Clebsch representation

In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field is:[1][2]

where the scalar fields and are known as Clebsch potentials[3] or Monge potentials,[4] named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and is the gradient operator.

BackgroundEdit

In fluid dynamics and plasma physics, the Clebsch representation provides a means to overcome the difficulties to describe an inviscid flow with non-zero vorticity – in the Eulerian reference frame – using Lagrangian mechanics and Hamiltonian mechanics.[5][6][7] At the critical point of such functionals the result is the Euler equations, a set of equations describing the fluid flow. Note that the mentioned difficulties do not arise when describing the flow through a variational principle in the Lagrangian reference frame. In case of surface gravity waves, the Clebsch representation leads to a rotational-flow form of Luke's variational principle.[8]

For the Clebsch representation to be possible, the vector field   has (locally) to be bounded, continuous and sufficiently smooth. For global applicability   has to decay fast enough towards infinity.[9] The Clebsch decomposition is not unique, and (two) additional constraints are necessary to uniquely define the Clebsch potentials.[1] Since   is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.[10]

VorticityEdit

The vorticity   is equal to[2]

 

with the last step due to the vector calculus identity   So the vorticity   is perpendicular to both   and   while further the vorticity does not depend on  

NotesEdit

ReferencesEdit

  • Aris, R. (1962), Vectors, tensors, and the basic equations of fluid mechanics, Prentice-Hall, OCLC 299650765
  • Bateman, H. (1929), "Notes on a differential equation which occurs in the two-dimensional motion of a compressible fluid and the associated variational problems", Proceedings of the Royal Society of London A, 125 (799): 598–618, Bibcode:1929RSPSA.125..598B, doi:10.1098/rspa.1929.0189
  • Benjamin, T. Brooke (1984), "Impulse, flow force and variational principles", IMA Journal of Applied Mathematics, 32 (1–3): 3–68, doi:10.1093/imamat/32.1-3.3
  • Clebsch, A. (1859), "Ueber die Integration der hydrodynamischen Gleichungen" (PDF), Journal für die Reine und Angewandte Mathematik, 56 (56): 1–10, doi:10.1515/crll.1859.56.1
  • Lamb, H. (1993), Hydrodynamics (6th ed.), Dover, ISBN 978-0-486-60256-1
  • Luke, J.C. (1967), "A variational principle for a fluid with a free surface", Journal of Fluid Mechanics, 27 (2): 395–397, Bibcode:1967JFM....27..395L, doi:10.1017/S0022112067000412
  • Morrison, P.J. (2006), "Hamiltonian Fluid Dynamics", Hamiltonian fluid mechanics (PDF), Encyclopedia of Mathematical Physics, Elsevier, 2, pp. 593–600, doi:10.1016/B0-12-512666-2/00246-7, ISBN 9780125126663
  • Rund, H. (1976), "Generalized Clebsch representations on manifolds", Topics in differential geometry, Academic Press, pp. 111–133, ISBN 978-0-12-602850-8
  • Salmon, R. (1988), "Hamiltonian fluid mechanics", Annual Review of Fluid Mechanics, 20: 225–256, Bibcode:1988AnRFM..20..225S, doi:10.1146/annurev.fl.20.010188.001301
  • Seliger, R.L.; Whitham, G.B. (1968), "Variational principles in continuum mechanics", Proceedings of the Royal Society of London A, 305 (1440): 1–25, Bibcode:1968RSPSA.305....1S, doi:10.1098/rspa.1968.0103
  • Serrin, J. (1959), Flügge, S.; Truesdell, C. (eds.), "Encyclopedia of Physics", Handbuch der Physik, Encyclopedia of Physics / Handbuch der Physik, VIII/1: 125–263, Bibcode:1959HDP.....8..125S, doi:10.1007/978-3-642-45914-6_2, ISBN 978-3-642-45916-0, MR 0108116, Zbl 0102.40503 |contribution= ignored (help)
  • Wesseling, P. (2001), Principles of computational fluid dynamics, Springer, ISBN 978-3-540-67853-3
  • Wu, J.-Z.; Ma, H.-Y.; Zhou, M.-D. (2007), Vorticity and vortex dynamics, Springer, ISBN 978-3-540-29027-8