Equations for the wave-averaged three-dimensional momentum equations have been published in this journal. It appears that these equations are not consistent with the known depth-integrated momentum balance, especially over a sloping bottom. These equations should thus be considered with caution as they can produce erroneous flows, in particular outside of the surf zone. It is suggested that the inconsistency in the equations may arise from the different averaging operators applied to the different terms of the momentum equation. It is concluded that other forms of the momentum equations, expressed in terms of the quasi-Eulerian velocity, are better suited for three dimensional modelling of wave-current interactions.
Deep Dive into Comments on "The Depth-Dependent Current and Wave Interaction Equations: A Revision".
Equations for the wave-averaged three-dimensional momentum equations have been published in this journal. It appears that these equations are not consistent with the known depth-integrated momentum balance, especially over a sloping bottom. These equations should thus be considered with caution as they can produce erroneous flows, in particular outside of the surf zone. It is suggested that the inconsistency in the equations may arise from the different averaging operators applied to the different terms of the momentum equation. It is concluded that other forms of the momentum equations, expressed in terms of the quasi-Eulerian velocity, are better suited for three dimensional modelling of wave-current interactions.
The wave-averaged conservation of momentum can take essentially two forms, one for the mean flow momentum only, and the alternative form for the full momentum, which includes the wave pseudo-momentum (hereinafter 'wave momentum ', see McIntyre 1981). This question is well known for depth-integrated equations (Longuet-Higgins and Stewart 1964;Garrett 1976;Smith 2006), but the vertical profiles of the mass and momentum balances are more complex. The pioneering effort of Mellor (2003, hereinafter M03) produced practical wave-averaged for the total momentum that, in principle, may be used in primitive equation models to investigate coastal flows, such as the wave-driven circulations observed by Lentz et al. (2008). The first formulation (Mellor 2003) was slightly inconsistent due to the improper approximation of wave motion with Airy wave theory, which is not enough on a sloping bottom, however small the slope may be. This question was discussed by Ardhuin et al. (2008), and a correction was given and verified. These authors acknowledged that these equations, when using the proper approximation, are not well suited for practical applications because very complex wave models are required for the correct estimation of the vertical fluxes of wave momentum, that are part of the fluxes of total momentum.
Although M03 gave correct wave-forcing expressionsin terms of velocity, pressure and wave-induced displacement, before any approximation - Mellor (2008, hereinafter M08) derived a new and different solution from scratch.
The two theories may be consistent over a flat bottom, but they differ at their lowest order over sloping bottoms, so that the M08 equations are likely to be flawed, given the analysis of M03 by Ardhuin et al. (2008), and the fact that their consistency was not verified numerically over sloping bottoms.
Instead, M08 asserted that the equations are consistent with the depth-integrated equations of Phillips (1977). Further, about the test case proposed by Ardhuin et al. (2008), M08 stated that the wave energy was unchanged along the wave propagation and that the resulting wave forcing whould be uniform over the depth. Here we show that the M08 equations do not yield the known depthintegrated equations (Phillips 1977) with a difference that produces very different mean sea level variations when waves propagate over a sloping bottom. As for the test case of proposed by Ardhuin et al. (2008), we show that a consistent analysis should take into account the small but significant change in wave energy due to shoaling. In the absence of dissipative processes, the M08 equations can produce spurious velocities of at least 30 cm/s, with 1 m high waves over a bottom slope of the order of 1% in 4 m water depth.
For simplicity we consider motions limited to a vertical plane (x, z) with constant water density and no Coriolis force nor wind stress or bottom friction. The wave-averaged momentum equation in M08 takes the form
and the continuity equation is
Where U and W are the Lagrangian mean velocity components, which contains the current and Stokes drift velocities, g is the acceleration due to gravity and η is the timeaveraged water level at the horizontal position x. The force given by M08 on the right hand side of (1) can be written as the sum
of a wave-induced pressure gradient
and the divergences of the horizontal flux of wave momentum,
where E is the wave energy, k is the wavenumber, u and w are respectively the horizontal and vertical wave-induced (orbital) velocities. E D is defined by
Using the mean water depth D, and bottom elevation -h, F CC , F SS and F SC are non-dimensional functions of kz and kD,
The depth-averaged mass-transport velocity is
with S P 77 xx given by Phillips (1977). However, for a depthuniform U , the depth integrated momentum equation in Phillips (1977) is
The forcing in the depth-integration of (1) differs from the forcing in ( 14), because the gradient is inside of the integral, namely,
The depth integral of M08 thus includes two extra term. In particular S M08 xx (z = -h)∂h/∂x = -2kE(∂h/∂x)/ sinh(2kD) can be dominant over a sloping bottom. As a result the momentum balance in M08, unlike M03, does not produce the known set-down and set-up. This is illustrated in figure 1. We take the case proposed by Ardhuin et al. (2008) with steady monochromatic waves shoaling on a slope without breaking nor bottom friction and for an inviscid fluid, conditions in which exact numerical solutions are known. The bottom slopes smoothly from a depth D = 6 to D = 4 m. We added a symmetric slope back down to 6 m to allow periodic boundary conditions if needed. For a wave period of 5.24 s the group velocity varies little from 4.89 to 4.64 m s -1 , giving a 2.7% increase of wave amplitude on the shoal. Contrary to statements in M08, ∂E/∂x is significant, with a 5.4% change of E over a few wavelengths.
From the Eulerian analysis of that situation (e.g. Longuet-Higgins 1967), the mean water
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
