Review Article The Ellipsoidal Vortex: A Novel …downloads.hindawi.com/journals/amp/2015/613683.pdfReview Article The Ellipsoidal Vortex: A Novel Approach to Geophysical Turbulence

Review ArticleThe Ellipsoidal Vortex A Novel Approach toGeophysical Turbulence

William J McKiver

ISMAR-CNR Arsenale-Tesa 104 Castello 2737F 30122 Venice Italy

Correspondence should be addressed to William J McKiver williammckiverveismarcnrit

Received 8 October 2014 Accepted 14 January 2015

Academic Editor Touvia Miloh

Copyright copy 2015 William J McKiver This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We review the development of the ellipsoidal vortex model within the field of geophysical fluid dynamics This vortex model isbuilt on the classical potential theory of ellipsoids and applies to large-scale fluid flows such as those found in the atmosphere andoceans where the dynamics are strongly affected by the Earthrsquos rotation In this large-scale limit the governing equations reduce tothe quasi-geostrophic system where all the dynamics depends on a single scalar field the potential vorticity which is a dynamicalmarker for vortices The solution of this system is achieved by the inversion of a Poisson equation that in the case of an ellipsoidalvortex can be solved exactly From this ellipsoidal solution equilibria have been determined and their stability properties have beenstudied Many studies have shown that this ellipsoidal vortex model while being conceptually simple is an extremely powerful toolin eliciting some of the fundamental characteristics of turbulent geophysical flows

1 Introduction

In turbulent flow there exist structures whose collective inter-actions are at the very heart of the dynamics of turbulencemdashvortices The vortex a word signifying a type of swirlingor spinning fluid has become a familiar concept in humansociety being associated with extreme and highly energeticweather phenomena such as typhoons tornados and hurri-canesThe relative infrequency of these most dramatic mani-festations of vortices could lead one to think that the conceptof the vortex itself is limited only to these extreme eventsbut on the contrary vortices are present in almost all fluidphenomena from the smallest scale flow of water in pipesto the largest scales within the atmosphere and the oceansBoth observations [1 2] and numerical simulations [3ndash5]have revealed the widespread appearance and dominance ofvortices in turbulent geophysical flows The prolific nature ofvortices has led to their study becoming a field in its ownright as a means to better understand and resolve the greatunsolved problem of classical physicsmdashturbulence

While vorticity has a clear mathematical definition beingdefined as a vector field which is the curl of the velocity flowfield (120596 = nabla times u) the definition of a vortex itself is far more

elusiveThis is due to the fact that a vortex does not have a par-ticular shape Ideas of tornadoes and hurricanes usually con-jure up an image of circular or cylindrical type shapes for thevortex but real vortices are not constrained to these perfectforms However when approaching them mathematically itis useful to apply a shape that is amenable to simple mathe-matical definition One of the earliest incarnations of such anapproach was that of Kirchhoff [6] who studied an isolatedtwo-dimensional patch of uniform vorticity bounded withinan ellipse He showed that this is an exact analytical solutionto the Euler equations which rotates with a constant angularvelocity Further work by Love [7] showed that there existedstable equilibria when this vortex is subjected to disturbances

The extension of these early works to the three-dimensional case of an ellipsoidal vortex had to wait almosta century for the works of Zhmur and Pankratov [8 9] andMeacham [10] who extended this approach to the quasi-geostrophic system [11] a set of equations relevant to large-scale geophysical fluid dynamics In fact the ellipsoidal formhas a long history stretching back (at least) to the seminalworks of MacClaurin [12] and Laplace [13] on gravitatingbodies (see [14 15] for a review of this subject) As the ellip-soidal form has an exact and simple mathematical definition

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 613683 9 pageshttpdxdoiorg1011552015613683

2 Advances in Mathematical Physics

that yet goes beyond the absolute symmetry of the sphereit is a powerful tool in determining shape structure andorientation of ldquobodiesrdquo concerned in many different fields ofapplication (see [16] for a review of the many applications)In the case of geophysical flows it gives a means to determinethe characteristics of vortices (such as typical height-to-widthratios orientations and rotation rates) that are the mostrobust and hence the most likely to occur in real turbulentflows

Here we will review the recent history of the developmentof the ellipsoidal vortex in quasi-geostrophic dynamics Thebeginning is in Section 2wherewe give a brief overview of thegoverning system of equations the quasi-geostrophic model[11] Following this we review the ellipsoidal vortex modelstarting in Section 3 with the case of the isolated ellipsoid asoriginally derived byMeacham [10] In Section 4 we considerthe case of an ellipsoidal vortex immersed in a backgroundshear flow following the matrix formulation approach ofMcKiver and Dritschel [17] In Section 5 we discuss theresults of studies of the equilibria and stability properties ofthis vortex system and what insights they provide for thefull solution to the equations in turbulence simulations InSection 6 we discuss the generalization of the single vortexellipsoidal theory to the multiple vortex case in particularoutlining the Hamiltonian method of Dritschel et al [18]Finally we conclude in Section 7 with an overview of theconsequences the ellipsoidal model has for the theory ofturbulence and discuss future avenues of research in this field

2 The Quasi-Geostrophic System

Here we summarize the quasi-geostrophic model (for adetailed derivation see [19]) Quasi-geostrophic (hereafterQG) theory is one of the simplest geophysical fluid dynamicalmodels that incorporates two crucial features underpinninglarge-scale fluid dynamics (1) the effect of the Earthrsquos rotationand (2) the effects of strong density stratification (lighterfluid lying on top of denser fluid) As a consequence ofthese features certain terms in the full equations of motion(Navier-Stokes equations) tend to dominate over othersStrong rotation leads to what is termed ldquogeostrophic balancerdquowhich is a balance between the horizontal pressure gradientand the term associated with the Earthrsquos rotation (the Coriolisterm)

119891k times uℎ = minus

nabla119901

1205880

(1)

where 119891 is the local vertical component of the planetaryrotation rate k is a vertical unit vector uℎ is the horizontalvelocity field 119901 is pressure and 1205880(119911) is the basic-statedensity distribution (1205880 is nearly constant in the oceans andexponential proportional to exp(minus119911119867) with 119867 asymp 7 km inthe atmosphere) Strong stratification leads to ldquohydrostaticbalancerdquo which is a balance between the vertical pressuregradient and buoyancy

120597119901

120597119911

= minus1205880119892(2)

where 119892 is the acceleration due to gravity The QG systemwhich incorporates both hydrostatic and geostrophic bal-ance at leading order is obtained from the full ldquoprimitiverdquoequations by an asymptotic expansion in three small nondi-mensional parameters (1) the height-width aspect ratio ofcharacteristic motions 119867119871 (constrained to be small due tothe shallow geometry in the oceans and atmosphere) (2) theRossby number 119877119900 equiv 119880119891119871 and (3) the Froude number119865119903 equiv 119880119873119867 where 119880 is a characteristic relative horizontalfluid speed 119871 is the horizontal scale of the flow 119867 is theaverage fluid depth and 119873 is the buoyancy frequency (theoscillation frequency that a small fluid volume would exhibitif it were displaced by a small vertical distance) The QGmodel is obtained assuming that119867119871 ≪ 1 and 1198652

119903≪ 119877119900 ≪ 1

Also in this model we consider the limit of infinite Reynoldsnumber Re equiv 119880119871] that is the effect of viscous dissipation] is negligible (in the real atmosphere Re sim 10

10ndash1012) Asdone inmany previous works one can consider the constant-coefficient (ldquooceanicrdquo) system when 119891 119873 and 1205880 are allconstant For an inviscid adiabatic fluid the QG equationsmay be expressed in terms of the conservation of a singlematerial scalar the ldquopotential vorticityrdquo 119902(x 119905) that is

119863119902

119863119905

equiv

120597119902

120597119905

+ 119906

120597119902

120597119909

+ V120597119902

120597119910

= 0 (3)

The potential vorticity (hereafter PV) is proportional tothe component of vorticity perpendicular to stratificationsurfaces (surfaces of constant density 1205880) Thus vortices areidentified as contiguous regions of PV In the QG model thevertical velocity 119908 = 0 to this order of approximation so 119902

is advected in a layerwise manner However the horizontalvelocity (119906 V) depends on all three spatial coordinates andthis is recovered from 119902 via the inversion of

nabla2120595 = 119902 (4)

for the stream function 120595 followed by the incompressibilityrelations

119906 = minus

120597120595

120597119910

V =120597120595

120597119909

(5)

Note that in this form the vertical coordinate 119911 is rescaledby the factor 119891119873 The three-dimensional distribution ofPV generates the velocity field through (4) and (5) whichcarries the PV conservatively to the next instant of timeWhile (4) is an isotropic relation between 119902 and 120595 theadvection remains anisotropic as there is no vertical motionThe QG equations are widely used in studies of atmosphericand oceanic phenomena [20ndash23] in part because they arefar easier to deal with both theoretically and numericallythan the complete equations But also the QG equationscapture many qualitative features of atmospheric and oceanicdynamics

3 Potential Theory of an IsolatedEllipsoidal Vortex

Now we consider the case of an isolated ellipsoidal vortexwith uniform potential vorticity as first considered in the

Advances in Mathematical Physics 3

works of Zhmur and Pankratov [8 9] (see also [24]) andindependently by Meacham [10] We will review the fullsolution to the QG ellipsoid as derived by Meacham [10] Weconsider the ellipsoid of PV having semiaxes lengths 119886 119887 and119888 where 119886 ge 119887 ge 119888 Without loss of generality we assume thevortex axes are aligned along the coordinate axis Thus theboundary of the ellipsoid is defined by the equation

119909

2

119886

2+

119910

2

119887

2+

119911

2

119888

2= 1

(6)

Inside the ellipsoid we have from (4)

nabla2120595119894 = 119902 (7)

where 119902 is the uniform potential vorticity A general solutionto this equation can be written as

120595119894 =

1

2

(119891119909

2+ 119892119910

2+ ℎ119911

2) + 119862119894

(8)

where 119891 119892 ℎ and 119862119894 are unknown coefficients to bedetermined Substituting this into (7) gives the condition

119891 + 119892 + ℎ = 119902 (9)

Outside the ellipsoid the potential vorticity is zero and theexternal potential satisfies

nabla2120595119890 = 0 (10)

With such a problem the natural approach is to move fromCartesian coordinates (119909 119910 119911) to an ellipsoidal coordinatesystem (120582 120583 ]) using the relations [25]

119909

2=

(120582 + 119886

2) (120583 + 119886

2) (] + 119886

2)

(119886

2minus 119887

2) (119886

2minus 119888

2)

(11a)

119910

2=

(120582 + 119887

2) (120583 + 119887

2) (] + 119887

2)

(119887

2minus 119888

2) (119887

2minus 119886

2)

(11b)

119911

2=

(120582 + 119888

2) (120583 + 119888

2) (] + 119888

2)

(119888

2minus 119886

2) (119888

2minus 119887

2)

(11c)

where the variables 120582 120583 and ] represent the roots to the cubicequation

119909

2

119886

2+ 120594

+

119910

2

119887

2+ 120594

+

119911

2

119888

2+ 120594

= 1 (12)

where 120582 gt minus119888

2gt 120583 gt minus119887

2gt ] gt minus119886

2 The 120582-surfacesare ellipsoids the 120583-surfaces are hyperboloids of one sheetand the ]-surfaces are hyperboloids of two sheets [16] Thelargest root 120582 is analogous to the radial variable in sphericalcoordinates with each 120582 representing a family of concentricellipsoids with 120582 = 0 being the value on the surface of theoriginal ellipsoid with semiaxes 119886 119887 and 119888 The boundaryconditions on the surface of the ellipsoid are the continuity

of the normal and tangential velocity components which canbe expressed in ellipsoidal coordinates as

120595119894

1003816100381610038161003816120582=0

= 120595119890

1003816100381610038161003816120582=0

(13a)

120597120595119894

120597120582

10038161003816100381610038161003816100381610038161003816120582=0

=

120597120595119890

120597120582

10038161003816100381610038161003816100381610038161003816120582=0

(13b)

Substituting (11a) (11b) and (11c) into the expression for theinternal field (8) we obtain

120595119894 =

119891 (120582 + 119886

2) [120583] + 119886

2(120583 + ]) + 119886

4]

2 (119886

2minus 119887

2) (119886

2minus 119888

2)

+

119892 (120582 + 119887

2) [120583] + 119887

2(120583 + ]) + 119887

4]

2 (119887

2minus 119888

2) (119887

2minus 119886

2)

+

ℎ (120582 + 119888

2) [120583] + 119888

2(120583 + ]) + 119888

4]

2 (119888

2minus 119886

2) (119888

2minus 119887

2)

+ 119862119894

(14)

A general solution for the external flow field can be written asan expansion in ellipsoidal harmonics [16 25 26]

120601119890 =

infin

sum

119898

2119898+1

sum

119896

F(119896)119898

(120582)E(119896)119898

(120583)E(119896)119898

(]) (15)

where E(119896)119898

denote Lame functions of the first kind while F(119896)119898

are Lame functions of the second kind defined by

F(119896)119898

(120582)

=

2119898 + 1

2

E(119896)119898

(120582)

sdot int

infin

0

119889119904

[E(119896)119898 (119904 + 120582)]

2

radic(119904 + 120582 + 119886

2) (119904 + 120582 + 119887

2) (119904 + 120582 + 119888

2)

(16)The dependence on the coordinates 120583 and ] for the inner fieldin (14) implies that the exterior solution needed in order tomatch the interior solution at the boundary has the form

120601119890 = 1198620F(1)

0(120582) + 119862

(1)

2F(1)2

(120582)E(1)2

(120583)E(1)2

(])

+ 119862

(2)

2F(2)2

(120582)E(2)2

(120583)E(2)2

(]) (17)

where 1198620 119862(1)

2 and 119862

(2)

2are unknown coefficients and the

second-order Lame functions of the first kind are

E(1)2

(120582) = 120582 + 1198891(18a)

E(2)2

(120582) = 120582 + 1198892(18b)

where

1198891 =

1

3

[ (119886

2+ 119887

2+ 119888

2)

+radic119886

4+ 119887

4+ 119888

4minus 119887

2119888

2minus 119888

2119886

2minus 119886

2119887

2]

(19a)

1198892 =

1

3

[ (119886

2+ 119887

2+ 119888

2)

minusradic119886

4+ 119887

4+ 119888

4minus 119887

2119888

2minus 119888

2119886

2minus 119886

2119887

2]

(19b)


Substituting these Lame functions into the external field givesus

120595119890 = 120583] [119862(1)2F(1)2

(120582) + 119862

(2)

2F(2)2

(120582)]

+ (120583 + ]) [1198891119862(1)

2F(1)2

(120582) + 1198892119862(2)

2F(2)2

(120582)]

+ 1198620F(1)

0(120582) + 119889

2

1119862

(1)

2F(1)2

(120582) + 119889

2

2119862

(2)

2F(2)2

(120582)

(20)

Substituting (14) and (20) into the boundary conditionsequations (13a) and (13b) provides 6 conditions since thereare terms proportional to 120583] and 120583 + ] as well as the constantterms These 6 conditions along with the condition given by(9) provide 7 equations in the 7 unknown coefficients andso can be solved as was shown by Meacham [10] Solvingthese simultaneous equationswe canwrite the interior streamfunction as

120595119894 =

1

2

(1205851198861199092+ 1205851198871199102+ 1205851198881199112) minus

3

2

120581V119877119865 (1198862 119887

2 119888

2) (21)

where 120585119886 120585119887 and 120585119888 are constant coefficients given by

120585119886 = 120581V119877119863 (1198872 119888

2 119886

2) (22a)

120585119887 = 120581V119877119863 (1198882 119886

2 119887

2) (22b)

120585119888 = 120581V119877119863 (1198862 119887

2 119888

2) (22c)

where 120581V = 1199021198861198871198883 is the vortex ldquostrengthrdquo and 119877119865 and 119877119863

are elliptic integrals of the first and second kind respectivelydefined as

119877119865 (119909 119910 119911) equiv

1

2

int

infin

0

119889119905

radic(119905 + 119909) (119905 + 119910) (119905 + 119911)

(23a)

119877119863 (119909 119910 119911) equiv

3

2

int

infin

0

119889119905

radic(119905 + 119909) (119905 + 119910) (119905 + 119911)

3

(23b)

This solution was originally determined by Laplace [13] forthe case of a gravitating ellipsoid of uniform mass [14] Theexterior stream function is

120595119890 =

1

2

(1205851205721199092+ 120585120573119910

2+ 1205851205741199112) minus

3

2

120581V119877119865 (1205722 120573

2 120574

2) (24)

where 1205722 = 119886

2+ 120582 1205732 = 119887

2+ 120582 and 1205742 = 119888

2+ 120582 and now we

have

120585120572 = 120581V119877119863 (1205732 120574

2 120572

2) (25a)

120585120573 = 120581V119877119863 (1205742 120572

2 120573

2) (25b)

120585120574 = 120581V119877119863 (1205722 120573

2 120574

2) (25c)

While the interior solution has quadratic dependence onspatial coordinates the exterior field has a more complicateddependence due to the coefficients being functions of thevariable 120582 However the flowfield that acts on the vortex itselfis the interior solution and so one can compute the flow fieldacting on the vortex itself from (5) This gives rise to a self-induced velocity field that is linear in spatial coordinates

4 Evolution of a Vortex in a BackgroundShear Flow

We now consider the case of an ellipsoidal vortex immersedin a background shear flow This background shear flowidealizes the effects of other vortices and so is the firststep towards understanding interactions between vorticesThis case was first considered by Meacham et al [27] fora number of specific types of background flow and latergeneralized byMeacham et al [28] where it was recast withinaHamiltonian frameworkHashimoto et al [29] extended theanalytical solutions of Meacham [10] for the isolated vortexby solving for exact equilibrium solutions for the case of theellipsoid in an external straining flow McKiver and Dritschel[17] introduced a reformulation of the system that greatlysimplified the equations to a simple 3 times 3 matrix problemIn this section we review their derivation of this matrixformulationThis approach is built on the simple fact that thestream function for the ellipsoid has quadratic dependenceon spatial coordinates and so the velocity field has a lineardependence Naturally any linear flow field imposed on theellipsoidal vortex while possibly altering the size (119886 119887 and 119888)and orientation of the vortex preserves the ellipsoidal formThus we consider a general velocity field of the form

u119887 (x 119905) = S119887 (119905) x (26)

where S119887 is a 3 times 3 matrix Within this flow we place asingle ellipsoidal vortex at the origin and require only that itsvelocity field has the same form as (26) that is uV = SVx Thetotal velocity field felt by the ellipsoid is

u = Sx (27)

where S = SV + S119887 and this depends only on time 119905 We referto S as the flow matrix subsequently

The ellipsoid is specified by its axis half lengths 119886 ge 119887 ge

119888 and the unit vectors a b and c directed along these axesThis geometric information may be encapsulated in a singlesymmetric matrix A defined by

A = MDM119879 (28)

where the superscript119879 denotes transposeM is an orthonor-mal rotation matrix (Mminus1 = M119879) defined by

M = (a b c) (29)

and D is a diagonal matrix whose terms are the inversesquared semiaxes lengths

D = (

1

119886

20 0

0

1

119887

20

0 0

1

119888

2

) (30)

The equation describing the surface of the ellipsoid is conve-niently written as

x119879Ax = 1 (31)


By taking a time derivative of this equation and using 119889x119889119905 =u(x 119905) = S(119905)x as well as the implied relation 119889x119879119889119905 =

x119879S119879(119905) we obtain

x119879 (AS + S119879A +

119889A119889119905

) x = 0 (32)

Since this must be true for all points x on the boundary of theellipsoid it follows that

119889A119889119905

= minus (AS + S119879A) (33)

which is simply the equation for the evolution of A in anarbitrary linear background flow Note that A remains asymmetric matrix for all time sinceAS+S119879A is automaticallysymmetric if A is

It proves convenient to rewrite (33) to evolve not A butits inverse B = Aminus1 Using the fact that the product AB is theidentity matrix it follows that

119889B119889119905

= minus B119889A119889119905

B

= B (AS + S119879A)B(34)

giving us the evolution equation (as originally derived in [17])

119889B119889119905

= SB + BS119879 (35)

where

B = MDminus1M119879 (36)

Dminus1 = (

119886

20 0

0 119887

20

0 0 119888

2

) (37)

In order to determine the evolution of the ellipsoid it isnecessary to be able to calculate the semiaxes lengths 119886 119887and 119888 and the orientation vectors a b and c of the ellipsoidfrom the matrix B But (36) right multiplied by M togetherwith (37) implies

Ba = 119886

2a (38a)

Bb = 119887

2b (38b)

Bc = 119888

2c (38c)

Hence the semiaxes lengths and the orientation vectors of theellipsoid can be found directly from the Bmatrix by solving asimple eigenvalue problem Finally one must specify the flowmatrix S = SV + S119887 The self-induced part is derived from thestream function solution to the inner potential problem 120595VThis solution can be written more generally for an arbitrarilyorientated ellipsoid as

120595V =1

2

x119879PVx minus3

2

120581V119877119865 (1198862 119887

2 119888

2) (39)

where

PV = MGM119879 (40)

and G is the diagonal matrix formed from the solutioncoefficients

G = (

120585119886 0 0

0 120585119887 0

0 0 120585119888

) (41)

From (5) it follows that

SV = LPV (42)

where L is defined as

L = (

0 minus1 0

1 0 0

0 0 0

) (43)

For the background flow matrix one can consider theeffect of a single distant vortex of strength 120581119887 centered atx = X119887 Let 119877 = |X119887 minus XV| be the distance between thetwo vortices then to leading order in 1119877 the vortices rotateabout each other at a rate

Ω =

120581119887 + 120581V

119877

3 (44)

At this order of approximation the vortices appear as pointswith no shape or internal structure If we adopt a frame ofreference rotating with the 119911-axis passing through the jointcentre of the two vortices so that their positionsXV andX119887 arefixed and assuming the original vortex is located at the originXV = 0 then in the vicinity of the origin the backgroundstream function takes the form (Taylor expanding)

120595119887 = minus

120581119887

119877

+

1

2

x119879P119887x + 119874(

120581119887 |x|3

119877

4) (45)

where notably the linear terms are absent and the matrix P119887is given by

P119887 =120581119887

119877

5(

119877

2minus 3119883

2

119887minus3119883119887119884119887 minus3119883119887119885119887

minus3119883119887119884119887 119877

2minus 3119884

2

119887minus3119884119887119885119887

minus3119883119887119885119887 minus3119884119887119885119887 119877

2minus 3119885

2

119887

)

minus(

Ω 0 0

0 Ω 0

0 0 0

)

(46)

The corresponding velocity field is given by u119887 = S119887x whereas before the flow matrix is S119887 = LP119887 leading to

S119887 = 120574(

3119883119884 3

119884

2minus 1 + 120573 3

119884119885

1 minus 120573 minus 3119883

2minus3

119883119884 minus3

119883119885

0 0 0

) (47)

where X equiv X119887119877

120574 equiv

120581119887

119877

3 (48)


hereinafter referred to as ldquothe strain raterdquo and

120573 equiv

Ω

120574

=

120581119887 + 120581V

120581119887

(49)

is a parameter depending only on the ratio of the vortexstrengths It can be simplified further by choosing either

119883

or 119884 to be zero leaving only three nonzero components Forexample if 119883 = 0

S119887 = 120574(

0

1

2

(1 + 3 cos 2120579) + 120573

3

2

sin 21205791 minus 120573 0 0

0 0 0

) (50)

where 119885 = sin 120579 119884 = cos 120579 and the range of the angle

120579 is restricted to [0

∘ 90

∘] because of symmetry This form

for the background flow matrix was originally considered byMeacham et al [27] but using a different notation

The complete model now consists of the evolution equa-tion (35) for the matrix B together with expressions (42) and(50) for the flow matrices SV and S119887 Note that 119889B33119889119905 = 0

in QG flowmdashthis follows because there is no vertical velocityThe elementB33 is the squared half-height of the vortex Notealso that the determinant |B| = |Dminus1| = (119886119887119888)

2 is proportionalto the squared vortex volume and is also invariant (becauseQG flow is incompressible)

5 Equilibria and Stability

Given the equations governing the evolution of an ellipsoidalvortex in a background shear flow one can then determineequilibrium vortices from (35) where they satisfy

SB + BS119879 = 0 (51)

Thiswas first exploited byReinaud et al [30] using an iterativelinear method to determine equilibria These equilibria haveimportant consequences for vortex interactions As a vortexevolves over its lifetime it may come into close proximitywith other vortices When this happens the strain exertedon one vortex by the others increases as the magnitude ofthis strain scales with the inverse cube of the separationdistance between vortices (see (48)) At some ldquocriticalrdquo valueof strain the vortex will begin to deform significantly andwill undergo a strong interaction If the vortices are closeenough they may partially or fully coalesce or ldquomergerdquo intoone forming a larger vortex along with filamentary debrisOften the onset of such strong interactions occurs at thepoint beyond which equilibria can no longer exist for strainsgreater than this critical value or beyondwhich equilibria areunstable to disturbancesThus the stability analysis of a vortexin equilibrium with a background shear is an importantindicator of when strong interactionsmay occurNextwe givea brief summary of the findings of a number of studies on thistopic

The stability of a freely rotating ellipsoid of potentialvorticity was first considered by Meacham [10] who foundinstabilities over a large range of the parameter space char-acterising the ellipsoid However a more recent study by

Dritschel et al [31] found that one of the unstable modesMeacham encountered is in fact stable and overall theirresults indicate that the freely rotating ellipsoid is widelystable

Miyazaki et al [32] studied the stability of a spheroid ofuniform PV tilted by some inclination angle from the verticalaxis They found that highly prolate spheroids are unstableif the inclination angle is large while oblate spheroids areunstable even if the inclination angle is very small In bothcases the instability is nonellipsoidal that is it destroys theellipsoidal form of the vortex

Hashimoto et al [29] derived the equations for the linearstability of an ellipsoid in a 2D strain field by expandingthe disturbing surface in Lame functions Using these theystudied the cases of a pure strain field (when there is nobackground rotation) and a simple shear flow (when thestrain rate is equal to the background rotation) They foundthat in a pure strain field highly elongated ellipsoids areunstable to modes whose order119898 is greater than 2 that is tononellipsoidalmodes In a simple shear flow they found that ahighly elongated ellipsoid whose major axis is perpendicularto the flow direction is unstable whereas any ellipsoidalvortex seems to be stable if the major axis is parallel to theflow direction

Meacham et al [27] derived equations for the evolutionof an ellipsoid of uniform PV in a background flow withboth horizontal strain and vertical shear They examinedthe stability for some special background flows either purehorizontal strain or pure vertical shear (in the absence ofhorizontal strain) A full linear stability analysis for the caseof a general linear background flow (50) was conducted byMcKiver and Dritschel [33] They characterized the systemby four parameters the height-to-width aspect ratio of thevortex ℎ119903 the strength ratio 120573 the angle 120579 and the strainrate 120574 Over nearly the entire parameter space examined theyfound that the ellipsoidal steady states are stable and whereinstability does occur the dominant modes encountered areellipsoidalThus the ellipsoidal vortex in a linear backgroundflow is a fairly robust modelThey also found that for vorticeswith aspect ratios greater than 08 the most unstable vortices(ie the ones which destabilize at the smallest strain values120574) are found in the ranges 55

∘le 120579 le 75

∘ and 20

∘le

120579 le 30

∘ for opposite- and like-signed vortices respectivelyBecause the strain is proportional to the inverse cube ofthe separation distances between two vortices this impliesthat vortices which are vertically offset by these 120579 valuesdestabilize from the greatest separation distances that is theyare least resilient This agrees with the findings of Reinaudand Dritschel [34] in which they examined the merger oftwo identical uniform PV vortices which are offset bothhorizontally and vertically and found that vortices whichare moderately offset in the vertical merge from a greaterseparation distance than vortices that are not offset verticallyIn McKiver and Dritschel [33] for opposite-signed vorticesthey found that the most stable vortices have an aspect ratiobetween 10 and 125 whereas for the like-signed vorticesthe most stable vortices have an aspect ratio near 08 Thisagreeswith the findings in a study byReinaud et al [30]wherethey performed simulations of a turbulent flow comprising


hundreds of QG vortices in order to determine the typicalshape of vortices in QG turbulence Remarkably their studyalso found that typically vortices are slightly oblate having aheight-to-width aspect ratio of119867119871 asymp 08

6 Multiple Ellipsoidal Vortices

While the single vortex case provides great insight into vortexbehaviour in order to understand some of the dynamics thatprecipitate strong interactions and vortex mergers one mustconsider a multivortex scenarioThere are a number of worksthat have generalized the single ellipsoidal vortex solution toamultivortex scenario [35ndash38] However because of the formof the external stream function (24) generated by a vortex ingeneral it can induce nonellipsoidal disturbances on externalvorticesThis means that assumptions must be made in orderto retain the ellipsoidal form Here we will briefly summarizean accurate approach introduced by Dritschel et al [18] Thisapproach referred to as ELM represents vortices as PV ellip-soids and filters higher-order nonellipsoidal deformationssimply by preserving only the linear part of the exterior fieldFollowing the matrix formulation in Section 4 each vortexcan be specified by the value of its potential vorticity 119902119894 thesymmetric matrix B119894 and the location of the center of thevortex X119894 = (119883119894 119884119894 119885119894) in terms of which the surface of eachellipsoid can be written as

(x minus X119894)119879 Bminus1119894(x minus X119894) = 1 (52)

where 119894 = 1 119873 where 119873 is the number of vortices Ifwe consider that the velocity field for a particular ellipsoid(119902119894X119894B119894) has a linear dependence on spatial coordinatesthat is

u119894 (x 119905) = U119894 (119905) + S119894 (119905) (x minus X119894) (53)

where U119894(119905) is an arbitrary velocity field and S119894(119905) is theflow matrix then this velocity field is the most general formwhich exactly preserves the ellipsoidal form of the vortexThe governing equation for ELM [18] is a finite Hamiltoniansystem where the equations of motion for the 119894th ellipsoid are

119889X119894119889119905

= minus

1

120581119894

L 120597119867120597X119894

(54a)

119889B119894119889119905

= S119894B119894 + B119894S119879

119894 (54b)

S119894 = minus

10

120581119894

L 120597119867120597B119894

(54c)

where 120581119894 is the vortex strength 119867 is the Hamiltonian ofthe system defined as the total energy of the system dividedby 4120587 and the flow matrix is decomposed into two partsS119894 = SV

119894+ S119887119894 the self-induced part SV

119894and the part induced

by all other ellipsoids S119887119894 For efficiency of the method

this background flow field is modelled for each ellipsoidas a finite sum of singular point vortices The positionand strength of the singularities are obtained such thatthe approximate stream function induced by them matches

(a)

(b)

Figure 1 The time evolution of two like-signed vortices solvedusing contour dynamics solution of the full QG equations (a) andusing the ellipsoidal model (b) The side view is from an angle of 60degrees Figures courtesy of httpwww-vortexmcsst-andacuksimdgdELM

exactly the function at a given order of accuracy in 1119877119877 being the distance from X119894 to the point of evaluationDritschel et al [18] find that high accuracy (119874(11198777)) cantypically be achieved by using 7 point singularities for eachellipsoid Comparisons of the ELM with full numericalsimulations (Figure 1) of the QG vortices show remarkableagreement (see httpwww-vortexmcsst-andacuksimdgdELM to see more comparisons)

The steady states for the case of a two-vortex ELM systemwere first analyzed by Reinaud and Dritschel [39] In thiscase the parameters considered were the volume ratios (120588119881 =11988121198811) the vertical offsets (Δ119911) and height-to-width aspectratios (ℎ119903) As with the single vortex case the method todetermine equilibria involves the solution of

S119894B119894 + B119894S119879

119894= 0 (55)

in the rotating reference frame where 119889X119894119889119905 = 0 A familyof equilibria for a given set of parameters (120588119881 Δ119911 ℎ119903) arefound for different horizontal distances between the vortices120575 where 120575 is defined as the horizontal gap between the twoinnermost edges of the ellipsoids specifically

120575 =

1003816100381610038161003816

X1 minus X21003816100381610038161003816

minus radic(B11)1 minus radic(B11)2 (56)

Below a critical value of 120575 the equilibria become unstableinducing strong interactions that may lead to merger ofthe two vortices Analyzing the stability of these equilibriaReinaud and Dritschel [39] discovered a ldquotiltrdquo instabilitythat results in the merger of prolate vortices from greaterseparation distances than seen in previous symmetric mergercases (ie no vertical offset) This tilt instability is the firstinstability able to precipitate the merger of prolate vorticesThe merger of corotating vortices was also studied in detailby Li et al [37] where they introduced a new set of nearlycanonical variables to solve the multivortex system The caseof two counterrotating ellipsoidal vortices having the samesize was considered by Miyazaki et al [40] They found that


in general the vortex pair propagate as a stable dipole exceptfor certain tall vortices when initially placed within a criticaldistance they begin to tilt and eventually destabilize

7 Conclusions

Here we reviewed the recent history of the theory of ellip-soidal vortices in the field of geophysical fluid dynamics Thestrength and utility of this model is that it provides analyticalvortex solutions whose equilibria and stability can easilybe determined Remarkably this simple model can provideinsights into the dynamics seen in the full solutions of thegoverning equations in complicated turbulence simulationsThis is clearly seen even for the single vortex in a backgroundshear flow where the most stable aspect ratio of sim08 [33] isthe ratio statistically most seen in complex QG turbulencesimulations [30]

An interesting recent study by Koshel et al [41] modifiedthe ellipsoidal model adding diffusion in order to studyhow passive scalar transport through the vortex boundaryis affected by both advection and diffusion They showedthat adding disturbances to the vortex can boost the rateof transport of the tracer out of the vortex relative to theundisturbed case and discuss the potential of this model as ameans to determine diffusivity values in the real ocean Thisstudy is one example of the potential applications that theellipsoidalmodel can have for studying processes in the oceanand atmosphere where it has the advantage of an analyticalsolution for the flow field

The application of the ellipsoidal vortex model discussedin this review has only been for the quasi-geostrophic equa-tions which applies to large-scale dynamics when the Rossbynumber is small However the QG equations are only thefirst order in a hierarchy of the so-called ldquobalancedrdquo models(ie based on geostrophic and hydrostatic balance) thatcapture the underlining effects of rotation and stratificationA very recent numerical study by Tsang and Dritschel [42]looked at the evolution of an ellipsoidal vortex for moregeneral dynamics beyond theQG limitThey found that theseellipsoidal vortices again are very robust structures being to alarge extent quasi-equilibria over long time integrationsThisprompted McKiver and Dritschel [43] to revisit the potentialproblem for an isolated ellipsoid using equations at the nextorder to theQG theory (so-calledQG+1) In this higher-ordermodel one encounters three Poisson equations as the velocityfield is now defined in terms of a vector potential instead ofa scalar stream function as in the QG case They showed thatan analytical solution can be found to this QG+1 system ofequations analogous to the QG caseThis is a powerful exten-sion of the ellipsoidal theory in geophysical fluid dynamicshaving the potential to capture many features not seen in theQG limit For one thing in the QG+1 model there is a verticalvelocity unlike the QG model Another limitation of the QGmodel is that there is complete symmetry between cyclonicand anticyclonic vortices whereas in reality the Earth rotatesin one direction giving rise to an asymmetry in the behaviourof cyclones and anticyclones The next order to QG picks upthis asymmetry and so is much more applicable to a widerrange of geophysical flows Potential future studies of these

higher-order solutions could determine equilibria and theirstability properties providing insights into turbulence beyondthe large-scale approximation Also the development of amultiellipsoidal vortex theory beyond QG would have hugebenefits in understandingmore realistic turbulence phenom-ena The exterior field in QG+1 is much more complicatedthan the QG case and so the projection of this onto a linearflow field that preserves ellipsoidal formwill filtermore of thedynamics However even understanding the limits of such amodel would be extremely revealing

The application of the ellipsoidal form to geophysicalvortex dynamics has only a short history beginning withthe works of Zhmur and Pankratov [8 9] and independentlyMeacham [10] Despite this relatively short period it alreadyhas yieldedmany insights into vortex interactions in geophys-ical flows These vortex interactions are crucial as they are ahuge driving force within the atmospheric and the oceaniccirculation There is still great potential for this subject toreveal more about the complex dynamics within geophysicalturbulence

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work has been funded by the Italian Flagship ProjectRITMARE The Italian Research for the Sea coordinated bythe Italian National Research Council and funded by theItalian Ministry of Education University and Research

References

[1] D B CheltonMG Schlax and RM Samelson ldquoGlobal obser-vations of nonlinear mesoscale eddiesrdquo Progress in Oceanogra-phy vol 91 no 2 pp 167ndash216 2011

[2] Z Zhang W Wang and B Qiu ldquoOceanic mass transport bymesoscale eddiesrdquo Science vol 345 pp 322ndash324 2014

[3] J C McWilliams J B Weiss and I Yavneh ldquoAnisotropy andcoherent vortex structures in planetary turbulencerdquo Science vol264 no 5157 pp 410ndash413 1994

[4] K Shaper Smith and G K Vallis ldquoThe scales and equilibrationof Midocean Eddies freely evolving flowrdquo Journal of PhysicalOceanography vol 31 no 2 pp 554ndash571 2001

[5] W J Mckiver and D G Dritschel ldquoBalance in non-hydrostaticrotating stratified turbulencerdquo Journal of Fluid Mechanics vol596 pp 201ndash219 2008

[6] G Kirchhoff Vorlesungen uber Matematische Physic Mechan-ik Teubner Leipzig1876

[7] A E Love ldquoOn the stability of certain vortex motionsrdquo Pro-ceedings of the London Mathematical Society vol 25 pp 18ndash421893

[8] V V Zhmur and K K Pankratov ldquoDynamics of a semi-ellipsoidal subsurface vortex in a nonuniform flowrdquo Okeanolo-gia vol 29 pp 150ndash154 1989

[9] V V Zhmur and K K Pankratov ldquoDynamics of mesoscaleeddy formation in the field currents of large intensive vortexrdquoOkeanologia vol 30 pp 124ndash129 1990


[10] S P Meacham ldquoQuasigeostrophic ellipsoidal vortices in a stra-tified fluidrdquo Dynamics of Atmospheres and Oceans vol 16 no3-4 pp 189ndash223 1992

[11] J G Charney ldquoOn the scale of atmosphericmotionsrdquoGeofysiskePublikasjoner vol 17 no 2 pp 3ndash17 1948

[12] C MacClaurin A Treatise on Fluxions vol 1-2 Printed by T Wand T Ruddimans Edinburgh UK 1742

[13] P S Laplace Laplace Pierre-Simon (1749-1827) Theorie duMouvement et de la Figure Elliptique Des Planetes ParisImprimerie de Ph-D Pierres 1784

[14] I TodhunterHistory of the Mathematical Theories of Attractionand the Figure of the Earth Constable London UK 1873

[15] S Chandrasekhar Ellipsoidal Figures of Equilibrium Dover1969

[16] G Dassios Ellipsoidal Harmonics Theory and ApplicationsCambridge University Press Cambridge UK 2012

[17] W J McKiver and D G Dritschel ldquoThe motion of a fluid ellip-soid in a general linear background flowrdquo Journal of FluidMech-anics vol 474 pp 147ndash173 2003

[18] D G Dritschel J N Reinaud and W J McKiver ldquoThe quasi-geostrophic ellipsoidal vortex modelrdquo Journal of Fluid Mech-anics vol 505 pp 201ndash223 2004

[19] J Pedlosky Geophysical Fluid Dynamics Springer New YorkNY USA 1979

[20] J G Charney ldquoGeostrophic turbulencerdquo Journal of the Atmo-spheric Sciences vol 28 no 6 pp 1087ndash1095 1971

[21] B J Hoskins M E McIntyre and A W Robertson ldquoOn theuse and significance of isentropic potential vorticity mapsrdquoQuarterly JournalmdashRoyal Meteorological Society vol 111 no470 pp 877ndash946 1985

[22] W R Holland ldquoQuasigeostrophic modelling of Eddy-resolvedocean circulationrdquo inAdvanced Physical Oceanographic Numer-ical Modelling J J OrsquoBrien Ed vol 186 of NATO ASI Series CMathematical and Physical pp 203ndash231 Springer DordrechtNetherlands 1986

[23] B Deremble G Lapeyre and M Ghil ldquoAtmospheric dynamicstriggered by an oceanic SST front in a moist quasigeostrophicmodelrdquo Journal of the Atmospheric Sciences vol 69 no 5 pp1617ndash1632 2012

[24] V V Zhmur and A F Shchepetkin ldquoEvolution of an ellipsoidalvortex in a stratified ocean Survivability of the vortex in aflow with vertical shearrdquo Izvestia Akademii Nauk SSSR FizikaAtmosfery i Okeana vol 27 pp 492ndash503 1991

[25] R A Lyttleton The Stability of Rotating Liquid Masses Cam-bridge University Paris France 1953

[26] E W Hobson Theory of Spherical and Ellipsoidal HarmonicsCambridge University Press 1931

[27] S P Meacham K K Pankratov A F Shchepetkin and V VZhmur ldquoThe interaction of ellipsoidal vorticeswith backgroundshear flows in a stratified fluidrdquo Dynamics of Atmospheres andOceans vol 21 no 2-3 pp 167ndash212 1994

[28] S P Meacham P J Morrison and G R Flierl ldquoHamiltonianmoment reduction for describing vortices in shearrdquo Physics ofFluids vol 9 no 8 pp 2310ndash2328 1997

[29] H Hashimoto T Shimonishi and T Miyazaki ldquoQuasigeo-strophic ellipsoidal vortices in a two-dimensional strain fieldrdquoJournal of the Physical Society of Japan vol 68 no 12 pp 3863ndash3880 1999

[30] J N Reinaud D G Dritschel and C R Koudella ldquoThe shapeof vortices in quasi-geostrophic turbulencerdquo Journal of FluidMechanics no 474 pp 175ndash192 2003

[31] D G Dritschel R K Scott and J N Reinaud ldquoThe stability ofquasi-geostrophic ellipsoidal vorticesrdquo Journal of FluidMechan-ics vol 536 pp 401ndash421 2005

[32] T Miyazaki K Ueno and T Shimonishi ldquoQuasigeostrophictilted spheroidal vorticesrdquo Journal of the Physical Society ofJapan vol 68 no 8 pp 2592ndash2601 1999

[33] W J McKiver and D G Dritschel ldquoThe stability of a quasi-geostrophic ellipsoidal vortex in a background shear flowrdquoJournal of Fluid Mechanics vol 560 pp 1ndash17 2006

[34] J N Reinaud and D G Dritschel ldquoThe merger of verticallyoffset quasi-geostrophic vorticesrdquo Journal of Fluid Mechanicsvol 469 pp 287ndash315 2002

[35] V V Zhmur and K K Pankratov ldquoDistant interaction of anensemble of quasigeostrophic ellipsoidal eddies HamiltonianformulationrdquoAtmospheric andOceanic Physics vol 26 pp 972ndash981 1990

[36] T Miyazaki Y Furuichi and N Takahashi ldquoQuasigeostrophicellipsoidal vortexmodelrdquo Journal of the Physical Society of Japanvol 70 no 7 pp 1942ndash1953 2001

[37] Y Li H Taira N Takahashi and T Miyazaki ldquoRefinementson the quasi-geostrophic ellipsoidal vortex modelrdquo Physics ofFluids vol 18 no 7 Article ID 076604 2006

[38] N Martinsen-Burrell K Julien M R Petersen and J BWeiss ldquoMerger and alignment in a reduced model for three-dimensional quasigeostrophic ellipsoidal vorticesrdquo Physics ofFluids vol 18 no 5 Article ID 057101 2006

[39] J N Reinaud and D G Dritschel ldquoThe critical merger distancebetween two co-rotating quasi-geostrophic vorticesrdquo Journal ofFluid Mechanics vol 522 pp 357ndash381 2005

[40] TMiyazakiM Yamamoto and S Fujishima ldquoCounter-rotatingquasigeostrophic ellipsoidal vortex pairrdquo Journal of the PhysicalSociety of Japan vol 72 no 8 pp 1948ndash1962 2003

[41] K V Koshel E A Ryzhov and V V Zhmur ldquoDiffusion-affectedpassive scalar transport in an ellipsoidal vortex in a shear flowrdquoNonlinear Processes in Geophysics vol 20 no 4 pp 437ndash4442013

[42] Y-K Tsang andD G Dritschel ldquoEllipsoidal vortices in rotatingstratified fluids beyond the quasi-geostrophic approximationrdquoJournal of Fluid Mechanics vol 762 pp 196ndash231 2015

[43] W J McKiver and D G Dritschel ldquoBalanced solutions for anellipsoidal vortex in a rotating stratified flowrdquo Journal of FluidMechanics submitted

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


that yet goes beyond the absolute symmetry of the sphereit is a powerful tool in determining shape structure andorientation of ldquobodiesrdquo concerned in many different fields ofapplication (see [16] for a review of the many applications)In the case of geophysical flows it gives a means to determinethe characteristics of vortices (such as typical height-to-widthratios orientations and rotation rates) that are the mostrobust and hence the most likely to occur in real turbulentflows

Here we will review the recent history of the developmentof the ellipsoidal vortex in quasi-geostrophic dynamics Thebeginning is in Section 2wherewe give a brief overview of thegoverning system of equations the quasi-geostrophic model[11] Following this we review the ellipsoidal vortex modelstarting in Section 3 with the case of the isolated ellipsoid asoriginally derived byMeacham [10] In Section 4 we considerthe case of an ellipsoidal vortex immersed in a backgroundshear flow following the matrix formulation approach ofMcKiver and Dritschel [17] In Section 5 we discuss theresults of studies of the equilibria and stability properties ofthis vortex system and what insights they provide for thefull solution to the equations in turbulence simulations InSection 6 we discuss the generalization of the single vortexellipsoidal theory to the multiple vortex case in particularoutlining the Hamiltonian method of Dritschel et al [18]Finally we conclude in Section 7 with an overview of theconsequences the ellipsoidal model has for the theory ofturbulence and discuss future avenues of research in this field

2 The Quasi-Geostrophic System

Here we summarize the quasi-geostrophic model (for adetailed derivation see [19]) Quasi-geostrophic (hereafterQG) theory is one of the simplest geophysical fluid dynamicalmodels that incorporates two crucial features underpinninglarge-scale fluid dynamics (1) the effect of the Earthrsquos rotationand (2) the effects of strong density stratification (lighterfluid lying on top of denser fluid) As a consequence ofthese features certain terms in the full equations of motion(Navier-Stokes equations) tend to dominate over othersStrong rotation leads to what is termed ldquogeostrophic balancerdquowhich is a balance between the horizontal pressure gradientand the term associated with the Earthrsquos rotation (the Coriolisterm)

119891k times uℎ = minus

nabla119901

1205880

(1)

where 119891 is the local vertical component of the planetaryrotation rate k is a vertical unit vector uℎ is the horizontalvelocity field 119901 is pressure and 1205880(119911) is the basic-statedensity distribution (1205880 is nearly constant in the oceans andexponential proportional to exp(minus119911119867) with 119867 asymp 7 km inthe atmosphere) Strong stratification leads to ldquohydrostaticbalancerdquo which is a balance between the vertical pressuregradient and buoyancy

120597119901

120597119911

= minus1205880119892(2)

where 119892 is the acceleration due to gravity The QG systemwhich incorporates both hydrostatic and geostrophic bal-ance at leading order is obtained from the full ldquoprimitiverdquoequations by an asymptotic expansion in three small nondi-mensional parameters (1) the height-width aspect ratio ofcharacteristic motions 119867119871 (constrained to be small due tothe shallow geometry in the oceans and atmosphere) (2) theRossby number 119877119900 equiv 119880119891119871 and (3) the Froude number119865119903 equiv 119880119873119867 where 119880 is a characteristic relative horizontalfluid speed 119871 is the horizontal scale of the flow 119867 is theaverage fluid depth and 119873 is the buoyancy frequency (theoscillation frequency that a small fluid volume would exhibitif it were displaced by a small vertical distance) The QGmodel is obtained assuming that119867119871 ≪ 1 and 1198652

119903≪ 119877119900 ≪ 1

Also in this model we consider the limit of infinite Reynoldsnumber Re equiv 119880119871] that is the effect of viscous dissipation] is negligible (in the real atmosphere Re sim 10

10ndash1012) Asdone inmany previous works one can consider the constant-coefficient (ldquooceanicrdquo) system when 119891 119873 and 1205880 are allconstant For an inviscid adiabatic fluid the QG equationsmay be expressed in terms of the conservation of a singlematerial scalar the ldquopotential vorticityrdquo 119902(x 119905) that is

119863119902

119863119905

equiv

120597119902

120597119905

+ 119906

120597119902

120597119909

+ V120597119902

120597119910

= 0 (3)

The potential vorticity (hereafter PV) is proportional tothe component of vorticity perpendicular to stratificationsurfaces (surfaces of constant density 1205880) Thus vortices areidentified as contiguous regions of PV In the QG model thevertical velocity 119908 = 0 to this order of approximation so 119902

is advected in a layerwise manner However the horizontalvelocity (119906 V) depends on all three spatial coordinates andthis is recovered from 119902 via the inversion of

nabla2120595 = 119902 (4)

for the stream function 120595 followed by the incompressibilityrelations

119906 = minus

120597120595

120597119910

V =120597120595

120597119909

(5)

Note that in this form the vertical coordinate 119911 is rescaledby the factor 119891119873 The three-dimensional distribution ofPV generates the velocity field through (4) and (5) whichcarries the PV conservatively to the next instant of timeWhile (4) is an isotropic relation between 119902 and 120595 theadvection remains anisotropic as there is no vertical motionThe QG equations are widely used in studies of atmosphericand oceanic phenomena [20ndash23] in part because they arefar easier to deal with both theoretically and numericallythan the complete equations But also the QG equationscapture many qualitative features of atmospheric and oceanicdynamics

3 Potential Theory of an IsolatedEllipsoidal Vortex

Now we consider the case of an isolated ellipsoidal vortexwith uniform potential vorticity as first considered in the



119909

2

119886

2+

119910

2

119887

2+

119911

2

119888

2= 1

(6)


nabla2120595119894 = 119902 (7)


120595119894 =

1

2

(119891119909

2+ 119892119910

2+ ℎ119911

2) + 119862119894

(8)


119891 + 119892 + ℎ = 119902 (9)


nabla2120595119890 = 0 (10)


119909

2=

(120582 + 119886

2) (120583 + 119886

2) (] + 119886

2)

(119886

2minus 119887

2) (119886

2minus 119888

2)

(11a)

119910

2=

(120582 + 119887

2) (120583 + 119887

2) (] + 119887

2)

(119887

2minus 119888

2) (119887

2minus 119886

2)

(11b)

119911

2=

(120582 + 119888

2) (120583 + 119888

2) (] + 119888

2)

(119888

2minus 119886

2) (119888

2minus 119887

2)

(11c)


119909

2

119886

2+ 120594

+

119910

2

119887

2+ 120594

+

119911

2

119888

2+ 120594

= 1 (12)


2gt 120583 gt minus119887




120595119894

1003816100381610038161003816120582=0

= 120595119890

1003816100381610038161003816120582=0

(13a)

120597120595119894

120597120582

10038161003816100381610038161003816100381610038161003816120582=0

=

120597120595119890

120597120582

10038161003816100381610038161003816100381610038161003816120582=0

(13b)


120595119894 =

119891 (120582 + 119886

2) [120583] + 119886

2(120583 + ]) + 119886

4]

2 (119886

2minus 119887

2) (119886

2minus 119888

2)

+

119892 (120582 + 119887

2) [120583] + 119887

2(120583 + ]) + 119887

4]

2 (119887

2minus 119888

2) (119887

2minus 119886

2)

+

ℎ (120582 + 119888

2) [120583] + 119888

2(120583 + ]) + 119888

4]

2 (119888

2minus 119886

2) (119888

2minus 119887

2)

+ 119862119894

(14)


120601119890 =

infin

sum

119898

2119898+1

sum

119896

F(119896)119898

(120582)E(119896)119898

(120583)E(119896)119898

(]) (15)

where E(119896)119898



F(119896)119898

(120582)

=

2119898 + 1

2

E(119896)119898

(120582)

sdot int

infin

0

119889119904

[E(119896)119898 (119904 + 120582)]

2

radic(119904 + 120582 + 119886

2) (119904 + 120582 + 119887

2) (119904 + 120582 + 119888

2)


120601119890 = 1198620F(1)

0(120582) + 119862

(1)

2F(1)2

(120582)E(1)2

(120583)E(1)2

(])

+ 119862

(2)

2F(2)2

(120582)E(2)2

(120583)E(2)2

(]) (17)

where 1198620 119862(1)

2 and 119862

(2)



E(1)2

(120582) = 120582 + 1198891(18a)

E(2)2

(120582) = 120582 + 1198892(18b)

where

1198891 =

1

3

[ (119886

2+ 119887

2+ 119888

2)

+radic119886

4+ 119887

4+ 119888

4minus 119887

2119888

2minus 119888

2119886

2minus 119886

2119887

2]

(19a)

1198892 =

1

3

[ (119886

2+ 119887

2+ 119888

2)

minusradic119886

4+ 119887

4+ 119888

4minus 119887

2119888

2minus 119888

2119886

2minus 119886

2119887

2]

(19b)



120595119890 = 120583] [119862(1)2F(1)2

(120582) + 119862

(2)

2F(2)2

(120582)]

+ (120583 + ]) [1198891119862(1)

2F(1)2

(120582) + 1198892119862(2)

2F(2)2

(120582)]

+ 1198620F(1)

0(120582) + 119889

2

1119862

(1)

2F(1)2

(120582) + 119889

2

2119862

(2)

2F(2)2

(120582)

(20)


120595119894 =

1

2

(1205851198861199092+ 1205851198871199102+ 1205851198881199112) minus

3

2

120581V119877119865 (1198862 119887

2 119888

2) (21)


120585119886 = 120581V119877119863 (1198872 119888

2 119886

2) (22a)

120585119887 = 120581V119877119863 (1198882 119886

2 119887

2) (22b)

120585119888 = 120581V119877119863 (1198862 119887

2 119888

2) (22c)



119877119865 (119909 119910 119911) equiv

1

2

int

infin

0

119889119905

radic(119905 + 119909) (119905 + 119910) (119905 + 119911)

(23a)

119877119863 (119909 119910 119911) equiv

3

2

int

infin

0

119889119905

radic(119905 + 119909) (119905 + 119910) (119905 + 119911)

3

(23b)


120595119890 =

1

2

(1205851205721199092+ 120585120573119910

2+ 1205851205741199112) minus

3

2

120581V119877119865 (1205722 120573

2 120574

2) (24)

where 1205722 = 119886

2+ 120582 1205732 = 119887

2+ 120582 and 1205742 = 119888


have

120585120572 = 120581V119877119863 (1205732 120574

2 120572

2) (25a)

120585120573 = 120581V119877119863 (1205742 120572

2 120573

2) (25b)

120585120574 = 120581V119877119863 (1205722 120573

2 120574

2) (25c)




u119887 (x 119905) = S119887 (119905) x (26)


u = Sx (27)




A = MDM119879 (28)


M = (a b c) (29)


D = (

1

119886

20 0

0

1

119887

20

0 0

1

119888

2

) (30)


x119879Ax = 1 (31)



x119879S119879(119905) we obtain

x119879 (AS + S119879A +

119889A119889119905

) x = 0 (32)


119889A119889119905

= minus (AS + S119879A) (33)



119889B119889119905

= minus B119889A119889119905

B

= B (AS + S119879A)B(34)


119889B119889119905

= SB + BS119879 (35)

where

B = MDminus1M119879 (36)

Dminus1 = (

119886

20 0

0 119887

20

0 0 119888

2

) (37)


Ba = 119886

2a (38a)

Bb = 119887

2b (38b)

Bc = 119888

2c (38c)


120595V =1

2

x119879PVx minus3

2

120581V119877119865 (1198862 119887

2 119888

2) (39)

where

PV = MGM119879 (40)


G = (

120585119886 0 0

0 120585119887 0

0 0 120585119888

) (41)


SV = LPV (42)


L = (

0 minus1 0

1 0 0

0 0 0

) (43)


Ω =

120581119887 + 120581V

119877

3 (44)


120595119887 = minus

120581119887

119877

+

1

2

x119879P119887x + 119874(

120581119887 |x|3

119877

4) (45)


P119887 =120581119887

119877

5(

119877

2minus 3119883

2

119887minus3119883119887119884119887 minus3119883119887119885119887

minus3119883119887119884119887 119877

2minus 3119884

2

119887minus3119884119887119885119887

minus3119883119887119885119887 minus3119884119887119885119887 119877

2minus 3119885

2

119887

)

minus(

Ω 0 0

0 Ω 0

0 0 0

)

(46)


S119887 = 120574(

3119883119884 3

119884

2minus 1 + 120573 3

119884119885

1 minus 120573 minus 3119883

2minus3

119883119884 minus3

119883119885

0 0 0

) (47)


120574 equiv

120581119887

119877

3 (48)



120573 equiv

Ω

120574

=

120581119887 + 120581V

120581119887

(49)


119883


S119887 = 120574(

0

1

2

(1 + 3 cos 2120579) + 120573

3

2

sin 21205791 minus 120573 0 0

0 0 0

) (50)



∘ 90








SB + BS119879 = 0 (51)







∘le 120579 le 75

∘ and 20

∘le

120579 le 30








u119894 (x 119905) = U119894 (119905) + S119894 (119905) (x minus X119894) (53)


119889X119894119889119905

= minus

1

120581119894

L 120597119867120597X119894

(54a)

119889B119894119889119905

= S119894B119894 + B119894S119879

119894 (54b)

S119894 = minus

10

120581119894

L 120597119867120597B119894

(54c)






(a)

(b)




S119894B119894 + B119894S119879

119894= 0 (55)


120575 =

1003816100381610038161003816

X1 minus X21003816100381610038161003816





7 Conclusions








Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909

2

119886

2+

119910

2

119887

2+

119911

2

119888

2= 1

(6)


nabla2120595119894 = 119902 (7)


120595119894 =

1

2

(119891119909

2+ 119892119910

2+ ℎ119911

2) + 119862119894

(8)


119891 + 119892 + ℎ = 119902 (9)


nabla2120595119890 = 0 (10)


119909

2=

(120582 + 119886

2) (120583 + 119886

2) (] + 119886

2)

(119886

2minus 119887

2) (119886

2minus 119888

2)

(11a)

119910

2=

(120582 + 119887

2) (120583 + 119887

2) (] + 119887

2)

(119887

2minus 119888

2) (119887

2minus 119886

2)

(11b)

119911

2=

(120582 + 119888

2) (120583 + 119888

2) (] + 119888

2)

(119888

2minus 119886

2) (119888

2minus 119887

2)

(11c)


119909

2

119886

2+ 120594

+

119910

2

119887

2+ 120594

+

119911

2

119888

2+ 120594

= 1 (12)


2gt 120583 gt minus119887




120595119894

1003816100381610038161003816120582=0

= 120595119890

1003816100381610038161003816120582=0

(13a)

120597120595119894

120597120582

10038161003816100381610038161003816100381610038161003816120582=0

=

120597120595119890

120597120582

10038161003816100381610038161003816100381610038161003816120582=0

(13b)


120595119894 =

119891 (120582 + 119886

2) [120583] + 119886

2(120583 + ]) + 119886

4]

2 (119886

2minus 119887

2) (119886

2minus 119888

2)

+

119892 (120582 + 119887

2) [120583] + 119887

2(120583 + ]) + 119887

4]

2 (119887

2minus 119888

2) (119887

2minus 119886

2)

+

ℎ (120582 + 119888

2) [120583] + 119888

2(120583 + ]) + 119888

4]

2 (119888

2minus 119886

2) (119888

2minus 119887

2)

+ 119862119894

(14)


120601119890 =

infin

sum

119898

2119898+1

sum

119896

F(119896)119898

(120582)E(119896)119898

(120583)E(119896)119898

(]) (15)

where E(119896)119898



F(119896)119898

(120582)

=

2119898 + 1

2

E(119896)119898

(120582)

sdot int

infin

0

119889119904

[E(119896)119898 (119904 + 120582)]

2

radic(119904 + 120582 + 119886

2) (119904 + 120582 + 119887

2) (119904 + 120582 + 119888

2)


120601119890 = 1198620F(1)

0(120582) + 119862

(1)

2F(1)2

(120582)E(1)2

(120583)E(1)2

(])

+ 119862

(2)

2F(2)2

(120582)E(2)2

(120583)E(2)2

(]) (17)

where 1198620 119862(1)

2 and 119862

(2)



E(1)2

(120582) = 120582 + 1198891(18a)

E(2)2

(120582) = 120582 + 1198892(18b)

where

1198891 =

1

3

[ (119886

2+ 119887

2+ 119888

2)

+radic119886

4+ 119887

4+ 119888

4minus 119887

2119888

2minus 119888

2119886

2minus 119886

2119887

2]

(19a)

1198892 =

1

3

[ (119886

2+ 119887

2+ 119888

2)

minusradic119886

4+ 119887

4+ 119888

4minus 119887

2119888

2minus 119888

2119886

2minus 119886

2119887

2]

(19b)



120595119890 = 120583] [119862(1)2F(1)2

(120582) + 119862

(2)

2F(2)2

(120582)]

+ (120583 + ]) [1198891119862(1)

2F(1)2

(120582) + 1198892119862(2)

2F(2)2

(120582)]

+ 1198620F(1)

0(120582) + 119889

2

1119862

(1)

2F(1)2

(120582) + 119889

2

2119862

(2)

2F(2)2

(120582)

(20)


120595119894 =

1

2

(1205851198861199092+ 1205851198871199102+ 1205851198881199112) minus

3

2

120581V119877119865 (1198862 119887

2 119888

2) (21)


120585119886 = 120581V119877119863 (1198872 119888

2 119886

2) (22a)

120585119887 = 120581V119877119863 (1198882 119886

2 119887

2) (22b)

120585119888 = 120581V119877119863 (1198862 119887

2 119888

2) (22c)



119877119865 (119909 119910 119911) equiv

1

2

int

infin

0

119889119905

radic(119905 + 119909) (119905 + 119910) (119905 + 119911)

(23a)

119877119863 (119909 119910 119911) equiv

3

2

int

infin

0

119889119905

radic(119905 + 119909) (119905 + 119910) (119905 + 119911)

3

(23b)


120595119890 =

1

2

(1205851205721199092+ 120585120573119910

2+ 1205851205741199112) minus

3

2

120581V119877119865 (1205722 120573

2 120574

2) (24)

where 1205722 = 119886

2+ 120582 1205732 = 119887

2+ 120582 and 1205742 = 119888


have

120585120572 = 120581V119877119863 (1205732 120574

2 120572

2) (25a)

120585120573 = 120581V119877119863 (1205742 120572

2 120573

2) (25b)

120585120574 = 120581V119877119863 (1205722 120573

2 120574

2) (25c)




u119887 (x 119905) = S119887 (119905) x (26)


u = Sx (27)




A = MDM119879 (28)


M = (a b c) (29)


D = (

1

119886

20 0

0

1

119887

20

0 0

1

119888

2

) (30)


x119879Ax = 1 (31)



x119879S119879(119905) we obtain

x119879 (AS + S119879A +

119889A119889119905

) x = 0 (32)


119889A119889119905

= minus (AS + S119879A) (33)



119889B119889119905

= minus B119889A119889119905

B

= B (AS + S119879A)B(34)


119889B119889119905

= SB + BS119879 (35)

where

B = MDminus1M119879 (36)

Dminus1 = (

119886

20 0

0 119887

20

0 0 119888

2

) (37)


Ba = 119886

2a (38a)

Bb = 119887

2b (38b)

Bc = 119888

2c (38c)


120595V =1

2

x119879PVx minus3

2

120581V119877119865 (1198862 119887

2 119888

2) (39)

where

PV = MGM119879 (40)


G = (

120585119886 0 0

0 120585119887 0

0 0 120585119888

) (41)


SV = LPV (42)


L = (

0 minus1 0

1 0 0

0 0 0

) (43)


Ω =

120581119887 + 120581V

119877

3 (44)


120595119887 = minus

120581119887

119877

+

1

2

x119879P119887x + 119874(

120581119887 |x|3

119877

4) (45)


P119887 =120581119887

119877

5(

119877

2minus 3119883

2

119887minus3119883119887119884119887 minus3119883119887119885119887

minus3119883119887119884119887 119877

2minus 3119884

2

119887minus3119884119887119885119887

minus3119883119887119885119887 minus3119884119887119885119887 119877

2minus 3119885

2

119887

)

minus(

Ω 0 0

0 Ω 0

0 0 0

)

(46)


S119887 = 120574(

3119883119884 3

119884

2minus 1 + 120573 3

119884119885

1 minus 120573 minus 3119883

2minus3

119883119884 minus3

119883119885

0 0 0

) (47)


120574 equiv

120581119887

119877

3 (48)



120573 equiv

Ω

120574

=

120581119887 + 120581V

120581119887

(49)


119883


S119887 = 120574(

0

1

2

(1 + 3 cos 2120579) + 120573

3

2

sin 21205791 minus 120573 0 0

0 0 0

) (50)



∘ 90








SB + BS119879 = 0 (51)







∘le 120579 le 75

∘ and 20

∘le

120579 le 30








u119894 (x 119905) = U119894 (119905) + S119894 (119905) (x minus X119894) (53)


119889X119894119889119905

= minus

1

120581119894

L 120597119867120597X119894

(54a)

119889B119894119889119905

= S119894B119894 + B119894S119879

119894 (54b)

S119894 = minus

10

120581119894

L 120597119867120597B119894

(54c)






(a)

(b)




S119894B119894 + B119894S119879

119894= 0 (55)


120575 =

1003816100381610038161003816

X1 minus X21003816100381610038161003816





7 Conclusions








Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120595119890 = 120583] [119862(1)2F(1)2

(120582) + 119862

(2)

2F(2)2

(120582)]

+ (120583 + ]) [1198891119862(1)

2F(1)2

(120582) + 1198892119862(2)

2F(2)2

(120582)]

+ 1198620F(1)

0(120582) + 119889

2

1119862

(1)

2F(1)2

(120582) + 119889

2

2119862

(2)

2F(2)2

(120582)

(20)


120595119894 =

1

2

(1205851198861199092+ 1205851198871199102+ 1205851198881199112) minus

3

2

120581V119877119865 (1198862 119887

2 119888

2) (21)


120585119886 = 120581V119877119863 (1198872 119888

2 119886

2) (22a)

120585119887 = 120581V119877119863 (1198882 119886

2 119887

2) (22b)

120585119888 = 120581V119877119863 (1198862 119887

2 119888

2) (22c)



119877119865 (119909 119910 119911) equiv

1

2

int

infin

0

119889119905

radic(119905 + 119909) (119905 + 119910) (119905 + 119911)

(23a)

119877119863 (119909 119910 119911) equiv

3

2

int

infin

0

119889119905

radic(119905 + 119909) (119905 + 119910) (119905 + 119911)

3

(23b)


120595119890 =

1

2

(1205851205721199092+ 120585120573119910

2+ 1205851205741199112) minus

3

2

120581V119877119865 (1205722 120573

2 120574

2) (24)

where 1205722 = 119886

2+ 120582 1205732 = 119887

2+ 120582 and 1205742 = 119888


have

120585120572 = 120581V119877119863 (1205732 120574

2 120572

2) (25a)

120585120573 = 120581V119877119863 (1205742 120572

2 120573

2) (25b)

120585120574 = 120581V119877119863 (1205722 120573

2 120574

2) (25c)




u119887 (x 119905) = S119887 (119905) x (26)


u = Sx (27)




A = MDM119879 (28)


M = (a b c) (29)


D = (

1

119886

20 0

0

1

119887

20

0 0

1

119888

2

) (30)


x119879Ax = 1 (31)



x119879S119879(119905) we obtain

x119879 (AS + S119879A +

119889A119889119905

) x = 0 (32)


119889A119889119905

= minus (AS + S119879A) (33)



119889B119889119905

= minus B119889A119889119905

B

= B (AS + S119879A)B(34)


119889B119889119905

= SB + BS119879 (35)

where

B = MDminus1M119879 (36)

Dminus1 = (

119886

20 0

0 119887

20

0 0 119888

2

) (37)


Ba = 119886

2a (38a)

Bb = 119887

2b (38b)

Bc = 119888

2c (38c)


120595V =1

2

x119879PVx minus3

2

120581V119877119865 (1198862 119887

2 119888

2) (39)

where

PV = MGM119879 (40)


G = (

120585119886 0 0

0 120585119887 0

0 0 120585119888

) (41)


SV = LPV (42)


L = (

0 minus1 0

1 0 0

0 0 0

) (43)


Ω =

120581119887 + 120581V

119877

3 (44)


120595119887 = minus

120581119887

119877

+

1

2

x119879P119887x + 119874(

120581119887 |x|3

119877

4) (45)


P119887 =120581119887

119877

5(

119877

2minus 3119883

2

119887minus3119883119887119884119887 minus3119883119887119885119887

minus3119883119887119884119887 119877

2minus 3119884

2

119887minus3119884119887119885119887

minus3119883119887119885119887 minus3119884119887119885119887 119877

2minus 3119885

2

119887

)

minus(

Ω 0 0

0 Ω 0

0 0 0

)

(46)


S119887 = 120574(

3119883119884 3

119884

2minus 1 + 120573 3

119884119885

1 minus 120573 minus 3119883

2minus3

119883119884 minus3

119883119885

0 0 0

) (47)


120574 equiv

120581119887

119877

3 (48)



120573 equiv

Ω

120574

=

120581119887 + 120581V

120581119887

(49)


119883


S119887 = 120574(

0

1

2

(1 + 3 cos 2120579) + 120573

3

2

sin 21205791 minus 120573 0 0

0 0 0

) (50)



∘ 90








SB + BS119879 = 0 (51)







∘le 120579 le 75

∘ and 20

∘le

120579 le 30








u119894 (x 119905) = U119894 (119905) + S119894 (119905) (x minus X119894) (53)


119889X119894119889119905

= minus

1

120581119894

L 120597119867120597X119894

(54a)

119889B119894119889119905

= S119894B119894 + B119894S119879

119894 (54b)

S119894 = minus

10

120581119894

L 120597119867120597B119894

(54c)






(a)

(b)




S119894B119894 + B119894S119879

119894= 0 (55)


120575 =

1003816100381610038161003816

X1 minus X21003816100381610038161003816





7 Conclusions








Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











x119879S119879(119905) we obtain

x119879 (AS + S119879A +

119889A119889119905

) x = 0 (32)


119889A119889119905

= minus (AS + S119879A) (33)



119889B119889119905

= minus B119889A119889119905

B

= B (AS + S119879A)B(34)


119889B119889119905

= SB + BS119879 (35)

where

B = MDminus1M119879 (36)

Dminus1 = (

119886

20 0

0 119887

20

0 0 119888

2

) (37)


Ba = 119886

2a (38a)

Bb = 119887

2b (38b)

Bc = 119888

2c (38c)


120595V =1

2

x119879PVx minus3

2

120581V119877119865 (1198862 119887

2 119888

2) (39)

where

PV = MGM119879 (40)


G = (

120585119886 0 0

0 120585119887 0

0 0 120585119888

) (41)


SV = LPV (42)


L = (

0 minus1 0

1 0 0

0 0 0

) (43)


Ω =

120581119887 + 120581V

119877

3 (44)


120595119887 = minus

120581119887

119877

+

1

2

x119879P119887x + 119874(

120581119887 |x|3

119877

4) (45)


P119887 =120581119887

119877

5(

119877

2minus 3119883

2

119887minus3119883119887119884119887 minus3119883119887119885119887

minus3119883119887119884119887 119877

2minus 3119884

2

119887minus3119884119887119885119887

minus3119883119887119885119887 minus3119884119887119885119887 119877

2minus 3119885

2

119887

)

minus(

Ω 0 0

0 Ω 0

0 0 0

)

(46)


S119887 = 120574(

3119883119884 3

119884

2minus 1 + 120573 3

119884119885

1 minus 120573 minus 3119883

2minus3

119883119884 minus3

119883119885

0 0 0

) (47)


120574 equiv

120581119887

119877

3 (48)



120573 equiv

Ω

120574

=

120581119887 + 120581V

120581119887

(49)


119883


S119887 = 120574(

0

1

2

(1 + 3 cos 2120579) + 120573

3

2

sin 21205791 minus 120573 0 0

0 0 0

) (50)



∘ 90








SB + BS119879 = 0 (51)







∘le 120579 le 75

∘ and 20

∘le

120579 le 30








u119894 (x 119905) = U119894 (119905) + S119894 (119905) (x minus X119894) (53)


119889X119894119889119905

= minus

1

120581119894

L 120597119867120597X119894

(54a)

119889B119894119889119905

= S119894B119894 + B119894S119879

119894 (54b)

S119894 = minus

10

120581119894

L 120597119867120597B119894

(54c)






(a)

(b)




S119894B119894 + B119894S119879

119894= 0 (55)


120575 =

1003816100381610038161003816

X1 minus X21003816100381610038161003816





7 Conclusions








Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120573 equiv

Ω

120574

=

120581119887 + 120581V

120581119887

(49)


119883


S119887 = 120574(

0

1

2

(1 + 3 cos 2120579) + 120573

3

2

sin 21205791 minus 120573 0 0

0 0 0

) (50)



∘ 90








SB + BS119879 = 0 (51)







∘le 120579 le 75

∘ and 20

∘le

120579 le 30








u119894 (x 119905) = U119894 (119905) + S119894 (119905) (x minus X119894) (53)


119889X119894119889119905

= minus

1

120581119894

L 120597119867120597X119894

(54a)

119889B119894119889119905

= S119894B119894 + B119894S119879

119894 (54b)

S119894 = minus

10

120581119894

L 120597119867120597B119894

(54c)






(a)

(b)




S119894B119894 + B119894S119879

119894= 0 (55)


120575 =

1003816100381610038161003816

X1 minus X21003816100381610038161003816





7 Conclusions








Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















u119894 (x 119905) = U119894 (119905) + S119894 (119905) (x minus X119894) (53)


119889X119894119889119905

= minus

1

120581119894

L 120597119867120597X119894

(54a)

119889B119894119889119905

= S119894B119894 + B119894S119879

119894 (54b)

S119894 = minus

10

120581119894

L 120597119867120597B119894

(54c)






(a)

(b)




S119894B119894 + B119894S119879

119894= 0 (55)


120575 =

1003816100381610038161003816

X1 minus X21003816100381610038161003816





7 Conclusions








Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











7 Conclusions








Acknowledgments


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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