3.1. Burgers model

To further simplify our wave problem, we consider in § 3 that the diffusion coefficient $\nu _t$ is constant ($\nu _t=\nu$). Equation (2.3) is then the well-known Burgers equation. It was introduced originally as a simple one-dimensional model to contribute to the study of turbulence (Burgers Reference Burgers1948). A synthesis on Burgers turbulence, also known as Burgulence, is presented in Frisch & Bec (Reference Frisch and Bec2002).

In this subsection, we consider freely decaying random waves $v(x,t)$ that are statistically homogeneous in space, with zero mean. The equation for the mean energy $\boldsymbol {E}_v=\langle v^2 \rangle$ is

(3.1)\begin{equation} \frac{\partial \boldsymbol{E}_v}{\partial t}={-}\boldsymbol{D}_v , \end{equation}

where $\boldsymbol {D}_v=2\nu \,\langle ({\partial v}/{\partial x})^2\rangle$ is the energy dissipation, and $\langle \,{\cdot}\, \rangle$ is the spatial mean.

A striking feature of Burgers solutions is the formation of shocks. Due to the nonlinear term, negative $v$ slopes are steepened in time until they build up into diffusive shocks, where nonlinear and diffusive effects are balanced. For initial random conditions, the wave field tends towards an irregular sawtooth profile, quite similar to that of SWs in the ISZ (figure 1).

Two main characteristic scales are involved in the SW regime: $\lambda _m$ is the mean distance between adjacent wave fronts, and $V_c$ is the characteristic scale of velocity jumps at wave fronts. The wave field is then controlled by two length scales: a macroscopic one, $\lambda _m$, and a small one, the average shock thickness $\delta \sim {\nu }/{V_c}$. Consequently, the problem is governed by one dimensionless parameter, the Burgers-type Reynolds number $R_B={V_c \lambda _m}/{\nu }$. It is well established that for wavenumbers ranging from $k_m=2{\rm \pi} /\lambda _m$ to the diffusive wavenumber $k_\nu$ ($k_\nu \sim \delta ^{-1}$), the Burgers energy spectrum follows a $k^{-2}$ power law (e.g. Tatsumi Reference Tatsumi1969). This $k^{-2}$ subrange is followed at high $k$ by a diffusive subrange where the energy decreases more rapidly.

We will see later that ISZ waves are characterized by moderate Reynolds numbers of approximately a few hundreds ($R_B \sim 100\unicode{x2013}500$). In this section, we analyse the spectral characteristics of Burgers waves for this $R_B$ range. It should be noted that most studies on Burgulence, unlike ours, focus on regimes with very high $R_B$. When referring to an ISZ wave, $R_B$ must be distinguished from the classical Reynolds number based on the kinematic viscosity. Before analysing random waves, we start by studying the nonlinear dynamics of periodic SWs. This idealized case is very useful, first for understanding basic nonlinear and dissipative processes in the spectral space, and second by serving as a basis for the development of a random SW theory.

3.2. Periodic sawtooth waves

In the non-diffusive case ($\nu =0$), the derivation of the periodic SW Burgers solution, $v_{i}(x,t)$, from the method of characteristics is straightforward and gives

(3.2)\begin{equation} v_{i}=\frac{V_J(t)}{2}\left(\frac{2x}{\lambda} -\text{sgn}(x) \right), \quad x \in\left[-\frac{\lambda}{2},\frac{\lambda}{2}\right], \end{equation}

where $\lambda$ is the wavelength, $V_J(t)={V_0}/({1+V_0t/\lambda })$ is the velocity jump across the inviscid shock (located in $x=0$), and $V_0$ is the velocity jump at $t=0$. Khokhlov derived an exact SW solution of the diffusive Burgers equation (see Gurbatov et al. (Reference Gurbatov, Rudenko and Saichev2012)):

(3.3)\begin{equation} v=\frac{V_J(t)}{2}\left(\frac{2x}{\lambda} -\tanh{\frac{V_J(t)\,x}{4\nu}} \right) . \end{equation}

This solution is not periodic, but for large Reynolds numbers it becomes quasi-periodic between $x=-\lambda /2$ and $x=\lambda /2$, with a diffusive front located in $x=0$. The dimensionless velocity jump $\delta v_b=\left (v({\lambda }/{2}^-)-v({\lambda }/{2}^+)\right )/V_J=1-\tanh ({{R_B}/{8}})$ is an exponentially decreasing function of $R_B$. For the $R_B$ values that concern us, we can consider the solution (3.3) as periodic due to the small value of $\delta v_b$ ($\delta v_b < 10^{-11}$).

For large $R_B$, an approximate relationship of total dissipation $\boldsymbol {D}_v$ can be derived. Using (3.3) and retaining only the leading terms in an expansion in powers of $R_B$ gives

(3.4)\begin{equation} \left\langle\left(\frac{\partial v }{\partial x}\right)^2\right\rangle=\frac{1}{12}\,\frac{V_J^3 }{\nu \lambda} \end{equation}

and then

(3.5)\begin{equation} \boldsymbol{D}_v=\frac{1}{6}\,\frac{V_J^3}{\lambda} . \end{equation}

This shows that for $R_B$ of interest, the total dissipation is virtually independent of $\nu$.

One of the main objectives of this paper is to analyse the spectral behaviour of SWs. In the case of periodic waves, it is possible to derive an expression for the energy spectral density from the Khokhlov solution (3.3) (see § A.1 in the supplementary material available at https://doi.org/10.1017/jfm.2023.878). This expression is written as

(3.6)\begin{equation} {\mathcal{E}}_{v_n}=2\nu^2 k_p^2\,\text{csch}^2\left( \frac{k_n}{k_\nu} \right), \end{equation}

where ${\mathcal {E}}_{v_n}={v_n^2}/{2}$ is the energy spectral density, $v_n$ is the $n$th Fourier coefficient of $v$, $k_p={2{\rm \pi} }/{\lambda }$, $k_n=n k_p$ and $k_\nu ={V_J}/{2{\rm \pi} \nu }$. For $k_n/k_\nu \ll 1$, ${\mathcal {E}}_{v_n}$ follows a $k_n^{-2}$ power law:

(3.7)\begin{equation} {\mathcal{E}}_{v_n}=\frac{2V_J^2 }{\lambda^2}\,k_n^{{-}2}. \end{equation}

This last relation is also the exact energy spectral density of the non-diffusive SW solution (3.2). For $k_n/k_\nu \gg 1$, ${\mathcal {E}}_{v_n}$ decreases exponentially with $k_n$. In the following, the inertial and diffusive subranges will be defined respectively as $k\in [k_p,k_\nu ]$ and $k\in [k_\nu,\infty ]$. The dimensionless width of the inertial subrange, $(k_\nu -k_p)/k_p$, increases linearly with $R_B$, since $k_\nu /k_p=R_B/(4{\rm \pi} ^2)$.

Substituting the Fourier series representation of $v$ into the Burgers equation, we obtain the spectral energy equation

(3.8)\begin{equation} \frac{{\rm d}}{{\rm d}\,t}{\mathcal{E}}_{v_n}+{\mathcal{T}}_{v_n}={-}{\mathcal{D}}_{v_n} , \end{equation}

where ${\mathcal {D}}_{v_n}=2\nu k_n^2 {\mathcal {E}}_{v_n}$ is the dissipation spectrum, and ${\mathcal {T}}_{v_n}$ the nonlinear transfer function. This last can be expressed as

(3.9)\begin{equation} {\mathcal{T}}_{v_n}=\frac{k_n}{4}\,v_n \sum_{m=1}^{n-1} v_m v_{n-m}-\frac{k_n}{2}\,v_n\sum_{m=1}^\infty v_m v_{n+m} , \end{equation}

where the first term on the right-hand side represents the triad sum interactions of components $(m,n-m)$, and the second term represents the triad difference interactions of components $(m,n+m)$.

In order to better understand the spectral behaviour of dissipation and nonlinear interactions, we analyse the different terms in (3.8). Substituting (3.6) into ${\mathcal {D}}_{v_n}=2\nu k_n^2 {\mathcal {E}}_{v_n}$, we get the dissipation spectrum

(3.10)\begin{equation} {\mathcal{D}}_{v_n}=4\nu^3 k_p^2 k_n^2\,\text{csch}^2\left( \frac{k_n}{k_\nu} \right) , \end{equation}

which is a decreasing function of $k_n$. For $k_n/k_\nu \ll 1$, the dissipation spectrum given by

(3.11)\begin{equation} {\mathcal{D}}_{v_n}=\frac{4\nu V_J^2 }{\lambda^2} \end{equation}

is constant. The nonlinear transfer function can be obtained from ${\mathcal {T}}_{v_n}=-(({{\rm d}}{\mathcal {E}}_{v_n}/{{\rm d}\,t}) + {\mathcal {D}}_{v_n})$, which results in

(3.12)\begin{equation} {\mathcal{T}}_{v_n}={-}2\nu k_n \left(k_n-k_p \coth \left( \frac{k_n}{k_\nu} \right) \right) {\mathcal{E}}_{v_n} . \end{equation}

Figure 3 illustrates the contribution of each term in the spectral energy equation (3.8). We consider two SW fields with the same $V_J$ and $\lambda$ but two contrasting Reynolds numbers $R_B=100$ and $R_B=500$ (respectively, minimum and maximum $R_B$ values observed in the ISZ experiments discussed in § 5). This means that we analyse two similar wave fields that have almost the same shape except at the wave front. Figure 3 shows that the temporal rate of change of the energy spectrum, ${{\rm d}}{\mathcal {E}}_{v_n}/{{\rm d}\,t}$, is virtually independent of $R_B$. This temporal rate of change follows the approximate relation for large $R_B$

(3.13)\begin{equation} \frac{{\rm d}}{{\rm d}\,t}{\mathcal{E}}_{v_n}={-}\frac{\boldsymbol{D}_v}{\boldsymbol{E}_v}\,{\mathcal{E}}_{v_n} , \end{equation}

where $\boldsymbol {E}_v={V_J^2}/{12}$, and $\boldsymbol {D}_v$ and ${\mathcal {E}}_{v_n}$ are given by (3.5) and (3.7), respectively. This explains why the energy spectrum shape is practically preserved as SWs evolve with time. Contrary to ${{\rm d}}{\mathcal {E}}_{v_n}/{{\rm d}\,t}$, the dissipation spectrum ${\mathcal {D}}_{v_n}$ and the nonlinear transfer function ${\mathcal {T}}_{v_n}$ are strongly dependent on $R_B$. For the highest $R_B$ ($R_B=500$), figure 3 shows that ${\mathcal {T}}_{v_1} \gg {\mathcal {D}}_{v_1}$, meaning that for the first wave mode (the most energetic), the rate of change of the energy spectrum is controlled mainly by energy transfer to higher wavenumbers. By contrast, for the lower $R_B$ ($R_B=100$), ${\mathcal {T}}_{v_1}$ is of the same order of magnitude as ${\mathcal {D}}_{v_1}$, thus the nonlinear energy transfer and the energy dissipation contribute nearly equally to the energy decrease with time. We can see in figure 3 that ${\mathcal {D}}_{v_n}$ is nearly constant in the inertial subrange following (3.11), and decreases strongly in the high wavenumber tail of the spectrum (i.e. the diffusive subrange). In the diffusive subrange, $\lvert {{\rm d}}{\mathcal {E}}_{v_n}/{{\rm d}\,t} \rvert$ is much smaller than ${\mathcal {D}}_{v_n}$ and $\lvert {\mathcal {T}}_{v_n}\lvert$. This means that for high wavenumbers, there is a balance between the dissipation and the nonlinear energy transfer from low wavenumbers. This result seems in line with surf zone field observations by Herbers, Russnogle & Elgar (Reference Herbers, Russnogle and Elgar2000).

Figure 3.

Dissipation spectrum and nonlinear transfer function for two SW solutions with the same $V_J$ and $\lambda$ but distinct Reynolds numbers: $R_B=100$ (continuous lines) and $R_B=500$ (dash-dotted lines). Red lines indicate ${\mathcal {D}}_{v_n}$ (3.10); black lines indicate ${\mathcal {T}}_{v_n}$ (3.12); blue lines indicate ${{\rm d}}{\mathcal {E}}_{v_n}/{{\rm d}\,t} =-({\mathcal {D}}_{v_n}+ {\mathcal {T}}_{v_n} )$; dotted lines indicate positions of $k_\nu /k_p$. For the sake of clarity, we use lines (continuous or dash-dotted) to represent the discrete spectra.

To summarize this part, ${{\rm d}}{\mathcal {E}}_{v_n}/{{\rm d}\,t}$ is governed primarily not by nonlinear interactions, even at low $k_n$, but rather by the competition between ${\mathcal {D}}_{v_n}$ and ${\mathcal {T}}_{v_n}$, which are both strongly dependent on diffusion processes at wave fronts. This conclusion, obtained for periodic SW, cannot be applied directly to ISZ random SW, which are a more complex phenomenon. However, these results may give ideas on how diffusive processes could affect the dissipation and the nonlinear interactions of ISZ SWs.

3.3. Random sawtooth waves

We now consider freely decaying random solutions $v(x,t)$ that are statistically homogeneous in space with zero mean. The power spectral density $\varPhi (k,t)$ is the Fourier transform of the autocorrelation function $R(r)$:

(3.14)\begin{equation} \varPhi(k,t)=\frac{1}{2{\rm \pi}} \int_{-\infty}^\infty R(r) \exp(-{\rm i}kr)\,{\rm d}r , \end{equation}

where $R(r,t)=\langle v(x,t)\,v(x+r,t)\rangle$. As $\varPhi (k,t)$ is an even function of $k$, we limit our analysis to $k \geq 0$, and denote $E_v(k,t)=2 \varPhi (k,t)$ the power spectral density function. With this notation, the total energy $\boldsymbol {E}_v=\langle v^2\rangle =\int _{-\infty }^\infty \varPhi (k)\, {\rm d}k$ can be expressed as

(3.15)\begin{equation} \boldsymbol{E}_v=\int_0^\infty E_v(k) \,{\rm d}k . \end{equation}

Throughout the paper, $E_v$ will denote the energy spectrum. In the spectral space, the energy equation is given by

(3.16)\begin{equation} \frac{\partial E_v }{\partial t}+T_v={-}D_v , \end{equation}

where $D_v(k,t)=2\nu k^2\,E_v(k,t)$ is the spectral energy dissipation, $T_v(k,t)=-k\int _{-\infty }^\infty \mathrm{Im} (B)\,{\rm d}l$ is the nonlinear transfer term, and $B(k,l,t)$ is the bispectrum.

Saffman (Reference Saffman1968) assumed that the periodic solution (3.3) reproduces the qualitative features of the small-scale behaviour of random SWs. From this hypothesis, he derived an expression for the energy spectrum, which is consistent with (3.6) obtained for the energy density function of periodic SWs. The equation for the energy spectrum is given by

(3.17)\begin{equation} E_v(k,t)=2\nu^2\,k_m(t)\,\text{csch}^2 \left(\frac{k}{k_\nu(t)} \right), \quad k\in[k_m,\infty] , \end{equation}

where $k_\nu ={V_c}/{2 {\rm \pi}\nu }$ is the diffusive wavenumber, $V_c$ is the characteristic scale of $v$-jumps at wave fronts, $k_m=2{\rm \pi} /\lambda _m$, and $\lambda _m$ is the mean distance between adjacent wave fronts. The Reynolds number $R_B={V_c\lambda _m}/{\nu }$ can be expressed as $R_B=4{\rm \pi} ^2({k_\nu }/{k_m})$. The derivation of Saffman (Reference Saffman1968) equation is presented in § A.2 of the supplementary material, where we have both clarified the definition of the characteristic horizontal length scale and corrected an error in the hyperbolic cosecant term. This theory applies to short waves (i.e. SW scale) and not to wavenumbers smaller than approximately $k_m$. For the sake of clarity, explicit reference to time has been omitted in the following equations.

For periodic SWs, the definition of the characteristic scale of $v$-jumps, $V_J$, is straightforward. It is the non-diffusive $v$-jump associated with the diffusive SW solution (3.3). The characteristic $v$-scale, $V_c$, in random SW fields is not explicit. It can be obtained implicitly from $\tilde {\boldsymbol {E}}_v=\int _{k_m}^\infty E_v(k) \,{\rm d}k$, the total energy over the wavenumber range $[k_m,\infty ]$. By integrating (3.17) from $k_m$ to $\infty$, we obtain

(3.18)\begin{equation} \tilde{\boldsymbol{E}}_v=2\nu^2 k_m k_\nu \left(\coth(k_m/k_\nu)-1\right) . \end{equation}

Given $k_m$ and $\tilde {\boldsymbol {E}}_v=\int _{k_m}^\infty E_v(k) \,{\rm d}k$, the implicit equation (3.18) yields $k_\nu$ and then $V_c$. Thus, for a given random SW field characterized by $k_m$ and $\tilde {\boldsymbol {E}}_v$, (3.17) can predict the energy spectrum.

For $k \ll k_\nu$, $E_v(k)$ follows a $k^{-2}$ power law independent of the diffusion coefficient $\nu$,

(3.19)\begin{equation} E_v(k)=\left(\frac{k_m V_c^2}{2{\rm \pi}^2} \right)k^{{-}2} , \end{equation}

and for $k \gg k_\nu$, $E(k)$ decreases exponentially with $k$,

(3.20)\begin{equation} E_v(k)=8\nu^2 k_m \exp \left({-}2\,\frac{k}{k_\nu} \right) . \end{equation}

The dissipation spectrum $D_v(k,t)=2\nu k^2\,E_v(k,t)$ is given by

(3.21)\begin{equation} D_v(k)=4\nu^3 k_m k^2\,\text{csch}^2 \left(\frac{k}{k_\nu} \right), \quad k\in[k_m,\infty] , \end{equation}

and its asymptotic form for small $k$ is

(3.22)\begin{equation} D_v(k)=\frac{\nu k_m V_c^2}{{\rm \pi}^2} . \end{equation}

It is worth noting that at large scale, the dissipation spectrum is constant, which means that there is equipartition of dissipation over the spectrum. The exponential decay of $E_v(k)$ at large wavenumbers guarantees that the $k$-integrated dissipation, $\tilde {\boldsymbol {D}}_v=\int _{k_m}^\infty D_v(k) \,{\rm d}k$, is finite. To my knowledge, the theoretical model (3.17) has never been validated. In order to test the validity of this model, we have computed numerically random SW solutions of the Burgers equation. It is solved with a spectral method where aliasing is avoided by using the so-called $3/2$ rule. The initial random $v$-field, $v_0(x)=v(x,0)$, is specified in the Fourier space by the energy spectrum

(3.23)\begin{equation} E_0(k)=E_0\,\frac{k}{k_p}\exp\left(-\frac{1}{2}\,((k/k_p)^2-1 )\right) , \end{equation}

where $k_p=2{\rm \pi} /\lambda _p$ is the peak wavenumber, and $E_0$ is the energy at $k_p$. The phase of each wavenumber is assigned a random value in the range $[0,2{\rm \pi} ]$. The length of the domain covers $2^8 \lambda _p$ and is discretized over $2^{16}$ grid points. The computed spectra $E_v(k,t)$ are estimated by an ensemble average over 1000 realizations.

Figure 4 illustrates the time evolution of the energy spectrum. The cascade of energy towards high wavenumbers, due to nonlinear triad interactions, leads to a rapid transition to an established SW regime. In this regime, the theoretical law (3.17) gives a very good description of the numerically computed spectra. The inertial and diffusive subranges are well represented by the $k^{-2}$ power law (3.19) and by an exponential decrease (3.20), respectively. However, (3.17) tends to slightly underestimate the numerically computed spectrum at large wavenumbers ($k>20 k_p$), whatever the numerical resolution. Contrary to the monochromatic case, the wavenumber $k_m$ is not constant and decreases with time due to shock merging. Consequently, the inertial subrange width, and thus the Reynolds number, decreases at a slower rate than in the monochromatic case. The time evolution of the dissipation spectrum is shown in figure 5. We observe very good agreement between the theoretical law (3.21) and the numerically computed dissipation spectra. In the inertial subrange, the dissipation is constant, in agreement with the non-diffusive SW theory (3.22). Then, in the diffusive subrange, the dissipation decreases exponentially. It is worth noting that for Burgers turbulence, the dissipation occurs mainly in the inertial subrange, unlike the classical hydrodynamic turbulence where dissipation occurs at smaller scale.

Figure 4.

Time evolution of the power spectral density for an initial condition given by (3.23). Dashed line indicates initial condition; cyan lines indicate numerical simulations at dimensionless times $t/t_*=1.45, 2.90, 4.85$ (corresponding to Reynolds numbers $R_B=443, 380, 363$), where $t_*=1/(k_p \sqrt {E_{T_0}})$; black line indicates theoretical model (3.17) starting from $k=k_m$; red line indicates $k^{-2}$ power law (3.19); red plus symbol indicates position of $k_\nu /k_p$. The length of the domain covers $2^8 \lambda _p$ and is discretized over $2^{16}$ grid points. Spectra have been averaged over 1000 realizations.

Figure 5.

Time evolution of the dissipation spectrum for an initial condition given by (3.23). Cyan lines indicate numerical simulations at dimensionless times $t/t_*=1.45, 2.90, 4.85$ (corresponding to Reynolds numbers $R_B=443, 380, 363$), where $t_*={1}/{k_p \sqrt {E_{T_0}}}$; black lines indicate the theoretical model (3.21) starting from $k=k_m$; red lines indicate (3.22). The length of the domain covers $2^8 \lambda _p$ and is discretized over $2^{16}$ grid points. Spectra have been averaged over 1000 realizations.

5.1. Description of datasets

The theoretical approach developed above is now evaluated against laboratory experiments on random wave propagation and breaking on a uniform slope. Six laboratory datasets are used: one from Mase & Kirby (Reference Mase and Kirby1993) (hereafter MK93), three from Bowen & Kirby (Reference Bowen and Kirby1994) (BK94), and two from van Noorloos (Reference van Noorloos2003) (vN03). These experiments cover a large variety of incident random wave conditions in terms of frequency peak $f_p$, wave height $H_{i}$, and spectral shape. The random waves were generated at the wave paddle using random realizations of analytical single-peaked spectra. Mase & Kirby (Reference Mase and Kirby1993) chose a Pierson–Moskowitz spectrum that can be seen in figure 6(c) (blue line). van Noorloos (Reference van Noorloos2003) used a JONSWAP spectrum with a peak-enhancement factor $\gamma =3.3$, which provides a much sharper frequency peak than that of the Pierson–Moskowitz spectrum (see blue lines in figures 6a,b). Bowen & Kirby (Reference Bowen and Kirby1994) chose a TMA spectrum, which is an extension of the JONSWAP spectrum describing waves in finite depth. In shallow water, the TMA spectrum follows an $\omega ^{-3}$ power law, which results in a broader spectrum than the JONSWAP spectrum (see blue lines in figures 6d–f). In all these experiments, the wave gauges are sampled at frequency 25 Hz. For our spectral analysis, we will consider frequency lower than half the Nyquist frequency, i.e. $\omega < \omega _{max}=40$ rad s$^{-1}$. The smallest wavelength, associated with $\omega _{max}$, is approximately 10 cm, a scale much larger than capillary wave scales. The experimental parameters of the six datasets are given in table 1.

Figure 6.

Energy spectra for (a) vN03-C3, (b) vN03-D3, (c) MK93, (d) BK94-7, (e) BK94-8, and ( f) BK94-9. Blue line indicates incident wave spectrum; grey line indicates spectrum at the breaking point; cyan line indicates ISZ spectrum in a mean water depth $h_0= 5.5$ cm; black line indicates (4.2); red line indicates (4.3). Blue cross indicates position of $\omega _m$; red cross indicates position of $\omega _\nu$.

Table 1.

Experimental parameters: $H_i$, incident wave height; $h_i$, water depth; $f_p$, peak frequency; $\gamma$, peak-enhancement factor; $s$, bottom slope; $f_s$, sampling rate.

As incoming waves shoal, energy is transferred from the most energetic portion of the energy spectrum (around the spectral peak) to both higher and lower frequencies (figure 6). At the breaking point, the spectrum shape depends strongly on the incident wave spectrum. For instance, narrow-band spectra (see figures 6a,b) develop harmonic peaks of the fundamental frequency and infragravity waves (i.e. waves with frequency lower than approximately $0.5f_p$) that are both significantly larger than those associated with broad-band incident wave spectra. By contrast, in the ISZ, the high-frequency part ($\omega > \omega _m$) of the wave spectrum always has a regular shape with an $\omega ^{-2}$ tendency in the inertial subrange, whatever the incident wave conditions. The $\omega ^{-2}$ tendency of ISZ wave spectra was already noticed by Kirby & Kaihatu (Reference Kirby and Kaihatu1997). While the shape of the high-frequency part of the ISZ spectrum is almost independent of the incident wave conditions, this is not the case for the frequency bounds of the inertial subrange. The lower bound $\omega _m$ is an increasing function of the frequency peak of the incident wave spectrum $\omega _p$. Moreover, the time evolution of $\omega _m$ is controlled in the surf zone by the wave front merging phenomenon. Tissier, Bonneton & Ruessink (Reference Tissier, Bonneton and Ruessink2017) showed that this phenomenon is favoured by the presence of infragravity waves whose intensity depends on incident wave conditions.

5.2. Assessment of the analytical wave energy spectrum

To apply the theoretical spectrum (4.2) to a laboratory dataset, it is necessary to know the three parameters $\omega _m$, $\omega _\nu$ and $\nu _c$. The parameter $\nu _c$ can be substituted by the energy $\tilde {\boldsymbol {E}}=\int _{\omega _m}^\infty E(\omega ) \,{\rm d}\omega$ using the equation

(5.1)\begin{equation} \tilde{\boldsymbol{E}}=\frac{8}{9}\,\frac{\nu_c^2}{g}\,\omega_m \omega_\nu \left(\coth(\omega_m/\omega_\nu)-1\right) . \end{equation}

For each dataset, the mean time $T_m$ between adjacent wave fronts is obtained from a wave-by-wave analysis that identifies each SW front. The energy $\tilde {\boldsymbol {E}}$ is then calculated by integrating the experimental energy spectrum from $\omega _m=2{\rm \pi} /T_m$ to the Nyquist frequency. The diffusive frequency $\omega _\nu$ is an unknown of the problem. To evaluate this frequency, we use a nonlinear least squares method to fit the measured spectrum with the theoretical one, rewritten in the form

(5.2)\begin{equation} \frac{E(\omega)}{\tilde{\boldsymbol{E}}}=\frac{\text{csch}^2\left( \dfrac{\omega}{\omega_\nu} \right)}{\omega_\nu \left(\coth(\omega_m/\omega_\nu)-1\right)} . \end{equation}

For the whole dataset, the diffusive frequency $\omega _\nu$ ranges from 14.5 to 25.8 rad s$^{-1}$ (i.e. $f_\nu =\omega _\nu /(2{\rm \pi} ) \in [2.3,4.1]$ Hz). Knowing $\omega _\nu$, we obtain finally $\nu _c$ from (5.1).

Figure 6 shows ISZ spectra measured in 5 cm water depth for the six contrasting datasets (see table 1). Whatever the random wave forcing, the ISZ spectral shape is well described by the theoretical spectrum (4.2). The $\omega ^{-2}$ tendency is identifiable on all measured spectra. However, this tendency is, of course, less marked for a small inertial subrange, i.e. small $R_B$ (e.g. see figure 6b) than for large $R_B$ (see figure 6f).

We are now interested in the evolution of the ISZ wave spectrum as waves propagate shoreward in decreasing water depth. The measured wave spectra are very well described by (4.2) irrespective of the dataset and the water depth (see figures 7 and 8, and in the supplementary material, figures 3–6). As illustrated in figure 9, by scaling measured wave spectra by $E_a={8\nu _c^2\omega _m}/{9g}$, data collapse on a single curve given by the dimensionless universal spectrum equation

(5.3)\begin{equation} \frac{E}{E_a}= \text{csch}^2\left( \frac{\omega}{\omega_\nu} \right) . \end{equation}

Spectra differ only in their dimensionless inertial range width $(\omega _\nu -\omega _m)/\omega _\nu$. We focus our analysis on two contrasting datasets: BK94-9 (figure 7) and vN03-C3 (figure 8). In both cases, we observe a decrease in $\omega _m$ as waves propagate (i.e. $h_0$ decreases), which is due to the bore merging phenomenon. This decrease is much stronger for vN03-C3 (from 3.53 to 1.99 rad s$^{-1}$) because large-amplitude infragravity waves favour bore merging (Tissier et al. Reference Tissier, Bonneton and Ruessink2017). However, as for the Burgers case (see figure 4), the shape of the energy spectrum at the SW scales, $\omega >\omega _m$, is not affected by low-frequency waves. The diffusive frequency $\omega _\nu$, which marks the transition between the inertial and diffusive subranges, increases significantly for BK94-9 (from 18.7 to 24.1 rad s$^{-1}$) and slightly for vN03-C3 (from 14.9 to 16.1 rad s$^{-1}$). For both experiments, we observe an increase in the inertial subrange width as waves propagate, and thus an increase in the Reynolds number $R_B$. This evolution is observable for all datasets, as can be seen in figure 10. This figure also shows that the Reynolds number is controlled primarily by the peak period $T_p$ of the incident wave field and is an increasing function of $T_p$. The increase of $R_B$ in decreasing water depth means that the relative importance of nonlinearities with respect to turbulent diffusion increases as waves propagate and thus as broken wave height decreases. This differs from what we have observed above for Burgers turbulence, where $R_B$ is an increasing function in $H_c$. It means that in the ISZ, the characteristic wave height $H_c$ decreases less rapidly than $\nu _c$ as $h_0$ decreases. Assuming that the turbulent diffusion coefficient is controlled mainly by $H_c$, dimensional analysis leads to the following relationship between $\nu _c$ and $H_c$:

(5.4)\begin{equation} \nu_c= \alpha_\nu g^{1/2}H_c^{3/2} , \end{equation}

where $\alpha _\nu$ is a dimensionless coefficient. Similar scalings are used commonly to estimated eddy viscosity in the surf zone (e.g. Svendsen, Schäffer & Hansen Reference Svendsen, Schäffer and Hansen1987). Figure 11 shows that the turbulent diffusion coefficient $\nu _c$ follows an $H_c^{3/2}$ power law in agreement with (5.4). However, the value of the coefficient $\alpha _\nu$ depends on incident wave conditions. The relation (5.4) implies that $R_B$ evolves approximately as $H_c^{-1/2}T_m$ and is indeed a decreasing function of $h_0$. It is worth noting that our approach applies only to the ISZ. Indeed, the increase in $\nu _c$ with $H_c$ should not be valid in the vicinity of the onset of wave breaking.

Figure 7.

Energy spectra at different locations in the ISZ for the BK94-9 experiment: (a) $x=24.50$ m, ${h_0=7.6}$ cm; (b) $x=24.72$ m, $h_0=6.8$ cm; (c) $x=24.97$ m, $h_0=6.2$ cm; (d) $x=25.22$ m, $h_0=5.5$ cm;(e) $x=25.50$ m, $h_0=4.9$ cm; and ( f) $x=25.76$ m, $h_0=4.0$ cm. Grey line indicates spectrum at the breaking point; cyan line indicates ISZ spectrum at water depth $h_0$; black line indicates (4.2); red line indicates (4.3). Blue dashed line indicates position of $\omega _m$; red dashed line indicates position of $\omega _\nu$.

Figure 8.

Energy spectra at different locations in the ISZ for the vN03-C3 experiment: (a) gauge 60, $h_0=8.4$ cm; (b) gauge 61, $h_0=7.6$ cm; (c) gauge 62, $h_0=6.8$ cm; (d) gauge 63, $h_0=6.1$ cm; (e) gauge 64, $h_0=5.3$ cm; ( f) gauge 65, $h_0=4.5$ cm; (g) gauge 66, $h_0=3.7$ cm; and (h) gauge 67, $h_0=3.0$ cm. Grey line indicates spectrum at the breaking point; cyan line indicates ISZ spectrum at water depth $h_0$; black line indicates (4.2); red line indicates (4.3). Blue dashed line indicates position of $\omega _m$; red dashed line indicates position of $\omega _\nu$.

Figure 9.

Dimensionless energy spectra $E/E_a$ ($E_a={8\nu _c^2\omega _m}/{9g}$) at different locations in the ISZ for the vN03-C3 experiment. Grey lines indicate measurements from gauges 60–67 ($h_0$ from 8.4 cm to 3 cm); black line indicates (5.3); red line indicates $\omega ^{-2}$ power law.

Figure 10.

Evolution of the Reynolds number $R_B$ as a function of water depth $h_0$.

Figure 11.

Evolution of the diffusion coefficient as a function of $g^{1/2}H_c^{3/2}$.