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where v j = (u j , v j ) , h j ,and j are velocity (radial,
azimuthal), thickness, and pressure normalized by density
(geopotential), respectively, in the j th layer (counted from
the top), j =1,2; f is the Coriolis parameter, f =2 ;and
D j denote Lagrangian derivatives in respective layers. The
boundary conditions are u =0at r = r 1 , r 2 .
By introducing the time scale 1 /f , the horizontal scale
r 0 = r 2
f
ΔΩ
H 1
ρ 1
r 1 , the vertical scale H 0 , and the velocity scale
V 0 = fr 0 , we use nondimensional variables from now on
without changing notation. By linearizing about a steady
statewithconstantazimuthalvelocities V 1
ρ 2
H 2
r 2 r
= V 2 ,weobtain
the following nondimensional equations (the ageostrophic
version of the Phillips model in cylindrical geometry):
t u j + V j
r 1
Figure 6.1. Schematic representation of a two-layer flow in the
annulus with a superrotating lid.
2 V j v j
r
r θ u j
v j
=
r π j ,
t v j + u j r V j + V j
r θ v j + u j + V j u j
θ π j
r
=
, (6.2)
total depth 2 H 0 . The radial width of the annulus is there-
fore r 2
r
r 1 , and the two layers have equal depths H 0 at
rest. The base and the lid are both horizontal and flat.
The angular velocity about the axis of symmetry is ,
and the upper lid is superrotating at + . This dif-
ferential rotation provides a vertical velocity shear of the
balanced basic state that is close to solid-body rotation of
each fluid layer with different angular velocities. Such a
state will be represented in the stability analysis that fol-
lows by a cyclogeostrophic equilibrium in each layer, with
linear radial profile of the azimuthal velocity, within the
rotating two-layer shallow-water model. In order to ful-
fil a complete linear stability analysis, we use below the
collocation method.
Our analysis is purely inviscid; however, in the exper-
iment the mean axisymmetric flow is controlled by fric-
tion. As is well known [see, e.g., Hart , 1972], Ekman,
Stewartson, and shear boundary layers are present in the
two-layer rotating fluid in the tank, and the related torques
are acting upon the quasi-inviscid interior. Moreover, the
interfacial layer has an internal structure depending on
whether two fluids are immiscible or not (see Chapter 11).
All this internal structure will be neglected in what follows,
and the layers will be considered to be in solid rotation.
r rH j u j r + 1
t h j + 1
r H j θ v j + V j
r θ h j =0,
where the pressure perturbations in the layers, π j ,are
related through the interface perturbation η as usual,
π 2
π 1 + s(π 2 + π 1 ) =Bu η ,
(6.3)
and s = 2
ρ 1 )/(ρ 2 + ρ 1 ) is the stratification parameter,
Bu = (R d /r 0 ) 2 is the Burger number, Ro = /( 2 ) is
the Rossby number (as used in experiments, cf. Flór
et al. [2011]), R d = (g H 0 ) 2 /( 2 ) is the Rossby deforma-
tion radius, and g =2 ρg/(ρ 1 + ρ 2 ) =2 sg is the reduced
gravity.
The depth profiles H j (r) and respective velocities V j (r)
in (6.2) correspond to a steady cyclogeostrophically bal-
anced state of the two-layer system that obeys the nondi-
mensional equations.
V j + V j
r
+ r
4 = r j ,
(6.4)
where the r/ 4 term corresponds to the centrifugal effect
at the interface, while the other terms correspond to the
classical cyclogeostrophic equilibrium.
The rotation rates of the layers lie in the interval
between the rotation rate of the base (0 in the rotating
frame) and that of the upper lid (Ro in the rotating frame).
Therefore, in general,
6.2.1. Equations of Motion, Basic States, and Linear
Stability Problem
Consider the two-layer rotating shallow-water model on
the plane rotating with constant angular velocity .The
momentum and continuity equations are written in polar
coordinates as
V 2 = α 2 r , V 1 = α 1 r
(6.5)
and we get the following expressions for the heights of the
layers in the state of cyclogeostrophic equilibrium for such
mean flow:
r v j
f + v j
r 2 =
D j u j
r j ,
r 2
2Bu
1 ) j
α 2 + α 2
α 1 ]
D j v j + f + v j
r u i =
H j = H j ( 0 ) + (
[
α 1
θ j
r
,
(6.1)
r 2
2Bu .
1 ) j s
α 2 + α 2 + α 1 + α 1 +1 / 2
+ (
[
]
(6.6)
D j h j + h j ∇·
v j =0,
 
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