Geoscience Reference
In-Depth Information
Table 8.1.
Parameters, nondimensional numbers, and length scales for each simulation. In the run labels, A, B, C denote runs with
approximately equal Re
b
; 1, 2, 3, etc., denote runs with equal
ν
and approximately equal Re. Dimensional quantities are given
in cgs units.
Run
N
ν
n
x
t
U
L
h
Fr
h
Re
Re
b
L
b
L
o
L
d
A1
0.1
1
128
7.4
0.25
1.6
150
0.11
240
2.7
100
32
16
A2
0.141
0.5
256
3.7
0.125
1.6
170
0.069
530
2.5
71
19
9.4
A3
0.2
0.25
512
1.8
0.0625
1.6
190
0.043
1200
2.3
51
11
5.7
A4
0.283
0.125
512
1.8
0.05
1.6
180
0.032
2400
2.4
37
6.5
3.4
A5
0.4
0.0625
1024
0.92
0.025
1.6
170
0.023
4300
2.3
25
3.8
2.0
A6
0.566
0.03125
1536
0.61
0.0167
1.7
180
0.017
9400
2.6
18
2.4
1.2
B1
0.2
1
128
7.4
0.25
1.6
170
0.046
270
0.58
50
11
16
B2
0.283
0.5
256
3.7
0.125
1.7
180
0.032
620
0.65
38
6.8
9.3
B3
0.4
0.25
512
1.8
0.05
1.7
190
0.022
1300
0.59
26
3.8
5.7
B4
0.566
0.125
512
1.8
0.05
1.7
190
0.016
2500
0.63
19
2.3
3.3
B5
0.8
0.0625
1024
0.92
0.02
1.7
180
0.011
4900
0.63
13
1.4
2.0
B6
1.13
0.03125
1536
0.61
0.0133
1.7
190
0.0081
10000
0.69
9.6
0.87
1.2
C1
0.4
1
128
7.4
0.2
1.7
210
0.020
370
0.15
27
3.9
16
C2
0.566
0.5
256
3.7
0.1
1.8
220
0.014
790
0.16
20
2.4
9.4
C3
0.8
0.25
512
1.8
0.04
1.8
230
0.0097
1600
0.15
14
1.4
5.6
C4
1.13
0.125
512
1.8
0.04
1.8
230
0.0069
3400
0.16
10
0.85
3.3
C5
1.6
0.0625
1024
0.92
0.015
2.0
270
0.0045
8600
0.17
7.7
0.51
1.9
C6
2.26
0.03125
1536
0.61
0.01
1.9
250
0.0034
15000
0.18
5.3
0.31
1.1
numbers. Simulations are designed to have roughly equal
velocity and length scales, so different Fr
h
,Re,andRe
b
are
obtained by varying
N
and
ν
. Three sets of six numerical
experiments are considered, corresponding to approxi-
mately equal buoyancy Reynolds numbers of
Accounting for this truncation, the effectivegrid spacing is
3
2
L
n
.
≡
≡
≡
x
y
z
(8.24)
The grid spacing for a given
ν
is chosen to resolve the Kol-
mogorov scale with
L
d
/x
2, 0.6,
and 0.2 (labeled A, B, and C, respectively). In each set
of simulations,
ν
is
v
aried by factors of
2
and
N
is var-
≈
2. The time step
t
was
selected to be roughly as large as possible while ensuring
stability.
Forcingwasappliedtoobtainstatisticallystationarytur-
bulence and avoid the rapid decay of unforced simulations
[as in, e.g.,
Riley and deBruynKops
, 2003]. We have mod-
ified the deterministic forcing approach of
Sullivan et al.
[1994] to excite vortical motion at large horizontal scales
(a similar approach was employed by
Bartello
[2000] to
force barotropic vortical motion). The forcing maintains
a fixed value
E
0
of horizontally rotational kinetic energy
in a cylindrical wave number shell with radius
k
f
,givenby
≈
ied by factors of
√
2 to obtain a spread of Re and Fr
h
with the same Re
b
. The overall ranges of Fr
h
and Re are
0.003-0.1 and 200-10,000, respectively, which are realiz-
able values in laboratory experiments. Parameters for all
runs are listed in Table 8.1.
The computational domain size is
L
×
×
(L/
4
)
, with
periodic boundary conditions on
u
and
b
in each direction.
The numerical model employs a Fourier-transform-based
spectral method with third-order Adams-Bashforth time
stepping, and viscous terms are treated implicitly with
a trapezoidal approach. The Fourier discretization uses
n
L
k
x
+
k
y
.
k
2
, w e
k
h
≡
|
k
h
−
k
f
|≤
(8.25)
(n/
4
)
wave numbers, yielding a wave number spac-
ing of
k
×
n
×
2
π/L
in each direction. Aliasing errors are
eliminated by the standard two-thirds rule [e.g.,
Durran
,
2010], which is implemented by truncating wave vectors
k
=
(k
x
,
k
y
,
k
z
)
outside the sphere,
≡
Over every time step, the energy in this shell naturally
evolves away from its initial value. The forcing is applied
by uniformly scaling the rotational modes in the shell
at the end of every time step to return their kinetic
energy to
E
0
. Rotational forcing is used to avoid exciting
large-amplitude internal gravity waves at the forcing scale
2
3
nπ
L
.
|
k
|≤
(8.23)
