... 1992a)2.1
In Eq. (10) of Tolman (1992a) the power of $10^{\rm -5}$ should be $10^{\rm -3}$; H. Tolman, personal communication, 1995. 
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... 1972)2.2
Collins (1972) contains an error in the expression due to an erroneous Jacobian transformation. See page A-16 of Tolman (1990). 
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... 2012)2.3
The time evolution of wave action of a wave packet is naturally described as evolving along a trajectory (or ray) through the phase space $(\vec{x},\vec{k})$. In this sense, the local wave number vector $\vec{k}$ conjugates to the position $\vec{x}$ owing to the linear dispersion relation of water waves, assuming a slowly varying medium in physical space (McDonald, 1988). The variables $\vec{x}$ and $\vec{k}$ are called the canonical coordinates in phase space with the components of $\vec{x}$ to be the usual Cartesian coordinates $(x,y)$ and the components of $\vec{k}$ to be the conjugate momenta $k_x$ and $k_y$. 
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... by2.4
Unless otherwise stated, integrals are with infinite limits. 
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... exists2.5
Likewise, the marginal $\int W(\vec{x},\vec{k})\, d\vec{x}$ yields the distribution $\left < {\hat \eta}\,{\hat \eta}^\star \right >(\vec{k})$ described in spectral space. 
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... elevation2.6
Strictly speaking, in the presence of the ambient current, the so-called action variable $\psi(\vec{x},t)$ should be employed instead of $\eta(\vec{x},t)$. This variable is characterized by its surface elevation $\eta(\vec{x},t)$ and surface potential $\phi(\vec{x},t)$. Without reproducing the rather involved definition of $\psi$ (see Akrish et al. (2020), their Eq. (2.2)), we just keep the notion of $\eta$ here as it is helpful to comprehend the rest of this section without further consequences. 
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... by2.7
A pseudo-differential operator is the inverse Fourier transform of the multiplication of ${\hat \eta}$ by a symbol (here $\omega$) in Fourier space and can be considered as a generalized differential operator. This is evidenced by the fact that the Fourier transform of a pseudo-differential operator acting on a function living in physical space can be expressed as the Fourier multiplier operator in Fourier space, that is, given the pseudo-differential operator $\Omega(-{\rm i}\nabla_{\vec{x}})$ (or a special case, the derivative $-{\rm i}\nabla_{\vec{x}}$) with symbol $\omega(\vec{k})$ (or $\vec{k}$ in the special case) we have ${\widehat {\Omega\,f}}(\vec{x}) = \omega(\vec{k})\,{\hat f}(\vec{k})$ for any smooth compactly supported function $f(\vec{x})$ and its Fourier transform ${\hat f}(\vec{k})$. The use of pseudo-differential operators is of great importance in describing the dynamics of water waves and wave functions in quantum mechanics. 
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... have2.8
We suppress the variable $t$ in the argument of $\eta$ and other functions for the convenience of presentation. 
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... identity2.9
If operators $A$ and $B$ do not commute, that is, $[A,B] \equiv AB - BA \neq0$, but they commute with their commutator, $[A,[A,B]] = [B,[A,B]] = 0$, then the BCH formula is given by $e^{A+B} = e^A\,e^B\,e^{-[A,B]/2}$. This is a commonly used formula in quantum mechanics. 
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... lengths2.10
As a matter of fact, the cross correlations are due to the interaction with a medium that varies slowly in space. 
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... shifting2.11
Wave refraction and Doppler shifting can be modelled explicitly through the term $-\nabla_{\vec{x}}\,\omega\cdot\nabla_{\vec{k}}\,W$ added to the left hand side of Eq. (2.225), as proposed in Smit et al. (2015a) using the local plane approximation, see pg. 1142 of their paper. However, for a number of reasons that will become clear in Section 3.9, we will not do so here. 
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... definition3.1
In the present context, we consider the discretization of the divergence operator applied to the energy flux. See Zijlema (2021) for details. 
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... frequency3.2
Waves can be blocked by the current at a relative high frequency. 
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... ones3.3
Although in SWAN the number of sweeps equals 4 and is hard-coded. 
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... 1993)3.4
The equivalent situation for such an equation is to have eigenvalues of very different magnitudes. 
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... 1992)3.5
It is noted here that the effective $\gamma$ used in SWAN is not equivalent to that of WAM: the former is a factor $2\pi$ larger. 
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... criterion3.6
The Lipschitz criterion is well known in the field of semi-Lagrangian schemes and its interpretation is that trajectories do not cross each other during one Lagrangian time step. See e.g. Smolarkiewicz and Pudykiewicz (1992) and Lin and Rood (1996). 
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... rate3.7
The spatial turning rate is the change in wave direction per unit forward distance $\ell$ that is travelled by the wave energy in a time interval $\ell/c_g$, and thus represents the curvature of the wave ray. This is equivalent to $c_\theta $, which is the turning rate of the wave direction per unit time. 
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... waves3.8
According to the Snel's law a wave direction with respect to the normal of a coastline within a directional bin $\Delta \theta$ can not turn more than $\Delta \theta$. 
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... software7.1
Available from http://www-unix.mcs.anl.gov/mpi/mpich. 
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