TY - JOUR

T1 - Existence of Corotating and Counter-Rotating Vortex Pairs for Active Scalar Equations

AU - Hmidi, Taoufik

AU - Mateu, Joan

N1 - Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - In this paper,we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the(SQG)αequations with α∈(0,1).From the numerical experiments implemented for Euler equations in Deem and Zabusky(Phys Rev Lett40(13):859–862,1978),Pierrehumbert(J Fluid Mech 99:129–144,1980),Saffman and Szeto(Phys Fluids 23(12):2339–2342,1980)it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs.There are some analytical proofs based on variational principle(Keady in J Aust Math Soc Ser B 26:487–502,1985;Turkington in Nonlinear Anal Theory Methods Appl9(4):351–369,1985);however,they do not give enough information about the pairs, such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the(SQG)α equation when α∈(0,1).The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem.

AB - In this paper,we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the(SQG)αequations with α∈(0,1).From the numerical experiments implemented for Euler equations in Deem and Zabusky(Phys Rev Lett40(13):859–862,1978),Pierrehumbert(J Fluid Mech 99:129–144,1980),Saffman and Szeto(Phys Fluids 23(12):2339–2342,1980)it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs.There are some analytical proofs based on variational principle(Keady in J Aust Math Soc Ser B 26:487–502,1985;Turkington in Nonlinear Anal Theory Methods Appl9(4):351–369,1985);however,they do not give enough information about the pairs, such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the(SQG)α equation when α∈(0,1).The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem.

UR - http://www.scopus.com/inward/record.url?scp=84991640184&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84991640184&partnerID=8YFLogxK

U2 - 10.1007/s00220-016-2784-7

DO - 10.1007/s00220-016-2784-7

M3 - Article

AN - SCOPUS:84991640184

SN - 0010-3616

VL - 350

SP - 699

EP - 747

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -