Applied Mathematics Letters 34 (2014) 1–6

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Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

Some nonlinear internal equatorial waves with a strongunderlying currentHung-Chu Hsu ∗

Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan 701, Taiwan

a r t i c l e i n f o

Article history:Received 19 December 2013Received in revised form 6 March 2014Accepted 6 March 2014Available online 17 March 2014

Keywords:Nonlinear internal waveGeophysical wavesExact solution

a b s t r a c t

We describe an exact solution of the nonlinear governing equations for the internalequatorial water waves with an underlying current in the f -plane approximation near theequator. Thesewaves travel westward at a constant speed, and a strong uniform horizontalcurrent underlies the flow.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Tropical dynamics has been a topic of interest for the past few decades because it is important for understanding theocean currents and climate system. The equatorial waves are trapped near the equator and travel along the thermocline,which separates warm (and less dense) surface water from colder deep water. Field data shows that these equatorial wavespresent nonlinear properties. They play a key role in the El Nino-Southern Oscillation (ENSO) phenomena (Cushman-Roisinand Beckers [1]).

There is a large literature on linear equatorial wave dynamics. The nonlinear literature, on the other hand, is muchless developed. Recently, some exact solutions describing nonlinear equatorial flows in the Lagrangian framework wereobtained. Lagrangian solutions present the great advantage that the fluid kinematics may be described explicitly (Alemanand Constantin [2], Bennett [3], Hsu et al. [4,5], Hsu [6]. In Constantin [7] equatorially trappedwindwaveswere presented inthe β-plane approximation—see also the discussion in Constantin and Germain [8] and Hsu [9]. In Constantin [10] internalwaves describing the oscillation of the thermocline as a density interface separating two layers of constant density, with thelower layer motionless, were presented. Hsu [11] extended the flow in [10] to study the equatorial internal waves with anunderlying current.

The presence of strong currents in the Equatorial Pacific is well-documented; cf. Philander [12]. Mollo-Christensen [13]studied Gerstner’s wavewith an underlying current in a number of physical settings. Henry [14] presented an exact solutionfor equatorial waves with an underlying current in the β-plane approximation. In this paper we present an exact solutionto the f -plane governing equations for the internal equatorial waves interacting with a uniform current. The unique featureof the f -plane approximation being considered is that it allows the waves to be westward propagating [15], unlike the moregeneral β-plane setting. This is interesting since the famous Equatorial Undercurrent (EUC) is indeedwestward propagatingon the surface. Constantin [16] and Henry and Matioc [17] present recent rigorous mathematical results for such westwardpropagating equatorial water waves with currents. The wave solution we construct corresponds to steady zonal waves,

∗ Tel.: +886 6 2371938.E-mail address: hchsu@thl.ncku.edu.tw.

http://dx.doi.org/10.1016/j.aml.2014.03.0050893-9659/© 2014 Elsevier Ltd. All rights reserved.

2 H.-C. Hsu / Applied Mathematics Letters 34 (2014) 1–6

travelling westward in the longitudinal direction with constant speed (Section 2). Section 3 is devoted to the presentationof the explicit solutions, while the last section is devoted to a discussion of the flow pattern.

2. Preliminaries

The Earth is taken to be a sphere of radius (R = 6371 km), rotating with constant rotational speed Ω = 7.29 ·

10−5 rad s−1 round the polar axis toward the east, in a rotating frame with the origin at a point on the Earth surface. In ourCartesian coordinates, (x, y, z) represent longitude, latitude, and the local vertical, respectively. The governing equations forgeophysical ocean waves are, cf. Pedlosky [18], the Euler equation

ut + uux + vuy + wuz + 2Ωw cosϕ − 2Ωv sinϕ = −1ρPx,

vt + uvx + vvy + wvz + 2Ωu sinϕ = −1ρPy,

wt + uwx + vwy + wwz − 2Ωu cosϕ = −1ρPz − g,

(1)

coupled with the equation of mass conservation

ρt + uρx + vρy + wρz = 0 (2)

and with the incompressibility constraint

ux + vy + wz = 0. (3)

Here t stands for time, ϕ for latitude, g = 9.81m s−2 is the (constant) gravitational acceleration at the Earth surface, and ρis the waters density, while P is the pressure.

Since we restrict our attention to a symmetric band of width of about 250 km on each side of the equator, we neglect thevariations of the Coriolis parameter and use the f -plane approximation. The justification and an in-depth discussion of thephysical considerations for using f -plane approximation are presented in (Constantin [19]). Within this regime, we replace

2Ω

w cosϕ − v sinϕ

u sinϕ−u cosϕ

(4)

by 2Ωw0

−2Ωu

. (5)

Consequently, the Euler equation (1) is replaced byut + uux + vuy + wuz + 2Ωw = −

1ρPx,

vt + uvx + vvy + wvz = −1ρPy,

wt + uwx + vwy + wwz − 2Ωu = −1ρPz − g.

(6)

We work with a two-layer model: two layers of constant densities, separated by a sharp interface—the thermocline.Let z = η(x, y, t) be the equation of the thermocline. We model the oscillations of this interface as propagating in thelongitudinal direction at constant speed c with an underlying current U . The upper boundary of the region M(t) above thethermocline and beneath the near-surface layer L(t) to which wind effects are confined is given by z = η+(x, y, t). Beneaththe thermocline thewater has a constant density ρ+ and is still: at every instant t we have u = v = w = 0 for z < η(x, y, t).From (6) we infer that

P = P0 − ρ+gz in the region z < η(x, y, t),

for some constant P0. We are interested in the westward propagation of internal geophysical waves with vanishingmeridional velocity (v ≡ 0) in the region M(t), and we do not address the interaction of geophysical waves and windwaves in the region L(t). We also neglect y-dependence. Throughout M(t) the water is assumed to have constant densityρ0 < ρ+, the typical value of the reduced gravity

g = gρ+ − ρ0

ρ0(7)

H.-C. Hsu / Applied Mathematics Letters 34 (2014) 1–6 3

being 6 · 10−3ms−2; cf. Fedorov and Brown [20]. Consequently, we seek travelling wave solutions (u(x − ct, z),w(x −

ct, z),η(x − ct, z) and η+(x − ct, z)) of the Euler equation in the formut + uux + wuz + 2Ωw = −

1ρPx,

wt + uwx + wwz − 2Ωu = −1ρPz − g,

(8)

in the region η(x − ct) < z < η+(x − ct), coupled with the incompressibility condition

ux + wz = 0 in η(x − ct) < z < η+(x − ct) (9)

and with the boundary condition

P = P0 − ρ+gz on z = η(x − ct) (10)

Moreover, we impose that the flow approaches a uniform current rapidly in the near-surface layer, that is

(u, v) → (−U, 0) as z → η+(x − ct). (11)

3. Main result

In this section, we will present the exact solution of the f -plane governing equation (6), describing internal wavestravelling westward in the longitudinal direction, at a constant speed of propagation, in the presence of a constant currentof strength U . For our solution, the Lagrangian positions (x, y, z) of fluid particles are given as functions of labeling variables(q, r, s) and time t by

x = q − Ut −1ke−kr sin[k(q − ct)],

y = s,

z = r −1ke−kr cos[k(q − ct)],

(12)

where k is the wave number. The parameter q runs over the whole real line, while s ∈ [−s0, s0], where s0 =√c0/β ≈ 250

km, cf. the discussion in Constantin [10], is the equatorial radius of deformation. As for the label r , we require r ∈ [−r0, r0],with the two positive numbers to be specified later on. The particle motion described by (12) imposes that the particlesdescribe circles. This resembles the situation encountered in the flow discovered by Gerstner [21]; see also the discussionin Constantin [22], Henry [23], Constantin [24]. This situation contrasts with that for the irrotational flow beneath a Stokeswave; cf. the discussion in Constantin [25], Henry [26], Constantin and Strauss [27]. This suggests a strongly sheared flow,an aspect that we discuss in Section 4 by describing the vorticity of the flow (12). At every fixed s, Eqs. (12) represent theflow beneath a Gerstner wave propagating westward at a constant speed

c =Ω −

Ω2 + k(g − 2ΩU)

k<0, (13)

so that (12) defines internal equatorial waves propagating westward. Note that for the case of c = [Ω +Ω2 + k(g − 2ΩU)]/k > 0, the internal equatorial waves propagating eastward also exist, which had been discussed

by Constantin [10] in the β-plane approximation.In order to verify that (12) is an exact solution, let us first observe that the determinant of the matrix

∂x∂q

∂y∂q

∂z∂q

∂x∂s

∂y∂s

∂z∂s

∂x∂r

∂y∂r

∂z∂r

=

1 − e−ξ cos θ 0 e−ξ sin θ0 1 0

e−ξ sin θ 0 1 + e−ξ cos θ

(14)

equals 1 − e−2ξ , where

ξ = kr, θ = k(q − ct). (15)

Thus (3) holds in the Eulerian framework of a fixed observer. We require that

r ≥ r∗ (16)

4 H.-C. Hsu / Applied Mathematics Letters 34 (2014) 1–6

for some r∗ > 0, to be specified later on. In Lagrangian variables, the kinematic boundary holds since, independently of thefixed value of s, the internal wave elevation is obtained by specifying a value of r (the label q being the free parameter of thecurve representing the wave profile at this latitude). The flow (12) presents no variation in the meridional direction.

We now write the Euler equation (6) as

DuDt

+ 2Ωw = −1ρPx,

Dv

Dt= −

1ρPy,

Dw

Dt− 2Ωu = −

1ρPz − g.

(17)

The formulas (12) yield the velocity and acceleration of a particle asu =

DxDt

= −U + ce−ξ cos θ,

v =DyDt

= 0,

w =DzDt

= −ce−ξ sin θ,

(18)

respectively,

DuDt

= kc2e−ξ sin θ,

Dv

Dt= 0,

Dw

Dt= kc2e−ξ cos θ.

(19)

We see that (17) takes the form

Px = −ρ0(kc2 − 2Ωc)e−ξ sin θ, (20)Py = 0, (21)

Pz = −ρ0[(kc2 − 2Ωc)e−ξ cos θ + (g + 2ΩU)]. (22)

The change of variables

PqPsPr

=

∂x∂q

∂y∂q

∂z∂q

∂x∂s

∂y∂s

∂z∂s

∂x∂r

∂y∂r

∂z∂r

PxPyPz

, (23)

enables us to write (20)–(22) as

Pq = −ρ0[kc2 − 2Ωc + (g + 2ΩU)]e−ξ sin θ, (24)

Ps = 0, (25)

Pr = −ρ0[kc2 − 2Ωc + (g + 2ΩU)]e−ξ cos θ − ρ0(kc2 − 2Ωc)e−2ξ− ρ0(g + 2ΩU). (26)

One can easily check that the gradient of the expression

P = ρ0(kc2 − 2Ωc)

2ke−2ξ

+ ρ0[kc2 − 2Ωc + (g + 2ΩU)]

ke−ξ cos θ − ρ0(g + 2ΩU)r + P0 (27)

with respect to the labeling variables is precisely the right-hand side of (24)–(26).Since r = r0 corresponds to the thermocline z = η(x − ct), the expression (27) evaluated at the thermocline matches

the boundary condition (10) if the dispersion equation

ρ0[kc2 − 2Ωc + (g + 2ΩU)] = ρ+g (28)

H.-C. Hsu / Applied Mathematics Letters 34 (2014) 1–6 5

holds. Note that if U = c then (28) gives c =g/k, and the geophysical effects have no influence on the dispersion relation

of the internal equatorial wave. We require c < 0 for the westward-propagating waves. This means that

c =Ω −

Ω2 + k(g − 2ΩU)

k. (29)

Therefore the choice (12) and (13) yields a solution to (8)–(11), the thermocline being determined by setting r = r0, wherer0 is the unique solution of the equation

P∗

0 = r +12k

exp (−2kr) (30)

with

P∗

0 =P0 − P0

ρ0γ(31)

where γ = g − 2ΩU . The function

r → r +12k

exp (−2kr) (32)

is a strictly increasing diffeomorphism from (0, ∞). This ensures that if

P∗

0 >12k

(33)

then we can find a unique solution r = r0 > 0 of the equation in (30). Note that the even function

r →12k

exp(−2kr) (34)

is strictly decreasing for r0 > 0.To complete the solution it remains to specify the boundary delimiting the two layersM(t) and L(t). This is obtained by

choosing some fixed constants

P0 > P∗

0 >12k

(35)

and setting r = r+ at any fixed value of s ∈ [−s0, s0], where r+ > 0 is the unique solution of the equation

P0 = r +12k

exp (−2kr) . (36)

The previous considerations show that P0 determines a unique number r+ > r0. We see that (16) holds with r∗

0 = r0. Thiscompletes the proof that (12) is an exact solution to (8)–(11). For fixed values of ρ0 < ρ+, we have constructed a familyof solutions with three degrees of freedom: the parameter k > 0 and the constants P0 and P∗

0 , subject to constraint (35).Setting U = 0, our solution (12) particularizes to the solution described in Hsu [28].

An important aspect of the solution is that there is no variation in the meridional direction. This is in stark contrast tothe equatorially trapped solutions (Constantin [10]), which were propagating eastward. Westward propagating waves arepossible in our setting, but not in the setting of Constantin [10], precisely because there are no variations in the meridionaldirection.

4. Discussion

We now investigate the vorticity of the flow (12). The explicit form of the matrix ∂(x, y, z)/∂(q, r, s) in (14) easily yieldsits inverse

∂q∂x

∂s∂x

∂r∂x

∂q∂y

∂s∂y

∂r∂y

∂q∂z

∂s∂z

∂r∂z

=

1 + e−ξ cos θ

1 − e−2ξ0 −

e−ξ sin θ

1 − e−2ξ

0 1 0

−e−ξ sin θ

1 − e−2ξ0

1 − e−ξ cos θ

1 − e−2ξ

. (37)

6 H.-C. Hsu / Applied Mathematics Letters 34 (2014) 1–6

Then we have∂u∂x

∂v

∂x∂w

∂x∂u∂y

∂v

∂y∂w

∂y∂u∂z

∂v

∂z∂w

∂z

=

∂q∂x

∂s∂x

∂r∂x

∂q∂y

∂s∂y

∂r∂y

∂q∂z

∂s∂z

∂r∂z

∂u∂q

∂v

∂q∂w

∂q∂u∂s

∂v

∂s∂w

∂s∂u∂r

∂v

∂r∂w

∂r

=

1 + e−ξ cos θ

1 − e−2ξ0 −

e−ξ sin θ

1 − e−2ξ

0 1 0

−e−ξ sin θ

1 − e−2ξ0

1 − e−ξ cos θ

1 − e−2ξ

−kce−ξ sin θ 0 −kce−ξ cos θ

0 0 0−kce−ξ cos θ 0 kce−ξ sin θ

=

−kce−ξ cos θ

1 − e−2ξ0

−kce−ξ cos θ − kce−2ξ

1 − e−2ξ

0 1 0

−kce−ξ cos θ + kce−2ξ

1 − e−2ξ0

kce−ξ sin θ

1 − e−2ξ

(38)

and the vorticity is

ω = (wy − vz, uz − wx, vx − uy)

=

0,

2kce−2ξ

1 − e−2ξ, 0

. (39)

We note that since the current is constant, it does not influence the vorticity of the flow. The middle component of ω isalways strictly negative, while the first and third components vanish. The vorticity contrasts with that of Constantin [10].

References

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[10] A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr. 43 (2013) 165–175.[11] H.C. Hsu, Some nonlinear internal equatorial flows, Nonlinear Anal. RWA (2014) http://dx.doi.org/10.1016/j.nonrwa.2013.12.011.[12] S.G.H. Philander, The Equatorial Undercurrent revisited, Annu. Rev. Earth Planet. Sci. 8 (1980) 191204.[13] E. Mollo-Christensen, Gravitational and geostrophic billows: some exact solutions, J. Atmos. Sci. 35 (1978) 1395–1398.[14] D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B/ Fluids 38 (2013) 1821.[15] H.C. Hsu, An Exact Solution for Nonlinear Internal Equatorial Waves in the f -Plane Approximation, J. Math. Fluid Mech. (2014) http://dx.doi.org/10.

1007/s00021-014-0168-3.[16] A. Constantin, On equatorial wind waves, Differential Integral equations 26 (2013) 237–252.[17] D. Henry, A. Matioc, On the existence of equatorial wind waves, Nonlinear Anal. 101 (2014) 113–123.[18] J. Pedlosky, Geophysical Fluid Dynamics, Springer, 1979, 742 pp.[19] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett. 39 (2012) L05602.[20] A.V. Fedorov, J.N. Brown, Equatorial waves, in: J. Steele (Ed.), Encyclopedia of Ocean Sciences, Academic Press, 2009, pp. 3679–3685.[21] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys. 2 (1809) 412–445. (in German).[22] A. Constantin, On the deep water wave motion, J. Phys. 34A (2001) 1405–1417.[23] D. Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys. 15 (2008) 87–95.[24] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, in: CBMS-NSF Regional Conference Series in

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