Monatsh MathDOI 10.1007/s00605-014-0618-2

An exact solution for equatorial waves

Hung-Chu Hsu

Received: 3 December 2013 / Accepted: 3 March 2014© Springer-Verlag Wien 2014

Abstract This study presents an explicit exact solution for nonlinear geophysicalequatorial waves in the f -plane approximation near the Equator. The solution describesin the Lagrangian framework equatorial waves propagating westward in a homogenousinviscid fluid.

Keywords Nonlinear geophysical wave · Free boundary · Exact solution

Mathematics Subject Classification 76B55 · 86A05 · 76E30

1 Introduction

The study of tropical dynamics is essential for understanding the climate system andfor improving climate and weather prediction models. It is a topic of interest for the pastfew decades. The equatorial waves exhibit particular dynamics due to the vanishingof the Coriolis parameter along the Equator. These equatorial waves are important indetermining large scale climatic processes such as the El Niño-Southern Oscillation(ENSO) phenomenon (Cushman-Roisin and Beckers [1]).

There is a large literature on equatorial wave dynamics. Explicit solutions for grav-ity fluid flows within the Lagrangian viewpoint are not numerous. It has been shownthat Lagrangian solutions have the great advantage that the fluid kinematics may bedescribed explicitly (Bennett [2], Aleman and Constantin [3], Hsu et al. [4,5], Hsu[6]). Exact solutions in Lagrangian variables were first found by Gerstner [7]. The very

Communicated by A. Constantin.

H.-C. Hsu (B)Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan 701, Taiwane-mail: hchsu@thl.ncku.edu.tw

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H.-C. Hsu

same flow was rediscovered by Froude [8], Rankine [9] and Reech [10]. Gerstner’swave is a periodic travelling wave with a specific vorticity distribution; see Constantin[11]; Henry [12] for modern treatments of Gerstner’s wave, and Yih [13], Constantin[14], Stuhlmeier [15], Matioc [16] for extensions to edge waves propagating alonga sloping bed. Constantin [17] showed that a similar approach provides a descrip-tion of equatorially trapped waves propagating eastwards; see also the discussion inConstantin and Germain [18]. Recently, the Gerstner-type approach was adapted toprovide internal equatorial wave propagating at the thermocline (Constantin [19,20]).

In this paper we present an exact solution to the f -plane governing equations forgeophysical water waves. The wave solution we construct corresponds to steady zonalwaves, travelling westwards in the longitudinal direction with constant speed (Sect. 2).Section 3 is devoted to the presentation of the explicit solutions, while the last sectionis devoted to a discussion of the flow pattern.

2 Preliminaries

In a rotating framework with the origin at a point on the Earth’s surface, with the x-axis chosen horizontally due east, the y-axis horizontally due north (latitude), and thez-axis upward, we consider waves travelling at the surface z = η(x, y, t) of a homo-geneous inviscid fluid of infinite depth. For a fluid layer localized near the Equator,the governing equations for geophysical ocean waves in the f -plane approximationare the Euler equation, cf. the discussion and the physical considerations presented inthe form (Constantin [21]),

⎧⎪⎨

⎪⎩

ut + uux + vuy + wuz + 2Ωw = − 1ρ

Px ,

vt + uvx + vvy + wvz = − 1ρ

Py,

wt + uwx + vwy + wwz − 2Ωu = − 1ρ

Pz − g,

(1)

and the equation of mass conservation

ux + vy + wz = 0. (2)

Here t is time, (u, v, w) is the fluid velocity, Ω = 7.29 ·10−5 rad s−1 is the (constant)rotational speed of the Earth (taken to be a sphere of radius R = 6371 km) round thepolar axis toward the east, g = 9.8 ms−2 is the (constant) gravitational accelerationat the earth’s surface, ρ is the (constant) water density, and P is the pressure. Weseek travelling waves with the fluid velocity field, the pressure and the free surfaceexhibiting an (x, t)-dependence of the form (x − ct), where c < 0 is the propagationspeed of the westward propagating wave. Moreover, we impose a vanishing meridionalvelocity v and no dependence on the y-variable. The boundary conditions are thereforethe kinematic boundary condition

w = (u − c)ηx on z = η(x − ct), (3)

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An exact solution for equatorial waves

together with the dynamic boundary condition

P = P0 on z = η(x − ct), (4)

where P0 stands for the (constant) atmospheric pressure. The boundary condition(3) expresses the fact that a particle on the surface remains confined to it, while (4)decouples the water flow from the motion of the air above it (Constantin [22]). Werequire the velocity field to decay rapidly with depth.

3 Main result

Let the Lagrangian position (x, y, z) of fluid particles be given as functions of thelabeling variables (q, r, s) and time t by

⎧⎨

⎩

x = q − 1k ekr sin[k(q − ct)],

y = s,z = r + 1

k ekr cos[k(q − ct)].(5)

Here k is the wave number and c < 0 is the wave speed. The labelling variables q ands run over the entire real line, while r ∈ (−∞, r0) for some fixed r0 < 0. At everyfixed s, the Eq. (5) represent the flow beneath a Gerstner wave propagating westwardat constant speed

c = −Ω − √Ω2 + kg

k, (6)

so that (5) defines equatorial waves propagating westward in a homogenous fluid withdensity ρ.

To check that (5) is an exact solution, let us first observe that the determinant of thematrix

⎛

⎜⎝

∂x∂q

∂y∂q

∂z∂q

∂x∂s

∂y∂s

∂z∂s

∂x∂r

∂y∂r

∂z∂r

⎞

⎟⎠ =

⎛

⎝1 − eξ cos θ 0 −eξ sin θ

0 1 0−eξ sin θ 0 1 + eξ cos θ

⎞

⎠ (7)

equals 1 − e2ξ , where we denoted

ξ = kr, θ = k(q − ct). (8)

Since the determinant is time independent, the flow is volume preserving and (2)holds. We require that the labelling variables satisfy

r ≤ r0 (9)

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H.-C. Hsu

for some constant r0 < 0. In Lagrangian variables, the kinematic boundary (3) holdssince, independently of the fixed value of s, the free surface is obtained by specifyinga value of r (the label q being the free parameter of the curve representing the waveprofile at this latitude). The flow (5) presents no variation in the meridional direction.

We now write the Euler Eq. (1) in the form⎧⎪⎨

⎪⎩

DuDt + 2Ωw = − 1

ρPx ,

DvDt = − 1

ρPy,

DwDt − 2Ωw = − 1

ρPz − g.

(10)

Using (5), we compute the velocity and acceleration of a particle as⎧⎪⎨

⎪⎩

u = DxDt = ceξ cos θ,

v = DyDt = 0,

w = DzDt = ceξ sin θ.

(11)

respectively,⎧⎪⎨

⎪⎩

DuDt = kc2eξ sin θ,

DvDt = 0,

DwDt = −kc2eξ cos θ.

(12)

We see that (10) takes the form

Px = −ρ(kc2 + 2Ωc)eξ sin θ, (13)

Py = 0, (14)

Pz = ρ(kc2 + 2Ωc)eξ cos θ − ρg. (15)

The change of variables

⎛

⎝Pq

Ps

Pr

⎞

⎠ =⎛

⎜⎝

∂x∂q

∂y∂q

∂z∂q

∂x∂s

∂y∂s

∂z∂s

∂x∂r

∂y∂r

∂z∂r

⎞

⎟⎠

⎛

⎝Px

Py

Pz

⎞

⎠, (16)

enables us to write (13)–(15) as

Pq = −ρ(kc2 + 2Ωc − g)eξ sin θ, (17)

Ps = 0, (18)

Pr = ρ(kc2 + 2Ωc − g)eξ cos θ − ρg + ρ(kc2 + 2Ωc)e2ξ . (19)

One can easily check that the gradient of the expression

P(q − ct, s, r) = ρkc2 + 2Ωc

2ke2ξ − ρgr + ρ

kc2 + 2Ωc − g

keξ cos θ + P0 (20)

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An exact solution for equatorial waves

with respect to the labeling variables is exactly the right-hand side of (17)–(19). Sincethe dynamic boundary condition (3) requires P to be constant on the free surface, theabove expression for P should be independent of time. This leads us to the dispersionrelation

kc2 + 2Ωc − g = 0. (21)

which amounts to the specification (6) for westward-propagating waves. Thus, (20)becomes, after choosing appropriately the value of the constant

P = ρg

(e2ξ

2k− r

)

+ P0 − ρg

(e2kr0

2k− r0

)

(22)

At any fixed latitude s, the free surface is the curve determined by setting r = r0in (5). This ensures the validity of (4).

An important aspect of the solution is that there is no variation in the meridionaldirection. This is in stark contrast to the equatorially trapped solutions (Constantin[17]), which were propagating eastwards. Westward propagating waves are possiblein our setting, but not in the setting of Constantin [17], precisely because there areno variations in the meridional direction. It is noted here that in β-plane westwardpropagating waves would present a growth in amplitude, increasing with the distancefrom the Equator. While mathematically their existence is not problematic, such waveform have to be neglected on physical ground as being unrealistic.

4 Discussion

In this section we perform a qualitative analysis of the flow pattern (5). A first observa-tion is that each particle moves in a vertical circle, with no meridional variation. Sincebeneath an irrotational water wave the particle paths are generally describing loopsthat are not closed, be that in finite or infinite depth, cf. the discussion in Constantin[23], Henry [24], Constantin and Strauss [25], we expect that the present flows havenon-zero vorticity. This observation is substantiated by the considerations in Sect. 4.1,while in Sect. 4.2 we discuss the issue of mass transport beneath the surface waves.

4.1 Vorticity

We calculate the vorticity of the flow prescribed by (5) explicitly. The explicit formof the matrix ∂(x, y, z)/∂(q, r, s) in (7) easily yields its inverse

⎛

⎜⎝

∂q∂x

∂s∂x

∂r∂x

∂q∂y

∂s∂y

∂r∂y

∂q∂z

∂s∂z

∂r∂z

⎞

⎟⎠ =

⎛

⎜⎝

1+eξ cos θ1−e2ξ

0eξ sin θ1−e2ξ

010

eξ sin θ1−e2ξ

01−eξ cos θ

1−e2ξ

⎞

⎟⎠ . (23)

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H.-C. Hsu

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+eξ cos θ1−e2ξ

0eξ sin θ1−e2ξ

010

eξ sin θ1−e2ξ

01−eξ cos θ

1−e2ξ

⎞

⎟⎠

⎛

⎝−kceξ sin θ 0 kceξ cos θ

0 0 0kceξ cos θ 0 kceξ sin θ

⎞

⎠

=⎛

⎜⎝

−kceξ sin θ1−e2ξ 0 kceξ cos θ+kce2ξ

1−e2ξ

0 0 0kceξ cos θ−kce2ξ

1−e2ξ 0 kceξ cos θ1−e2ξ

⎞

⎟⎠

(24)

and the vorticity is

ω = (wy − vz, uz − wx , vx − uy)

=(

0, − 2kce2ξ

1−e2ξ , 0)

.. (25)

Note that the middle component of ω is constant and always strictly positive, whilethe first and third component vanish.

4.2 Mass transport beneath the equatorial waves

The Lagrangian mean velocity vanishes:

〈u〉L = 1

T

T∫

0

u(q − ct, s, r)dt = 0 (26)

since the time average of the horizontal fluid velocity u, given by (11), over a waveperiod T = L/c is clearly zero. Note that the crest/trough levels of the surface wavethat propagates westward are z± = r0 ± 1

k ekr0 , while its mean level is

l = r0 + 1

L

L∫

0

1

kekr0 cos[k(q − ct)]dx

= r0 + 1

L

L∫

0

1

kekr0 cos[k(q − ct)](1 − ekr0 cos[k(q − ct)])dq

= r0 − 1

2ke2kr0 < 0,

(27)

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An exact solution for equatorial waves

as one can see by using in the second step the outcome of the differentiation of thex component of (5) with respect to the q-variable. A fixed depth z = z0 beneath thelevel z = z− of the surface wave troughs is characterized in Lagrangian variables bymeans of the relation

z0 = r + 1

keξ cos θ, (28)

which yields a functional dependence

r = α(q − ct, s, z0). (29)

By differentiating (28) with respect to the q variable we obtain

0 = αq + αqeξ cos θ − eξ sin θ, (30)

so that

αq = eξ sin θ

1 + eξ cos θ. (31)

Therefore the Eulerian mean velocity 〈u〉E (z0) at the fixed depth z0 can be expressedby means of

c + 〈u〉E (z0) = c

L

L/

c∫

0

[c + u(x − ct, s, z0)]dt = 1

L

L∫

0

[c + u(x − ct, s, z0)]dt

= 1

L

L∫

0

{c + u(q − ct, s, α(q − ct, s, z0))}∂x

∂qdq

= 1

L

L∫

0

{c + u(q − ct, s, α(q − ct, s, z0))} × (1 − eξ αq sin θ − eξ cos θ)dq

= 1

L

L∫

0

c(1 + eξ cos θ) × (1−eξ cos θ− e2ξ sin2θ

1+eξ cos θ)dq = 1

L

L∫

0

c(1 − e2ξ )dq,

(32)

due to (5), (11), (29) and (31). Consequently

〈u〉E (z0) = − c

L

L∫

0

e2kα(q,z0)dq ∈ (−c, 0), (33)

so that the Eulerian mean flow is eastward. Therefore the Stokes drift, defined byLonguet-Higgins [26] as the difference between the Lagrangian and the Eulerian mean

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H.-C. Hsu

velocities, is westward (the Lagrangian mean flow being zero). Differentiating (28)with respect to z0, we get

1 = αz0 + αz0 eξ cos θ, (34)

so that

αz0 = 1

1 + eξ cos θ> 0. (35)

This relation in combination with (33) leads to

∂z0〈u〉E (z0) = −2kc

L

L∫

0

e2ξ

1 + eξ cos θdq > 0. (36)

Therefore, while the Lagrangian mean velocity 〈u〉L vanishes, the Eulerian meanvelocity 〈u〉E depends monotonically on the depth, with the eastward flow of sig-nificance only in the near-surface region and barely noticeable at great depths. Thisindicates a non-uniform wave-induced eastward current.

Recall from Longuet-Higgins [26] that 〈u〉L is sometimes called the mass-transportvelocity, being the mean velocity of a marked particle. To further elucidate the flow(5), we investigate the mass transport past a fixed point at a fixed latitude. The massflux past x = x0, with x0 fixed, representing the instantaneous zonal transport acrossa fixed longitude s, is given by

m(x0 − ct) =η(x0−ct)∫

−∞u(x0 − ct, y, z)dz. (37)

From (11) and differentiating with respect to r the third component of (5), evaluatedat q = β(r, t, s) determined by the constraint

x0 = q − 1

kekr sin[k(q − ct)], (38)

we deduce that

m(x0 − ct) =r0∫

−∞ceξ cos θ(1 + eξ cos θ − βr eξ sin θ)dr . (39)

Differentiating (38) with respect to r yields

0 = βr − eξ sin θ − βr eξ cos θ, (40)

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An exact solution for equatorial waves

so that

βr = eξ sin θ

1 − eξ cos θ, (41)

and (39) becomes

m(x0 − ct) = c

r0∫

−∞eξ cos θ

1 − e2ξ

1 − eξ cos θdr . (42)

When the wave troughs/crests lie on the line x = x0, we have cos θ = ±1, andfrom (42) we infer that mass is carried backward/forward, respectively. However, thetime average of the mass flux over a wave period T is zero. This can be seen at oncefrom (42) by noticing that (38) yields

βt = − ceξ cos θ

1 − eξ cos θ(43)

and (42) takes on the form

m(x0 − ct) = −c

r0∫

−∞βt (1 − e2ξ )dr . (44)

for the T -periodic function t �→ β(r, t, s). Therefore, the time average of the massflux over a wave period T is zero. This was expected since 〈u〉L = 0.

Acknowledgments The author would like to acknowledge the insightful critiquing of the two referees.The author acknowledges the support of the International Wave Dynamics Research Center in Taiwan (NSC103-2911-I-006-302).

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