Stokes Drift and Net Transport for Two-Dimensional Wave Groups in Deep Water

Stokes drift and net transport for two-dimensional wave groups in deep water T.S. van den Bremer & P.H. Taylor Department of Engineering Science, University of Oxford [email protected], [email protected] Introduction This paper explores Stokes drift and net subsurface transport by non-linear two-dimensional wave groups with realistic underlying frequency spectra in deep water. It combines analytical expressions from second- order random wave theory with higher order approxi- mate solutions from Creamer et al (1989) to give accu- rate subsurface kinematics using the H-operator of Bate- man, Swan & Taylor (2003). This class of Fourier series based numerical methods is extended by proposing an M-operator, which enables direct evaluation of the net transport underneath a wave group, and a new conformal Figure 1: Illustration of the localized irrotational mass mapping primer with remarkable properties that removes circulation moving with the passing wave group. The the persistent problem of high-frequency contamination four fluxes, the Stokes transport in the near surface re- for such calculations. gion and in the direction of wave propagation (left to Although the literature has examined Stokes drift in right); the return flow in the direction opposite to that regular waves in great detail since its first systematic of wave propagation (right to left); the downflow to the study by Stokes (1847), the motion of fluid particles right of the wave group; and the upflow to the left of the transported by a (focussed) wave group has received con- wave group, are equal. siderably less attention. Focussed wave groups are rele- vant, because they can be used to describe the average shape of an extreme wave crest in a random sea (Tro- mans, Anaturk & Hagemeijer 1991). In practice, the Non-linear wave kinematics model wave field on the open sea often has a group-like struc- A two-dimensional body of water of infinite depth and ture. indefinite lateral extent is assumed with a coordinate sys- From mass conservation it is clear that Stokes drift by tem (x,y), where x denotes the horizontal coordinate and a wave group and Stokes drift by regular waves behave in y the vertical coordinate measured from the undisturbed a fundamentally different way. In fact, the wave-induced water level upwards. Inviscid, incompressible and irro- Stokes drift and its associated return flow at depth tational flow is assumed and the usual boundary condi- should be locally confined to the wave group and only tions (kinematic and dynamic at the free surface and a regular waves (infinite wave trains) can transport mass no flow boundary condition at infinite depth) apply. In and momentum indefinitely (McIntyre 1981). Second- order to study the drift beneath large waves, we adopt order wave theories (e.g. Dalzell 1999) based on the the NewWave wave group of Tromans et al (1991) and interaction between waves of different frequencies (cf. set each amplitude term to be proportional to its share Longuet-Higgins 1962), predict an irrotational return in the total discretized wave energy spectrum S(!): flow at depth that is equal and opposite to the flow S(!n) associated with Stokes drift at the surface, as illustrated an = aL P ; (1) N S(! ) in figure 1. We show that these results can be extended n=1 n to higher order by retaining more of the non-linearity of so that the total maximum amplitude is aL and the the underlying equations using the results of Creamer et shape is that of the auto-correlation function for the sea al (1989) with significant additional drift near the focus state. In particular, we consider a (discretized) Jonswap point. spectrum for fetch-limited seas (Hasselmann, Dunckel x0 = 0 focus point x0 = 0 focus point 20 20 x =−1980 m FC 0 15 FT 15 x =0 m 0 10 10 5 5 ] ] m m [ [ y 0 y 0 −5 −5 −10 −10 −15 −15 −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 x[m] ∆x[m] Figure 2: Particles trajectories at the free surface for a Figure 4: Particles trajectories at the free surface for a focussed wave crest (FC) and a focussed wave trough focussed wave crest (FC) with a linear wave amplitude (FT) with a linear maximum wave amplitude aL = 15m. aL = 15m for two different particles located at x0 = The particles are at x0 = 0 at the time of focus. −1980 m and x0 = 0 m (focus point) at the time of focus. 160 Creamer et al 140 160 Dalzell Creamer et al 150 120 Linear Dalzell 140 Linear 100 ] 130 m [ x 80 ] 120 ∆ m [ 60 x ∆ 110 40 100 20 90 0 80 0 5 10 15 aL[m] 70 −2000 −1500 −1000 −500 0 500 1000 1500 2000 x m Figure 3: Net horizontal displacement ∆x = x(t = 0[ ] −∞) − x(t = 1) by a wave group travelling past as a Figure 5: Net horizontal displacement ∆x = x(t = −∞ − 1 function of the maximum linear wave amplitude aL (i.e. ) x(t = ) by a wave group with a linear max- increasing steepness) for focussed crests (top line) and imum wave amplitude aL = 15 m of particles at the free focussed troughs (bottom line). surface as a function of their location at the time of focus x0 = x(t = 0). & Ewing 1980) and we set Tp = 12:4 and the peak- enhancement factor γ = 3:3. A random wave solution mations of surface elevation ηn and the surface potential to the governing equation and the boundary conditions φs;n on this non-linear free surface are given by (Creamer that is linear in wave steepness (O(kη)) takes the form: et al 1989): Z 1 XN 1 − iknη~L(x) − iknx η (x; t) = a cos(k x − ! t + µ ) ηn = e 1 e dx; L n n n n jknj −∞ n=1 Z 1 (3) (2) 1 0 − N iknη~L(x) ~ iknx X φn = e φL(x)e dx; kny jknj −∞ φL(x; y; t) = an!ne sin(knx − !nt + µn); n=1 whereη ~L(x) is the Hilbert transform of the linear free ~0 To include non-linear effects, we rely on the formula- surface signal ηL(x) and φL is the Hilbert transform of tion of Creamer et al (1989), who transform the surface the spatial gradient of the linear velocity potential. elevation η and the surface potential φs, the two canonical variables that capture the (Hamiltonian) dynamics Conformal mapping of the entire problem, into a new set of canonical vari- We introduce the following conformal mapping to reduce ables. These new variables exactly reproduce the second- the steepness of the surface profile to in turn improve order non-linear behaviour of surface waves and provide the convergence behaviour of the H-operator of Bateman, a remarkably good approximation to higher-order non- Swan & Taylor (2003) to calculate subsurface kinematics linearity. The nth Fourier component of the transfor- and eliminate persistent high-frequency contamination without having to resort to filtering: The coefficients An and hence the value of the volume nX=N flux s(x; t) can now be obtained from a procedure (the −i(knz−!nt+µn) ξ = z + iA − i ane ; (4) M-operator) similar to the H-operator of Bateman, Swan n=0 & Taylor (2003). Making use of the properties of the where z = x + iy are the complex coordinates in real complex potential across conformally mappable spaces, space and ξ = α + iβ are the complex coordinates in the M-operator can be applied in ξ-space to avoid high-frequency contamination and ensure convergence. transformed space. The coefficients an and kn are, re- spectively, the wave amplitude and the wave number of the first N components of the underlying linear sig- Result 1: Stokes drift nal. We demonstrate that (4), when applied inversely, As the particles undergoes its otherwise circular motion, for regular waves almost exactly maps a straight line in two factors contribute to Stokes drift at the free surface: ξ-space (β = 0) to the free surface corresponding to a deviation from the still water level z = 0 and devia- fifth-order Stokes expansion (cf. Fenton 1985) in z-space. tion from the initial horizontal position x0. Using linear For wave groups, the mapping (4) applied inversely maps kinematics, the net horizontal displacement by a pass- a straight line into a wavy surface which exactly repro- ing wave group ∆x = x(t = 1) − x(−∞) of a particle duces the features of a second-order free surface (e.g. at the surface can thus be obtained by integrating the Dalzell 1999) excluding the rather small effect of the fre- horizontal velocity u(x; η(x; t); t) with respect to time: quency difference terms. Z 1 dx XN a2 ! k From the definition of the complex potential Ψ = φ + ∆x = dt = n n n T ∗ + O(3); (8) i , a relationship between the velocities in z and ξ-space −∞ dt 2 n=1 can be found: ∗ dΨ dΨ dξ dξ where T = 2π=∆! is the repeat period of the discretized u − iv = = = u − iu : (5) 1 dz dξ dz α β dz (with intervals ∆!) frequency spectrum and only up to second-order in wave steepness are included. By first mapping to a reduced steepness surface, apply- By numerically solving the two simultaneous differen- ing the operators in this space and then mapping back, tial equations for the position of a particle that is initially the persistent problem of high-frequency contamination at the surface (and thus stays there): is effectively eliminated removing the need for inevitably arbitrary numerical filtering.

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