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Register a new userWeakly non-linear transient waves on a shear current: Ring waves and skewed Langmuir rolls
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Simen {\\AA}. Adnoy Ellingsen
Publication date
2018
fields
Physics
and research's language is
English
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We investigate the weakly nonlinear dynamics of transient gravity waves at infinite depth under the influence of a shear current varying linearly with depth. An analytical solution is permitted via integration of the Euler equations. Although similar problems were investigated in the 1960s and 70s for special cases of resonance, this is to our knowledge the first general wave interaction (mode coupling) solution derived to second order with a shear current present. Wave interactions are integrable in a spectral convolution to yield the second order dynamics of initial value problems. To second order, irrotational wave dynamics interacts with the background vorticity field in a way that creates new vortex structures. A notable example is the large parallel vortices which drive Langmuir circulation as oblique plane waves interact with an ocean current. We also investigate the effect on wave pairs which are misaligned with the shear current. In contrast to a conjecture by Leibovich (1983) we find similar, but skewed, vortex structures in every case except when the mean wave direction is perpendicular to the direction of the current. Similar nonlinear wave-shear interactions are found to also generate near-field vortex structures in the Cauchy-Poisson problem with an initial surface elevation. These interactions create further groups of dispersive ring waves in addition to those present in linear theory. The second order solution is derived in a general manner which accommodates any initial condition through mode coupling over a continuous wave spectrum. It is therefore applicable to a range of problems including special cases of resonance. As a by--product of the general theory, a simple expression for the Stokes drift due to a monochromatic wave propagating at oblique angle with a current of uniform vorticity is derived, for the first time to our knowledge.
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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.
We study the waves and wave-making forces acting on ships travelling on currents which vary as a function of depth. Our concern is realism; we consider a real current profile from the Columbia River, and model ships with dimensions and Froude numbers typical of three classes of vessels operating in these waters. To this end we employ the most general theory of waves from free-surface sources on shear current to date, which we derive and present here. Expressions are derived for ship waves which satisfy an arbitrary dispersion relation and are generated by a wave source acting on the free surface, with the sources shape and time-dependence is also being arbitrary. Practical calculation procedures for numerically calculating dispersion on a shear current which may vary arbitrarily with depth both in direction and magnitude, are indicated. For ships travelling at oblique angle to a shear-current, the ship wave pattern is asymmetrical, and wave-making radiation forces have a lateral component in addition to the conventional wave resistance, the sternward component. No corresponding lateral force exists in the absence of shear. We consider the dependence of wave resistance and lateral force for upstream, downstream and cross-stream motion on the Columbia River current, both in steady motion and during two different maneouvres: a ship suddenly set in motion, and a ship turning through 360 deg. We find that for smaller ships (tugboats, fishing-boats) the wave resistance can differ drastically from that in quiescent water, and depends strongly on Froude number and direction of motion. For Froude numbers typical of such boats, wave resistance can vary by a factor 3 between upstream and downstream motion, and the strong Froude number dependence is made more complicated by interference effects. The lateral radiation force ... [abstract truncated due to ArXiVs space restrictions]
An approximate dispersion relation is derived and presented for linear surface waves atop a shear current whose magnitude and direction can vary arbitrarily with depth. The approximation, derived to first order of deviation from potential flow, is shown to produce good approximations at all wavelengths for a wide range of naturally occuring shear flows as well as widely used model flows. The relation reduces in many cases to a 3D generalization of the much used approximation by Skop [1987], developed further by Kirby & Chen [1989], but is shown to be more robust, succeeding in situations where the Kirby & Chen model fails. The two approximations incur the same numerical cost and difficulty. While the Kirby & Chen approximation is excellent for a wide range of currents, the exact criteria for its applicability have not been known. We explain the apparently serendipitous success of the latter and derive proper conditions of applicability for both approximate dispersion relations. Our new model has a greater range of applicability. A second order approximation is also derived. It greatly improves accuracy, which is shown to be important in difficult cases. It has an advantage over the corresponding 2nd order expression proposed by Kirby & Chen that its criterion of accuracy is explicitly known, which is not currently the case for the latter to our knowledge. Our 2nd order term is also arguably significantly simpler to implement, and more physically transparent, than its sibling due to Kirby & Chen.
We investigate analytically the linearized water wave radiation problem for an oscillating submerged point source in an inviscid shear flow with a free surface. A constant depth is taken into account and the shear flow increases linearly with depth. The surface velocity relative to the source is taken to be zero, so that Doppler effects are absent. We solve the linearized Euler equations to calculate the resulting wave field as well as its far-field asymptotics. For values of the Froude number $F^2=omega^2 D/g$ ($omega$: oscillation frequency, $D$ submergence depth) below a resonant value $F^2_text{res}$ the wave field splits cleanly into separate contributions from regular dispersive propagating waves and non-dispersive critical waves resulting from a critical layer-like street of flow structures directly downstream of the source. In the sub-resonant regime the regular waves behave like sheared ring waves while the critical layer wave forms a street of a constant width of order $Dsqrt{S/omega}$ ($S$ is the shear flow vorticity) and is convected downstream at the fluid velocity at the depth of the source. When the Froude number approaches its resonant value, the the downstream critical and regular waves resonate, producing a train of waves of linearly increasing amplitude contained within a downstream wedge.
In this Letter we regard nonlinear gravity-capillary waves with parameter of nonlinearity being $varepsilon sim 0.1 div 0.25$. For this nonlinearity time scale separation does not occur and kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in emph{Kartashova, emph{EPL} textbf{97} (2012), 30004.} We compute for the first time an analytical expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function depending on the ratio of surface tension to the gravity acceleration. It is shown that its two limits - pure capillary and pure gravity waves on a fluid surface - coincide with the previously obtained results. We also discuss relations of the model of D-cascade with a few known models used in the theory of nonlinear waves such as Zakharovs equation, resonance of the modes with nonlinear Stokes corrected frequencies and Benjamin-Feir index. These connections are crucial in the understanding and forecasting specifics of the energy transport in a variety of multi-component wave dynamics, from oceanography to optics, from plasma physics to acoustics.
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