Geoscience Reference
In-Depth Information
L
β
=
U
rms
β
1
/
2
Re =
Ud/ν
. At Re above about 50 one would expect the
formation of the vortex street behind the cylinder. In our
experiment the Reynolds number is quite large, Re
,
(5.29)
600,
yet the vortex street does not form. Another relevant con-
trol parameter in this problem is again
R
β
=
βd
2
/U
, which
characterizes the relative importance of the
β
-effect on
the scale of the cylinder. Exploration of the parameter
space (Re,
R
β
)
reveals that the vortex street does form
in the eastward flow when the Reynolds number is suf-
ficiently high. In fact, the boundary between the Rossby
wave regime and the vortex street regime is described by
an empirical linear relation
≈
where
U
rms
is the root mean square (rms) velocity of the
flow. This scale was obtained by finding a length scale
at which a field of vortices (eddies) in two-dimensional
isotropic turbulence is synchronized with linear Rossby
waves on the
β
-plane. We equate the frequency of a vor-
tex (its rotation rate) to the frequency of the Rossby wave
obtained from the dispersion relation. The wavelength of
the Rossby wave is determined by the distance between the
vortices in the vortex arrangement. The original Rhines
scale was obtained for
β
-plane flow, and it can be mod-
ified for the polar
β
-plane as follows (see also
Slavin
and Afanasyev
[2012]). Consider vortices arranged in an
approximately checkerboard pattern such that the oppo-
sitely signed vortices are located next to each other. This
arrangement specifies the wave number
k
=
π/d
,where
d
is a typical diameter of a vortex. Taking the rotation
rate of a vortex to be
v
, we require the synchronization
between the vortex arrangement and the Rossby wave to
be such that
ω
=
v
,where
ω
is the frequency of the wave
obtained from the dispersion relation (5.26). We further
require a match between the wave number of the vortex
arrangement and that of the Rossby wave. The quan-
tity
α
mn
/R
in (5.26) represents the isotropic wave number
and is analogous to the wave number
k
=
(k
x
+
k
y
)
1
/
2
defined for a two-dimensional Fourier transform in Carte-
sian coordinates. We take
α
mn
/R
=
k
andalsotake
the azimuthal wave number
m
at “midlatitudes” of the
domain (at
R/
2) to be equal to
kR/
2. Then solving (5.26)
for
k
, we obtain
Re = 1100
R
β
+ 50.
(5.28)
The vortex street regime is then predicted to occur above
the line given by (5.28). Another surprising effect of this
flow is that the observed vortex street is different from its
classical counterpart: The order of vortices in the street is
opposite to that in the regular (classical) street.
In this flow anticyclonic vorticity is generated on the
northern side of the cylinder while cyclonic vorticity is
generated on its southern side. In a regular vortex street
the vortices are expected to maintain their relative posi-
tions. However, in this flow, immediately after being shed
from the cylinder, the vortices exchange their positions
such that the anticyclones move to the southern side while
the cyclones move to the northern side. As a result of this
“castling,” the vortex street resembles a so-called reverse
von Karman-Benard vortex street, which is normally
formed behind a self-propelled body (a body providing
a thrust force on the fluid). The reverse vortex street can
be clearly seen in Figures 5.6b and 5.6d, which show the
velocity and relative vorticity in the flow induced by a
cylinder moving westward (this is equivalent to eastward
mean flow). This anomalous vortex street can be com-
pared with the regular vortex street shown in Figures 5.6a
and 5.6c where the cylinder moves eastward (westward
mean flow).
Another interesting and geophysically relevant effect is
the interaction between vortex streets and Rossby waves.
In fact, each vortex moves fluid parcels in the north-south
direction changing their relative vorticity and thus cre-
ating Rossby waves. The generation of Rossby waves by
individual vortices and eddies has been widely discussed
in the literature [e.g.,
Korotaev
, 1997;
Reznik et al.
, 2001;
Flor and Eames
, 2002;
McWilliams
, 2006]. Multiple vor-
tices that are arranged in a certain organized pattern can
be highly synchronized with Rossby waves. As a result,
the dynamics can be something in between that of a vor-
tical flow and of a wave-dominated flow. The boundary
between these two regimes is provided by the Rhines scale
[
Rhines
, 1975], which can be written as
f
0
Ro
+
(γ R)
2
γ R
1
R
b
k
=
−
.
(5.30)
f
0
Ro
2
Here Ro = 2
v
/f
0
is the Rossby number of a vortex. Note
that
R
is the radius of the domain, which is the radius of
the tank for laboratory flows or the radius of a planet for
oceanic/atmospheric flows. The (modified) Rhines scale
can then be introduced as a wavelength of the vortex
arrangement/wave pattern
L
γ
=2
π/k
. When the radius of
deformation is relatively large,
R
b
> f
0
Ro
/(γ R)
,wecan
neglect the second term under the square root in (5.30),
which gives
L
γ
=
πf
0
Ro
γ R
.
(5.31)
Note that Eq. (5.31) is analogous to the classic form of the
Rhines scale (5.29) but written for the polar
β
-plane. Thus,
when the typical size of the vortices created either by the
flow around an obstacle or by some other process (a typ-
ical oceanographic example is the field of eddies created
by a baroclinic instability) reaches
L
γ
, the emission of
Rossby waves becomes very significant. For larger scales,
