Wave Action Flux Research Articles

Inverse Cascade Spectrum of Gravity Waves in the Presence of a Condensate: A Direct Numerical Simulation.

During the set of direct numerical simulations of the forced isotropic turbulence of surface gravity waves in the framework of primordial dynamical equations, the universal inverse cascade spectrum was observed. The slope of the spectrum is the same (in the margin of error) for different levels of pumping and nonlinearity as well as dissipation present in the system. In all simulation runs formation of the inverse cascade spectrum was accompanied by the appearance of a strong long wave background (condensate). The observed slope of the spectrum ∼k^{-3.07} is different from the constant wave action flux solution predicted by the wave turbulence theory ∼k^{-23/6}.

Upscale transfer of waves in one-dimensional rotating shallow water

We study the inverse flux of waves in one of the simplest geophysical fluid dynamics models: one-dimensional rotating shallow water equations. Based on direct numerical integration of the governing equations, we find that waves injected at small scales get transferred upscale predominantly via resonant quartic interactions between wave modes. The waves’ upscale transfer is non-local and involves turbulent transfer between disparate scales of the flow. Our analysis reveals that the upscale transfer of waves is extremely intermittent and is a result of localized-in-time bursts in wave action flux. These intermittent events of flux bursts lead to shallower waves’ spectrum and relatively higher amplitude wave fields in physical space. On examining statistics of the flow fields, we find that low-energy high wavenumbers more or less comply with the assumptions used in wave turbulence theory, such as uniformly distributed wave phases and the Gaussian distribution of fields, while non-uniform distribution of wave phases and non-Gaussian statistics dominate at large scales or low wavenumbers that contain a major share of the flow energy. Our findings point out that the one-dimensional rotating shallow water equations, despite being a simple geophysical fluid dynamic model, harbour complex and intricate features associated with the upscale transfer of waves that have not been recognized in the past.

The complex interplay between tidal inertial waves and zonal flows in differentially rotating stellar and planetary convective regions

Context.Quantifying tidal interactions in close-in two-body systems is of prime interest since they have a crucial impact on the architecture and the rotational history of the bodies. Various studies have shown that the dissipation of tides in either body is very sensitive to its structure and to its dynamics. Furthermore, solar-like stars and giant gaseous planets in our Solar System experience differential rotation in their outer convective envelopes. In this respect, numerical simulations of tidal interactions in these objects have shown that the propagation and dissipation properties of tidally excited inertial waves can be strongly modified in the presence of differential rotation.Aims.In particular, tidal inertial waves may strongly interact with zonal flows at the so-called co-rotation resonances, where the wave’s Doppler-shifted frequency is cancelled out. The energy dissipation at such resonances could deeply modify the orbital and spin evolutions of tidally interacting systems. In this context, we aim to provide a deep physical understanding of the dynamics of tidal waves at co-rotation resonances in the presence of differential rotation profiles that are typical of low-mass stars and giant planets.Methods.In this work, we have developed an analytical local model of an inclined shearing box that describes a small patch of the differentially rotating convective zone of a star or a planet. We investigate the propagation and the transmission of free inertial waves at co-rotation, and more generally at critical levels, which are singularities in the governing wave differential equation. Through the construction of an invariant called the wave action flux, we identify different regimes of wave transmission at critical levels, which are confirmed with a one-dimensional three-layer numerical model.Results.We find that inertial waves can be fully transmitted, strongly damped, or even amplified after crossing a critical level. The occurrence of these regimes depends on the assumed profile of differential rotation, on the nature as well as the latitude of the critical level, and on wave parameters such as the inertial frequency and the longitudinal and vertical wavenumbers. Waves can thus either deposit their action flux in the fluid when damped at critical levels, or they can extract action flux from the fluid when amplified at critical levels. Both situations can lead to significant angular momentum exchange between the tidally interacting bodies.

Physics-Capturing Discretizations for Spectral Wind-Wave Models

This paper discusses the discretization methods that have been commonly employed to solve the wave action balance equation, and that have gained a renewed interest with the widespread use of unstructured grids for third-generation spectral wind-wave models. These methods are the first-order upwind finite difference and first-order vertex-centered upwind finite volume schemes for the transport of wave action in geographical space. The discussion addresses the derivation of these schemes from a different perspective. A mathematical framework for mimetic discretizations based on discrete calculus is utilized herein. A key feature of this algebraic approach is that the process of exact discretization is segregated from the process of interpolation, the latter typically involved in constitutive relations. This can help gain insight into the performance characteristics of the discretization method. On this basis, we conclude that the upwind finite difference scheme captures the wave action flux conservation exactly, which is a plus for wave shoaling. In addition, we provide a justification for the intrinsic low accuracy of the vertex-centred upwind finite volume scheme, due to the physically inaccurate but common flux constitutive relation, and we propose an improvement to overcome this drawback. Finally, by way of a comparative demonstration, a few test cases is introduced to establish the ability of the considered methods to capture the relevant physics on unstructured triangular meshes.

Approximation of wave action conservation in vertically sheared mean flows

We develop asymptotic expressions for wave action density and action flux, using an extension of Kirby & Chen (1989)’s perturbation solution for weakly-sheared currents allowing for a basic flow with Froude number F=U∕gh=O(1) but with weak vertical shear. The accuracy of the expressions for action density and flux is established by comparison to analytic results for a current with constant shear, and to numerical results for a field case involving a buoyant ebb-tidal plume with strong vertical shear and for a case involving a numerically determined profile for a wind-driven current. We compare our results to those from recent work of Quinn et al. (2017), and find unresolved discrepancies in that prior work. We provide additional suggestions for efficiently implementing the required extensions in coupled wave/circulation models using a Taylor series expansion based on conditions at peak frequency and direction. These results generalize the previous work of Banihashemi et al. (2017) to motions in two horizontal dimensions, and cover the determination of the wave action.

Flux singularities in multiphase wavetrains and the Kadomtsev‐Petviashvili equation with applications to stratified hydrodynamics

AbstractThis paper illustrates how the singularity of the wave action flux causes the Kadomtsev‐Petviashvili (KP) equation to arise naturally from the modulation of a two‐phased wavetrain, causing the dispersion to emerge from the classical Whitham modulation theory. Interestingly, the coefficients of the resulting KP are shown to be related to the associated conservation of wave action for the original wavetrain, and therefore may be obtained prior to the modulation. This provides a universal form for the KP as a dispersive reduction from any Lagrangian with the appropriate wave action flux singularity. The theory is applied to the full water wave problem with two layers of stratification, illustrating how the KP equation arises from the modulation of a uniform flow state and how its coefficients may be extracted from the system.

Approximation of wave action flux velocity in strongly sheared mean flows

Spectral wave models based on the wave action equation typically use a theoretical framework based on depth uniform current to account for current effects on waves. In the real world, however, currents often have variations over depth. Several recent studies have made use of a depth-weighted current U˜ due to [Skop, R. A., 1987. Approximate dispersion relation for wave-current interactions. J. Waterway, Port, Coastal, and Ocean Eng. 113, 187–195.] or [Kirby, J. T., Chen, T., 1989. Surface waves on vertically sheared flows: approximate dispersion relations. J. Geophys. Res. 94, 1013–1027.] in order to account for the effect of vertical current shear. Use of the depth-weighted velocity, which is a function of wavenumber (or frequency and direction) has been further simplified in recent applications by only utilizing a weighted current based on the spectral peak wavenumber. These applications do not typically take into account the dependence of U˜ on wave number k, as well as erroneously identifying U˜ as the proper choice for current velocity in the wave action equation. Here, we derive a corrected expression for the current component of the group velocity. We demonstrate its consistency using analytic results for a current with constant vorticity, and numerical results for a measured, strongly-sheared current profile obtained in the Columbia River. The effect of choosing a single value for current velocity based on the peak wave frequency is examined, and we suggest an alternate strategy, involving a Taylor series expansion about the peak frequency, which should significantly extend the range of accuracy of current estimates available to the wave model with minimal additional programming and data transfer.

Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg-de Vries equation.

Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg–de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically.

The Linear Stability of a Wavetrain Propagating on Water of Variable Depth

A narrow-banded spectrum of gravity waves propagating in one horizontal direction at the surface of water with finite depth may be governed by a nonlinear Schroźdinger equation (NLSE). If the depth is nonuniform, then the coefficients in the NLSE are variable and an additional linear term due to conservation of wave action flux is present, which may act as a dissipative term that causes the wave envelope to broaden and decrease in amplitude or which may act as a growth term that causes the wave envelope to focus and increase in amplitude. The addition of linear, viscous dissipation either enhances the dissipative process or competes with the growth effect. Here we consider such a variable-coefficient, dissipative NLSE. We find a solution that has uniform amplitude in the appropriate reference frame and examine its linear stability. We find that for waves propagating into shallower water, both bathymetry and viscous dissipation act to stabilize the modulational instability. For waves propagating into deeper water, bathymetry acts to enhance instability, but viscous dissipation eventually causes stabilization for small enough initial perturbation amplitudes.

Phase dynamics of periodic waves leading to the Kadomtsev–Petviashvili equation in 3+1 dimensions

The Kadomstev–Petviashvili (KP) equation is a well-known modulation equation normally derived by starting with the trivial state and an appropriate dispersion relation. In this paper, it is shown that the KP equation is also the relevant modulation equation for bifurcation from periodic travelling waves when the wave action flux has a critical point. Moreover, the emergent KP equation arises in a universal form, with the coefficients determined by the components of the conservation of wave action. The theory is derived for a general class of partial differential equations generated by a Lagrangian using phase modulation. The theory extends to any space dimension and time, but the emphasis in the paper is on the case of 3+1. Motivated by light bullets and quantum vortex dynamics, the theory is illustrated by showing how defocusing NLS in 3+1 bifurcates to KP in 3+1 at criticality. The generalization to N >3 is also discussed.

Self-similar formation of an inverse cascade in vibrating elastic plates.

The dynamics of random weakly nonlinear waves is studied in the framework of vibrating thin elastic plates. Although it has been previously predicted that no stationary inverse cascade of constant wave action flux could exist in the framework of wave turbulence for elastic plates, we present substantial evidence of the existence of a time-dependent inverse cascade, opening up the possibility of self-organization for a larger class of systems. This inverse cascade transports the spectral density of the amplitude of the waves from short up to large scales, increasing the distribution of long waves despite the short-wave fluctuations. This dynamics appears to be self-similar and possesses a power-law behavior in the short-wavelength limit which significantly differs from the exponent obtained via a Kolmogorov dimensional analysis argument. Finally, we show explicitly a tendency to build a long-wave coherent structure in finite time.

Bifurcation from rolls to multi-pulse planforms via reduction to a parabolic Boussinesq model

A mechanism is presented for the bifurcation from one-dimensional spatially periodic patterns (rolls) into two-dimensional planar states (planforms). The novelty is twofold: the planforms are solutions of a Boussinesq partial differential equation (PDE) on a periodic background and secondly explicit formulas for the coefficients in the Boussinesq equation are derived, based on a form of planar conservation of wave action flux. The Boussinesq equation is integrable with a vast array of solutions, and an example of a new planform bifurcating from rolls, which appears to be generic, is presented. Adding in time leads to a new time-dependent PDE, which models the nonlinear behaviour emerging from a generalization of Eckhaus instability. The class of PDEs to which the theory applies is evolution equations whose steady part is a gradient elliptic PDE. Examples are the 2+1 Ginzburg–Landau equation with real coefficients, and the 2+1 planar Swift–Hohenberg equation.

Diagnosis and application of potential shear deformation wave-activity density in the torrential rain of Typhoon Morokat (2009)

Based on previous studies of the interaction between wave and flow, combining the vertical component of disturbance convective vorticity vector, the disturbance horizontal divergence and the vertical gradient of disturbance generalized potential temperature, We define a new disturbance thermodynamic shear advection parameter in this paper. This parameter means a typical wave action density, its equation of tendency is a typical wave action equation, which could describe the development or evolution of mesoscale disturbance. With the wave action density and the wave action equation, we have conducted a diagnostic analysis of the heavy-rainfall event caused by landfall typhoon "Morakot" in 2009. Results show that the abnormal values of the wave action density (namely the disturbance thermodynamic shear advection parameter) will always change with the development of the observed precipitation regions, both their horizontal distribution and their temporal evolution are quite similar. Statistical analysis reveals a certain correspondence between the wave action density and the observed 6-hour accumulated surface rainfall in the summer of 2009, and they are intimately related to each other. The wave action density may be used to describe the typical vertical structure of dynamic and thermodynamic fields of a precipitation system, so it is closely related to the occurrence and development of the precipitation system and may have certain relation with the surface rainfall regions. The calculation of the wave action flux divergence shows that at the time around the landing of typhoon "Morakot" in the southeastern China coastal region, the potential vorticity of disturbance ageostrophic wind is the main forcing term affecting the local variation of wave action density. The contribution of the coupled term between the first-order disturbance advection and disturbance shear to the local variation of wave action density is secondary. While the exchange of disturbance momentum and disturbance heat between the basic state and disturbance has relatively weak influence on the development and evolution of disturbance. After the landing of typhoon "Morakot" in the southeastern China coastal region, these forcing terms and the wave action itself are apparently weakened.

Emergence of unsteady dark solitary waves from coalescing spatially periodic patterns

A dark solitary wave, in one space dimension and time, is a wave that is bi-asymptotic to a periodic state, with a phase shift, and with localized modulation in between. The most well-known case of dark solitary waves is the exact solution of the defocusing nonlinear Schrödinger equation. In this paper, our interest is in developing a mechanism for the emergence of dark solitary waves in general, and not necessarily integrable, Hamiltonian PDEs. The focus is on the periodic state at infinity as the generator. It is shown that a natural mechanism for the emergence is a transition between one periodic state that is (spatially) elliptic and another one that is (spatially) hyperbolic. It is shown that the emergence is governed by a Korteweg–de Vries (KdV) equation for the perturbation wavenumber on a periodic background. A novelty in the result is that the three coefficients in the KdV equation are determined by the Krein signature of the elliptic periodic orbit, the curvature of the wave action flux and the slope of the wave action, with the last two evaluated at the critical periodic state.

A Spectral Parameterization of Drag, Eddy Diffusion, and Wave Heating for a Three-Dimensional Flow Induced by Breaking Gravity Waves

AbstractThere are three distinct processes by which upward-propagating gravity waves influence the large-scale dynamics and energetics of the middle atmosphere: (i) nonlocalized transport of momentum through wave propagation in three dimensions that remotely redistributes atmospheric momentum in both zonal and meridional directions from wave generation to wave dissipation regions; (ii) localized diffusive transport of momentum, heat, and tracers due to mixing induced by wave breaking; and (iii) localized transport of heat by perturbing wave structures due to dissipation that redistributes the thermal energy within a finite domain. These effects become most significant for breaking waves when momentum drag, eddy diffusion, and wave heating— the “breaking trinity”—are all imposed on the background state. This paper develops a 3D parameterization scheme that self-consistently includes the breaking trinity in large-scale numerical models. The 3D parameterization scheme is developed based on the general relationship between the wave action flux and the subgrid-scale momentum and heat fluxes developed by Zhu in 1987 and a mapping approximation between the wave source spectrum and momentum deposition distribution developed by Alexander and Dunkerton in 1999. For a set of given input wind and temperature profiles at each model grid, the parameterization scheme outputs the vertical profiles of the subgrid-scale force terms together with the eddy diffusion coefficients in the momentum and energy equations for a 3D background flow.

Modeling the impact of the gravity wave source strength on the thermal structure of the middle atmosphere

The role of gravity wave (GW) forcing in the zonal mean circulation of the middle atmosphere (16–120 km) is investigated on the base of a two-dimensional, time-dependent, mechanistic model. The globally averaged prescribed profiles of O 3 and CO 2 are used to calculate net diabatic heating basing on the detailed non-LTE algorithm for cooling due to CO 2 15-μm bands [Fomichev, V.I., Kutepov, A.A., Akmaev, R.A., Shved, G.M., 1993. Parameterization of the 15 micron CO 2 band cooling in the middle atmosphere (15–115 km). Journal of Atmospheric and Terrestrial Physics 55, 7–18] and O 3 9.6-micron band [Fomichev, V.I., Shved, G.M., 1985. Parameterization of the radiative divergence in the 9.6 micron O 3 band. Journal of Atmospheric and Terrestrial Physics 47, 1037–1049]. The GW parameterization scheme has been developed to account for momentum deposition, heating, and turbulent heat transfer associated to the GW breaking. Motivated by possible changes in atmospheric GW climatology, we have investigated the sensitivity of the middle atmosphere structure to tropospheric GW source strength by prescribing the appropriate value of the vertical wave action flux at the lower boundary of the model domain. The results show, that the associated temperature response varies greatly with height, latitude, and season, and attains its maximum in the mesosphere and lower thermosphere (MLT) region primary due to the adiabatic effect caused by the corresponding changes in the vertical velocity field. Thus, the long-term variability of tropospheric GW source strength can produce substantial effect on MLT temperatures, which should be accounted for in the analysis of the observed climatological changes in the thermal structure of the middle atmosphere.

Secondary criticality of water waves. Part 1. Definition, bifurcation and solitary waves

A generalization of criticality – called secondary criticality – is introduced and applied to finite-amplitude Stokes waves. The theory shows that secondary criticality signals a bifurcation to a class of steady dark solitary waves which are biasymptotic to a Stokes wave with a phase jump in between, and synchronized with the Stokes wave. We find the that the bifurcation to these new solitary waves – from Stokes gravity waves in shallow water – is pervasive, even at low amplitude. The theory proceeds by generalizing concepts from hydraulics: three additional functionals are introduced which represent non-uniformity and extend the familiar mass flux, total head and flow force, the most important of which is the wave action flux. The theory works because the hydraulic quantities can be related to the governing equations in a precise way using the multi-symplectic Hamiltonian formulation of water waves. In this setting, uniform flows and Stokes waves coupled to a uniform flow are relative equilibria which have an attendant geometric theory using symmetry and conservation laws. A flow is then ‘critical’ if the relative equilibrium representation is degenerate. By characterizing successively non-uniform flows and unsteady flows as relative equilibria, a generalization of criticality is immediate. Recent results on the local nonlinear behaviour near a degenerate relative equilibrium are used to predict all the qualitative properties of the bifurcating dark solitary waves, including the phase shift. The theory of secondary criticality provides new insight into unsteady waves in shallow water as well. A new interpretation of the Benjamin–Feir instability from the viewpoint of hydraulics, and the connection with the creation of unsteady dark solitary waves, is given in Part 2.

Numerical modeling of the constant flux spectra for surface gravity waves in a case of angular anisotropy

The process of constant flux spectra formation is studied numerically, for the case of angular anisotropy. The method of investigation is based on a numerical solution of the nonlinear four-wave kinetic equation for gravity waves, added with the terms of a source and a sink for wave energy. Stationary solutions of such an equation are just the constant flux spectra under investigation. Two principally different situations were considered: the case of a constant energy flux to the high frequencies (the case of “flux up”) and the case of a constant wave action flux to the low frequencies (the case of “flux down”). It was found that, for the case of flux down, the stationary two-dimensional spectra are angularly isotropic, independently of the initial anisotropy of the system. But for the case of flux up, the stationary spectra have specific angular anisotropy. For both cases, corresponding one-dimensional spectra have shapes close to ones analytically derived in [Dokl. Akad. Nauk SSSR, 170 (1996) 1292–1295 (in Russian); Izv. Akad. S. USSR, Atmos. Ocean Phys. 18 (1982) 970–979], for the case of angular isotropy. Numerous features of the process considered are specified, and the mentioned peculiarities of the results are discussed.

A Spectral Parameterization of Mean-Flow Forcing due to Breaking Gravity Waves

A spectral parameterization of mean-flow forcing due to breaking gravity waves is described for application in the equations of motion in atmospheric models. The parameterization is based on linear theory and adheres closely to fundamental principles of conservation of wave action flux, linear stability, and wave‐mean-flow interaction. Because the details of wave breakdown and nonlinear interactions are known to be very complex and are still poorly understood, only the simplest possible assumption is made: that the momentum fluxes carried by the waves are deposited locally and entirely at the altitude of linear wave breaking. This simple assumption allows a straightforward mapping of the momentum flux spectrum, input at a specified source altitude, into vertical profiles of mean-flow force. A coefficient of eddy diffusion can also be estimated. The parameterization can be used with any desired input spectrum of momentum flux. The results are sensitive to the details of this spectrum and also realistically sensitive to the background vertical shear and stability profiles. These sensitivities make the parameterization ideally suited for studying both the effects of gravity waves from unique sources like topography and convection as well as generalized broad input spectra. Existing constraints on input parameters are also summarized from the available observations. With these constraints, the parameterization generates realistic variations in gravity-wave-driven, mean-flow forcing.

A simulated spectrum of convectively generated gravity waves: Propagation from the tropopause to the mesopause and effects on the middle atmosphere

This work evaluates the interaction of a simulated spectrum of convectively generated gravity waves with realistic middle atmosphere mean winds. The wave spectrum is derived from the nonlinear convection model described by Alexander et al. [1995] that simulated a two‐dimensional midlatitude squall line. This spectrum becomes input to a linear ray tracing model for evaluation of wave propagation as a function of height through climatological background wind and buoyancy frequency profiles. The energy defined by the spectrum as a function of wavenumber and frequency is distributed spatially and temporally into wave packets for the purpose of estimating wave amplitudes at the lower boundary of the ray tracing model. A wavelet analysis provides an estimate of these wave packet widths in space and time. Without this redistribution of energies into wave packets the Fourier analysis alone inaccurately assumes the energy is evenly distributed throughout the storm model domain. The growth with height of wave amplitudes is derived from wave action flux conservation coupled to a convective instability saturation condition. Mean flow accelerations and wave energy dissipation profiles are derived from this analysis and compared to parameterized estimates of gravity wave forcing, providing a measure of the importance of the storm source to global gravity wave forcing. The results suggest that a single large convective storm system like the simulated squall line could provide a significant fraction of the zonal mean gravity wave forcing at some levels, particularly in the mesosphere. The vertical distributions of mean flow acceleration and energy dissipation do not much resemble the parameterized profiles in form because of the peculiarities of the spectral properties of the waves from the storm source. The ray tracing model developed herein provides a tool for examining the role of convectively generated waves in middle atmosphere physics.