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and (2) the continuity equation evaluated at r = r 1 + L .
Hence, the only constraint to be imposed at the front for
the upper layer is the regularity of (u 1 , v 1 , h 1 + h 2 ) .
We also have to ensure the continuity of pressure of the
lower layer across the front. In the region r > r 1 + L with
no upper layer, the lower layer obeys the one-layer rotat-
ing shallow-water equations with (hydrostatic) pressure
proportional to the height of the fluid column. In what
follows we consider an outer cylinder to be far enough
from the front ( r 2 >> r 0 ) so that its influence is negligi-
ble. Moreover, below we will limit ourselves, for technical
simplicity, only by the balanced component of π 2 , which,
in the leading order, in polar coordinates, satisfies the
equation [cf., e.g., Reznik et al. , 2001]
f
2
α
H 1
ρ 1
ρ 2
H 2
γ
r
r 1
r 2
Figure 6.9. Schematic representation of a two-layer outcrop-
ping flow in the annulus with linearly sloping bottom.
r r (r∂ r π 2 ) + k 2
π 2 = 0,
k 2
r 2
1
1
R d 2
(6.14)
Here δ s = ρ 1 2 is the density ratio, s = 2
ρ 1 )/(ρ 2 +
ρ 1 ) is the stratification parameter, and Bu = (R d /r 0 ) 2 the
Burger number.
The depth profiles H j (r) and respective velocities V j (r)
in (6.2) correspond to steady cyclogeostrophically bal-
anced states in each layer:
where R d 2 = gH 2 /f is the Rossby deformation radius
of the lower layer. We thus impose the continuity of the
full solution for π 2 in the inner region, r < r 1 + L , with the
decaying balanced solution in the outer region, r > r 1 + L ,
at r = r 1 + L . By this choice an unbalanced part of the
one-layer flow beyond the front, consisting of freely prop-
agating surface inertia-gravity waves, is discarded. We thus
loose possible resonances of the eigenmodes of the inner
flow with the outer inertia-gravity wavefield and related
radiative instabilities. For small to moderate Rossby num-
bers, which is the case of existing experiments, and our
case below, these instabilities are weak [cf. Zeitlin , 2008].
As we will see later, the stability analysis under these
assumptions reproduces the experiments well, which gives
an a posteriori justification.
Injecting (6.11) into (6.2) and (6.9), we obtain an eigen-
value problem of order 6 that can be solved by applying
the spectral collocation method along the same lines as
in the previous section. In what follows, we will first con-
sider the simplest case of a bottom layer initially at rest
( U 2 = 0) and an upper flow with a constant rotation rate
U 1 = αr .
V j
r
+ r
4 = Bu
2 s r j s H 1 + H 2 ) .
V j +
(6.10)
We look for solutions harmonic in the azimuthal
direction:
(u j (r , θ) , v j (r , θ) , π j (r , θ))
= (
˜
u j (r) ,
˜
v j (r) ,
π j (r)) exp
˜
[
ik(θ
ct)
]
+ c.c. (6.11)
The boundary condition of no normal flow at the coast is
the same as in the previous case for both layers, u j (r 1 ) =0.
The boundary conditions at the front for the upper layer
are
H 1 (r) + h 1 (r , θ , t) =0, D t L R = v at r = L R (θ) ,
(6.12)
6.3.2. Resonances and Instabilities
where r = r 1 + L is the location of the free streamline of the
basic state, L R , t) is the position of the perturbed free
streamline, and D t = t + u∂ r + v/r∂ θ is the Lagrangian
derivative. Physically, they correspond to the conditions
that the fluid terminates at the boundary, which is a mate-
rial line. The linearized boundary conditions give (1) the
relation between the perturbation of the position of the
free streamline and the value of the height perturbation,
As in the configuration of Section 6.2, the instabili-
ties in the outcropping case originate from resonances
between the eigenmodes of the linearized problem. As
in the previous section, the wave species are Poincaré
(inertia-gravity) modes, Rossby modes (if PV gradients
are present), and unidirectional Kelvin modes trapped at
the boundary. Additional ingredients in the outcropping
configuration are the frontal modes trapped in the vicinity
of the free streamlines (outcropping lines). These modes
are described in Iga [1993] as mixed Rossby-gravity waves
r = r 1 + L
h 1
H 1 r
L R =
,
(6.13)
 
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