Abstract
The conservation of the enstrophy (\(L^2\) norm of the vorticity \(\omega \)) plays an essential role in the physics and mathematics of two-dimensional (2D) Euler fluids. Generalizing to compressible ideal (inviscid and barotropic) fluids, the generalized enstrophy \(\int _{\varSigma (t)}\) f(\(\omega /\rho )\rho \mathrm {d}^2 x\) (f an arbitrary smooth function, \(\rho \) the density, and \(\varSigma (t)\) an arbitrary 2D domain co-moving with the fluid) is a constant of motion, and plays the same role. On the other hand, for the three-dimensional (3D) ideal fluid, the helicity \(\int _{M}\) V \(\cdot \varvec{\omega }\,\mathrm {d}^3x\) (\(\varvec{V}\) the flow velocity, \(\varvec{\omega }=\nabla \times \varvec{V}\), and M the three-dimensional domain containing the fluid) is conserved. Evidently, the helicity degenerates in a 2D system, and the (generalized) enstrophy emerges as a compensating constant. This transition of the constants of motion is a reflection of an essential difference between 2D and 3D systems, because the conservation of the (generalized) enstrophy imposes stronger constraints, than the helicity, on the flow. In this paper, we make a deeper inquiry into the helicity-enstrophy interplay: the ideal fluid mechanics is cast into a Hamiltonian form in the phase space of Clebsch parameters, generalizing 2D to a wider category of epi-2D flows (2D embedded in 3D has zero-helicity, while the converse is not true – our epi-2D category encompasses a wider class of zero-helicity flows); how helicity degenerates and is substituted by a new constant is delineated; and how a further generalized enstrophy is introduced as a constant of motion applying to epi-2D flow is described.
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Notes
- 1.
In this work, we do not argue for the existence of regular solutions of the model equations. The conservation laws discussed are, therefore, a priori relations satisfied by all regular solutions if they exist.
- 2.
We note that \(\partial _z=0\) does not mean that the system extends in the z-direction homogeneously; instead, we consider a thin system in which variation of physical quantities in the z-direction is much larger than in the x and y directions. Thus, \(\partial _z\) can be separated from \(\partial _x\) and \(\partial _y\).
- 3.
Fukumoto [3] points out that the helicity and the generalized enstrophy can be unified by the concept of cross helicity.
- 4.
Here the phase space (function space) X may be viewed as a cotangent bundle of \(X_q =\{ (\xi _2, \xi _4, \xi _6)^{\mathrm {T}} ;\, \xi _j \in \mathrm {\Omega }^0(M)\}\). For \(F\in C^\infty (X)\), \(\partial _{\varvec{\xi }} F\in X^*\) (to be defined in (12)) may be regarded as a “1-form” on X. Hence, the duality of \(X^*\) and X corresponds to that of “co-vectors” and “vectors.” At the same time, the components of the field \(\varvec{\xi }\in X\) are differential forms (0-forms and n-forms) on the “base space” M; the “Hodge-duality” of \(X^*\) and X is in the sense of the differential forms on M, while the duality (10) is in the sense of “co-vectors” and “vectors” on the function space X.
- 5.
Here we denote by \(\alpha ^*\) the Hodge-dual of a differential form \(\alpha \).
- 6.
- 7.
When considering a relativistic fluid, we generate a diffeomorphism group \(\mathrm {e}^{\tau U}\) (\(\tau \) the proper time), acting on a space-time manifold \(\tilde{M}=\mathbb {R}\times {M}\), by a space-time velocity \(U\in T\tilde{M}\). Then, the space-time derivative \(\widetilde{\mathscr {L}}_{\varvec{V}}\) is replaced by the natural Lie derivative \(\mathscr {L}_U\). When \(\mathscr {L}_U\) applies to a differential form \(\alpha \) on \(\tilde{M}\), the temporal and spatial components are mixed up (cf. [17]).
- 8.
- 9.
While we have started the discussion with the assumption of the boundary-less domain \(M=T^3\) (in order to avoid complexities in formulating the Hamiltonian mechanics), the definition of the epi-2D flow is independent of the boundary condition, so we may consider an arbitrary domain.
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Acknowledgements
ZY acknowledges discussions with Professor Y. Fukumoto and Y. Kimura. The work of ZY was supported by JSPS KAKENHI Grant Number 23224014 and 15K13532, while that of PJM was supported by USDOE contract DE-FG02-04ER-54742.
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Yoshida, Z., Morrison, P.J. (2017). Epi-Two-Dimensional Flow and Generalized Enstrophy. In: Maekawa, Y., Jimbo, S. (eds) Mathematics for Nonlinear Phenomena — Analysis and Computation. MNP2015 2015. Springer Proceedings in Mathematics & Statistics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-319-66764-5_13
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