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Stokes drift and net transport for two-dimensional 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 spectra in deep wa- ter. It combines analytical expressions from - 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 (left to Although the literature has examined Stokes drift in right); the return flow in the direction opposite to that regular in great detail since its first systematic of wave propagation (right to left); the downflow to the study by Stokes (1847), the 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 (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 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 (cf. set each term to be proportional to its share Longuet-Higgins 1962), predict an irrotational return in the total discretized wave energy spectrum S(ω): flow at depth that is equal and opposite to the flow S(ωn) associated with Stokes drift at the surface, as illustrated an = aL P , (1) N S(ω ) 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 (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 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 = ∞) by a wave group travelling past as a Figure 5: Net horizontal displacement ∆x = x(t = −∞ − ∞ 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 ∞  XN 1 − iknη˜L(x) − iknx η (x, t) = a cos(k x − ω t + µ ) ηn = e 1 e dx, L n n n n |kn| −∞ n=1 Z ∞ (3) (2) 1 0 − N iknη˜L(x) ˜ iknx X φn = e φL(x)e dx, kny |kn| −∞ φL(x, y, t) = anωne sin(knx − ωnt + 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 potential. elevation η and the surface potential φs, the two canon- ical 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 ψs(x, t) can now be obtained from a procedure (the −i(knz−ωnt+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 , 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 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 = ∞) − 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 ∞ dx XN a2 ω k From the definition of the complex potential Ψ = φ + ∆x = dt = n n n T ∗ + O(3), (8) iψ, a relationship between the in z and ξ-space −∞ dt 2 n=1 can be found: ∗ dΨ dΨ dξ dξ where T = 2π/∆ω is the repeat period of the discretized u − iv = = = u − iu . (5) 1 dz dξ dz α β dz (with intervals ∆ω) 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. dx dy = u(x, y, t), = w(x, y, t), (9) dt dt The M(ass flux)-operator To directly evaluate the mass flux beneath a given wave we also evaluate the Stokes drift in a framework in which profile, we propose a new operator based on the principle kinematics are described by Dalzell’s (1999) second-order of the H-operator of Bateman, Swan & Taylor (2003), wave theory and the Creamer transform. Figure 2 traces which we term the M-operator. The general solution out the orbital paths that are described by a particle for the velocity potential as a solution to the governing that is located at the focus point at the time of focus Laplace equation ∇2φ = 0 can be given by the sum of N t = 0 for a focussed crest and a focussed trough based Fourier components (cf. equation 2). From the Cauchy- on the non-linear kinematics obtained from the Creamer Riemann equations for an irrotational and incompress- transform. A large part of the transport is performed by ible two-dimensional flow field, we have ∂φ/∂x = ∂ψ/∂y. just a few large orbits close to the focus point, where the The partial derivative of the velocity potential ∂φ/∂x can surface elevation is large and the waves are at their most be evaluated from (2) to give a solution for the stream non-linear. function ψ expressed as an indefinite integral: Figure 3 then compares the net horizontal displace- XN ment by a passing wave group ∆x of a particle at the kny surface located at the focus point at the time of focus ψ(x, y, t) = Ane cos(knx−ωnt+µn)+f(x, t), (6) n calculated by three different methods: the analytical ex- where the stream function (6) is now defined up to an pression based on linear wave theory (8) and the numeri- arbitrary function of x and t, f(x, t). We can define cal solutions based on Dalzell’s (1999) second-order ran- dom wave theory and the Creamer transform. It is evi- ψs(x, t) = ψ(x, η(x), t) − ψ(x, −∞, t) as the total mass flux (per unit width and with unit density) below the dent from this figure that the horizontal displacement in- free surface: 1Note that (8) is not a function of the number of components −1 ∗ XN N as N becomes large. To see this, note an ∼ N and T ∼ N −1 knη(x,t) resulting in the individual terms in the sum (8) to behave as N . ψs(x, t) = Ane cos(knx − ωnt + µn). (7) Summing over all the N terms then cancels the dependency on N. n

Linear of fluid in the direction of wave propagation by Stokes 300 Dalzell Creamer et al drift at and near the surface must therefore be compen- 200 sated by a return flow at depth in the opposite direction

] to ensure conservation of mass. Creamer et al (1989), al- /s

2 100 m [

s beit only for a narrow-banded wave group, show that this ψ 0 is indeed the case for the non-linear wave profile obtained using their transform. For second-order random wave −100 theories, the frequency difference terms (k1 − k2) of the −200 horizontal velocity are responsible for this return flow. −40 −30 −20 −10 0 10 20 30 40 t[s] These terms have a repeat period of (2π/∆ω), where ∆ω is the interval of the discrete frequency spectrum. In Figure 6: Net depth-integrated mass flux at the fo- the limit ω → 0, which corresponds to an infinite num- cus point as a function of time evaluated from linear ber of representative Fourier components N → ∞ and theory, using the second-order random wave theory of the limit in which the wave group is truly localized and Dalzell(1999) and the M-operator applied to the more not periodic with a very long repeat period, these terms non-linear signal obtained from the Creamer transform therefore correspond to a net flow, which only decays (aL = 15 m). very slowly with depth (cf. exp(|k1 − k2|y)). The M-operator is applied in figure 6 showing the zero- mean depth-integrated mass flux underneath a focussed creases with increasing wave amplitude and hence, keep- crest from linear theory, Dalzell (1999) second-order ran- ing the other parameters of the spectrum fixed, increas- dom wave theory and the Creamer transform as a func- ing wave steepness and that non-linear effects are impor- tion of time. It is evident from this figure that non-linear tant. effects account for a larger local transport of mass by the It is evident from (8) that displacement predicted largest crest, which is compensated by a greater return based on linear theory is not a function of the initial flow underneath the troughs. A second feature is that the position of the particle. Close to the focus point, the time integral of the depth-integrated flux for all cases is net displacement will be produced by a few large orbits, zero, to within the numerical accuracy of the computa- whereas away from the focus point, where the wave tions, as would be expected from continuity. group is dispersed, the equivalent net displacement is produced by a large number of small orbits. Comparing, References on the other hand, the trajectories of two particles, Bateman, W.J.D., Swan, C. & Taylor, P.H., 2003. On the calcu- one located at the focus point at the time of focus lation of the water particle kinematics arising in a directionally and one very far away from it, obtained from the spread wavefield. J. Comput. Phys. 186, 70-92. non-linear kinematics and surface elevation through Creamer, D.B., Henyey F., Schult, R. & Wright, J., 1989. Improved the Creamer transform, figure 4 shows very different linear representation of surface waves. J. Fluid Mech. 205, 135-161. behaviour. Clearly, a particle at the focus point at the Dalzell, J.R., 1999. A note on finite depth second-order wave-wave time of focus is transported further mainly by one or interactions. Appl. Ocean Res. 21, 3, 105-111. two large orbits corresponding to the highest few peak Fenton, J.D., 1985. A fifth-order Stokes theory for steady waves. J. of the wave group. Figure 5 shows the net displacement Waterway Port Coastal and Ocean Engineering, 111, 216-234. ∆x as a function of the position of the particle at the Hasselmann, D.E., Dunckel, M., Ewing, J.A., 1980. Directional Wave Spectra Observed during JONSWAP 1973. J. Phys. time of focus x0 confirming an additional drift effect Oceanogr., 10, 1264-1280. for particles within ∼ 2λp of the focus point at the Longuet-Higgins, M.S., 1962. Resonant interaction between two time of focus, where here λp ≈ 240 m. For a relatively trains of waves. J. Fluid Mech., 12, 321-332. steep wave considered herein (cf. k a ≈ 0.39), the net Mcintyre, M.E., 1981. On the ‘wave momentum’ myth. J. Fluid p L Mech., 106, 331-347. displacement ∆x in this region is approximately two Stokes, G.G., 1847. On the theory of oscillatory waves. Trans. the net displacement predicted by linear theory. Camb. Philos. Soc. 8, 441-455. Reprinted in: Stokes, G.G. (1880). Mathematical and Physical Papers, Volume I, Cam- bridge University Press. 197-229. Result 2: Return flow and net transport Taylor, P.H., 1992. On the kinematics of large ocean waves. Pro- One of the fundamental properties of wave groups is that ceedings 6th International Conference of the Behaviour of Off- the surface elevation returns to the still water η(x) → 0 shore Structures (BOSS.) 1, 134-145. as x → ∞. The same applies to the kinematics. In fact, Tromans, P.S., Anaturk, A.H.R., Hagemeijer, P.M., 1991. New model for the kinematics of large ocean waves - application as a close to focus, non-zero surface elevation and kinematics design wave. Proceedings of the 1st International Offshore and are confined to ≈ 4λp of the focus point. The transport Polar Engineering Conference, 64–71.