A New Integro-Differential Equation for Rossby Solitary

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 597807 8 pageshttpdxdoiorg1011552013597807

Research ArticleA New Integro-Differential Equation for Rossby Solitary Waveswith Topography Effect in Deep Rotational Fluids

Hongwei Yang1 Qingfeng Zhao1 Baoshu Yin23 and Huanhe Dong1

1 Information School Shandong University of Science and Technology Qingdao 266590 China2 Institute of Oceanology China Academy of Sciences Qingdao 266071 China3 Key Laboratory of Ocean Circulation and Wave Chinese Academy of Sciences Qingdao 266071 China

Correspondence should be addressed to Baoshu Yin baoshuyin126com

Received 7 May 2013 Accepted 2 September 2013

Academic Editor Rasajit Bera

Copyright copy 2013 Hongwei Yang et al 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

From rotational potential vorticity-conserved equation with topography effect and dissipation effect with the help of the multiple-scale method a new integro-differential equation is constructed to describe the Rossby solitary waves in deep rotational fluidsBy analyzing the equation some conservation laws associated with Rossby solitary waves are derived Finally by seeking thenumerical solutions of the equation with the pseudospectral method by virtue of waterfall plots the effect of detuning parameterand dissipation on Rossby solitary waves generated by topography are discussed and the equation is compared with KdV equationand BO equation The results show that the detuning parameter 120572 plays an important role for the evolution features of solitarywaves generated by topography especially in the resonant case a large amplitude nonstationary disturbance is generated in theforcing region This condition may explain the blocking phenomenon which exists in the atmosphere and ocean and generated bytopographic forcing

1 Introduction

Among the many wave motions that occur in the ocean andatmosphere Rossby waves play one of the most importantrolesThey are largely responsible for determining the oceanrsquosresponse to atmospheric and other climate changes [1] Inthe past decades the research on nonlinear Rossby solitarywaves had been givenmuch attention in themathematics andphysics and some models had been constructed to describethis phenomenon Based upon the pioneering work of Long[2] and Benney [3] on barotropic Rossby waves there hadbeen remarkably exciting developments [4ndash11] and formedclassical solitary waves theory and algebraic solitary wavestheoryThe so-called classical solitary waves indicate that theevolution of solitary waves is governed by the Korteweg-deVries (KdV) type model while the behavior of solitary wavesis governed by the Benjamin-Ono (BO) model it is calledalgebraic solitary waves After the KdVmodel and BOmodela more general evolution model for solitary waves in a finite-depth fluid was given by Kubota and the model was called

intermediate long-wave (ILW) model [12 13] Many math-ematicians solved the above models by all kinds of methodand got a series of results [14ndash19] We note that most of theprevious researches about solitary waves were carried out inthe zonal area and could not be applied directly to the spheri-cal earth and little attention had been focused on the solitarywaves in the rotational fluids [20] Furthermore as everyoneknows the real oceanic and atmospheric motion is a forcedand dissipative system Topography effect as a forcing factorhas been studied by many researchers [21ndash25] on the otherhand dissipation effect must be considered in the oceanicand atmospheric motion otherwise the motion would growexplosively because of the constant injecting of the externalforcing energy Our aim is to construct a new model todescribe the Rossby solitary waves in rotational fluid withtopography effect and dissipation effect It has great differencefrom the previous researches

In this paper from rotational potential vorticity-con-served equationwith topography effect and dissipation effect

2 Abstract and Applied Analysis

with the help of the multiple-scale method we will first con-struct a new model to describe Rossby solitary waves in deeprotational fluids Then we will analyse the conservation rela-tions of themodel and derive the conservation laws of Rossbysolitarywaves Finally themodel is solved by the pseudospec-tral method [26] Based on the waterfall plots the effect ofdetuning parameter and dissipation on Rossby solitary wavesgenerated by topography are discussed the model is com-pared with KdVmodel and BOmodel and some conclusionsare obtained

2 Mathematics Model

According to [27] taking plane polar coordinates (119903 120579) 119903pointing to lower latitude is positive and the positive rotationis counter-clockwise and then the rotational potential vortic-ity-conserved equation including topography effect and tur-bulent dissipation is in the nondimensional form given by

[

120597

120597119905

+ (

1

119903

120597Ψ

120597119903

120597

120597120579

minus

1

119903

120597Ψ

120597120579

120597

120597119903

)]

times [

1

119903

120597

120597119903

(119903

120597Ψ

120597119903

) +

1

1199032

1205972Ψ

1205971205792

+ ℎ (119903 120579)] +

120573

119903

120597Ψ

120597120579

= minus1205820[

1

119903

120597

120597119903

(119903

120597Ψ

120597119903

) +

1

1199032

1205972Ψ

1205971205792] + 119876

(1)

where Ψ is the dimensionless stream function 120573 =

(12059601198770) cos120601

0(1198712119880) in which 119877

0is the Earthrsquos radius 120596

0

is the angular frequency of the Earthrsquos rotation 1206010is the

latitude 119871 and 119880 are the characteristic horizontal lengthand velocity scales ℎ(119903 120579) expresses the topography effect1205820[(1119903)(120597120597119903)(119903(120597Ψ120597119903))+(1119903

2)(1205972Ψ1205971205792)]denotes the vor-

ticity dissipation which is caused by the Ekman boundarylayer and 120582

0is a dissipative coefficient 119876 is the external

source and the form of 119876 will be given in the latterIn order to consider weakly nonlinear perturbation on a

rotational flow we assume

Ψ = int

119903

(Ω (119903) minus 119888 + 120576120572) 119903119889119903 + 120576120595 (119903 120579 119905) (2)

where 120572 is a small disturbance in the basic flow and reflectsthe proximity of the system to a resonate state 119888 is a constantwhich is regarded as a Rossby waves phase speed 120595 denotesdisturbance stream function Ω(119903) expresses the rotationalangular velocity In order to consider the role of nonlinearitywe assume the following type of rotational angular velocity

Ω (119903) =

120596 (119903) 1199031le 119903 le 119903

2

1205961

119903 gt 1199032

(3)

where1205961is constant and120596(119903) is a function of 119903 For simplicity

120596(119903) is assumed to be smooth across 119903 = 1199032

In the domain [1199031 1199032] in order to achieve a balance among

topography effect turbulent dissipation and nonlinearityand to eliminate the derivative term of dissipation we assume

ℎ (119903 120579) = 1205762119867(119903 120579) 120582

0= 12057632

120582

119876 = 12057632

120582

1

119903

120597

120597119903

[1199032(120596 minus 119888

0+ 120576120572)]

(4)

Substituting (2) (3) and (4) into (1) leads to the followingequation for the perturbation stream function 120595

[

120597

120597119905

+ (120596 minus 119888 + 120576120572)

120597

120597120579

+ 120576 (

1

119903

120597120595

120597119903

120597

120597120579

minus

1

119903

120597120595

120597120579

120597

120597119903

)]

times [

1

119903

120597

120597119903

(119903

120597120595

120597119903

) +

1

1199032

1205972120595

1205971205792+ 120576119867 (119903 120579)]

+

1

119903

[120573 minus

119889

119889119903

(

1

119903

1198891199032120596

119889119903

)]

120597120595

120597120579

= minus12057632

120582[

1

119903

120597

120597119903

(119903

120597120595

120597119903

) +

1

1199032

1205972120595

1205971205792]

(5)

In the domain [1199032infin] the parameter 120573 is smaller than that

in the domain [1199031 1199032] and we assume 120573 = 0 for [119903

2infin] Fur-

thermore the turbulent dissipation and topography effect areabsent in the domain and only consider the features of distur-bances generated Substituting (2) and (3) into (1) we have thefollowing governing equations

[

120597

120597119905

+ (1205961minus 119888 + 120576120572)

120597

120597120579

+ 120576 (

1

119903

120597120595

120597119903

120597

120597120579

minus

1

119903

120597120595

120597120579

120597

120597119903

)]

times [

1

119903

120597

120597119903

(119903

120597120595

120597119903

) +

1

1199032

1205972120595

1205971205792] = 0

(6)

For (5) we introduce the following stretching transforma-tions

Θ = 12057612

120579 119903 = 119903 119879 = 12057632

119905 (7)

and the perturbation expansion of 120595 is in the following form

120595 = 1205951(Θ 119903 119879) + 120576120595

2(Θ 119903 119879) + sdot sdot sdot (8)

Substituting (7) and (8) into (5) comparing the samepower of120576 term we can obtain the 120576

12 equation

L1205951= 0 (9)

where the operatorL is defined as

L =

1

119903

120597

120597Θ

(120596 minus 119888) [

120597

120597119903

(119903

120597

120597119903

)] + [120573 minus

119889

119889119903

(

1

119903

1198891199032120596

119889119903

)]

(10)

Assume the perturbation at boundary 119903 = 1199031does not exist

that is

1205951= 1205952= sdot sdot sdot = 0 (11)

and the perturbation at boundary 119903 = 1199032is determined by (6)

For the linear solution to be separable assuming the solutionof (9) in the form

1205951= 119860 (Θ 119879) 120601 (119903) (12)

thus 120601(119903) should satisfy the following equation

(120596 minus 119888)

119889

119889119903

(119903

119889120601 (119903)

119889119903

) + [120573 minus

119889

119889119903

(

1

119903

1198891199032120596

119889119903

)]120601 (119903) = 0

(13)

Abstract and Applied Analysis 3

On the other hand we proceed to the 12057632 equation

L1205952+

120596 minus 119888

1199032

12059731205951

120597Θ3

+ (

120597

120597119879

+ 120572

120597

120597Θ

+

1

119903

1205971205951

120597119903

120597

120597Θ

minus

1

119903

1205971205951

120597Θ

120597

120597119903

+ 120582)

times [

1

119903

120597

120597119903

(119903

1205971205951

120597119903

)] + (120596 minus 119888)

120597119867 (119903 Θ)

120597Θ

= 0

(14)

Multiplying the both sides of (14) by 119903120601(120596 minus 119888) and integrat-ing it with respect to 119903 from 119903

1to 1199032 employing the boundary

conditions (11) we get

120597

120597Θ

119903(120601

120597

120597119903

1205952minus

119889120601

119889119903

1205952)

10038161003816100381610038161003816100381610038161003816119903=1199032

+ 119860

120597119860

120597Θ

int

1199032

1199031

1206013

120596 minus 119888

119889

119889119903

times [

120573 minus (119889119889119903) ((1119903) (1198891199032120596119889119903))

120601 (120596 minus 119888)

] 119889119903

+ (

120597119860

120597119879

+ 120572

120597119860

120597Θ

+ 120582119860)int

1199032

1199031

120601

120596 minus 119888

119889

119889119903

(119903

119889120601

119889119903

) 119889119903

+

1205973119860

120597Θ3int

1199032

1199031

1206012

119903

119889119903 + int

1199032

1199031

119903120601

120597119867

120597Θ

119889119903 = 0

(15)

In (15) if the boundary conditions on 120601 and 1205952are known

the equation governing the amplitude 119860 will be determinedAssuming the solution of (5)matches smoothlywith the solu-tion of (6) at 119903 = 119903

2 we can solve (6) to seek the solution at

119903 = 1199032

For (6) we adopt the transformations in the forms

120588 = 120579 119903 = 119903 119879 = 12057632

119905 (16)

and the perturbation function is shown then by substitut-ing (16) into (6) we can get the 120576

0 equation

(1205961minus 119888)

120597

120597120588

[

1

119903

120597

120597119903

(119903

120597

120597119903

) +

1

1199032

1205972

1205971205882] = 0 (17)

It is easy to find that (17) can reduce to

1

119903

120597

120597119903

(119903

120597

120597119903

) +

1

1199032

1205972

1205971205882

= 0 119903 ge 1199032

997888rarr 0 119903 997888rarr infin

(18)

Obviously the solution of (18) is

(120588 119903 119879 120576) =

1

2120587

int

2120587

0

(1199032minus 1199032

2) (120588

1015840 1199032 119879 120576)

1199032

2minus 21199032119903 cos (120588 minus 120588

1015840) + 11990321198891205881015840 (19)

Taking the derivative with respect to 119903 for both sides of (19)leads to

120597

120597119903

=

1199032

120587

int

2120587

0

(1205881015840 1199032 119879 120576)

[21199032119903 minus (119903

2+ 1199032

2) cos (120588 minus 120588

1015840)]

[1199032

2minus 21199032119903 cos (120588 minus 120588

1015840) + 1199032]2

1198891205881015840

(20)

Because the solution of (5) matches smoothly with the solu-tion of (6) at 119903 = 119903

2 we obtain

1205951(Θ 1199032 119879) + 120576120595

2(Θ 1199032 119879) = (120588 119903

2 119879 120576) + 119874 (120576

2) (21)

1205971205951

120597119903

(Θ 1199032 119879) + 120576

1205971205952

120597119903

(Θ 1199032 119879) =

120597

120597119903

(120588 1199032 119879 120576) + 119874 (120576

2)

(22)

From (21) we have

119860 (Θ 119879) 120601 (1199032) = (120588 119903

2 119879 120576) 120595

2(Θ 1199032 119879) = 0 (23)

Substituting (23) into (20) leads to

120597

120597119903

(120588 1199032 119879 120576) = 120576120601 (119903

2)

1205972J (119860 (Θ 119879))

120597Θ2

(24)

where J(119860(Θ 119879)) = (11990322120587) int

2120587

0119860(Θ1015840 119879) ln | sin((Θ minus

Θ1015840)2)|119889Θ

1015840 Then based on (22) and (24) we get

1206011015840(1199032) = 0

1205971205952

120597119903

(Θ 1199032 119879) = 120601 (119903

2)

1205972J (119860 (Θ 119879))

120597Θ2

(25)

Substituting the boundary conditions (23) and (25) into (15)yields

120597119860

120597119879

+ 120572

120597119860

120597Θ

+ 1198861119860

120597119860

120597Θ

+ 1198862

1205973119860

120597Θ3

+ 1198863

1205973

120597Θ3J (119860 (Θ 119879)) + 120582119860 =

120597119866

120597Θ

(26)

Equation (26) can be rewritten as follows

120597119860

120597119879

+ 120572

120597119860

120597Θ

+ 1198861119860

120597119860

120597Θ

+ 1198862

1205973119860

120597Θ3

+ 1198863

1205972

120597Θ2H (119860 (Θ 119879)) + 120582119860 =

120597119866

120597Θ

(27)

whereH(119860(Θ 119879)) = (11990324120587) int

2120587

0119860(Θ1015840 119879)cot((ΘminusΘ

1015840)2)119889Θ

1015840

and 119886 = int

1199032

1199031

(120601(120596 minus 119888))(119889119889119903)(119903(119889120601119889119903))119889119903 1198861= int

1199032

1199031

(1206013(120596 minus

119888))(119889119889119903)[120573 minus (119889119889119903)((1119903)(1198891199032120596119889119903))120601(120596 minus 119888)]119889119903119886 119886

2=

int

1199032

1199031

(1206012119903)119889119903119886 119886

3= 11990321206012(1199032)119886 119866 = int

1199032

1199031

119903120601119867119889119903119886 Equation(27) is an integro-differential equation and 120582119860 expressesdissipation effect and has the same physical meaning with theterm 120597

2119860120597Θ

2 in Burgers equation When 1198863= 120582 = 119867 = 0


the equation degenerates to the KdV equation When 1198862

=

120582 = 119867 = 0 the equation degenerates to the so-called rota-tional BO equation Here we call (27) forced rotational KdV-BO-Burgers equationAswe know the forced rotationalKdV-BO-Burgers equation as a governing model for Rossby soli-tary waves is first derived in the paper

3 Conservation Laws

In this section the conservation laws are used to explore somefeatures of Rossby solitary waves In [7] Ono presented fourconservation laws of BO equation and we extend Onorsquos workto investigate the following questions Has the rotationalKdV-BO-Burgers equation also conservation laws withoutdissipation effect Has it four conservation laws or moreHow to change of these conservation quantities in the pres-ence of dissipation effect

In this section topography effect is ignored that is 119867 istaken zero in (27) Based on periodicity condition we assumethat the values of 119860 119860

Θ 119860ΘΘ

119860ΘΘΘ

at Θ = 0 equal that atΘ = 2120587 Then integrating (27) with respect to Θ over (0 2120587)we are easy to obtain the following conservation relation

1198761= int

2120587

0

119860119889Θ = exp (minus120582119879)int

2120587

0

119860 (Θ 0) 119889Θ (28)

From (28) it is obvious that 1198761decreases exponentially with

the evolution of time 119879 and the dissipation coefficient 120582 Byanalogy with the KdV equation 119876

1is regarded as the mass

of the solitary waves This shows that the dissipation effectcauses the mass of solitary waves decrease exponentiallyWhen dissipation effect is absent the mass of the solitarywaves is conserved

In what follows (27) has another simple conservationlaw which becomes clear if we multiply (27) by 119860(Θ 119879) andcarry the integration by using the property of the operatorH int

2120587

0119891(Θ)H(119891(Θ))119889Θ = 0 then we get

1198762= int

2120587

0

1198602119889Θ = exp (minus2120582119879)int

2120587

0

1198602(Θ 0) 119889Θ (29)

Similar to the mass 1198761 1198762is regarded as the momentum of

the solitary waves and is conserved without dissipation Themomentumof the solitary waves also decreases exponentiallywith the evolution of time 119879 and the increasing of dissipativecoefficient 120582 in the presence of dissipation effect Further-more the rate of decline of momentum is faster than the rateof mass

Next wemultiply (27) by (1198602minus(11988631198861)H(119860

Θ)) and obtain

(

1

3

1198603)

119879

minus

1198863

1198861

H (119860Θ) 119860119879

+ (120572 + 1198861119860)119860Θ(1198602minus

1198863

1198861

H (119860Θ))

+ 1198862119860ΘΘΘ

(1198602minus

1198863

1198861

H (119860Θ))

+ 1198863(1198602minus

1198863

1198861

H (119860Θ)) (H (119860))

ΘΘ

+ 120582(1198602minus

1198863

1198861

H (119860Θ))119860 = 0

(30)

Then taking the derivative of (27) with respect toΘ andmul-tiplying (minus(2119886

21198861)119860Θ+ (11988631198861)H(119860)) lead to

(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))119860Θ119879

+ [120572119860ΘΘ

+ 1198861(119860119860Θ)Θ] (minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))

+ 1198862119860ΘΘΘΘ

(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))

+ 1198863(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))H(119860)ΘΘΘ

+ 120582119860(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860)) = 0

(31)

Adding (30) to (31) by virtue of the property of operatorH

H(119860)ΘΘ

= H (119860ΘΘ

) int

2120587

0

119906HV119889Θ = minusint

2120587

0

VH119906119889Θ

(32)

we have

(

1

3

1198603minus

1198862

1198861

1198602

Θ+

1198863

1198861

119860ΘH (119860))

119879

+ 120572[

1

3

1198603minus

1198862

1198861

1198602

Θ+

1198863

1198861

H (119860)119860Θ]

Θ

+ (

1198861

4

1198604)

Θ

+

1198862

3

21198861

[119867 (119860)119867(119860)ΘΘ

]Θ

+

11988621198863

1198861

(119860ΘΘΘ

119867(119860) minus 2119860Θ119867(119860)ΘΘ

)Θ

minus

21198862

1198861

(119860Θ119860ΘΘΘ

minus

1

2

1198602

ΘΘ)

Θ

+ 1198862(1198602119860ΘΘ

minus 21198601198602

Θ)Θ

+ 1198863[119860(119860H (119860))

Θ]Θ+ 120582(119860

3minus

1198862

1198861

1198602

Θ+

1198863

1198861

119860Θ119867(119860))

= 0

(33)

Taking1198763= int

2120587

0((13)119860

3minus(11988621198861)1198602

Θ+(11988631198861)119860ΘH(119860))119889Θ

we are easy to see that when the dissipation effect is absentthat is 120582 = 0 119876

3is a conserved quantity and regarded as

the energy of the solitary waves So we can conclude that theenergy of solitary waves is conserved without dissipation Byanalysing (33) we can find the decreasing trend of energy ofsolitary waves


Finally let us consider a quantity related to the phase ofsolitary waves

1198764=

119889

119889119879

int

2120587

0

Θ119860119889Θ (34)

and we can get 1198891198764119889119879 = 0 without dissipation According

[7] we present the velocity of the center of gravity forthe ensemble of such waves 119876

4= 11987641198761 by employing

1198891198761119889119879 = 0 and 119889119876

4119889119879 = 0 we have 119889119876

4119889119879 = 0 which

shows that the velocity of the center of gravity is conservedwithout dissipation

After the four conservation relations are given we canproceed to seek the fifth conservation quantity In fact aftertedious calculation we can also verify that

1198765=int

2120587

0

(

1

4

1198604minus

31198862

1198861

1198601198602

Θ+

91198862

1198862

1

1198602

ΘΘ+

1198863

41198861

1198602H (119860)) 119889Θ

(35)

is also conservation quantity According the idea we canobtain the sixth conservation quantity 119876

6and the seventh

conservation quantity 1198767 so we can guess that similar to

the KdV equation the rotational KdV-BO-Burgers equationwithout dissipation also owns infinite conservation laws butit needs to be verified in the future

4 Numerical Simulation and Discussion

In this section we will take into account the generation andevolution feature of Rossby solitarywaves under the influenceof topography and dissipation so we need to seek the solu-tions of forced rotational KdV-BO-Burgers equation But weknow that there is no analytic solution for (27) and here weconsider the numerical solutions of (27) by employing thepseudospectral method

The pseudo-spectral method uses a Fourier transformtreatment of the space dependence together with a leap-forgscheme in time For ease of presentation the spatial period isnormalized to [0 2120587] This interval is divided into 2119873 pointsand then Δ119879 = 120587119873 The function 119860(119883 119879) can be trans-formed to the Fourier space by

119860 (V 119879) = 119865119860 =

1

radic2119873

2119873minus1

sum

119895=0

119860 (119895Δ119883 119879) 119890minus120587119894119895V119873

V = 0 plusmn1 plusmn119873

(36)

The inversion formula is

119860 (119895Δ119883 119879) = 119865minus1119860 =

1

radic2119873

sum

V119860 (V 119879) 119890

120587119894119895V119873 (37)

These transformations can use Fast Fourier Transform algo-rithm to efficiently performWith this scheme 120597119860120597119883 can beevaluated as 119865minus1119894V119865119860 1205973119860120597119883

3 as minus119894119865minus1V3119865119860 120597119867120597119883 as

119865minus1119894V119865119867 and so on Combined with a leap-frog time step

(27) would be approximated by

119860 (119883 119879 + Δ119879) minus 119860 (119883 119879 minus Δ119879) + 119894120572119865minus1

V119865119860Δ119879

+ 1198941198861119860119865minus1

V119865119860Δ119879 minus 1198862119894119865minus1

V3119865119860Δ119879

minus 1198863119865minus1

V2119865H (119860) Δ119879 + 120582119860 = 119894119865minus1

V119865119866Δ119879

(38)

The computational cost for (38) is six fast Fourier transformsper time step

Once the zonal flow Ω(119903) and the topography function119867(119903 Θ) as well as dissipative coefficient 120582 are given it is easyto get the coefficients of (27) by employing (13) In order tosimplify the calculation and to focus attention on the timeevolution of the solitary waves with topography effect anddissipation effect and to show the difference among the KdVmodel BO model and rotational KdV-BO model we take1198861= 1 119886

2= minus1 and 119886

3= minus1 As an initial condition we take

119860(119883 0) = 0 In the present numerical computation thetopography forcing is taken as 119866 = 119890

minus[30(Θminus120587)]24

41 Effect of Detuning Parameter 120572 and Dissipation InFigure 1 we consider the effect of detuning parameter 120572 onsolitary waves The evolution features of solitary waves gen-erated by topography are shown in the absence of dissipationwith different detuning parameter 120572 It is easy to find fromthese waterfall plots that the detuning parameter 120572 plays animportant role for the evolution features of solitary wavesgenerated by topography

When 120572 gt 0 (Figure 1(a)) a positive stationary solitarywave is generated in the topographic forcing region anda modulated cnoidal wave-train occupies the downstreamregion There is no wave in the upstream region A flat bufferregion exists between the solitary wave in the forcing regionand modulated cnoidal wave-train in the downstream Withthe detuning parameter 120572 decreasing the amplitudes of bothsolitary wave in the forcing region and modulated cnoidalwave-train in the downstream region increase and themodu-lated cnoidal wave-train closes to the forcing region graduallyand the flat buffer region disappears slowly

Up to 120572 = 0 (Figure 1(b)) the resonant case forms Inthis case a large amplitude nonstationary disturbance is gen-erated in the forcing region To some degree this conditionmay explain the blocking phenomenon which exists in theatmosphere and ocean and generated by topographic forcing

As 120572 lt 0 from Figure 1(c) we can easy to find that anegative stationary solitary wave is generated in the forcingregion and this is great difference with the former two condi-tions Meanwhile there are both wave-trains in the upstreamand downstream region The amplitude and wavelength ofwave-train in the upstream region are larger than those inthe downstream regions Similar to Figure 1(b) and unlikeFigure 1(a) the wave-trains in the upstream and downstreamregions connect to the forcing region and the flat buffer regiondisappears

Figure 2 shows the solitary waves generated by topogra-phy in the presence of dissipation with dissipative coefficient120582 = 03 and detuning parameter 120572 = 25 The conditions of120572 = 0 and120572 lt 0 are omitted Compared to Figure 1(a) wewill


00

5

10

15

T

minus05

0

05

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) 120572 = 25

00

5

10

15

T

minus05

0

05

1

15

2

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) 120572 = 0

00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(c) 120572 = minus25

Figure 1 Solitary waves generated by topography in the absence of dissipation

00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

Figure 2 Solitary waves generated by topography in the presence ofdissipation (120582 = 03 120572 = 25)

find that there is also a solitary wave generated in the forcingregion but because of dissipation effect the amplitude of soli-tary wave in the forcing region decreases as the dissipativecoefficient 120582 increases (Figures omitted) and time evolutionMeanwhile the modulated cnoidal wave-train in the down-stream region is dissipated When 120582 is big enough the mod-ulated cnoidal wave-train in the downstream region disap-pears

42 Comparison of KdV Model BO Model and KdV-BOModel We know that the rotation KdV-BO equation reducesto the KdV equation as 119886

3= 0 and to the BO equation as

1198862= 0 so in this subsection by comparing Figure 1(a) with

Figure 3 we will look for the difference of solitary waveswhich is described by KdV-BO model KdV model and BOmodel The role of detuning parameter 120572 and dissipationeffect has been studied in the former subsection so here weonly consider the condition of 120582 = 0 120572 = 25

At first we can find that a positive solitary wave is all gen-erated in the forcing region in Figures 1(a) 3(a) and 3(b) butit is stationary in Figures 1(a) and 3(a) and is nonstationary inFigure 3(b) By surveying carefully we find that the amplitudeof stationary wave in the forcing region in Figure 1(a) is largerthan that in Figure 3(a) Additionally a modulated cnoidalwave-train is excited in the downstream region in Figures 1(a)and 3(a) and in both downstream and upstream region inFigure 3(b) The amplitude of modulated cnoidal wave-trainin downstream region in Figure 3(b) is the largest and inFigure 1(a) is the smallest among the three models Further-more in Figure 3(a) the wave number of modulated cnoidalwave-train is more than that in Figures 1(a) and 3(b) Ina word by the above analysis and comparison it is easyto find that Figure 1(a) is similar to Figure 3(a) and hasgreat difference with Figure 3(b) This indicates that theterm 119886

2(1205973119860120597Θ

3) plays more important role than the term

1198863(1205972120597Θ2)H(119860) in rotational KdV-BO equation


00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) KdV model (1198863 = 0 120572 = 25 120582 = 0)

00

5

10

15

T

minus05

0

05

1

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) BO model (1198862 = 0 120572 = 25 120582 = 0)

Figure 3 Comparison of KdV model Bo model and KdV-BO model

5 Conclusions

In this paper we presented a newmodel rotational KdV-BO-Burgers model to describe the Rossby solitary waves gener-ated by topography with the effect of dissipation in deep rota-tional fluids By analysis and computation five conservationquantities of KdV-BO-Burgers model were derived and cor-responding four conservation laws of Rossby solitary waveswere obtained that is mass momentum energy and velocityof the center of gravity of Rossby solitary waves are conservedwithout dissipation effect Further we presented that therotational KdV-BO-Burgers equation owns infinite conserva-tion quantities in the absence of dissipation effect Detailednumerical results obtained using pseudospectral method arepresented to demonstrate the effect of detuning parame-ter 120572 and dissipation By comparing the KdV model BOmodel and KdV-BO model we drew the conclusion thatthe term 119886

2(1205973119860120597Θ

3) plays more important role than the

term 1198863(1205972120597Θ2)H(119860) in rotational KdV-BO equation More

problems onKdV-BO-Burgers equation such as the analyticalsolutions integrability and infinite conservation quantitiesare not studied in the paper due to limited space In fact thereare many methods carried out to solve some equations withspecial nonhomogenous terms [28] as well asmultiwave solu-tions and other form solution [29 30] of homogenous equa-tion These researches have important value for understand-ing and realizing the physical phenomenon described by theequation and deserve to carry out in the future

Acknowledgments

This work was supported by Innovation Project of Chi-nese Academy of Sciences (no KZCX2-EW-209) NationalNatural Science Foundation of China (nos 41376030 and11271107) Nature Science Foundation of Shandong Provinceof China (no ZR2012AQ015) Science and Technology planproject of the Educational Department of Shandong Provinceof China (no J12LI03) and SDUST Research Fund (no2012KYTD105)

References

[1] S G H Philander ldquoForced oceanic wavesrdquo Reviews of Geophys-ics and Space Physics vol 16 no 1 pp 15ndash46 1978

[2] R R Long ldquoSolitary waves in the westerliesrdquo Journal of theAtmospheric Sciences vol 21 pp 197ndash200 1964

[3] D J Benney ldquoLong non-linear waves in fluid flowsrdquo Journal ofMathematical Physics vol 45 pp 52ndash63 1966

[4] L G Redekopp ldquoOn the theory of solitary Rossby wavesrdquo Jour-nal of Fluid Mechanics vol 82 no 4 pp 725ndash745 1977

[5] J I Yano and Y N Tsujimura ldquoThe domain of validity of theKdV-type solitary Rossby waves in the shallow water BETA-plane modelrdquo Dynamics of Atmospheres and Oceans vol 11 no2 1987

[6] S Jian and Y Lian-Gui ldquoModified KdV equation for solitaryrossby waves with 120573 effect in barotropic fluidsrdquo Chinese PhysicsB vol 18 no 7 pp 2873ndash2877 2009

[7] H Ono ldquoAlgebraic solitary waves in stratified fluidsrdquo vol 39 no4 pp 1082ndash1091 1975

[8] R Grimshaw ldquoSlowly varying solitary waves in deep fluidsrdquoProceedings of the Royal Society A vol 376 no 1765 pp 319ndash3321981

[9] L Dehai and J Liren ldquoAlgebraic rossby solitary wave and block-ing in the atmosphererdquoAdvances in Atmospheric Sciences vol 5no 4 pp 445ndash454 1988

[10] L Meng and K L Lv ldquoDissipation and algebraic solitary long-waves excited by localized topographyrdquoChinese Journal of Com-putational Physics vol 19 pp 159ndash167 2002

[11] X B Su GWei and SQDai ldquoTwo-dimensional algebraic soli-tary wave and its vertical structure in stratified fluidrdquo AppliedMathematics andMechanics vol 26 no 10 pp 1255ndash1265 2005

[12] TKubotaD R S Ko andLDDobbs ldquoWeakly-nonlinear longinternal gravity waves in stratified fluids of finite depthrdquo Journalof Hydronautics vol 12 no 4 pp 157ndash165 1978

[13] Q P Zhou ldquoSecond-order solitary waves in stratified fluids offinite depthrdquo Science China Mathematics vol 10 pp 924ndash9341984

[14] W X Ma ldquoComplexiton solutions to the Korteweg-de Vriesequationrdquo Physics Letters A vol 301 no 1-2 pp 35ndash44 2002


[15] W-X Ma ldquoWronskians generalized Wronskians and solutionsto the Korteweg-de Vries equationrdquo Chaos Solitons amp Fractalsvol 19 no 1 pp 163ndash170 2004

[16] W-X Ma and Y You ldquoSolving the Korteweg-de Vries equationby its bilinear form wronskian solutionsrdquo Transactions of theAmerican Mathematical Society vol 357 no 5 pp 1753ndash17782005

[17] T C Xia H Q Zhang and Z Y Yan ldquoNew explicit and exacttravellingwave solutions for a class of nonlinear evolution equa-tionsrdquo Applied Mathematics and Mechanics vol 22 no 7 pp788ndash793 2001

[18] H-Y Wei and T-C Xia ldquoSelf-consistent sources and conser-vation laws for nonlinear integrable couplings of the Li solitonhierarchyrdquo Abstract and Applied Analysis vol 2013 Article ID598570 10 pages 2013

[19] S I A El-Ganaini ldquoNew exact solutions of some nonlinearsystems of partial differential equations using the first integralmethodrdquo Abstract and Applied Analysis vol 2013 Article ID693076 13 pages 2013

[20] D H Luo ldquoNonlinear schrodinger equation in the rotationalatmosphere and atmospheric blockingrdquoActaMeteor Sinica vol48 pp 265ndash274 1990

[21] O E Polukhina and A A Kurkin ldquoImproved theory of nonlin-ear topographic Rossby wavesrdquo Oceanology vol 45 no 5 pp607ndash616 2005

[22] L G Yang C J Da J Song H Q Zhang H L Yang and YJ Hou ldquoRossby waves with linear topography in barotropic flu-idsrdquoChinese Journal of Oceanology and Limnology vol 26 no 3pp 334ndash338 2008

[23] J Song and L-G Yang ldquoForce solitary Rossby waves with betaeffect and topography effect in stratified flowsrdquoActa Physica Sin-ica vol 59 no 5 pp 3309ndash3314 2010

[24] H-W Yang B-S Yin D-Z Yang and Z-H Xu ldquoForced soli-tary Rossby waves under the influence of slowly varying topog-raphy with timerdquo Chinese Physics B vol 20 no 12 Article ID120203 2011

[25] H W Yang Y L Shi B S Yin and Q B Wang ldquoForced ILW-Burgers equation as amodel for Rossby solitarywaves generatedby topography in finite depth fluidsrdquo Journal of Applied Mathe-matics vol 2012 Article ID 491343 17 pages 2012

[26] B FornbergAPractical Guide to PseudospectralMethods Cam-bridge University Press Cambridge UK 1996

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

[28] W X Ma and B Fuchssteiner ldquoExplicit and exact solutions to aKolmogorov-Petrovskii-Piskunov equationrdquo International Jour-nal of Non-Linear Mechanics vol 31 no 3 pp 329ndash338 1996

[29] M G Asaad and W-X Ma ldquoPfaffian solutions to a (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equa-tion and its modified counterpartrdquo Applied Mathematics andComputation vol 218 no 9 pp 5524ndash5542 2012

[30] B G Zhang ldquoAnalytical and multishaped solitary wave solu-tions for extended reduced ostrovsky equationrdquo Abstract andApplied Analysis vol 2013 Article ID 670847 8 pages 2013


with the help of the multiple-scale method we will first con-struct a new model to describe Rossby solitary waves in deeprotational fluids Then we will analyse the conservation rela-tions of themodel and derive the conservation laws of Rossbysolitarywaves Finally themodel is solved by the pseudospec-tral method [26] Based on the waterfall plots the effect ofdetuning parameter and dissipation on Rossby solitary wavesgenerated by topography are discussed the model is com-pared with KdVmodel and BOmodel and some conclusionsare obtained

2 Mathematics Model

According to [27] taking plane polar coordinates (119903 120579) 119903pointing to lower latitude is positive and the positive rotationis counter-clockwise and then the rotational potential vortic-ity-conserved equation including topography effect and tur-bulent dissipation is in the nondimensional form given by

[

120597

120597119905

+ (

1

119903

120597Ψ

120597119903

120597

120597120579

minus

1

119903

120597Ψ

120597120579

120597

120597119903

)]

times [

1

119903

120597

120597119903

(119903

120597Ψ

120597119903

) +

1

1199032

1205972Ψ

1205971205792

+ ℎ (119903 120579)] +

120573

119903

120597Ψ

120597120579

= minus1205820[

1

119903

120597

120597119903

(119903

120597Ψ

120597119903

) +

1

1199032

1205972Ψ

1205971205792] + 119876

(1)

where Ψ is the dimensionless stream function 120573 =

(12059601198770) cos120601

0(1198712119880) in which 119877

0is the Earthrsquos radius 120596

0

is the angular frequency of the Earthrsquos rotation 1206010is the

latitude 119871 and 119880 are the characteristic horizontal lengthand velocity scales ℎ(119903 120579) expresses the topography effect1205820[(1119903)(120597120597119903)(119903(120597Ψ120597119903))+(1119903

2)(1205972Ψ1205971205792)]denotes the vor-

ticity dissipation which is caused by the Ekman boundarylayer and 120582

0is a dissipative coefficient 119876 is the external

source and the form of 119876 will be given in the latterIn order to consider weakly nonlinear perturbation on a

rotational flow we assume

Ψ = int

119903

(Ω (119903) minus 119888 + 120576120572) 119903119889119903 + 120576120595 (119903 120579 119905) (2)

where 120572 is a small disturbance in the basic flow and reflectsthe proximity of the system to a resonate state 119888 is a constantwhich is regarded as a Rossby waves phase speed 120595 denotesdisturbance stream function Ω(119903) expresses the rotationalangular velocity In order to consider the role of nonlinearitywe assume the following type of rotational angular velocity

Ω (119903) =

120596 (119903) 1199031le 119903 le 119903

2

1205961

119903 gt 1199032

(3)

where1205961is constant and120596(119903) is a function of 119903 For simplicity

120596(119903) is assumed to be smooth across 119903 = 1199032

In the domain [1199031 1199032] in order to achieve a balance among

topography effect turbulent dissipation and nonlinearityand to eliminate the derivative term of dissipation we assume

ℎ (119903 120579) = 1205762119867(119903 120579) 120582

0= 12057632

120582

119876 = 12057632

120582

1

119903

120597

120597119903

[1199032(120596 minus 119888

0+ 120576120572)]

(4)

Substituting (2) (3) and (4) into (1) leads to the followingequation for the perturbation stream function 120595

[

120597

120597119905

+ (120596 minus 119888 + 120576120572)

120597

120597120579

+ 120576 (

1

119903

120597120595

120597119903

120597

120597120579

minus

1

119903

120597120595

120597120579

120597

120597119903

)]

times [

1

119903

120597

120597119903

(119903

120597120595

120597119903

) +

1

1199032

1205972120595

1205971205792+ 120576119867 (119903 120579)]

+

1

119903

[120573 minus

119889

119889119903

(

1

119903

1198891199032120596

119889119903

)]

120597120595

120597120579

= minus12057632

120582[

1

119903

120597

120597119903

(119903

120597120595

120597119903

) +

1

1199032

1205972120595

1205971205792]

(5)

In the domain [1199032infin] the parameter 120573 is smaller than that

in the domain [1199031 1199032] and we assume 120573 = 0 for [119903

2infin] Fur-

thermore the turbulent dissipation and topography effect areabsent in the domain and only consider the features of distur-bances generated Substituting (2) and (3) into (1) we have thefollowing governing equations

[

120597

120597119905

+ (1205961minus 119888 + 120576120572)

120597

120597120579

+ 120576 (

1

119903

120597120595

120597119903

120597

120597120579

minus

1

119903

120597120595

120597120579

120597

120597119903

)]

times [

1

119903

120597

120597119903

(119903

120597120595

120597119903

) +

1

1199032

1205972120595

1205971205792] = 0

(6)

For (5) we introduce the following stretching transforma-tions

Θ = 12057612

120579 119903 = 119903 119879 = 12057632

119905 (7)

and the perturbation expansion of 120595 is in the following form

120595 = 1205951(Θ 119903 119879) + 120576120595

2(Θ 119903 119879) + sdot sdot sdot (8)

Substituting (7) and (8) into (5) comparing the samepower of120576 term we can obtain the 120576

12 equation

L1205951= 0 (9)

where the operatorL is defined as

L =

1

119903

120597

120597Θ

(120596 minus 119888) [

120597

120597119903

(119903

120597

120597119903

)] + [120573 minus

119889

119889119903

(

1

119903

1198891199032120596

119889119903

)]

(10)

Assume the perturbation at boundary 119903 = 1199031does not exist

that is

1205951= 1205952= sdot sdot sdot = 0 (11)

and the perturbation at boundary 119903 = 1199032is determined by (6)

For the linear solution to be separable assuming the solutionof (9) in the form

1205951= 119860 (Θ 119879) 120601 (119903) (12)

thus 120601(119903) should satisfy the following equation

(120596 minus 119888)

119889

119889119903

(119903

119889120601 (119903)

119889119903

) + [120573 minus

119889

119889119903

(

1

119903

1198891199032120596

119889119903

)]120601 (119903) = 0

(13)



L1205952+

120596 minus 119888

1199032

12059731205951

120597Θ3

+ (

120597

120597119879

+ 120572

120597

120597Θ

+

1

119903

1205971205951

120597119903

120597

120597Θ

minus

1

119903

1205971205951

120597Θ

120597

120597119903

+ 120582)

times [

1

119903

120597

120597119903

(119903

1205971205951

120597119903

)] + (120596 minus 119888)

120597119867 (119903 Θ)

120597Θ

= 0

(14)




120597

120597Θ

119903(120601

120597

120597119903

1205952minus

119889120601

119889119903

1205952)

10038161003816100381610038161003816100381610038161003816119903=1199032

+ 119860

120597119860

120597Θ

int

1199032

1199031

1206013

120596 minus 119888

119889

119889119903

times [

120573 minus (119889119889119903) ((1119903) (1198891199032120596119889119903))

120601 (120596 minus 119888)

] 119889119903

+ (

120597119860

120597119879

+ 120572

120597119860

120597Θ

+ 120582119860)int

1199032

1199031

120601

120596 minus 119888

119889

119889119903

(119903

119889120601

119889119903

) 119889119903

+

1205973119860

120597Θ3int

1199032

1199031

1206012

119903

119889119903 + int

1199032

1199031

119903120601

120597119867

120597Θ

119889119903 = 0

(15)




119903 = 1199032


120588 = 120579 119903 = 119903 119879 = 12057632

119905 (16)


0 equation

(1205961minus 119888)

120597

120597120588

[

1

119903

120597

120597119903

(119903

120597

120597119903

) +

1

1199032

1205972

1205971205882] = 0 (17)


1

119903

120597

120597119903

(119903

120597

120597119903

) +

1

1199032

1205972

1205971205882

= 0 119903 ge 1199032

997888rarr 0 119903 997888rarr infin

(18)


(120588 119903 119879 120576) =

1

2120587

int

2120587

0

(1199032minus 1199032

2) (120588

1015840 1199032 119879 120576)

1199032

2minus 21199032119903 cos (120588 minus 120588

1015840) + 11990321198891205881015840 (19)


120597

120597119903

=

1199032

120587

int

2120587

0

(1205881015840 1199032 119879 120576)

[21199032119903 minus (119903

2+ 1199032

2) cos (120588 minus 120588

1015840)]

[1199032

2minus 21199032119903 cos (120588 minus 120588

1015840) + 1199032]2

1198891205881015840

(20)


2 we obtain

1205951(Θ 1199032 119879) + 120576120595

2(Θ 1199032 119879) = (120588 119903

2 119879 120576) + 119874 (120576

2) (21)

1205971205951

120597119903

(Θ 1199032 119879) + 120576

1205971205952

120597119903

(Θ 1199032 119879) =

120597

120597119903

(120588 1199032 119879 120576) + 119874 (120576

2)

(22)

From (21) we have

119860 (Θ 119879) 120601 (1199032) = (120588 119903

2 119879 120576) 120595

2(Θ 1199032 119879) = 0 (23)


120597

120597119903

(120588 1199032 119879 120576) = 120576120601 (119903

2)

1205972J (119860 (Θ 119879))

120597Θ2

(24)

where J(119860(Θ 119879)) = (11990322120587) int

2120587

0119860(Θ1015840 119879) ln | sin((Θ minus

Θ1015840)2)|119889Θ


1206011015840(1199032) = 0

1205971205952

120597119903

(Θ 1199032 119879) = 120601 (119903

2)

1205972J (119860 (Θ 119879))

120597Θ2

(25)


120597119860

120597119879

+ 120572

120597119860

120597Θ

+ 1198861119860

120597119860

120597Θ

+ 1198862

1205973119860

120597Θ3

+ 1198863

1205973

120597Θ3J (119860 (Θ 119879)) + 120582119860 =

120597119866

120597Θ

(26)


120597119860

120597119879

+ 120572

120597119860

120597Θ

+ 1198861119860

120597119860

120597Θ

+ 1198862

1205973119860

120597Θ3

+ 1198863

1205972

120597Θ2H (119860 (Θ 119879)) + 120582119860 =

120597119866

120597Θ

(27)

whereH(119860(Θ 119879)) = (11990324120587) int

2120587

0119860(Θ1015840 119879)cot((ΘminusΘ

1015840)2)119889Θ

1015840

and 119886 = int

1199032

1199031

(120601(120596 minus 119888))(119889119889119903)(119903(119889120601119889119903))119889119903 1198861= int

1199032

1199031

(1206013(120596 minus

119888))(119889119889119903)[120573 minus (119889119889119903)((1119903)(1198891199032120596119889119903))120601(120596 minus 119888)]119889119903119886 119886

2=

int

1199032

1199031

(1206012119903)119889119903119886 119886

3= 11990321206012(1199032)119886 119866 = int

1199032

1199031


2119860120597Θ




=


3 Conservation Laws



Θ 119860ΘΘ

119860ΘΘΘ


1198761= int

2120587

0

119860119889Θ = exp (minus120582119879)int

2120587

0

119860 (Θ 0) 119889Θ (28)






2120587

0119891(Θ)H(119891(Θ))119889Θ = 0 then we get

1198762= int

2120587

0

1198602119889Θ = exp (minus2120582119879)int

2120587

0

1198602(Θ 0) 119889Θ (29)




Θ)) and obtain

(

1

3

1198603)

119879

minus

1198863

1198861

H (119860Θ) 119860119879

+ (120572 + 1198861119860)119860Θ(1198602minus

1198863

1198861

H (119860Θ))

+ 1198862119860ΘΘΘ

(1198602minus

1198863

1198861

H (119860Θ))

+ 1198863(1198602minus

1198863

1198861

H (119860Θ)) (H (119860))

ΘΘ

+ 120582(1198602minus

1198863

1198861

H (119860Θ))119860 = 0

(30)


21198861)119860Θ+ (11988631198861)H(119860)) lead to

(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))119860Θ119879

+ [120572119860ΘΘ

+ 1198861(119860119860Θ)Θ] (minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))

+ 1198862119860ΘΘΘΘ

(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))

+ 1198863(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))H(119860)ΘΘΘ

+ 120582119860(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860)) = 0

(31)


H(119860)ΘΘ

= H (119860ΘΘ

) int

2120587

0


2120587

0

VH119906119889Θ

(32)

we have

(

1

3

1198603minus

1198862

1198861

1198602

Θ+

1198863

1198861

119860ΘH (119860))

119879

+ 120572[

1

3

1198603minus

1198862

1198861

1198602

Θ+

1198863

1198861

H (119860)119860Θ]

Θ

+ (

1198861

4

1198604)

Θ

+

1198862

3

21198861

[119867 (119860)119867(119860)ΘΘ

]Θ

+

11988621198863

1198861

(119860ΘΘΘ

119867(119860) minus 2119860Θ119867(119860)ΘΘ

)Θ

minus

21198862

1198861

(119860Θ119860ΘΘΘ

minus

1

2

1198602

ΘΘ)

Θ

+ 1198862(1198602119860ΘΘ

minus 21198601198602

Θ)Θ

+ 1198863[119860(119860H (119860))

Θ]Θ+ 120582(119860

3minus

1198862

1198861

1198602

Θ+

1198863

1198861

119860Θ119867(119860))

= 0

(33)

Taking1198763= int

2120587

0((13)119860

3minus(11988621198861)1198602

Θ+(11988631198861)119860ΘH(119860))119889Θ






1198764=

119889

119889119879

int

2120587

0

Θ119860119889Θ (34)



4= 11987641198761 by employing

1198891198761119889119879 = 0 and 119889119876

4119889119879 = 0 we have 119889119876

4119889119879 = 0 which



1198765=int

2120587

0

(

1

4

1198604minus

31198862

1198861

1198601198602

Θ+

91198862

1198862

1

1198602

ΘΘ+

1198863

41198861

1198602H (119860)) 119889Θ

(35)


6and the seventh






119860 (V 119879) = 119865119860 =

1

radic2119873

2119873minus1

sum

119895=0

119860 (119895Δ119883 119879) 119890minus120587119894119895V119873


(36)


119860 (119895Δ119883 119879) = 119865minus1119860 =

1

radic2119873

sum

V119860 (V 119879) 119890

120587119894119895V119873 (37)






V119865119860Δ119879

+ 1198941198861119860119865minus1

V119865119860Δ119879 minus 1198862119894119865minus1

V3119865119860Δ119879


V2119865H (119860) Δ119879 + 120582119860 = 119894119865minus1

V119865119866Δ119879

(38)













00

5

10

15

T

minus05

0

05

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) 120572 = 25

00

5

10

15

T

minus05

0

05

1

15

2

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) 120572 = 0

00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(c) 120572 = minus25


00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579








2(1205973119860120597Θ




00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) KdV model (1198863 = 0 120572 = 25 120582 = 0)

00

5

10

15

T

minus05

0

05

1

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) BO model (1198862 = 0 120572 = 25 120582 = 0)


5 Conclusions


2(1205973119860120597Θ




Acknowledgments


References


































L1205952+

120596 minus 119888

1199032

12059731205951

120597Θ3

+ (

120597

120597119879

+ 120572

120597

120597Θ

+

1

119903

1205971205951

120597119903

120597

120597Θ

minus

1

119903

1205971205951

120597Θ

120597

120597119903

+ 120582)

times [

1

119903

120597

120597119903

(119903

1205971205951

120597119903

)] + (120596 minus 119888)

120597119867 (119903 Θ)

120597Θ

= 0

(14)




120597

120597Θ

119903(120601

120597

120597119903

1205952minus

119889120601

119889119903

1205952)

10038161003816100381610038161003816100381610038161003816119903=1199032

+ 119860

120597119860

120597Θ

int

1199032

1199031

1206013

120596 minus 119888

119889

119889119903

times [

120573 minus (119889119889119903) ((1119903) (1198891199032120596119889119903))

120601 (120596 minus 119888)

] 119889119903

+ (

120597119860

120597119879

+ 120572

120597119860

120597Θ

+ 120582119860)int

1199032

1199031

120601

120596 minus 119888

119889

119889119903

(119903

119889120601

119889119903

) 119889119903

+

1205973119860

120597Θ3int

1199032

1199031

1206012

119903

119889119903 + int

1199032

1199031

119903120601

120597119867

120597Θ

119889119903 = 0

(15)




119903 = 1199032


120588 = 120579 119903 = 119903 119879 = 12057632

119905 (16)


0 equation

(1205961minus 119888)

120597

120597120588

[

1

119903

120597

120597119903

(119903

120597

120597119903

) +

1

1199032

1205972

1205971205882] = 0 (17)


1

119903

120597

120597119903

(119903

120597

120597119903

) +

1

1199032

1205972

1205971205882

= 0 119903 ge 1199032

997888rarr 0 119903 997888rarr infin

(18)


(120588 119903 119879 120576) =

1

2120587

int

2120587

0

(1199032minus 1199032

2) (120588

1015840 1199032 119879 120576)

1199032

2minus 21199032119903 cos (120588 minus 120588

1015840) + 11990321198891205881015840 (19)


120597

120597119903

=

1199032

120587

int

2120587

0

(1205881015840 1199032 119879 120576)

[21199032119903 minus (119903

2+ 1199032

2) cos (120588 minus 120588

1015840)]

[1199032

2minus 21199032119903 cos (120588 minus 120588

1015840) + 1199032]2

1198891205881015840

(20)


2 we obtain

1205951(Θ 1199032 119879) + 120576120595

2(Θ 1199032 119879) = (120588 119903

2 119879 120576) + 119874 (120576

2) (21)

1205971205951

120597119903

(Θ 1199032 119879) + 120576

1205971205952

120597119903

(Θ 1199032 119879) =

120597

120597119903

(120588 1199032 119879 120576) + 119874 (120576

2)

(22)

From (21) we have

119860 (Θ 119879) 120601 (1199032) = (120588 119903

2 119879 120576) 120595

2(Θ 1199032 119879) = 0 (23)


120597

120597119903

(120588 1199032 119879 120576) = 120576120601 (119903

2)

1205972J (119860 (Θ 119879))

120597Θ2

(24)

where J(119860(Θ 119879)) = (11990322120587) int

2120587

0119860(Θ1015840 119879) ln | sin((Θ minus

Θ1015840)2)|119889Θ


1206011015840(1199032) = 0

1205971205952

120597119903

(Θ 1199032 119879) = 120601 (119903

2)

1205972J (119860 (Θ 119879))

120597Θ2

(25)


120597119860

120597119879

+ 120572

120597119860

120597Θ

+ 1198861119860

120597119860

120597Θ

+ 1198862

1205973119860

120597Θ3

+ 1198863

1205973

120597Θ3J (119860 (Θ 119879)) + 120582119860 =

120597119866

120597Θ

(26)


120597119860

120597119879

+ 120572

120597119860

120597Θ

+ 1198861119860

120597119860

120597Θ

+ 1198862

1205973119860

120597Θ3

+ 1198863

1205972

120597Θ2H (119860 (Θ 119879)) + 120582119860 =

120597119866

120597Θ

(27)

whereH(119860(Θ 119879)) = (11990324120587) int

2120587

0119860(Θ1015840 119879)cot((ΘminusΘ

1015840)2)119889Θ

1015840

and 119886 = int

1199032

1199031

(120601(120596 minus 119888))(119889119889119903)(119903(119889120601119889119903))119889119903 1198861= int

1199032

1199031

(1206013(120596 minus

119888))(119889119889119903)[120573 minus (119889119889119903)((1119903)(1198891199032120596119889119903))120601(120596 minus 119888)]119889119903119886 119886

2=

int

1199032

1199031

(1206012119903)119889119903119886 119886

3= 11990321206012(1199032)119886 119866 = int

1199032

1199031


2119860120597Θ




=


3 Conservation Laws



Θ 119860ΘΘ

119860ΘΘΘ


1198761= int

2120587

0

119860119889Θ = exp (minus120582119879)int

2120587

0

119860 (Θ 0) 119889Θ (28)






2120587

0119891(Θ)H(119891(Θ))119889Θ = 0 then we get

1198762= int

2120587

0

1198602119889Θ = exp (minus2120582119879)int

2120587

0

1198602(Θ 0) 119889Θ (29)




Θ)) and obtain

(

1

3

1198603)

119879

minus

1198863

1198861

H (119860Θ) 119860119879

+ (120572 + 1198861119860)119860Θ(1198602minus

1198863

1198861

H (119860Θ))

+ 1198862119860ΘΘΘ

(1198602minus

1198863

1198861

H (119860Θ))

+ 1198863(1198602minus

1198863

1198861

H (119860Θ)) (H (119860))

ΘΘ

+ 120582(1198602minus

1198863

1198861

H (119860Θ))119860 = 0

(30)


21198861)119860Θ+ (11988631198861)H(119860)) lead to

(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))119860Θ119879

+ [120572119860ΘΘ

+ 1198861(119860119860Θ)Θ] (minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))

+ 1198862119860ΘΘΘΘ

(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))

+ 1198863(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))H(119860)ΘΘΘ

+ 120582119860(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860)) = 0

(31)


H(119860)ΘΘ

= H (119860ΘΘ

) int

2120587

0


2120587

0

VH119906119889Θ

(32)

we have

(

1

3

1198603minus

1198862

1198861

1198602

Θ+

1198863

1198861

119860ΘH (119860))

119879

+ 120572[

1

3

1198603minus

1198862

1198861

1198602

Θ+

1198863

1198861

H (119860)119860Θ]

Θ

+ (

1198861

4

1198604)

Θ

+

1198862

3

21198861

[119867 (119860)119867(119860)ΘΘ

]Θ

+

11988621198863

1198861

(119860ΘΘΘ

119867(119860) minus 2119860Θ119867(119860)ΘΘ

)Θ

minus

21198862

1198861

(119860Θ119860ΘΘΘ

minus

1

2

1198602

ΘΘ)

Θ

+ 1198862(1198602119860ΘΘ

minus 21198601198602

Θ)Θ

+ 1198863[119860(119860H (119860))

Θ]Θ+ 120582(119860

3minus

1198862

1198861

1198602

Θ+

1198863

1198861

119860Θ119867(119860))

= 0

(33)

Taking1198763= int

2120587

0((13)119860

3minus(11988621198861)1198602

Θ+(11988631198861)119860ΘH(119860))119889Θ






1198764=

119889

119889119879

int

2120587

0

Θ119860119889Θ (34)



4= 11987641198761 by employing

1198891198761119889119879 = 0 and 119889119876

4119889119879 = 0 we have 119889119876

4119889119879 = 0 which



1198765=int

2120587

0

(

1

4

1198604minus

31198862

1198861

1198601198602

Θ+

91198862

1198862

1

1198602

ΘΘ+

1198863

41198861

1198602H (119860)) 119889Θ

(35)


6and the seventh






119860 (V 119879) = 119865119860 =

1

radic2119873

2119873minus1

sum

119895=0

119860 (119895Δ119883 119879) 119890minus120587119894119895V119873


(36)


119860 (119895Δ119883 119879) = 119865minus1119860 =

1

radic2119873

sum

V119860 (V 119879) 119890

120587119894119895V119873 (37)






V119865119860Δ119879

+ 1198941198861119860119865minus1

V119865119860Δ119879 minus 1198862119894119865minus1

V3119865119860Δ119879


V2119865H (119860) Δ119879 + 120582119860 = 119894119865minus1

V119865119866Δ119879

(38)













00

5

10

15

T

minus05

0

05

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) 120572 = 25

00

5

10

15

T

minus05

0

05

1

15

2

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) 120572 = 0

00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(c) 120572 = minus25


00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579








2(1205973119860120597Θ




00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) KdV model (1198863 = 0 120572 = 25 120582 = 0)

00

5

10

15

T

minus05

0

05

1

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) BO model (1198862 = 0 120572 = 25 120582 = 0)


5 Conclusions


2(1205973119860120597Θ




Acknowledgments


References


































=


3 Conservation Laws



Θ 119860ΘΘ

119860ΘΘΘ


1198761= int

2120587

0

119860119889Θ = exp (minus120582119879)int

2120587

0

119860 (Θ 0) 119889Θ (28)






2120587

0119891(Θ)H(119891(Θ))119889Θ = 0 then we get

1198762= int

2120587

0

1198602119889Θ = exp (minus2120582119879)int

2120587

0

1198602(Θ 0) 119889Θ (29)




Θ)) and obtain

(

1

3

1198603)

119879

minus

1198863

1198861

H (119860Θ) 119860119879

+ (120572 + 1198861119860)119860Θ(1198602minus

1198863

1198861

H (119860Θ))

+ 1198862119860ΘΘΘ

(1198602minus

1198863

1198861

H (119860Θ))

+ 1198863(1198602minus

1198863

1198861

H (119860Θ)) (H (119860))

ΘΘ

+ 120582(1198602minus

1198863

1198861

H (119860Θ))119860 = 0

(30)


21198861)119860Θ+ (11988631198861)H(119860)) lead to

(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))119860Θ119879

+ [120572119860ΘΘ

+ 1198861(119860119860Θ)Θ] (minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))

+ 1198862119860ΘΘΘΘ

(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))

+ 1198863(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860))H(119860)ΘΘΘ

+ 120582119860(minus

21198862

1198861

119860Θ+

1198863

1198861

H (119860)) = 0

(31)


H(119860)ΘΘ

= H (119860ΘΘ

) int

2120587

0


2120587

0

VH119906119889Θ

(32)

we have

(

1

3

1198603minus

1198862

1198861

1198602

Θ+

1198863

1198861

119860ΘH (119860))

119879

+ 120572[

1

3

1198603minus

1198862

1198861

1198602

Θ+

1198863

1198861

H (119860)119860Θ]

Θ

+ (

1198861

4

1198604)

Θ

+

1198862

3

21198861

[119867 (119860)119867(119860)ΘΘ

]Θ

+

11988621198863

1198861

(119860ΘΘΘ

119867(119860) minus 2119860Θ119867(119860)ΘΘ

)Θ

minus

21198862

1198861

(119860Θ119860ΘΘΘ

minus

1

2

1198602

ΘΘ)

Θ

+ 1198862(1198602119860ΘΘ

minus 21198601198602

Θ)Θ

+ 1198863[119860(119860H (119860))

Θ]Θ+ 120582(119860

3minus

1198862

1198861

1198602

Θ+

1198863

1198861

119860Θ119867(119860))

= 0

(33)

Taking1198763= int

2120587

0((13)119860

3minus(11988621198861)1198602

Θ+(11988631198861)119860ΘH(119860))119889Θ






1198764=

119889

119889119879

int

2120587

0

Θ119860119889Θ (34)



4= 11987641198761 by employing

1198891198761119889119879 = 0 and 119889119876

4119889119879 = 0 we have 119889119876

4119889119879 = 0 which



1198765=int

2120587

0

(

1

4

1198604minus

31198862

1198861

1198601198602

Θ+

91198862

1198862

1

1198602

ΘΘ+

1198863

41198861

1198602H (119860)) 119889Θ

(35)


6and the seventh






119860 (V 119879) = 119865119860 =

1

radic2119873

2119873minus1

sum

119895=0

119860 (119895Δ119883 119879) 119890minus120587119894119895V119873


(36)


119860 (119895Δ119883 119879) = 119865minus1119860 =

1

radic2119873

sum

V119860 (V 119879) 119890

120587119894119895V119873 (37)






V119865119860Δ119879

+ 1198941198861119860119865minus1

V119865119860Δ119879 minus 1198862119894119865minus1

V3119865119860Δ119879


V2119865H (119860) Δ119879 + 120582119860 = 119894119865minus1

V119865119866Δ119879

(38)













00

5

10

15

T

minus05

0

05

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) 120572 = 25

00

5

10

15

T

minus05

0

05

1

15

2

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) 120572 = 0

00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(c) 120572 = minus25


00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579








2(1205973119860120597Θ




00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) KdV model (1198863 = 0 120572 = 25 120582 = 0)

00

5

10

15

T

minus05

0

05

1

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) BO model (1198862 = 0 120572 = 25 120582 = 0)


5 Conclusions


2(1205973119860120597Θ




Acknowledgments


References


































1198764=

119889

119889119879

int

2120587

0

Θ119860119889Θ (34)



4= 11987641198761 by employing

1198891198761119889119879 = 0 and 119889119876

4119889119879 = 0 we have 119889119876

4119889119879 = 0 which



1198765=int

2120587

0

(

1

4

1198604minus

31198862

1198861

1198601198602

Θ+

91198862

1198862

1

1198602

ΘΘ+

1198863

41198861

1198602H (119860)) 119889Θ

(35)


6and the seventh






119860 (V 119879) = 119865119860 =

1

radic2119873

2119873minus1

sum

119895=0

119860 (119895Δ119883 119879) 119890minus120587119894119895V119873


(36)


119860 (119895Δ119883 119879) = 119865minus1119860 =

1

radic2119873

sum

V119860 (V 119879) 119890

120587119894119895V119873 (37)






V119865119860Δ119879

+ 1198941198861119860119865minus1

V119865119860Δ119879 minus 1198862119894119865minus1

V3119865119860Δ119879


V2119865H (119860) Δ119879 + 120582119860 = 119894119865minus1

V119865119866Δ119879

(38)













00

5

10

15

T

minus05

0

05

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) 120572 = 25

00

5

10

15

T

minus05

0

05

1

15

2

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) 120572 = 0

00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(c) 120572 = minus25


00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579








2(1205973119860120597Θ




00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) KdV model (1198863 = 0 120572 = 25 120582 = 0)

00

5

10

15

T

minus05

0

05

1

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) BO model (1198862 = 0 120572 = 25 120582 = 0)


5 Conclusions


2(1205973119860120597Θ




Acknowledgments


References

































00

5

10

15

T

minus05

0

05

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) 120572 = 25

00

5

10

15

T

minus05

0

05

1

15

2

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) 120572 = 0

00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(c) 120572 = minus25


00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579








2(1205973119860120597Θ




00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) KdV model (1198863 = 0 120572 = 25 120582 = 0)

00

5

10

15

T

minus05

0

05

1

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) BO model (1198862 = 0 120572 = 25 120582 = 0)


5 Conclusions


2(1205973119860120597Θ




Acknowledgments


References

































00

5

10

15

T

minus04

minus02

0

02

04

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(a) KdV model (1198863 = 0 120572 = 25 120582 = 0)

00

5

10

15

T

minus05

0

05

1

A(120579T

)

1205874 1205872 31205874 120587 51205874 31205872 71205874 2120587

120579

(b) BO model (1198862 = 0 120572 = 25 120582 = 0)


5 Conclusions


2(1205973119860120597Θ




Acknowledgments


References

















































Documents

A New Integro-Differential Equation for Rossby Solitary