Geoscience Reference
In-Depth Information
2
Primary Flow Transitions in the Baroclinic Annulus:
Prandtl Number Effects
Gregory M. Lewis, Nicolas Périnet, and Lennaert van Veen
2.1. INTRODUCTION
Früh and Read , 1997] and geostrophic turbulence [ Read ,
2001].
The Taylor number
Thebaroclinicannulusexperiments,alsocalleddifferen-
tially heated rotating fluid annulus experiments, have been
used extensively to study baroclinic instability, which is
an important mechanism in the process of cyclogenesis in
the midlatitudes [ Holton , 2004]. The experiments consist
of observing the flow of a differentially heated fluid that
is contained between two coaxial cylinders that are both
rotated at rate . The differential heating can be generated
by holding the inner cylinder at a different temperature,
usually colder, than the outer cylinder, but other mecha-
nisms have been used, for example, internal heating. See,
e.g., Read [2001] for a review. The differential heating and
rotation provide the necessary conditions to observe baro-
clinic instability; in fact, it has been shown that over a
wide range of parameters, it is the primary mechanism of
instability[see,e.g., Williams ,1971; Hide andMason ,1975].
For small values of the forcing (rate of rotation and dif-
ferential heating), a steady axisymmetric flow is observed.
However, as the parameters are varied, the steady flow
becomes unstable to baroclinic perturbations, and a
steadily rotating wave equilibrates. These waves are often
referred to as steady waves [ Hide and Mason , 1975] or
baroclinic waves because they result as a consequence of
baroclinic instability [ Williams , 1971]. A wide variety of
other complex wave flows may also be observed in differ-
ent regions of the parameter space, for example, modu-
lated waves (e.g., amplitude vacillation [ Hignett , 1985]),
mixed azimuthal mode flows (also called wave dispersion
[ Pfeffer and Fowlis , 1968] or interference vacillation [ Früh
and Read , 1997]), as well as more complicated flows such
as modulated amplitude vacillation [ Read et al. , 1992;
T
and the thermal Rossby num-
ber
are generally regarded as the two most important
dimensionless parameters that determine the nature of the
observed flow [ Hide and Mason , 1975], where the Taylor
number is given by
R
= 4 2 R 4
ν 2
T
,
(2.1)
where is the rate of rotation, R = r b
r a is the differ-
ence between the outer radius r b and inner radius r a of the
annulus, and ν is the kinematic viscosity of the fluid, and
the thermal Rossby number is given by
= αgDT
2 R 2
R
,
(2.2)
where T is the imposed horizontal temperature gradient,
α is the coefficient of thermal expansion, D is the depth of
the fluid, and g is the gravitational acceleration. Indeed, if
all other parameters are held fixed, the thermal Rossby
number and Taylor number together have a one-to-one
relationship with the rotation rate and differential heat-
ing T , which are the physical parameters that are varied
in a particular experiment. Thus, experimental results are
usually presented as a regime diagram on a log-log graph
of Taylor number versus thermal Rossby number. How-
ever, the dynamics of the fluid in fact depend on no less
than four other dimensionless parameters. One possible
set of dimensionless parameters consists of the thermal
Rossby number, the Taylor number, the vertical aspect
ratio δ = D/R , the horizontal aspect ratio η = r a /R ,a
rotational Froude number F = 2 R/g , and the Prandtl
number Pr = ν/κ ,where κ is the thermal diffusivity. This
set is not unique, and other complete sets with different
dimensionless parameters can be constructed.
Institute of Technology, University of Ontario, Oshawa,
Ontario, Canada.
 
 
Search WWH ::




Custom Search