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),
 
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