不同三维辐射应力公式在近岸波生流模拟中的效果评估

周浩洋; 李锐; 宋德海; 陈宇

doi:10.12284/hyxb2026000

不同三维辐射应力公式在近岸波生流模拟中的效果评估

doi: 10.12284/hyxb2026000

周浩洋^{1, 3,},
李锐²,
宋德海^{1, 3, ,},
陈宇^{1, 3}

1.
中国海洋大学物理海洋教育部重点实验室，山东青岛 266100
2.
自然资源部北海预报减灾中心，山东青岛 266061
3.
中国海洋大学海洋与大气学院，山东青岛 266100

基金项目: 国家自然科学基金（42476158）；中央高校基本科研业务费专项（202562003）；泰山学者工程专项经费资助（tsqn202211056）。

详细信息

作者简介:
周浩洋（2001—），男，安徽省淮北市人，主要从事浅海动力学研究。E-mail：3wzwhwy@stu.ouc.edu.cn

通讯作者:
宋德海，教授，主要从事近海环流与物质输运研究。E-mail：songdh@ouc.edu.cn

中图分类号: P000.0
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出版历程
- 收稿日期: 2026-01-10
- 录用日期: 2026-04-29
- 修回日期: 2026-04-14
- 网络出版日期: 2026-05-06

Performance of different 3D wave radiation stress formulations in modelling nearshore wave-induced current

Zhou Haoyang^{1, 3
,},
Li Rui²,
Song Dehai^{1, 3
, ,},
Chen Yu^{1, 3}

1.
Key Laboratory of Physical Oceanography, Ministry of Education, at Ocean University of China, Qingdao 266100, China
2.
North China Sea Marine Forecasting and Disaster Mitigation Center, Ministry of National Resources, Qingdao 266061, China
3.
College of Oceanic and Atmospheric Sciences, Ocean University of China, Qingdao 266100, China

摘要

摘要: 为准确模拟近岸波生流的变化过程，本文提出了一种垂向分布权重函数的构建方法，据此整合了现有的部分深度依赖型水平辐射应力公式，并推导出一个垂向分布特征可变的水平辐射应力公式。基于浪流耦合数值模式，对不同辐射应力公式的适用性开展对比评估，并选取四组不同条件的水槽实验数据进行验证。结果表明，不同辐射应力公式对波生流结构的模拟效果存在显著差异；本文提出的公式在模拟波浪增减水、有效波高及跨岸流速等方面表现更优，可降低跨岸垂向剖面流速的均方根误差。针对斜坡地形下水平辐射应力公式存在的垂向动量不平衡问题，依托Roseau型地形模型开展动量平衡诊断分析，结果证实，采用本文提出的水平辐射应力公式模拟坡度地形波生流是可行的，其动量平衡特性与考虑垂向分量的三维辐射应力高阶公式基本一致，同时阐释了其他水平辐射应力公式的垂向动量平衡特征，指出部分公式存在明显的动量不平衡缺陷。
- 波浪辐射应力 /
- 波生流 /
- 浪流耦合数值模式 /
- 权重函数 /
- 动量平衡
Abstract: To accurately simulate nearshore wave-induced currents, this study proposes a general construction method for vertical weighting functions. Based on this method, existing depth-dependent horizontal wave radiation stress formulations are unified, and a modified formulation (Z04m) with tunable vertical distribution characteristics is derived. This formulation features a smooth and continuous vertical profile and allows for flexible adaptation to varying wave conditions and sloping topographies by adjusting a single parameter. Using a developed two-way coupled three-dimensional coastal wave-current interaction model, the applicability of different formulations is comprehensively evaluated. The model is validated against four laboratory flume experiments with varying conditions. The results show that the performance of different radiation stress formulations varies considerably in reproducing wave-induced current structures. The modified formulation outperforms the existing ones in simulating wave setup/setdown, significant wave height, and cross-shore vertical velocity profiles, with a significantly reduced root mean square error in cross-shore velocities compared with other formulations. Regarding the potential vertical momentum imbalance associated with using depth-dependent horizontal radiation stress formulations over sloping bottoms, momentum balance diagnostics based on the Roseau-type topography confirm the feasibility of the modified formulation, which includes only horizontal components. Its momentum balance characteristics are comparable to those of higher-order formulations that incorporate vertical components. The vertical momentum balance characteristics of other depth-dependent horizontal radiation stress formulations are also presented, and some are found to be imbalanced.
- wave radiation stress /
- wave-induced current /
- wave-current coupled model /
- weighting function /
- momentum balance

HTML全文

图 1 标准化辐射应力（S_ii/kE）随相对深度kD和垂向坐标σ的分布

分别以下列公式和条件计算：（a）M15；（b）M08；（c）M21，kH_s = 0.4；（d）Z04；（e）Z04m，n = 0.2；（f）Z04m，n = 5.0；垂向第一个σ层厚度dσ₁取0.05；辐射应力标准化后于不同计算场景下的可比性增强

Fig. 1 Distribution of normalized wave radiation stress (S_ii/kE) with respect to the relative depth kD and the vertical coordinate σ

Results calculated by the following formulations and conditions: (a) M15; (b) M08; (c) M21, kH_s = 0.4; (d) Z04; (e) Z04m, n = 0.2; (f) Z04m, n = 5.0; dσ₁ = 0.05; Standardizing radiation stress improves comparability across different scenarios

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图 2 标准化的${\mathcal{J}(\sigma )} $（${\mathcal{J}(\sigma )} $/kE）随相对深度kD和垂向坐标σ的分布

计算条件和标准化意义同图1

Fig. 2 Distribution of normalized ${\mathcal{J}(\sigma )} $ (${\mathcal{J}(\sigma )} $/kE) with respect to the relative depth kD and the vertical coordinate σ

The calculation formulations and conditions and the meanings of standardization are the same as in Figure 1

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图 3 TK94实验的地形及有效波高、水位的模拟和实测结果

注意部分绘图线条有重叠

Fig. 3 Simulated and observed results of topography, significant wave height and surface elevation for experiment TK94

Note that some plot lines overlap in the figure

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图 4 T01实验的地形及有效波高、水位的模拟和实测结果

Fig. 4 Simulated and observed results of topography, significant wave height and surface elevation for experiment T01

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图 5 RR95实验的地形及有效波高、水位的模拟和实测结果

Fig. 5 Simulated and observed results of topography, significant wave height and surface elevation for experiment RR95

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图 6 CIEM实验的地形及有效波高、水位的模拟和实测结果

Fig. 6 Simulated and observed results of topography, significant wave height and surface elevation for experiment CIEM

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图 7 TK94实验模拟所得跨岸流速剖面及实测流速分布

三个离岸较远剖面未使用；注意部分绘图线条有重叠

Fig. 7 Simulated and observed cross-shore velocity profiles for experiment TK94

Three offshore profiles farthest from the shore have not been used; Note that some plot lines overlap in the figure

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图 8 系数Γ的计算方案示意图

Fig. 8 The calculation method of the coefficient Γ

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图 9 T01实验模拟所得跨岸流速剖面及实测流速分布

Fig. 9 Simulated and observed cross-shore velocity profiles for experiment T01

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图 10 RR95实验模拟所得跨岸流速剖面及实测流速分布

Fig. 10 Simulated and observed cross-shore velocity profiles for experiment RR95

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图 11 CIEM实验模拟所得跨岸流速剖面及实测流速分布

Fig. 11 Simulated and observed cross-shore velocity profiles for experiment CIEM

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图 12 R76实验的地形及有效波高、水位的模拟结果

Fig. 12 Simulations of topography, significant wave height and surface elevation for experiment R76

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图 13 实验R76模拟的（a）标准化净波浪力和（b）减去M15后的标准化净波浪力的垂向分布

Fig. 13 The vertical distribution of (a) normalized net wave force and (b) normalized net wave force (after subtracting M15) simulated in experiment R76

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表 1 模型信息及变量设置

Tab. 1 Numerical experiments setup

变量	TK94	T01	RR95	CIEM
模型跨岸长度x/m	18	29.5	180	50
水平网格间距dx/m	0.2	0.5	2.0	0.5
垂向σ层数kb	41	41	20	41
水深/m	0.4	0.46	4.1	2.65
最大相对深度kD	0.79	1.0	0.87	0.93
地形坡度m	1:35	1:35	1:60 ~ 1:10	−1:4 ~ 1:7
底粗糙度/m	2.5×10⁻³	2.5×10⁻³	2.0×10⁻⁵	2.0×10⁻⁴
入射波有效波高H_s0/m	0.127	0.1524	0.96	0.85
入射波周期T₀/s	2.0	2.0	5.0	4.0
入射波波陡s₀	0.020	0.024	0.025	0.034
入射波Iribarren数ξ₀	0.20	0.18	0.11	0.44
破波系数γ或计算方案	S15	S15	0.78	C22
破波生湍系数α	0.4	0.4	0.4	0.4
POM外模时间步长/s	0.005	0.01	0.02	0.00625
POM内外模时间步长比	5	5	5	4
SWAN时间步长/s	0.05	1.0	1.0	1.0
数据交换时间步长/s	0.05	1.0	1.0	1.0
注：深水Iribarren数ξ₀ = m/s₀^0.5为描述波浪破碎类型的无量纲数^[72]。

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表 2 不同辐射应力公式模拟各实验所得剖面流速的决定系数（R²）和均方根误差（RMSE）

Tab. 2 The coefficients of determinations (R²) and the root mean square errors (RMSEs) for cross-shore velocity profiles simulated by different wave radiation stress formulations

实验	R²/RMSE (m·s⁻¹)
实验	M15	M08	M21	Z04	Z04m
TK94	−/0.079	0.527/0.041	0.525/0.041	0.395/0.046	0.558/0.040
T01	−/0.040	0.270/0.010	0.294/0.010	−/0.021	0.501/0.008
RR95	−/0.103	0.278/0.058	0.276/0.059	−/0.072	0.312/0.057
CIEM	−/0.269	0.328/0.165	0.132/0.187	0.346/0.163	0.412/0.154
注：“−”表示R²为负，加粗为同比最优。‘−’ denotes negative R² values; bolded values indicate comparatively better performance

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A1 不同辐射应力公式模拟的各剖面流速的均方根误差（RMSE）

A1 RMSEs for cross-shore velocity profiles simulated by different wave radiation stress formulations

实验	剖面位置	RMSE（10⁻¹ m·s⁻¹）
实验	剖面位置	M15	M08	M21	Z04	Z04m
TK94	x = 7.275 m	0.700	0.402	0.390	0.301	0.472
	x = 7.885 m	0.811	0.588	0.592	0.307	0.362
	x = 8.495 m	0.948	0.413	0.411	0.659	0.455
	x = 9.110 m	0.725	0.182	0.179	0.492	0.275
	x = 9.725 m	0.720	0.213	0.251	0.525	0.345
T01	d = 13.72 cm	0.361	0.101	0.100	0.188	0.0862
	d = 9.39 cm	0.419	0.101	0.0984	0.212	0.0746
	d = 6.25 cm	0.434	0.0855	0.0836	0.232	0.0782
RR95	x = 65 m	0.300	0.280	0.291	0.277	0.284
	x = 115 m	0.418	0.523	0.522	0.365	0.417
	x = 130 m	0.804	0.479	0.479	0.527	0.424
	x = 138 m	1.45	0.584	0.584	1.04	0.725
	x = 152 m	1.24	0.781	0.782	0.851	0.708
	x = 156 m	1.37	0.738	0.739	0.923	0.706
CIEM	x = 51 m	0.573	0.565	0.543	0.631	0.743
	x = 53 m	1.12	0.758	0.744	0.785	0.693
	x = 54.5 m	2.14	1.09	1.10	1.61	1.16
	x = 55 m	2.69	1.12	1.13	1.92	1.17
	x = 55.5 m	3.94	1.62	1.64	2.09	1.66
	x = 56 m	5.63	3.01	3.02	3.10	3.00
	x = 56.5 m	5.82	3.14	3.03	3.33	2.97
	x = 57 m	3.29	1.08	0.918	1.45	0.938
	x = 58.1 m	1.63	1.51	2.04	0.493	1.49
	x = 59 m	1.99	2.22	2.73	1.59	1.90
	x = 60.2 m	1.95	1.98	2.56	1.62	1.67
	x = 63 m	1.32	0.626	0.777	1.14	0.456
加粗为同比最优 Bolded values indicate comparatively better performance

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B1 部分变量符号说明

B1 Symbols and abbreviations of some variables

变量符号	说明
$\mathcal{J} $	水平辐射应力公式第二项，可由垂向分布函数构造
Φ	垂向分布函数
F、G	基函数，用于构造垂向分布函数
f、g	基函数F、G的导函数
Γ	M21公式系数，控制式中M08、M15组合比

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