# Solution of Shallow-Water Equations by a Layer-Integrated Hydrostatic Least-Squares Finite-Element Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Governing Equations and Numerical Method

#### 2.1. Shallow-Water Equations

_{i}= z

_{i}− z

_{i}

_{−1}, i = 1, …, M, and ${z}_{0}$ = −d(x,y) and ${z}_{M}$ = η(x,y).

_{i}and V

_{i}are the depth-averaged velocity of layer i, i.e., ${U}_{i}=\frac{1}{{h}_{i}}{\displaystyle {\int}_{-{z}_{i-1}}^{-{z}_{i}}udz}$, ${V}_{i}=\frac{1}{{h}_{i}}{\displaystyle {\int}_{-{z}_{i-1}}^{-{z}_{i}}vdz}$, the layer thickness h

_{i}= −z

_{i}+ z

_{i−1}, and S

_{u}and S

_{v}are source terms in x and y momentum equations, respectively. They include surface stress (τ

^{s}) on the top layer, bottom friction (τ

^{b}) on the bottom layer, interface viscos terms, etc.

#### 2.2. θ Time Integration Method

^{n}) and future (t = t

^{n+1}= t

^{n}+ Δt) with time increment Δt = t

^{n+1}− t

^{n}. $\theta $ is a weight between 0.0 and 1.0. $\theta $ = 0.0 corresponds to the fully explicit scheme, $\theta $ = 1.0 corresponds to the fully implicit scheme, and $\theta $ = 0.5 corresponds to the Crank–Nicolson scheme, correspondingly. Integrate Equations (5)–(7) with $\theta $ method from t = t

^{n}to t = t

^{n}+ Δt, and we have:

^{n}and t = t

^{n+1}, and time increment Δt = t

^{n+1}− t

^{n}.

#### 2.3. Least-Squares Finite Element Method

## 3. Results and Discussions

#### 3.1. Von Karman Vortex Shedding

^{−1}, 0.67 ms

^{−1}, and 1.00 ms

^{−1}in the bottom, middle, and top layer, respectively. Free slip boundary condition was employed for both top and bottom boundaries, and a non-slip boundary condition was enforced on the cylinder surface. The resulting Reynolds number was Re = 100, 200, and 300 in the bottom, middle, and top layer, respectively.

#### 3.2. Green Island Vortex Shedding

^{2}, with a diameter of about 13.5 km. The smaller one is Green Island, which is located at 22°40′ N 121°28′ E, about 45 km off the southeast coast of Taiwan. It covers an area about 15 km

^{2}, with a diameter about 5.5 km.

^{−1}), a swirling recirculation was formed behind the island. The size of the recirculated water was about 1–1.5 times the size of the island, as shown in Figure 7b. This value is smaller than the value of 2–3 times the size of the island that was reported in [25] by analyzed satellite images. The speed of the Kuroshio current on the left-hand side downwind of the island was large, reaching 2–3 ms

^{−1}. However, the surface current speed in the recirculation zone was less than 0.6 ms

^{−1}because of the blockage and shield of the island. The edge of the vortex street was identified by edge detection approach of the time-averaged radar images (see Figure 7c). The swing edge of the vortex shedding was time varying (see Figure 7d). The period of the swinging vortex shedding was estimated to be about 13 h.

#### 3.3. Numerical Study

^{−1}in the top layer (0 m to −200 m), 0.5 ms

^{−1}in the middle layer (−200 m to −400 m), and 0.3 ms

^{−1}in the bottom layer (−400 m to −700 m). The influence range of the Kuroshio current is up to −800 m: water below −800 m is essentially motionless.

^{−1}in the top layer, 0.67 ms

^{−1}in the middle layer, and 0.33 ms

^{−1}in the bottom layer. This information was used in the simulations. Kuroshio enters the study area from the south boundary and exits the north boundary freely. On the west boundary, to the southeast of Taiwan, slip boundary condition was employed, and on the east boundary no normal flux condition was employed. A non-slip boundary condition was specified for both Orchid Island and Green Island.

#### 3.4. Discussion

## 4. Conclusions

^{−1}) was small, due to the blockage and shield of the island (agreeing with the ADCP measurement [22] and the X-band radar measurement [64]). The streamwise and spanwise size of the Green Island vortex street of the top layer were 50–60 km and 10–15 km (agreeing with the value of 50–60 km and 10–20 km by the analysis of satellite imageries [24,25], and larger than value of 40–50 km and 10–20 km by the one-layer modelling results of [28,29,30,34]). The period of Green Island vortex shedding was 14.5 h (larger than the value of 13 h by the X-band radar measurement [64]). It was revealed that Green Island vortices were affected by the upstream Orchid Island vortices. This result was supported by the field measurement of [22].

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Study domain and computational meshes: (

**a**) global view and (

**b**) close-up local view around the cylinder. Point P is the monitor point where flow variables are recorded. (Unit measurement for x and y axis is m).

**Figure 2.**Contours of U (

**a**–

**c**) and V (

**d**–

**f**) of the von Karman vortex street of the top, middle, and bottom layer, respectively (U–contours range from −0.008 to 0.35 ms

^{−1}with an interval 0.001 ms

^{−1}; V–contours range from −0.016 to 0.015 ms

^{−1}with an interval 0.001 ms

^{−1}).

**Figure 3.**Contours of vorticity (each plot consisted of 30 contours): (

**a**) ranges from −0.165 to 0.122 of the top layer; (

**b**) ranges from −0.089 to 0.088 of the middle layer; and (

**c**) ranges from −0.032 to 0.025 of the bottom layer, respectively.

**Figure 4.**Time history of η, U and V of the monitor point P (see Figure 1b) of (

**a**) top, (

**b**) middle, and (

**c**) bottom layers, respectively.

**Figure 5.**(

**a**) A composite ERS-1 SAR image of Kuroshio-induced ocean vortex of Orchid Island and Green Island taken at 17:01 UT 25 September (lower, with lighter colour background) and 19:02 UT 2 October 1996 (top, with darker colour background). (

**b**) Zoomed-in Green Island vortex (source: modified from Figure 2 of Zheng and Zheng [24]).

**Figure 6.**Kuroshio-induced Green Island vortex by MODIS Aqua images at (

**a**) 05:10 UTC on 4 April 2015, (

**b**) 05:05 UTC on 16 June 2015, (

**c**) 05:15 UTC on 30 June 2015, and (

**d**) 05:05 UTC on 2 July 2015. (Source: Figure 7 of Hsu, et al. [25]).

**Figure 7.**(

**a**) Photo of the X-band marine radar observation system located to the northwest of Green Island. (

**b**) Surface current speed near Green Island at 18:00, 19 August 2014. (

**c**) Edge of vortex shedding detected from time averaged radar images. (

**d**) Time varying edge of vortex shedding on 21 August 2014.

**Figure 8.**(

**a**) Schematic illustration of the study area. (

**b**) Computational meshes of top, middle and bottom layer. (

**c**) Close-up view of fine meshes near Orchid Island and Green Island where variations of flow and waves are significant.

**Figure 9.**Contours of (

**a**) U (upper panel) and (

**b**) V (lower panel) of Orchid Island vortex and Green Island vortex of the (1) top (left panels), (2) middle (central panels), and (3) bottom (right panels) layer. (U–contours range from −1.5 to 1.5 ms

^{−1}with an interval 0.1 ms

^{−1}; V-contours range from −1.2 to 2.6 ms

^{−1}with an interval 0.1 ms

^{−1}).

**Figure 10.**Streamlines of Orchid Island vortex and Green Island vortex at (

**a**) the top, (

**b**) middle, and (

**c**) bottom layer.

**Figure 11.**Contours of vorticity of the Orchid Island vortex and Green Island vortex at (

**a**) the top layer with ranges from −0.0022 to 0.0033, (

**b**) the middle layer with ranges from −0.0013 to 0.0020, and (

**c**) the bottom layer with ranges from −0.0006 to 0.0014 (there are 30 contours in each plot).

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Liang, S.-J.; Doong, D.-J.; Chao, W.-T.
Solution of Shallow-Water Equations by a Layer-Integrated Hydrostatic Least-Squares Finite-Element Method. *Water* **2022**, *14*, 530.
https://doi.org/10.3390/w14040530

**AMA Style**

Liang S-J, Doong D-J, Chao W-T.
Solution of Shallow-Water Equations by a Layer-Integrated Hydrostatic Least-Squares Finite-Element Method. *Water*. 2022; 14(4):530.
https://doi.org/10.3390/w14040530

**Chicago/Turabian Style**

Liang, Shin-Jye, Dong-Jiing Doong, and Wei-Ting Chao.
2022. "Solution of Shallow-Water Equations by a Layer-Integrated Hydrostatic Least-Squares Finite-Element Method" *Water* 14, no. 4: 530.
https://doi.org/10.3390/w14040530