201 research outputs found
Tropical Atmospheric Circulations: Dynamic Stability and Transitions
In this article, we present a mathematical theory of the Walker circulation
of the large-scale atmosphere over the tropics. This study leads to a new
metastable state oscillation theory for the El Nino Southern Oscillation
(ENSO), a typical inter-annual climate low frequency oscillation. The
mathematical analysis is based on 1) the dynamic transition theory, 2) the
geometric theory of incompressible flows, and 3) the scaling law for proper
effect of the turbulent friction terms, developed recently by the authors
Dynamic Transitions and Baroclinic Instability for 3D Continuously Stratified Boussinesq Flows
The main objective of this article is to study the nonlinear stability and
dynamic transitions of the basic (zonal) shear flows for the three-dimensional
continuously stratified rotating Boussinesq model. The model equations are
fundamental equations in geophysical fluid dynamics, and dynamics associated
with their basic zonal shear flows play a crucial role in understanding many
important geophysical fluid dynamical processes, such as the meridional
overturning oceanic circulation and the geophysical baroclinic instability. In
this paper, first we derive a threshold for the energy stability of the basic
shear flow, and obtain a criteria for nonlinear stability in terms of the
critical horizontal wavenumbers and the system parameters such as the Froude
number, the Rossby number, the Prandtl number and the strength of the shear
flow. Next we demonstrate that the system always undergoes a dynamic transition
from the basic shear flow to either a spatiotemporal oscillatory pattern or
circle of steady states, as the shear strength of the basic flow
crosses a critical threshold . Also we show that the dynamic
transition can be either continuous or catastrophic, and is dictated by the
sign of a transition parameter , fully characterizing the nonlinear
interactions of different modes. A systematic numerical method is carried out
to explore transition in different flow parameter regimes. We find that the
system admits only critical eigenmodes with horizontal wave indices .
Such modes, horizontally have the pattern consisting of -rolls aligned
with the x-axis. Furthermore, numerically we encountered continuous transitions
to multiple steady states, continuous and catastrophic transitions to
spatiotemporal oscillations.Comment: 20 pages, 7 figure
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