The Effects of Wave-Induced Stokes Drift and Mixing Induced by Nonbreaking Surface Waves on the Ocean in a Climate System Ocean Model

Abstract

Oceanic general circulation models (OGCMs) are important tools used to investigate mechanisms for ocean climate variability and predict the ocean change in the future. However, in most current ocean models, the impact of sea surface waves as one of the most significant dynamic processes in the upper ocean is absent. In this study, the Stokes drift and the vertical mixing induced by nonbreaking surface waves derived from the wave model (WAVEWATCH III) are incorporated into a Climate System Ocean Model, and their effects on an ocean climate simulation are analyzed. Numerical experiments show that both physical processes can improve the simulation of sea surface temperature (SST) and mixed layer depth (MLD) in the Southern Hemisphere. The introduction of Stokes drift effectively reduces the subsurface warm bias in the equatorial tropics, which is caused by the weakening of vertical mixing in the equatorial region. The nonbreaking surface wave mainly reduces the temperature bias in the Southern Ocean by enhancing mixing in the upper ocean. For the MLD, the Stokes drift mainly improves the simulation of the winter MLD, and the nonbreaking surface wave improves the summer MLD. For MLD south of 40° S in summer, the introduction of nonbreaking surface waves resulted in a reduction of 11.86 m in MLD bias and 7.8 m in root mean square errors (RMSEs), respectively. For winter subtropical MLD in the Southern Hemisphere, considering the Stokes drift, the MLD bias and RMSEs were reduced by 2.49 and 5.39 m, respectively. Adding these two physical processes simultaneously provides the best simulation performance for the structure of the upper layer. The introduction of sea surface waves effectively modulates the vertical mixing of the upper ocean and then improves the simulation of the MLD. Thus, sea surface waves are very important for ocean simulation, so we will further couple a sea waves model in the Chinese Academy of Sciences Earth System Model (CAS-ESM) as part of their default model component.

1. Introduction

The ocean general circulation model (OGCM), as an important component of Earth system models, is an essential science tool used to understand how the ocean works within the Earth’s climate system and project ocean climate change in the future. Although numerical ocean models have achieved great success in recent decades, ongoing development is still needed to improve simulation fidelity and capability, especially for improved parameterization schemes of unresolved physical processes, such as vertical mixing schemes [1]. The well-known vertical mixing schemes applied in ocean climate models include the Pacanowski and Philander scheme [2], the Mellor–Yamada scheme [3], the K-profile parameterization scheme [4], and the Canuto scheme [5,6]. A common problem with ocean models using these vertical parameterization schemes is the underestimation of vertical mixing in the upper ocean [7,8], which results in a simulation with a higher SST and shallower MLD in the Southern Ocean summer [9,10].

Currently, many studies have shown that sea surface wave processes, especially Stokes drift and nonbreaking surface waves, can improve the simulation of vertical mixing in the upper ocean. Stokes drift velocity has a significant impact on the Ekman spiral and temperature structure in the upper ocean by interacting with planetary vorticity [11,12]. Li et al. investigated the structure of the upper ocean circulation and found that Stokes drift could enhance the shear instability of the upper ocean, thereby intensifying the mixing of the ocean [13]. Reichl et al. found that Stokes drift could enhance upwelling and cool SST in typhoons [14]. McWilliams et al. reported that the effect of Stokes drift could enhance the turbulence of the upper ocean and deepen the mixed layer [15].

Another important physical process is vertical mixing caused by nonbreaking surface waves [16]. The influence of nonbreaking surface waves on the ocean can reach tens of meters, thus greatly affecting the vertical mixing of the upper layer of the ocean [16,17,18]. Previous studies have shown that the upper temperature structure and MLD can be effectively improved by considering the influence of nonbreaking surface waves [19,20,21,22,23]. In addition, breaking waves also generate turbulence, which then leads to mixing near the surface compared to the nonbreaking waves [24,25,26]. Moreover, the existing parameterization schemes for the breaking wave turbulence down to the bottom of the mixing layer may have serious physical problems [18,27]. Therefore, we only consider the impact of vertical mixing caused by nonbreaking surface waves in this study. It is noted that the role of sea surface wave processes is absent in the state-of-the-art Earth system models. For example, there are only four climate models [23,28,29,30] in CMIP6 that consider sea surface wave processes. Therefore, it is necessary to use more Earth system models and ocean models to study the impact of sea surface wave processes [31].

CAS-ESM2.0 is an Earth system model from the Institute of Atmospheric Physics, Chinese Academy of Sciences (IAP/CAS), which has a long history of developing climate models. Since the 1980s, scientists at IAP/CAS have been devoted to developing climate models that include an atmospheric general circulation model (AGCM), OGCM, and land surface model (LSM) [32,33,34], and these models have been used for short-term climate predictions in China and have been included in CMIP6 [35,36,37,38]. It has been noted that the sea surface wave process is absent. Currently, there is also a lack of research that simultaneously considers the effects of both Stokes drift and wave-induced mixing on the upper ocean. Therefore, this study focused on the effects of Stokes drift and mixing induced by nonbreaking surface waves on the simulation of the upper ocean in the ocean component of CAS-ESM.

The remainder of the paper is structured as follows. In the following section, the models and experimental design are introduced. Section 3 provides the simulation results with and without Stokes drift/nonbreaking surface wave mixing. Section 4 provides a summary.

2. Model Description and Experiments

2.1. The Description of Two Physical Processes of Sea Surface Waves

In our study, we mainly considered the effects of Stokes drift velocity and nonbreaking surface waves on the ocean model. The descriptions of the two processes are briefly introduced in the following section.

• The surface Stokes drift

The surface Stokes drift velocity ${\mathit{u}}_{\mathit{s}}$ produced by sea surface waves can be calculated based on the following formula from the WAVEWATCH III wave model [39]:

$${\mathit{u}}_{\mathit{s}}=\iint \omega \cosh 2kd\frac{\left(k\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta \right),k\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta \right)\right)}{{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^{2}kd}F\left(k,\theta \right)dkd\theta ,$$

(1)

where $\omega$ is the angular frequency, $k$ is the wavenumber, $d$ is the depth of the water, $\theta$ is the wave direction, and $F\left(k,\theta \right)$ is the wave energy spectrum. In addition, $k\cos \left(\theta \right)$ is used to calculate Stokes drift velocity in the zonal direction and $k\sin \left(\theta \right)$ in the meridional direction. The effects of the surface Stokes drift velocity on the ocean circulation model are mainly focused on two aspects: one is added to the Coriolis–Stokes force term and the other is considered in the air–sea turbulence scheme.

The surface Stokes drift is considered in the momentum budget equation, and the corresponding formulation is modified as follows [40,41]:

$$\frac{\partial {\mathit{u}}_h}{\partial t}+{\left[\mathit{u}·\nabla \mathit{u}\right]}_h+f\widehat{z}\times \left(\mathit{u}+{\mathit{u}}_{\mathit{s}}\right)=-\nabla \frac{p}{{\rho }_w}+{\mathit{D}}_h+b\widehat{z}+\frac{\partial }{\partial z}\left(A_v^o\frac{\partial {\mathit{u}}_h}{\partial z}\right),$$

(2)

Here, the subscript $h$ represents a 2D horizontal velocity vector (zonal and meridional), $f$ is the Coriolis parameter, $\mathit{u}$ is the 3D current velocity vector, ${\mathit{u}}_{\mathit{s}}$ is the 2D Stokes drift velocity, and $p$, ${\rho }_w$, and $b$ are the pressure, reference ocean density, and buoyancy, respectively. ${\mathit{D}}_h$ represents the parameterization of sub-grid physics and $A_v^o$ is the vertical eddy viscosity.

The effect of the surface Stokes drift velocity on the air–sea turbulence flux (including momentum flux ($\overrightarrow{\tau }$), sensible heat flux ($Q_H$), and latent heat flux ($Q_E$)) is mainly represented by the relative velocity between the 10 m wind and sea surface current. The corresponding detailed formulas are as follows:

$$\overrightarrow{\tau }={\rho }_A|∆\overrightarrow{u}|C_d∆\overrightarrow{u},$$

(3)

$$Q_H={\rho }_A|∆\overrightarrow{u}|C_h∆\theta ,$$

(4)

$$Q_E={\rho }_A\left|∆\overrightarrow{u}\right|C_e∆q,$$

(5)

where ${\rho }_A$ is the air density, and $C_d$, $C_h$, and $C_e$ are the transfer coefficients for momentum, sensible heat, and latent heat, respectively. $∆\theta$ and $∆q$ are the differences in temperature and humidity between the atmosphere and the ocean, respectively. The relative velocity is $∆\overrightarrow{u}$, i.e., $∆\overrightarrow{u}={\overrightarrow{u}}_A-{\overrightarrow{u}}_O-{\overrightarrow{u}}_S$ [23].

• Nonbreaking surface wave mixing

Qiao et al. [16,19] proposed a calculation method for the vertical mixing caused by nonbreaking surface waves and proved that it is important for the temperature structure of the upper ocean. The calculation formula of the wave-induced mixing coefficient is as follows:

$$Bv=\alpha {\iint }_{\overrightarrow{k}}F\left(\overrightarrow{k}\right)exp\left\{2kz\right\}d\overrightarrow{k}\frac{\partial }{\partial z}{\left({\iint }_{\overrightarrow{k}}{\omega }^{2}F\left(\overrightarrow{k}\right)exp\left\{2kz\right\}d\overrightarrow{k}\right)}^{1/2},$$

(6)

where $\alpha$ is a constant coefficient and set to 1 in this study, $\omega$ is the angular frequency, $k$ is the wavenumber, $F\left(\overrightarrow{k}\right)$ is the wave energy spectrum, and $z$ is the vertical coordinate axis upward positive with z = 0 at the ocean surface.

Then, $Bv$ is added to the original vertical viscosity $akm$ and diffusivity $akt$ from the Canuto scheme of the ocean model. The new vertical mixing coefficients are as follows:

$$k_m=akm+Bv,$$

(7)

$$k_h=akt+Bv,$$

(8)

2.2. Model Description

The OGCM in this paper was based on a revised version of LICOM2.0 derived from the State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics and the Institute of Atmospheric Physics (LASG/IAP) Climate System Ocean Model [42]. The main features of LICOM2.0 include η-coordinates, free surfaces, and primitive equations. The model domain is located between 78.5° S and 87.5° N with a 1° zonal resolution. The meridional resolution is refined to 0.5° between 10° S and 10° N and is increased gradually from 0.5° to 1° between 10° and 20°. There are 30 levels in the vertical direction with 10 m per layer in the upper 150 m and the vertical turbulent mixing scheme is the Canuto scheme [5,6]. Based on the original version of LICOM2.0, key modifications were made: (1) a new sea surface salinity boundary condition was introduced that is based on the physical process of air–sea flux exchange at the actual air–sea interface [43]; (2) a new formulation of the turbulent air–sea fluxes was introduced [44]. The revised LICOM2.0 is the OGCM of the CAS-ESM that participated in CMIP6 [36,38].

The wave model used in this study is the WAVEWATCH III spectral wave model version 6.07 [45]. We configured the wave model with a horizontal grid spacing of 1° longitude × 1° latitude. The surface wave spectrum was discretized in 24 directions and 25 intrinsic (relative) frequencies in the range of 0.041~0.406 Hz, and the logarithmic increment was $f(n+1)=1.1f\left(n\right)$, where $f\left(n\right)$ is the nth frequency. For the wave growth and dissipation terms, we used the parameterization scheme proposed by Ardhuin et al., ST4 in WAVEWATCH III [46]. The nonlinear wave–wave interactions use the Discrete Interaction Approximation (DIA) package according to Hasselmann et al. [47]. In addition, we considered physical processes such as bottom friction following Hasselmann et al. [48] and depth-induced wave breaking by Battjes et al. [49] during wave propagation.

2.3. Numerical Experimental Design

To investigate the effects of Stokes drift and nonbreaking surface waves on ocean climate simulations, four experiments were conducted with details of the setups shown in

Table 1. Experimental setups of four simulations.

Experiment ID	Stokes Drift	Nonbreaking Surface Wave
con	no	no
uss	yes	no
bv	no	yes
bv_uss	yes	yes

Note: the con experiment represents the control run, uss experiment represents the addition of Stokes drift, bv experiment represents the addition of nonbreaking surface wave mixing, bv_uss experiment represents the addition of both of these physical processes.

. The default revised LICOM2.0 that did not consider Stokes drift and nonbreaking surface waves was the control experiment (denoted as “con” hereafter). The experiment with only Stokes drift effect was denoted as the uss experiment. The experiment in which only nonbreaking surface wave mixing was added was denoted as the bv experiment. Both of the above physical processes were applied simultaneously, which was called the bv_uss experiment. The difference between the experiments was whether Stokes drift or nonbreaking surface waves were considered.

For the con experiment, the Coordinated Ocean-ice Reference Experiments-I (CORE I) protocol was employed, the repeating annual cycle of climate atmospheric forcing from Large and Yeager [50] was used, and the model was run for 300 years to reach quasi-equilibrium. For the other three experiments, the 6-hourly ERA5 wind fields, monthly SODA3.12.2 current fields, and weekly NOAA OISST V2 ice fields were used as the WAVEWATCH III wave model forced field for integration from 1996 to 2006, and the climatological daily mean Stokes drift velocity and wave-mixing coefficients were obtained offline. Then, the Stokes drift velocity and wave-mixing coefficients were incorporated into the ocean model by the method described in the previous section. These four experiments started in the quasi-equilibrium state and were integrated under the same CORE I forcing field for 50 years, and the last decade was used for analysis.

3. Results

Currently, the simulated overheating SST and the too-shallow MLD in summertime are common biases in most ocean models [7,51,52]. Wave-mixing parameterization schemes have been proposed to improve these biases [16,53,54]. The vertical mixing effect of sea surface waves is very prominent in summer but not in winter [55], this is because the impact of sea surface waves on the vertical mixing effect is related to the stratification of the water column. In summer, the heated sea surface water corresponds to the relatively low density results in stable stratification, and in winter, the cooling water can weaken the stratification. In addition, waves have a greater effect on the Southern Ocean due to the larger wind speed ranges from 10 to 17 m/s. Therefore, to explore the effects of waves on the ocean, we mainly focused on the Southern Hemisphere summer in this study.

We first investigated the performance and importance of Stokes drift and nonbreaking surface waves in January during austral summer simulated by WAVEWATCH III. At the ocean surface, Stokes drift is the Lagrangian-averaged fluid velocity caused by surface wave motion. The fifth generation ECMWF daily reanalysis data were used as observations for the comparison of the Stokes drift simulation [56]. The climatological velocities of Stokes drift can exceed 0.1 m/s in the Southern Ocean and mid-high latitudes of the Northern Hemisphere (Figure 1a). The corresponding climatological Stokes drift simulated by WAVEWATCH III was basically similar in pattern and magnitude relative to the observations (Figure 1b). Figure 1c shows the spatial distribution of sea surface current velocity, with large velocity values mainly located at the equator, the western boundary current and its extension, and the Southern Ocean. As a part of sea surface water movement, Stokes drift velocity accounts for more than 60% of sea surface current velocity in most of the surface ocean (Figure 1d), indicating that it plays a crucial role in sea surface circulation.

Figure 2 shows the distribution of $Bv$ averaged over the upper 75 m in January. Large values of $Bv$ were distributed in the North Atlantic, North Pacific, and westerly belt of the Southern Ocean (Figure 2a). The maximum reached 70 cm²/s in the North Atlantic, which is consistent with the results of Qiao et al. [16]. The vertical mixing coefficients produced by the Canuto scheme are shown in Figure 2b,c. There were two large value centers located in the North Atlantic and North Pacific, with values up to 100 cm²/s. By comparison, $Bv$ was smaller relative to the vertical mixing coefficient north of 40° S. However, $Bv$ was important for vertical mixing in the Southern Ocean region, and it had a larger value than both the vertical viscosity coefficient $akm$ and the vertical diffusivity coefficient $akt$ computed by the Canuto scheme.

To investigate the impact of Stokes drift and nonbreaking surface waves on the sea temperature, Figure 3 presents the SST anomalies simulated by the con experiment compared to the WOA13 dataset [57] and the differences in the uss, bv, and bv_uss experiments relative to the con experiment (Figure 3b,d). In comparison with the observations, the con experiment showed that the cold biases were mainly located in the Northwest Pacific, Northeast Atlantic, and mid-latitude belt of the Southern Ocean (30° S–40° S). The warm temperature biases were pronounced at high latitudes in the Southern Hemisphere (Figure 3a), which is consistent with the results of Meijers et al. [58]. Both Stokes drift and nonbreaking surface waves can reduce the warm biases of the high-latitude zonal belt of the Southern Ocean presented by the con experiment (Figure 3b,c). The cooling anomalies in the bv experiment with a maximum of ~1.0 °C were almost three times as strong as the cooling anomalies of the uss experiment. The average SST of the con experiment, uss experiment, bv experiment, bv_uss experiment, and observation south of 40° S were 8.01, 7.92, 7.39, 7.37, and 6.71 °C, respectively. The RMSEs between the four experiments and the observations were 1.75, 1.67, 1.30, and 1.29 °C, respectively (

Table 2. The average SST of the con experiment, uss experiment, bv experiment, bv_uss experiment, and observation south of 40° S, and their RMSEs with observation (units: °C).

Experiment ID	Average SST	RMSEs
observation	6.71
con	8.01	1.75
uss	7.92	1.67
bv	7.39	1.30
bv_uss	7.37	1.29

). The comparison showed that both the nonbreaking surface waves and the Stokes drift contributed to the improvement of the simulated SST in the Southern Ocean, the effect of nonbreaking surface waves is primary, and the Stokes drift is secondary. The former is mainly due to the strengthening of the upper mixing and the latter is mainly caused by the strong upwelling according to previous studies [55,59] In addition, adding the nonbreaking surface waves also improved the simulation of SST in the extended regions of the Gulf Stream and Kuroshio. Both physical processes are considered simultaneously, and the simulated SST is the closest to the observation (Figure 3d). The result exhibited a reduction of 0.64 and 0.46 °C in the mean SST and RMSEs relative to the con experiment in the Southern Ocean through exerting the two physical processes.

The biases of the zonally averaged temperature in different basins in the upper 300 m are shown in Figure 4. Consistent with the SST, there was a warm bias in the simulation of the con experiment in the upper 20 m of the Southern Ocean (Figure 4a), which can be well relieved by the addition of the nonbreaking surface wave process (Figure 4c). It is noted that a strong warm bias in the con experiment was located in the subsurface of the equatorial regions of the Indian Ocean, Pacific Ocean, and Atlantic Ocean, especially for the equatorial Pacific Ocean with a warm bias larger than 3 °C (Figure 4a). The uss experiment revealed cold temperature anomalies with respect to the con experiment in the equatorial region (Figure 4b) and decreased the warm bias in the con experiment in this region. However, the bv experiment exhibited a warm deviation below 20 m and exacerbated these equatorial warm biases (Figure 4c), which is consistent with previous research results [55,60,61]. This mainly results from the strengthening of the upper layer mixing due to nonbreaking surface waves. Therefore, we can conclude that nonbreaking surface waves improved the simulation of ocean temperature in the Southern Ocean, and Stokes drift improved the simulation of ocean temperature in the equatorial region. By including these two physical processes simultaneously in the bv_uss experiment, the simulated temperature is more consistent with the observation (Figure 4d).

To further investigate the influence of the Stokes drift velocity on the subsurface layer in the equatorial region, the temperature structure of the upper equatorial Pacific Ocean was shown in Figure 5. Compared with the observations, there was a strong warm bias within 200 m of the upper layer in the con experiment (Figure 5a). The introduction of the Stokes drift can alleviate the warm bias (Figure 5b), which mainly results from reducing the vertical mixing coefficient in the upper layer relative to the con experiment. The weakening of the mixing process in the upper layer leads to a decrease in the heat transfer from the surface to the subsurface layer, which cools the temperature of the subsurface layer, thus alleviating the warm bias in the con experiment. This is inconsistent with the results of McWilliams et al. [15]. Their results suggested that the Stokes drift velocity enhances mixing in the upper layers, which may be associated with the absence of the effect of the surface Stokes drift velocity on the air–sea turbulence flux, especially for the wind stress.

The differences between the simulated summer MLD and observed MLD (observational data from the World Ocean Atlas 2013) are shown in Figure 6 (September and January for the Northern and Southern Hemispheres, respectively). MLD is defined as the depth at which the density increases from that at the surface by an amount equal to the increase in density caused by the given (the 0.5 °C) decrease in temperature [62,63,64,65]. In comparison to the observations, the con experimental results showed that the shallow biases were mainly distributed in the middle and high latitudes (North Pacific, North Atlantic, and Southern Ocean), while the deep biases were located in the low latitudes (especially for the equatorial region) (Figure 6a). This is generally consistent with SST biases: warm SST bias corresponds to a shallow mixed layer, and vice versa. The maximum bias zone simulated by the con experiment was located in the Southern Ocean and was generally shallower than the observations, with a bias greater than 30 m in most regions, which is a common problem in current ocean models [66]. The reason for the low simulation results of the MLD in the Southern Ocean in summer is that the vertical mixing rate is insufficient [67,68], which prevents heat transfer from the surface layer to the subsurface layer. This causes the simulated SST of the Southern Ocean to be warmer. Both Stokes drift and nonbreaking surface waves tended to deepen the summer Southern Ocean MLD, where the con experiment showed negative biases (Figure 6b,c). The bv experiment deepened the mixed layer far more than the uss experiment. The mean MLD of the con experiment, uss experiment, bv experiment, and bv_uss experiment south of 40° S were 21.44, 24.04, 33.30, and 34.08 m, respectively (

Table 3. The average summer MLD of the con experiment, uss experiment, bv experiment, bv_uss experiment, and observation south of 40° S, and their RMSEs with observation (units: m).

Experiment ID	Average MLD	RMSEs
observation	36.16
con	21.44	26.16
uss	24.04	23.55
bv	33.30	18.36
bv_uss	34.08	18.28

). The corresponding observed value was 36.16 m. The RMSEs between the four experiments were 26.16, 23.55, 18.36, and 18.28 m, respectively. The results showed that compared to the con experiment, the simulation effect is the best using the two physical processes simultaneously, and the MLD bias and RMSEs were reduced by 12.64 and 7.88 m, respectively. Thus, this result suggests that the introduction of surface waves has led to larger turbulent mixing in the Southern Ocean.

There is still debate about the role of wave processes in simulating the winter MLD, with some studies indicating that the impact of waves is negligible [29], while others suggest that simulations of the MLD can be improved [22]. Here, we verified the influence of sea waves on the simulation of the mixed layer in winter. Figure 7 shows the winter MLD (January and September for the Northern and Southern Hemispheres, respectively). The negative biases of MLD in the con experiment relative to the observations were mainly located in the North Pacific Ocean and Southern Ocean, while the positive biases were located in the subtropical regions of the Northern and Southern Hemispheres (especially in the Southern Hemisphere) (Figure 7a), which are consistent with the results of Chen et al. [22]. The simulation of the global MLD was improved after taking Stokes drift into account. Compared with the con experiment, the uss experiment had a negative difference in the subtropical regions of the Northern and Southern Hemispheres. In addition, there was a positive difference to improve the simulation of the MLD for the shallow area of the Indian Ocean sector in the con experiment (Figure 7b). However, the improvement in MLD by nonbreaking surface waves was minimal (Figure 7c), which is consistent with the conclusion of Fan et al. [29]. This may be related to the reasonable vertical mixing in winter provided by the Canuto scheme. The mean MLD of the con experiment, uss experiment, bv experiment, and bv_uss experiment in the subtropical regions of the Southern Hemisphere were 137.16, 134.67, 142.36, and 139.69 m, respectively. The corresponding observed value was 92.13 m. The RMSEs between the four experiments and observations were 90.08, 84.69, 88.67, and 83.81 m, respectively (

Table 4. The average winter MLD of the con experiment, uss experiment, bv experiment, bv_uss experiment, and observation in the subtropical regions of the Southern Hemisphere, and their RMSEs with observation (units: m).

Experiment ID	Average MLD	RMSEs
observation	92.13
con	137.16	90.08
uss	134.67	84.69
bv	142.36	88.67
bv_uss	139.69	83.81

). The comparison showed that the contribution of the Stokes drift was larger than that of nonbreaking surface waves for simulations of the mixed layer of the Southern Hemisphere in winter. Fan et al. [29] pointed out that in the winter of the Southern Hemisphere, the effect of nonbreaking surface waves on the MLD is not very obvious, mainly because the MLD is much deeper than the depth affected by the mixing of nonbreaking surface waves. Through the comparison of Figure 7b,d, both the uss experiment and the bv_uss experiment show a similar characteristic of MLD difference in the Southern Ocean, the simulation of MLD in the bv_uss experiment was improved in the North Pacific Ocean, while in the uss experiment, it aggravated the simulation bias of the depth of the mixing layer in the con experiment. Therefore, we concluded that nonbreaking surface waves had a greater effect on the simulation of the MLD in summer, while Stokes drift was more important in winter. By adopting the two physical processes simultaneously, the simulation of MLD is the best and closest to the observation (Figure 6d and Figure 7d).

4. Summary and Discussion

In this study, to improve the simulation of the upper ocean, two new physical processes related to sea surface waves in the OGCM were introduced. Four experiments (con, uss, bv, and bv_uss) were conducted to investigate the influence of sea surface waves on the climate system ocean model by incorporating Stokes drift and the vertical mixing parameterization of nonbreaking surface waves into an ocean model.

We demonstrate that both physical processes have the potential to improve simulations of the temperature structure of the upper ocean, especially for the summer hemisphere, where stable stratification exists. Stokes drift can improve the upper layer temperature structure in the tropical equatorial region. This improvement is achieved by changing the intensity of mixing in the equatorial regions. The Stokes drift weakens vertical mixing in equatorial regions, preventing heat transfer from the surface to the subsurface, and thus alleviates the warm bias of the subsurface in the con experiment. In addition, for the Southern Ocean in summer, the Stokes drift improves the simulation of SST, but the effect is not as strong as that of nonbreaking surface waves. In the bv experiment, the reduction in the warm bias of the Southern Ocean SST is mainly due to the deepening of the MLD in summer compared with that in the con experiment. Considering both physical processes simultaneously, the simulation of ocean temperature structure is the closest to the observation.

For the summer MLD, the improvement effect of nonbreaking surface waves is better than the effect of Stokes drift, and the simulated MLD in the Southern Ocean and North Pacific are more similar to observations, which is consistent with the results of Chen et al. [22]. The result shows that the introduction of nonbreaking surface waves makes MLD bias and RMSEs significantly reduced by 11.86 and 7.8 m. For the winter MLD, the simulation effect of Stokes drift is larger than that of nonbreaking surface waves. After taking Stokes drift into account, the MLD bias and RMSEs decreased by 2.49 and 5.39 m in the subtropical MLD of the Southern Hemisphere. The effect of nonbreaking surface waves on MLD in winter is not significant, which is consistent with the results of Fan et al. [31]. This behavior of the nonbreaking surface waves is caused by the dependence of the mixing length on the wave attenuation scale. In summer, the attenuation scale of waves is comparable to that of the MLD, and the extra mixing coefficient $Bv$ resulting from nonbreaking surface waves can affect the entire mixing layer and deepen the mixed layer. During winter, the attenuation scale is smaller than that of the mixed layer, so $Bv$ has little effect on the MLD. Through the numerical experimental comparison of Stokes drift and nonbreaking surface waves, we conclude that considering two physical processes simultaneously, the simulation performance of the mixed layer is the best and closest to the observation.

Although our analysis in this work was performed for one specific ocean model and one specific wave model, the significant improvements in upper ocean temperatures clearly indicate the importance of Stokes drift and nonbreaking surface wave mixing processes. Therefore, we propose considering the influence of waves in other oceanic climates as well. In addition, whether sea surface waves can reduce the simulation bias in Earth system models needs further research. Therefore, incorporating a wave model in CAS-ESM 2.0 is part of our further work.