Clebsch representation

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

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

where the scalar fields ${\displaystyle \varphi ({\boldsymbol {x}})}$${\displaystyle ,\psi ({\boldsymbol {x}})}$ and ${\displaystyle \chi ({\boldsymbol {x}})}$ are known as Clebsch potentials ^[3] or Monge potentials, ^[4] named after Alfred Clebsch (1833–1872) and Gaspard Monge (1746–1818), and ${\displaystyle {\boldsymbol {\nabla }}}$ 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 ${\displaystyle {\boldsymbol {v}}}$ has (locally) to be bounded, continuous and sufficiently smooth. For global applicability ${\displaystyle {\boldsymbol {v}}}$ 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. ^[1] Since ${\displaystyle \psi {\boldsymbol {\nabla }}\chi }$ is in general not solenoidal, the Clebsch representation does not in general satisfy the Helmholtz decomposition. ^[10]

Vorticity

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

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

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

Notes

1. Lamb (1993 , pp. 248–249)
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)

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