TY - JOUR
T1 - A unified view of topological invariants of barotropic and baroclinic fluids and their application to formal stability analysis of three-dimensional ideal gas flows
AU - Fukumoto, Yasuhide
AU - Sakuma, Hirofumi
N1 - Funding Information:
We are grateful to Profs. D. D. Holm and M. V. Kurgansky for their invaluable and inspiring comments. Y. F. was supported in part by the Joint Research Project between the Japan Society of the Promotion of Science and the Royal Society of London.
PY - 2013
Y1 - 2013
N2 - Noether's theorem associated with the particle relabeling symmetry group leads us to a unified view that all the topological invariants of a barotropic fluid are variants of the cross helicity. The same is shown to be true for a baroclinic fluid. A cross-helicity representation is given to the Casimir invariant, a class of integrals including an arbitrary function of the specific entropy and the potential vorticity. We then develop a new energy-Casimir convexity method for three-dimensional stability of equilibria of general rotating flows of an ideal baroclinic fluid, without appealing to the Boussinesq approximation. By fully exploiting the Casimir invariant, we have succeeded in ruling out a term including the gradient of a dependent variable from the energy-Casimir function and have established a sharp linear stability criterion, being an extension of the Richardson-number criterion.
AB - Noether's theorem associated with the particle relabeling symmetry group leads us to a unified view that all the topological invariants of a barotropic fluid are variants of the cross helicity. The same is shown to be true for a baroclinic fluid. A cross-helicity representation is given to the Casimir invariant, a class of integrals including an arbitrary function of the specific entropy and the potential vorticity. We then develop a new energy-Casimir convexity method for three-dimensional stability of equilibria of general rotating flows of an ideal baroclinic fluid, without appealing to the Boussinesq approximation. By fully exploiting the Casimir invariant, we have succeeded in ruling out a term including the gradient of a dependent variable from the energy-Casimir function and have established a sharp linear stability criterion, being an extension of the Richardson-number criterion.
UR - http://www.scopus.com/inward/record.url?scp=84876497387&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84876497387&partnerID=8YFLogxK
U2 - 10.1016/j.piutam.2013.03.025
DO - 10.1016/j.piutam.2013.03.025
M3 - Conference article
AN - SCOPUS:84876497387
VL - 7
SP - 213
EP - 222
JO - Procedia IUTAM
JF - Procedia IUTAM
SN - 2210-9838
T2 - IUTAM Symposium on Topological Fluid Mechanics II
Y2 - 23 July 2012 through 27 July 2012
ER -