scholarly journals The set-down and set-up of directionally spread and crossing surface gravity wave groups

2017 ◽  
Vol 835 ◽  
pp. 131-169 ◽  
Author(s):  
M. L. McAllister ◽  
T. A. A. Adcock ◽  
P. H. Taylor ◽  
T. S. van den Bremer
Keyword(s):  
Free Surface ◽  
Gravity Wave ◽  
Standing Wave ◽  
Wave Pattern ◽  
Surface Gravity ◽  
Linear Wave ◽  
Surface Gravity Wave ◽  
Wave Groups ◽  
Single Wave ◽  
Set Up

For sufficiently directionally spread surface gravity wave groups, the set-down of the wave-averaged free surface, first described by Longuet-Higgins and Stewart (J. Fluid Mech. vol. 13, 1962, pp. 481–504), can turn into a set-up. Using a multiple-scale expansion for two crossing wave groups, we examine the structure and magnitude of this wave-averaged set-up, which is part of a crossing wave pattern that behaves as a modulated partial standing wave: in space, it consists of a rapidly varying standing-wave pattern slowly modulated by the product of the envelopes of the two groups; in time, it grows and decays on the slow time scale associated with the translation of the groups. Whether this crossing wave pattern actually enhances the surface elevation at the point of focus depends on the phases of the linear wave groups, unlike the set-down, which is always negative and inherits the spatial structure of the underlying envelope(s). We present detailed laboratory measurements of the wave-averaged free surface, examining both single wave groups, varying the degree of spreading from small to very large, and the interaction between two wave groups, varying both the degree of spreading and the crossing angle between the groups. In both cases, we find good agreement between the experiments, our simple expressions for the set-down and set-up, and existing second-order theory based on the component-by-component interaction of individual waves with different frequencies and directions. We predict and observe a set-up for wave groups with a Gaussian angular amplitude distribution with standard deviations of above $30{-}40^{\circ }$ ($21{-}28^{\circ }$ for energy spectra), which is relatively large for realistic sea states, and for crossing sea states with angles of separation of $50{-}70^{\circ }$ and above, which are known to occur in the ocean.

scholarly journals Lagrangian transport for two-dimensional deep-water surface gravity wave groups

2016 ◽  
Vol 472 (2192) ◽  
pp. 20160159 ◽  
Author(s):  
T. S. van den Bremer ◽  
P. H. Taylor
Keyword(s):  
Gravity Wave ◽  
Frequency Dispersion ◽  
Wave Theory ◽  
Stokes Drift ◽  
Surface Gravity ◽  
Return Flow ◽  
Surface Gravity Wave ◽  
Wave Groups ◽  
Lagrangian Transport ◽  
Separation Of Scales

The Lagrangian trajectories of neutrally buoyant particles underneath surface gravity wave groups are dictated by two physical phenomena: the Stokes drift results in a net displacement of particles in the direction of propagation of the group, whereas the Eulerian return flow, as described by the multi-chromatic wave theory of Longuet-Higgins & Stewart (1962 J. Fluid Mech. 13 , 481–504. ( doi:10.1017/S0022112062000877 )), transports such particles in the opposite direction. By pursuing a separation of scales expansion, we develop simple closed-form expressions for the net Lagrangian displacement of particles. By comparing the results from the separation of scales expansion at different orders in bandwidth, we study the effect of frequency dispersion on the local Lagrangian transport, which we show can be ignored for realistic sea states.

scholarly journals Experimental study of particle trajectories below deep-water surface gravity wave groups

2019 ◽  
Vol 879 ◽  
pp. 168-186 ◽  
Author(s):  
T. S. van den Bremer ◽  
C. Whittaker ◽  
R. Calvert ◽  
A. Raby ◽  
P. H. Taylor
Keyword(s):  
Gravity Wave ◽  
Deep Water ◽  
Water Waves ◽  
Wave Equations ◽  
Stokes Drift ◽  
Surface Gravity ◽  
Return Flow ◽  
Surface Gravity Wave ◽  
Wave Groups ◽  
Wave Induced

Owing to the interplay between the forward Stokes drift and the backward wave-induced Eulerian return flow, Lagrangian particles underneath surface gravity wave groups can follow different trajectories depending on their initial depth below the surface. The motion of particles near the free surface is dominated by the waves and their Stokes drift, whereas particles at large depths follow horseshoe-shaped trajectories dominated by the Eulerian return flow. For unidirectional wave groups, a small net displacement in the direction of travel of the group results near the surface, and is accompanied by a net particle displacement in the opposite direction at depth. For deep-water waves, we study these trajectories experimentally by means of particle tracking velocimetry in a two-dimensional flume. In doing so, we provide visual illustration of Lagrangian trajectories under groups, including the contributions of both the Stokes drift and the Eulerian return flow to both the horizontal and the vertical Lagrangian displacements. We compare our experimental results to leading-order solutions of the irrotational water wave equations, finding good agreement.

The Set-Down and Set-Up of Directionally Spread and Crossing Surface Gravity Wave Groups in Severe North Sea Storms

2018 ◽  
Author(s):  
Mark L. McAllister ◽  
Thomas A. A. Adcock ◽  
Ton S. van den Bremer ◽  
Paul H. Taylor
Keyword(s):  
Free Surface ◽  
North Sea ◽  
Water Depth ◽  
Order Theory ◽  
Relative Magnitude ◽  
Second Order ◽  
Experimental Results ◽  
Surface Gravity Wave ◽  
Wave Groups ◽  
Set Up

Recent work by McAllister et al. (2018) [1] has experimentally confirmed that the set-down of the wave-averaged free surface, first described by Longuet-Higgins and Stewart (1962) [2], can turn into a set-up when wave groups are sufficiently spread or cross at large angles. Experimental results were shown to agree well with second-order theory, including frequency-sum and frequency-difference terms, where the latter are responsible for the wave-averaged free surface. In this paper, we review these experimental results and examine theoretically the magnitude of the wave-averaged free surface in realistic extreme North Sea conditions. Specifically, we examine the role of the shape of the spectrum, water depth, and the relative magnitude of the peak frequencies of the two crossing groups. We find that having a realistic spectrum (JONSWAP vs. Gaussian) considerably enhances the magnitude of the second-order contribution, the total second-order signal increases with decreasing depth and can display a maximum provided the water depth is shallow enough for small to moderate degrees of spreading or crossing angles and is larger for spectral peaks that are further apart.

scholarly journals Effect of free surface on submerged stratified shear instabilities

2018 ◽  
Vol 843 ◽  
pp. 98-125 ◽  
Author(s):  
Mihir H. Shete ◽  
Anirban Guha
Keyword(s):  
Growth Rate ◽  
Free Surface ◽  
Shear Layer ◽  
Gravity Wave ◽  
Stability Boundary ◽  
Boussinesq Approximation ◽  
Surface Gravity ◽  
Line Profiles ◽  
Surface Gravity Wave ◽  
Shear Instabilities

In this paper, we have considered the effects of the shallowness of the domain as well as the air–water free surface on the stratified shear instabilities of the fluid underneath. First, we numerically solve the non-Boussinesq Taylor–Goldstein equation for smooth velocity and density profiles of a model shear layer with a free surface. When the depth of the fluid is relatively shallow compared to the shear layer thickness, the surface gravity waves existing at the free surface come closer to the waves existing in the shear layer. This can lead to resonant wave interactions, making the flow unstable to more varieties of modal instabilities. In order to obtain a deeper understanding of the instability mechanisms, we have performed analytical studies with broken-line profiles (profiles for which vorticity and density are piecewise constant). Furthermore, reduced-order broken-line profiles have also been developed, based on which dispersion diagrams are constructed. Through these diagrams we have underpinned the resonantly interacting waves leading to each type of instability. Two new instabilities have been found; one of them, referred to as the ‘surface gravity – interfacial gravity (SG-IG) mode’, arises due to the interaction between a surface gravity wave and an interfacial gravity wave, and would therefore be absent if there is no internal density stratification. The other one – the ‘surface gravity – lower vorticity (SG-LV) mode’, which arises due to the interaction between a surface gravity wave and the lower vorticity wave, surpasses Kelvin–Helmholtz (KH) instability to become the most unstable mode, provided the system is significantly shallow. Stability boundary of the SG-LV mode is found to be quite different from that of KH. In fact, KH becomes negligible for relatively shallow flows, while SG-LV’s growth rate is significant – comparable to the growth rate of KH for unbounded domains (${\approx}0.18$). Moreover, the SG-LV mode is found to be analogous to the barotropic mode observed in two-layer quiescent flows. We have found that the effect of a free surface on the Holmboe instability is not appreciable. Holmboe in the presence of a free surface is found to be analogous to the baroclinic mode observed in two-layer quiescent flows. Except for the Holmboe instability, remarkable differences are observed in all other instabilities occurring in shallow domains when the air–water interface is replaced by a rigid lid. We infer that the rigid-lid approximation is valid for large vertical domains and should be applied with caution otherwise. Furthermore, we have also shown that if shear is absent at the free surface, our problem can be modelled using Boussinesq approximation, that is, $O(1)$ density variations in the inertial terms can still be neglected.

Non-stationary regimes of surface gravity wave turbulence

JETP Letters ◽  
2013 ◽  
Vol 97 (8) ◽  
pp. 459-465 ◽  
Author(s):  
R. Bedard ◽  
S. Lukaschuk ◽  
S. Nazarenko
Keyword(s):  
Gravity Wave ◽  
Surface Gravity ◽  
Wave Turbulence ◽  
Surface Gravity Wave
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