First, the problem of the generation of waves in a numerical wave flume is formulated and solved. In the present study, a potential flow wave theory is used to obtain the solution within the Eulerian frame of reference. Weakly nonlinear analytical and higher-order numerical methods are employed to determine the wave fields in the rectangular domain, in which the water elements are defined by the horizontal x and vertical z coordinates of the Cartesian system. The origin of the system is located at the intersection of the wavemaker zero position and free-surface level corresponding to the hydrostatic conditions. The mechanically driven oscillation of the free surface represented by elevation function η(x,t) is induced by the piston-like motion of the wavemaker paddle according to the displacement function χ(t). The flume bottom is assumed horizontal, and the water depth is h=const. The general presentation of the computational domain and the location of the coordinate system are depicted in Fig. .

According to the potential flow assumptions, the motion of an inviscid and incompressible fluid is irrotational. Moreover, the solid boundaries are impervious. The scalar velocity potential function ϕ(x,z,t) may be introduced to determine the velocity vector field v(x,z,t) such that v=∇ϕ. The wavemaker boundary-value problem is then formulated as 1∇2ϕ=0,-h≤z≤η,2ηt+ϕxηx-ϕz=0,z=η,3ϕt+12(ϕx2+ϕz2)+gη=0,z=η,4ϕz=0,z=-h,5χt-ϕx=0,x=χ, where g is the acceleration due to gravity.

The first part of a pseudo-spectral solution involves expanding the kinematic free-surface boundary condition (), the dynamic free-surface boundary condition () and the kinematic wavemaker boundary condition () in a Taylor series about a mean position corresponding to the still water level (z=0) for Eqs. ()–() and wavemaker paddle zero position (x=0) for Eq. (): 6∑m=0ηmm!∂m∂zm(ηt+ϕxηx-ϕz)=0,z=0,7∑m=0ηmm!∂m∂zmϕt+12(ϕx2+ϕz2)+gη=0,z=0,8∑m=0χmm!∂m∂zm(χt-ϕx)=0,x=0. In this way, a simple rectangular form of the computational domain is preserved, and the solution procedure is advanced further using either the perturbation or spectral approach. In the present study both weakly nonlinear (perturbation expansions) and fully nonlinear (spectral expansions) solutions are briefly presented and applied to calculate the velocity field for wave-induced mixing calculations. The order of nonlinearity depends on the order of nonlinear terms of Taylor series expansions of boundary conditions. Second-order terms are retained for weakly nonlinear solutions, while the fully nonlinear solution is provided with no upper limit (arbitrary order).

The numerical solution to a two-dimensional advection–diffusion equation with relevant boundary conditions is used to predict the evolution of the temperature field under mechanically generated regular waves in the flume. Assuming that the changes in temperature are solely due to wave-induced and diffusion processes, the following boundary-value problem is formulated as in the work of : 30Tt+UTx+WTz=∂x(κTx)+∂z(κTz),31Tz=0,z=0,32Tx=0,x=0,33Tz=0,z=-h,34Tx=0,x=b, where T is the water temperature, and κ is the diffusion coefficient represented as a sum of the molecular (κm) and wave-induced (κv) diffusivities. 35κ=κm+κv

In the advection–diffusion equation () the velocities U an W are, in fact, Lagrangian velocities which describe the mean transport velocity field (the Stokes drift and a return current in this case). The inclusion of the Stokes drift (without return current) to the advection equation was discussed explicitly for the case of ocean waves by .

The molecular diffusivity κm=1.4×10-7 m s-2, while the wave-induced diffusivity is calculated by the formula reported by as 36κv(z)=αa3kσsinh⁡2(k(z+h))cosh⁡(k(z+h))sinh⁡3(kh), where α=0.002 is the dimensionless coefficient, which is evaluated based on measurements . The derivation of κv is based on the weakly nonlinear theory. More information is provided in the work of , where the comparison between the weakly nonlinear and more general forms is provided. The parameter α was estimated based on the experiments only for the presented form of κv. In order to use a more general formula, new values of α must be determined using experimental data from a wider range of wave conditions.

The advection–diffusion equation (Eq. ) holds in the entire fluid domain, while the heat radiation is assumed zero at the water surface (Eq. ), the bottom (Eq. ) and the lateral boundaries (Eqs.  and ). The length b is sufficiently long in order to reduce the effect of the finite domain on the results in the area of interest, which is limited to the region of several water depths from the wavemaker paddle. The omitted procedure of incorporating the non-zero heat radiation at the water surface is discussed in the paper by .

The solution of the advection–diffusion equation (Eq. ) is achieved by employing a similar methodology to that used to solve the wavemaker problem admitting higher-order nonlinearities presented in the previous section. Accordingly, the scalar temperature field function is expanded into a double Fourier series of the form 37T(x,z,t)=∑i=1∑j=1dijcos⁡(λi(x-b))cos⁡(μj(z+h)).

Again, the solution of Eq. () is achieved by a time-stepping procedure, which, in this case, consists of an application of the spectral expansion method to describe T and the Runge–Kutta formulas to proceed in time (see e.g. ). Accordingly, the wave-induced diffusivity is determined using Eq. (), while the velocity field is calculated by means of either the weakly nonlinear solution or a higher-order approach for selected wave parameters. The initial condition of given vertical temperature distribution T0(z) is used to start up the time-stepping procedure. The time derivative of temperature T is calculated from Eq. () with the aid of the expansion Eq. () after the coefficients dij are determined by applying a two-dimensional cosine fast Fourier transform. The implementation of this procedure is freely available in .

The waves are generated by a monochromatic wavemaker motion in a numerical flume. The transitional and shallow water wave cases are considered for three values of the depth-relative dimensionless parameter kh of 0.5, 1 and 2. The effect of wave height H is also studied and corresponds to the dimensionless steepness parameter H/h of 0.05, 0.1 and 0.2. In the case of the higher-order solution, the waves are generated starting from rest with the ramp function applied to the stroke of the wavemaker motion for the first five wave periods. As previously stated, the longitudinal size of the flume b is sufficiently large to exclude wave reflections from the analysis, and it corresponds to 10 wavelengths (L) for kh=2 and 20L in the remaining longer wave cases. The selected progressive wave parameters are the same as in previous studies on the modelling of wave-induced mixing in wave flumes . After the fully developed state is achieved, the time-independent wave velocity field is determined. The resulting horizontal and vertical velocity components U and W are then substituted in Eq. (), and the temperature field evolution is predicted. A summary of wave parameters is provided in Table  together with wave-induced diffusivity values at the surface in relation to its molecular counterpart.

It should be noted that the model may be applied to deep-water as well as shallow-water conditions, which are presented in the work of . Here, the cases correspond to a limited range of kh from 0.5 to 2.0. In this way, the results from the previous work may be directly compared. Moreover, for deep-water conditions corresponding to kh>π, the mixing would completely sweep away the warmer water from the analysed part of the fluid domain for the selected time frame, and this case was omitted in the present study.

The accurate assessment of the time-independent velocity field is of significant importance in the modelling of temperature changes under an undulating water surface. The increasing steepness of generated waves intensifies wave-induced mixing processes but also changes the structure of heat flux distribution in the region occupied by the fluid. The analysis of phase-averaged velocity distribution in the direct vicinity of the wavemaker paddle helps to identify the effect of advection terms of Eq. () on the temperature evolution driven by waves.

In Figs. –, time-independent velocity fields calculated by weakly nonlinear and higher-order methods are presented. The arrows represent the vectors of a phase-averaged velocity corresponding to mass transport induced by waves. Black arrows correspond to a weakly nonlinear solution, while green and blue arrows correspond to a higher-order solution and two methods of averaging – EMTV and LMTV, respectively (Figs. –). The way the results are presented is in line with the expected velocity vector field pattern in the part of the fluid domain adjacent to the wavemaker (see , for a more detailed description). The fluid flow forms half of the circulation cell limited by the air–water interface (z/h=0), the wavemaker paddle (x/h=0) and the bottom (z/h=-1). The water mass flows with the direction of wave propagation near the surface, while the adverse flow pushes the water to the wavemaker paddle in the lower part of the water column. At the wavemaker paddle, the water flows vertically, forming the lateral boundary of the general circulation in the flume. Far away from the paddle, the vertical velocity components vanish, and the typical mass-transport velocity profile over depth emerges as a sum of the Stokes drift Eq. () and the return current velocity Eq. (). Thus, the figures are complemented by the weakly nonlinear horizontal mass-transport velocity profile UL(z) calculated outside the direct vicinity of the wavemaker paddle Eq. (), where the effect of evanescent standing waves generated by the wavemaker (see e.g. ) may be neglected (x>3h). UL(z) is plotted at the right outer edge of the graph at x=3h for convenience of comparison. Vector plots are complemented by bubble charts of relative differences between corresponding vector magnitudes as presented in Figs. –, where black circles denote the differences between LMTV and weakly nonlinear results, while filled green circles refer to the differences between LMTV and EMTV.

The results presented in Fig.  provide information on the time-independent velocity field predicted in the direct vicinity of the wavemaker paddle and the differences between the weakly nonlinear approach and the solution admitting higher-order terms in the most extreme nonlinear regime of shallow water (kh=0.5). Due to the fact that the considered wave cases are characterized by the highest values of the Ursell number, the differences between the two wavemaker models immediately appear even for the lowest amplitude of free-surface oscillations of H/h=0.05 (Fig. a). The highest differences in the range between 10 % and 20 % correspond to the velocities near the wavemaker paddle and the surface, respectively (Fig. a). By increasing the magnitude of the surface oscillations to H/h=0.1 and H/h=0.2 (Fig. b and c), the differences become higher in the larger area of the wavemaker paddle vicinity. The higher-order model predicts more intensive mass transport near the bottom and the surface. The increase in the subsurface velocity and the magnitude of the current near the bottom may even reach from 15 % to 30 % for H/h=0.1 and from 20 % to 40 % for H/h=0.2 (Fig. b and c, respectively).

Some important information may also be acquired from the comparison between two methods of averaging corresponding to either Lagrangian particle tracking (LMTV) or Eulerian averaging (EMTV) based on Eqs. () and (), respectively. Although both methods of averaging provide consistent results for lower wave heights of the generated waves, EMTV results are less reliable in the case of the highest waves H/h=0.2 (Fig. c). It can be seen that the velocity vectors near the corner point determined by the intersection of the wavemaker paddle mean position and the surface (x/h=0, z/h=0) are unnaturally large as the velocity in this area (green arrows around the point (0,0) in Figs. c and c) is expected to vanish (see , for a comprehensive discussion on numerical methods' inaccuracy at corner points of a fluid domain with moving boundaries). The problem still persists when considering the subsurface velocity at some distance away from the wavemaker. The differences between the LMTV and EMTV may even reach 20 % in the velocity magnitude near the intersection between the wavemaker and the surface, while an even greater discrepancy is visible at corner points in most extreme longer-wave cases (Fig. c).

With increasing relative water depth (Figs.  and ), the differences between weakly nonlinear and higher-order time-independent velocity are less pronounced but still significant for steeper waves. In the case characterized by kh=1, the discrepancies are generally less then 10 % for H/h=0.05 and 0.1, ranging from 10 % to 20 % for the highest waves (Fig. ). Again the more intensive mass circulation is apparent when higher-order terms are taken into account for the case of kh=1 and H/h=0.2 (Fig. c). Similar conclusions may be drawn for the deeper water kh=2 (Fig. ). However, in the case of the mass-transport velocity profile relatively far away from the wavemaker paddle (x=3h) the differences are twice as high as in the wave cases corresponding to kh=1. It should be noted that, for higher kh (Fig. ), the results corresponding to two methods of averaging are more consistent; however the highest difference may still reach 20 % as in the case of longer waves (Fig. c).

Putting aside the problems with averaging of the velocity field, the general conclusion from the analysis of the results depicted in Figs. – is that admitting higher-order terms of the Taylor series expansions of the boundary conditions imposed at the moving boundaries leads to more accurate prediction of the time-independent velocity near the wavemaker as expected. This is of significant importance especially for the modelling of wave-induced mixing and the evolution of the water temperature field as the higher-order approach predicts enhanced streaming in layers adjacent to the bottom and the surface. This leads to more intense mixing processes and heat exchange.

The modelling of wave-induced vertical mixing in terms of the temperature redistribution in the water column relies on the solution to the two-dimensional boundary-value problem defined by Eqs. ()–(). The input parameters being introduced to the governing advection–diffusion equation (Eq. ) are the diffusion coefficient κv calculated using Eqs. () and () and the time-independent wave velocity field (U, W). The modelling procedure remains in accordance with previously published results with different velocity field input. The temperature T0(z) determines the initial thermal state of the fluid (Fig. ).

In Figs. –, the temperature spatial distributions representing the thermal states of the undulating water body after 100 s is provided for the test cases listed in Table . The results in the plots correspond to a weakly nonlinear solution and higher-order model predictions of the time-independent wave velocity field. Both methods are compared in order to highlight the effects of nonlinearity on vertical mixing due to waves.

In the direct vicinity of the wavemaker paddle, the initial state of the water temperature is uniformly stratified according to T0. The moving wavemaker paddle generates regular waves, and the layers of water of equal temperature are deformed by the oscillatory motion of the water body according to the time-independent velocity field presented in Figs. –. Advection plays an important role in wave-induced mixing processes as even small changes in the velocity strongly affect the evolution of the temperature field. This implies major differences between weakly nonlinear and higher-order predictions of the resultant temperature when the steepness of waves increases. This effect is clearly seen in Figs. –. As previously stated, the discrepancy between weakly nonlinear velocities and the velocity predictions admitting higher-order terms in the free-surface boundary conditions grows with increasing steepness and wave nonlinearity (Ursell number) and may even reach 40 % in the subsurface layer.

Although the highest relative differences are relevant for the longest waves, the effect of including higher-order terms is most apparent for waves of the largest kh characterized by the highest time-independent velocities. It may be seen in Fig.  that after 100 s the colder water is moved to the surface close to the wavemaker only for the steepest waves (Fig. f) when predicted by the higher-order model, while for the corresponding weakly nonlinear temperature field this process is less intensive (Fig. e). Additionally, the higher-order results exhibit some variation in mixing along the flume (Fig. f).

In the case of waves characterized by kh=1, the wave-induced mixing is more intensive (see Table ), and for the steepest waves (Fig. e, f) the warmer water is moved away from the wavemaker. It should be noted that the higher-order solution predicts enhanced streaming near the surface (Fig. f). In this way, the decreased water surface temperature affects regions located further away from the wavemaker when compared to weakly nonlinear results (Fig. ). The similar temperature field modification affects the waves of kh=2, but for the lower analysed height (cf. Figs. e, f and c, d). In the case of the highest waves of kh=2, the warmer water is almost completely swept away from the region of direct wavemaker action (Fig. e, f).

Energy input from wind to ocean surface waves is tremendous and exceeds 60 TW . A large amount of energy dissipates, which implies stirring and mixing processes in the oceanic mixed layer. A parameterization scheme for surface wave-induced mixing was proposed, and numerical experiments indicate that this parameterization affects the performance and outcome of ocean circulation models as well as climate predictions . Although field observations suggest that surface waves can generate vertical mixing , it is still difficult to distinguish the mixing originating from the variety of processes that accompanied ocean waves. Although the problem of wave-induced vertical mixing is of significant importance for physical oceanographers and climatologists, the research on this subject is still in its infancy. Wave-induced mixing is a very important process from a practical point of view and is a challenging problem for theoretical investigations. The understanding of wave-induced mixing is of fundamental importance for the modelling and accurate prediction of ocean transport processes and climate changes. The problem is that the parameterization scheme applied in the modelling and prediction of ocean surface wave-induced mixing is based on drastic simplifications. Wave models derived using simplifying assumptions cannot be applied to predict ocean surface wave-induced mixing with sufficient accuracy. The present studies clearly show that it is necessary to apply at least a weakly nonlinear wave theory to obtain reasonable results (no wave-induced mass transport in linear theory means no heat exchange along the direction of wave propagation), while an increased accuracy is possible using advanced nonlinear wave models admitting higher-order effects. It should also be noted that in the present studies the effects of turbulence and the development of viscous boundary layers are neglected, while these processes may have some impact on larger timescales. For example, the mass-transport velocity profile changes due to the effects of viscosity near the boundaries (the undulating free surface and the bottom) for a sufficiently long time interval, which was studied theoretically by and also investigated experimentally in a wave flume (see e.g. ).

There is one more critical outcome from the point of view of engineers and scientists working on the modelling of ocean transport processes, wave-induced mixing and climate changes. Namely, since it is difficult to distinguish the mixing that accompanied ocean waves, the only chance to provide reliable insight into the wave-induced mixing processes is to conduct laboratory experiments in a wave flume. The repeatable experiments in the well-controlled environment of a wave flume enable us to perform an accurate investigation that is essential in the analysis of the physics of the wave-induced mixing phenomenon. Moreover, laboratory investigations provide useful data for the analysis of the correlations between spectral and statistical characteristics of wave regimes and wave-induced mixing processes. Finally, laboratory experiments enable us to avoid various side effects and separate the mixing from other processes that accompanied ocean waves such as wave breaking. This is of fundamental importance for understanding of mixing and accurate calibration and verification of numerical models. The problem is that, in addition to progressive laboratory waves, the moving wavemaker represented by the kinematic wavemaker boundary condition enforces the return current that affects transport processes and wave-induced mixing.

It is the idea behind the present study, which employs a numerical wave flume model, to thoroughly analyse wave-induced mixing effects using the derived numerical approach and assist further works on experimental fluid mechanics aiming at better understanding of transport processes in the open ocean. To the best of the authors' knowledge, the derived wave-induced mixing approach is the only available numerical solution that, in addition to nonlinear free-surface boundary conditions, also satisfies the kinematic wavemaker boundary condition. Accordingly, the wavemaker model admits a return current and may be applied to quantify and separate the effects of the return flow on wave-induced mixing processes. The presented results should hopefully improve simple models based on the Stokes drift applicable to random ocean waves . As previously mentioned, the derived modelling framework may be modified to cover open-ocean conditions for the periodic domains and quasi-random sea states using other forms of wave excitation .