## Abstract

With the method of a wind tank experiment, the real scenario of lakes with horizontal and vertical circulation of wind-induced flows is considered, and the features of wind wave height and its distribution in the different conditions of wind blowing distance, wind speed and water depth are studied systematically. Afterwards, comparison of the wave height distributions derived directly from experiment and the typical wave height distribution models show that some defects exist in typical wave height distribution models when describing wind wave height distribution in the wave growth stage. On this basis, we propose a new distribution model which is suitable for the description of wind wave height during the growth stage, and the model parameters are acquired with the programming solution method. Finally, the model is further optimized by relating *B* to *σ*_{a}, and *H*_{s} to *σ*_{a}. Comparison results of the optimized model and the typical ones show that the optimized model has advantages in calculation accuracy and convenience of use.

## INTRODUCTION

Shallow lakes, as one of the most fragile ecosystems, commonly have water environmental problems (Qin *et al.* 2009; Islam *et al.* 2012; Havens & Ji 2018). Wind-induced waves and wind-induced currents (vertical and horizontal circulation) are important hydrodynamic processes of shallow lakes, which affect the suspension and transport of sediment, cycling of nutrients and pollutants (Suntoyo *et al.* 2007; Longo 2009). A wind-induced wave is a special water wave as it is generated by wind stress, and its characteristics are determined by the coupling process in boundary layers between the air and water. Surface wave motion, the local water surface wind-drift and water turbulence are important elements of the wind-induced wave, which makes the wind-induced wave complex, especially for shallow lakes, where the limited depth enhances the interaction between free surface and lakebed, and increases the wave breaking rates (Babanin *et al.* 2001; Longo *et al.* 2002).

Wind-induced waves in shallow lakes are always young for the limited blowing distance (lake area), which is named as a growing wave in a wind-driven wave regime by Sullivan & McWilliams (2010). A wind-induced wave, consisting of various frequency wavelets, is a multiscale system, and its components gradually propagate with blowing distance under the effect of non-linearity and wave–wave interaction (Longo *et al.* 2002). Generally speaking, the lower frequency components (large scale) of the water elevation spectrum show significant growth, while relative high frequency components (small scale) generally retain their amplitude along the blowing distance (Liberzon & Shemer 2011).

There are many aspects to understanding wave characteristics in shallow lakes (Banner & Song 2002), of which the wave height statistical distribution is one of those of interest (Chalikov & Bulgakov 2017). Research on the wave height statistical distribution of shallow lakes is fundamental to exploring the hydrodynamic process of shallow lakes (Longo 2012), which also has theoretical and engineering consequence for the aspect of building design and boating safety in shallow lakes (Katsardi *et al.* 2013).

Triad interactions and depth-induced wave breaking become relevant in shallow water, which causes profile distortion (bound spectral components) with steep crests and shallow troughs, in contrast to the Gaussian waves in deep water (Clavero *et al.* 2016). Thus, the surface elevation in shallow water can no longer be regarded as a narrow-banded linear Gaussian process, which makes wave behaviour in shallow water more complicated and knowledge of wave height statistical characteristics more limited (Battjes & Groenendijk 2000).

Many efforts have been made to model wave height distribution characteristics of shallow water waves. Forristall (1978) presented the Weibull distribution using field-observed data (later called the Forristall distribution). Glukhovskii (1966) took into account the effect of depth-limiting on wave height and modelled the shallow water wave height with a new Weibull distribution. Klopman (1996) optimized the Weibull distribution presented by Glukhovskii based on experimental data (later called the Grukhovskii distribution). However, whether these models work well for young waves in the wind-driven wave regime, where the wind-induced wave gradually propagates with the wind blowing distance, is yet to be investigated.

Understanding the fundamental processes that connect winds, waves, and currents is partly inhibited by the challenging task of acquiring outdoor observations: sensor technology, cost, and broad band variability of the atmosphere all limit the acquisition of wave observational data sets in the wave surface layers (Sullivan & McWilliams 2010). Therefore, many scholars have used wind tanks to study the characteristics of wind waves (e.g., Waseda *et al.* 2001; Caulliez *et al.* 2008; Longo 2012; Longo *et al.* 2012a, 2012b; Latheef & Swan 2013; Banerjee *et al.* 2015; Clavero *et al.* 2016). However, only vertical circulation of the water body was considered in their experimental design, and the horizontal circulation that exists in a wide range of natural lakes is ignored (England *et al.* 2014), which makes the hydrodynamic process different from a real lake scenario, e.g., add the water surface average slope and amplify the role of vertical circulation, etc.

The objective of the current study is to study wind-induced wave height characteristics in a purpose-designed wind wave tank, where the vertical and horizontal circulations can both be simulated simultaneously. The contents of this article are arranged as follows. Experimental facilities and manner are introduced in the next section. Wave height distribution characteristics are presented in the first part of the section after that; on this basis, a comparison is carried out between experiment and the typical wave distribution model, and a detailed analysis of the comparison results are implemented in the following subsection; next a modified wave height Weibull distribution model (Md-Weibull) is introduced based on experimental data and the programming solution method; and then, the Md-Weibull model is used to describe the wave height distribution, and the comparison results between the Md-Weibull model and typical models are discussed. Conclusions are given in the final section.

## MATERIALS AND METHODS

### Wind wave tank

The study was carried out in the wind wave tank at the Hydraulic Engineering Laboratory of Nanjing Hydraulic Research Institute. The effective length of the tank is 22.5 m (width 1 m, height 1.2 m) with a flat bottom. The diagrammatic sketch of the tank is shown in Figure 1. The cross-section of the water is rectangular, and its top is closed by a circular-shaped cover. A vertical partition is set up at the axis of the tank, which divides the tank horizontally into two flow channels with the same width. We set a cover on the upper side of one of the flow channels, and opened an overflow hole at the end of the vertical partition. Under the induction of the wind, a wind-induced wave and flow is formed in one side of the flow channel, and an opposite flow is formed in another flow channel. The design achieves the simulation of the vertical and horizontal circulations simultaneously in the scenario of a real lake wind wave flow. An air suction fan is located at the end of the tank. A wave-breaking plate is arranged at the end of tank to absorb the incoming waves and avoid wave reflection.

### Instrumentation

Water surface elevation was monitored using capacitance wave gauges, with a measuring accuracy of 1 mm, and a sampling frequency of 100 Hz. In view of the non-uniform wind at the transition section of wind import and export, the water surface fluctuation measuring points were set from F6 to F21 (Figure 1). Wind speed was collected by hot wire anemometers, with a measuring accuracy of 0.01 m/s, and a sampling frequency of 2 Hz. The anemometer was placed at the F10 section. The lowest measurement point was 0.2 m from the initial water surface, and measurement points were set with an interval of 0.1 m in the vertical direction.

### Experiment group

Designed orthogonal test groups are shown in Table 1. In order to study the restrictive effect of the bottom wall on the development of wind wave patterns, three initial water depths were selected. The average wind speed at 0.2 m above the initial water depth (*d*_{0}) is used as the representative wind speed (*V*_{s}) of the experiment group. Preliminary experiments show that the time of wave stabilization is generally within 5 minutes, so we set the test time of each group as 10 minutes.

Case . | d_{0} (m)
. | V_{s} (m/s)
. | ||
---|---|---|---|---|

1 | 0.15 | 5.11 | 8.42 | 11.60 |

2 | 0.30 | 5.37 | 8.96 | 11.93 |

3 | 0.45 | 5.61 | 9.21 | 12.28 |

Case . | d_{0} (m)
. | V_{s} (m/s)
. | ||
---|---|---|---|---|

1 | 0.15 | 5.11 | 8.42 | 11.60 |

2 | 0.30 | 5.37 | 8.96 | 11.93 |

3 | 0.45 | 5.61 | 9.21 | 12.28 |

### Data preprocessing

Wave height (*H*) is acquired with the upper zero point method, and the average wave height (*H*_{a}) and the significant wave height (*H*_{s}) are also calculated. In addition, each wave height is subdivided into the crest height (*H*_{c}) above the average water level and the trough depth (*H*_{t}) below the average water level, and their corresponding average values *H*_{ca}, *H*_{ta} are calculated, wherein, *H*_{a} = *H*_{ca} + *H*_{ta}.

## RESULTS AND DISCUSSION

### Wave height characteristics

Due to the limitation of the wind blowing distance (*S*), the wave height distributions at different blowing distances are different. Taking the *d*_{0} = 0.30 m, *V*_{s} = 5.37 m/s experiment group as an example, the cumulative probability distributions (*P*) of *H*, *H*_{c} and *H*_{t} for different blowing distances are shown in Figure 2.

Figure 2(a) shows that the wave height cumulative frequency distribution rises gradually with the increase of the blowing distance, indicating that the wave scale increases gradually with the blowing distance. Figure 2(b) shows that the crest height cumulative frequency distribution also rises gradually with the increase of the blowing distance. Nevertheless, in Figure 2(c), the trough depth cumulative frequency distribution increases in the early stage (before section F15) as the blowing distance increases, while being basically the same after section F15.

In order to analyze relations of *H*, *H*_{c} and *H*_{t} with the blowing distance, statistical averages (*H*_{a}, *H*_{ca}, *H*_{ta}) are summarized in Table 2. It can be seen that *H*_{a} and *H*_{ca} simultaneously increase along the whole blowing distance, while *H*_{ta} gradually increases before section F15 and then remains virtually constant at larger blowing distances*.* This phenomenon indicates that the inhibition effect of the bottom wall on the vertical scale of wind waves is mainly reflected in the constraints on the deep development of the wave trough, and this restriction only emerges when the wave evolves to a certain scale.

S (m)
. | 6 . | 7.5 . | 9 . | 10.5 . | 12 . | 15 . | 18 . | 21 . |
---|---|---|---|---|---|---|---|---|

N_{waves} (#) | 3,799 | 3,184 | 2,722 | 2,156 | 2,056 | 1,935 | 1,760 | 1,530 |

H_{a} (mm) | 11.6(4.4) | 13.6(5.6) | 14.9(5.8) | 16.1(6.5) | 18.1(7.3) | 20.9(7.2) | 22.9(7.9) | 25.4(8.1) |

H_{ca} (mm) | 6.9(2.8) | 8.0(3.5) | 8.8(3.6) | 9.2(4.0) | 11.5(4.6) | 13.0(4.6) | 15.1(5.1) | 17.6(5.3) |

H_{ta} (mm) | 4.7(2.0) | 5.6(2.5) | 6.2(2.7) | 6.9(3.0) | 6.6(3.2) | 7.9(3.3) | 7.9(3.5) | 7.8(3.6) |

S (m)
. | 6 . | 7.5 . | 9 . | 10.5 . | 12 . | 15 . | 18 . | 21 . |
---|---|---|---|---|---|---|---|---|

N_{waves} (#) | 3,799 | 3,184 | 2,722 | 2,156 | 2,056 | 1,935 | 1,760 | 1,530 |

H_{a} (mm) | 11.6(4.4) | 13.6(5.6) | 14.9(5.8) | 16.1(6.5) | 18.1(7.3) | 20.9(7.2) | 22.9(7.9) | 25.4(8.1) |

H_{ca} (mm) | 6.9(2.8) | 8.0(3.5) | 8.8(3.6) | 9.2(4.0) | 11.5(4.6) | 13.0(4.6) | 15.1(5.1) | 17.6(5.3) |

H_{ta} (mm) | 4.7(2.0) | 5.6(2.5) | 6.2(2.7) | 6.9(3.0) | 6.6(3.2) | 7.9(3.3) | 7.9(3.5) | 7.8(3.6) |

### Contrastive analysis for the distribution of wind-induced wave height characteristics

We select the maximum and minimum values of water depth and wind speed as representatives, analyze the characteristics of wave distribution, and compare them with the Rayleigh distribution, Forristall distribution and Grukhovskii distribution.

The Rayleigh distribution has the largest deviation from experimental data than the other two models. As shown at the test sections F7.5 and F12 in Figure 3(a), the Rayleigh distribution can characterize the wave height distribution well when the wave height is small (*P* > 0.2), but for larger wave heights, the deviation is brought out (*P* < 0.2). In particular, for waves with larger blowing distance and more mature shapes, the Rayleigh distribution also underestimates the probability of a smaller wave height. As shown from the test section F21 in Figure 3(a), when *P* < 0.17, the Rayleigh distribution overestimates the probability distribution of large wave height; but when *P* > 0.17, the Rayleigh distribution underestimates the probability distribution of small wave height. This indicates that the wave form is gradually affected by nonlinearity during its development, and the wave height can no longer be fully described using the Rayleigh distribution.

The Forristall distribution and Grukhovskii distribution, which consider the effect of water depth, have better conformity of wave height. As shown at F12 in Figure 3(a), the two distributions fit well with the experimental data. However, as for the waves which are more mature at long blowing distance, some deviations still exist, which indicate that both distribution models have disadvantages of underestimating the probability of exceeding the value of the smaller wave height. For instance, the phenomenon can be clearly seen from section F21 in Figure 3. In addition, when water depth is consistent, the larger the wind speed, the larger the estimation error of the models (F21 in Figure 3(a) and 3(b)). Furthermore, it can be found that the shallower the water depth, the larger the estimation deviations of the models (F21 in Figure 3(a) and 3(c)). This suggests that with higher wind speed, shallower water depth and larger blowing distance, the nonlinear feature of the wave is more significant, and the Forristall distribution and the Grukhovskii distribution still have disadvantages when estimating wind-induced wave height distribution in their growth stage.

### Establishment of wave height distribution model

*A*and

*B*is key to the building of model (1). We use the method of the nonlinear programming solution, with Equation (2) serving as the objective function, and

*A*and

*B*as the decision variables, to calculate each wave height measuring point

*A*and

*B*in all experiment groups:

where *n* is the total number of groupings to the wave height sequence, is the cumulative total frequency of the *i*-th wave height corresponding to the experimental data, and is the cumulative total frequency of the *i-*th wave height calculated by Equation (1).

*A*and

*B*with the mean value of surface fluctuation (

*σ*

_{a}). We find that

*A*is between −1.8 and −2.0, and is not sensitive to the change of

*σ*

_{a}, therefore, the average value −1.94 is used as the optimal value of

*A*, which is a little bit bigger than −2 (Rayleigh distribution value). Scatter data of

*B*and

*σ*

_{a}are plotted in Figure 4(b), and their relational expression is acquired based on the particle swarm optimization method:

Note that *B* and *σ*_{a} are positively correlated, for the light wind region where water surface fluctuation is weak, and the wave is relatively small, which can be considered as a deep water wave, then, the *B* value in Equation (3) tends to be 2.23.

*H*

_{s}and

*σ*

_{a}is established, as shown in Figure 4(c) and Equation (4). This

*H*

_{s}/

*σ*

_{a}value is almost the same as the wind-induced waves in Nagoya port (Goda 1979:

*H*

_{s}/

*σ*

_{a}≈ 3.79) and hurricane-generated waves in the Gulf of Mexico (Forristall 1978:

*H*

_{s}/

*σ*

_{a}≈ 3.77):

### Discussion of wave height distribution model

In order to verify the applicability of the Md-Weibull, four extremes of the maximum and minimum values of water depth and wind speed are taken into account (Figure 5). Results show that the Md-Weibull distribution model can be used to estimate the distribution characteristics of wave heights. Nevertheless, the deviations in the estimation of larger wind speed and wind blowing distance still exist; as shown in section F21 from Figure 5(b) and 5(d), the Md-Weibull distribution overestimates the probability of smaller wave height. However, practically speaking, a larger wave is absolutely dominant over a smaller wave for hydrodynamics since the larger wave contains the dominant energy and more easily breaks (Clavero *et al.* 2016), therefore, the estimation accuracy of Md-Weibull is relatively acceptable.

Table 3 shows the comparison results of several models (error function *f*) in representative experimental conditions. The overall Md-Weibull distribution model is obviously more accurate than the other three models. Thus, from the perspective of model accuracy, the Md-Weibull distribution model has better applicability. Another advantage of the Md-Weibull model is that it just has one parameter, *σ*_{a}, which only needs to average the measured surface fluctuations. In contrast, both the Forristall distribution and Grukhovskii distribution need to calculate *H*_{s}. Therefore, the Md-Weibull distribution model is much more convenient to use.

. | V_{s} (m/s)
. | |||
---|---|---|---|---|

5.11 . | 11.60 . | |||

S (m) | 7.5 | 21 | 7.5 | 21 |

Rayleigh | 0.98 | 1.85 | 2.79 | 8.15 |

Forristall | 1.33 | 2.29 | 3.58 | 8.39 |

Glukhovskii | 1.13 | 1.95 | 2.62 | 6.99 |

Md-Weibull | 0.17 | 0.11 | 0.32 | 0.56 |

. | V_{s} (m/s)
. | |||
---|---|---|---|---|

5.11 . | 11.60 . | |||

S (m) | 7.5 | 21 | 7.5 | 21 |

Rayleigh | 0.98 | 1.85 | 2.79 | 8.15 |

Forristall | 1.33 | 2.29 | 3.58 | 8.39 |

Glukhovskii | 1.13 | 1.95 | 2.62 | 6.99 |

Md-Weibull | 0.17 | 0.11 | 0.32 | 0.56 |

It is known that for a narrow-banded frequency spectrum wave (regular wave), characteristic wave heights are theoretically proportional to the standard deviation of the water surface elevation, e.g., *H*_{s}/*σ*_{a} ≈ 4. For shallow water waves in the wind-driven regime, wind action, depth-limited, wave breaking, wave–wave interaction and wave-turbulence transformations make the waves significantly nonlinear (Clavero *et al.* 2016; Zhou *et al.* 2017), the frequency spectrum is of finite bandwidth, and wave height distribution characteristics and some ratios change accordingly.

The material presented above indicates that the nonlinearity effect on the wind-induced wave height distribution is to reduce the larger waves but at the same time to enhance the smaller ones (see Figure 3). The nonlinearity of the wave profile tends to increase the value of *A* (see Figure 5(a)), nevertheless, this value holds constant and has no correlation with wind speed and water depth.

In addition, nonlinearity reduces the value of *H*_{s}/*σ*_{a} from 4 to 3.8, as can be seen in Figure 5(c). This value is also constant and depth-independent and wind-independent for it retains almost stable at different water depths and wind speeds, even for field data of wind-induced waves in deep water (Goda 1979) and hurricane-generated waves (Forristall 1978). Therefore, we can attribute *H*_{s}/*σ*_{a} ≈ 3.8 to the inherent limitation nature of wind-induced waves, which reflects the coupling process in boundary layers between air and water.

Note that the lower limit of *B* is 2.23 (see Equation (3) and Figure 5(b)), which is also larger than the value in the Rayleigh distribution (*B* = 2) and in the Forristall distribution (*B* = 2.126). The positive correlation between *B* and *σ*_{a} means that as the wave scale increases, the deviation of *B* between the Md-Weibull distribution and Rayleigh distribution gradually increases, while the increase rate gradually decreases since the exponent value is 0.64 in Equation (3), which is between 0 and 1.

## CONCLUSIONS

Wind-induced waves in a shallow lake are subject to nonlinearity, which results in a relevant change in the form of the wave height distribution. A wind wave tank considering the real scenario of a lake in terms of the horizontal and vertical circulation is used as the experimental facility. The effects of wind blowing distance, wind speed and water depth on the wave height distribution and the wave characteristics during the wave growth stage are analyzed.

The applicability of some typical distribution models is tested, and their disadvantages are analyzed. Therefore, we propose a new wave height distribution model based on the Weibull distribution model to describe the wind-induced wave height distribution in depth-limited water. Its parameters are further modified by establishing the relationship between *B* and *σ*_{a}, *H*_{s} and *σ*_{a}. Some parameter properties of wind-induced waves are estimated, e.g., *A* = −1.94, *H*_{s}/*σ*_{a} ≈ 3.8 and the lower limit of *B* is 2.23.

This article yields a model that predicts the local wind-induced wave height distribution in depth-limited water for a given local surface fluctuation *σ*_{a} with significantly greater accuracy than existing models.

## ACKNOWLEDGEMENTS

This work was jointly funded by the National Key Research and Development Program of China (2018YFC0407200), the Projects of National Natural Science Foundation of China (51679146; 51479120), and Research Projects of Nanjing Hydraulic Research Institute (Y117009; Y118009; Y118012). The authors thank Mr Haipeng Luo from University of Victoria, Canada, for his comments about this manuscript.

## CONFLICTS OF INTEREST

The authors declare no conflict of interest.