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Clebsch representation

In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field v ( x ) {\displaystyle {\boldsymbol {v}}({\boldsymbol {x}})} is:

v = ∇ φ + ψ ∇ χ , {\displaystyle {\boldsymbol {v}}={\boldsymbol {\nabla }}\varphi +\psi \,{\boldsymbol {\nabla }}\chi ,}

where the scalar fields φ ( x ) {\displaystyle \varphi ({\boldsymbol {x}})} , ψ ( x ) {\displaystyle ,\psi ({\boldsymbol {x}})} and χ ( x ) {\displaystyle \chi ({\boldsymbol {x}})} are known as Clebsch potentials or Monge potentials, named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and ∇ {\displaystyle {\boldsymbol {\nabla }}} is the gradient operator.

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Background

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.567 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 v {\displaystyle {\boldsymbol {v}}} has (locally) to be bounded, continuous and sufficiently smooth. For global applicability v {\displaystyle {\boldsymbol {v}}} 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.10 Since ψ ∇ χ {\displaystyle \psi {\boldsymbol {\nabla }}\chi } is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.11

Vorticity

The vorticity ω ( x ) {\displaystyle {\boldsymbol {\omega }}({\boldsymbol {x}})} is equal to12

ω = ∇ × v = ∇ × ( ∇ φ + ψ ∇ χ ) = ∇ ψ × ∇ χ , {\displaystyle {\boldsymbol {\omega }}={\boldsymbol {\nabla }}\times {\boldsymbol {v}}={\boldsymbol {\nabla }}\times \left({\boldsymbol {\nabla }}\varphi +\psi \,{\boldsymbol {\nabla }}\chi \right)={\boldsymbol {\nabla }}\psi \times {\boldsymbol {\nabla }}\chi ,}

with the last step due to the vector calculus identity ∇ × ( ψ A ) = ψ ( ∇ × A ) + ∇ ψ × A . {\displaystyle {\boldsymbol {\nabla }}\times (\psi {\boldsymbol {A}})=\psi ({\boldsymbol {\nabla }}\times {\boldsymbol {A}})+{\boldsymbol {\nabla }}\psi \times {\boldsymbol {A}}.} So the vorticity ω {\displaystyle {\boldsymbol {\omega }}} is perpendicular to both ∇ ψ {\displaystyle {\boldsymbol {\nabla }}\psi } and ∇ χ , {\displaystyle {\boldsymbol {\nabla }}\chi ,} while further the vorticity does not depend on φ . {\displaystyle \varphi .}

Notes

References

  1. Lamb (1993, pp. 248–249) - Lamb, H. (1993), Hydrodynamics (6th ed.), Dover, ISBN 978-0-486-60256-1 ↩

  2. Serrin (1959, pp. 169–171) - Serrin, J. (1959), "Mathematical principles of classical fluid mechanics", in Flügge, S.; Truesdell, C. (eds.), Strömungsmechanik I [Fluid Dynamics I], Encyclopedia of Physics / Handbuch der Physik, vol. VIII/1, pp. 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 https://ui.adsabs.harvard.edu/abs/1959HDP.....8..125S ↩

  3. Benjamin (1984) - Benjamin, T. Brooke (1984), "Impulse, flow force and variational principles", IMA Journal of Applied Mathematics, 32 (1–3): 3–68, Bibcode:1984JApMa..32....3B, doi:10.1093/imamat/32.1-3.3 https://ui.adsabs.harvard.edu/abs/1984JApMa..32....3B ↩

  4. Aris (1962, pp. 70–72) - Aris, R. (1962), Vectors, tensors, and the basic equations of fluid mechanics, Prentice-Hall, OCLC 299650765 https://search.worldcat.org/oclc/299650765 ↩

  5. Clebsch (1859) - Clebsch, A. (1859), "Ueber die Integration der hydrodynamischen Gleichungen", Journal für die Reine und Angewandte Mathematik, 1859 (56): 1–10, doi:10.1515/crll.1859.56.1, S2CID 122730522 https://zenodo.org/record/1448884 ↩

  6. Bateman (1929) - 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 https://ui.adsabs.harvard.edu/abs/1929RSPSA.125..598B ↩

  7. Seliger & Whitham (1968) - 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, S2CID 119565234 https://ui.adsabs.harvard.edu/abs/1968RSPSA.305....1S ↩

  8. Luke (1967) - 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, S2CID 123409273 https://ui.adsabs.harvard.edu/abs/1967JFM....27..395L ↩

  9. Wesseling (2001, p. 7) - Wesseling, P. (2001), Principles of computational fluid dynamics, Springer, ISBN 978-3-540-67853-3 ↩

  10. Lamb (1993, pp. 248–249) - Lamb, H. (1993), Hydrodynamics (6th ed.), Dover, ISBN 978-0-486-60256-1 ↩

  11. Wu, Ma & Zhou (2007, p. 43) - Wu, J.-Z.; Ma, H.-Y.; Zhou, M.-D. (2007), Vorticity and vortex dynamics, Springer, ISBN 978-3-540-29027-8 ↩

  12. Serrin (1959, pp. 169–171) - Serrin, J. (1959), "Mathematical principles of classical fluid mechanics", in Flügge, S.; Truesdell, C. (eds.), Strömungsmechanik I [Fluid Dynamics I], Encyclopedia of Physics / Handbuch der Physik, vol. VIII/1, pp. 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 https://ui.adsabs.harvard.edu/abs/1959HDP.....8..125S ↩