Geoscience Reference
In-Depth Information
8
Direct Numerical Simulations of Laboratory-Scale
Stratified Turbulence
Michael L. Waite
8.1. INTRODUCTION
to the atmospheric mesoscale (
O(
100
)
to
O(
1
)
km) and
oceanic sub-mesoscale (
O(
10
)
km to
O(
100
)
m). The cas-
cade of energy through this scale range forms a key
link in the global energy budget, as it connects large-
scale QG turbulence with microscale isotropic turbulence
and ultimately dissipation. In the case of the atmo-
sphere, various attempts have been made to explain the
observed mesoscale energy spectrum with a stratified tur-
bulence hypothesis, which posits that the essential non-
linear dynamics of the mesoscale regime are captured by
stratified turbulence [
Gage
, 1979;
Lilly
, 1983;
Lindborg
,
2006]. While its details have evolved, this hypothesis has
motivated much of the research on stratified turbulence
over the last 30 years. We will use primarily atmospheric
terminology in this chapter (e.g., “mesoscale”), but the
stratified turbulence hypothesis has also been advanced to
explain the sub-mesoscale cascade in the ocean [
Riley and
Lindborg
, 2008].
Since the early experiments of the 1960s and 1970s
[reviewed by, e.g.,
Lin and Pao
, 1979] and the pioneering
computations of
Riley et al.
[1981], laboratory experi-
ments and numerical simulations have played important
roles in investigating the dynamics of stratified turbulence.
But even today there are major differences between the
parameter regimes of the atmosphere and ocean and those
accessible numerically and in the laboratory. Stratified
turbulence is characterized by the Froude and Reynolds
numbers. Distinguishing between horizontal and verti-
cal scales (denoted by subscripts
h
and
v
), the Froude
numbers are
As already discussed in previous chapters, stable density
stratification is a fundamental property of atmospheric
and oceanic flows. Stratification creates buoyancy forces,
which inhibit vertical motions and can have a pro-
found effect on dynamics across a wide range of length
scales. This chapter is about turbulence in such fluids,
which is commonly called stratified turbulence. Strati-
fied turbulence differs from other types of turbulence,
such as isotropic three-dimensional, two-dimensional,
and quasi-geostrophic (QG) turbulence, in several impor-
tant ways. It is characterized by approximately horizontal
velocities, thin layers of intense vertical shear, quasi-
two-dimensional vortices, and interactions with internal
gravity waves (for reviews of stratified turbulence, see
Lin
and Pao
[1979],
Hopfinger
[1987], and
Riley and Lelong
[2000]). We will make a few common idealizations and
consider stratified turbulence that is statistically homoge-
neous, nonrotating, and not dominated by internal gravity
waves. For a recent discussion of rotating stratified turbu-
lence,see
Bartello
[2010], and forgravity waves,see
Staquet
and Sommeria
[2002].
Stratified turbulence has relevance for geophysical flows
because, over large regions of the atmosphere and ocean,
the Brunt-Väisälä frequency
N
is much larger than the
Coriolis parameter
f
; typical midlatitude values of
N/f
are
O(
100
)
. As a result, there is a range of length
scales in both fluids over which buoyancy forces are
thought to dominate the Coriolis effect. This range is
between larger scales where both rotation and strati-
fication are important and smaller scales where both
effects are negligible. These scales correspond roughly
U
NL
h
U
NL
v
Fr
h
≡
, F
v
≡
,
(8.1)
where
U
is the horizontal velocity scale and
L
h
and
L
v
are horizontal and vertical length scales. The Reynolds
number is given by
Department of Applied Mathematics, University of
Waterloo, Waterloo, Ontario, Canada.
