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whose theoretical development and efficient numerical
implementation were discussed thoroughly by
Wedi and
Smolarkiewicz
[2004]. The transformation (9.3) allows for
time-dependent upper,
H(x
,
y
,
t)
, and lower,
z
s
(x
,
y
,
t)
,
boundary forcing without small-amplitude approxima-
tions.
The governing equations (9.2) are discretized in
the transformed space using a second-order-accurate,
optionally semi-Lagrangian or Eulerian, nonoscillatory
forward-in-time (NFT) approach, broadly documented
in the literature (cf.
Smolarkiewicz and Prusa
[2002] and
Smolarkiewicz
[2006] for a recent review). Here we employ
the flux form Eulerian, semi-implicit version of the algo-
rithm unless stated otherwise. All prognostic equations
in (9.2) are integrated using the trapezoidal rule, treat-
ing all inviscid forcing on the right-hand side implicitly.
The viscous and diffusive terms are computed explic-
itly to first-order accuracy, (see Section 3.5.4 in
Smo-
larkiewicz and Margolin
[1998]). Together with the curvi-
linearity of the coordinates, this leads to a complicated
elliptic problem for pressure (see Appendix A in
Prusaand
Smolarkiewicz
[2003] for the complete description) solved
iteratively using a preconditioned nonsymmetric Krylov
subspace method [
Smolarkiewicz and Margolin
, 2000].
The bottom boundary in the latter simulations was
assumed rigid no slip. The length of the oscillation period
has been found to be moderately sensitive to the choice of
the bottom boundary condition [
Wediand Smolarkiewicz
,
2006].
For the initial condition a static state was assumed with
buoyancy frequency
N
=1.57s
−
1
. The integration time
was several hours with a time step
dt
= 0.05s. QBO-like
oscillations are a canonical example of long-time behav-
ior resulting from short-term fluctuations. Therefore, one
may expect the initial conditions to be of minor relevance
to the long-term solution. Nevertheless, the choice of ini-
tial conditions can be important for practical reasons. The
zonal mean zonal flow oscillation was seeded with a zonal
background flow
u
e
in the near-membrane layers using
u
e
=
u
0
{
with
u
0
=0.02m
/
s,
δ
= 0.9999999,
d
0
= 0.06 m, and
γ
=
z
.However,
it was found later that this is not necessary, as all that
is needed to start an oscillation is the addition of
zon-
ally asymmetric
noise. If this is not done, the simulation
may not develop an oscillation for many hours of simula-
tion time, after which finally numerical truncation errors
have sufficiently accumulated to provide the same effect
(see also Section 3b in
Wedi and Smolarkiewicz
[2006]).
Figure 9.3 illustrates the overall agreement of the
solution structure between the laboratory experiment
(adapted from Figure 10 of
Plumb and McEwan
[1978])
and the numerical simulation of the laboratory analogue.
The detailed chronology of the zonal mean zonal flow
reversal in the DNS of the laboratory analogue is sketched
in Figure 9.4. It describes the turbulent breakdown of
gravity wave trains with viscous dissipation and critical
layer absorption chronologically secondary to instabili-
ties from nonlinear flow interactions. The latter motivated
FetecauandMuraki
[2011] to refine the modulation theory
of the time evolution of slowly varying wave trains in a
density-stratified fluid with coupling to the mean flow (as,
e.g., applied in
Plumb
[1977]). The authors derived high-
order corrections to the modulation theory and found
self-regularizing properties of these corrections, whereby
the growth of unstable modes is essentially tempered by
nonlinearity. Taking into account these corrections result-
ing from nonlinearity, they found excellent agreement
between solutions of the corrected modulation system
and corresponding variables extracted from the numeri-
cal solutions of the nonlinear Boussinesq equations. The
corrected modulation system thus provides a mathemati-
cal pathway commensurate with the original explanation
of the laboratory experiment and the DNS results. In
agreement with the original analysis of the laboratory
experiment, the most dominant sensitivity on the length
of the zonal mean zonal flow oscillation period is by the
forcing wave number and thus the shape of the oscillating
membrane [
Wedi and Smolarkiewicz
, 2006].
1
−
0.5
δ
[
1+tanh
(z
−
d
0
)/γ
]}
9.2.2.2. DNS Simulation.
In the numerical simulation
an initially stagnant fluid is forced by an oscillating lower
boundary. The cylindrical laboratory tank is represented
by a zonally periodic, rectangular computational domain
of 639
×
38
×
188 grid intervals with
L
x
=2
π(a
+
b)/
2,
L
y
=
b
a
,
L
z
= 0.43 m. The impermeable lower and lateral
boundaries are no slip. At the upper boundary a free-slip
rigid lid is assumed. The lower surface is prescribed as a
linear combination of elementary shape functions
−
η
ξ
η
(x)
sin
π
L
y
y
sin
(ω
0
t
+
φ
η
)
, (9.4)
z
s
(x
,
y
,
t)
=
N
+1
η
=0
with individual amplitudes
η
and zonal profiles
ξ
η
=
cos
2
(πr
η
/
2
L
η
)
if
r
η
/L
η
≤
1,
(9.5)
0
otherwise,
where
L
η
=
L
x
/
0.5
)L
η
.Ingen-
eral, equations (9.4) and (9.5) describe a subdivision of the
periodic domain into
N
and
r
η
=
x
+
(η
−
N
independently oscillating cham-
bers. For
η
,and
φ
η
=
π
for even
η
(
φ
η
=0forodd
η
), the equation collapses into the sym-
metric solution of the 2D wave equation employed in the
original Plumb and McEwan paper. In the original exper-
iment,
H
=
H
0
. In the case of the Kyoto experiment, the
equivalent numerical simulations are obtained by setting
z
s
= 0 in (9.3) and using
H(x
,
y
,
t)
=
H
0
−
N
= 16,
η
= 0.008 m
∀
z
s
(x
,
y
,
t)
with
z
s
from (9.4).
