Spectral action balance equation

All information about the sea surface is contained in the wave variance spectrum or energy density $E(\sigma,\theta)$, distributing wave energy over (radian) frequencies $\sigma$ (as observed in a frame of reference moving with current velocity) and propagation directions $\theta$ (the direction normal to the wave crest of each spectral component).


Usually, wave models determine the evolution of the action density $N(\vec{x},t;\sigma,\theta)$ in space $\vec{x}$ and time $t$. The action density is defined as $N = E/\sigma$ and is conserved during propagation along its wave characteristic in the presence of ambient current, whereas energy density $E$ is not (Whitman, 1974). Wave action is said to be adiabatic invariant.


The rate of change of the action density $N$ at a single point in space $(\vec{x};\sigma,\theta)$ is governed by the action balance equation, which reads (e.g., Mei, 1983; Komen et al., 1994)

  $$\frac{\partial N}{\partial t} + \nabla_{\vec{x}} \cdot [({\vec{c}... ...a} + \frac{\partial c_\theta N}{\partial \theta} = \frac{S_{\rm tot}}{\sigma}$$ (2.16)

The left hand side is the kinematic part of this equation. The second term denotes the propagation of wave energy in two-dimensional geographical $\vec{x}$-space, including wave shoaling, with the group velocity ${\vec{c}}_g = \partial \sigma /\partial \vec{k}$ following from the dispersion relation ${\sigma}^2 = g\vert\vec{k}\vert\tanh(\vert\vec{k}\vert d)$. The third term represents the effect of shifting of the radian frequency due to variations in depth and mean currents. The fourth term represents depth-induced and current-induced refraction. The quantities $c_\sigma$ and $c_\theta$ are the propagation velocities in spectral space $(\sigma,\theta)$. Notice that the second, third and fourth terms are divergence terms representing the amount of flux entering or leaving a point, and hence, they act as source (negative divergence, i.e. flux entering a point) or sink (positive divergence, i.e. flux leaving a point) terms. The right hand side contains $S_{\rm tot}$, which is the non-conservative source/sink term that represents all physical processes which generate, dissipate, or redistribute wave energy at a point. They are defined for energy density $E(\sigma,\theta)$ (i.e. not wave action). Details are given in Section 2.3.


At deep water without ambient current, Equation (2.16) is reduced to

  $$\frac{\partial E}{\partial t} + \nabla_{\vec{x}} \cdot ({\vec{c}}_g E) = S_{\rm tot}$$ (2.17)

which can be considered as a ray equation for a wave packet propagating along its wave ray. In the absence of the generation and dissipation of waves, wave energy is conserved along its propagation path, which implies that the net flux of wave energy along this path is conserved (i.e. the divergence of this flux is zero). This is known as the law of constant energy flux along the wave ray (Burnside, 1915; Whitham, 1974, pg. 245). This law is essentially the bedrock on which the discretization of the action balance equation has been built. This will be discussed in Section 3.2.1.


It must be noted that the second term in the left hand side of Eq. (2.17) should not be interpreted as the transport of $E$ (being a transported quantity) with a transport velocity ${\vec{c}}_g$. The underlying reason is that the group velocity is generally not divergence free. Instead, we rewrite Eq. (2.17) as follows

  $$\frac{\partial E}{\partial t} + {\vec{c}}_g \cdot \nabla_{\vec{x}} \, E + E\, \nabla_{\vec{x}} \cdot {\vec{c}}_g = S_{\rm tot}$$ (2.18)

The second term in the left hand side represents the actual transport of $E$ along the wave ray with velocity ${\vec{c}}_g$ and the third term can be considered as a source or sink term with respect to energy density $E$; this density can be created (shoaling) or destroyed (de-shoaling) along the wave ray. This is due to the change in the group velocity along this ray. The correct interpretation of the second term of Eq. (2.17) is the divergence of the energy flux ${\vec{c}}_g E$, i.e. the net energy flux per unit square (it measures the flux source or sink at a point). The space discretization, as will be described in Section 3.2.1, is based on this interpretation.


Equation (2.16) can be recasted in Cartesian or spherical co-ordinates. For small scale applications the spectral action balance equation may be expressed in Cartesian co-ordinates as given by

  $$\frac{\partial N}{\partial t} + \frac{\partial c_x N}{\partial x}... ...a} + \frac{\partial c_\theta N}{\partial \theta} = \frac{S_{\rm tot}}{\sigma}$$ (2.19)

With respect to applications at shelf sea or oceanic scales the action balance equation may be recasted in spherical co-ordinates as follows

  $$\frac{\partial \tilde{N}}{\partial t} + \frac{\partial c_\lambda ... ...ial \tilde{c}_\theta \tilde{N}}{\partial \theta} = \frac{S_{\rm tot}}{\sigma}$$ (2.20)

with action density $\tilde{N}$ with respect to longitude $\lambda$ and latitude $\varphi$. Note that $\theta$ is the wave direction taken counterclockwise from geographic East. The propagation velocities are reformulated as follows. On a sphere, we have

$$dx = R \cos \varphi d\lambda$$ (2.21)

$$dy = R d\varphi$$

with $R$ the radius of the earth. The propagation velocities in geographic space are then given by

$$\frac{d\lambda}{dt} = c_\lambda = \frac{1}{R\cos \varphi} \left [... ...igma \vert\vec{k}\vert \cos \theta}{{\vert\vec{k}\vert}^2} + u_\lambda \right ]$$ (2.22)

$$\frac{d\varphi}{dt} = c_\varphi = \frac{1}{R} \left [ \frac{1}{2}... ...igma \vert\vec{k}\vert \sin \theta}{{\vert\vec{k}\vert}^2} + u_\varphi \right ]$$

with $u_\lambda$ and $u_\varphi$ the ambient currents in longitude and latitude direction, respectively. The propagation velocity in $\sigma-$space remain unchanged. To rewrite the propagation velocity $\tilde{c}_\theta$ in terms of spherical co-ordinates, we use the so-called Clairaut's equation that states that on any geodesic, the following expression holds:

  $$R \cos \varphi \cos \theta = \mbox{constant}$$ (2.23)

Differentiation of Eq. (2.23) with respect to a space co-ordinate $s$ in wave direction gives

  $$-R \sin \varphi \cos \theta \frac{d\varphi}{ds} - R \cos \varphi \sin \theta \frac{d\theta}{ds} = 0$$ (2.24)

Since, $dy = ds \sin \theta$, we have $d\varphi/ds = \sin\theta/R$. Substitution into Eq. (2.24) and using $ds=(c_x \cos \theta + c_y \sin \theta)dt$ yields

  $$\frac{d\theta}{dt} = -\frac{c_x \cos \theta + c_y \sin \theta}{R} \cos \theta \tan \varphi$$ (2.25)

This term (2.25) accounts for the change of propagation direction relative to true North when travelling along a great circle. This holds for deep water and without currents. Hence,

  $$\tilde{c}_\theta = c_\theta - \frac{c_x \cos \theta + c_y \sin \theta}{R} \cos \theta \tan \varphi$$ (2.26)

In Eq. (2.20), $\tilde{N}$ is related to the action density $N$ in a local Cartesian frame $(x,y)$ through $\tilde{N} d\sigma d\theta d\varphi d\lambda = N d\sigma d\theta dx dy$, or $\tilde{N} = NR^2 \cos \varphi$. Substitution into (2.20) yields:

  $$\frac{\partial N}{\partial t} + \frac{\partial c_\lambda N}{\part... ...ac{\partial \tilde{c}_\theta N}{\partial \theta} = \frac{S_{\rm tot}}{\sigma}$$ (2.27)