Wave-induced Mean Flow Research Articles

Along-slope bottom currents driven by dissipation of internal tides in the northeastern South China Sea

Recent mooring observations on the continental slope on the east side of the Dongsha Island in the northeastern South China Sea (SCS) showed that an along-slope bottom current can be generated when internal tides obliquely incident to the slope are dissipated near the seafloor. In this study, new mooring data collected on the south side of the Dongsha Island are used to explore the universality of internal wave driven the bottom currents and test the ability of the previous theory in estimating the along-slope current. The data show strong near-bottom energy dissipation due to the critical reflection of diurnal internal tides on the continental slope, with a time-mean depth-integrated dissipation rate of ~4.8×10-3 W/m2. Because of the obliquely incident of diurnal internal tides to the slope, near-bottom dissipation of internal tides generates a southwestward along-slope current, with the maximum velocity exceeding 6 cm/s. By comparing the observations, the previous theory for internal wave induced mean flows developed by Thorpe (1999) shows a good ability to estimate the along-slope bottom current velocity. Based on the theory, as well as modelled energy dissipation on the entire continental slope in the northeastern SCS, a map is obtained to quantitatively describe the along-slope bottom flow caused by internal tide breaking on the slope.

Hydroelastic lumps in shallow water

Hydroelastic solitary waves propagating on the surface of a three-dimensional ideal fluid through the deformation of an elastic sheet are studied. The problem is investigated based on a Benney–Luke-type equation derived via an explicit non-local formulation of the classic water wave problem. The normal form analysis is carried out for the newly developed equation, which results in the Benney–Roskes–Davey–Stewartson (BRDS) system governing the coupled evolution of the envelope of a carrier wave and the wave-induced mean flow. Numerical results show three types of free solitary waves in the Benney–Luke-type equation all of which are predicted by the BRDS system: plane solitary wave, lump (i.e., fully localized traveling waves in three dimensions), and transversally periodic solitary wave. They are linked together by a dimension-breaking bifurcation where plane solitary waves and lumps can be viewed as two limiting cases, and transversally periodic solitary waves serve as intermediate states. The stability and interaction of solitary waves are investigated via a numerical time integration of the Benney–Luke-type equation. For a localized load moving on the elastic sheet with a constant speed, it is found that there exists a transcritical regime of forcing speed for which there are no steady solutions. Instead, periodic shedding of lumps can be observed if the forcing moves at speed in this range.

Laboratory study of the wave-induced mean flow and set-down in unidirectional surface gravity wave packets on finite water depth

We derive a multiple-scales solution for the Eulerian mean flow under wavepackets, which is driven by the divergence of Stokes drift and the setdown. This solution is valid for all water depths and is validated by flume experiments and recovers all previous solutions in their respective limits.

Fully nonlinear simple internal waves over subcritical slopes in continuously stratified fluids: Theoretical development

The dynamics of fully nonlinear internal waves in continuously stratified fluids is investigated using a new simple-wave (or Riemann-wave) theory with 2nd-order topographic effects. Unlike previous fully nonlinear internal-wave theories, the proposed theory is applicable to large water-depth change over a long distance, provided that the local bottom slopes are mild. An essential step in the theoretical development is the separation of “adiabatic” and “diabatic” topographic effects. First, adiabatic topographic effects are incorporated into the previous simple-wave theory, in such a way that one-directional, fully nonlinear, nondispersive waves become conservative over topography. Since the Riemann variable (or invariant) remains constant along the characteristic, this extends the applicability of the solution method based on characteristic curves from flat bottom to mild slopes. Then, 2nd-order diabatic topographic effects are added to the adiabatic simple-wave solution by a perturbation approach. The application of the proposed theory to Wunsch’s subcritical wedge problem suggests that the proposed theory is applicable to subcritical slopes, although the error unavoidably increases as the slope angle approaches the propagation angle of internal wave rays (i.e., the critical angle). An important finding is the relatively rapid growths of wave-induced isopycnal setup and mean flow under the combined effects of strong nonlinearity and topography, in contrast to temporally stationary setup and mean flow under weak topographic effects. This implies, for example, that large-amplitude internal waves on continental shelves have an inherent tendency to modify the “background” stratification and currents without mixing and that it could occur within a spring-neap tidal cycle.

Evolution and Stability of Two-Dimensional Anelastic Internal Gravity Wave Packets

The weakly nonlinear evolution, stability, and overturning of horizontally and vertically localized internal gravity wave packets is examined for a nonrotating, anelastic atmosphere that is stationary in the absence of waves. The weakly nonlinear evolution is examined through the derivation of their wave-induced mean flow, which is used to formulate a nonlinear Schrödinger equation. The induced flow is manifest as a long, hydrostatic, bow wake-like disturbance, whose flow direction transitions from positive on the leading flank of the wave packet to negative on the trailing flank of the wave packet. As such, two-dimensional wave packets are always modulationally unstable. This instability results in enhanced amplitude growth confined to either the leading or trailing flank. Hence, when combined with anelastic growth predicted by linear theory, we anticipate two-dimensional waves will overturn either somewhat below or just above the heights predicted by linear theory. Numerical solutions of the Schrödinger equation are compared with the results of fully nonlinear simulations to establish the validity of the weakly nonlinear theory. Actual wave overturning heights are determined quantitatively from a range of fully nonlinear simulations.

Propagation of non-stationary acoustic-gravity waves at thermospheric temperatures corresponding to different solar activity

Numerical simulation of non-stationary nonlinear acoustic-gravity waves (AGWs) propagating from the surface wave source to the thermosphere reveals that their propagation conditions and parameters depend on changes in background temperature, density, composition, molecular viscosity and heat conduction caused by changes in solar activity (SA). At small wave source amplitudes, AGW amplitudes, momentum fluxes and wave accelerations of the mean flow are slightly larger at altitudes about 150 km at low SA because of smaller mean density and ρ0−1/2dependence of wave amplitudes at low dissipation. Larger kinematic coefficients of molecular viscosity and heat conduction lead to stronger decrease of wave amplitudes and momentum fluxes at altitudes above 150 km at low SA. At large amplitudes of surface wave excitation, AGW breaking and smaller-scale inhomogeneities appear at altitudes 100–150 km, which are stronger at low SA. Increased dissipation of breaking AGWs may produce wave-induced jet streams with velocities close to the wave horizontal phase speed and near-critical layers at altitudes 110–150 km, which dramatically decrease amplitudes and momentum fluxes of the primary AGW mode propagating from the surface wave source. The wave-induced horizontal wind becomes smaller above altitude of 150 km and allows growing amplitudes of the primary wave mode partially penetrating through the near-critical layer and of secondary AGW modes, possibly generating in the wave induced jet stream. The wave amplitude grows at altitudes higher than 150 km is larger at high SA due to smaller velocities of wave-induced mean wind and smaller molecular viscosity and heat conduction. Accelerations of the mean flow by dissipating AGWs are generally larger at low SA. This determines faster grows of wave-induced jet streams in time at low SA. In almost all simulated cases, velocities of the wave-induced mean flows are higher at low SA compared to high SA. Resulting SA impact at a given thermospheric altitude depends on competition between AGW amplitude increase due to smaller molecular dissipation and smaller energy transfer to the wind-induced mean flow and amplitude decrease caused by larger density and stronger reflection at higher SA.

Wave-induced mean flows in rotating shallow water with uniform potential vorticity

Theoretical and numerical computations of the wave-induced mean flow in rotating shallow water with uniform potential vorticity are presented, with an eye towards applications in small-scale oceanography where potential-vorticity anomalies are often weak compared to the waves. The asymptotic computations are based on small-amplitude expansions and time averaging over the fast wave scale to define the mean flow. Importantly, we do not assume that the mean flow is balanced, i.e. we compute the full mean-flow response at leading order. Particular attention is paid to the concept of modified diagnostic relations, which link the leading-order Lagrangian-mean velocity field to certain wave properties known from the linear solution. Both steady and unsteady wave fields are considered, with specific examples that include propagating wavepackets and monochromatic standing waves. Very good agreement between the theoretical predictions and direct numerical simulations of the nonlinear system is demonstrated. In particular, we extend previous studies by considering the impact of unsteady wave fields on the mean flow, and by considering the total kinetic energy of the mean flow as a function of the rotation rate. Notably, monochromatic standing waves provide an explicit counterexample to the often observed tendency of the mean flow to decrease monotonically with the background rotation rate.

The wave-induced flow of internal gravity wavepackets with arbitrary aspect ratio

We examine the wave-induced flow of small-amplitude, quasi-monochromatic, three-dimensional, Boussinesq internal gravity wavepackets in a uniformly stratified ambient. It has been known since Bretherton (J. Fluid Mech., vol. 36 (4), 1969, pp. 785–803) that one-, two- and three-dimensional wavepackets induce qualitatively different flows. Whereas the wave-induced mean flow for compact three-dimensional wavepackets consists of a purely horizontal localized circulation that translates with and around the wavepacket, known as the Bretherton flow, such a flow is prohibited for a two-dimensional wavepacket of infinite spanwise extent, which instead induces a non-local internal wave response that is long compared with the streamwise extent of the wavepacket. One-dimensional (horizontally periodic) wavepackets induce a horizontal, non-divergent unidirectional flow. Through perturbation theory for quasi-monochromatic wavepackets of arbitrary aspect ratio, we predict for which aspect ratios which type of induced mean flow dominates. We compose a regime diagram that delineates whether the induced flow is comparable to that of one-, two- or compact three-dimensional wavepackets. The predictions agree well with the results of fully nonlinear three-dimensional numerical simulations.

Three-dimensional evolution of internal waves reflected from a submarine seamount

The interaction between internal waves and submarine seamounts is three-dimensional (3D). In this work, we conduct laboratory experiments to investigate the role of the three-dimensional seamount on the reflection of internal waves. Experimental results show that the energy of internal waves is focused after the reflection and the phase of the internal wave front become curved. The energy of the focusing region was nearly 2 times as that of the scattering region. There is a strong internal wave induced mean flow whose strength is nearly as high as 50% of that of the reflected internal wave. The mean flow was generated by the variation of the velocity amplitude in both the x axis and y axis. These results indicate that the reflection from the 3D seamount could lead to a complex spatial distribution of the internal wave and a different energy conversion process.

Interaction between mountain waves and shear flow in an inertial layer

Mountain-generated inertia–gravity waves (IGWs) affect the dynamics of both the atmosphere and the ocean through the mean force they exert as they interact with the flow. A key to this interaction is the presence of critical-level singularities or, when planetary rotation is taken into account, inertial-level singularities, where the Doppler-shifted wave frequency matches the local Coriolis frequency. We examine the role of the latter singularities by studying the steady wavepacket generated by a multiscale mountain in a rotating linear shear flow at low Rossby number. Using a combination of Wentzel–Kramers–Brillouin (WKB) and saddle-point approximations, we provide an explicit description of the form of the wavepacket, of the mean forcing it induces and of the mean-flow response. We identify two distinguished regimes of wave propagation: Regime I applies far enough from a dominant inertial level for the standard ray-tracing approximation to be valid; Regime II applies to a thin region where the wavepacket structure is controlled by the inertial-level singularities. The wave–mean-flow interaction is governed by the change in Eliassen–Palm (or pseudomomentum) flux. This change is localised in a thin inertial layer where the wavepacket takes a limiting form of that found in Regime II. We solve a quasi-geostrophic potential-vorticity equation forced by the divergence of the Eliassen–Palm flux to compute the wave-induced mean flow. Our results, obtained in an inviscid limit, show that the wavepacket reaches a large-but-finite distance downstream of the mountain (specifically, a distance of order$(k_{\ast }\unicode[STIX]{x1D6E5})^{1/2}\unicode[STIX]{x1D6E5}$, where$k_{\ast }^{-1}$and$\unicode[STIX]{x1D6E5}$measure the wave and envelope scales of the mountain) and extends horizontally over a similar scale.

Numerical modeling of surf zone dynamics under weakly plunging breakers with SPH method

The wave breaking of weak plungers over a relatively mild slope is investigated in this paper. Numerical modeling aspects are studied, concerning the propagation and breaking of shore-normal, nonlinear and regular waves. The two-dimensional (2-D) kinematics and dynamics (fluctuating flow features and large 2-D eddies) of the wave-induced flow on a vertical cross-section over the entire surf zone are simulated with the use of Smoothed Particle Hydrodynamics (SPH). The academic ‘open source’ code SPHysics v.2 is employed and the viscosity treatment is based on a Sub-Particle Scale (SPS) approach, similarly to the Large Eddy Simulations (LES) concept. Thorough analysis of the turbulent flow scales determines the necessary refinement of the spatial resolution. The initial particle discretization reaches down to the demarcation point between integral turbulence length scales and Taylor micro-scales. A convolution-type integration method is implemented for the transformation of scattered Lagrangian particle data to Eulerian values at fixed gauges. A heuristic technique of ensemble-averaging is used for the discrimination of the fluctuating flow components from coherent structures and ordered wave motion. Comparisons between numerical and experimental data give encouraging results for several wave features. The wave-induced mean flows are simulated plausibly, and even the ‘streaming’ effect near the bed is reproduced. The recurring vorticity patterns are derived, and coherent 2-D structures inside the surf zone are identified. Fourier spectral analysis of velocities reveals isotropy of 2-D fluctuating dynamics up to rather high frequencies in shear intensified regions. The simulated Reynolds stresses follow patterns that define the characteristic mechanism of wave breaking for weak plungers. Persisting discrepancies at the incipient breaking region confirm the need for fine, massively ‘parallel’ 3-D SPS-SPH simulations.

The Effect Of Randomness On The Stability Of Capillary Gravity Waves In The Presence Of Air Flowing Over Water

A nonlinear spectral transport equation for the narrow band Gaussian random surface wave trains is derived from a fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves. The effect of randomness on the stability of deep water capillary gravity waves in the presence of air flowing over water is investigated. The stability is then considered for an initial homogenous wave spectrum having a simple normal form to small oblique long wave length perturbations for a range of spectral widths. An expression for the growth rate of instability is obtained; in which a higher order contribution comes from the fourth order term in the evolution equation, which is responsible for wave induced mean flow. This higher order contribution produces a decrease in the growth rate. The growth rate of instability is found to decrease with the increase of spectral width and the instability disappears if the spectral width increases beyond a certain critical value, which is not influenced by the fourth order term in the evolution equation.

Dynamical and thermal effects of nonsteady nonlinear acoustic-gravity waves propagating from tropospheric sources to the upper atmosphere

We performed numerical simulations of nonlinear AGW propagation to the middle and upper atmosphere from a plane wave forcing at the Earth’s surface with period τ=2×103s. After activating the surface wave forcing, initial pulse of acoustic and very long gravity modes in a few minutes can reach altitudes above 100km. Dissipation of this initial pulse produces substantial mean heating and wave-induced mean winds at altitudes above 200km. This may influence AGW propagation and produce enhanced vertical gradients of temperature, horizontal velocity and increased wave dissipation in the lower part of the wave-induced mean flows helping their downward expansions. Later, AGWs may produce layers of convective instability and peaks of the wave-induced jets at altitudes 100–120km. Shorter AGWs with smaller horizontal wave speeds produce smaller mean heating and wave-induced mean velocities in the upper atmosphere at fixed amplitudes and periods of the surface wave excitation. Numerical simulation of nonlinear AGW propagation helps better understanding the details of dynamical and thermal influence of waves coming from the troposphere on the mean temperature and wind in the middle and upper atmosphere.

Anelastic internal wave reflection and transmission in uniform retrograde shear

We perform fully nonlinear simulations in two dimensions of a horizontally periodic, vertically localized, anelastic internal wavepacket in order to examine the effects of weak and strong nonlinearity upon wavepackets approaching a reflection level in uniform retrograde shear. Transmission, reflection, and momentum deposition are measured in terms of the horizontal momentum associated with the wave-induced mean flow. These are determined in part as they depend upon the initial wavenumber vector, \documentclass[12pt]{minimal}\begin{document}$\vec{k}=(k,m)$\end{document}k⃗=(k,m), which determines the modulational stability (if |m/k| ≳ 0.7) or instability (if |m/k| ≲ 0.7) of moderately large amplitude quasi-monochromatic internal wavepackets. Whether modulationally stable or unstable, the evolution of the wavepacket is determined by the height of the reflection level predicted by linear theory, zr, relative to the height, zΔ, at which weak nonlinearity becomes significant, and the height, \documentclass[12pt]{minimal}\begin{document}$z_b>z_\Delta$\end{document}zb>zΔ, at which linear theory predicts anelastic waves first overturn in the absence of shear. If \documentclass[12pt]{minimal}\begin{document}$z_r<z_\Delta$\end{document}zr<zΔ, the amplitude remains sufficiently small and the waves reflect as predicted by linear theory. If zr is moderately larger than \documentclass[12pt]{minimal}\begin{document}$z_\Delta$\end{document}zΔ, a fraction of the momentum associated with the wavepackets transmits past the reflection level. This is because the positive shear associated with the wave-induced mean flow can partially shield the wavepacket from the influence of the negative background shear enhancing its transmission. The effect is enhanced for weakly nonlinear modulational unstable wavepackets that narrow and grow in amplitude faster than the anelastic growth rate. However, as nonlinear effects become more pronounced, a significant fraction of the momentum associated with the wavepacket is irreversibly deposited to the background below the reflection level. This is particularly the case for modulationally unstable wavepackets, whose enhanced amplitude growth leads to overturning below the predicted breaking level. Because the growth in the amplitude envelope of modulationally stable wavepackets is retarded by weakly nonlinear effects, reflection is enhanced and transmission retarded relative to their modulationally unstable counterparts. Applications to mountain wave propagation through the stratosphere in the winter hemisphere are discussed.

The Zakharov equation with separate mean flow and mean surface

AbstractUsing the Hamiltonian approach of Krasitskii (J. Fluid Mech., vol. 272, 1994, pp. 1–20), we derive a variant of the Zakharov equation in which the wave-induced mean surface elevation and the surface potential of the wave-induced mean flow are represented as separate variables governed by separate evolution equations. The kernel function of this new variant is simpler, and in particular also well defined in the uniform-wave-train limit for waves on finite depth. This form of the Zakharov equation may be advantageous in some applications. One example is the derivation of nonlinear Schrödinger equations in the narrow-band limit, where the handling of the mean flow and mean surface is significantly simpler than when starting from the original Zakharov equation. In this paper we have used the alternative form of the Zakharov equation to derive a Hamiltonian nonlinear Schrödinger equation for directional waves on arbitrary depth, valid to one order higher in bandwidth than the Hamiltonian equation recently presented by Craig, Guyenne and Sulem (Wave Motion, vol. 47, 2010, pp. 552–563).

Range of validity of an extended WKB theory for atmospheric gravity waves: one-dimensional and two-dimensional case

AbstractA computational model of the pseudo-incompressible equations is used to probe the range of validity of an extended Wentzel–Kramers–Brillouin theory (XWKB), previously derived through a distinguished limit of a multiple-scale asymptotic analysis of the Euler or pseudo-incompressible equations of motion, for gravity-wave packets at large amplitudes. The governing parameter of this analysis had been the scale-separation ratio$\varepsilon $between the gravity wave and both the large-scale potential-temperature stratification and the large-scale wave-induced mean flow. A novel feature of the theory had been the non-resonant forcing of higher harmonics of an initial wave packet, predominantly by the large-scale gradients in the gravity-wave fluxes. In the test cases considered a gravity-wave packet is propagating upwards in a uniformly stratified atmosphere. Large-scale winds are induced by the wave packet, and possibly exert a feedback on the latter. In the limit$\varepsilon \ll 1$all predictions of the theory can be validated. The larger$\varepsilon $is the more the transfer of wave energy to the mean flow is underestimated by the theory. The numerical results quantify this behaviour but also show that, qualitatively, XWKB remains valid even when the gravity-wave wavelength approaches the spatial scale of the wave-packet amplitude. This includes the prevalence of first and second harmonics and the smallness of harmonics with wave number higher than two. Furthermore, XWKB predicts for the vertical momentum balance an additional leading-order buoyancy term in Euler and pseudo-incompressible theory, compared with the anelastic theory. Numerical tests show that this term is relatively large with up to$30\hspace{0.167em} \% $of the total balance. The practical relevance of this deviation remains to be assessed in future work.

Large Scale Energy Transfer from an Internal Gravity Wave Reflecting on a Simple Slope

Several processes lead to mixing and transport in the ocean, among those being the interaction of the internal gravity wave field with bottom topography. The latter process is considered in the present work, through joint laboratory experiments and numerical simulations. The basic configuration is a plane wave of finite extent reflecting onto a sloping bottom in a uniformly stratified fluid. As expected, the interaction between the incident and reflected waves produces harmonic waves, but an irreversible wave-induced mean flow grows in the interacting region between those waves, whose amplitude may be larger than that of the incident wave. This mean flow appears to be controlled by nonlinear and dissipative effects associated with the reflected wave component. Unlike in the atmosphere, the role of this wave-induced mean flow has been completely overlooked in the ocean.

Implementation of the vortex force formalism in the coupled ocean-atmosphere-wave-sediment transport (COAWST) modeling system for inner shelf and surf zone applications

The coupled ocean-atmosphere-wave-sediment transport modeling system (COAWST) enables simulations that integrate oceanic, atmospheric, wave and morphological processes in the coastal ocean. Within the modeling system, the three-dimensional ocean circulation module (ROMS) is coupled with the wave generation and propagation model (SWAN) to allow full integration of the effect of waves on circulation and vice versa. The existing wave-current coupling component utilizes a depth dependent radiation stress approach. In here we present a new approach that uses the vortex force formalism. The formulation adopted and the various parameterizations used in the model as well as their numerical implementation are presented in detail. The performance of the new system is examined through the presentation of four test cases. These include obliquely incident waves on a synthetic planar beach and a natural barred beach (DUCK’ 94); normal incident waves on a nearshore barred morphology with rip channels; and wave-induced mean flows outside the surf zone at the Martha’s Vineyard Coastal Observatory (MVCO).Model results from the planar beach case show good agreement with depth-averaged analytical solutions and with theoretical flow structures. Simulation results for the DUCK’ 94 experiment agree closely with measured profiles of cross-shore and longshore velocity data from Garcez Faria et al. (1998, 2000). Diagnostic simulations showed that the nonlinear processes of wave roller generation and wave-induced mixing are important for the accurate simulation of surf zone flows. It is further recommended that a more realistic approach for determining the contribution of wave rollers and breaking induced turbulent mixing can be formulated using non-dimensional parameters which are functions of local wave parameters and the beach slope. Dominant terms in the cross-shore momentum balance are found to be the quasi-static pressure gradient and breaking acceleration. In the alongshore direction, bottom stress, breaking acceleration, horizontal advection and horizontal vortex forces dominate the momentum balance. The simulation results for the bar/rip channel morphology case clearly show the ability of the modeling system to reproduce horizontal and vertical circulation patterns similar to those found in laboratory studies and to numerical simulations using the radiation stress representation. The vortex force term is found to be more important at locations where strong flow vorticity interacts with the wave-induced Stokes flow field. Outside the surf zone, the three-dimensional model simulations of wave-induced flows for non-breaking waves closely agree with flow observations from MVCO, with the vertical structure of the simulated flow varying as a function of the vertical viscosity as demonstrated by Lentz et al. (2008).

Conservation Laws, Hamiltonian Structure, Modulational Instability Properties and Solitary Wave Solutions for a Higher-Order Model Describing Nonlinear Internal Waves

Recent theoretical advances in connecting the wave-induced mean flow with the conserved pseudomomentum per unit mass has permitted the first rational derivation of a model that describes the weakly nonlinear propagation of internal gravity plane waves in a continuously stratified fluid. Depending on the particular parameter regime examined the new model corresponds to an extended bright or dark derivative nonlinear Schrödinger equation or an extended complex-valued modified Korteweg-de Vries or Sasa–Satsuma equation. Mass, momentum, and energy conservation laws are derived. A noncanonical infinite-dimensional Hamiltonian formulation of the model is introduced. The modulational stability characteristics associated with the Stokes wave solution of the model are described. The bright and dark solitary wave solutions of the model are obtained.

Lagrangian approach to weakly nonlinear stability of elliptical flow

Rotating flows with elliptically strained streamlines suffer from parametric resonance instability between a pair of Kelvin waves whose azimuthal wavenumbers are separated by two. We address the weakly nonlinear amplitude evolution of three-dimensional (3D) Kelvin waves, in resonance, on a flow confined in a cylinder of elliptic cross-section. In a traditional Eulerian approach, derivation of the mean flow induced by nonlinear interaction of Kelvin waves stands as an obstacle. We show how a topological idea, or the Lagrangian approach, facilitates calculation of the wave-induced mean flow. A steady incompressible Euler flow is characterized as a state of the maximum of the total kinetic energy with respect to perturbations constrained to an isovortical sheet, and the isovortical perturbation is handled only in terms of the Lagrangian variables. The criticality in energy of a steady flow allows us to calculate the wave-induced mean flow only from the linear Lagrangian displacement. With the mean flow at hand, the Lagrangian approach provides us with a shortcut to enter into a weakly nonlinear amplitude evolution regime of 3D disturbances. Unlike the Eulerian approach, the amplitude equations are available directly in the Hamiltonian normal form.