TY - JOUR
T1 - Normal forms for three-dimensional parametric instabilities in ideal hydrodynamics
AU - Knobloch, Edgar
AU - Mahalov, Alex
AU - Marsden, Jerrold E.
N1 - Funding Information:
In the last few years it has become recognized that elliptical distortion of a circular hydrodynamic flow can lead to instability \[36,2,42,43\]. The inclusion of precession can also lead to instability \[20,29\ ]. The unstable modes are characterized by an azimuthal wavenumber m, an axial wavenum-bet k and a frequency oJ. Linear stability theory for Hamiltonian systems shows that an instability can only occur for wavenumbers and frequencies corresponding to intersections of dispersion curves for two distinct modes of oscillation or deformation of the circular flow, i.e., when kl = k2 ~ 0 and ~ol = co2, with the latter perhaps both zero. Generically such a situation arises for a discrete set of axial wavenumbers. If the corresponding modes are coupled by distortion or precession, instability can result. For the elliptical distortion, this requires that the corresponding azimuthal wavenumbers i Research partially supported by NSF Contract DMS 89-19074 and CTS 89-06343. 2 Research partially supported by DOE Contract DE-FGO3-92ER25129.
PY - 1994/5/15
Y1 - 1994/5/15
N2 - We derive and analyze several low dimensional Hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics. The equations depend on two parameters (ε{lunate}, λ), where ε{lunate} is the strength of a system symmetry breaking perturbation and λ is a detuning parameter. In many cases the resulting equations are completely integrable and have an interesting Hamiltonian structure. Our work is motivated by three-dimensional instabilities of rotating columnar fluid flows with circular streamlines (such as the Burger vortex) subjected to precession, elliptical distortion or off-center displacement.
AB - We derive and analyze several low dimensional Hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics. The equations depend on two parameters (ε{lunate}, λ), where ε{lunate} is the strength of a system symmetry breaking perturbation and λ is a detuning parameter. In many cases the resulting equations are completely integrable and have an interesting Hamiltonian structure. Our work is motivated by three-dimensional instabilities of rotating columnar fluid flows with circular streamlines (such as the Burger vortex) subjected to precession, elliptical distortion or off-center displacement.
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U2 - 10.1016/0167-2789(94)90225-9
DO - 10.1016/0167-2789(94)90225-9
M3 - Article
AN - SCOPUS:0011581571
SN - 0167-2789
VL - 73
SP - 49
EP - 81
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
IS - 1-2
ER -