Geoscience Reference
In-Depth Information
(a)
(b)
0.040
0.040
0.030
0.030
0.020
0.020
0.11
0.069
0.043
0.032
0.023
0.017
0.046
0.032
0.022
0.016
0.011
0.0081
0.010
0.010
0.000
0.000
0
300
600
900
1200
1500
1800
0
300
600
900
1200
1500
1800
t
(s)
t
(s)
(c)
0.040
0.030
0.020
0.020
0.014
0.0097
0.0069
0.0045
0.0034
0.010
0.000
0
300
600
900
1200
1500
1800
t
(s)
Figure 8.1.
Time series of kinetic energy dissipation in simulations with Re
b
≈
(a) 2, (b) 0.6, and (c) 0.2 (simulation sets A, B, and
C). Dash patterns denote different Fr
h
.
(forcing total kinetic energy was found to produce wave-
dominated flows for certain parameter values). Relatively
large horizontal scales are forced by setting
k
f
=3
k
.
The dimensional domain size
L
is set to 2
π
meters for
simplicity, so the wave number spacing is
k
=1m
−
1
.
The forcing level
E
0
is chosen to produce velocities com-
parable to those of laboratory experiments. We let
E
0
=1
cm
2
/s
2
, which yields rms velocities of
U
transfers, etc.) are averaged over the last 800 s of each sim-
ulation. Values of Fr
h
,Re,andRe
b
for each simulation,
alongwithvelocityandlengthscales,aregiveninTable8.1.
These values are computed using the rms velocity
U
,the
time-averaged dissipation
,and
L
h
from Taylor's relation.
8.3.2. Energy Spectra
≈
1-2 cm/s
0.02-0.03 cm
2
/s
3
.Charac-
teristic horizontal scales calculated from Taylor's relation
L
h
≡
and dissipation rates of
≈
Horizontal wave number spectra of kinetic energy
E
K
(k
h
)
are computed by integrating the energy in each
wave vector
k
over cylindrical wave number shells of width
k
[as in, e.g.,
Waite
, 2011]. In other words,
E
K
(k
h
)k
gives the total kinetic energy in modes with horizontal
wave numbers within
k/
2of
k
h
.Otherspectra,suchas
the potential energy
E
P
(k
h
)
and the transfer and buoyancy
flux spectra discussed below, are computed similarly.
Over the range of parameters considered here, the
kinetic energy spectra are extremely sensitive to Fr
h
and
Re. Figure 8.2a shows
E
K
(k
h
)
from three simulations
with the same stratification Fr
h
U
3
/
are around 2 m, yielding turnover time scales
L
h
/U
of
O(
100
)
s. The Kolmogorov scales of the different
simulations range between 1 and 16 cm.
Lower-resolutionsimulationswith
n
≤
512areinitialized
with low-level noise and run for 2000 s. Higher-resolution
simulations with
n
1024 are first spun up to
t
= 1000
s with
n
= 512 and correspondingly lower Re and then
continued to
t
= 2000 s at full resolution. Time series of
(Figure8.1)showthatthesimulationsstartedfromresttake
several hundred seconds to spin up, while those restarted
at
t
= 1000 s adjust much more quickly. The turbulence
appears reasonably stationary by
t
= 1200 s in all cases.
The various quantities presented below (energy spectra,
≥
0.02 and differ-
ent Re. All the spectra are peaked at
k
h
/k
=3, where
kinetic energy is injected by forcing. For the largest
Reynolds number (which has Re = 9400 and Re
b
=2.6),
≈