a note on the symmetric and antisymmetric constituents of weakly nonlinear solutions of a classical...

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Eur. Phys. J. Plus (2014) 129: 142 DOI 10.1140/epjp/i2014-14142-y

A note on the symmetric and antisymmetricconstituents of weakly nonlinear solutions of aclassical wind-driven ocean circulation model

Fulvio Crisciani and Gualtiero Badin

DOI 10.1140/epjp/i2014-14142-y

Regular Article

Eur. Phys. J. Plus (2014) 129: 142 THE EUROPEANPHYSICAL JOURNAL PLUS

A note on the symmetric and antisymmetric constituents ofweakly nonlinear solutions of a classical wind-driven oceancirculation model

Fulvio Crisciani1 and Gualtiero Badin2,a

1 Department of Physics, University of Trieste, Trieste, Italy2 Institute of Oceanography, University of Hamburg, Hamburg, Germany

Received: 2 April 2014 / Revised: 20 May 2014

Published online: 2 July 2014 – c© Societa Italiana di Fisica / Springer-Verlag 2014

Abstract. A classical model of wind-driven ocean circulation is studied in the weakly nonlinear approx-imation. An asymptotic expansion for small Rossby number is applied to the separate symmetric andasymmetric components of the stream function, where the symmetry refers to a north-south reflectiontransformation. The asymptotic expansion allows for the formulation of a coupled set of nonlinear partialdifferential equations for the two components. Results show that the asymmetric component is responsi-ble for the formation of steady cyclones and anticyclones that cause the deformation of the total streamfunction of the system. Higher-order components of the stream function in the asymptotic expansion areforced by an effective wind stress arising from lower-order entries in the Jacobian term, and these effectivestresses act only to redistribute vorticity.

1 Introduction

Western intensified wind-driven currents are an ubiquitous feature of the World oceans. Using a vertically homogeneousmodel in a square domain with flat bottom, Stommel [1] was the first to ascribe the western intensification to themeridional gradient of the planetary vorticity. The analytical model by Stommel was however based on the assumptionof linearity of the system, which results in a North-South symmetry of the solution that was unable to explain thenorthward intensification of the Gulf Stream, which is responsible for the propagation of the current toward Europe.

The North-South symmetry of the solution can be broken through the inclusion of the nonlinear advection ofvorticity. As summarized in, e.g., [2] the effect of the nonlinearity can give rise to different dynamical regimes, extendingfrom the weakly nonlinear regime, characterized by a western instensification of the current and with solution thatcan be expressed as a sum of a boundary layer and an inertial contribution, till a full inertial solution, characterizedby the formation of Fofonoff gyres.

The pioneering work of Veronis [3,4] was the first one to point out the role of the nonlinear advection of vorticityin deforming the shape of the streamlines in the classical models of wind-driven ocean circulation with bottom stressdissipation retaining both a North-South and a East-West asymmetry. Subsequently, first [5] and then [6] analyzed therole of the relative importance of bottom and lateral stress. The first application to asymmetric boundary conditionand to domains with irregular shape is due to [7].

One of the consequences of the nonlinearities of the vorticity equation is to break the property of symmetry reflectionof the system. This particular symmetry was first studied for the Niiler model by Crisciani and Ozgokmen [8], whoshowed that the solution can be approximated by the superposition of three components, each having definite symmetryproperties under North-South and East-West reflections. The North-South symmetry for the weakly nonlinear regimewas instead studied in [9] and [10] to prove, respectively, the symmetry/asymmetry of the linear Munk’s model whensymmetric or asymmetric boundary conditions are applied.

The aim of this paper is to explore the mathematical property of the mirror symmetry of the weakly nonlinearregime in order to study the role of the advection term to deflect the streamlines of the flow in the Boening’s modelof wind-driven ocean circulation [6].

a e-mail: [email protected]

of 12 Eur. Phys. J. Plus (2014) 129: 142

2 Boening’s model in a weakly nonlinear regime

Boening’s model [6] is the basis of an extended numerical investigation on the influence of frictional parameterization inwind-driven circulation of a constant-density ocean. Its non-dimensional formulation resorts to the vorticity equation,

∂

∂t∇2ψ + RJ(ψ,∇2ψ) +

∂ψ

∂x= − sin(y) + εL∇

4ψ − εB∇2ψ, (1)

governing the fluid motion in the square fluid domain,

D = [0 ≤ x ≤ π] × [0 ≤ y ≤ π], (2)

with the boundary conditionsψ = 0 and ∇2ψ = 0, (3)

along ∂D. Equation (1) yields the evolution of the geostrophic current,

u = k ×∇ψ, (4)

subject to the steady, anticyclonic single-gyre wind stress

τ = − cos(y)i, (5)

under the dissipative influence of lateral diffusion of relative vorticity, i.e. εL∇2(∇2ψ) = εL∇

4ψ, and bottom friction,i.e. −εB∇2ψ. As shown by the author by means of many numerical experiments, the model solutions are criticallydependent on the Rossby number R, the horizontal Ekman number εL and the vertical Ekman number εB ; in particular,hereafter we will concern with the so-called “experiment 3”, in which the steady model solution follows from the choice

R = 2.33 × 10−3, εL = 9.00 × 10−4, εB = 0. (6)

We stress that steady circulation demands that at least one of Ekman numbers be positive. The values of R and of εL

reported in (6) satisfy the condition for the formation of a weakly nonlinear regime, the latter being defined in generalby the relationship

R1/2 < ε1/3L . (7)

The flow pattern resulting from (6) is reported in fig. 4 of the original paper by Boening [6] and in fig. 1(a); in bothfigures, the most striking feature appears in the N-W corner of the domain, where the westward intensification tendsto extend as an eastward jet before meandering towards a Sverdrup-like interior. This phenomenology lies betweenthat of the strictly linear model, in which neither the eastward jet nor the meandering take place, and that of markedlynonlinear models, in which an intense recirculation flow entirely confined into the N-W corner of the domain is observed.Owing to the link between nonlinearity and antisymmetry in the context of problem (1)–(3), together with the factthat weakly nonlinear regimes are amenable of a partially analytic treatment, the phenomenology of the circulationassociated to regime (6) is explained from a new point of view.

3 Symmetric and antisymmetric components of a perturbative expansion of the modelsolution

We anticipate the following result (the proof is reported in the appendix). With reference to the steady vorticityequation,

RJ(ψ,∇2ψ) +∂ψ

∂x= curlzτ(x, y) + εL∇

4ψ − εB∇2ψ, (8)

the fluid domain (2) and boundary conditions (3), assume

curlz τ(x, y) = curlzτ (x, π − y). (9)

ThenR = 0 ⇔ ψ(x, y) = ψ(x, π − y). (10)

Note that the wind-stress curl of (1) is a special case of (9). Relationship (10) suggests the possibility to investigatenonlinearity by exploring the behavior of the model solution under the mirror reflection with respect to the mid-basinlatitude, that is (x, y) → (x, π − y). Based on (10), the defect of symmetry ψa of ψ can be usefully introduced setting

ψa(x, y) = [ψ(x, y) − ψ(x, π − y)]/2.

Eur. Phys. J. Plus (2014) 129: 142 Page 3 of 12

(a) {, 2R1/2

=gL

1/3

0 pi/2 pi0

pi/2

pi

(d) {s

(0)+R{

a

(1), 2R

1/2=g

L

1/3

0 pi/2 pi0

pi/2

pi

(b) {a, 2R

1/2=g

L

1/3

0 pi/2 pi0

pi/2

pi

(e) R{a

(1), 2R

1/2=g

L

1/3

0 pi/2 pi0

pi/2

pi

(c) {s, 2R

1/2=g

L

1/3

0 pi/2 pi0

pi/2

pi

(f) {s

(0), 2R

1/2=g

L

1/3

0 pi/2 pi0

pi/2

pi

Fig. 1. Streamlines for (a) the total model stream function ψ, (b) ψA = (ψ(x, y)−ψ(x, π−y))/2, (c) ψS = (ψ(x, y)+ψ(x, π−y))/2,

(d) ψ(0)S + Rψ

(1)A , (e) Rψ

(1)A and (f) ψ

(0)S for Boening’s model with 2R1/2 = ε

1/3L . Full and dashed lines represent positive and

negative values, respectively.

of 12 Eur. Phys. J. Plus (2014) 129: 142

Thus, R �= 0 ⇔ ψa �= 0, so ψa is the antisymmetric component of ψ that breaks the N-S symmetry because ofnonlinearity. On the other hand, one easily ascertains that ψs = ψ − ψa is the symmetric component of ψ which isinvariant under the N-S mirror reflection. A set of streamlines of ψa and ψs inferred from the model solution of (1)–(3)with (6) is reported in fig. 1(b) and (c), respectively.

After these preliminaries, reconsider the steady version of (1)–(3) with parameters (6). The smallness of R allowsto expand the solution in powers of R

ψ =∑

n≥0

Rnψ(n), (11)

whose substitution into

RJ(ψ,∇2ψ) +∂ψ

∂x= − sin(y) + εL∇

4ψ (12)

yields, for n = 0,∂ψ(0)

∂x= − sin(y) + εL∇

4ψ(0) (13)

and, for n ≥ 1,∑

i+j=n−1

J(ψ(i),∇2ψ(j)) +∂ψ(n)

∂x= εL∇

4ψ(n). (14)

Note that eq. (13) is nothing but Munk’s linear model with the same anticyclonic sinusoidal wind-stress curl as in (1).Now, define for each n = 0, 1, 2 . . .,

ψ(n)s (x, y) =

1

2

[

ψ(n)(x, y) + ψ(n)(x, π − y)]

(15)

and

ψ(n)a (x, y) =

1

2

[

ψ(n)(x, y) − ψ(n)(x, π − y)]

. (16)

Relationship (10) assures that ψ(n)a �= 0 at least for some integer n. Note that ψ

(n)s and ψ

(n)a satisfy the same boundary

conditions (3) as ψ. Positions (15) and (16) imply

ψ(n) = ψ(n)s + ψ(n)

a , (17)

so, under the mirror reflection(x, y) → (x, π − y), (18)

the transformation rule,ψ(n)

s + ψ(n)a → ψ(n)

s − ψ(n)a , (19)

holds true. Because of (19), ∂∂y → − ∂

∂y and, hence, also J → −J . The results obtained above are now put together.

By using (17) into (13) one obtains

∂

∂xψ(0)

s +∂

∂xψ(0)

a = − sin(y) + εL∇4ψ(0)

s + εL∇4ψ(0)

a . (20)

Then, application of (19) to (20) yields

∂

∂xψ(0)

s −∂

∂xψ(0)

a = − sin(y) + εL∇4ψ(0)

s − εL∇4ψ(0)

a . (21)

Addition of (20) to (21) results in the equation

∂

∂xψ(0)

s = − sin(y) + εL∇4ψ(0)

s , (22)

while subtraction of (21) from (20) gives∂

∂xψ(0)

a = εL∇4ψ(0)

a . (23)

Integration of (22) with (3) shows that ψ(0)s ∝ sin(y). On the other hand, integration of (23) shows that

ψ(0)a = 0. (24)


Equation (14), with n = 1, becomes

J(ψ(0)s ,∇2ψ(0)

s ) +∂ψ

(1)s

∂x+

∂ψ(1)a

∂x= εL∇

4ψ(1)s + εL∇

4ψ(1)a (25)

and, by applying (19) to (25), one obtains

−J(ψ(0)s ,∇2ψ(0)

s ) +∂ψ

(1)s

∂x−

∂ψ(1)a

∂x= εL∇

4ψ(1)s − εL∇

4ψ(1)a . (26)

Then, the same procedure as above implies

∂ψ(1)a

∂x= −J(ψ(0)

s ,∇2ψ(0)s ) + εL∇

4ψ(1)a (27)

and∂ψ

(1)s

∂x= εL∇

4ψ(1)s . (28)

Integration of (27) shows that ψ(1)a ∝ sin(2y), while integration of (28) yields

ψ(1)s = 0. (29)

Quite in general, starting from (14) and using (17) together with (19), the equations

∑

i+j=n−1

[

J(ψ(i)s ,∇2ψ(j)

a ) + J(ψ(i)a ,∇2ψ(j)

s )]

+∂ψ

(n)s

∂x= εL∇

4ψ(n)s (30)

and∑

i+j=n−1

[

J(ψ(i)s ,∇2ψ(j)

s ) + J(ψ(i)a ,∇2ψ(j)

a )]

+∂ψ

(n)a

∂x= εL∇

4ψ(n)a (31)

follow. Based on (24), (29), (30) and (31), the induction principle can be invoked to prove that

ψ(n)a = 0, for even n and ψ(n)

s = 0, for odd n. (32)

To prove (32), assume ψ(k)a = 0 ∀ even k ≤ n−1 (this is true for k = 0 according to (24)) and ψ

(k)s = 0 ∀ odd k ≤

n− 1 (this is true for k = 1 according to (29)). Consider, first, eq. (30) and take odd n. If n is odd, then n− 1 is even,

that is i + j is even. Hence, i and j are both even or both odd. If i and j are both even, ψ(i)a = ψ

(j)a = 0 while, if i and

j are both odd, ψ(i)s = ψ

(j)s = 0. Thus, in any case, eq. (30) simplifies into

∂ψ(n)s

∂x= εL∇

4ψ(n)s , (33)

whence the conclusion is ψ(n)s = 0 for odd n. Consider now eq. (31) and take even n. If n is even, then n − 1 is odd,

that is i + j is odd. Hence, i is odd while j is even or i is even while j is odd. The first possibility implies ψ(i)s = 0 and

ψ(j)a = 0; on the other hand, the second possibility implies ψ

(i)a = 0 and ψ

(j)s = 0. Thus, in any case, eq. (31) simplifies

into∂ψ

(n)a

∂x= εL∇

4ψ(n)a , (34)

whence the conclusion is ψ(n)a = 0 for even n. Statement (32) is so proved and expansion (11) can be written in the

formψ = ψ(0)

s + Rψ(1)a + R2ψ(2)

s + O(R3), (35)

where the law of the full sequence is established by (32).Reconsider (30), that is

∂ψ(n)s

∂x= −

∑

i+j=n−1

[

J(ψ(i)s ,∇2ψ(j)

a ) + J(ψ(i)a ,∇2ψ(j)

s )]

+ εL∇4ψ(n)

s . (36)

of 12 Eur. Phys. J. Plus (2014) 129: 142

By using the identityJ(ψ, q) = curlz(ψ∇q), (37)

eq. (36) is equivalent to

∂ψ(n)s

∂x= curlz

⎧

⎨

⎩

−∑

i+j=n−1

[

ψ(i)s ∇(∇2ψ(j)

a ) + ψ(i)a ∇(∇2ψ(j)

s )]

⎫

⎬

⎭

+ εL∇4ψ(n)

s . (38)

Thus, each symmetric term of (35) satisfies a linear equation analogous to that of Munk’s, but forced by the stress

τ (n) = −∑

i+j=n−1

[

ψ(i)s ∇(∇2ψ(j)

a ) + ψ(i)a ∇(∇2ψ(j)

s )]

(even n ≥ 2), (39)

while τ (0) is given by (5). By using again (37), quite analogously eq. (31) yields

∂ψ(n)a

∂x= curlz

⎧

⎨

⎩

−∑

i+j=n−1

[

ψ(i)s ∇(∇2ψ(j)

s ) + ψ(i)a ∇(∇2ψ(j)

a )]

⎫

⎬

⎭

+ εL∇4ψ(n)

a , (40)

so also every antisymmetric term of (35) satisfies a linear Munk-like equation forced by the stress

τ (n) = −∑

i+j=n−1

[

ψ(i)s ∇(∇2ψ(j)

s ) + ψ(i)a ∇(∇2ψ(j)

a )]

(odd n ≥ 1). (41)

The conclusion is that the vertical components of the curls of (39) and (41) are the forcing of the successive partialstream functions of expansion (35).

Note that, irrespectively of the symmetric or antisymmetric structure of the partial stream functions, ψ(n)· = 0 and

τ (n) = 0 along the boundary of the fluid domain (2). Thus, the integration of

∂ψ(n)·

∂x= curlzτ

(n) + εL∇4ψ

(n)· (n ≥ 1) (42)

on (2) gives∫

D∇2ψ

(n)· dxdy = 0, that is to say the flux Φ of relative vorticity across the boundary ∂D is zero

Φ(∇2ψ(n)) ≡

∮

∂D

∇(∇2ψ(n)) · n dℓ = 0 ∀n ≥ 1. (43)

Equation (43) means that each partial stream function redistributes the relative vorticity while conserving its integral

value. This, unlike the linear solution ψ(0)s (and the total solution ψ), whose governing eq. (13) determines the non-

vanishing outgoing vorticity flux

Φ(∇2ψ) = Φ(∇2ψ(0)s ) = 2π/εL.

Remark on the additional boundary condition

Expansion (35) relies on the integration of eqs. (23), (28), (33) and (34) by using the no mass-flux boundary conditionplus an additional one. In Boening’s model, the latter is free-slip. Other additional conditions can be used and theoutputs are expected to differ, also significantly, fron those of Boening. However, if the symmetry (9) of the wind-stresscurl is retained, the integration of (23), (28), (33) and (34) leads again to (35) also by applying additional conditionsdifferent of free-slip but satisfying a definite relationship, as shown below.

Irrespectively of the symmetry or the asymmetry of the partial stream function, each of (23), (28), (33) and (34)has the form

∂ψ

∂x= εL∇

4ψ. (44)

Multiplication of (44) yields, after little algebra,

1

2

∂

∂xψ2 = εL

{

div[

ψ∇(∇2ψ)]

−∇ψ · ∇(∇2ψ)}

, (45)


and integration of (45) on D gives

∮

∂D

∇2ψ∇ψ · ndℓ −

∫

D

(∇2ψ)2 dxdy = 0. (46)

Because of (46), every additional condition satisfying the relationship

∮

∂D

∇2ψ∇ψ · ndℓ = 0 (47)

implies∫

D(∇2ψ)2dxdy = 0. Thus, ψ is the unique solution of problem

{

∇2ψ = 0 ∀(x, y) ∈ D

ψ = 0 ∀(x, y) ∈ ∂D,

that isψ = 0 ∀(x, y) ∈ D. (48)

Equation (48), suitably referred to the symmetric or to the antisymmetric part of each partial stream function, allowsto conclude as in (32). Typical additional conditions consistent with (47) are the no-slip condition,

∇ψ · n = 0 ∀(x, y) ∈ ∂D,

and the mixed condition, for instance with reference to (2),

{

∇2ψ = 0 in y = 0 and in y = π

∂ψ/∂x = 0 in x = 0 and in x = π.

To summarize, given the quasi-geostrophic steady circulation model governed by the equation

– RJ(ψ,∇2ψ) + ∂ψ/∂x = curlzτ + εL∇4ψ,

such that

– R ≪ 1,– curlzτ(x, y) = curlzτ(x, π − y),– ψ = 0 ∀(x, y) ∈ ∂D,– endowed further on with any additional boundary condition satisfying (47)

Then the model solution ψ can be expanded in symmetric and antisymmetric partial stream functions as in (35).The same conclusion as above can be drawn in the presence of the more general bottom dissipation reported in (1).

Obviously, if only bottom dissipation is considered, no additional boundary condition is requested to single out themodel solution. In any case, eqs. (32) hold true.

4 Results from the numerical evaluation of partial and total solutions

A numerical code is implemented to solve the full time-dependent version of the vorticity equation (12). The JacobianJ(ψ,∇2ψ) is evaluated using the Arakawa scheme [11], which conserves kinetic energy and enstrophy, while therelative vorticity is inverted using the method of successive overrelaxations [12]. The task of this section is to discussthe comparison between the output of the numerical integration of eq. (12), with the approximated solutions obtainedfrom the numerical integration of the time-dependent version of eqs. (22) and (27). Integrations are performed on thedomain (2), with no mass-flux and free-slip boundary conditions and with the choice of parameters (6). Figure 1(a)shows a set of streamlines of eq. (12) which exhibit the main features arising from nonlinearity. They are a (small)northward displacement of the centre of the gyre, the tendency of intensification to extend from the western to thenorthern boundary and the formation of a meandering in the N-W corner of the domain, which connects the intensifiedcurrent with that of the interior. The antisymmetric (ψa) and the symmetric (ψs) components of the total solution,introduced in (A.2) and in (A.1), respectively, are reported in fig. 1(b) and (c), respectively. As fig. 1(b) shows, theeffect of nonlinearity mainly consists in pairs of cyclonic-anticyclonic vortices localized in the western area of thedomain. On the other hand, in fig. 1(c) the spread of the symmetric part of the total solution into the whole domainis quite evident. Indeed, positions (A.1) and (A.2) do not explain the structure of ψs and ψa however, in the regime

of weak nonlinearity, one can approximate ψs with ψ(0)s and ψa with ψ

(1)a . In this way one is able to explain, within a

of 12 Eur. Phys. J. Plus (2014) 129: 142

(a) J(ψs

(0),∇

2ψ

s

(0)), 2R

1/2=ε

L

1/3

0 pi/2 pi0

pi/2

pi

0 pi/2 pi−1

0

1

2

3

4

5

(b) ψ along x=x*, 2R

1/2=ε

L

1/3

Rψa

(1)

ψs

(0)+Rψ

a

(1)

ψ

Fig. 2. (a) Streamlines for the forcing term J(ψ(0)S ,∇2ψ

(0)S ). Full and dashed lines represent positive and negative values,

respectively. (b) Meridional section along x = x∗, corresponding to the longitude at which the total stream function ψ acquires

maximum value, for Rψ(1)A (dashed black line), ψ

(0)S + Rψ

(1)A (full black line) and the total model stream function ψ (gray line).

Both panels refer to Boening’s model with 2R1/2 = ε1/3L .

certain degree of approximation, the behavior of ψs and ψa in terms of ψ(0)s and ψ

(1)a , whose governing equations (22)

and (27) have a definite and autonomous physical meaning. The integration of (22) yields the well-known Munk’ssolution (indeed a special case of it, given that additional boundary conditions alternative to free-slip can be applied).

The streamlines given by ψ(0)s (x, y) = const. are depicted in fig. 1(f). Due to the simple form of the forcing term,

analytical solutions of the same problem can be obtained as well, for instance, by means of boundary layer methods [2,13,14]. These solutions have the noticeable property that, in the interior and eastern part of the fluid domain, theyare close to that of Sverdrup

ψSv = (π − x) sin(y). (49)

Streamlines of (49) are quite evident far from the western boundary and at all latitudes in the lower panel of fig. 1(f)and in fig. 3 that shows the zonal section of the stream function at y = π/2, y = 3π/4, y = π/4. The comparison

between the streamlines of ψ(0)s (fig. 1(f)) and the streamlines of the symmetric component of the total stream function

(fig. 1(c)) show good agreement.


0 pi/2 pi−1

0

1

2

3

4

(a) ψ along y=π /2, 2R1/2

=εL

1/3

Rψa

(1)

ψs

(0)+Rψ

a

(1)

ψψ

SV

0 pi/2 pi−1

0

1

2

3

4

(b) ψ along y=3 π /4, 2R1/2

=εL

1/3

Rψa

(1)

ψs

(0)+Rψ

a

(1)

ψψ

SV

0 pi/2 pi−1

−0.5

0

0.5

1

1.5

2

2.5

(c) ψ along y=π /4, 2R1/2

=εL

1/3

Rψa

(1)

ψs

(0)+Rψ

a

(1)

ψψ

SV

Fig. 3. Zonal section along (a) y = π/2, (b) y = 3π/4 and (c) y = π/4 for Rψ(1)A (dashed black line), ψ

(0)S + Rψ

(1)A (full black

line), the total stream function ψ (full gray line) and the stream function corresponding to the Sverdrup solution ψSv (dashed

gray line) for Boening’s model with 2R1/2 = ε1/3L .

of 12 Eur. Phys. J. Plus (2014) 129: 142

Once ψ(0)s is given, eq. (27) can be integrated. The structure of the forcing −J(ψ

(0)s ,∇2ψ

(0)s ) of (27) is interesting for

two reasons: first, it is antisymmetric with respect to y just because of the symmetry of ψ(0)s ; second, the relationship

∇2ψSv = −ψSv, which trivially comes from (49), implies −J(ψ(0)s ,∇2ψ

(0)s ) = 0 in the interior and eastern part of the

fluid domain. Thus, the forcing works only in a region of the western side. Both these features are visible in fig. 2(a),that shows the streamlines of the Jacobian and where full lines indicate positive forcing and dashed lines indicate

negative forcing. Results show that the forcing −J(ψ(0)s ,∇2ψ

(0)s ) gives rise to an anticyclonic vortex in the N-W corner

of the domain, included between y = π/2 and y = π. A secondary (less intense) cyclonic vortex is present at itseastern side. A cyclonic vortex appears in the S-W corner of the domain, with a secondary anticyclonic vortex atits eastern side. The non-dimensional longitudinal extension of the whole structure is relatively small, less than π/4.

These contributions act to shape the partial solution ψ(1)a , whose streamlines are shown in fig. 1(e), and which plays

a basic role to shape ψ. In the northern side of the domain, the interplay between Munk’s solution and the counter-rotating vortices goes as follows: 1) the superposition of the eastward currents associated to both the anticyclonicvortex and Munk’s solution in the N-W corner of the domain contributes to the formation of a zonal eastward jet,of small longitudinal extent. This is the reason why nonlinearity favours the continuation of westward intensificationalong the northern boundary. 2) At the longitude where the currents generated by the anticyclonic vortex in the N-Wcorner of the domain and the cyclonic vortex at its eastern side interact, they deflect the currents from the Munk’ssolution southward. 3) Finally, Munk’s solution is deflected northward again at the eastern side of the cyclonic vortex.This complex pattern of deflections of the symmetric Munk’s solution by the interaction between vortices along the

northern boundary is responsible for the meandering of ψ(0)s + Rψ

(1)a (fig. 1(d)).

The panels of fig. 3 show the longitudinal profiles of some stream functions involved in the model, at the centrallatitude y = π/2 (upper panel), at the latitude y = 3π/4 (middle panel) and at the latitude y = π/4 (lower panel).

Because of the antisymmetric structure of ψ(1)a , we have ψ

(1)a (x, π/2) = 0 as the dotted grey line shows. Moreover,

the first two terms (only the first one, indeed) of expansion (35) yield a slight overestimate of the full solution inthe western part of the domain. On the other hand, for increasing longitudes, both the partial and the full solution

converge towards the Sverdrup’s solution (49). In the middle panel, the contribution of ψ(1)a to enhance the modulation

of the partial and total solutions is pointed out by the common phase exhibited by ψ(1)a , ψ

(0)s + Rψ

(1)a and ψ in the

western part of the domain. In the remaining part, the situation is quite similar as that of the upper panel. Unlike the

latter case, in the lower panel the oscillations of ψ(1)a (x, π/4) have a phase opposite to ψ

(1)a (x, 3π/4) and, hence, the

model solution has a minor (evanescent, indeed) meandering, as already shown in fig. 1(d) Here, the first two termsof expansion (35) slightly underestimate the full solution.

In fig. 2(b) a set of stream functions is plotted along the longitude x∗, where ψ acquires the maximum value. The

dotted line comes from the proportionality ψ(1)a ∝ sin(2y), whose superposition with ψ

(0)s ∝ sin(y) produces the plot

of the first two terms of (35) (black line). The latter plot is openly asymmetric with the maximum in the upper half

of the fluid domain. In spite of the crude approximation ψ ≈ ψ(0)s + Rψ

(1)a , the maxima of ψ and of ψ

(0)s + Rψ

(1)a are

very close each other, with an error of about 1.5%

5 Summary

The role of nonlinear advection of vorticity was studied using a classical model of wind-driven ocean circulation,originally formulated by Boening. The model was studied in the weakly nonlinear approximation, in order to allowfor analytical solutions of the problem. After a mirror reflection in the east-west direction, the stream function wasseparated in a symmetric and asymmetric component. An asymptotic expansion for small Rossby number led to theformulation of a coupled set of evolution equations for the two symmetric components. Results show that the asym-metric component is responsible for the formation of steady cyclones and anticyclones that allows for the deformationof the total stream function of the system. It should be noted that higher-order components of the stream function areforced by an effective wind stress arising from lower-order entries in the Jacobian term, and that these effective stressesact only to redistribute vorticity. The solution that was found in this work is reminiscent of the cyclone-anticyclonepair that is found is a perturbation analysis is applied, for example, to the model of Veronis [3] (see also [15]) but ishere reformulated in a rigorous mathematical formalism based on the reflection symmetry of the system.

The homogeneous model of the wind-driven ocean circulation that was employed here, is based on a number ofapproximations that make these models only qualitative descriptors of the system. For example, the application of thehomogeneous model to an ocean basin with flat bottom but realistic North Atlantic basin boundaries and wind stresscurl is able to correctly represent the emergence of the subtropical and subpolar gyres, but the western instensificationappears to be too diffuse and the Gulf Stream appears to separate too much on the north [16]. However, the studyof the mathematical properties of these models is in qualitative agreement with the observations and gives invaluableinformation about the relative importance of the dynamics of the wind-driven circulation.


F.C. wishes to thank Dr. R. Purini for encouraging comments on this investigation.

Appendix A. Proof of (10)

Define

ψs(x, y) =1

2[ψ(x, y) + ψ(x, π − y)] (A.1)

and

ψa(x, y) =1

2[ψ(x, y) − ψ(x, π − y)] . (A.2)

Hence

ψ = ψs + ψa. (A.3)

Substitution of (A.3) into (8) gives

RJ(ψs + ψa,∇2ψs + ∇2ψa) +∂ψs

∂x+

∂ψa

∂x= curlzτ + ℘(ψs + ψa), (A.4)

where we put in short ℘ = εL∇4 − εB∇2.

Under the mid-latitude mirror reflection (x, y) → (x, π − y) we have ψs+ψa → ψs−ψa, ∂/∂y → −∂/∂y so J → −Jand ℘ → ℘, while the forcing is left unchanged because of (9). Hence, eq. (A.4) transforms into

−RJ(ψs − ψa,∇2ψs −∇2ψa) +∂ψs

∂x−

∂ψa

∂x= curlzτ + ℘(ψs − ψa). (A.5)

Subtraction of (A.5) from (A.4) results in equation

R[J(ψs,∇2ψs) + J(ψa,∇2ψa)] +

∂ψa

∂x= ℘ψa. (A.6)

Equation (A.6) shows that R = 0 implies the equation

∂ψa

∂x= εL∇

4ψa − εB∇2ψa. (A.7)

Multiplication of (A.7) by ψa and the subsequent integration on the fluid domain with the aid of boundary condi-tions (3) or of any other boundary condition satisfying (47), together with the divergence theorem, yields

εL

∫

D

(∇2ψa)2dxdy + εB

∫

D

|∇ψa|2dxdy = 0. (A.8)

From (A.8), and the no mass-flux boundary condition, the problems for ψa, given by

{

∇ψa = 0 ∀(x, y) ∈ D

ψa = 0 ∀(x, y) ∈ ∂D

{

∇2ψa = 0 ∀(x, y) ∈ D

ψa = 0 ∀(x, y) ∈ ∂D,

follow. Both of them have the unique solution ψa = 0 and therefore, according to (A.2), ψ(x, y) = ψ(x, π − y).On the other hand, if one assumes ψa = 0, eq. (A.6) implies RJ(ψ,∇2ψ) = 0, whence the alternative

R = 0 (A.9)

or

J(ψ,∇2ψ) = 0 (A.10)

is established. Equation (A.10) must be ruled out by the fact that stable solutions of this equation are symmetric withrespect to the central longitude (here x = π/2) of the fluid domain [17] but this event is inconsistent with westwardintensification of wind-driven ocean circulation; thus eq. (A.9) necassarily holds.

of 12 Eur. Phys. J. Plus (2014) 129: 142

References

1. H.M. Stommel, Trans. Amer. Geophys. Union 29, 202 (1948).2. J. Pedlosky, Ocean Circulation Theory (Springer-Verlag, 1996) pp. 453.3. G. Veronis, Deep-Sea Res. 13, 17 (1966).4. G. Veronis, Deep-Sea Res. 13, 31 (1966).5. R.R. Blandford, Deep-Sea Res. 18, 739 (1971).6. C.W. Boening, Dyn. Atmos. Oceans 10, 63 (1986).7. K. Bryan, J. Atmos. Sci. 20, 594 (1963).8. F. Crisciani, T. Ozgokmen, Dyn. Atmos. Oceans 33, 135 (2001).9. G. Badin, F. Cavallini, F. Crisciani, Nuovo Cimento B 124, 653 (2009).

10. G. Badin, A.M Barry, F. Cavallini, F. Crisciani, Eur. Phys. J. Plus 127, 45 (2012).11. A. Arakawa, J. Comput. Phys. 1, 119 (1966).12. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd

edition (Cambridge University Press, Cambridge, 2007) pp. 1256.13. J. Pedlosky, Geophysical Fluid Dynamics, 2nd edition (Springer-Verlag, 1987) pp. 710.14. F. Cavallini, F. Crisciani, Quasi-Geostrophic Theory of Oceans and Atmosphere (Springer, 2012) pp. 385.15. G.K. Vallis, Atmospheric and Oceanic Fluid Dynamics (Cambridge University Press, Cambridge, 2006) pp. 74516. R. Zhang, G.K. Vallis, J. Phys. Oceanogr. 37, 2053 (2007).17. F. Crisciani, R. Mosetti, R. Purini, Eur. Phys. J. Plus 128, 18 (2013).

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