On Irrotational Flows Beneath Periodic Traveling Equatorial Waves

. We discuss some aspects of the velocity ﬁeld and particle trajectories beneath periodic traveling equatorial surface waves over a ﬂat bed in a ﬂow with uniform underlying currents. The system under study consists of the governing equations for equatorial ocean waves within a non-inertial frame of reference, where Euler’s equation of motion has to be suitably adjusted, in order to account for the inﬂuence of the earth’s rotation. obtained by reﬂection about the y -axis. The observations rely on the strict monotonicity of u along streamlines in D + ∪ S + ∪ B + and its monotonicity in y -direction along the crest and trough line, c.f. Proposition 3.4 and the proof of Proposition 3.7.

This paper deals with some qualitative properties of certain geophysical waves, which are not governed by Euler's equations of motion (which apply for inertial systems), but by suitable extensions that account for the influence of the earth's rotation. More precisely we study irrotational flows and particle paths beneath symmetric periodic traveling surface waves in regions close to the equator. In the case of equatorial surface waves, which propagate practically unidirectionally in the East-West direction due to the prevailing wind pattern (known as trade winds), it is justifiable 1 to consider the f -plane approximation for two-dimensional flows instead of the full geophysical governing equations in three dimensions. This reduction and the simplifying assumption of irrationality, 2 implying that underlying currents are uniform, make it possible to analyze qualitative properties of the flow with the aid of well-known tools from complex and harmonic analysis, and conformal mapping theory. The qualitative techniques developed in [4,10] to study Stokes waves can be adapted for the investigation of irrotational equatorial waves.
For recent studies of equatorial waves under the influence of the Coriolis effect we refer to [2,3,7,8,13,14,16,18,20,22] and the references therein. In particular, the influence of the Coriolis force on the dispersion relation is brought to light in the papers [2,13] dealing with surface waves; we refer moreover to [5] for effects on internal waves, and [17,21] for edge waves.
The outline of the paper is the following. First we introduce the governing equations for equatorial waves in Sect. 2. After transforming this system to moving frame coordinates, we provide two alternative reformulations of the problem: a stream function formulation and its transformation on a strip via a conformal hodograph mapping. Section 3 is devoted to the study of the velocity field beneath a surface wave, which provides the basis for the qualitative description of particle trajectories in Sect. 4.

The Governing Equations for Equatorial Waves
The governing equations for gravity water waves in the equatorial f -plane are given by where P atm is the constant atmospheric pressure, and We restrict our considerations on flows being irrotational at the initial time T = 0. The corresponding vorticity equation 3 for such two-dimensional flows implies that the curl will remain zero for all times. We may therefore assume that (U, V ) is curl-free throughout the fluid domain:

Moving Frame Re-formulation for Traveling Waves
The goal of this paper is the analysis of flows beneath traveling periodic surface waves. Let us therefore consider the corresponding moving frame coordinates for a given wave speed c > 0: 3 The details are provided in the Appendix.
The corresponding boundary conditions on the free surface and on the flat bed are given by and v = 0 on y = −d. (2.8) We assume that u, v, P and η are periodic in the x-direction. The period L > 0 corresponds to the wavelength of the surface wave.

Stream Function Formulation
Equation (2.4) permits the definition of a stream function satisfying We see that ψ is unique up to an additive constant and observe that ψ is constant on both parts of the boundary of D η : on y = −d due to (2.8) and also on the surface, since its derivative along the free surface η is zero by (2.6): With the choice This formula tells us in particular, that ψ inherits the x-periodicity from u. The constant m is called relative mass flux ; its value is given by Indeed, m is an invariant of the flow; differentiating the right hand side of (2.11) with respect to the x-variable and employing (2.4), (2.6) and (2.8) gives zero. Due to (2.5) and (2.9) we obtain

Therefore (2.3) yields Bernoulli's law for irrotational equatorial flows
in other words, the expression is constant throughout the fluid. By means of (2.7) and (2.10) we infer that for some physical constant Q, called head. In order to ensure positivity of the coefficient of y in the above relation, we impose the following realistic upper bound for the wave speed c: In summary we obtained the following reformulation of (2.3)-(2.8): (2.14) Note that the pressure P does not appear explicitly in (2.14). It can be recovered by means of (2.12) and (2.7): Since we are interested in solutions different from the trivial solution, that would be ψ ≡ 0 (with m = 0) throughout the strip D 0 , it follows from the strong maximum principle applied to the harmonic function ψ, that m = 0. Without loss of generality we take m > 0, then ψ > 0 throughout D η and because ψ attains a minimum at every surface point (x, η(x)) and a maximum at every point (x, −d) on the flat bad (by Hopf's lemma, this strict inequality holds everywhere on the boundary ∂D η of the fluid domain D η , therefore the strong maximum principle applied on the harmonic function ψ y implies (2.15)). The irrotationality condition (2.5) and L-periodicity imply that at all depths y 0 below the trough level η(L/2) by means of Green's theorem. In other words: the mean of u at any depth below the trough level takes the constant value κ < c referred to as mean current: (2.16)

Flows Beneath Equatorial Waves
We distinguish between three cases: if κ is positive, there is a uniform underlying current moving with the wave, if κ is negative, it moves against the wave and κ = 0 indicates the absence of an underlying current. In the latter case we have that: An immediate consequence of (2.16) is that In the remaining sections we investigate flows beneath symmetric periodic traveling ocean waves. More precisely, we study qualitative properties of smooth periodic solutions (η, ψ) of (2.14) (for a given wave speed c > 0, relative muss flux m > 0 and mean current κ), having period L in the x-variable and mean level y = 0, i.e. Moreover η and ψ are supposed to be symmetric about the crest line, i.e. the vertical line from (0, η(0)) to (0, −d), where η attains its maximum. Furthermore we assume that the surface wave η has only one crest per period. Its wave profile is supposed to be strictly increasing from the trough at x = −L/2, where the minimum is attained, to the crest at x = 0. In terms of the velocity field (u, v) and the pressure P , the symmetry of ψ means that u and P are symmetric, whereas v is anti-symmetric about the crest line. We refer the reader to [15] for the existence of solutions of (2.14) with these properties.
Let us note that the approach developed in [6] can be adapted to our setting for equatorial waves, yielding that the free surface is actually real-analytic.

Hodograph Transform
We introduce new coordinates q and p, which will turn out to be of great use in the further analysis.
Due to (2.5), there exists a potential φ for the velocity field (u − c, v): The two integrals on the right hand side are equal, since u is even, which gives (2.21). The stream function ψ and the potential φ are harmonic conjugates: the mapping x + iy → φ(x, y) + iψ(x, y) is holomorphic throughout the fluid domain. Let us consider the orientation preserving conformal hodograph transform which transforms the free boundary value problem (2.14) into a nonlinear boundary value problem for the harmonic function in a fixed strip; c.f. Fig. 2. The corresponding system for h reads We will frequently make use of the following identities: (2.29)

Flows Beneath Equatorial Waves
Note for instance, that we obtained the boundary condition in (2.24) by multiplying the corresponding condition in (2.14) by 2g v 2 +(c−u) 2 ((c−u) 2 +v 2 ) 2 to get This is precisely the boundary condition in (2.24) in view of (2.28), (2.23), (2.22), (2.25) and the fact, that y = η(x) translates into p = 0 due to the definition of ψ.

Properties of the Velocity Field
This section is dedicated to the study of basic properties of the velocity field (u, v) of solutions (ψ, η) to (2.14) as we introduced them in the last paragraph of Sect. 2.2. In particular we will determine the zero level sets of u and v, where sign changes occur. This will permit a detailed qualitative analysis of fluid particle paths in the physical frame, see Sect. 4.
Due to periodicity we may restrict our considerations to the particular periodicity window We denote by D the closure of D in R 2 ; its left and right boundary 4 are referred to as trough lines. The crest line is the intersection of D with the vertical line {x = 0}. In view of the symmetry of ψ it is convenient to distinguish between the right half D + of D and its left half D − : Let us furthermore denote by We start our analysis by determining the sign of the vertical velocity component v in D, c.f. Fig. 3. As a consequence we obtain the following corollary, which tells us that all streamlines except the flat bed y = −d replicate the shape of the free surface. Proof. Recall that ψ has no critical points due to (2.15). Let us identify an arbitrary streamline in D with the function It is clear, that y is analytic, since it is a parametrization of a level set of the harmonic map ψ, i.e. ψ(x, y(x)) is constant for all x ∈ (−L/2, L/2). Therefore we have that Thus we see, in view of the sign of v together with (2.15), that all streamlines within D are symmetric and strictly decreasing between the wave crest and the wave trough.  Proof. Let us first observe that the pressure P is superharmonic throughout the entire fluid region, in particular within D. By (2.3) and (2.5) we have that Let us now consider the function Q defined on the entire fluid domain and given by The map Q is superharmonic as a sum of the superharmonic function P and a multiple of the harmonic function ψ. Therefore we have that the minimum of Q is attained on boundary of the fluid domain, i.e. on the flat bed or on the free surface. The maximum principle tells us that the minimum of Q is attained on the boundary of the fluid domain. On the bottom we find that because of (2.8) and the second component of (2.3). Hence Q decreases in y-direction on the bottom, therefore the minimum can only be attained on the surface, where Q takes the constant value ρ −1 P atm ; i.e. every point (x, η(x)) is a minimum point. Since the positive x-direction, as well as the positive ydirection, point outwards the fluid domain along S + we infer that both Q x (x, η(x)) and Q y (x, η(x)) are strictly negative for all x ∈ (0, L/2). By (2.3) and the definition of Q we obtain for all x ∈ (0, L/2). In terms of the (p, q)-coordinates this inequality reads for all q ∈ (0, λ/2); here we employed (2.26). Moreover, since v = 0 on the crest and trough lines, (2.4) guarantees that also u x = 0 there and hence (2.26) tells us that On the flat bed we have that .
since v is minimal along the bottom B + , and therefore we infer that In summary we have that u q ≤ 0 all along the boundary ofD + ; the inequality being strict on the upper and lower boundary. The harmonicity of u is preserved under (2.22), and so we can apply the strong maximum principle in oder to find that Since u is even, we obtain that u q > 0 in D − . Thus we achieved the desired monotonicity along horizontal lines in the conformal frame, which correspond to streamlines in the moving frame. The transformation to which determines the claimed monotonicity of u along streamlines, since h p > 0 due to (2.28) and (2.15).
In order to establish the claimed monotonicity along the crest and trough lines we recall first that u y = (c − u)u p on these lines because of (2.27) and the fact that v vanishes there due to Proposition 3.1. We recall that u p = v q as a consequence of (2.4), (2.5) and (2.26). We recall from Proposition 3.1 that the harmonic function v is positive in D + and vanishes on the crest line and the trough lines, hence these points are minimizers of v in D + . Therefore Hopf's lemma tells us that v x (0, y) > 0 for y ∈ (−d, η(0)) and v x (L/2, y) < 0 for y ∈ (−d, η(L/2)).
The claim follows since v q = h p v x on the images of the crest and trough lines under the conformal mapping (2.22) and due to (2.15). Proof. The maximum of u can only be attained somewhere along the crest line and its minimum has to be found on the trough lines, since u x < 0 throughout the region Due to Proposition 3.4 we know that u increases in y-direction on the crest line and decreases in y-direction on the trough lines, which proves the assertion of the corollary.   on the trough line with some surface point (x 1 , η(x 1 )), 0 < x 1 < L/2. (c) κ ≥ −u 0 (L/2, η(L/2)). In this case, u is nonnegative throughout D + , i.e. no change of sign occurs.

The Particle Path Pattern
In this section we study the trajectory t → (X(t), Y (t)) of a fluid particle initially located at (X 0 , Y 0 ), which satisfies the system with initial condition (X(0), Y (0)) = (X 0 , Y 0 ). In the moving frame the above time dependent problem is transformed to the autonomous system with initial data (x(0), y(0)) = (X 0 , Y 0 ). We note that ψ is a Hamiltonian for this system, i.e. ψ satisfies and moreover ψ is an integral of motion, since it does not depend on time.
Let us assume, that the fluid particle is initially (at time t = 0) located at the point (L/2, Y 0 ) and observe that this is no restriction of the general case, since any fluid particle would cross the vertical line {x = L/2} after some finite time in view of the moving frame. Indeed, due to (2.15) and (4.2) there exists a δ > 0 such that x = u(x, y) − c ≤ −δ < 0 for all (x, y) in the entire fluid domain, and this means that x runs from ∞ to −∞ when t goes from −∞ to ∞. Since u is periodic in x, there exists a uniquely determined positive time (depending only on the initial height Y 0 ) it takes a fluid particle to traverse one periodicity window of length L: Remark 4.2 (Viewpoint q,p-coordinates). Let us denote by (q(t), p(t)) the images of points (x(t), y(t)) under the conformal transformation of variables (2.22) and recall that p(t) = p is an integral of motion because of (2.22) and (4.3), whereas 19) and (4.2). So (q(t), p(t)) moves along straight horizontal lines from right to left in the (q, p)plane. This shows (by keeping in mind that q(0) = λ/2) that there exists a unique time θ = θ(p) such 7 Note that we require −κ < g 2Ω − c to ensure that c 0 satisfies (2.13).

R. Quirchmayr
JMFM that q(θ) = −λ/2. By construction we have that θ coincides with the elapsed time: θ(p) = τ (Y 0 ) and consistently we obtain where the last equality holds true because v is odd in the q-variable.

Proposition 4.3. The elapsed time is given by
. (4.6) In order to recognize the second representation of θ we recall that by means of (2.28), and calculate the derivative of q along streamlines: Let us now consider the divergence-free vector field (v, c−u) (recall (2.5)) restricted to the region D yY 0 ⊆ D beneath the streamline y = y Y0 (x), above the flat bed y = −d and between the trough lines x = ±L/2. The divergence theorem implies that where we exploited the anti-symmetry of v and the fact that y Y0 has the same shape as the free surface, see Corollary 3.2. By exploiting (2.21) and the periodicity of u in the x-variable, we infer from (4.7) that

Flows Beneath Equatorial Waves
Due to the Cauchy-Schwarz inequality we get We recall that the streamline y Y0 satisfies (see Corollary 3.2) ∂ x y Y0 (x) = 0 for x ∈ (−L/2, L/2)\{0}, and hence (4.10) In summery we get from (4.6), (4.9), (4.10) and (4.8) that In order to see that θ(p) > L/(c − κ) holds still true for p = −m, we argue by contradiction: if we assume the contrary, namely that θ(−m) = L/(c − κ), we enforce equality in the Cauchy-Schwarz inequality (4.7): which means that the functions x →  u(x, −d)) are linearly dependent. But this would only be possible if u was constant on the flat bed, but this is not the case, since we know that u x < 0 on (0, L/2) by Proposition 3.4. And so we conclude that (4.5) is satisfied. Proof. We show that θ (−m) = 0 and θ (p) > 0 for p ∈ (−m, 0]. Differentiation of (4.4) with respect to p gives Since h is periodic the q-variable with period λ, we obtain , since h q is constantly zero on the flat bed by (2.8). The second derivative of θ is given by for all p ∈ [−m, 0]; in order to establish the second equality, we used once more, that h is harmonic and λ-periodic in q: Combining  Proof. If a particle trajectory is closed, there exists some time T > 0 such that (X(T ), Y (T )) = (X 0 , Y 0 ). From Y (T ) = Y 0 it follows that T = nθ for some nonzero integer n. Then it holds that 0 = X(nτ )−X(0) = n(X(τ ) − X(0)) which implies that D(Y 0 ) = 0. If on the other hand D(Y 0 ) = 0, then X(τ ) = X(0), Y (τ ) = Y (0) and τ = L/c. By x-periodicity both t → (X(t), Y (t)) and t → (X(t+τ ), Y (t+τ )) solve (4.1) with the initial condition (X(0), Y (0)) = (X 0 , Y 0 ). Since (U, V ) is real analytic and bounded, thus in particular Lipschitz continuous, we have uniqueness of solutions, i.e. X(t + τ ) = X(t) and Y (t + τ ) = Y (t) for all t ∈ R. In particular, the drift is strictly positive for each fluid particle, if κ ≥ 0.
As an immediate consequence of Propositions 4.6 and 4.7, we obtain the following  We collect all possible scenarios concerning the sign of the particle drift in case of a negative κ in the following for which the drift vanishes on B and S respectively. In view of the notation introduced in Remark 3.8 and the formula (4.13) for the particle drift, we have that where θ(−m) = and due to Proposition 4.3. Therefore, as desired, both κ B and κ S are negative. Additionally, Corollary 4.9 tells us that So in view of the strict monotonicity of the particle drift D(p), see Corollary 4.9, we end up with the following cases. 1. κ B ≤ κ < 0. The drift is positive from the surface S down to the bed B. In the borderline case κ = κ B , we have that D(p) > 0 for p ∈ (−m, 0] and D(−m) = 0. 2. κ S < κ < κ B . In this situation, D(0) > 0 and D(−m) < 0, thus Corollary 4.9 yields the existence of a unique streamline Y in D where D vanishes (along which particles move in circles due to Proposition 4.6), and the drift becomes negative for particles below Y (Fig. 6).
The drift is negative from the surface to the bed. In the borderline situation κ = κ S it holds that D(p) < 0 for p ∈ [−m, 0) and D(0) = 0.
We exploit now the insights gained from studying the velocity field in Sect. 3 and the particle drift to describe the particle trajectories in a qualitative manner. The discussion contains the three different scenarios of no underlying current (κ = 0), a favorable current (κ > 0) and an adverse current (κ < 0).

Particle Trajectories in Flows Without Underlying Currents
In the situation of the absence of an underlying current, we proved (see Proposition 3.7) that the level set {u = 0} in D + consists of a smooth curve C + , which connects B + with S + and intersects each streamline Let us distinguish between the position (x(t), y(t)) of the particle at time t in the moving frame and the corresponding position (X(t), Y (t)) in the physical frame. Let γ = γ Y0 : [0, θ] → R 2 , t → (x(t), y(t)) be the trajectory of the particle in the moving frame (Fig. 7) and let Γ = Γ Y0 : be the corresponding path in the physical frame (Fig. 8).
As before we assume that the particle is initially located at Furthermore let B, C, D and E be the corresponding points in the physical frame. We have that E = Γ(θ) = (−L/2 + cθ, Y 0 ) lies to the right of A since D(Y 0 ) > 0. Since the images γ(I ab ) and γ(I de ) lie within regions where u < 0, the corresponding horizontal displacement Γ(t) has to be backwards during these time intervals. The image γ(I bd ) lies in the region where u > 0, hence Γ(t) moves forward during this time interval. Moreover γ(I ac ) lies in D + , where v is strictly positive, thus Γ(t) moves upwards within the time window I ac and Γ(t) moves downwards for t ∈ I ce since γ(I ce ) lies in D − , where v is strictly negative. This description holds for all particles being initially located above the flat bed. A particle which is initially located on the flat bed, i.e. Y 0 = −d, will always remain there, since v = 0. We have that Γ −d (t) moves forward at times t ∈ I bd , it moves backwards when t lies in the time intervals I ab and I de . This means that the particle oscillates backward-forward-backward, mirroring the projection of the loops of Γ Y0 , Y 0 > −d to the flat bed.

Particle Trajectories in Favorable Currents
As in the previous case, the drift is positive for all particles; see Proposition 4.7. We recall that the sign of the vertical velocity component v is not effected by the presence of an underlying current, see Proposition 3.1. We only have to take into account the qualitative changes of the horizontal velocity component u for an increasing mean current κ in order to give a qualitative description of the motion of fluid particles. According to Remark 3.8, we distinguish three different cases, regarding the sign of u in D. Fig. 9. Particle paths in the presence of a favorable current: the first trajectory corresponds to that of a particle experiencing a small favorable current or that of a particle in a moderate one above the critical streamline Y (the loops are smaller in comparison with Fig. 8), the second describes the wavelike path on or below Y, or that of the particles experiencing a strong favorable current

Flows Beneath Equatorial Waves
In the case of a small favorable current, where κ ≤ −u 0 (L/2, −d), the particle path pattern is the same as in the case κ = 0. Only in the borderline case κ = −u 0 (L/2, −d) the situation changes for particles at the flat bed: there is no backward motion at all.
In the case of a moderate favorable current, where one finds that −u 0 (L/2, −d) < κ < −u 0 (L/2, η(L/2)), the qualitative picture of the particle paths depends on the depth: a critical streamline Y separates D into two layers. Particles in the upper layer follow the κ = 0 pattern, whereas particles on or below Y do not move backwards.
If κ ≥ −u 0 (L/2, η(L/2)), we say that a strong favorable current is present and all particles experience a pure forward motion.
Particles which are located at the flat bed move to the right with a periodic change of velocity in the latter two sub-cases.
We visualized our considerations in Fig. 9.

Particle Trajectories in Adverse Currents
In the situation of an adverse current, the possibility of a negative particle drift has to be taken additionally into account in order to describe particle paths. We combine the results in Remarks 3.8 and 4.10 to classify the particle path pattern. First we observe, that Proposition 3.4 on the monotonicity of u together with (4.14), (4.15) and (4.16) allow us to relate κ S , κ B , −u 0 (0, η(0)) and −u 0 (0, −d): we keep also in mind that κ S < κ B as well as −u 0 (0, η(0)) < −u 0 (0, −d). Note that in general we can not determine a priori, whether κ S < −u 0 (0, −d) or −u 0 (0, −d) ≤ κ S . Therefore the following scenarios for the motion of particles might occur when an adverse current is present. there is a streamline Y 2 which splits D into an upper region, where u changes signs, and a lower region, where u is negative. Observe, that it can not happen, that Y 1 lies below Y 2 , since this would entail particles with a pure backward motion but a positive drift. Moreover it is not possible, that Y 1 and Y 2 coincide: closed loops enforce a periodic occurrence of backward and forward motion. Therefore Y 1 lies above Y 2 , and D splits into three different layers. In the lowest one, particles move wavelike to the left, c.f. Fig. 12. Particles in the middle layer are looping to the left, and particles in the top layer are looping to the right. 5. κ < κ S and −u 0 (0, η(0)) < κ < −u 0 (0, −d). The drift is negative for all particles and there exists a streamline Y that splits D into an upper layer, where u changes sign along streamlines, and a lower layer, where u is negative. Particles in the upper layer loop to the left, whereas particles below Y move as depicted in Fig. 12.
Here both D and u are negative throughout D. All particles move wavelike to the left.
Remark 4.11. We have demonstrated that the movement of particles within irrotational flows beneath periodic traveling equatorial surface waves follows the same qualitative pattern as it is the case for Stokes waves; c.f. the results in [4,10]. We could essentially reproduce these results for irrotational equatorial waves. Let us point out once again, that the bounds we have imposed on c and κ are not restrictive from a physical point of view.
Nevertheless, it is remarkable, that even in the presence of Coriolis effects in the f -plane, we can preserve the analysis without any restrictions on the wave amplitude; in particular our results do not rely on approximations.
Let us finally point out, that the effects of underlying (uniform) currents on particle paths in our setting is quite different from that in the explicit equatorial waves obtained in [2,5,13,16,17,21] for flows Flows Beneath Equatorial Waves with non-constant vorticity. These papers start by imposing a circular particle path in the absence of a current, changed to a trochoid by an underlying current, whereas in our setting for irrotational flows, closed particle paths are only possible, if there is a suitable adverse current.

Appendix A. Vorticity Equation for Equatorial Waves
We provide a derivation of the vorticity equation for three-dimensional flows beneath equatorial waves, which describes the evolution of the vorticity in a flow.
Since the divergence of ω is generally zero and due to the second equation ( Here, D/Dt = ∂/∂t + (u · ∇) denotes the so-called material derivative. For the special situation of a two-dimensional flow, which can be identified with a three dimensional flow satisfying u 2 = 0 with u 1 and u 3 being independent of the x 2 -variable, it holds that (ω · ∇)u = ω 2 ∂ x2 u = 0, and therefore equation This means, that the local spin of each fluid particle is preserved, as it moves with the flow. In particular, particles having no local spin at the initial time, will never acquire it.