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Azimuthal equatorial capillary–gravity flows in spherical coordinates
Nonlinear Analysis: Real World Applications 36 (2017) 278–286
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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
Azimuthal equatorial capillary–gravity flows in spherical coordinates Hung-Chu Hsu a , Calin Iulian Martin b,∗ a
Tainan Hydraulics Laboratory, National Cheng Kung University, 5th F., No.500, Sec. 3, Anming Rd., Tainan 70995, Taiwan b Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
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info
Article history: Received 14 June 2016 Accepted 31 January 2017 Available online 21 February 2017 Keywords: Azimuthal flows Spherical coordinates Coriolis force Implicit function theorem
abstract We work in the setting of spherical coordinates to prove existence of free-surface capillary–gravity azimuthal equatorial flows which allow for variations in the vertical direction (so currents can be accommodated) but with no meridional flow. We perform the task by deriving an implicit equation, which determines the pressure at the surface if the free surface is known, and vice-versa (given the current profile). The solutions of this equation are found by using the implicit function theorem in a proper setting, which incorporates the observed symmetry about the Equator and takes advantage of the freedom in choosing the reference for the length of the Earth’s radius at the Equator. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Our motivation with this study lies in the necessity of providing analytical studies of the plethora of phenomena encountered in the Earth’s oceans. An impressive effort dedicated to the rigorous mathematical treatment of the flows occurring in the equatorial Pacific has began with the work of Constantin [1] on the modeling of such flows. Supplying the understanding of the dynamics of the typical flows, the same author derived in [2] and exact and explicit solution (in the Lagrangian framework) to the β-plane governing equations. The availability of sufficient field data, gathered over the last decades, is an invitation for in depth analyses of the peculiar flows occurring in band of about 2◦ latitude from the Equator. One aspect is a significant fluid stratification, giving rise to a pycnocline/thermocline that separates a layer of relatively warm water near the surface from a deeper layer of colder, and therefore denser water. A supplement to the oddity of the Equatorial Pacific is the presence of depth-dependent currents which alternate from a westward flow near ∗ Corresponding author. E-mail addresses: [email protected] (H.-C. Hsu), [email protected] (C.I. Martin).
http://dx.doi.org/10.1016/j.nonrwa.2017.01.014 1468-1218/© 2017 Elsevier Ltd. All rights reserved.
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279
the surface to a quite notable flow to the East, situated below the surface. Concerning the details, while in a near-surface layer, within about 150 km on each side of the Equator, there is, cf. [3–5], a westward current (driven by the prevailing trade winds), at depths of no more than 100 m, lies the Equatorial Undercurrent (EUC), which is an eastward flowing jet residing on the thermocline. According to [6–8], the EUC profoundly influences the geophysical ocean dynamics in the equatorial region, being made responsible for the formation of the El Ni˜ no and La Ni˜ na phenomena, which in turn are believed to have impact on climate change and related phenomena [9]. A remarkable characteristic of the EUC is its symmetry about the Equator, feature that we also incorporate in the equatorial flows that we consider here. The latter aspects constitute compelling reasons for a thorough understanding of equatorial flows from a multitude of perspectives ranging from equatorially trapped waves that decay exponentially away from the Equator [10,2,11,12], internal waves of large amplitude [13–15], issues pertaining to stability of geophysical flows [11,16–18]. Moreover, a number of explicit exact solutions have been derived for different physical settings [2,14,13,12,19]. As far as our paper is concerned, we aim here at exact solutions, in a spherical coordinate system, depicting flows that propagate only in the azimuthal direction driven by gravity and surface tension. Our study is stimulated by the recent works using cylindrical [20,21], and spherical coordinates [20,22]. Cylindrical and spherical coordinates are somewhat favored because of their compatibility with the (almost) spherical shape of the Earth. 2. The governing equations We introduce in this section the governing equations for an inviscid, incompressible fluid, written in spherical coordinates which are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions. Working in spherical coordinates (r, θ, ϕ), (where r is the distance from the center of the Earth, θ (with 0 ≤ θ ≤ π) is the polar angle, and ϕ (with 0 ≤ ϕ < 2π) is the azimuthal angle), denoting with (er , eθ , ez ) the corresponding orthonormal vectors and with u = uer + veθ + weϕ , the velocity field with respect to the orthonormal system (er , eθ , eϕ ), we are led, cf. [20], to the Euler’s equations of motion, which, written in a coordinate system with its origin at the center of the sphere, read as v w 1 1 ut + uur + uθ + uϕ − (v 2 + w2 ) = − pr + Fr r r sin θ r ρ v w 1 11 2 vt + uvr + vθ + vϕ + (uv − w cot θ) = − pθ + Fθ r r sin θ r ρr v w 1 1 1 wt + uwr + wθ + wϕ + (uw + vw cot θ) = − pϕ + Fϕ , r r sin θ r ρ r sin θ
(2.1)
(where p(r, θ, ϕ) denotes the pressure in the fluid and (Fr , Fθ , Fϕ ) is the body-force vector) and ρ denotes the constant pressure. The equation of mass conservation in spherical coordinates is 1 ∂ 2 1 ∂ 1 ∂w (r u) + (v sin θ) + = 0. 2 r ∂r r sin θ ∂θ r sin θ ∂ϕ
(2.2)
The governing equations (2.1) and (2.2) are supplemented by the boundary conditions as follows. At the free surface, given as r = R + h(θ, ϕ), where R ≈ 6378 km denotes the radius of the Earth, we impose the
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surface pressure condition p = P (θ, ϕ) + σ∇ · n,
(2.3)
(where σ is the coefficient of surface tension and n is the outward pointing unit normal vector) and the kinematic condition u=
w ∂h v ∂h + , r ∂θ r sin θ ∂ϕ
(2.4)
respectively. On the bottom of the ocean, which is an impermeable, solid boundary, described by the equation r = d(θ, ϕ), the kinematic condition is u=
v ∂d w ∂d + . r ∂θ r sin θ ∂ϕ
(2.5)
In order to encapsulate the effects of the Earth’s rotation we associate now (er , eθ , eϕ ) with a point fixed on the sphere which is rotating about its polar axis. Consequently, we need to add on the left column of (2.1) the Coriolis force 2Ω × u and the centripetal acceleration Ω × (Ω × r), with Ω = Ω (er cos θ − eθ sin θ), u = uer + veθ + weϕ , and r = rer , where Ω ≈ 7.29 × 10−5 rad s−1 is the constant rate of rotation of the Earth. The contribution of the Coriolis and centripetal acceleration is therefore equal to 2Ω (−w sin θ, −w cos θ, u sin θ + v cos θ) − rΩ 2 (sin2 θ, sin θ cos θ, 0).
(2.6)
3. Existence and analysis of special solutions We are concerned here with finding solution flows that are purely in the azimuthal direction and which do not vary in this direction. However, the azimuthal velocity component will prove to have an arbitrary variation with depth (i.e. radius). Therefore, this approach is suitable for the incorporation of a EUC. Moreover, we will aspire to find a formula for the pressure function (in the case of flow as described before) relating its variations at the free surface to variations of the shape of the free surface. By resorting to the implicit function theorem we will then perform an analysis of the link between the two mentioned quantities. The requirements about the particular flows we consider are encapsulated in the conditions u=v=0
everywhere,
(3.1)
and w = w(r, θ).
(3.2)
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The latter two conditions together with (2.1) and the terms from w2 − 2Ω w sin θ − rΩ 2 sin2 θ − r w2 cot θ − 2Ω w cos θ − rΩ 2 sin θ cos θ − r 0
281
(2.6) give rise to the system 1 = − pr − g, ρ 1 = − pθ , ρr 1 =− pϕ . ρr sin θ
(3.3)
We see at once that a flow satisfying (3.1) and (3.2) also verifies (identically) the equation of mass conservation (2.2). Elimination of p between the first two equations of (3.3) leads, cf. [20,22], to r sin θ 2 w (y) 1 dy p(r, θ) = A − ρgr + 2ρΩΨ (r sin θ) + ρr2 Ω 2 sin2 θ + ρ 2 y a sin θ π/2 −ρ w2 (a sin θ′ ) cot θ′ dθ′ , (3.4) θ
where A, a > 0 are constants and Ψ is a function with w(r, θ) = Ψ ′ (r sin θ) and which additionally satisfies Ψ (0) = 0. The independence of ϕ of the general exact solution from (3.4) allows us to write the free surface as r = R + h(θ). Consequently, we see that the kinematic boundary condition on the free surface (2.4) is automatically satisfied, as it is also the kinematic boundary condition (2.5) on the bottom, given as r = d(θ). We would like now to exploit the surface pressure condition (2.3). To this end we need to explicitate the term proportional to the surface tension coefficient in (2.3), task that we perform in the sequel. A normal vector to a surface given implicitly as H(r, θ, ϕ) = 0 is 1 1 N = Hr er + Hθ eθ + Hϕ eϕ . r r sin θ For H(r, θ, ϕ) := r − h(θ) − R a normal vector to the free surface is given as N = er −
hθ eθ . r
Consequently, the outward pointing unit normal is n=
r hθ er − eθ . 2 2 + hθ r + h2θ
r2
Therefore, the divergence of n is 1 1 1 ∂r (r2 nr ) + ∂θ (nθ sin θ) + ∂ϕ (nϕ ) 2 r r sin θ r sin θ 1 r3 1 −hθ sin θ = 2 ∂r + ∂θ r r sin θ r2 + h2θ r2 + h2θ
∇·n =
= =
2r2 + 3h2θ
r2 hθθ sin θ + r2 hθ cos θ + h3θ cos θ 1 3/2 − r sin θ · 3/2 r2 + h2θ r2 + h2θ 2(R + h(θ))2 + 3h2θ (R + h(θ))2 + h2θ
3/2
h3θ cos θ 1 (R + h(θ))2 hθθ sin θ + (R + h(θ))2 hθ cos θ − + 3/2 . 3/2 (R + h(θ)) sin θ (R + h(θ))2 + h2θ (R + h(θ))2 + h2θ
(3.5)
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Recalling now that the solution for p(r, θ) is 1 p(r, θ) = A − ρgr + 2ρΩΨ (r sin θ) + ρr2 Ω 2 sin2 θ + ρ 2 π/2 −ρ w2 (a sin θ′ ) cot θ′ dθ′ ,
r sin θ
a sin θ
w2 (y) dy y (3.6)
θ
and using the dynamic boundary condition p(r, θ) = P (θ) + σ(∇ · n)
on
r = R + h(θ),
we obtain 1 2 2 P (θ) = A − ρ[R + h(θ)] g − (R + h(θ))Ω sin θ 2 [R+h(θ)] sin θ 2 w (y) dy + 2ρΩΨ ([R + h(θ)] sin θ) + ρ y a sin θ π/2 −ρ w2 (a sin θ′ ) cot θ′ dθ′ − σ(∇ · n).
(3.7)
θ
We see at once that the pressure required to maintain the free surface undisturbed (and consequently following the curvature of the Earth) is given as 1 P0 (θ) = A − ρgR + ρR2 Ω 2 sin2 θ + 2ρΩΨ (R sin θ) 2 R sin θ 2 π/2 w (y) 2σ +ρ dy − ρ , w2 (a sin θ′ ) cot θ′ dθ′ − y R a sin θ θ
(3.8)
relation obtained by setting h ≡ 0 in (3.7). Assuming that the pressure at the Equator (given as θ = π2 ) equals the atmospheric pressure Pa then we have R 2 w (y) 2σ 1 dy − . (3.9) Pa = A − ρgR + ρR2 Ω 2 + 2ρΩΨ (R) + ρ 2 y R a Dividing relation (3.7) by Pa we obtain its non-dimensional form α − β[1 + h(θ)] + γ[1 + h(θ)]2 sin2 θ + δf [1 + h(θ)] sin θ [1+h(θ)]R sin θ 2 π/2 ρ w (y) ρ + dy − w2 (a sin θ′ ) cot θ′ dθ′ − P(θ) Pa a sin θ y Pa θ σ 2(1 + h)2 + 3h2θ (1 + h)(hθθ + hθ cot θ) σ h3θ cot θ − − + · 3/2 3/2 3/2 RPa RPa (1 + h)2 + h2θ (1 + h)2 + h2θ (1 + h) (1 + h)2 + h2θ = 0,
(3.10)
with the non-dimensional functions h(θ) , R P (θ) P(θ) := , Pa Ψ (Rs) f(s) := , Ψ (R) h(θ) :=
and the constants α, β, γ, δ > 0, defined by α=
A , Pa
β=
ρgR , Pa
γ=
ρR2 Ω 2 , 2Pa
δ=
2ρΩΨ (R) . Pa
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Note that Eq. (3.10) can be written as F(h, P) = 0,
(3.11)
with the functional F given by the left hand side of (3.10). This makes it amenable to a functional analytic study, once we identify suitable function spaces and a sufficiently simple but also physically meaningful solution to (3.11). To this end, note first that F(0, P0 ) = 0 where P0 is the pressure required to maintain the free surface undisturbed and is equal to 2σ P0 (θ) := α − β + γ sin2 θ + δf(sin θ) − RPatm R sin θ 2 π/2 ρ w (y) ρ w2 (a sin θ′ ) cot θ′ dθ′ . (3.12) + dy − Pa a sin θ y Pa θ It is also a matter of observation that F defines a continuously differentiable map π π π π π π C2 − ε, + ε × C − ε, + ε → C − ε, + ε , 2 2 2 2 2 2 where the spaces above are endowed with the usual supremum norms which make them into Banach spaces. To capture the symmetry of our problem in the definition of spaces we will further refine them in due course of the paper. In order to prove existence of non-trivial solutions to (3.11) we will rely on the implicit function theorem (see [23]). We will thus need to compute the derivative Dh F(0, P0 )(h) = lim
s→0
F(sh, P0 ) − F(0, P0 ) . s
(3.13)
To this end note that the derivative of the term [1+h(θ)]R sin θ 2 π/2 ρ ρ w (y) dy − w2 (a sin θ′ ) cot θ′ dθ′ Pa a sin θ y Pa θ at h ≡ 0 equals ρw2 (R sin θ) h(θ), Pa cf. [22]. Moreover, the derivative at h ≡ 0 of the term (1 + h)
(1 + h)2 + h2θ
3/2
equals lim
3/2 (1 + th)2 − (1 + th)2 + t2 h2θ t
t→0
= lim
3/2 (1 + th)3 − (1 + th)2 + t2 h2θ − th(1 + th)2 t
t→0
1 + th −
(1 + th)2 + t2 h2θ
= −h + lim t→0 t 2 × · lim (1 + th) + (1 + th) (1 + th)2 + t2 h2θ + (1 + th)2 + t2 h2θ t→0
= −h.
(3.14)
Altogether we have ρw2 (R sin θ) 2σ + h Dh F(0, P0 )(h) = −β + 2γ sin2 θ + δf′ (sin θ) sin θ + Pa RPa σ σ + hθ cot θ + hθθ . RPa RPa
(3.15)
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Remark 3.1. In order to reflect the symmetry with respect of the line of the Equator (given in spherical coordinates as θ = π2 ) of the flows under consideration we will work with functions h satisfying π π h(θ) = h(π − θ), for all θ ∈ − ε, + ε . 2 2 We will be moreover concerned only with solutions h satisfying h(π/2) = 0. This means that we will take the Earth’s radius to be the reference value. The features of the solutions as discussed above will be encapsulated in the definitions of the functions spaces we will work with. As such, we define Csym π/2 − ε, π/2 + ε (3.16) := h ∈ C π/2 − ε, π/2 + ε , h(θ) = h(π − θ) for all θ ∈ π/2 − ε, π/2 + ε . Moreover, we set 2 Csym,0 π/2 − ε, π/2 + ε := h ∈ C 2 π/2 − ε, π/2 + ε , h(θ) = h(π − θ) for all θ ∈ π/2 − ε, π/2 + ε and h(π/2) = 0 .
(3.17)
With the preceding preparations we are in the position to state one of the main results of this paper. Proposition 3.2. Dh F(0, P0 ) defines a map 2 Csym,0 π/2 − ε, π/2 + ε → Csym π/2 − ε, π/2 + ε , which is in fact a linear homeomorphism. Proof. Note that the mapping property holds true in view of the symmetry with respect to θ = π/2 of 2 functions in Csym,0 π/2 − ε, π/2 + ε . 2 To prove the injectivity of Dh F(0, P0 ) let q ∈ Csym,0 π/2 − ε, π/2 + ε be a solution of the homogeneous equation Dh F(0, P0 )q = 0. We then have that q = c1 q1 + c2 q2 ,
(3.18)
where c1 , c2 are real constants and {q1 , q2 } is a basis of fundamental solutions. The symmetry condition for q implies that q′ (π/2) = 0, which together with q(π/2) = 0 gives that c1 = c2 = 0. Thus, we infer from (3.18) that q = 0 and the injectivity is proved. Let now g ∈ Csym π/2 − ε, π/2 + ε . Then, cf. [24], there is r ∈ C 2 π/2 − ε, π/2 + ε such that Dh F(0, P0 )r = g. It is now easy to see that the function θ → r(π − θ),
θ ∈ π/2 − ε, π/2 + ε
is also a solution of the previous non homogeneous equation. Therefore, the function, θ→
r(θ) + r(π − θ) =: h(θ), 2
θ ∈ π/2 − ε, π/2 + ε
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is a solution of the non homogeneous equation satisfying h(θ) = h(π − θ) for all θ ∈ π/2 − ε, π/2 + ε . Moreover, h can be written as θ
θ W1 (t)g(t) W2 (t)g(t) dt + q2 (θ) dt W (q , q )(t) W (q1 , q2 )(t) 1 2 π/2 π/2 + c1 q1 (θ) + c2 q2 (θ), θ ∈ π/2 − ε, π/2 + ε ,
h(θ) = q1 (θ)
(3.19)
where {q1 , q2 } is a basis for the solutions of the homogeneous equation, W (q1 , q2 ) is the Wronskian of the basis {q1 , q2 }, Wk (k = 1, 2) is the determinant obtained from W by substituting the kth column (q1 q2 )t through (0 1)t , and c1 , c2 are constants. In the sequel we will determine the constants c1 , c2 by requiring the 2 solution h to belong to Csym,0 π/2 − ε, π/2 + ε . We notice that, the symmetry condition h(θ) = h(π − θ), implies h′ (π/2) = 0. Therefore, we have 0 = h′ (π/2) =
2 qk (π/2)Wk (π/2)g(π/2)
W (q1 , q2 )(π/2)
k=1
+ c1 q′1 (π/2) + c2 q′2 (π/2),
= c1 q′1 (π/2) + c2 q′2 (π/2),
(3.20)
where, the last line of the equality, follows from W1 = −q2
and W2 = q1 .
From the latter equality, from h(π/2) = 0 and using also the linear independence of {q1 , q2 } we infer that c1 = c2 = 0. Thus, the first order linear non-homogeneous equation Dh F(0, P0 )r = g 2 has a unique solution which belongs to Csym,0 π/2 − ε, π/2 + ε . Moreover, this unique solution can be represented as h(θ) =
2
θ
qj (θ)
π/2
j=1
Wj (s)g(s) ds, W (q1 , q2 )(s)
θ ∈ π/2 − ε, π/2 + ε .
(3.21)
θ ∈ π/2 − ε, π/2 + ε
(3.22)
Using also the equality q1 W1 + q2 W2 = 0 we find that ′
h (θ) =
2
q′j (θ)
j=1
θ
π/2
Wj (s)g(s) ds, W (q1 , q2 )(s)
and ′′
h (θ) =
2 j=1
q′′j (θ)
θ
π/2
q′j (θ)Wj (θ)g(θ) Wj (s)g(s) ds + , W (q1 , q2 )(s) W (q1 , q2 )(θ)
θ ∈ [π/2 − ε, π/2 + ε].
We conclude from (3.21)–(3.23) that Dh F(0, P0 )−1 is also continuous. This completes the proof. From the implicit function theorem [23], the previous proposition and since F(0, P0 ) = 0 we conclude the existence of genuine wave solutions to the equation F(h, P) = 0, fact that we state in the following theorem.
(3.23)
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Theorem 3.3. Given a sufficiently small deviation P of P0 there exists a unique h ∈ C[π/2 − ε, π/2 + ε] such that (3.10) holds. References [1] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett. 39 (2012) L05602. [2] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans 117 (2012) C05029. [3] R.S. Johnson, An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics, J. Nonlinear Math. Phys. 22 (4) (2015) 475–493. [4] G.C. Johnson, M.J. McPhaden, E. Firing, Equatorial Pacific ocean horizontal velocity, divergence, and upwelling, J. Phys. Oceanogr. 31 (2001) 839–849. [5] J.P. McCreary, Modeling equatorial ocean circulation, Annu. Rev. Fluid Mech. 17 (1985) 359–409. [6] A. Constantin, R.S. Johnson, The dynamics of waves interacting with the equatorial undercurrent, Geophys. Astrophys. Fluid Dyn. 109 (4) (2015) 311–358. [7] A.V. Fedorov, J.N. Brown, Equatorial waves, in: J. Steele (Ed.), Encyclopedia of Ocean Sciences, Academic Press, New York, 2009, pp. 3679–3695. [8] S. Philander, Equatorial waves in the presence of the equatorial undercurrent, J. Phys. Oceanogr. 9 (1979) 254–262. [9] T. Izumo, The equatorial undercurrent, meridional overturning circulation, and their roles in mass and heat exchanges during El Ni˜ no events in the tropical Pacific Ocean, 55, 2005, pp. 110–123. [10] A. Constantin, On equatorial wind waves, Differential Integral Equations 26 (3–4) (2013) 237–252. [11] A. Constantin, P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans 118 (2013) 2802–2810. [12] D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids 38 (2013) 18–21. [13] A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr. 43 (1) (2013) 165–175. [14] A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr. 44 (2) (2014) 781–789. [15] D. Henry, Internal equatorial water waves in the f-plane, J. Nonlinear Math. Phys. 22 (4) (2015) 499–506. [16] D. Henry, H.-C. Hsu, Instability of equatorial water waves in the f-plane, Discrete Contin. Dyn. Syst. 35 (3) (2015) 909–916. [17] D. Henry and, H.-C. Hsu, Instability of internal equatorial water waves, J. Differential Equations 258 (4) (2015) 1015–1024. [18] D. Ionescu-Kruse, Instability of edge waves along a sloping beach, J. Differential Equations 256 (12) (2014) 3999–4012. [19] A.-V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A 45 (2012) 10. 365501. [20] A. Constantin, R.S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr. 46 (2016) 1935–1945. [21] H.-C. Hsu, C.I. Martin, Free surface capillary–gravity azimuthal equatorial flows, Nonlinear Anal. 144 (2016) 1–9. [22] C.I. Martin, On the existence of free-surface azimuthal equatorial flows, Appl. Anal. (in press). http://dx.doi.org/10.1080/ 00036811.2016.1180370. [23] M.S. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977. [24] E.A. Coddington, An Introduction to Ordinary Differential Equations, Dover, New York, 1961.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
Azimuthal equatorial capillary–gravity flows in spherical coordinates Hung-Chu Hsu a , Calin Iulian Martin b,∗ a
Tainan Hydraulics Laboratory, National Cheng Kung University, 5th F., No.500, Sec. 3, Anming Rd., Tainan 70995, Taiwan b Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
article
info
Article history: Received 14 June 2016 Accepted 31 January 2017 Available online 21 February 2017 Keywords: Azimuthal flows Spherical coordinates Coriolis force Implicit function theorem
abstract We work in the setting of spherical coordinates to prove existence of free-surface capillary–gravity azimuthal equatorial flows which allow for variations in the vertical direction (so currents can be accommodated) but with no meridional flow. We perform the task by deriving an implicit equation, which determines the pressure at the surface if the free surface is known, and vice-versa (given the current profile). The solutions of this equation are found by using the implicit function theorem in a proper setting, which incorporates the observed symmetry about the Equator and takes advantage of the freedom in choosing the reference for the length of the Earth’s radius at the Equator. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Our motivation with this study lies in the necessity of providing analytical studies of the plethora of phenomena encountered in the Earth’s oceans. An impressive effort dedicated to the rigorous mathematical treatment of the flows occurring in the equatorial Pacific has began with the work of Constantin [1] on the modeling of such flows. Supplying the understanding of the dynamics of the typical flows, the same author derived in [2] and exact and explicit solution (in the Lagrangian framework) to the β-plane governing equations. The availability of sufficient field data, gathered over the last decades, is an invitation for in depth analyses of the peculiar flows occurring in band of about 2◦ latitude from the Equator. One aspect is a significant fluid stratification, giving rise to a pycnocline/thermocline that separates a layer of relatively warm water near the surface from a deeper layer of colder, and therefore denser water. A supplement to the oddity of the Equatorial Pacific is the presence of depth-dependent currents which alternate from a westward flow near ∗ Corresponding author. E-mail addresses: [email protected] (H.-C. Hsu), [email protected] (C.I. Martin).
http://dx.doi.org/10.1016/j.nonrwa.2017.01.014 1468-1218/© 2017 Elsevier Ltd. All rights reserved.
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the surface to a quite notable flow to the East, situated below the surface. Concerning the details, while in a near-surface layer, within about 150 km on each side of the Equator, there is, cf. [3–5], a westward current (driven by the prevailing trade winds), at depths of no more than 100 m, lies the Equatorial Undercurrent (EUC), which is an eastward flowing jet residing on the thermocline. According to [6–8], the EUC profoundly influences the geophysical ocean dynamics in the equatorial region, being made responsible for the formation of the El Ni˜ no and La Ni˜ na phenomena, which in turn are believed to have impact on climate change and related phenomena [9]. A remarkable characteristic of the EUC is its symmetry about the Equator, feature that we also incorporate in the equatorial flows that we consider here. The latter aspects constitute compelling reasons for a thorough understanding of equatorial flows from a multitude of perspectives ranging from equatorially trapped waves that decay exponentially away from the Equator [10,2,11,12], internal waves of large amplitude [13–15], issues pertaining to stability of geophysical flows [11,16–18]. Moreover, a number of explicit exact solutions have been derived for different physical settings [2,14,13,12,19]. As far as our paper is concerned, we aim here at exact solutions, in a spherical coordinate system, depicting flows that propagate only in the azimuthal direction driven by gravity and surface tension. Our study is stimulated by the recent works using cylindrical [20,21], and spherical coordinates [20,22]. Cylindrical and spherical coordinates are somewhat favored because of their compatibility with the (almost) spherical shape of the Earth. 2. The governing equations We introduce in this section the governing equations for an inviscid, incompressible fluid, written in spherical coordinates which are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions. Working in spherical coordinates (r, θ, ϕ), (where r is the distance from the center of the Earth, θ (with 0 ≤ θ ≤ π) is the polar angle, and ϕ (with 0 ≤ ϕ < 2π) is the azimuthal angle), denoting with (er , eθ , ez ) the corresponding orthonormal vectors and with u = uer + veθ + weϕ , the velocity field with respect to the orthonormal system (er , eθ , eϕ ), we are led, cf. [20], to the Euler’s equations of motion, which, written in a coordinate system with its origin at the center of the sphere, read as v w 1 1 ut + uur + uθ + uϕ − (v 2 + w2 ) = − pr + Fr r r sin θ r ρ v w 1 11 2 vt + uvr + vθ + vϕ + (uv − w cot θ) = − pθ + Fθ r r sin θ r ρr v w 1 1 1 wt + uwr + wθ + wϕ + (uw + vw cot θ) = − pϕ + Fϕ , r r sin θ r ρ r sin θ
(2.1)
(where p(r, θ, ϕ) denotes the pressure in the fluid and (Fr , Fθ , Fϕ ) is the body-force vector) and ρ denotes the constant pressure. The equation of mass conservation in spherical coordinates is 1 ∂ 2 1 ∂ 1 ∂w (r u) + (v sin θ) + = 0. 2 r ∂r r sin θ ∂θ r sin θ ∂ϕ
(2.2)
The governing equations (2.1) and (2.2) are supplemented by the boundary conditions as follows. At the free surface, given as r = R + h(θ, ϕ), where R ≈ 6378 km denotes the radius of the Earth, we impose the
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surface pressure condition p = P (θ, ϕ) + σ∇ · n,
(2.3)
(where σ is the coefficient of surface tension and n is the outward pointing unit normal vector) and the kinematic condition u=
w ∂h v ∂h + , r ∂θ r sin θ ∂ϕ
(2.4)
respectively. On the bottom of the ocean, which is an impermeable, solid boundary, described by the equation r = d(θ, ϕ), the kinematic condition is u=
v ∂d w ∂d + . r ∂θ r sin θ ∂ϕ
(2.5)
In order to encapsulate the effects of the Earth’s rotation we associate now (er , eθ , eϕ ) with a point fixed on the sphere which is rotating about its polar axis. Consequently, we need to add on the left column of (2.1) the Coriolis force 2Ω × u and the centripetal acceleration Ω × (Ω × r), with Ω = Ω (er cos θ − eθ sin θ), u = uer + veθ + weϕ , and r = rer , where Ω ≈ 7.29 × 10−5 rad s−1 is the constant rate of rotation of the Earth. The contribution of the Coriolis and centripetal acceleration is therefore equal to 2Ω (−w sin θ, −w cos θ, u sin θ + v cos θ) − rΩ 2 (sin2 θ, sin θ cos θ, 0).
(2.6)
3. Existence and analysis of special solutions We are concerned here with finding solution flows that are purely in the azimuthal direction and which do not vary in this direction. However, the azimuthal velocity component will prove to have an arbitrary variation with depth (i.e. radius). Therefore, this approach is suitable for the incorporation of a EUC. Moreover, we will aspire to find a formula for the pressure function (in the case of flow as described before) relating its variations at the free surface to variations of the shape of the free surface. By resorting to the implicit function theorem we will then perform an analysis of the link between the two mentioned quantities. The requirements about the particular flows we consider are encapsulated in the conditions u=v=0
everywhere,
(3.1)
and w = w(r, θ).
(3.2)
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The latter two conditions together with (2.1) and the terms from w2 − 2Ω w sin θ − rΩ 2 sin2 θ − r w2 cot θ − 2Ω w cos θ − rΩ 2 sin θ cos θ − r 0
281
(2.6) give rise to the system 1 = − pr − g, ρ 1 = − pθ , ρr 1 =− pϕ . ρr sin θ
(3.3)
We see at once that a flow satisfying (3.1) and (3.2) also verifies (identically) the equation of mass conservation (2.2). Elimination of p between the first two equations of (3.3) leads, cf. [20,22], to r sin θ 2 w (y) 1 dy p(r, θ) = A − ρgr + 2ρΩΨ (r sin θ) + ρr2 Ω 2 sin2 θ + ρ 2 y a sin θ π/2 −ρ w2 (a sin θ′ ) cot θ′ dθ′ , (3.4) θ
where A, a > 0 are constants and Ψ is a function with w(r, θ) = Ψ ′ (r sin θ) and which additionally satisfies Ψ (0) = 0. The independence of ϕ of the general exact solution from (3.4) allows us to write the free surface as r = R + h(θ). Consequently, we see that the kinematic boundary condition on the free surface (2.4) is automatically satisfied, as it is also the kinematic boundary condition (2.5) on the bottom, given as r = d(θ). We would like now to exploit the surface pressure condition (2.3). To this end we need to explicitate the term proportional to the surface tension coefficient in (2.3), task that we perform in the sequel. A normal vector to a surface given implicitly as H(r, θ, ϕ) = 0 is 1 1 N = Hr er + Hθ eθ + Hϕ eϕ . r r sin θ For H(r, θ, ϕ) := r − h(θ) − R a normal vector to the free surface is given as N = er −
hθ eθ . r
Consequently, the outward pointing unit normal is n=
r hθ er − eθ . 2 2 + hθ r + h2θ
r2
Therefore, the divergence of n is 1 1 1 ∂r (r2 nr ) + ∂θ (nθ sin θ) + ∂ϕ (nϕ ) 2 r r sin θ r sin θ 1 r3 1 −hθ sin θ = 2 ∂r + ∂θ r r sin θ r2 + h2θ r2 + h2θ
∇·n =
= =
2r2 + 3h2θ
r2 hθθ sin θ + r2 hθ cos θ + h3θ cos θ 1 3/2 − r sin θ · 3/2 r2 + h2θ r2 + h2θ 2(R + h(θ))2 + 3h2θ (R + h(θ))2 + h2θ
3/2
h3θ cos θ 1 (R + h(θ))2 hθθ sin θ + (R + h(θ))2 hθ cos θ − + 3/2 . 3/2 (R + h(θ)) sin θ (R + h(θ))2 + h2θ (R + h(θ))2 + h2θ
(3.5)
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Recalling now that the solution for p(r, θ) is 1 p(r, θ) = A − ρgr + 2ρΩΨ (r sin θ) + ρr2 Ω 2 sin2 θ + ρ 2 π/2 −ρ w2 (a sin θ′ ) cot θ′ dθ′ ,
r sin θ
a sin θ
w2 (y) dy y (3.6)
θ
and using the dynamic boundary condition p(r, θ) = P (θ) + σ(∇ · n)
on
r = R + h(θ),
we obtain 1 2 2 P (θ) = A − ρ[R + h(θ)] g − (R + h(θ))Ω sin θ 2 [R+h(θ)] sin θ 2 w (y) dy + 2ρΩΨ ([R + h(θ)] sin θ) + ρ y a sin θ π/2 −ρ w2 (a sin θ′ ) cot θ′ dθ′ − σ(∇ · n).
(3.7)
θ
We see at once that the pressure required to maintain the free surface undisturbed (and consequently following the curvature of the Earth) is given as 1 P0 (θ) = A − ρgR + ρR2 Ω 2 sin2 θ + 2ρΩΨ (R sin θ) 2 R sin θ 2 π/2 w (y) 2σ +ρ dy − ρ , w2 (a sin θ′ ) cot θ′ dθ′ − y R a sin θ θ
(3.8)
relation obtained by setting h ≡ 0 in (3.7). Assuming that the pressure at the Equator (given as θ = π2 ) equals the atmospheric pressure Pa then we have R 2 w (y) 2σ 1 dy − . (3.9) Pa = A − ρgR + ρR2 Ω 2 + 2ρΩΨ (R) + ρ 2 y R a Dividing relation (3.7) by Pa we obtain its non-dimensional form α − β[1 + h(θ)] + γ[1 + h(θ)]2 sin2 θ + δf [1 + h(θ)] sin θ [1+h(θ)]R sin θ 2 π/2 ρ w (y) ρ + dy − w2 (a sin θ′ ) cot θ′ dθ′ − P(θ) Pa a sin θ y Pa θ σ 2(1 + h)2 + 3h2θ (1 + h)(hθθ + hθ cot θ) σ h3θ cot θ − − + · 3/2 3/2 3/2 RPa RPa (1 + h)2 + h2θ (1 + h)2 + h2θ (1 + h) (1 + h)2 + h2θ = 0,
(3.10)
with the non-dimensional functions h(θ) , R P (θ) P(θ) := , Pa Ψ (Rs) f(s) := , Ψ (R) h(θ) :=
and the constants α, β, γ, δ > 0, defined by α=
A , Pa
β=
ρgR , Pa
γ=
ρR2 Ω 2 , 2Pa
δ=
2ρΩΨ (R) . Pa
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Note that Eq. (3.10) can be written as F(h, P) = 0,
(3.11)
with the functional F given by the left hand side of (3.10). This makes it amenable to a functional analytic study, once we identify suitable function spaces and a sufficiently simple but also physically meaningful solution to (3.11). To this end, note first that F(0, P0 ) = 0 where P0 is the pressure required to maintain the free surface undisturbed and is equal to 2σ P0 (θ) := α − β + γ sin2 θ + δf(sin θ) − RPatm R sin θ 2 π/2 ρ w (y) ρ w2 (a sin θ′ ) cot θ′ dθ′ . (3.12) + dy − Pa a sin θ y Pa θ It is also a matter of observation that F defines a continuously differentiable map π π π π π π C2 − ε, + ε × C − ε, + ε → C − ε, + ε , 2 2 2 2 2 2 where the spaces above are endowed with the usual supremum norms which make them into Banach spaces. To capture the symmetry of our problem in the definition of spaces we will further refine them in due course of the paper. In order to prove existence of non-trivial solutions to (3.11) we will rely on the implicit function theorem (see [23]). We will thus need to compute the derivative Dh F(0, P0 )(h) = lim
s→0
F(sh, P0 ) − F(0, P0 ) . s
(3.13)
To this end note that the derivative of the term [1+h(θ)]R sin θ 2 π/2 ρ ρ w (y) dy − w2 (a sin θ′ ) cot θ′ dθ′ Pa a sin θ y Pa θ at h ≡ 0 equals ρw2 (R sin θ) h(θ), Pa cf. [22]. Moreover, the derivative at h ≡ 0 of the term (1 + h)
(1 + h)2 + h2θ
3/2
equals lim
3/2 (1 + th)2 − (1 + th)2 + t2 h2θ t
t→0
= lim
3/2 (1 + th)3 − (1 + th)2 + t2 h2θ − th(1 + th)2 t
t→0
1 + th −
(1 + th)2 + t2 h2θ
= −h + lim t→0 t 2 × · lim (1 + th) + (1 + th) (1 + th)2 + t2 h2θ + (1 + th)2 + t2 h2θ t→0
= −h.
(3.14)
Altogether we have ρw2 (R sin θ) 2σ + h Dh F(0, P0 )(h) = −β + 2γ sin2 θ + δf′ (sin θ) sin θ + Pa RPa σ σ + hθ cot θ + hθθ . RPa RPa
(3.15)
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Remark 3.1. In order to reflect the symmetry with respect of the line of the Equator (given in spherical coordinates as θ = π2 ) of the flows under consideration we will work with functions h satisfying π π h(θ) = h(π − θ), for all θ ∈ − ε, + ε . 2 2 We will be moreover concerned only with solutions h satisfying h(π/2) = 0. This means that we will take the Earth’s radius to be the reference value. The features of the solutions as discussed above will be encapsulated in the definitions of the functions spaces we will work with. As such, we define Csym π/2 − ε, π/2 + ε (3.16) := h ∈ C π/2 − ε, π/2 + ε , h(θ) = h(π − θ) for all θ ∈ π/2 − ε, π/2 + ε . Moreover, we set 2 Csym,0 π/2 − ε, π/2 + ε := h ∈ C 2 π/2 − ε, π/2 + ε , h(θ) = h(π − θ) for all θ ∈ π/2 − ε, π/2 + ε and h(π/2) = 0 .
(3.17)
With the preceding preparations we are in the position to state one of the main results of this paper. Proposition 3.2. Dh F(0, P0 ) defines a map 2 Csym,0 π/2 − ε, π/2 + ε → Csym π/2 − ε, π/2 + ε , which is in fact a linear homeomorphism. Proof. Note that the mapping property holds true in view of the symmetry with respect to θ = π/2 of 2 functions in Csym,0 π/2 − ε, π/2 + ε . 2 To prove the injectivity of Dh F(0, P0 ) let q ∈ Csym,0 π/2 − ε, π/2 + ε be a solution of the homogeneous equation Dh F(0, P0 )q = 0. We then have that q = c1 q1 + c2 q2 ,
(3.18)
where c1 , c2 are real constants and {q1 , q2 } is a basis of fundamental solutions. The symmetry condition for q implies that q′ (π/2) = 0, which together with q(π/2) = 0 gives that c1 = c2 = 0. Thus, we infer from (3.18) that q = 0 and the injectivity is proved. Let now g ∈ Csym π/2 − ε, π/2 + ε . Then, cf. [24], there is r ∈ C 2 π/2 − ε, π/2 + ε such that Dh F(0, P0 )r = g. It is now easy to see that the function θ → r(π − θ),
θ ∈ π/2 − ε, π/2 + ε
is also a solution of the previous non homogeneous equation. Therefore, the function, θ→
r(θ) + r(π − θ) =: h(θ), 2
θ ∈ π/2 − ε, π/2 + ε
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is a solution of the non homogeneous equation satisfying h(θ) = h(π − θ) for all θ ∈ π/2 − ε, π/2 + ε . Moreover, h can be written as θ
θ W1 (t)g(t) W2 (t)g(t) dt + q2 (θ) dt W (q , q )(t) W (q1 , q2 )(t) 1 2 π/2 π/2 + c1 q1 (θ) + c2 q2 (θ), θ ∈ π/2 − ε, π/2 + ε ,
h(θ) = q1 (θ)
(3.19)
where {q1 , q2 } is a basis for the solutions of the homogeneous equation, W (q1 , q2 ) is the Wronskian of the basis {q1 , q2 }, Wk (k = 1, 2) is the determinant obtained from W by substituting the kth column (q1 q2 )t through (0 1)t , and c1 , c2 are constants. In the sequel we will determine the constants c1 , c2 by requiring the 2 solution h to belong to Csym,0 π/2 − ε, π/2 + ε . We notice that, the symmetry condition h(θ) = h(π − θ), implies h′ (π/2) = 0. Therefore, we have 0 = h′ (π/2) =
2 qk (π/2)Wk (π/2)g(π/2)
W (q1 , q2 )(π/2)
k=1
+ c1 q′1 (π/2) + c2 q′2 (π/2),
= c1 q′1 (π/2) + c2 q′2 (π/2),
(3.20)
where, the last line of the equality, follows from W1 = −q2
and W2 = q1 .
From the latter equality, from h(π/2) = 0 and using also the linear independence of {q1 , q2 } we infer that c1 = c2 = 0. Thus, the first order linear non-homogeneous equation Dh F(0, P0 )r = g 2 has a unique solution which belongs to Csym,0 π/2 − ε, π/2 + ε . Moreover, this unique solution can be represented as h(θ) =
2
θ
qj (θ)
π/2
j=1
Wj (s)g(s) ds, W (q1 , q2 )(s)
θ ∈ π/2 − ε, π/2 + ε .
(3.21)
θ ∈ π/2 − ε, π/2 + ε
(3.22)
Using also the equality q1 W1 + q2 W2 = 0 we find that ′
h (θ) =
2
q′j (θ)
j=1
θ
π/2
Wj (s)g(s) ds, W (q1 , q2 )(s)
and ′′
h (θ) =
2 j=1
q′′j (θ)
θ
π/2
q′j (θ)Wj (θ)g(θ) Wj (s)g(s) ds + , W (q1 , q2 )(s) W (q1 , q2 )(θ)
θ ∈ [π/2 − ε, π/2 + ε].
We conclude from (3.21)–(3.23) that Dh F(0, P0 )−1 is also continuous. This completes the proof. From the implicit function theorem [23], the previous proposition and since F(0, P0 ) = 0 we conclude the existence of genuine wave solutions to the equation F(h, P) = 0, fact that we state in the following theorem.
(3.23)
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Theorem 3.3. Given a sufficiently small deviation P of P0 there exists a unique h ∈ C[π/2 − ε, π/2 + ε] such that (3.10) holds. References [1] A. Constantin, On the modelling of equatorial waves, Geophys. Res. Lett. 39 (2012) L05602. [2] A. Constantin, An exact solution for equatorially trapped waves, J. Geophys. Res.: Oceans 117 (2012) C05029. [3] R.S. Johnson, An ocean undercurrent, a thermocline, a free surface, with waves: a problem in classical fluid mechanics, J. Nonlinear Math. Phys. 22 (4) (2015) 475–493. [4] G.C. Johnson, M.J. McPhaden, E. Firing, Equatorial Pacific ocean horizontal velocity, divergence, and upwelling, J. Phys. Oceanogr. 31 (2001) 839–849. [5] J.P. McCreary, Modeling equatorial ocean circulation, Annu. Rev. Fluid Mech. 17 (1985) 359–409. [6] A. Constantin, R.S. Johnson, The dynamics of waves interacting with the equatorial undercurrent, Geophys. Astrophys. Fluid Dyn. 109 (4) (2015) 311–358. [7] A.V. Fedorov, J.N. Brown, Equatorial waves, in: J. Steele (Ed.), Encyclopedia of Ocean Sciences, Academic Press, New York, 2009, pp. 3679–3695. [8] S. Philander, Equatorial waves in the presence of the equatorial undercurrent, J. Phys. Oceanogr. 9 (1979) 254–262. [9] T. Izumo, The equatorial undercurrent, meridional overturning circulation, and their roles in mass and heat exchanges during El Ni˜ no events in the tropical Pacific Ocean, 55, 2005, pp. 110–123. [10] A. Constantin, On equatorial wind waves, Differential Integral Equations 26 (3–4) (2013) 237–252. [11] A. Constantin, P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans 118 (2013) 2802–2810. [12] D. Henry, An exact solution for equatorial geophysical water waves with an underlying current, Eur. J. Mech. B Fluids 38 (2013) 18–21. [13] A. Constantin, Some three-dimensional nonlinear equatorial flows, J. Phys. Oceanogr. 43 (1) (2013) 165–175. [14] A. Constantin, Some nonlinear, equatorially trapped, nonhydrostatic internal geophysical waves, J. Phys. Oceanogr. 44 (2) (2014) 781–789. [15] D. Henry, Internal equatorial water waves in the f-plane, J. Nonlinear Math. Phys. 22 (4) (2015) 499–506. [16] D. Henry, H.-C. Hsu, Instability of equatorial water waves in the f-plane, Discrete Contin. Dyn. Syst. 35 (3) (2015) 909–916. [17] D. Henry and, H.-C. Hsu, Instability of internal equatorial water waves, J. Differential Equations 258 (4) (2015) 1015–1024. [18] D. Ionescu-Kruse, Instability of edge waves along a sloping beach, J. Differential Equations 256 (12) (2014) 3999–4012. [19] A.-V. Matioc, An exact solution for geophysical equatorial edge waves over a sloping beach, J. Phys. A 45 (2012) 10. 365501. [20] A. Constantin, R.S. Johnson, An exact, steady, purely azimuthal equatorial flow with a free surface, J. Phys. Oceanogr. 46 (2016) 1935–1945. [21] H.-C. Hsu, C.I. Martin, Free surface capillary–gravity azimuthal equatorial flows, Nonlinear Anal. 144 (2016) 1–9. [22] C.I. Martin, On the existence of free-surface azimuthal equatorial flows, Appl. Anal. (in press). http://dx.doi.org/10.1080/ 00036811.2016.1180370. [23] M.S. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977. [24] E.A. Coddington, An Introduction to Ordinary Differential Equations, Dover, New York, 1961.