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Register a new userSurface waves on arbitrary vertically-sheared currents
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Benjamin Smeltzer Benjamin Smeltzer
Publication date
2016
fields
Physics
and research's language is
English
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We study dispersion properties of linear surface gravity waves propagating in an arbitrary direction atop a current profile of depth-varying magnitude using a piecewise linear approximation, and develop a robust numerical framework for practical calculation. The method has been much used in the past for the case of waves propagating along the same axis as the background current, and we herein extend and apply it to problems with an arbitrary angle between the wave propagation and current directions. Being valid for all wavelengths without loss of accuracy, the scheme is particularly well suited to solve problems involving a broad range of wave vectors, such as ship waves and Cauchy-Poisson initial value problems for example. We examine the group and phase velocities over different wavelength regimes and current profiles, highlighting characteristics due to the depth-variable vorticity. We show an example application to ship waves on an arbitrary current profile, and demonstrate qualitative differences in the wake patterns between concave down and concave up profiles when compared to a constant shear profile with equal depth-averaged vorticity. We also discuss the nature of additional solutions to the dispersion relation when using the piecewise-linear model. These are vorticity waves, drifting vortical structures which are artifacts of the piecewise model. They are absent for a smooth profile and are spurious in the present context.
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The propagation of surface water waves interacting with a current and an uneven bottom is studied. Such a situation is typical for ocean waves where the winds generate currents in the top layer of the ocean. The role of the bottom topography is taken into account since it also influences the local wave and current patterns. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types. The arising KdV equation with variable coefficients, dependent on the bottom topography, is studied numerically when the initial condition is in the form of the one soliton solution for the initial depth. Emergence of new solitons is observed as a result of the wave interaction with the uneven bottom.
A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are presented.
We present a theoretical and numerical framework -- which we dub the Direct Integration Method (DIM) -- for simple, efficient and accurate evaluation of surface wave models allowing presence of a current of arbitrary depth dependence, and where bathymetry and ambient currents may vary slowly in horizontal directions. On horizontally constant water depth and shear current the DIM numerically evaluates the dispersion relation of linear surface waves to arbitrary accuracy, and we argue that for this purpose it is superior to two existing numerical procedures: the piecewise-linear approximation and a method due to textit{Dong & Kirby} [2012]. The DIM moreover yields the full linearized flow field at little extra cost. We implement the DIM numerically with iterations of standard numerical methods. The wide applicability of the DIM in an oceanographic setting in four aspects is shown. Firstly, we show how the DIM allows practical implementation of the wave action conservation equation recently derived by textit{Quinn et al.} [2017]. Secondly, we demonstrate how the DIM handles with ease cases where existing methods struggle, i.e. velocity profiles $mathbf{U}(z)$ changing direction with vertical coordinate $z$, and strongly sheared profiles. Thirdly, we use the DIM to calculate and analyse the full linear flow field beneath a 2D ring wave upon a near--surface wind--driven exponential shear current, revealing striking qualitative differences compared to no shear. Finally we demonstrate that the DIM can be a real competitor to analytical dispersion relation approximations such as that of textit{Kirby & Chen} [1989] even for wave/ocean modelling.
Simulations of strongly stratified turbulence often exhibit coherent large-scale structures called vertically sheared horizontal flows (VSHFs). VSHFs emerge in both two-dimensional (2D) and three-dimensional (3D) stratified turbulence with similar vertical structure. The mechanism responsible for VSHF formation is not fully understood. In this work, the formation and equilibration of VSHFs in a 2D Boussinesq model of stratified turbulence is studied using statistical state dynamics (SSD). In SSD, equations of motion are expressed directly in the statistical variables of the turbulent state. Restriction to 2D turbulence makes available an analytically and computationally attractive implementation of SSD referred to as S3T, in which the SSD is expressed by coupling the equation for the horizontal mean structure with the equation for the ensemble mean perturbation covariance. This second order SSD produces accurate statistics, through second order, when compared with fully nonlinear simulations. In particular, S3T captures the spontaneous emergence of the VSHF and associated density layers seen in simulations of turbulence maintained by homogeneous large-scale stochastic excitation. An advantage of the S3T system is that the VSHF formation mechanism, which is wave-mean flow interaction between the emergent VSHF and the stochastically excited large-scale gravity waves, is analytically understood in the S3T system. Comparison with fully nonlinear simulations verifies that S3T solutions accurately predict the scale selection, dependence on stochastic excitation strength, and nonlinear equilibrium structure of the VSHF. These results facilitate relating VSHF theory and geophysical examples of turbulent jets such as the oceans equatorial deep jets.
Nonlinear dynamics of surface gravity waves trapped by an opposing jet current is studied analytically and numerically. For wave fields narrowband in frequency but not necessarily with narrow angular distributions the developed asymptotic weakly nonlinear theory based on the modal approach of (V. Shrira, A. Slunyaev, J. Fluid. Mech, 738, 65, 2014) leads to the one-dimensional modified nonlinear Schr{o}dinger equation of self-focusing type for a single mode. Its solutions such as envelope solitons and breathers are considered to be prototypes of rogue waves; these solutions, in contrast to waves in the absence of currents, are robust with respect to transverse perturbations, which suggests potentially higher probability of rogue waves. Robustness of the long-lived analytical solutions in form of the modulated trapped waves and solitary wave groups is verified by direct numerical simulations of potential Euler equations.
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