Explicit wave-averaged primitive equations using a generalized Lagrangian mean

Abstract

The generalized Langrangian mean theory provides exact equations for general wave-turbulence-mean flow interactions in three dimensions. For practical applications, these equations must be closed by specifying the wave forcing terms. Here an approximate closure is obtained under the hypotheses of small surface slope, weak horizontal gradients of the water depth and mean current, and weak curvature of the mean current profile. These assumptions yield analytical expressions for the mean momentum and pressure forcing terms that can be expressed in terms of the wave spectrum. A vertical change of coordinate is then applied to obtain glm2z-RANS equations (55) and (57) with non-divergent mass transport in cartesian coordinates. To lowest order, agreement is found with Eulerian-mean theories, and the present approximation provides an explicit extension of known wave-averaged equations to short-scale variations of the wave field, and vertically varying currents only limited to weak or localized profile curvatures. Further, the underlying exact equations provide a natural framework for extensions to finite wave amplitudes and any realistic situation. The accuracy of the approximations is discussed using comparisons with exact numerical solutions for linear waves over arbitrary bottom slopes, for which the equations are still exact when properly accounting for partial standing waves. For finite amplitude waves it is found that the approximate solutions are probably accurate for ocean mixed layer modelling and shoaling waves, provided that an adequate turbulent closure is designed. However, for surf zone applications the approximations are expected to give only qualitative results due to the large influence of wave nonlinearity on the vertical profiles of wave forcing terms.

Effects of waves on coastal water dispersion in a small estuarine bay

“…The obtained set of equations is nondivergent, thanks to a transformation of the vertical coordinate. The A08 formulation is consistent with Mc Williams et al [2004] to the considered order of approximation in the limit of a weak mean-current vertical shear [Ardhuin et al, 2008b]. Bennis et al [2011] confirmed the ability of A08 formulation to provide the vertical structure of the flow by numerical modeling of an adiabatic configuration.…”

“…The new code version solves the glm2z asymptotic formulation of the GLM wave-current equations for the quasi-Eulerian momentum, as proposed by Ardhuin et al [2008b] and further adapted by Bennis et al [2011]. The quasi-Eulerian velocity field, noted u5ðû 1 ;û 2 ;ŵÞ, is defined as [Jenkins, 1989] …”

“…It should be noted that the vertical coordinate change used by Ardhuin et al [2008b] corrects the quasi-Eulerian free surface position z5ĝ for the Stokes correction of g, so that it is equal to the local phaseaveraged free surface position z5g.…”

Morphological and sedimentological response of a mixed-energy barrier island tidal inlet to storm and fair-weather conditions

“…The ongoing debate about the vertical distribution of wave-induced radiation stresses that generally split up into a surface component, a bottom component and a body force and their implementation within 3-D momentum equations (discussed in, for example, Ardhuin and Roland, 2013;Ardhuin et al, 2008;Bennis et al, 2011) reflects on and indicates that important wave-induced processes interacting with the flow circulation may still be inadequately implemented in Delft3D. These model limitations are, however, accepted in the present study assuming minor effects on the sedimentology and morphology at the tidal inlet.…”

Thickness-Weighted Mean Theory for the Effect of Surface Gravity Waves on Mean Flows in the Upper Ocean

“…2 We show here 54 that the VWS is contained in the Favre-filtered momentum equations. Indeed, we are able to reproduce 55 the equations used by all these previous authors and are able to point out the different surface fluxes of 56 momentum that are implicit in these papers and that account for the differences between the different 57 2 To our knowledge no previous studies, except for an attempt by Ardhuin et al (2008), have used the Generalized Lagrangian Mean equations of Andrews and McIntyre (1978) to explain the VWS. ), will be discussed in a later paper in the context of the Favre-filtered equations in the VL coordinate 61…”