Flow Stability. (In collaboration with María J. Olascoaga.)
The Lie-Poisson Hamiltonian formulation of the fluid equations in their
Eulerian form provides a unifying framework wherein symmetry properties are
readily apparent and can be connected to conservation laws through Noether's
theorem. The existence of these conservation laws can then be used to derive
stability theorems using Arnold's method. The existence of a nonlinear
(Lyapunov) stable state, in turn, allows for the establishment of nonlinear
saturation bounds on the growth of perturbations to an unstable basic state in
terms of the “distance” to the stable state. A common feature of these topics
is that they combine challenging mathematical problems with direct relevance to
practical questions of geophysical fluid dynamics interest. For instance,
nonlinear saturation bounds are potentially useful in the architecture of
closures for eddy parameterizations in circulation models of terrestrial oceans
and atmosphere, and planetary atmospheres.
Lagrangian Dynamics. (In collaboration with María J. Olascoaga, Michael G. Brown, Huseyin Kocak, Irina I. Rypina and
Ilya A. Udovydchenkov.) We have been studying Lagrangian dynamics of
atmospheric zonal jets in connection to the impermeability of the stratospheric
polar vortex. We have argued that a twistless KAM torus (invariant material
closed curve for which the standard nondegeneracy condition used in KAM theory
is violated) accounts for the sharp boundary of the Antarctic ozone hole at the
perimeter of the stratospheric polar vortex in the austral spring. We have
demonstrated that the remarkable stability of twistless tori under perturbation
is linked to very small resonance widths near these tori, which we have termed strong
KAM stability.
Transport barriers in geophysical
flows have been traditionally associated with sharp gradients in the
distribution of the ambient potential vorticity (PV). Theoretical and numerical
work has demonstrated that quasigeostrophic beta plane (or hemispherical)
turbulence tends to evolve to a
strongly anisotropic state dominated by alternating narrow eastward and broad
westward zonal jets with almost piecewise constant PV between adjacent eastward
jets. The atmosphere of Jupiter provides a vivid example of a mean flow pattern
with these qualitative features. Observations of Jupiter strongly suggest that
both eastward and westward jets act as robust meridional transport barriers,
which cannot be explained using the traditional PV-gradient criterion.
Contrarily, our strong KAM stability mechanism successfully predicts that both
eastward and westward zonal jets should act as robust meridional transport
barriers. Numerical simulations of an
idealized model of Jupiter's midlatitude circulation at the cloud top level
show that both eastward and westward zonal jets act as robust meridional
transport barriers, consistent with the strong KAM stability explanation of the
transport barrier mechanism.
We have been also studying
Lagrangian dynamics on the West Florida Shelf in connection to harmful algal
blooms and the effects of Lagrangian mixed phase space dynamics and population
dynamics in plankton patchiness generation.
We are currently completing a
review article on the dynamics associated with nonautonomous quasiperiodic
perturbations to integrable autonomous Hamiltonians. This includes a proof of
the admission of invariant tori in the case that the unperturbed Hamiltonian
does not satisfy the standard Kolmogorov nondegeneracy (or twist) condition,
but rather the weaker condition due to Russmann. The issues of nonautonomous
quasiperiodic perturbations and degeneracy are central in all our Lagrangian
dynamics work.
Ocean Dynamics and Thermodynamics. I have been working also on
the generalization of Ripa’s inhomogeneous-density single-layer
primitive-equation model to include an arbitrary number of layers. Ripa’s model
represents a very important improvement over bulk mixed layer models. Namely,
those that consider velocity and buoyancy fields as vertically uniform or
slablike, or that have an implicit representation of the vertical velocity
shear through the thermal wind balance with the lateral buoyancy gradient. In
addition to allowing arbitrary velocity and buoyancy variations in horizontal
position and time, in Ripa’s model these fields are also allowed explicitly to
vary linearly with depth. As a consequence, Ripa’s model enjoys a number of
properties which make it to behave much better than bulk mixed layer models. In
collaboration with María
J. Olascoaga and Javier Zavala-Garay we showed that substantially more
accurate overall results in free boundary QG baroclinic instability than those
produced by the single layer version of Ripa’s model can be attained by
considering only a few more layers. Furthermore, preliminary results for free
boundary ageostrophic baroclinic instability look very promising.
I also study long-range sound propagation in the ocean using ray methods, mainly in collaboration with Michael G. Brown. Hamiltonian ray methods, in particular, provide insight into the physics underlying wave propagation through an ocean filled with internal waves that is difficult (if not impossible) to attain by any other means. For instance, these methods have been found very useful in showing that the background sound speed structure largely controls both ray and travel time stability. A challenging goal is to develop a theory of wave propagation in random media that includes the important role played by the background sound speed structure.