Miles' mechanism for generating surface water waves by wind, in finite water depth and subject to constant vorticity flow
Norbert Kern, Christophe Chaubet, Roberto Kraenkel, Miguel Manna
aa r X i v : . [ phy s i c s . a o - ph ] F e b Miles’ mechanism for generating surface water waves by wind, infinite water depth and subject to constant vorticity flow
N. Kern , C. Chaubet , R. A. Kraenkel , and M. A. Manna Université Montpellier, Laboratoire Charles Coulomb UMR 5221,CNRS UM, France, F-34095, Montpellier, France. and Instituto de Física Téorica-UNESP, Universidade Estadual Paulista,Rua Dr. Bento Teobaldo Ferraz 271 Bloco II, 01140-070, São Paulo, Brazil
The Miles’ theory of wave amplification by wind is extended to the case of finitedepth h and a shear flow with (constant) vorticity Ω. Vorticity is characterisedthrough the non-dimensional parameter ν = Ω U /g , where g the gravitational ac-celeration, U a characteristic wind velocity and k the wavenumber. The notion of’wave age’ is generalised to account for the effect of vorticity. Several widely usedgrowth rates are derived analytically from the dispersion relation of the wind/waterinterface, and their dependence on both water depth and vorticity is derived anddiscussed. Vorticity is seen to shift the maximum wave age, similar to what waspreviously known to be the effect of water depth. At the same time, a novel effectarises and the growth coefficients, at identical wave age and depth, are shown toexperience a net increase or decrease according to the shear gradient in the waterflow. I. INTRODUCTION : WAVE GENERATION BY WIND IN WATER OF FINITEDEPTH
How wind generates ocean waves is a formidable problem, many aspects of which are stillnot understood today. It starts from the Navier-Stokes equations [1], applied to two layersof fluid. Solving these is a classic exercise, but the presence of wind requires a successionof subtle approximations and assumptions. In particular, the interface between water andair requires careful thought. The pioneering works are those due to Jeffreys [2, 3] and toMiles [4]. The theory by Jeffreys assumes that, on the lee-side of the surface wave, theondulated water-air interface is sheltered from the air flowing over it surface waves. Thiskinematic scenario produces a pressure gradient which performs work on the wave. Henceenergy is transferred from the wind to the wave (Jeffrey’s sheltering mechanism ). Miles’theory of wave generation by wind assumes that ocean surface waves are generated by aresonance phenomenon. Resonance appears between the wave-induced pressure gradient onthe inviscid airflow and the surface waves. This leads to a growing wave amplitude whenthe phase velocity of the surface wave equals the speed of the airflow (see reference [5] for athorough discussion). In Jeffreys’ and Miles’ theories the viscosity is neglected both in airand in water, and furthermore water is considered as infinitely deep and irrotational. Boththeories rely on linearising the equations of motions.Historically the first experiments and numerical studies concerning finite depth wind-wave growth were conducted by Thijsse [6] and Bretschneider [7]. Particularly important, inorder to understand the physics of wind-wave dynamics in finite depth, were the experimentscarried out in Lake George (Australia) by Young and Verhagen [8, 9]. For a full account ofthe Young and Verhagen works see reference [10].Recently, in reference [11], Miles’ theory was extended to water of finite depth, and inreferences [12, 13] the wind-wave amplification was studied as a possible mechanism forfinite-time blow-up of solitary wind waves and steep wave events (rogue waves) in finitewater depth. In reference [14] we have established a fully nonlinear model equation ofsurface wind-wave generation. There, the interaction between air and water is describedfrom a quasi-linear point of view, in which the water obeys the fully nonlinear Serre-Green-Naghdi model [15–17] while, as in Miles’ theory, the air flow is kept linear and obeys thelinear Euler equation of motion.Net currents in the water are absent in all these approaches. In coastal waters, underlyingcurrents may be present, and this raises the question how the propagation of wind-drivensurface waves is affected. Such currents may range from weak to very strong, and can begenerated through various mechanisms such as tidal flow, oceanic circulation, wind action orbreaking of waves. Depending on how they are generated the currents are often observed tovary with depth, and thus carry an underlying vorticity . Such background vorticity, knownto be present for example in strong tidal currents [18] or in wind driven currents [19], can beimportant and should be taken into account when modeling the propagation of water waves[20]. Vorticity is especially observed in shallow water environments. For instance, linearshear due to strong currents has been observed in the surf zone, in strong rip currents, insitu [21] or in laboratory experiments [22]. More recently, a similar depth dependence ofcurrents was observed over coral reefs [23].From a theoretical point of view, after the pioneering works of Benjamin [24] and Freemanand Johnson [25], the role played by constant or variable vorticity (almost exponentially de-creasing in depth) constitutes a classical but vast subject in fluid mechanics. Many theories,exact or approximate, were formulated for steady periodic waves and for progressive waves,in finite or infinite depth, under the action of vorticity [26–31]. The combined action of vor-ticity and surface tension were studied in references [32] [33]. The modulational instability(Benjamin-Feir instability) in finite depth under the action of vorticity was investigated in[34].The aim of this work is to provide a theory for the growth of surface wind waves, in waterof finite depth and in presence of constant vorticity currents. The purpose is twofold: onone hand it intends to establish the mathematical laws able to qualitatively reproduce atleast some crucial features of the field experiments. On the other hand the intention is tosupply a theoretical framework, thus going beyond empirical laws.To carry out this task we build upon the theory by Miles in a finite depth context, asintroduced in reference [11], which we extend in order to account for the action of a linearshear current. The paper is organized as follows.Section (II) lays out the mathematical framework. In subsection (II A) we introduce thenonlinear Euler equation governing the dynamics of surface water waves under the actionof constant vorticity. In subsection (II B) the air domain is introduced and coupled to thewater domain, leading to an air pressure which is no longer an overall constant, but insteadFIG. 1: Sketch to illustrate the reference frame, the geometry as well as the flowconditions in water and in air. Both water currents and wind are taken towards the right.The example has positive vorticity, meaning that the current decreases with depth. Thesystem is invariant in the y direction.varies in space and over time. Solving the linear problem at the interface we derive thelinear dispersion relation of wave amplification, in finite depth and in presence of a constantshear in the flow field. In section (II D) we introduce dimensionless variables and scalings,to derive various adequate growth rates which are commonly considered in the field. Theresulting linear dynamics is described analytically, in terms of explicit results, which we thenexploit numerically. Finally, section (IV) draws the conclusions to our findings. II. MATHEMATICAL ANALYSIS
Here we lay out the equations describing the system consisting of a water domain (includ-ing the underlying shear current which induces vorticity) and the air domain (including theflow field due to wind), before coupling these at the interface on which a wave propagates.In the absence of a propagating wave the situation is as follows, also represented in thesketch in Fig. (1). We consider the system to be invariant in the y-direction. Thus waterand air particles are located in a two-dimensional Cartesian coordinate system with axes x, z . The origin is at x = z = 0, and the horizontal domain is x ∈ ] − ∞ , + ∞ [. At aheight z = 0 is the water-air interface, the domain of positive z ∈ ]0 , ∞ [ corresponds to the(unperturbed) air, and z ∈ [ − h,
0] is the (unperturbed) water domain. The bottom of thewater domain, at z = − h , is considered unpermeable. Both water and air are taken to beinviscid and incompressible, and surface tension effects at the interface are not accounted for.We assume the water to flow in the positive x direction, the velocity varying linearlywith depth. In an earth-bound reference frame the velocities therefore interpolate between v s ~e x at the water surface ( z = 0) and v b ~e x at the bottom ( z = − h ), where ~e x is the unitvector. The surface velocity v s , bottom velocity v b as well as the water depth h are takento be constants. In the following we will work in the reference frame in which the watersurface is static, i.e. the frame which moves at the surface velocity v s ~e x with respect to theearth-bound reference frame. This is a natural choice, since the interface is the support forthe propagating wave, although there are some implications which we will return to later.In the reference frame of the water surface the water flow profile due to vorticity is thusgiven by ~U (Ω)0 ( z ) = Ω z ~e x for − h ≤ z ≤ , (1)where the parameter Ω = v s − v b h (2)is in fact the (y-component of the) vorticity attributed to the steady shear flow ( ~ω = ∇ × ~U (Ω)0 ( z ) = Ω ~e y = const ).For the air domain, a steady-state airflow is prescribed to represent the effect of wind.To this end the mean horizontal velocity profile is of the form ~U ( z ) = U ( z ) ~e x = U ˆ U ( z ) ~e x , (3)where U is a characteristic wind speed and ˆ U is a dimensionless function characterising thewind profile in terms of the vertical position.Two points require careful thought. First, different wind profiles may be considered [35].We will later focus on the most common choice, a logarithmic wind profile, but for the timebeing the calculations are general. Second, matching the flow fields in water and air is ahighly non-trivial matter, due to the presence of a turbulent layer between both media,which develops when the wind blows over the ondulated water surface. However, it as beenshown that one may account for this in a phenomenological way by requiring that the airflow profile (3) match the water velocity not at the top of the water volume ( z = 0), butat a height z > z is known as the roughness length . For a wave of wavelength 2 π/k propagating with a phase velocity c the roughness length has been shown to obey theso-called Charnock relation [35] z = Ω CH k (cid:18) U c (cid:19) , (4)where Ω CH is a phenomenological constant [36], the value of which is to be adapted tothe specific wind profile and to the choice of the characteristic wind velocity U . In thisapproach, the air flow then vanishes at the top of the roughness layer z , rather than at theaverage position of the water-air interface ( z = 0), i.e. we have U ( z ) = 0 . (5)This is a widely used approximation, first proposed in Ref. [37], and it is indeed appropriatefor the ranges of wind speed considered here [38].Note that the empirical expression (4) for the roughness length z has been established ininfinite deep water and on static water [35, 37]. Here we assume that this relation extendsto our situation.In order to analyse the propagation of a wave at the water-air interface in our systemwe now establish the hydrodynamic equations in each domain, before coupling them via theappropriate boundary conditions. A. The water domain
As a wave propagates on the water surface, for now without accounting for the effect ofwind, the flow field is ~U w ( x, z, t ) = (cid:16) U w ( x, z, t ) , , W w ( x, z, t ) (cid:17) = (cid:16) U (Ω)0 ( z ) + u ( x, z, t ) , , w ( x, z, t ) (cid:17) (6)in the reference frame of the water surface. Here U (Ω)0 ( z ) is the stationary flow field referredto in (1), to which perturbations are added as the wave propagates. This velocity fieldsatisfies the Euler equations: U w,x + W w,z = 0 (7) U w,t + U w U w,x + W w U w,z = − ρ w P x (8) W w,t + U w W w,x + W w W w,z = − ρ w P z − g , (9)where P is the pressure, g is the gravitational acceleration and ρ w the water density. Sub-scripts to U w ( x, z, t ), W w ( x, z, t ) and P ( x, z, t ) denote partial derivatives.The boundary conditions to this flow are to be imposed at z = η ( x, t ) and at z = − h .They are P = P a , at z = η (10) η t + U w η x − W w = 0 , at z = η (11) W w = 0 , at z = − h . (12)Here P a ( x, z, t ) is the variable air pressure, and η x and η t are partial derivatives of η ( x, t ).Thus equation (10) expresses the continuity of the pressure across the air/water interface.We introduce the dynamic pressure P ( x, z, t ), relative to the hydrostatic pressure in theundisturbed initial state, as P ( x, z, t ) = P ( x, z, t ) + ρ w gz − P , (13)where P is a constant.The perturbation to the free surface is η ( x, t ), and u ( x, z, t ), w ( x, z, t ) as well as P ( x, z, t )are those to the horizontal and vertical velocities as well as the pressure, respectively. Lin-earizing equations (7)-(12) around the unperturbed state (1), expressed in terms of P andthe perturbations u , w and η , we have u x + w z = 0 (14) u t + Ω z u x + Ω w = − ρ w P x (15) w t + Ω z w x = − ρ w P z . (16)The boundary conditions are also to be linearised, and they become P ( x, η, t ) = P a ( x, η, t ) + ρ w gη − P (17) η t = w (0) , (18) w ( − h ) = 0 . (19) c / c f d (cid:1) (tanh(kh)/(kg)) FIG. 2: Effect of vorticity on a wave propagating on the water surface. The graph showsthe multiplicative factor by which the phase velocity c is modified as compared to thesame system with no vorticity. The graph is based on Eq. (28), and vorticity is representedthrough a dimensionless expression Ω q tanh( kh ) / √ kg , proportional to vorticity. Byconstruction, the value 1 is attained for vanishing vorticity, indicated by a circle. Negativevorticity increases the phase velocity, whereas positive vorticity slows the wave down.The system of linear equations (14)-(19) can be solved assuming normal mode solutionsas P = P ( z ) exp ( iϕ ) , u = U ( z ) exp ( iϕ ) , w = W ( z ) exp ( iϕ ) , η = η exp ( iϕ ) , (20)with ϕ = k ( x − ct ), where k is the wavenumber, c the phase speed and η is a constant.Using equations (20), (14), (15), (16), (17) and (19) we obtain u ( x, z, t ) = 2 ai exp ( − kh ) exp ( iϕ ) cosh k ( z + h ) , (21) w ( x, z, t ) = 2 a exp ( − kh ) exp ( iϕ ) sinh k ( z + h ) , (22)1 ρ w P ( x, z, t ) = 1 ρ w [ P a ( x, η, t ) − P ] + gη − aic exp ( − kh ) exp ( iϕ ) n cosh ( kh ) (23) − cosh k ( z + h ) } + 2 ai Ω exp ( − kh ) exp ( iϕ ) {− z cosh k ( z + h ) − k sinh ( kh ) + 1 k sinh k ( z + h ) o with some constant a . Now we calculate (1 /ρ w ) P x from (23), and substituting into (15) weobtain ikgη + 2 a exp ( − kh ) exp ( iϕ ) n kc cosh ( kh ) + Ω sinh ( kh ) o = − P a,x ρ w . (24)From (18) and (22) we obtain − ikηc − a exp ( − kh ) exp ( iϕ ) sinh( kh ) = 0 . (25)Finally, it follows from (24) and (25), integrating over x , that η n g − kc cosh ( kh )sinh ( kh ) − c Ω o = − P a + P ρ w . (26)For P a = P , Equation (26) leads to the very well known phase velocity c = c c = − Ω2 k tanh ( kh ) ± s Ω k tanh ( kh ) + 4 gk tanh ( kh ) , (27)where we introduce the subscript ’0’ to refer to the reference state of a propagating wave onthe water surface in the presence of shear flow, but without air flow.In order to clarify the discussion we choose to consider a wave propagating to the right inthe remainder of the manuscript, which amounts to selecting the positive branch of Equation(27): c = − Ω2 k tanh ( kh ) + 12 s Ω k tanh ( kh ) + 4 gk tanh ( kh ) (28)= c fd s tanh ( kh )4 kg − Ω q tanh ( kh )2 √ kg , (29)where c fd is the finite depth velocity defined by c fd = ( g tanh ( kh ) /k ) / . This phase velocityplays an important role when discussing the effect of wind, and we therefore illustrate itsdependence on vorticity in Fig. 2 for later reference.In the complete problem, however, in which we must account for the wind profile, theair pressure can no longer be taken to be constant: P a = P . In order to determine thephase velocity c in the presence of wind, from (26), we thus need to determine the dynamicperturbation to the air pressure P a , and its value at the interface position z = η ultimatelyenters the balance equations at the water-air interface. B. The air domain
Let us consider the linearized governing equation of a steady air flow, with a prescribedmean horizontal velocity U ( z ), as stated in (3). We are going to study perturbations to thismean flow in the x and the z components, where the subscript a stands for air : u a ( x, z, t ), w a ( x, z, t ), as well as the air pressure field P a ( x, z, t ), are the dynamic contributions as thewave propagates. ρ a is the air density, and we define P a ( x, z, t ) = P a ( x, z, t ) + ρ a gz − P .0In the reference frame where there is no surface flow we have, from continuity and from theNavier Stokes equations: u a,x + w a,z = 0 (30) u a,t + U ( z ) u a,x + U z ( z ) w a = − ρ a P a,x (31) w a,t + U ( z ) w a,x = − ρ a P a,z . (32)To these equations governing the flow field we must add the appropriate boundary conditions.The first one is the kinematic boundary condition η t + U ( z ) η x = w a ( z ) . (33)As discussed above, it is to be evaluated at the aerodynamic sea surface roughness z .Exploiting the fact that the wind flow vanishes at the roughness height, Eq. (5), the kineticboundary condition (33) finally reduces to η t = w a ( z ) . (34)Next we introduce normal mode expressions, as in (20), also for the air flow, P a = P a ( z ) exp ( iϕ ) , u a = U a ( z ) exp ( iϕ ) , w a = W a ( z ) exp ( iϕ ) , (35)and we add the following boundary conditions on W a and P a :lim z → + ∞ ( W ′ a + k W a ) = 0 (36)lim z → z W a = W (37)lim z → + ∞ P a = 0 . (38)The first condition imposes that, far from the interface, the perturbation of air flow mustdecrease exponentially. The remaining conditions set the vertical component of the windspeed in terms of the wave motion at the sea surface, and require pressure continuity acrossthe water-air interface.Finally, using equations (30)-(32) and (38) we have w a ( x, z, t ) = W a exp ( iϕ ) (39) u a ( x, z, t ) = ik W a,z exp ( iϕ ) (40) P a ( x, z, t ) = ikρ a exp ( iϕ ) Z ∞ z h U ( z ′ ) − c i W a ( z ′ ) dz ′ . (41)1By eliminating the pressure from the Euler equations we obtain the well-known Rayleighequation [39], which holds ∀ z \ z < z < + ∞ h U ( z ) − c i ( W a,zz − k W a ) − V zz W a = 0 . (42)This is also known as the inviscid Orr-Sommerfeld equation. It is singular at the ’critical’or ’matched’ height z c = z e cκ/u ∗ > z >
0, where U ( z c ) = c . We recall that this modelassumes any eddies or other non-linear phenomena to be accounted for by the roughnessheight z , and therefore the bulk of the air flow is non-turbulent. C. Matching the flow at the interface
Recall that, in equations (39)-(42), neither the function determining the perturbation tothe air flow W a ( z ) nor the phase velocity c are known: indeed, finding the phase velocityin the presence of wind, as well as the shear flow in water, is precisely the object of ourcalculation. This is achieved by applying the appropriate boundary conditions across atinterface. In order to do so we have to determine the air pressure field P a ( x, η, t ). We obtain P a ( x, η, t ) = P − ρ a gη + ikρ a exp ( iϕ ) Z ∞ z h U ( z ) − c i W a ( z ) dz , where the lower bound for integration may be taken to be the constant roughness height z instead of z = η , since we are studying the linear problem.Finally, using equation (34) to eliminate the term ikρ a exp( iϕ ) and substituting P a givenby equation (43) into (26) we obtain what is effectively the dispersion relation of the problem,as it fixes the phase velocity c : g (1 − ǫ ) + c ( sk W I − Ω ) − c ( sk W I + k coth( kh ) ) = 0 . (43)Here s = ρ a /ρ w and the integrals I and I are defined as I = Z ∞ z U W a dz, I = Z ∞ z W a dz . (44)The density of air being small compared to that of water, relation (43) may be expanded asa series in terms of s = ρ a /ρ w ∼ − : c = c + sc + O ( s ) . (45)2An explicit expression for the first order term c can be established using a perturbationmethod, which consists in solving the Rayleigh equation (42) based on the zero-order phasevelocity c . This approach, which follows [35], is pursued in the next section.The dispersion relation can be deduced from (43), once we have evaluated the integrals I and I given by (44): once it is known, its imaginary part carries the information on wavegrowth. For this, in turn, the profile W a ( z ) is to be established as a solution to the Rayleighequation (42). Before doing this numerically, we pursue to introduce the growth coefficientswe intend to analyse. D. Wave growth during propagation in the presence of vorticity and wind
Once the function W a ( z ) is known, its imaginary part intervenes in the dispersion relation(43), and sets the complex part to the phase velocity c . This directly yields the growth rate γ of the wave amplitude η ( x, t ), defined as γ = k ℑ ( c ) , (46)where ℑ stands for the imaginary part.The theoretical and numerical results for the growth rate γ are traditionally studied andcomputed in terms of dimensionless parameters. To the established parameters δ , θ dw (seeYoung and Verhagen [8], Young and Verhagen [9] and Montalvo et al. [11]) we now add athird parameter ν characterising vorticity. These are defined as δ = ghU , θ dw = 1 U r gk , ν = Ω U g (47)It is thus immediately clear that, in contrast to deep water, where the growth rate canbe characterised by as a single graph showing the growth rate as a function of wave age,the situation is more complex here: the presence of two additional dimensionless parametersimply that we will instead be dealing with a two-parameter family of curves .Note that the parameters defined in (47) have direct physical meaning. The (square rootof) dimensionless parameter δ relates the phase velocity of a shallow water wave to the windvelocity. The parameter θ dw is the analogous ratio of the deep-water phase velocity andthe wind speed. Finally, ν is a measure for the importance of the velocity gradient due tovorticity.3The parameter θ dw is often referred to as the ’wave age’, or at least a theoretical equivalentof this observational quantity, and is therefore important for making contact with fieldobservations and literature. It has been put forward recently [11] that this notion can begeneralized to water of finite depth, by defining the finite depth wave age θ fd as θ fd = 1 U r gk q tanh( kh ) = θ dw T / , (48)where T = tanh( δθ dw ) . (49)Expression (48) is a depth dependent parameter, which is constructed to interpolate be-tween deep water and shallow water situations: indeed, for a finite and constant θ dw , wehave θ fd ∼ θ dw when δ → ∞ and θ fd ∼ δ / = √ gh/U if δ →
0. Therefore θ fd reduces tothe ratio of the appropriate phase velocity and the wind speed, in both limiting cases. Inthe following section we will extend this concept to flow with constant vorticity.We can now proceed to determine the dispersion relation and deduce the parameterscharacterising wave growth. The following non-dimensional variables and scalings will beused in order to non-dimensionalise the growth coefficients: U = U ˆ U , W a = W ˆ W a , z = ˆ zk , c = U ˆ c, t = U g ˆ t , (50)where ’hats’ indicate dimensionless quantities.Using (47) and (50) in equations (27) and (43), and retaining only first order terms in s ,we obtain expressions for ˆ c and ˆ c . The zero-order terms reproduce Eq. (28), as expected.The linear term yieldsˆ c = ˆ c + s − ˆ c − ν ˆ c + ˆ c θ dw [2 − ν ˆ c ] [ ˆ I − ˆ c ˆ I ] ! . (51)which is the desired dispersion relation under the effect of wind.Finally the growth of the amplitude η with time is given byexp γt = exp ˆ γ ˆ t (52)with the corresponding growth rateˆ γ = s ℑ (ˆ c ) θ dw = s ˆ c θ dw [2 − ν ˆ c ] (cid:16) ℑ ( ˆ I ) − ˆ c ℑ ( ˆ I ) (cid:17) . (53)4Hence we have established an expression for the growth rate ˆ γ for a given physical situation,i.e. a given set of parameters ( δ, θ dw , ν ).Our results also explore the β -Miles parameter, related by β = 2ˆ γs θ dw ˆ c , (54)where we have taken β as it is usually defined, via ℑ ( c ) = ( s/ βc ( U c ) .Using (8) in (54) we obtain β = 2 ˆ c θ dw [2 − ν ˆ c ] (cid:16) ℑ ( ˆ I ) − ˆ c ℑ ( ˆ I ) (cid:17) . (55)Finally, another important parameter to be considered is the dimensionless fractionalenergy increase per radian, defined asˆΓ = c ,g ω E dEdx = 2 γ c ,g ω c , (56)with c ,g , c the group and phase velocity of the dominant (spectral peak) waves of frequency ω . For Ω = 0 the ration c ,g /c is given by the very well known expression c ,g c = 12 " kh sinh (2 kh ) , for Ω = 0 . (57)Here we are interested in the generalization of (57) to the presence of vorticity, which isgiven by c ,g c = 12 " kh sinh (2 kh ) + " − kh sinh (2 kh ) × s Ω T Ω T + 4 gk ! , (58)for all Ω. From this equation (58) yields ˆΓ, which readsˆΓ = θ dw T / ˆ γ δθ dw sinh ( δθ dw ) + − δθ dw sinh ( δθ dw ) × vuut ν θ fd ν θ fd − νθ fd + r (cid:16) νθ fd (cid:17) , (59)which is the dimensionless fractional energy increase per radian, accounting for the effect ofvorticity. III. RESULTS AND DISCUSSION
In this section we exploit the results we have obtained above, based on the analyticalexpressions but aided by a numerical approach when required. To this end we follow the5approach by Beji and Nadaoka [35], which consists in numerically solving the Rayleighequation in order to determine the velocity profile W a ( z ) in the air domain. This requireshandling the singularity of the Rayleigh equation (42), which is done by establishing ananalytical approximation valid close to the singularity, i.e. around the height z c where thewind velocity equals the phase velocity. From this, starting values are deduced which serveto initialise a solver for differential equations.At this stage we now focus our analysis on a specific air flow field U ( z ), which is thelogarithmic wind profile with U ( z ) = U ln( zz ) . (60)This relation is commonly used to describe the vertical dependency of the horizontal meanwind speed [40]. This can be justified based on scaling arguments and solution matchingbetween the near-surface air layer and the geostrophic air layer (see Tennekes [41]). A. Accounting for vorticity in the wave age
The first point we address is the notion of a generalised wave age, which is the parameterˆ c already defined via (50) as the ratio θ = c U = ˆ c , (61)with the phase velocity c given by Eq. (28). To see how this extends the existing definition,recall that the notion of wave age was defined originally, for deep water, as the ratio c ,dw /U .It thus measures the phase velocity of a wave propagating without wind as compared tothe wind speed. A similar parameter ˆ c ,fd = c ,fd /U has been introduced in [11] as thegeneralisation of the wave age to finite water depth. Setting θ fd = c ,fd /U , this characteriseswave propagation on the surface of a finite depth water body. No vorticity was consideredat this stage. In our analysis, the expression for the phase velocity as established in Eq. (28)now suggests that the parameter defined in Eq. (61) directly generalises the notion of waveage further, by accounting for the presence of vorticity.With this definition, the wave age in the presence of vorticity is deduced directly fromEq. (28) as θ = c /U = θ fd − νθ fd vuut νθ fd ! . (62)6As an aside, it may be worth pointing out that the choice of reference is important here:the ratio (61) is to be taken with velocities expressed in the reference frame of the watersurface, which we have already adopted here. Defining the wave age in this way is onlymeaningful with respect to this particular reference frame. This is because the variable U defines the typical wind scale in the reference frame of the water surface. It cannot simply betransformed to a different frame, as transforming constant velocity to the wind profile (60)adds an overall constant, which is not equivalent to modifying U . Rather, the definitionof U is intrinsically based on the wind speed relative to the water surface , as this is whatthe Charnock relation (4) matches the wind profile to, and therefore U . Note that thisconsideration is new to the system combining waves with both wind and surface flow.All other physical quantities concerning the growth of waves ( γ and β for instance) remainunchanged through a Galilean transformation from one reference frame to another. Ratherexpectedly, calculations in the earth-bound frame are found to lead to the identical resultswhen the relative velocity between reference frames is added to the phase velocity. B. Miles growth parameter β We first discuss the effect of vorticity on the Miles growth parameter β . In Fig. 3, resultsare shown for the Miles parameter β as a function of the vorticity-corrected wave age θ fd, Ω .All graphs are calculated for the same depth parameter δ = 25, but correspond to differentvalues of the vorticity parameter ν . In all our plots the black curve corresponds to zerovorticity, for easy reference, and therefore the black graph of Fig. 3 reproduces the result ofFig. 2 in reference [11].The most remarkable feature of β ( θ ) in the absence of vorticity is the steep drop to zeroas the wave age approaches √ δ = 5. This has been discussed in [11], and implies a maximalwave age in shallow water: it is directly related to the fact that there is a maximum phasevelocity, which a propagating wave cannot exceed in finite water depth. In Fig. 3 it is clearthat this maximum wave age is modified by vorticity: it is increased for negative values of ν , whereas it is reduced for positive values.One way of visualising this is to plot the generalised wave age θ fd, Ω , which accounts forvorticity, as a function of the zero-vorticity wave age θ fd . This is shown in Fig. 4. Thedashed black line indicates that θ fd, Ω and θ fd are identical for zero vorticity. The effect of7vorticity is then quite different according to its sign. To discuss this, we first return to thecase of zero vorticity, and recall that the upper bound to the wave ages which are accessiblein water of finite depth (see [11]) is θ fd ≤ θ ( max ) fd = √ δ = √ ghU . (63)Consequently, the maximum possible wave age decreases with depth, and it is only forinfinitely deep water that a potentially unlimited domain of accessible wave ages is to beconsidered.This is no longer the case in the presence of positive vorticity. Indeed, for ν > /ν in deep water or for strong vorticity. Therefore one impact of positivevorticity is to set a maximum wave age even for a deep water wave. For negative vorticity,no such upper bound is set in deep water; instead, the vorticity correction increases thewage age.To complete the argument, we state the expression of the upper bound in the velocity-corrected wave age. Using Eq. (62) and (63) we obtain, θ ( max ) fd, Ω = θ ( max ) fd − νθ fd,c s ν θ fd,c = √ δ − ν √ δ s ν δ . (64)Note that this correctly reduces to δ / for ν = 0, as it must. From this expression it followsdirectly that negative vorticity increases the maximum wave age, and hence the maximumwave propagation speed. Therefore, negative velocity produces older seas while positivevorticity has the opposite effect. The corresponding graph is shown in Fig. 5.To complete the discussion of the Miles growth coefficient β , we now contrast Fig. 3by similar plots in Fig. 6, obtained for two different values of δ . The values used for thevorticity parameter ν are identical in all three graphs. We observe the same tendencies, witha major difference worth pointing out. Indeed, the vorticity parameter ν has a weaker effectat small depth: curves for identically spaced values of ν are closer one to another for δ = 4.In comparison, the effect of vorticity is larger in greater depth, exemplified by δ = 81.As concerns practical implications, however, one must keep in mind that an identicalvorticity parameter does not mean an identical current at the water surface. Rather, it isthe combination of vorticity and depth which sets the shear current, through the difference8 M il e s g r o w t h c o e (cid:0) c i e n t (cid:2) wave age (cid:3) fd, (cid:4) (cid:5) =-1 (cid:6) =-0.5 (cid:7) =-0.25 (cid:8) =-0.1 (cid:9) = 0 (cid:10) = 0.1 (cid:11) = 0.25 (cid:12) = 0.5 (cid:13) = 1 FIG. 3: The Miles’ growth coefficient β , as defined in (55) and obtained by numericallysolving the Rayleigh equation (42). All plots are for the same depth parameter δ = 25.Each line corresponds to a given value of vorticity, according to the parameter ν indicatedin the legend. The black line, for ν = 0, reproduces the known result for wave growth infinite depth [11]. Negative vorticity increases the growth coefficient, whereas positivevorticity reduces it.of surface and bottom currents. Indeed, non-dimensionalising this current by the wind speed U we have ( U s − U b ) /U = Ω h/U = νδ , and thus both ν and δ intervene.Apart from this important difference, all families of curves for a given value of δ showthe same behavior, and the same enhancement for negative vorticity. More specifically, amaximum is observed for β as soon as negative vorticity is present. This maximum increaseswith increasing (negative) vorticity, a novel feature: changes in water depth cannot provokethis effect. In Fig. 7 we analyse the shift in β max with respect to the plateau value at zerovorticity, plotting it as a function of ν . The shift is clearly linear for (negative) vorticitiesup to ν ≤ − .
5, as shown in the inset of Fig. 7. This is followed by β saturating at around ν ≈ β max ≈ .
5. The implication is that there is an upper bound to theincrease of β which can be provoked by vorticity.The increase of β is confirmed by the evolution of the amplitude growth coefficient ˆ γ withvorticity. In Fig.8 we have plotted three families of curves, corresponding to the same values9 w a v e a g e θ f d , Ω ( a cc o un t i n g f o r v o r t i c i t y ) wave age θ fd (no vorticity) ν =-3.0 ν =-1.0 ν =-0.3 ν =-0.1 ν =0 ν =0.1 ν =0.3 ν =1.0 ν =3.0 FIG. 4: The generalised wave age θ fd, Ω , which incorporates the effect of vorticity, can bewritten as a function of θ fd = θ fd, Ω=0 , the wave age without vorticity. The mathematicalrelation is given by Eq. (62). Colours represent different values of vorticity; solid linescorrespond to positive vorticity whereas dashed lines of the same colour correspond tonegative vorticity of the same negative values. Note that only in infinite depth all waveages are accessible: in a finite depth system the wave age has an upper bound of θ ( max ) fd = √ δ . This entails a maximum value θ ( max ) fd, Ω also in the presence of vorticity, but thismaximum accessible wave age is reduced for positive vorticity and increased for negativevorticity.of the depth parameter ( δ = 4 , ,
81) used in the previous figure. First, for each familyof curves we confirm the previous observations: negative vorticity increases the maximumwave age, whereas positive vorticity has the opposite effect. Interestingly, this behaviour issimilar to what is observed in the absence of vorticity when the water depth is varied (seeFig.2 of ref. [11]). Second, we observe a global enhancement of ˆ γ for negative vorticity, whichreplicates the behavior of β . However, this contrasts with the effect of water depth [11], whichonly displaces the maximum wave age. The enhancement of the actual growth coefficient ˆ γ due to vorticity is an entirely novel feature.0 m(cid:14)x(cid:15)(cid:16)(cid:17)(cid:18) w a v e a g e (cid:19) f(cid:20)(cid:21)(cid:22) ( (cid:23)(cid:24)(cid:25) ) d(cid:26)(cid:27)(cid:28)(cid:29) (cid:30)(cid:31)r !" ν=-1.0ν=-0.3ν=-0.1ν=0ν=0.1ν=0.3ν=1.0ν=3.0 FIG. 5: The maximum achievable value for the wave age (Eq. (64)) as a function of thedepth parameter δ , for selected vorticity values of both signs. For a static sea ( ν = 0, blackcurve) we have θ ( max ) fd = √ δ , as is known from [11]. For positive vorticity, the maximumvalue is reduced; in turn, it is increased for negative vorticity. C. Energy growth rate
Another commonly used way to assess the growth of waves is to represent the energygrowth rate Γ as a function of the inverse wave age 1 /θ , see e.g. in [35]. This is shown inFig. 9, based on Eq. (59). Three choices for the depth parameter δ are juxtaposed, usingthe same values as in Figs. 3, 6 and 8. The behaviour is again reminiscent of what happenswhen the water depth is finite (see ref. [11]), i.e. the location of the vertical drop of theenergy growth rate is displaced, here under the effect of vorticity. For positive vorticity theshift is towards the right, for negative vorticity it is towards the left. This again reflectsthe existence of a maximum wave age, which decreases with positive vorticity but increaseswith negative vorticity.On the left side of this graph, for older seas, curves do not merge and differences are wellvisible. Towards the right, i.e. for very young seas, all curves appear to merge asymptoti-cally. However, small differences are still present between different graphs corresponding todifferent vorticities, although they are hardly visible on the logarithmic scale.An alternative view of the same quantities is therefore presented in Fig. 10, where we use1a linear scale. It highlights the effect of vorticity by plotting Γ( ν ), for three values of thewave age θ (panels a-c). For each of these, graphs for several values of the depth parametersare superposed.Each graph presents a maximum, i.e. there is a specific vorticity for which the energytransfer is maximum. In deep water it is located at zero vorticity, but occurs at negativevorticity in finite depth. For young seas (panel (b)), the depth does not play a significantrole, and the effect of vorticity is almost independent of the depth parameter δ . In this casethe energy transfer is maximal for zero vorticity, i.e. the presence of any shear flow willreduce the energy transfer. Therefore, as long as the sea is not well developed, vorticityof any sign will hinder wave growth. This changes as the sea develops, see panels (a) and(c). For higher wave ages, the energy growth coefficient is still maximum for a specificvalue of the vorticity, which is always negative in finite depth. Moreover, this value is nowclearly dependent on the depth parameter δ : the smaller the depth parameter, the morenegative the vorticity for which the energy transfer is maximum, and the higher the energytransfer which can be achieved. Note also that, whatever the wave age, the graphs remainasymptotically universal for large depth parameters δ : in deep water, the maximum energytransfer is always achieved in the absence of vorticity. Consequently, vorticity of any signwill always hinder energy transfer in deep water. IV. CONCLUSIONS AND OUTLOOK
We have presented a full theoretical treatment of wind-generated surface waves, on waterof finite depth, when the water body is subject to a shear flow of constant vorticity. Thiswork extends the theory of Miles to include vorticity, a topic with great practical implicationssince water currents are expected to be present in oceans, and may lead to particularly strongvorticity in near-coastal waters.Generalising the approach by Beji and Nadaoka [35] has yielded expressions for all growthcoefficients characterising the wave growth due to wind, such as the Miles growth coefficient β , the amplitude growth coefficient γ and the energy growth coefficient Γ, known fromprevious studies in the absence of water currents [11].As a first result, the notion of a generalised wave age has emerged, defined in the samespirit as in [11], which accounts for the effect of both finite depth and vorticity in the water2flow on the phase velocity of wind-generated waves. It shows that, in otherwise identicalconditions, vorticity alters the wave age according to its sign. The wave age is reduced forpositive vorticity, which implies that in the natural reference frame of the water surface theflow velocity is directed against the wind, as sketched in Fig. 1. Negative vorticity, on theother hand, leads to more developed seas.Interestingly, there is an intrinsic limit to this wave age, i.e. an upper bound whichcannot be exceeded. This is already known to arise in water of finite depth, but now themaximum wave age is modified by the presence of vorticity. Positive vorticity reduces thismaximum value, and therefore leads to ’younger’ seas. Negative vorticity, however, makesit possible to attain a more developed sea.More specifically, the state towards which the sea evolves under the effect of wind isquantified by various growth coefficients. We have determined several such coefficients, bynumerically solving the Rayleigh equation, again following the strategy by Beji and Nadaoka[35], which has revealed new features with respect to the case of static water.First, we have considered the Miles growth coefficient β , related to the perturbation inthe water pressure [42]. In deep water it is known to decrease exponentially to zero beyondthe wave age corresponding to the developed sea [5, 35]. Recall the effect of a finite waterdepth: the vanishing of the Miles growth coefficient is pre-empted by the aforementionedmaximum wave age, thereby reducing the wave age of the developed sea. Nevertheless, atsufficiently small wave ages the growth coefficient is unaffected by the water depth. Theaction of vorticity is now twofold.At a first level, increasing (positive) vorticity reduces this maximum wave age, and there-fore modifies the wave age of the developed sea; negative vorticity has the opposite effect.In this respect a change in vorticity is similar to a change in water depth. This implies alsothat a shift in maximum wave age cannot unambiguously be attributed to either finite waterdepth or to the presence of shear currents, as both can compensate.On a second level, however, vorticity also affects the value of the growth coefficient forsmall wave ages: it is diminished for positive vorticity but enhanced for negative vorticity.This effect has no counterpart due to water depth, it is entirely novel in the present context.The growth parameter ˆ γ which directly characterises the growth of the wave amplitudealso mirrors this behaviour.Finally, the energy transfer from the wind to the wave, characterised by the coefficient ˆΓ,3shows a more complex behaviour. For young seas, we have illustrated numerically that thepresence of vorticity necessarily leads to a decrease in ˆΓ: irrespective of the water depth,the energy transfer attains its maximum as vorticity approaches zero. For larger wave agesthough, this is no longer true: the maximum in the energy transfer is achieved for a negativevalue of vorticity. This value furthermore depends both on wave age and water depth. ACKNOWLEDGMENTS
This work has been partially funded by the MUSE program of the University of Mont-pellier, in the framework of the international program ’KIM Sea and Coast’. [1] G. B. Whitham,
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FIG. 7: The maximum value of Miles’ growth coefficient β as a function of (negative)vorticity ν . We report the difference between the maximum and the plateau value at zerovorticity. Negative vorticity increases the Miles coefficient, until it saturates. The insetshows a zoom on the same data, for small values of vorticity, indicating that thedependency is initially linear.8 VW XY -5 -4 -3 -2 -1
0 2 4 6 8 10 12 G r o w t h c o e ffi c i e n t γ wave age θ fd,Ω ν= Z[ ν= \] .5ν= ^_ .25ν= ‘abc ν= e ν= ghi ν= jk ln o (a) δ = 25 pqsuvwyz{}~(cid:127)(cid:128)(cid:129)(cid:130)(cid:131)(cid:132)(cid:133)(cid:134)(cid:135)(cid:136)(cid:137)(cid:138)(cid:139)(cid:140)(cid:141)(cid:142)
2 4 (cid:143) (cid:144)(cid:145) (cid:146)(cid:147) (b) δ = 4 -7 -6 -5 -4 -3 -2 -1
0 2 4 6 8 10 12 (c) δ = 81 FIG. 8: The amplitude growth coefficient ˆ γ , defined by Eq. (53), as a function of the waveage θ for three values of the depth parameter. In (b) and (c) the data ranges, as well assymbols, are identical to those indicated in (a).9 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 (cid:148)(cid:149) e r (cid:150)(cid:151) t (cid:152)(cid:153)(cid:154)(cid:155) f e r (cid:156)(cid:157)(cid:158)(cid:159)(cid:160)¡¢£⁄ Γ ¥ƒ§¤'“« wav ‹ ›fifl (cid:176)– θ fd , Ω ν =-1 ν =-0.5 ν =-0.25 ν =-0.1 ν = 0 ν = 0.1 ν = 0.25 ν = 0.5 ν = 1 (a) δ = 25 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 (b) δ = 9 -6 -5 -4 -3 -2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 (c) δ = 81 FIG. 9: The energy transfer parameter Γ, defined by Eq. (59), as a function of the inversewave age 1 /θ . The choices for the depth and vorticity parameters δ and ν are as in Figs. 3,6 and 8. In (b) and (c) the data ranges, as well as symbols, are identical to those indicatedin (a).0 E n e r g y t r a n s f e r P a r a m e t e r † Vorticity parameter ν δ=4δ=9δ=25δ=49δ=81δ ‡·(cid:181)¶ (a) θ = 4 •‚„ ”» …‰(cid:190)¿(cid:192) `´ ˆ˜ ¯˘˙¨(cid:201) ˚¸(cid:204)˝ -1 ˛ˇ—(cid:209) (cid:210) (cid:211)(cid:212)
5 1 (b) θ = 1 (cid:213)(cid:214)(cid:215)(cid:216)(cid:217) (cid:218)(cid:219)(cid:220)(cid:221)(cid:222)(cid:223)(cid:224)Æ (cid:226)ª(cid:228)(cid:229)(cid:230)(cid:231)ŁØ Œº(cid:236)(cid:237)(cid:238)(cid:239)(cid:240)æ (cid:242)(cid:243)(cid:244)ı(cid:246)(cid:247)łø œß(cid:252)(cid:253) -1 (cid:254)(cid:255)-(cid:0) (cid:1)(cid:2)