The QC approximation

To establish a criterion with which the truncation of the exponential series function in (2.177) is carried out, we define the following length scales. Let $L$, $L_{\rm m}$ and $L_{\rm s}$ denote the characteristic wave length, the characteristic length scale of medium variation and the characteristic length scale over which second order wave statistics vary in physical space, respectively. Since our starting point in the present analysis is Eq. (2.163), thus assuming the medium changing slowly, it is required that the ratio $\epsilon = L/L_{\rm m}$ is small. Hence, the physical validity of Eq. (2.177) remains unaffected by assuming $\epsilon \ll 1$. In addition, the statistics of the wave field (including cross correlations) is assumed to vary weakly on distances of the order of ${\cal O}(100-1000)$ wave lengths^2.10, that is, the ratio $\mu = L/L_{\rm s} \ll 1$.


Next, we introduce the correlation (or coherent) length scale of the wave field $L_{\rm c}$. It measures the correlation between two wave components separated by the distance $\vec{\xi}$. Generally, the correlation function $\Gamma (\vec{x},\vec{\xi})$ tends to zero as $\vert\vec{\xi}\vert \rightarrow L_{\rm c}$ (Smit et al., 2015a). In particular, for narrow-band waves, the wave field remains correlated over many wave lengths, whereas the coherent radius of directionally spread sea states is relatively small. So, $L_{\rm c}$ is directly related to the characteristic width of the spectrum $\Delta k$ as $L_{\rm c} = 2\pi/\Delta k$ (Smit and Janssen, 2013; Smit et al., 2015a). We note that the spectrum width $\Delta k$ may be measured by the standard deviation of the incident wave field, which is usually statistically quasi-homogeneous.


Finally, to relate the correlation length scale to the medium variation length scale, we consider the ratio $\beta = L_{\rm c}/L_{\rm m}$. This is the key for establishing a suitable approximation to Eq. (2.177). We address two cases of interest, namely, (i) $\beta \ll 1$, in which the wave field de-correlates over distances short compared to the medium variations; and (ii) $\beta = {\cal O}(1)$ where seabed or current varies rapidly within the coherent radius of the wave field.


The step in getting explicit approximations is to truncate appropriately the expansion of the exponential function in Eq. (2.177). To this end, consider a wave field at point $\vec{x}_0$ and described by the carrier wave number $\vec{k}_0$. As parameter $\beta$ indicates the relative importance of the medium variation (related to $\nabla_{\vec{x}_0}\,\omega$) with respect to the coherent radius of the wave field (related to $\nabla_{\vec{k}_0}W$), the first term in the exponential function is scaled by

  $$\beta\,\nabla_{\vec{x}}\,\omega\cdot\nabla_{\vec{k}}W\,\vert _{\vec{x}=\vec{x}_0,\vec{k}=\vec{k}_0}$$ (2.177)

Similarly, parameter $\mu$ expresses the significance of the spatial variation of the spectrum with respect to the wave length scale (related to $\nabla_{\vec{x}_0}\,W$), so that the second term of the exponential scaled as

  $$\mu\,\nabla_{\vec{k}}\,\omega\cdot\nabla_{\vec{x}}W\,\vert _{\vec{x}=\vec{x}_0,\vec{k}=\vec{k}_0}$$ (2.178)

(Note that the group velocity $\nabla_{\vec{k}}\,\omega = {\cal O}(1)$.)


Now, under the conditions that $\mu \ll 1$ and also that the wave field de-correlates on shorter scales than the scale of the medium variations, that is, $\beta \ll 1$, the Taylor series expansion around the phase space point $(\vec{x}_0,\vec{k}_0)$ can be expressed as

  $$\exp \left ( \frac{{\rm i}}{2} \overleftarrow{\nabla}_{\vec{x}}\c... ...cdot\overrightarrow{\nabla}_{\vec{x}} + {\cal O} \left ( \beta^2,\mu^2 \right)$$ (2.179)

On substitution into Eq (2.177) and neglecting higher order terms we obtain the familiar form for the action balance equation

  $$\frac{\partial W}{\partial t} + \nabla_{\vec{k}}\,\omega\cdot\nabla_{\vec{x}}\,W - \nabla_{\vec{x}}\,\omega\cdot\nabla_{\vec{k}}\,W = 0$$ (2.180)

which is thus the lowest order approximation of Eq. (2.177).


At this point, the above analysis is only limited to cases in which $\beta \ll 1$ as the truncated expansion in $\beta$ is no longer valid for $\beta = {\cal O}(1)$. A different path is presented here, paved by Smit and Janssen (2013), to extend the range of applicability of the approximate model to values of $\beta = {\cal O}(1)$.


We recall the Weyl symbol $\omega$ that reads

$$\omega(\vec{x}+{\rm i}{\nabla_{\vec{k}}}/2,\vec{k}-{\rm i}\nabla_{\vec{x}}/2) = \int \int {\hat \omega}(\vec{q},\vec{p})\,e^{{\rm i}\vec{q}\cdot\... ...ac{1}{2}\vec{q}\cdot\overrightarrow{\nabla}_{\vec{k}} \right)\,d\vec{q}d\vec{p}$$

$$= \int {\hat \omega}(\vec{q},\vec{k})\,e^{{\rm i}\vec{q}\cdot\vec{x... ...t (-\frac{1}{2}\vec{q}\cdot\overrightarrow{\nabla}_{\vec{k}} \right )\,d\vec{q}$$

with

  $${\hat \omega}(\vec{q},\vec{k}) = \int {\hat \omega}(\vec{q},\vec{p})\,e^{{\rm i}\vec{p}\cdot\vec{k}}\,d\vec{p}$$ (2.181)

Consequently, the Moyal product yields

  $$\omega(\vec{x}+{\rm i}{\nabla_{\vec{k}}}/2,\vec{k}-{\rm i}\nabla_... ...}\cdot\overrightarrow{\nabla}_{\vec{k}} \right )\,W(\vec{x},\vec{k})\,d\vec{q}$$ (2.182)

Observing that the assumption $\mu \ll 1$ still holds and that the term $\exp (-\frac{1}{2}\vec{q}\cdot\overrightarrow{\nabla}_{\vec{k}})\,W(\vec{x},\vec{k})$ equals the Taylor series of $W(\vec{x},\vec{k}-\vec{q}/2)$, the phase-space equation is given by

  $$\frac{\partial W}{\partial t}(\vec{x},\vec{k}) = -{\rm i}\int {\h... ...c{x}} \right)\, W(\vec{x},\vec{k}-\frac{\vec{q}}{2})\,d\vec{q}\, + \rm {c.c.}$$ (2.183)

Note that the integral expression is treated as a convolution between ${\hat \omega}$ and $W$ (including their derivatives). Eq. (2.184) is the central result of the paper of Smit and Janssen (2013) (see their Eq. (15). But see also Akrish et al. (2020), their Eq. (2.19), where the mean currents have been included.) By virtue of the assumption $\mu \ll 1$, they refer to this approximation as the (first order) quasi-coherent (QC) approximation.


The purpose of the rest of the work presented here is to make the QC approximation numerically more amenable by recasting Eq. (2.184) in the following form

  $$\frac{\partial W}{\partial t} + \nabla_{\vec{k}}\,\omega\cdot\nabla_{\vec{x}}\,W = S_{\rm qc}$$ (2.184)

where $S_{\rm qc}$ is a scattering source term that accounts for the generation and propagation of inhomogeneous wave field induced by medium variations, including wave refraction, Doppler shifting^2.11 and wave interference.


We revisit the convolution integral in Eq. (2.184) in order to find a form for $S_{\rm qc}$ that can be efficiently computed; it is denoted by $G(\vec{x},\vec{k})$. We consider a point $\vec{x}^\prime$ in the neighborhood of the origin. Then let ${\hat \omega}(\vec{q},\vec{k})$ be the Fourier transform of $\omega(\vec{x}^\prime,\vec{k})$, as follows

  $${\hat \omega}(\vec{q},\vec{k}) = \frac{1}{4\pi^2} \int \omega(\vec{x}^\prime,\vec{k})\,e^{-{\rm i}\vec{q}\cdot\vec{x}^\prime}d\vec{x}^\prime$$ (2.185)

Furthermore, we have (cf. Eq. (2.156))

  $$W(\vec{x},\vec{k}-\frac{\vec{q}}{2}) = \frac{1}{4\pi^2} \int \Gamma (\vec{x},\vec{\xi}) \, e^{-{\rm i}(\vec{k}-\vec{q}/2)\cdot\vec{\xi}}d\vec{\xi}$$ (2.186)

Substituting both of these Fourier transforms in the integral yields

$$G = \int {\hat \omega}(\vec{q},\vec{k})\,e^{{\rm i}\vec{q}\cdot\v... ...ow{\nabla}_{\vec{x}} \right)\, W(\vec{x},\vec{k}-\frac{\vec{q}}{2})\,d\vec{q} =$$

$$\frac{1}{16\pi^4} \int \int \int \omega(\vec{x}^\prime,\vec{k})\,... ...}/2)}\,e^{-{\rm i}\vec{k}\cdot\vec{\xi}}\,d\vec{\xi}\,d\vec{x}^\prime\,d\vec{q}$$

Since the inverse Fourier transform of the (complex) exponential is a shifted Dirac delta, the triple integral is then rewritten as

$$G = \frac{1}{16\pi^4} \int \int \omega(\vec{x}^\prime,\vec{k})\, \lef... ...(\vec{x}-\vec{x}^\prime+\frac{\vec{\xi}}{2} \Bigr)\,d\vec{\xi}\,d\vec{x}^\prime$$

$$= \frac{1}{4\pi^2} \int \omega(\vec{x}+\frac{\vec{\xi}}{2},\vec{k})... ...t)\, \Gamma (\vec{x},\vec{\xi})\,e^{-{\rm i}\vec{k}\cdot\vec{\xi}} \,d\vec{\xi}$$

The last line essentially implies that function $\omega(\vec{x}^\prime,\vec{k})$ must have a compact support in $\vert\vec{x}^\prime\vert \leq \vert\vec{\xi}\vert/2$. Since, by nature, the correlation function is compactly supported, that is, $\Gamma(\vec{\xi}) = 0$ for $\vert\vec{\xi}\vert > L_{\rm c}$ (note $L_{\rm c} \sim \Delta k^{-1}$ is finite), it is concluded that the local wave statistics can only be affected by the medium within a radius $L_{\rm c}/2$ around point $\vec{x}$.


Following Akrish et al. (2020), the dispersion relation at the point $\vec{x}+\vec{\xi}/2$ is expressed as a superposition of the local value at $\vec{x}$ and a remainder, $\omega(\vec{x}+\vec{\xi}/2,\vec{k}) = \omega(\vec{x},\vec{k})+\Delta \omega(\vec{x}+\vec{\xi}/2,\vec{k})$. Then substitution gives

$$G = \frac{1}{4\pi^2} \int \omega(\vec{x},\vec{k})\, \left (1-\frac{{\... ...\, \Gamma (\vec{x},\vec{\xi})\,e^{-{\rm i}\vec{k}\cdot\vec{\xi}} \,d\vec{\xi} +$$

$$\frac{1}{4\pi^2} \int \Delta \omega(\vec{x}+\frac{\vec{\xi}}{2},\... ...t)\, \Gamma (\vec{x},\vec{\xi})\,e^{-{\rm i}\vec{k}\cdot\vec{\xi}} \,d\vec{\xi}$$

$$= \omega(\vec{x},\vec{k})\, \left (1-\frac{{\rm i}}{2}\overleftarro... ..._{\vec{k}}\cdot\overrightarrow{\nabla}_{\vec{x}} \right)\, W(\vec{x},\vec{k}) +$$

$$\frac{1}{4\pi^2} \int \Delta \omega(\vec{x}+\frac{\vec{\xi}}{2},\... ...t)\, \Gamma (\vec{x},\vec{\xi})\,e^{-{\rm i}\vec{k}\cdot\vec{\xi}} \,d\vec{\xi}$$

We recall Eq. (2.184) and substitute the final expression with the result

  $$\frac{\partial W}{\partial t} + \nabla_{\vec{k}}\,\omega\cdot\nab... ...c{x},\vec{\xi})\,e^{-{\rm i}\vec{k}\cdot\vec{\xi}} \,d\vec{\xi}\, + \rm {c.c.}$$ (2.187)

Finally, the source term in its suitable form is obtained by transforming back

$$S_{\rm qc} = \frac{-{\rm i}}{4\pi^2} \int \Delta \omega(\vec{x}+\frac{\vec{\xi... ...ec{x},\vec{\xi})\,e^{-{\rm i}\vec{k}\cdot\vec{\xi}} \,d\vec{\xi}\, + \rm {c.c.}$$

$$= -{\rm i}\int \Delta {\hat \omega}(\vec{q},\vec{k})\,e^{{\rm i}\ve... ...vec{x}} \right)\, W(\vec{x},\vec{k}-\frac{\vec{q}}{2})\,d\vec{q}\, + \rm {c.c.}$$

with $\Delta {\hat \omega}(\vec{q},\vec{k})$ the Fourier transform of $\Delta \omega(\vec{x}^\prime,\vec{k})$. By shifting the origin to the point $\vec{x}$, the Fourier transform of $\Delta \omega(\vec{x}+\vec{x}^\prime,\vec{k})$ is then given by

  $$e^{{\rm i} \vec{q}\cdot\vec{x}}\,\Delta {\hat \omega}(\vec{q},\vec{k})$$ (2.188)

This Fourier transform is denoted by $\Delta {\hat \omega}(\vec{x},\vec{q},\vec{k})$ and is computed over a square domain with a fixed size $L_{\rm c}/2$ around the point $\vec{x}$. It should be noted that the source term $S_{\rm qc}$ is evaluated by means of integration over $\vec{q}$ and not over $\vec{\xi}$.


In summary, the evolution equation for the Wigner distribution is given by Eq. (2.185), whereas the QC scattering term that describes the evolution of the coherent structures in the wave field reads

$$S_{\rm qc} = -{\rm i}\int \Delta {\hat \omega}(\vec{x},\vec{q},\vec{k})\, \lef... ...{\nabla}_{\vec{x}} \right)\, W(\vec{x},\vec{k}-\frac{\vec{q}}{2})\,d\vec{q}\, +$$

$$+{\rm i}\int \Delta {\hat \omega}(\vec{x},\vec{q},\vec{k})\, \lef... ...rrow{\nabla}_{\vec{x}} \right)\, W(\vec{x},\vec{k}+\frac{\vec{q}}{2})\,d\vec{q}$$ (2.189)