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In physics and mathematics, the Clebsch representation of an arbitrary three-dimensional vector field ${\boldsymbol {v}}({\boldsymbol {x}})$ is:^[1]^[2]

${\boldsymbol {v}}={\boldsymbol {\nabla }}\varphi +\psi \,{\boldsymbol {\nabla }}\chi ,$

where the scalar fields $\varphi ({\boldsymbol {x}})$ $,\psi ({\boldsymbol {x}})$ and $\chi ({\boldsymbol {x}})$ are known as Clebsch potentials^[3] or Monge potentials,^[4] named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and ${\boldsymbol {\nabla }}$ is the gradient operator.

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.^[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 ${\boldsymbol {v}}$ has (locally) to be bounded, continuous and sufficiently smooth. For global applicability ${\boldsymbol {v}}$ 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.^[1] Since $\psi {\boldsymbol {\nabla }}\chi$ is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition.^[10]

Vorticity

The vorticity ${\boldsymbol {\omega }}({\boldsymbol {x}})$ is equal to^[2]

${\boldsymbol {\omega }}={\boldsymbol {\nabla }}\times {\boldsymbol {v}}={\boldsymbol {\nabla }}\times \left({\boldsymbol {\nabla }}\varphi +\psi \,{\boldsymbol {\nabla }}\chi \right)={\boldsymbol {\nabla }}\psi \times {\boldsymbol {\nabla }}\chi ,$

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

Notes

^ ^a ^b Lamb (1993, pp. 248–249)
^ ^a ^b Serrin (1959, pp. 169–171)
^ Benjamin (1984)
^ Aris (1962, pp. 70–72)
^ Clebsch (1859)
^ Bateman (1929)
^ Seliger & Whitham (1968)
^ Luke (1967)
^ Wesseling (2001, p. 7)
^ Wu, Ma & Zhou (2007, p. 43)

References

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, Bibcode:1984JApMa..32....3B, doi:10.1093/imamat/32.1-3.3
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
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, S2CID 123409273
Morrison, P.J. (2006). "Hamiltonian Fluid Dynamics" (PDF). Hamiltonian fluid mechanics. Encyclopedia of Mathematical Physics. Vol. 2. Elsevier. 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, S2CID 119565234
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
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

[Lamb-1] Lamb (1993, pp. 248–249)

[Serrin-2] Serrin (1959, pp. 169–171)

[3] Benjamin (1984)

[4] Aris (1962, pp. 70–72)

[5] Clebsch (1859)

[6] Bateman (1929)

[7] Seliger & Whitham (1968)

[8] Luke (1967)

[9] Wesseling (2001, p. 7)

[10] Wu, Ma & Zhou (2007, p. 43)

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