Skip to main content
SpringerLink
Log in

Point Vortex Dynamics for the 2D Boussinesq Equations Over the Tropics

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

The aim of this paper is to find some vortex solutions of finite core size for plane Boussinesq equations in a weighted subspace of \(L^2(\mathbb {R}^2)\). Here, the solution of the vorticity and temperature equations are separated into \(N\) components and derive separate evolution equations for each component. Next, the solutions are expanded into series of Hermite eigenfunctions. Hence, it is obtained the coefficients of the series and a set of \(2n\) PDEs which govern the evolution of the vorticity of each vortex structure and a set of \(4n\) ODEs governing the motion of the centers of each vortex structure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Baghani H, Alzabut J, Nieto JJ (2020) Further Results on the Existence of Solutions for Generalized Fractional Basset–Boussinesq–Oseen Equation. Iran J Sci Technol Trans A Sci 44(5):1461–1467

    Article  MathSciNet  Google Scholar 

  • Batchelor GK (1967) An introduction to fluid mechanics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Constantin P, Doering C (1996) Heat transfer in convective turbulence. Nonlinearity 9:1049–1060

    Article  MathSciNet  Google Scholar 

  • Gill AE (2016) Atmosphere ocean dynamics. Elsevier, New York

    Google Scholar 

  • Helmholtz HV (1858) Über Integrale der hydrodynamischen Gleichungen. J Reine Angew Math 55:25–55

    MathSciNet  Google Scholar 

  • Islam Sh, Nur Alam Md, Fayz-Al-Asad Md, Tunç C (2021) An analytical technique for solving new computational of the modified Zakharov–Kuznetsov equation arising in electrical engineering. J Appl Comput Mech 7(2):715–726

    Google Scholar 

  • Kelvin L (1868) On vortex motion. Trans Roy Soc Edinburgh 25:217–60

    Article  Google Scholar 

  • Lamb H (1932) Hydrodynamics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Lappa M (2010) Thermal convection: patterns evolutions and stability. Wiley Publication, New York

    MATH  Google Scholar 

  • Ma T, Wang S (2007) Rayleigh Bénard convection: dynamics and structure in the physical space. Commun Math Sci 5(3):553–574

    Article  MathSciNet  Google Scholar 

  • Ma T, Wang S (2009) Tropical atmospheric circulations: dynamic stability and transitions. Discrete Contin Dynam Systems 26(4):1399–1417

    Article  MathSciNet  Google Scholar 

  • Ma T, Wang Sh. (2014a) Phase transition dynamics. Springer, Berlin

    Book  Google Scholar 

  • Ma T, Wang S (2014b) Astrophysical dynamics and cosmology. J Math Study 47(4):305–378

    Article  MathSciNet  Google Scholar 

  • Majda A (2003) Introduction to PDEs and waves for the atmosphere and ocean. American Mathematical Soc, New York

    Book  Google Scholar 

  • Majda A, Bertozzi A (2002) Vorticity and incompressible flow. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Milne-Thomson LM (1968) Theoretical hydrodynamics. Dover Publications, New York

    Book  Google Scholar 

  • Nagem RJ, Sandri G, Uminsky D (2007) Vorticity dynamics and sound generation in two-dimensional fluid flow. J Acoust Soc Am 122:128–134

    Article  Google Scholar 

  • Nagem RJ, Sandri G, Uminsky D, Wayne CE (2009) Generalized Helmholtz–Kirchhoff model for two-dimensional distributed vortex motion. SIAM J Appl Dyn Syst 8:160–179

    Article  MathSciNet  Google Scholar 

  • Nur Alam Md, Tunç C (2016) An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator-prey system. Alex Eng J 11(1):152–161

    Google Scholar 

  • Nur Alam Md, Tunç C (2020a) New solitary wave structures to the (2+1)-dimensional KD and KP equations with spatio-temporal dispersion. J King Saud Univ Sci 32:3400–3409

    Article  Google Scholar 

  • Nur Alam Md, Tunç C (2020b) The new solitary wave structures for the (2+1)-dimensional time-fractional Schrodinger equation and the space-time nonlinear conformable fractional Bogoyavlenskii equations. Alex Eng J 59:2221–2232

    Article  Google Scholar 

  • Özer S, Şengül T (2018) Transitions of spherical thermohaline circulation to multiple equilibria. J Math Fluid Mech 20:499–515

    Article  MathSciNet  Google Scholar 

  • Pedlosky J (1981) Geophysical fluid dynamics I. Springer, Berlin

    MATH  Google Scholar 

  • Prandtl L (1952) Essentials of fluid dynamics. Hafner Publishing Company, New York

    MATH  Google Scholar 

  • Richardson LF (1922) Weather prediction by numerical process. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Saffman PG (1992) Vortex dynamics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Sharifi M, Raesi B (2020) Vortex theory for two dimensional Boussinesq equations. Appl Math Nonlinear Sci 5(2):67–84

    Article  MathSciNet  Google Scholar 

  • Uminsky, DT (2009) The viscous N vortex problem: a generalized Helmholtz/Kirchhoff approach. Ph.D. thesis, Boston University

  • Uminsky D, Wayne CE, Barbaro A (2012) A multi-moment vortex method for 2D viscous fluids. J Comput Phys 231:1705–1727

    Article  MathSciNet  Google Scholar 

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Mathematics, Shahed University, Tehran, Iran

    Mahdi Kamandar & Behruz Raesi

Authors
  1. Mahdi Kamandar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Behruz Raesi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Behruz Raesi.

Ethics declarations

Conflict of interest

The authors have not disclosed any competing interests.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kamandar, M., Raesi, B. Point Vortex Dynamics for the 2D Boussinesq Equations Over the Tropics. Iran J Sci Technol Trans Sci 46, 839–848 (2022). https://doi.org/10.1007/s40995-022-01305-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-022-01305-6

Keywords

Mathematics Subject Classification

Search

Navigation