Nonlinear Analysis: Real World Applications 18 (2014) 69–74

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Nonlinear Analysis: Real World Applications

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

Some nonlinear internal equatorial flowsHung-Chu Hsu ∗

Department of Mathematics, King’s College London, Westminster, UKTainan Hydraulics Laboratory, National Cheng Kung University, Tainan 701, Taiwan

a r t i c l e i n f o

Article history:Received 25 November 2013Accepted 27 December 2013

a b s t r a c t

We present an exact solution of the nonlinear governing equations for geophysical waterwaves in the β-plane approximation near the equator.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Recently, some exact solutions describing nonlinear equatorial flows in the Lagrangian framework were obtained. InConstantin [1] equatorially trapped wind waves were presented—see also the discussion in Constantin & Germain [2], andHenry [3] showed that one can also include a uniform underlying current. In Constantin [4] internal waves describingthe oscillation of the thermocline as a density interface separating two layers of constant density, with the lower layermotionless, were presented. Our aim is to extend the solution in Constantin [4] to include an underlying uniform current.The presence of strong currents in the Equatorial Pacific is well-documented, cf. Philander [5]. The present extension of theflow in Constantin [4], while showing that one can accommodate an underlying current, differs from the flow presentedrecently in Constantin [6].

2. Preliminaries

The Earth is taken to be a sphere of radius, R = 6371 km, rotating with constant rotational speedΩ = 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’s surface, so that theCartesian coordinates (x, y, z) represent longitude, latitude, and the local vertical, respectively. The governing equationsfor geophysical ocean waves are, cf. Pedlosky [7], 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)

∗ Correspondence to: Tainan Hydraulics Laboratory, Tainan, Taiwan. Tel.: +886 6 2371938.E-mail address: hchsu@thl.ncku.edu.tw.

1468-1218/$ – see front matter© 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.nonrwa.2013.12.011

70 H.-C. Hsu / Nonlinear Analysis: Real World Applications 18 (2014) 69–74

Here t stands for time, φ for latitude, g = 9.81m s−2 is the (constant) gravitational acceleration at the Earth’s surface, andρ is the water’s 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, theapproximations sinφ ≈ φ and cosφ ≈ 1 can be used, cf. Vallis [8]. This approximation, called the equatorial β-planeapproximation, approximates the Coriolis force

2Ω

w cosφ − v sinφ

u sinφ−u cosφ

by 2Ωw − βyvβyu

−2Ωu

with β = 2Ω/R = 2.28 ·10−11 m−1 s−1, cf. Cushman-Roisin and Becker [9]. Consequently, the Euler equation (1) is replacedby

ut + uux + vuy + wuz + 2Ωw − βyv = −1ρPx,

vt + uvx + vvy + wvz + βyu = −1ρPy,

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

(4)

We work with a two-layer model: two layers of constant densities, separated by a sharp interface—the thermocline. Letz = 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 . The upper boundary of the region M(t) above the thermocline and beneath thenear-surface layer L(t) to which wind effects are confined is given by z = η+(x, y, t). Beneath the thermocline the waterhas a constant density ρ+ and is still: at every instant t we have u = v = w = 0 for z < η(x, y, t). From (4) we infer that

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

for some constant P0. We investigate the eastward propagation of geophysical waves with vanishing meridional velocity(v ≡ 0) in the region M(t), without discussing the interaction of geophysical waves and wind waves in the region L(t).ThroughoutM(t) the water is assumed to have constant density ρ0 < ρ+, the typical value of the reduced gravity

g = gρ+ − ρ0

ρ0(5)

being 6 · 10−3 ms−2, cf. Fedorov and Brown [10]. Consequently, we seek solutions (u(x, y, z, t), w(x, y, z, t), η(x, y, t) andη+(x, y, t)) of the Euler equations in the form

ut + uux + wuz + 2Ωw = −1ρPx,

βyu = −1ρPy,

wt + uwx + wwz − 2Ωu = −1ρPz − g,

(6)

in η(x, y, t) < z < η+(x, y, t), coupled with the incompressibility condition

ux + wz = 0 in η(x, y, t) < z < η+(x, y, t), (7)

and with the boundary condition

P = P0 − ρ+gz on z = η(x, y, t). (8)

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, y, t). (9)

H.-C. Hsu / Nonlinear Analysis: Real World Applications 18 (2014) 69–74 71

3. Main result

In this section, wewill present the exact solution, representingwaves traveling in the longitudinal direction at a constantspeed of propagation c > 0, in the presence of a constant current of 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−k[r+f (s)] sin[k(q − ct)],

y = s,

z = r −1ke−k[r+f (s)] cos[k(q − ct)],

(10)

where k is the wave number, and

f (s) =cβ2γ

s2, (11)

with γ = g−2ΩU . The function f (s) determines the decay of particle oscillation in themeridional direction. The parameterq runs over the whole real line, while s ∈ [−s0, s0], where s0 =

√c0/β ∼ 250 km, cf. the discussion in Constantin [4],

is the equatorial radius of deformation. As for the label r , for every fixed s, we require r ∈ [r0(s), r+(s)], with the twopositive numbers to be specified by later on. The particle motion described by (10) prescribes that the particles do notmove in the meridional direction, describing circles in a vertical plane, parallel to that along the Equator. This resembles thesituation encountered in the flow discovered by Gerstner [11]; see also the discussion in Constantin [12], Henry [13], andConstantin [14]. This situation contrasts with that occurring in the irrotational flow beneath a Stokeswave, cf. the discussionin Constantin [15], Henry [16], and Constantin and Strauss [17]. This is therefore indicative of a strongly sheared flow, anaspect that we discuss in Section 4 by describing the vorticity of the flow (10). Note that while in Gerstner’s wave thediameters of the circles decrease as we descend into the fluid, in the present setting, the diameter decreases as we ascendabove the thermocline. In particular, as we approach the near-surface layer, the flow acquires more and more the characterof a pure uniform current.

In order to verify that (10) 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 θ

fse−ξ sin θ 1 fse−ξ cos θ

e−ξ sin θ 0 1 + e−ξ cos θ

(12)

equals 1 − e−2ξ , where for convenience we set

ξ = k[r + f (s)], θ = k(q − ct). (13)

The time-independence of the determinant expresses the fact that the corresponding fluid flow is volume-preserving, thatis, (3) holds in the Eulerian framework of a fixed observer. We require that the labeling variables satisfy

r + f (s) ≥ r∗ (14)

for some r∗ > 0. We can write the Euler equation (4) as

DuDt

+ 2Ωw = −1ρPx,

Dv

Dt+ βyu = −

1ρPy,

Dw

Dt− 2Ωu = −

1ρPz − g.

(15)

The formulas (10) yield the velocity and acceleration of a particle asu =

DxDt

= −U + ce−ξ cos θ,

v =DyDt

= 0,

w =DzDt

= −ce−ξ sin θ,

(16)

72 H.-C. Hsu / Nonlinear Analysis: Real World Applications 18 (2014) 69–74

respectively,

DuDt

= kc2e−ξ sin θ,

Dv

Dt= 0,

Dw

Dt= kc2e−ξ cos θ.

(17)

Since

PqPsPr

=

∂x∂q

∂y∂q

∂z∂q

∂x∂s

∂y∂s

∂z∂s

∂x∂r

∂y∂r

∂z∂r

PxPyPz

, (18)

and, due to (16)–(17), (15) can be written as

Px = −ρ0(kc2 − 2Ωc)e−ξ sin θ, (19a)

Py = −ρ0βs(−U + ce−ξ cos θ), (19b)

Pz = −ρ0[kc2e−ξ cos θ − 2Ωce−ξ cos θ + (g + 2ΩU)], (19c)

we have that (15) is equivalent to

Pq = −ρ0[kc2 − 2Ωc + (g + 2ΩU)]e−ξ sin θ, (20)

Ps = −ρ0fs(kc2 − 2Ωc)e−2ξ− ρ0[cβs + fs(g + 2ΩU)]e−ξ cos θ + ρ0βsU, (21)

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

For

f (s) =β

2(kc − 2Ω)s2

we have

fs =β

(kc − 2Ω)s,

and the gradient of the expression

P = ρ0[kc2 − 2Ωc + (g + 2ΩU)]

ke−ξ cos θ + ρ0

(kc2 − 2Ωc)2k

e−2ξ− ρ0(g + 2ΩU)r + ρ0(kc − 2Ω)Uf (s) + P0 (23)

with respect to the labeling variables is precisely the right-hand side of (20)–(22). Since r = r0(s) corresponds to thethermocline z = η(x − ct), the expression (23) evaluated at the thermocline matches the boundary conditions (8) if thedispersion equation

ρ0[kc2 − 2Ωc + (g + 2ΩU)] = ρ+g (24)

holds. We require c > 0 for the eastward-propagating waves. This means that

c =

Ω2 + k(g − 2ΩU) + Ω

k. (25)

Therefore the choice (11) and (25) yields a solution to (6)–(9), the thermocline being determined by setting r = r0(s), wherer0(s) is the unique solution of the equation

P∗

0 = r +Ucf (s) +

12k

exp

−2kr −kβcγ

s2

, (26)

at a fixed value of s ∈ [−s0, s0], with

P∗

0 =P0 − P0

ρ0γ.

H.-C. Hsu / Nonlinear Analysis: Real World Applications 18 (2014) 69–74 73

For every fixed s ∈ [−s0, s0], the function

r → r +Ucf (s) +

12k

exp

−2kr −kβcγ

s2

is a strictly increasing diffeomorphism from (0, ∞) ontoUcf (s) +

12k

exp

−kβcγ

s2

, ∞

.

This ensures that if

P∗

0 >Uβs202γ

+12k

, (27)

then for every s ∈ [−s0, s0] we can find a unique solution r = r0(s) > 0 of the equation in (26). By the implicit functiontheorem, r0(s) is a smooth and even function of s. Evaluating (26) at r = r0(s) and differentiating the outcome with respectto s yield

r ′

0(s) =βsγ

ce−2ξ− U

1 − e−2ξ. (28)

The relation (28) shows that r0(s) increase as s > 0 increases, provided that U < ce−2r0(s). From (28) we can infer

r ′

0(s) +βcsγ

=βcsγ

1 − U/c1 − e−2ξ

(29)

so that the even function

s →12k

exp

−2kr0(s) −kβcγ

s2

(30)

is strictly decreasing for s > 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 constant

P0 > P∗

0 >12k

(31)

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

P0 = r +Ucf (s) +

12k

exp

−2kr −kβcγ

s2

. (32)

The previous considerations show that P0 determines a unique number r+(s) > r0(s). The function s → r+(s) presents thesame features as the function s → r0(s), namely, it is even, smooth, and strictly increasing for s > 0, with

s →12k

exp

−2kr+(s) −kβcγ

s2

strictly decreasing for s > 0. (33)

We see that (14) holds with r∗= r0(0). This completes the proof that (10) is an exact solution to (6)–(9). For fixed values

of ρ0 < ρ+, we have constructed a family of solutions with three degrees of freedom: the parameter k > 0 and theconstants P0 and P∗

0 , subject to the constraint (31). Setting U = 0, our solution (10) particularizes to the solution describedin Constantin [4].

4. Discussion

We now calculate the vorticity of the flow prescribed by (6). The explicit form of the matrix ∂(x, y, z)/∂(q, r, s) in (12)easily yields its 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

−fse−ξ sin θ

1 − e−2ξ1 −fs

e−ξ cos θ − e−2ξ

1 − e−2ξ

−e−ξ sin θ

1 − e−2ξ0

1 − e−ξ cos θ

1 − e−2ξ

. (34)

74 H.-C. Hsu / Nonlinear Analysis: Real World Applications 18 (2014) 69–74

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

−fse−ξ sin θ

1 − e−2ξ1 −fs

e−ξ cos θ − e−2ξ

1 − e−2ξ

−e−ξ sin θ

1 − e−2ξ0

1 − e−ξ cos θ

1 − e−2ξ

−kce−ξ sin θ 0 −kce−ξ cos θ

−kfsce−ξ cos θ 0 kfsce−ξ sin θ

−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

kcfse−2ξ

1 − e−2ξ+

kcfse−ξ cos θ

1 − e−2ξ1

kcfse−ξ sin θ

1 − e−2ξ

−kce−ξ cos θ + kce−2ξ

1 − e−2ξ0

kce−ξ sin θ

1 − e−2ξ

(35)

and the vorticity is

ω = (wy − vz, uz − wx, vx − uy)

=

skc2βγ

e−ξ sin θ

1 − e−2ξ,

2kce−2ξ

1 − e−2ξ, s

kc2βγ

e−ξ cos θ − e−2ξ

1 − e−2ξ

. (36)

We note that since the current is constant, it does not influence the vorticity of the flow, and the prescription of the vorticitymatches that of Constantin [4].

Acknowledgment

The authors would like to acknowledge the insightful critiquing of the referee.

References

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http://dx.doi.org/10.1175/JPO-D-13-0174.1.[7] J. Pedlosky, Geophysical Fluid Dynamics, Springer, 1979, p. 742.[8] G.K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge Univ. Press, 1979, p. 772.[9] B. Cushman-Roisin, J.-M. Beckers, Introduction to Geophysical Fluid Dynamics, Academic Press, 2011, p. 320.

[10] A.V. Fedorov, J.N. Brown, Equatorial waves, in: J. Steele (Ed.), Encyclopedia of Ocean Sciences, Academic Press, 2009, pp. 3679–3685.[11] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys. 2 (1809) 412–445 (in German).[12] A. Constantin, On the deep water wave motion, J. Phys. 34A (2001) 1405–1417.[13] D. Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys. 15 (2008) 87–95.[14] A. Constantin, Nonlinear Water Waves with Applications to Wave–Current Interactions and Tsunamis, in: CBMS-NSF Regional Conference Series in

Applied Mathematics, vol. 81, SIAM, Philadeplhia, PA, 2011, p. 321.[15] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006) 523–535.[16] D. Henry, On the deep-water Stokes flow, Int. Math. Res. Not. 22 (2008). http://dx.doi.org/10.1093/imrn/rnn071.[17] A. Constantin, W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. 53 (2010) 533–557.