Mean flow velocities and mass transport for Equatorially-trapped water waves with an underlying current

arXiv:2409.08714v1 [math.AP] 13 Sep 2024 Mean flow velocities and mass transport for Equatorially-trapped water waves with an underlying current David Henry and Silvia Sastre-Gómez Abstract In this paper we present an analysis of the mean flow velocities, and related mass transport, which are induced by certain Equatorially-trapped water waves. In particular, we examine a recently-derived exact and explicit solution to the geophysical governing equations in the β−plane approximation at the Equator which incorporates a constant underlying current. 1 Introduction The question of determining the fluid drift induced by the propagation of surface water waves is a fascinating issue and, despite pioneering work on this subject being instigated by Stokes as far back as the mid-1800’s, it is still a highly curious and perplexing matter at even the most fundamental level. For instance, in the setting of periodic surface gravity water waves Stokes demonstrated by way of approximations [39] that fluid particles experience a (mean) forward drift to the order of ǫ2, where ǫrelates to the wave steepness. This drift is in a mean sense, whereby an average is taken over the wave period, and it is an inherently nonlinear phenomenon with regard to the order of wave amplitude. The subtleties of these drift properties may be illustrated by considering the classical assumption that for periodic irrotational wave motion it was assumed, at the linear level, that fluid particles follow closed trajectories [30], whereas according to the Stokes drift phenomenon at the order of expansion ǫ2it is implied that at least some particle paths are non-closed. It is noteworthy that, with regard to particle trajectories for periodic irrotational water waves, it was recently proven by various methods that all particle paths throughout the fluid domain are indeed non-closed for flows induced by a wide-range of gravity (and capillary-gravity) waves, both in the approximate linear regime and for exact solutions of the fully nonlinear governing equations [6, 7, 8, 12, 15, 21, 22, 27, 33]. In recent decades, following the work of Longuet-Higgins, the study of mean drift velocities induced by surface wave motion was placed on a firmer theoretical footing for a broad range of fluid motions [1, 3, 31, 32]. It was observed that key features of the mean fluid drift velocity, or so-called Stokes’ drift velocity, could be characterised in terms of the mean Eulerian flow velocity and the mean Lagrangian flow velocity, whereby: Lagrange = Euler + Stokes. In spite of recent progress, determining the mean fluid flow velocities remains a highly complex and intricate issue from both a theoretical, and experimental [37, 41], viewpoint. 1 In this paper we present an analysis of the mean flow velocities, and related mass transport, induced by certain Equatorially-trapped water waves. In particular, we examine a recently-derived [23] exact solution for the geophysical governing equations in the β−plane approximation [14, 16, 17] at the Equator. The form of this solution is explicit in terms of Lagrangian variables, and a benefit inherent in employing the Lagrangian framework is that fluid kinematics may often be described explicitly and with (relative) ease, [2, 4, 5, 9, 10, 11, 13, 18, 19, 20, 23, 25, 26, 28, 38, 40]. A significant complicating factor for the analysis undertaken in this paper, particularly with regard to determining the mean Eulerian flow velocity and subsequently the Stokes drift velocity, is the presence of a constant underlying current term in the solution given in [23]. This is in spite of the underlying current assuming a relatively simple manifestation in the Lagrangian formulation of the solution, along the lines of the underlying current terms which were introduced by Mollo-Christensen [36] into Gerstner-type solutions in an attempt to model billows and various other complicating effects in both atmospheric and oceanographic situations. We also note that it is well established that currents play a vital role in Equatorial dynamics [9, 11, 14, 16, 29], and interestingly a transverse Equatorial current can be incorporated into a Gerstner-like exact solution in the Equatorial f−plane formulation, cf. [24]. The paper is concluded with a brief discussion of some mass-transport properties of these Equatorially trapped waves. 2 The Equatorially trapped wave solution 2.1 Governing equations We consider geophysical waves in the Equatorial region, where we assume that the earth is a perfect sphere of radius R= 6378 km, and work in a reference frame rotating with the earth whose origin is fixed at the earth’s surface, with the {x, y, z}-coordinate frame chosen so that the x-axis is pointing horizontally due east (the zonal direction), the y-axis is due north (meridional direction), and the z-axis is pointing vertically upwards and perpendicular to the earth’s surface. The governing equations for geophysical ocean waves are given by ut+uux+vuy+wuz+ 2Ωwcos Φ −2Ωvsin Φ = −1 ρPx(2.1a) vt+uvx+vvy+wvz+ 2Ωusin Φ = −1 ρPy(2.1b) wt+uwx+vwy+wwz−2Ωucos Φ = −1 ρPz−g, (2.1c) together with the mass conservation equation ρt+uρx+vρy+wρz= 0 (2.2a) and the equation of incompressibility ux+vy+wz= 0.(2.2b) 2 Here Φrepresents the latitude, (u, v, w)is the fluid velocity, Ω = 73.10−6rad/s is the (constant) rotational speed of earth [16], g= 9.8m/s−2is the gravitational constant, ρ is the water density, and Pis the pressure. We are interested in Equatorial waves, that is, geophysical ocean waves in a region which is within 2olatitude of the Equator. Since the latitude is small, we may use the approximations sin Φ ≈Φ, and cos Φ ≈1, and thus linearising the Coriolis force leads to the β-plane approximation to equations (2.1) given by ut+uux+vuy+wuz+ 2Ωw−βyv =−1 ρPx vt+uvx+vvy+wvz+βyu =−1 ρPy wt+uwx+vwy+wwz−2Ωu=−1 ρPz−g, (2.2c) where β= 2Ω/R = 2.28·10−11 m−1s−1. The relevant boundary conditions are the kinematic boundary conditions w=ηt+uηx+vηyon z=η(x, y, t),(2.2d) P=Patm on z=η(x, y, t),(2.2e) where Patm is the (constant) atmospheric pressure, and η(x, y, t)is the free surface. The boundary condition (2.2d) states that all the particles in the surface will stay in the surface for all time t, and the boundary condition (2.2e) decouples the water flow from the motion of the air above. Finally, we assume the water to be infinitely deep, with the flow converging rapidly with depth to a uniform zonal current, that is, (u, v, w)→(−c0,0,0) as z→ −∞.(2.2f) The set of equations (2.2) comprises the governing equations for the β−plane approximation of geophysical ocean waves with a constant underlying current. 2.2 Exact solution In this section we briefly describe the exact solution of the β-plane governing equations (2.2) which was presented in [23]. This solution prescribes a three-dimensional eastwardpropagating steady geophysical wave in the presence of a constant underlying current of magnitude |c0|. The wave-like term is periodic in the zonal direction and it has a constant phasespeed c > 0. Furthermore, the wave is Equatorially trapped, exhibiting a strong exponential decay away from the Equator. Equatorially trapped waves which are symmetric about the Equator and propagate eastward are known to exist, and they are regarded as an important factor in a possible explanation of the El Niño phenomenon (cf. [14, 16, 17]). The solution of (2.2) we present is formulated in the Lagrangian framework, whereby the evolution in time of individual fluid particles is prescribed [2]. In this Lagrangian formulation the Eulerian coordinates of fluid particles (x, y, z)are expressed as functions of the 3 Lagrangian labelling variables (q, r, s)∈(R,(−∞, r0),I), and time t, as follows: x=q−c0t−1 kek[r−f(s)] sin [k(q−ct)],(2.3a) y=s, (2.3b) z=r+1 kek[r−f(s)] cos [k(q−ct)],(2.3c) where r0<0and kis the wavenumber defined by k= 2π/L, and where Lis the (fixed) wavelength. For c0>0the underlying current is adverse, while for c0<0the current is following, and we see below that the sign of the current determines whether Iis the real line Ror a finite interval. The function f(s)determines the decay of the particle oscillations in the latitudinal direction away from the equator and it is given by f(s) = cβ 2γs2,(2.4) where γ:= 2Ωc0+g(>0) is a “modified gravity” term and we make the (physically reasonable) assumption that c0>−g 2Ω . For notational convenience let us choose ξ=k(r−f(s)) , θ =k(q−ct). Then the Jacobian matrix of the transformation (2.3) is given by ∂(x, y, z) ∂(q, s, r)=  1−eξcos θ0−eξsin θ fseξsin θ1−fseξcos θ −eξsin θ0 1 + eξcos θ  ,(2.5) which has the time-independent determinant 1−e2ξ. Consequently the flow defined by (2.3) is volume preserving, ensuring that (2.2b) holds in the Eulerian setting [2]. Since the solution (2.3) is explicit in the Lagrangian formulation, we may immediately discern some qualitative properties of the physical fluid motion. Indeed, a significant benefit of working in the Lagrangain framework is that the fluid kinematics can often be described explicitly and with relative ease. In the case above we calculate the velocity field directly from (2.3) to get u(q, r, s;t) = Dx Dt =ceξcos θ−c0,(2.6a) v(q, r, s;t) = Dy Dt = 0,(2.6b) w(q, r, s;t) = Dz Dt =ceξsin θ, (2.6c) where D/Dt is the material (or convective) derivative with respect to Eulerian variables. For fixed latitudes, that is for every fixed s, the system (2.3) describes the flow beneath a surface wave propagating eastwards at constant speed cdetermined by the dispersion relation (2.10) 4 below. Additionally, for fixed latitudes the free surface z=η(x, y, t)is obtained by setting r=r0(s)in (2.3c), where r0(s)< r0is the unique solution to e2k[r(s)−cβ 2γs2] 2k−r(s) + c0β 2γs2−e2kr0 2k+r0= 0,(2.7) The existence of a unique solution r(s)to (2.7) for |s|>0is equivalent to the condition e2k[r0−cβ 2γs2] 2k+c0β 2γs2<e2kr0 2k,(2.8) cf. [23] for details. For c0≤0, it is easy to see that condition (2.8) holds for all s∈R. For c0>0, condition (2.8) will hold for restricted values of son a finite interval Iwhich depends on the magnitude |c0|of the current. For our present purposes we remark that, given a current with c0>0, for a unique solution of (2.7) to exist it is necessary that c0< ce2kr0,(2.9) and accordingly (2.3) represents a dynamically possible solution of (2.2). Since c06=c(by (2.9)) it follows that the dispersion relation for the wave takes the form c=pΩ2+kγ −Ω k=pΩ2+k(2Ωc0+g)−Ω k>0.(2.10) We remark that if c0=cthen the dispersion relation for the wave would take the form c=pg/k. Hence, in this situation geophysical Coriolis effects have no bearing on the dispersion relation, which instead matches that of the celebrated Gerstner’s wave solution [5, 7, 20] for deep-water gravity waves. This observation leads us to infer that precluding the case c0=c, as is consistent with condition (2.9), is natural in the context of geophysical water waves (cf. [23] for details on the dispersion relations). Finally, we note that at fixed latitudes s=s∗the crest and trough levels of the wave surface profile are prescribed in terms of the Lagrangian parameters by z±(s∗) = r0(s∗)±1 kek[r0(s∗)−f(s∗)]. 3 Mean velocities and Stokes drift In this section we analyse the effect that the constant underlying current has with respect to both the mean Lagrangian and Eulerian flow velocities induced by the exact solution (2.3). In [13] it was shown that in the absence of a current, that is for c0= 0, the mean Lagrangian velocity is zero and the mean Eulerian velocity flows westwards. Hence, in the absence of the current the Stokes drift (or mean Stokes flow velocity), which which is the difference in the mean Lagrangian and Eulerian velocities [31, 32], is eastwards. Here we show that the situation is far more complex in the presence of an underlying current, in particular when determining the mean Eulerian velocity. Throughout the following considerations we fix the latitude by setting s=s∗. 5 3.1 Mean Lagrangian flow velocity The mean Lagrangian flow velocity (also known as the mass-transport velocity [31]) at a point in the fluid domain is the mean velocity over a wave period of a marked fluid particle which originates at that point. For the exact solution (2.3) we may calculate the average of the horizontal velocity uin (2.6a) over a wave period T=L/c as follows: huiL=1 TZT 0 u(q−ct, s, r)dt =ceξ TZT 0 cos [k(q−ct)] dt −1 TZT 0 c0dt =−c0, (3.1) where we have used the fact that the first integral on the left-hand side above vanishes. It is immediately apparent that the mean Lagrangian flow velocity is either westwards or eastwards, depending on whether the sign of c0is positive or negative respectively. When c0= 0 the mean Lagrangian velocity is zero, which concurs with the result of [13], and in this light the form of the mean Lagrangian flow velocity above is not particularly surprising considering the explicit manner in which c0appears in the expression for the Lagrangian velocity (2.6a). We note that the expression for the mean Lagrangian velocity is independent of both the latitude s, and the location in the fluid domain where the fluid parcel originates. 3.2 Mean Eulerian flow velocity When working in the Eulerian setting matters are greatly complicated by the presence of the underlying current. The mean Eulerian flow velocity at a fixed-point in the fluid domain is the Eulerian fluid velocity at that fixed-point averaged over a wave period. In the case of the velocity field (2.6) the mean Eulerian flow velocity may be computed by taking the mean over a wave period of the horizontal velocity (2.6a) at any fixed-depth beneath the wave trough. Letting z=z−(s∗)denote the vertical position of the wave trough level, we fix a depth z=z0< z−(s∗). This fixed depth z=z0may be characterised in terms of Lagrangian variables, using (2.3c), by the equation z0=R+1 keξ(R)cos θ, (3.2) where we denote by r=R(q−ct;s∗, z0)the functional relationship induced by relation (3.2) between the otherwise independent variables rand q, as follows from the implicit function theorem. We note that a consequence of (3.2) is that Ris periodic in the q−variable, with period L. Differentiating (3.2) with respect to qyields 0 = Rq+Rqeξ(R(q)) cos θ−eξ(R(q)) sin θ, that is Rq=eξsin θ 1 + eξcos θ.(3.3) 6 We note from (3.3) that Ris maximised or minimised with respect to qwhenever sin θ= 0, and therefore for a fixed-depth z0the maximal and minimal values achieved by Rare given implicitly by the relations z0=R±1 keξ(R), where the positive (negative) sign corresponds to the minimal (maximal) value of R, respectively. To compute the Eulerian mean velocity huiE(s∗, z0)at latitude s∗and depth z0≤z−(s∗)we examine c+huiE(s∗, z0) = 1 TZT 0 [c+u(x−ct, y, z0)] dt. =1 LZL 0 [c+u(x−ct, y, z0)] dx, which upon transforming, by way of (2.3), to the labelling variables (q, s, r), and invoking functional periodicity with respect to the q−variable, we get c+huiE(s∗, z0) = 1 LZL 0 [c+u(q−ct, s∗, R(q−ct;s∗, z0))] ∂x ∂q dq. By differentiating xin (2.3a) with respect to q, using (2.6a), and taking into account (3.3), we obtain c+huiE(s, z0) = 1 LZL 0hc+ceξ(R(q)) cos θ−c0ih1−eξ(R(q)) cos θ−eξ(R(q))Rqsin θidq =1 LZL 0 c1 + eξ(R(q)) cos θ1−e2ξ(R(q)) 1 + eξ(R(q)) cos θdq −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos θdq =c−c LZL 0 e2ξ(R(q))dq −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos θdq. Therefore the mean Eulerian velocity is given by the relation huiE(s∗, z0) = −c LZL 0 e2ξ(R(q))dq −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos (k[q−ct])dq. (3.4) The presence of a non-zero underlying current c0adds a significant complicating factor to expression (3.4), and in particular the sign (and hence direction) of the mean Eulerian velocity is not easily discernible from the above expression in general. Nevertheless, depending on the size and direction of the current c0, we may obtain estimates which determine the direction of the mean Eulerian velocity following from the inequalities ZL 0 1−e2ξ 1 + eξdq ≤ZL 0 1−e2ξ 1 + eξcos θdq ≤ZL 0 1−e2ξ 1−eξdq. (3.5) 7 3.2.1 The case c0>0: First of all let us study the case when c0is positive, which represents an underlying adverse current in the Lagrangian variables. The second integral term on the right-hand side of inequality (3.4) satisfies −c0 LZL 0 1−e2ξ 1−eξdq ≤ −c0 LZL 0 1−e2ξ 1 + eξcos θdq ≤ −c0 LZL 0 1−e2ξ 1 + eξdq. (3.6) Since 0< c0< ce2kr0< c from (2.9), equation (3.6) yields huiE≤ − c LZL 0 e2ξdq −c0 LZL 0 1−e2ξ 1 + eξdq ≤ −c0 LZL 0 1 + e3ξ 1 + eξdq < 0.(3.7) Therefore the mean Eulerian flow velocity is westwards for all admissible values of c0for which (2.9) holds. To get an idea of the range of the mean Eulerian flow we note that huiE≥ − c LZL 0 e2ξdq −c0 LZL 0 1−e2ξ 1−eξdq ≥ − c LZL 0 1−e3ξ 1−eξdq. (3.8) Hence, since ξ≤kR < kr0<0, we see that for all latitudes sand depths z0< z−(s)the mean Eulerian flow velocity is in the range huiE(s, z0)∈−c1−e3kr0 1−ekr0,0.(3.9) That the mean Eulerian flow is westward for an adverse current is not surprising, since in the absence of the current the mean Eulerian flow is westward (cf. [13]) and the presence of the adverse current term in (3.4) merely serves to exacerbate this effect. 3.2.2 The case c0≤0: The case when c0is nonpositive, c0≤0, represents an underlying following current. In this case the influence that the current has on the mean Eulerian flow in (3.4) is complex and difficult to discern, and it is not generally possible to analytically determine its effect directly from expression (3.4). Nonetheless, we can deduce some broad characteristics of the flow by working as follows. The mean Eulerian velocity (3.4) is westwards, that is huiE(s∗, z0)<0, if −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos θdq ≤ −c0 LZL 0 1−e2ξ(R(q)) 1−eξ(R(q)) dq ≤ −c0max q∈[0,L] 1−e2ξ(R(q)) 1−eξ(R(q)) < c min q∈[0,L]e2ξ(R(q)) ≤c LZL 0 e2ξ(R(q))dq. These series of inequalities hold, and accordingly huiE(s∗, z0)<0, if c0>−cmin q∈[0,L] e2k(R(q;z0)−f(s∗))1−ek(R(q;z0)−f(s∗)) 1−e2k(R(q;z0)−f(s∗)) .(3.10) 8 We note that in the absence of an underlying current, that is when c0= 0, condition (3.10) always holds and so the resulting mean Eulerian velocity is always in the westerly direction, an observation which accords with [13]. The mean Eulerian flow (3.4) is eastwards, huiE(s∗, z0)>0, if −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos θdq ≥ −c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) dq ≥ −c0min q∈[0,L] 1−e2ξ(R(q)) 1 + eξ(R(q)) > c max q∈[0,L]e2ξ(R(q)) ≥c LZL 0 e2ξ(R(q))dq. These inequalities hold, and hence huiE(s∗, z0)>0, if c0<−cmax q∈[0,L] e2k(R(q;z0)−f(s∗))1 + ek(R(q;z0)−f(s∗)) 1−e2k(R(q;z0)−f(s∗)) .(3.11) 3.3 Stokes drift The Stokes drift (or mean Stokes) velocity US(z0)is defined (cf. [1, 13, 31, 32, 37, 39]) by the relation huiL(z0) = huiE(z0) + US(z0). We derive an expression for the Stokes drift by computing US=huiL− huiE=c LZL 0 e2ξ(R(q))dq +c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos (k[q−ct])dq −c0. For an adverse current, c0≥0, it follows from (2.9) that US=1 LZL 0ce2ξ(R(q)) −c0dq +c0 LZL 0 1−e2ξ(R(q)) 1 + eξ(R(q)) cos (k[q−ct])dq > 0. Therefore for c0≥0the Stokes drift is eastwards throughout the fluid domain. In the case a following current, c0<0, the expression for Stokes drift is altogether more complicated and intractable. Nevertheless we remark that, for c0<0, if the magnitude of the current is such that (3.11) holds then the Stokes drift must be westwards. 4 Mass flux We conclude with a brief discussion of mass-transport properties of the flow (2.6), where we recall that huiL, being the mean velocity of a marked particle, is sometimes called the mass-transport velocity. For a non-zero underlying current, c06= 0, we intuitively expect the total mass flux below the free-surface wave past a point x=x0fixed in Eulerian coordinates to be infinite. To see this directly we compute the integral m(x0−ct, s) = Zη(x0−ct,s) −∞ u(x0−ct, s, z)dz. (4.1) 9