  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 $\Lambda$ of the basic flow
crosses a critical threshold $\Lambda_c$. Also we show that the dynamic
transition can be either continuous or catastrophic, and is dictated by the
sign of a transition parameter $A$, 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 $(0,m_y)$.
Such modes, horizontally have the pattern consisting of $m_y$-rolls aligned
with the x-axis. Furthermore, numerically we encountered continuous transitions
to multiple steady states, continuous and catastrophic transitions to
spatiotemporal oscillations.
