and DH3) in the ECS.

2.2 Eddy Viscosity Estimation Schemes

The measured current velocity was used to estimate the eddy viscosity because the mean velocity profiles are de- termined by the mean turbulent mixing in the bottom boundary layer where the turbulent mixing term is dominant in the momentum equations. The governing equation for velocity in the bottom Ekman boundary layer is

where=+iis the complex horizontal current vector,is the time,is the Coriolis frequency,is the height above the seafloor, andis the eddy viscosity coefficient. Represent a tidal current of frequencyby

According to Yoshikawa and Endoh (2015),

where>0 corresponds to the counterclockwise-rotating current component and<0 corresponds to the clockwise-rotating component;A(A) andg(g) are the conventional tidal harmonic coefficients of the zonal (meridional) velocities. If the tides are barotropic in the interior layer, the boundary layer component() can be calculated as follows:

The discretized form of Eq. (5) is

whererepresents the tidal component (,=1 is the clockwise component of the M2tide) andis the vertical level. The first two schemes used to estimate the eddy viscosity are based on the above equation and the least squares method.

2.2.1 Scheme 1

Scheme 1 was first used by Yoshikawa(2010). It assumes that errors are included in Eq. (6):

Note thatε,is a complex number. The sum of the squaredε,is defined as

where*denotes the complex conjugate. To minimize, it is required that

The above equations are solved to obtainμfor all vertical levels.

2.2.2 Scheme 2

Scheme 2 was proposed by Yoshikawa and Endoh (2015). In this scheme, Eq. (6) is integrated fromztoz=, and the errors are assumed in the integrated equations:

Taking partial derivatives ofwith respect toμand setting them equal to zero,μcan be obtained by solving the resulting equations:

2.2.3 Scheme 3

The third scheme used in the study is based on the adjoint method. Details can be found in Cao(2017), and this scheme is also summarized as follows. The scheme is composed of the bottom Ekman boundary layer model and its adjoint model, as well as a minimization algorithm. The eddy viscosity profiles in the bottom Ekman boundary layer can be estimated by assimilating observations (harmonic constants of tidal currents).

The bottom Ekman boundary layer model is introduced above (Eqs. (1)–(6)).

The adjoint model development begins with the cost function calculation:

The Lagrangian function is constructed as

whereλ=λ,+iλ,is the complex adjoint variable ofW.

with boundary conditions

Assuming thatWis a single-valued function of, when the simulatedWapproaches the observation, the estimatedshould be close to its real value. Therefore, there exists a minimization problem, min:(), which can be solved by adopting the gradient descent algorithm. The gradient of the cost function with respect to the eddy viscosity coefficient is

where Re indicates the real component. Considering that the eddy viscosity coefficients can vary by several orders of magnitudes in the bottom Ekman boundary layer, the gradient of the cost function with respect to the logarithm of the eddy viscosity coefficient is used in the minimization algorithm:

According to the gradient descent algorithm,

After 1000 iterations, eddy viscosity coefficientsare greatly optimized as the cost function decreases.

3 Results

3.1 Tidal Characteristics

Harmonic analysis was first conducted on current velocity data to describe the characteristics of tides and to choose the tidal constitutes suitable for estimating the eddy viscosity. The t_tide toolbox (Pawlowicz, 2002) was applied to perform harmonic analysis (HA), which resolved 35 constituents based on the automated decision tree of Foreman (1977). Notably, only the M2and O1constituents were further analyzed according to the observing periods (for approximately 2–4 months). Additionally, the results of O1at DH3 were ignored because of their small magnitudes and relatively large confidence intervals. Fig.2 shows the vertical structures of the M2and O1tidal ellipses at the three stations. At first glance, we find that the M2tide is much stronger than the O1tide, which is consistent with previous studies (, Kang, 1998; Fang, 2004). The M2tide is dominated by the barotropic component, displaying a uniform character throughout the water column except near the bottom, which is consistent with previous observations (Wang, 1999). The tidal ellipses at stations DH1 and DH2 are circular and aligned in the northwest-southeast directions. The M2tide at station DH3 is weaker than those at the other two stations and shows a more circular shape. The O1tide shows a different feature from the M2tide. The O1tides are surface intensified, and the variance of their Greenwich phase with depths is relatively large. The O1tide is more rectilinear compared to the M2tide. The major axis orientation of the O1tide at station DH1 is mainly north-south. At station DH2, the direction changes downward from northeast-southwest to northwest-southeast. In a word, the O1tides are much weak- er and show more changes in the vertical direction.

Fig.2 Vertical structures of the M2 and O1 tidal ellipses at the three stations. The ellipses are centered at their corresponding measurement depths. The lines from the center show the Greenwich phases. Note that the x-axis and y-axis limits in these subfigures are different.

Fig.3 shows the vertical structures of the M2, O1, and mean currents (time average of the observed velocity during the entire period) at the three stations. The tidal velocity vector is displayed when the velocity at a 25-m height reaches its maximum. The classic Ekman solutions are also plotted for reference. They are calculated with a constant eddy viscosity of 10?3m2s?1as follows:

where

The features of the M2tidal currents are consistent with the classical Ekman theory in the bottom boundary layer. The currents rotate clockwise downward, and their magnitudes change slightly above around 20m and decrease near the bottom. The O1tidal currents show a counterclockwise rotation with depth, which can be explained by the classical Ekman theory. However, their magnitude is much weaker than the M2tide and deviates further from the corresponding theoretical solution. The mean currents are also feeble, which are less than 5cms?1within 20m height from the bottom. At stations DH1 and DH2, the mean current rotates clockwise with the increase of water depth, which is opposite to the direction predicted by the Ekman theory and can be attributed to small values (Yoshikwa, 2010). At station DH3, the rotation of the mean current is consistent with the theory.

3.2 Eddy Viscosity in the Bottom Boundary Layer

According to Yoshikawa and Endoh (2015) and Cao(2017), the use of a stronger tidal constituent improves the eddy viscosity estimation. We use the M2tide to estimate the eddy viscosity because it is strong and shows a typical bottom Ekman spiral.

Fig.3 Vertical structure of the (top) mean currents, (middle) M2, and (bottom) O1 at stations (left) DH1, (middle) DH2, and (right) DH3. The tidal velocity vectors at various heights (label) are displayed when the velocity reaches its maximum at the height of 25m. The analytical Ekman solutions are also shown by black lines, with colored dots representing the height from the bottom.

The boundary layer height,, needs to be given a priori to calculate the Ekman boundary layer velocity. Although depths at the three stations vary from 30 to 60m, the vertical structures of the M2tide in Figs.2 and 3 show that the velocity varies above the 20m height, indicating thatat all these stations is similar. As suggested by Yoshikawa and Endoh (2015), we tried various values forand set the top of the boundary layer at 25m for all three stations. Fig.4 shows eddy viscosity profiles estimated fromthe three schemes at the ADCP stations. Overall, these profiles have a similar pattern–they show the maximum values at approximately 5m height and decrease exponentially with the height (except that estimated by scheme 3 at EH3, which increases gradually above 20m). The exponent decay rates are different because of various density profiles. At stations DH1 and DH2, results of the three schemes are close below the height of 20m; however, near the top of the boundary layer, the eddy viscosity estimated by scheme 1 decreases to a value smaller than the other two schemes by 1–3 orders of magnitude. At station DH3, profiles estimated from the three schemes depart from each other above the height of 15m, and their difference is larger compared to the other two stations. According to Yoshikawa and Endoh (2015), scheme 2 is susceptible to the balance error, which is relatively large in ADCP data observed in the ECS and which can account for the difference between estimation using scheme 2 and other schemes. The eddy viscosity profiles, even if estimated using the same schemes, differ at each station and from previous studies in the ECS (, the first two schemes are used in Yoshikawa and Endoh (2015) to estimate eddy viscosity profiles at two stations in the ECS, the one located at 31?45?N, 127?25?E and the other at 31?45?N, 125?30?E), indicating that turbulence characteristics in the bottom boundary layer are site-independent and it is important to make more quantifications based on observations.

Velocity profiles were reproduced with the estimated eddy viscosity (according to Eq. (6)) and compared with the observations for validation because the true values of the eddy viscosity are unknown. The magnitudes of the M2tide are shown in Fig.5. All the reproduced velocities show a decreasing trend with height, which is similar to the observations. The profiles estimated with schemes 1 and 3 are much closer to the observations than those estimated with scheme 2. The depth-averaged root mean square of the errors of schemes 1 and 3 are smaller than that of scheme 2 by one order of magnitude at DH1 and DH2 (, 0.27, 1.33, and 0.45cms?1for the three schemes, respectively, at DH1). At DH3, the error of scheme 2 is also the largest. The results indicate that the eddy viscosity profiles estimated from schemes 1 and 3 are close to the true values.

Based on the eddy viscosity estimated by scheme 3, the shear production of the turbulent kinetic energy of the M2and O1tides is calculated as follows:

The results are displayed in Fig.6. Both show a periodic character, which is consistent with the corresponding tidal constituents. However, their characteristics are different from each other. In terms of magnitude, the M2shear production is larger than that of the O1tide by 1 order of magnitude at each station. In terms of the vertical structure, the M2shear production intensifies as it approaches the bottom, which is consistent with the theory of the bottom Ekman boundary layer. In contrast, the O1shear production is larger within the layer of height 20–25m.

Fig.4 Eddy viscosity profiles estimated from three schemes at stations (a) DH1, (b) DH2, and (c) DH3.

Fig.5 Magnitude of the M2 tide reproduced with schemes 1, 2, and 3 at three stations.

Fig.6 Shear production of the TKE of the M2 and O1 tides in a tidal period at three stations.

4 Summary

In this study, tidal characteristics at three stations in the ECS are analyzed, and the velocity profiles are used to estimate the eddy viscosity in the bottom boundary layer. The M2tide takes a dominant role and shows a barotropic feature, whereas the O1tide is much weaker and shows more changes in the vertical direction. Based on the M2tide, three schemes for estimating the eddy viscosity are compared. The results show that schemes 1 and 3 perform well in the western ECS near the Yangtze River. The estimated eddy viscosity shows the maximum values at approximately 5m height and decreases exponentially with the height. The values vary within 10?4–10?2m2s?1. Shear production of the turbulent kinetic energy of the M2and O1tides shows unique characteristics in terms of structure and magnitude. The shear production of the M2tide is much stronger than the diurnal tide and bottom intensified, whereas that of the O1tide shows larger values within the 15–25m height range.

Acknowledgements

This study was supported by the Zhejiang ProvincialNatural Science Foundation of China (No. LY21D060005), the Shandong Provincial Natural Science Foundation (No. ZR2022MD082), the Joint Project of Zhoushan Municipality and Zhejiang University (No. 2019C810060), the Open Fund Project of Key Laboratory of Marine Environmental Information Technology and the Strategic PriorityResearch Program of Chinese Academy of Sciences (No. XDA19060201).

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(May 14, 2021; revised August 23, 2021; accepted October 27, 2021)

? Ocean University of China, Science Press and Springer-Verlag GmbH Germany 2023

Corresponding author. E-mail: jfwang2013@qdio.ac.cn

(Edited by Xie Jun)