THE EFFECTS OF DOUBLE-DIFFUSION ON
A BAROCLINIC VORTEX
by
Wendy Marie Smith
B. S., Rensselaer Polytechnic Institute
(1983)
SUBMITTED TO THE DEPARTMENT OF
EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
IN PHYSICAL OCEANOGRAPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 1987
@Massachusetts Institute of Technology 1987
Signature of Author
I I
/I
MIT/WHOI Joint Program in Oceanography & Oceanographic Engineering
January 15, 1986
Certified by
Raymond W. SAmitt Jr.
Thesis Supervisor
Physical Oceanography Department
Woods Holy Oceanographic Institution
Accepted by
William F. Brace
Chairman, Committee on Departmental Graduate Students
~4.
THE EFFECTS OF DOUBLE-DIFFUSION ON A BAROCLINIC VORTEX
by
WENDY MARIE SMITH
Submitted to the Department of Earth, Atmospheric, and Planetary Sciences
on January 16, 1986
in partial fulfillment of the requirements for the degree of
Master of Science in Physical Oceanography
Abstract
Laboratory experiments were performed to study the combined effects of double-diffusion
and rotation on an oceanic intrusion. Intrusions are driven across density-compensated
fronts by the divergence of the double-diffusive buoyancy flux. The increased momentum
transport across a double-diffusive interface, however, acts to oppose the action of the
buoyancy flux. Turbulent double-diffusive Ekman layers could be a means of redistributing
momentum.
A model of an intrusion was made by injecting salt or sugar solution at the surface of a
denser layer of sugar or salt solution in a rotating tank to form a baroclinic vortex. The size
and shape of the vortex and the velocity structure of the intrusion were measured as functions of time. The double-diffusive vortex spread more quickly and had slower azimuthal
velocities than a non-double-diffusive one. This effect increased as the density ratio approached unity. These results indicate that momentum transport across a double-diffusive
interface is larger than that across a non-double-diffusive one; thus, the parameterization
of friction in an intrusion model should be considered carefully.
Thesis Supervisor: Dr. Raymond W. Schmitt Jr.
Title: Associate Scientist, Woods Hole Oceanographic Institution
Acknowledgements
I am very grateful to Dan Kelley and Eric Kunze, without whose help this thesis never
could have been written. They assisted during experiments, answered stupid questions,
and patiently read and reread earlier drafts. Throughout this work, they have taught and
encouraged me. Thanks, eh?
I would like to thank my advisor, Ray Schmitt, and the other members of my committee,
Jack Whitehead, John Toole, and Glenn Flierl, for giving me both answers and advice. With
little prior laboratory experience, I was greatly helped by suggestions from Jack Whitehead
and Karl Helfrich. Technical assistance was provided by Bob Frazel.
I have enjoyed working with the other students in the Joint Program, whom I like and
respect immensely. Special thanks go to those who lent a hand during experiments-Esther
Brady, Steve Pierce, Tammy Wood, and Barry Klinger.
I am grateful to the people who got me started-Eric Mandel and Rick Harnden of
the Harvard-Smithsonian Center for Astrophysics, and RPI professors Peter Watt and Sam
Katz. I'd also like to thank the ones who helped me get finished-Steve Pierce and Warren
Sass (both of whom keep rather odd houra).
My family and friends have given me the support and encouragement necessary to persevere. In particular, my parents have set me an example. I thank Lucia Susani, for putting
up with me, and Matt Leo, for all his love and support.
This work was supported by NSF Grant # OCE84-09323.
Contents
1
Introduction
2 Intrusions
3 Previous experimental investigations
4 Mechanisms for momentum transport
4.1
Transport of momentum by double-diffusion ....
4.2
The interfacial Ekman layer ................
5 Dynamics of the baroclinic vortex
5.1
Inviscid solution .............
5.2
Double-diffusive buoyancy fluxes
5.3
5.4
15
. . . . . . . . . . . .
15
. . . . . . . . . . . .
17
Viscous effects ..............
. . . . . . . . . . . .
22
Instabilities of the baroclinic vortex.
. . . . . . . . . . . .
24
. .
6 Experiments
6.1
Experimental design .........
. . . . . . . . . . . . . . . . . . . . . .
25
6.2
Choice of parameters ..........
. . . . . . . . . . . . . . . . . . . . . .
27
6.3
Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
6.4
Observations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
6.4.1
Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
6.4.2
General character of interface . . . . . . . . . . . . . . . . . . . . . .
30
6.4.3
Quantitative measurements . . . . . . . . . . . . . . . . . . . . . . .
32
Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
6.5.1
Interface shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
6.5.2
Azimuthal velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
6.5
7 Discussion
8
25
Conclusions
1
Introduction
Double-diffusion is an important oceanic mixing process. One place it is likely to occur
is at the boundary between two water masses, where interleaving layers are often seen.
Interleaving sharpens the gradients of temperature and salinity and increases the surface
area available to smaller scale mixing processes, such as double-diffusion (Joyce, 1977).
The formation of interleaving intrusions therefore enhances the mixing of water masses.
Rotation, buoyancy flux and friction all influence the growth of an intrusion.
The effective diffusivities of heat and salt across a double-diffusive interface are much
greater than the molecular diffusivities. In addition, double-diffusion increases the effective
viscosity (Maxworthy, 1983; Ruddick, 1985). I have studied the buoyancy and momentum
flux in a rotating double-diffusive intrusion by performing a series of laboratory experiments.
Salt and sugar solutions were used instead of cold, fresh and warm, salty water, respectively.
This system is analogous to the heat-salt system because of the unequal diffusivities of salt
and sugar. The basic experiment is the release of salt water in the center of a rotating
tank filled with more dense sugar water to make a diffusive interface, or sugar solution into
salt to make a fingering one. Double-diffusion has a dramatic effect on the evolution of the
resulting vortex. The effect is greatest when the density ratio is close to one, i.e. the salt and
sugar contribute nearly equally to the density difference. The vortex "spins down" much
more quickly in the double-diffusive case, indicating that the effective viscosity is greater.
Direct measurements of stress are difficult, but the magnitude of the momentum flux
can be inferred from the the time evolution of the interface depth and radial velocity.
Momentum transport in a rotating system is modified by the presence of Ekman layers.
The Ekman layer at a double-diffusive interface will be turbulent and therefore thicker; the
corresponding increase in transport speeds the decay of the vortex.
Oceanic intrusions are discussed in Section 2. Many experiments have been performed to
study intrusions and double-diffusion; the most relevant ones are summarized in Section 3.
The transfer of momentum across an interface is considered in Section 4, then the concepts
of the preceding two sections are applied to the baroclinic vortex (Section 5). Section 6
contains the description of the experiments on the double-diffusive vortex. The data are
presented in Section 6, and the results applied to the problem of estimating momentum
transport in Section 7.
2
Intrusions
The mixing of large water masses in the ocean is accomplished on very small scales. Various
mechanisms exist for effecting this mixing; one particularly interesting one is the combination of interleaving intrusions and double-diffusion. Double-diffusion is a small-scale mixing
process which draws on the potential energy of an unstable salinity or temperature gradient which is compensated by the other component so that the overall density gradient is
stable. The conversion of potential to kinetic energy (which is then dissipated by molecular
processes) is possible because of the different diffusivities of heat and salt.
Double-diffusive instability has two manifestations: "salt fingering", when warm salty
water overlies (more dense) cold fresh water, and "diffusive" instability, when cold water
overlies salty water. Some of the places salt fingering has been observed are under the
Mediterranean outflow (Williams, 1974; Magnell, 1976), in the Tyrrhenian Sea (Williams,
1975), and in the North Pacific Central Water (Gargett and Schmitt, 1982).
Diffusive
instability is possible, for example, in polar regions. It has been observed in the ocean by
Neshyba, Neal and Denner (1971), Horne (1978), Middleton and Foster (1980), and Perkin
and Lewis, (1984), among others. Both salt fingering and diffusive convection occur in
thermohaline intrusions.
Salt fingers are formed when a density gradient which is stable because of the temperature gradient is destabilized by the salt gradient. If a parcel of water is displaced
vertically, it will lose or gain heat faster than salt, and the driving force on it will increase.
The convection driven by the resulting buoyancy differences takes the form of finger-like
columns. Diffusive convection, in contrast to the "direct" salt finger convection, is characterized by an "overstable" oscillation. In this case, the growing restoring force takes energy
from the destabilizing temperature gradient. The condition for vigorous double-diffusion
is that the density ratio R, = (aT,)/(#S.) be close to unity, where a = -p-1(8p/8T),,s,
P = p- 1 (aP/aS),T, and T, and S, are the vertical temperature and salinity gradients
(Schmitt, 1981).
Both kinds of double-diffusion cause upgradient buoyancy flux, which results in layer
formation. Overturning layers are separated by relatively thin interfaces; several layers form
a "thermohaline staircase". This step-like vertical structure in temperature and salinity is an
indication of double-diffusion in the ocean. Because the double-diffusive vertical transport
of the un8table component is greater, the net density flux is upgradient. This buoyancy
flux can cause small-scale convective overturning on either side of the interface.
While
molecular diffusion acts to smooth vertical gradients in temperature and salinity, doublediffusion tends to make them steppier. It reduces the length scale of the gradients so that
molecular diffusion can act more efficiently.
Horizontal temperature and salinity gradients in the ocean are formed by large-scale
differences in surface buoyancy flux. The boundary between two water masses is often
sharpened by oceanic flow to form a front, i.e. a region of large horizontal property gradients.
Observations (Stommel and Fedorov, 1967; Joyce, Zenk, and Toole, 1978; Schmitt and
Georgi, 1982) show that on intermediate scales (1-50 km horizontally, 5-50 m vertically)
such large scale fronts are distorted by a series of interleaving intrusions. These intrusions
increase the surface area over which double-diffusive mixing occurs, and increase the effective
lateral diffusivity by exchanging heat and salt (Joyce, 1977; see Figure 1). The intrusions
therefore increase the rate at which property gradients are reduced to small scales and
cross-frontal mixing is accomplished.
Often frontal boundaries are density compensated; that is, there are horizontal gradients
in salt and heat but little or none in density. Stern (1967), Ruddick and Turner (1979),
Toole and Georgi (1981), and McDougall (1985a,b) have discussed the mechanism by which
these intrusions might form at a double-diffusively unstable front. The vertical divergence of
Figure 1: Figure from Joyce (1977) showing the scales of motion near a thermohaline front.
Large-scale water masses, top; medium-scale interleaving and intrusions, center; small-scale
mechanical and double-diffusive vertical mixing, bottom.
8
the buoyancy flux gives rise to horizontal pressure gradient forces which drive the intrusion
across the front, in geostrophic balance (Stern, 1967). Toole and Georgi (1981) suggest
that friction is important in defining a vertical scale for the fastest-growing intrusion, while
Ruddick and Turner (1979) argue that the scale is set by the strength of the vertical and
horizontal gradients. Thus, there is evidence that double-diffusive buoyancy flux, friction,
and rotation are all important elements of the dynamics of oceanic intrusions.
In a rotating system, there is the possibility that friction can act through the mechanism
of the Ekman layer. In the ocean, internal Ekman layers may form at fronts (Horne,
Bowman, and Okubo, 1978; Garrett and Loder, 1981; Garrett, 1982). Stommel and Federov
(1967) discussed interfacial Ekman layers in the context of intrusions. The Ekman transport
in the intrusion can cause it to grow across the front. Interfacial Ekman layers are considered
in greater detail in Section 4.2.
Experiments on double-diffusive intrusions have been performed by several investigators.
These are discussed in Section 3. Of particular relevance to the present problem are the
experiments of Yoshida, Nagashima and Ma (1986, unpublished).
They compared the
opposing effects of double-diffusive buoyancy flux, which drives the cross-front motion, and
momentum flux, which slows it. In my experiments, I used a two-layer baroclinic vortex
to investigate the effects of double-diffusion.
The dynamics of the two-layer baroclinic
vortex are sufficiently simple and well-understood that the effects of double-diffusion can
be isolated.
The experiments described in Section 6 are intended to model the effects of doublediffusion on a single intrusion, or on one interface of such an intrusion, rather than a
series of interleaving layers. Care must be used when drawing conclusions about oceanic
intrusions from such a simple geometry, but it is hoped that the results will contribute to an
understanding of the basic dynamics. In particular, they begin to address the question of
how the increased turbulent momentum transport expected at a double-diffusive interface
might affect a rotating intrusion, and how this compares to the acceleration of intrusions
by the buoyancy flux.
3
Previous experimental investigations
Several investigators have studied double-diffusion in the laboratory since the pioneering
experiments of Turner (1965, 1967). This mixing process is well suited to laboratory study
because of the small scales on which it takes place. Often, experiments are done with the
sugar-salt instead of the salt-heat system because it is easier to work with. The diffusivity
ratio of salt to sugar is about 3, as compared to 100 for the heat-salt system. Empirical
flux laws for the two systems have the same form, with quantitative differences.
Mixing involves transfer of both mass and momentum across the double-diffusive interface. Turner (1965, 1967) and Stern (1969) discuss the flux laws governing buoyancy
transfer. Turner's (1965, 1967) experiments were performed with nonrotating stratified
doubly-diffusive fluids. He found flux laws that describe the transfer of heat and salt across
fingering and diffusive interfaces. The flux laws, discussed in Section 5.2, have been refined
by several observers. Ruddick and Turner (1979) investigated intrusions formed at the front
between stratified salt and sugar solutions. Experiments similar to their "dam-break" experiment, but using two homogenous layers, were done by Maxworthy (1983) and Yoshida,
Nagashima, and Ma (unpublished manuscript). Maxworthy also performed experiments in
which the upper layer fluid was released at a constant rate. These investigators attempted to
estimate the momentum flux across the double-diffusive interface. They found that double
diffusion acts to slow the cross-front motion. Yoshida et al. compared the opposing effects
of double-diffusive buoyancy flux, which drives the cross-front motion, and momentum flux,
which slows it.
In a rotating system, the flow is predominantly along rather than across density gradients. Chereskin and Linden (1986) formed intrusions into a rotating, continuously stratified
salt solution by heating an interior wall. Rudels (1979) injected sugar solution at its own
density level into a linear salt stratification. The effects of double-diffusion in these rotating
intrusion experiments are quite dramatic, but unfortunately only qualitative results were
obtained because the dynamics are rather complicated.
The two-layer baroclinic vortex is a simpler system, in which the effects of doublediffusion can be found by comparison of double-diffusive and non-double-diffusive vortices.
Analytical studies of such vortices in the non-double-diffusive case have been made by Gill et
al. (1979), Flierl (1979), and Griffiths and Linden (1981). One disadvantage of this system is
the presence of instabilities, which are interesting in their own right but serve to complicate
the dynamics.
The experiment described in Section 6 is essentially a rotating version of Maxworthy's
(1983) experiment. Maxworthy measured the advance of the front as a function of time, and
estimated the velocity by the rate of change of the front position. In the baroclinic vortex,
the radial velocity is analogous to the cross-front velocity, but it is much smaller than the
azimuthal velocity and therefore less important in the momentum balance. Keeling (1978)
performed similar experiments, and found that the azimuthal velocity was diminished in the
double-diffusive case. The experiment could be considered a model of the upper or lower
interface of an intrusion at a front. It may also be relevant to the study of double diffusion
in warm core rings.
4
Mechanisms for momentum transport
4.1
Transport of momentum by double-diffusion
Momentum flux by double-diffusion was studied in the nonrotating system experimentally
by Maxworthy (1983) and Yoshida et al. (1986, unpublished), and theoretically by Ruddick
(1980, 1985).
Maxworthy and Yoshida et al. discuss the drag on a density current due
to double-diffusive convection on the interface. Maxworthy (1983) found that as doublediffusion begins to act, the intrusion slowed and, in the case of a constant volume release,
eventually stopped.
Yoshida et al. considered a similar experiment, but with a smaller
density difference. They found a steady velocity in all cases, whether or not double-diffusive;
however, double diffusion generally resulted in a slower flow. In the double-diffusive case,
the pressure-gradient driving force due to the density difference Ap(t) increases with time.
This is opposed by the enhanced momentum transfer across the interface.
Yoshida et al. also studied the dependence of the velocity on the initial density difference. Although the across front velocity decreased with decreasing Ap, as expected for a
density current, it vanished as Ap - 0 only in the non-double-diffusive single component
experiments. This is because the driving force increased with time as the initially very small
density difference became bigger. In Yoshida et al.'s experiments, small density difference
corresponds to a density ratio close to one, and thus vigorous double-diffusion and large
fluxes of salt and sugar across the interface. For larger Ap, then, the double-diffusion slows
the across-front spread of the intrusion, while for small Ap it speeds it.
Maxworthy and Yoshida et al.estimate the momentum transport by the double-diffusion
as the product of the across-front intrusion velocity and the convective vertical velocity. The
vertical velocity, due to the double-diffusive buoyancy flux, is estimated from the empirical
flux laws (Turner, 1967, and others.) This is only one possible parameterization of the
frictional effect of the convection. Ruddick (1985) studied the form of dissipation and eddy
viscosity in thermohaline staircases. He suggests that although the finger Reynolds stresses
are negligible, the convection in the mixed layers can generate internal waves which transfer
momentum. An interesting consequence is that the stress is proportional to the shear across
the interface, unlike the usual quadratic drag.
4.2
The interfacial Ekman layer
In a rotating system, friction can act more efficiently because of the Ekman layer (Tritton,
1977). The divergence of boundary layer transport produces vertical velocity ("Ekman
pumping") which allow the friction at the boundary to affect the interior flow. In the ocean,
internal Ekman layers may form at fronts (Horne, Bowman, and Okubo, 1978; Garrett
and Loder, 1981; Garrett, 1982). In the interfacial Ekman layer, the Ekman transport is
upward along the front on the lighter side of the front, and downward on the denser side (see
Figure 2). This divergence in the upper layer gives rise to an horizontal diffusion equation
for the depth of the interface in the two-layer case or for the depth of isopycnals in the
case of continuous stratification (Garrett and Loder, 1981; Gill, 1981; Garrett, 1982). In
other words, the vertical friction acts, through the mechanism of the Ekman layers, like a
horizontal diffusion.'
Stommel and Federov (1967) suggested that the Ekman transport
in an intrusion drives the growth of the intrusion across the front (see Figure 3).
Ekman layer thickness is
6
, =
The
/2vfT-1 (or V/2Af-, for a turbulent layer with an eddy
viscosity A). Using this, Stommel and Federov derive a diffusion equation for the intrusion
thickness, ht = g'6Ef ~1h.., where g' is the reduced gravity between the layers and
f
is
'This result is independent of the parameterization of the friction, e.g. as linear or quadratic, although the
details of the similarity solution will be changed (Garrett and Loder, 1981).
transport
Figure 2: Interfacial Ekman layers at a front
the Coriolis parameter. The Ekman transport, and hence the effective lateral diffusivity
9'SE/f, depends on the Ekman layer thickness; they suggest that the maximum diffusivity
occurs for an intermediate value of 6E, i.e. E = (0.55)h. Double-diffusive convection is one
mechanism that might tend to thicken the Ekman layer and increase the spreading rate of
the front.
The possible effects of interfacial Ekman transport in my experiment are determined
by applying the interfacial Ekman model to a baroclinic vortex. Csanady (1979) uses an
energy argument and a quadratic drag law to demonstrate that the frictional decay of a
warm core ring by Ekman spin-down is much more efficient than that due to entrainment
of adjacent fluid. The circulation in the vortex and the Ekman layers is considered by Gill
et al. (1979), Griffiths and Linden (1981), and Flierl and Mied (1985). Flierl and Mied show
that, when the density gradient and shear are concentrated at the interface, the circulation
is outward in the upper Ekman layer and inward in the lower (see Figure 4). Griffiths and
Linden (1981) created a baroclinic vortex in the laboratory and observed that the radial
flow was mostly in the Ekman layer, outside an inviscid core. The divergence in the upper
layer causes the vortex to be broader and shallower than predicted by an inviscid model
(Gill et al., 1979; Griffiths and Linden, 1981).
Figure 3: Interfacial Ekman layers acting on an intrusion
Figure 4: Interfacial Ekman layers in a baroclinic vortex
5
Dynamics of the baroclinic vortex
5.1
Inviscid solution
The experiments described in this thesis involve injection at a constant flow rate at the
surface of a fluid of higher density. The inviscid theory for this case has been worked out by
Griffiths and Linden (1981), who were interested in the transition to instability in the nondouble-diffusive case. A surface vortex always becomes unstable eventually as it spreads
out laterally. Their work was used as a guide in making a vortex that remains stable as long
as possible, in order to isolate the effects of double-diffusion. In this section the solution
for an inviscid, axisymmetric two-layer vortex with no buoyancy flux across the interface
is derived, following Griffiths and Linden (1981). In the following sections the effects of
buoyancy and momentum flux across the interface are discussed.
The experimental parameters are listed here for reference (see Figures 5 and 6).
H
depth of water in tank
h
maximum depth of vortex
R
radius of vortex
6
depth ratio h/H
f
Coriolis parameter
P2
density of environment
Ap
density difference between vortex and environment
Rz
vortex Rossby radius (Apgh/f2)I
go
Rossby radius of environment (Apg H/f2)2
Q
inflow rate
The two-layer system is described with cylindrical coordinates (r, e, z); radial and azimuthal velocity components are (u, v). The interface depth, measured from the surface,
is r (r, t). The upper and lower layers are denoted by the subscripts 1 and 2, respectively.
The fluid in the upper layer, entering from the confined source with no initial angular
momentum, has velocity
fr
The l2 21
The lower layer conserves potential vorticity
(1)
11,=so
=
the relative vorticity, assuming
t
H
R
Figure 5: injection experiment of Griffiths and Linden (1981)
azimuthal symmetry, is given by
2=
(rv2 ), =
(2)
In either layer, the momentum balance in the radial direction is
r
v2 -
f
=--
Pr
(3)
p
Subtracting the momentum equations for the two layers and using the hydrostatic relation
yields
r- I(V2 - V2) + f (V1 - V2) = g'r7,
(4)
where g' = gAp/po. If the depth ratio 6 is small, the motion in the lower layer is weak and
the centrifugal term (v2 ) 2 /r can be neglected relative to fv 2 . The centrifugal term in the
upper layer, however, must be retained (Flierl, 1979).
Using (2) and (1) in (4), an equation for v2 (r) is obtained
r2 V2rr + rv2, - V2
1+
)=
(5)
The solution has the form
v2 =
(6)
cIi(r/Ro) - fr/4
where I, is a modified Bessel function and c is a constant.
The interface depth is obtained from (2):
=
H
H
2cIo(r/Ro)]
fr
2
'Rof
The constant c is then determined using r;(R) = 0 at the edge of the vortex r = R; this
yields
C=
R*
2Io(R/)Ro)
So finally the interface depth is given by
t H [1
Io(r/Ro)
(8)
2Io(R/RZo)(8
And the lower layer velocity by
V2 =IRo
Ii(r/o)
2 Io(R/Ro)
fr
4
(9)
The radial extent of the vortex R is found from conservation of the vortex volume; the
volume is
V = Qt = 27rj
0
rirdr= rHR2 1 - 2oII(R/o)
2
RIo(R/R)J
(10)
(
which is an implicit equation for R(t).
Using appropriate flux laws for the fingering and diffusive interfaces, the governing
equations could be integrated in time to give a model of the evolution of the vortex. In the
following sections, however, the effects of buoyancy and momentum flux will be considered
separately.
5.2
Double-diffusive buoyancy fluxes
In the double-diffusive experiments, the density difference across the interface can change
with time because of the differential fluxes of salt and sugar. The equation of state is
assumed to be linear,
p = po(1+ aAT +#AS)
where ciAT is the density anomaly due to salt and #AS that due to sugar; po is the density
of fresh water. The density difference between the two layers is (initially)
Ap(O) = polaAT - #AS|.
As double diffusion acts, the density contrast becomes sharper, since the net mass transport
is upgradient. In this section, the change of density with time and its effects are discussed.
The equations for the conservation of sugar, salt, and density in a double-diffusive system
are
D(#AS)
Dt
#lFs
=V - Ps
D(aAT)
Dt
Dp
o
aFs
ho
(11)
#Fs
_
=haV-FT
-r
ho
|IaAT + #ASI A O(1
-
rf)#Fs
(12)
(13)
where Fs and FT are the double-diffusive fluxes of sugar and salt (or salt and heat), and
ho(t) is an average depth. The vortex is assumed to be well-mixed. The flux ratio is
rf = cFT/#Fs.
The molecular diffusion of heat and salt has been neglected compared to the double-diffusive
flux. Although the density is changing, the change is small relative to the average density,
so the Boussinesq approximation holds. The fluxes depend on the density ratio and on AS
or AT, and therefore change with time.
The laws governing the fluxes of heat and salt (or salt and sugar) across a doublediffusive interface have been extensively studied theoretically and in the laboratory. Turner
(1965) used a dimensional argument to show that the flux of heat across a diffusive interface
is proportional to the I power of the temperature across the interface. Various theoretical
refinements of this simple "S law" have been proposed; here I am concerned only with
empirical determinations of its form.
The effect of rotation on salt fingers was studied experimentally by Schmitt and Lambert
(1979). They measured an increase in the height of salt fingers under rotation, indicating
increased stability. Various theoretical studies (Worthem, Mollo-Christensen and Ostapoff,
1983; Masuda, 1978; Pearlstein, 1981; and Flierl, unpublished) have considered the effect
of rotation on double-diffusive stability. These authors, however, do not discuss the effect
of rotation on the double-diffusive fluxes once the convection has been established. Kelley
(1986, unpublished) presents an argument that rotation has no effect on the fluxes, either
in the ocean or the laboratory. None of the above studies includes the effects of velocity
shear. In particular, the interfacial Ekman layer in my experiment is accompanied by strong
nonparallel shear. In the absence of data on rotating double-diffusive fluxes, the nonrotating
empirical flux laws will be used.
Flux measurements across the fingering interface in the heat-salt system were made by
Turner (1967), Schmitt (1979), and McDougall and Taylor (1984); and across the diffusive
interface by Turner (1965), Crapper (1975), and Marmorino and Caldwell (1976).
The
finger fluxes in the salt-sugar case were measured by Lambert and Demenkow (1972) and
Griffiths and Ruddick (1980); the diffusive fluxes by Shirtcliffe (1973).
In general, the heat, salt, (or salt, sugar,) and buoyancy fluxes depend on the density ratio R,,, the Prandtl number g,
and the diffusivity ratio g,
where v is kinematic
viscosity, res is the diffusivity of salt (or sugar, in the sugar/salt experiments) and
XT
is
the diffusivity of heat (or salt). The salt-sugar system has a much larger diffusivity ratio
(S s 0.3) and correspondingly high flux ratio (rf s 0.9). My experiments were performed
at quite low density ratios, R, = 1.02-1.09. For the fingering salt-sugar interface at small
R, = aAT/#3AS, Griffiths and Ruddick (1980) give
#Fs = (0.226cms-1 )R, 6~(#AS) 4 /3
(14)
for the sugar flux, and rf = 0.94 at R,, = 1.02. For the diffusive interface, R,, will be defined
as #AS/aAT, and Shirtcliffe (1973) gives
aFT = (0.181cms-1 )(R,)-1 2 .6 (iAT) 4/3
(15)
and rf = 0.6 for R,, = 1.1-2.
Yoshida et al. (unpublished) discuss the effect of buoyancy flux on the density in a
double-diffusive intrusion. Assuming that the change in density is uniform over the layers,
they write (see Equation (13)
)
6-
Ifs
PO
-
aFT6t
(16)
ho
where ho is the ratio of volume to area of the intrusion. In the fingering case this is
eSp
P
PO
#
= - 1
F(1- r1 )St
ho
.
(17)
The change in sugar is given by Equation (11):
#Fs6t
(18)
ho
(8
S(#AS)=
and using the flux law
(19)
#Fs = C(R,, -,r)(#AS)4/3
gives
8(flAS)
C(# AS) 4/ 36t
ho(20)
=
This can be integrated if ho and C (i.e. R,) are held constant. Since the flux will decrease
with time as R,, increases away from one, this assumption will result in an overestimate of
the effect of the double-diffusion on the sugar concentration. If double-diffusive modification
of the sugar concentration is not important except on long time scales, this simple model
will be adequate.
The integration yields
(21)
t
# AS
[
#8ASo
rto
where So is the initial salinity and
1 3
/
ro = 3ho/C(#AS)
is the time scale for the sugar concentration to change. Using (19) and (21) in (17) gives
an equation that can be integrated to find the density
6P = Po
ho
11+ -
(ASo)4/3
Iro
4
St.
(22)
For small t/ro the result is
p(t) - p(0)
PO
_
t
(23)
r
where the time scale for density changes is
r
ho
('8'&S0- 4 / 3
C(1- r)
The diffusive interface case is completely analogous, with aAT and aFT replacing #AS
and
#Fs.
The importance of double-diffusion to the density change can be seen in Table 1, in which
the time scales for changing the sugar concentration (ro) and the density (r) are given for
diffusive
fingering
R,,
#ASo
aATo
P2 (kg m- 3 )
g' (m s- 2 )
C (m s-1)
1.02
.1122
.1100
1112.2
.01939
1.411 x10- 3
133.1
1008.
1.045
.0511
.0489
1051.1
.02051
1.040 x10~ 3
236.6
4033.
1.09
.0266
.0244
1026.6
.02100
0.611 x10- 3
507.5
17333.
1.02
.1100
.1122
1112.2
.01939
1.983 x10~ 3
94.72
2870.
1.045
.0489
.0511
1051.1
.02051
1.735 x10- 3
141.9
9670.
1.09
.0244
.0266
1026.6
.02100
1.348 x10~ 3
230.2
31450.
To (s) r (s)
Table 1: Time scales for double-diffusive fluxes
various density ratios, using the flux laws (14) and (15) and assuming po = 1000 kg m- 3 and
ho = 3 cm. This table indicates the rapid increase in fluxes as the density ratio approaches
one.
Because the fluxes depend so strongly on R,,, they will decrease rapidly as the density
difference changes. In a fingering experiment, for example, the sugar concentration changes
more quickly than the salt concentration, so the appropriate time scale would be ro, the
scale for changes in S. Over the course of the experiment-about ten minutes-one would
expect to see the the effects of double-diffusion when the density ratio is less than or equal
to about 1.04.
Increasing g' causes the inviscid vortex to spread out. The ideal vortex formed by injection from a point source has zero potential vorticity; conservation of angular momentum
implies that v = -fr/2 (see Equation (1) above). In order to maintain cyclogeostrophic
balance, the interface slope must decrease. The effect is easily calculated in the case of an
inviscid vortex above an infinitely deep layer. This "1j layer model" approximates the experimental conditions when the vortex is much shallower than the lower layer. Substituting
(1) into (4) above gives
f--2 r, =
4
-g 1,
the solution, using q(R) = 0, is
=
(R2
-
r2 ).
R is found by conserving volume, as before:
V = 27 R
fo
rdr
2rf 2 R 4
32g'
Therefore, R oc
(g')1/4
and h =
2 2
=f
-()
R /8g' oc (g')- 1/ 2 .
So, as g' increases, the vortex
spreads out, but rather slowly.
Although an explicit solution for R(g') and h(g') in the two-layer model is not possible,
it is clear that increasing g' will increase the radial length scale Ro of the vortex and cause
it to spread. The change in g' will be estimated from the data in Section 7.
5.3
Viscous effects
Gill et al. (1979) consider the effect of friction on the (non-double-diffusive) baroclinic vortex. They find a formulation of Ekman friction in the linear case, then apply it to the
cyclogeostrophically balanced azimuthal velocity using a numerical model. They estimate
a radial scale R, at which friction becomes important; this scale is used in Section 6.5.2
to scale plots of velocity as a function of radius. The friction radius R, is determined by
balancing the advection time scale with the frictional (Ekman) time scale:
RI,
H.
where H. is a depth scale and U is a radial velocity scale. From continuity,
Q
~ H*RU,
which yields a scale for the radius of the the "inviscid core"
R ,
~Q1/
2
(f y)-1/4
Griffiths and Linden (1981) observed the flow in the baroclinic vortex by introducing
dye into the injected fluid after the experiment had started. They observed flow out from
the center in a boundary layer on the upper side of the interface, with flow inward in a
boundary layer on the lower side. The inflow in the lower layer resulted in a convergence
at the tip of the vortex. In addition, an inviscid core was observed to grow outward from
the center. These observations support the interfacial Ekman model, with divergence of the
Ekman transport resulting in the spreading of the vortex with time.
Friction is most important at the edges of the vortex, where the interface depth is small.
The velocity in the lower layer is small at the edges because there is less vortex squashing;
the assumption of no flow in the lower layer is therefore best at the edges of the vortex. In
addition, the centrifugal term is small here-in a vortex acted on by friction, the velocity
decays beyond the friction radius, and so v 2 /r becomes small as r -- oo (see Gill et al.,
1979, Griffiths and Linden, 1981, and Section 6.5.2.) Therefore, we can write the equations
for the spin-down of an isolated vortex (Q = 0) as (see Garrett and Loder, 1981)
fv = g'r,
r7t =
r
r
(24)
(-Av)
(25)
where A is a linear friction coefficient. The change in interface depth is simply given by the
divergence of the Ekman transport -Av/f. The above equations can be combined to give
18
r ar
(26)
-rt= CH 1a(rtrr)
a diffusion equation for r7, with
IH =9 A/f
2
.
This "horizontal diffusion" of the interface depth (or, in the continuously stratified case,
of the isopycnals) was discussed by Stommel and Federov (1967), Garrett and Loder (1981),
Gill (1981), Garrett (1982) and Flierl and Mied (1985). In the continuously stratified ocean,
the Ekman model might be inappropriate, but it should work well for a two-layer baroclinic
vortex, in which the shear as well as the density gradient is concentrated at the interface
(Flierl and Mied, 1985).
Equation (26) has the similarity solution
t(r,t) = h(t)e_, 2/2a2(t)
(27)
where
h(t) =
(28)
V is the vortex volume, and
a2 (t) =
2
xCHt.
(29)
The geostrophic velocity is then given by
v(r,t) = -rwo(t)e-r/2a2(t)
(30)
where wo(t) = g'h(t)/f a2 (t). Equation (29) shows that, if the friction A (or XH) is larger at
the double-diffusive interface, the length scale a will also be larger, i.e. the vortex will be
broader and the azimuthal velocity smaller.
Flierl and Mied (1985) show that the meridional flow is not confined to the Ekman
layers; there is an interior return flow. Some of the Ekman divergence is balanced by the
divergence of this interior flow. It is weaker than the boundary layer flow by a factor of
(Ekman number) 1/ 2 . The slower interior flow, however, can be associated with a transport
of the same order of magnitude as that in the much thinner Ekman layer. The effect of the
interior flow is to slow down the spread of the vortex, so the friction parameter A estimated
from the vortex spin-down will be an underestimate.
5.4
Instabilities of the baroclinic vortex
Laboratory studies of the effect of rotation on oceanic intrusions are complicated by the
possibility of baroclinic instability causing formation of secondary vortices (Griffiths and
Linden, 1981, 1982). Saunders (1973) found, for the non-double-diffusive case, that a collapsing vortex was stable as long as its initial radius was smaller than the internal Rossby
radius. Otherwise, the vortex broke up into several smaller, cyclonic vortices. This result
was extended by Keeling (1978), who found that, when double-diffusion was possible, the
initial radius must be even smaller to avoid instability.
Griffiths and Linden (1981) observed instabilities of a vortex when lighter fluid was
injected at the surface. Almost always, the instability was of wavenumber n = 2; i.e. the
initial anticyclonic vortex broke up into two cyclones.
The driving mechanism for the
instability depends on the aspect ratio of the vortex. It is baroclinic in the case of a deep
vortex, releasing the potential energy of the mean flow. Shallow vortices are subject to
barotropic instability, which gets its energy from the mean flow's shear. Occasionally, for
large
f, an n =
1 instability, or "vortex wandering", was seen. Griffiths and Linden suggest
that this is a centrifugal instability due to the large surface curvature; unlike the other
instabilities, it does not release energy.
Griffiths and Linden found that the vortex became unstable to an n = 2 instability
when the parameter
R
6
became less than about 0.02, provided 5 > 0.2. This value of Eis always reached eventually;
as the volume injected increases, R and h increase, so (R/R) 2 decreases and 6 (and hence
6(1 - 6)-1/2) increases, decreasing e. Even if the flow is stopped, friction acts to flatten
m I
Camera
Camera
Q
Mirror
H
f/2
Figure 6: The injection experiment
the vortex and decrease e. Another effect of friction is to make the vortex flatter than the
inviscid prediction, so instability is reached sooner.
If the depth ratio 6 < 0.2, the vortex becomes unstable at a larger value of the instability
parameter, i.e. when e i 0.06. For small 6, e n (Ro/R) 2 ; the vortex should therefore become
unstable when R > 42o.
Rudels (1979) found, in his continuously stratified experiments, that double diffusion
destabilizes the baroclinic vortex. In Keeling's (1978) experiments, the double-diffusion was
found to change both the scale and the character of the instability.
6
Experiments
6.1
Experimental design
The experimental apparatus is shown schematically in Figure 6. The experiments were
performed in a square tank, 61 x 61 x 38 cm, on the one meter rotating table in the
Geophysical Fluid Dynamics Laboratory at WHOI. Although the tank is square, side-wall
effects can be ignored; with a radius of about 6.5 cm, the vortex was much smaller than the
tank. The use of a square tank allowed side view photographs.
The tank was filled with either salt (NaCl) or sugar (sucrose) solution, and dyed salt or
sugar solution was injected. Both one-component and double-diffusive runs were performed.
The one-component runs were repeated using the same density difference but different values
of the background density. In the double-diffusive experiments, the density ratio was varied,
keeping the density difference the same. In order to avoid complicating thermal effects, the
solutions were allowed to stand overnight to come to room temperature, which could vary
between experiments in the range 21*-23* C, but was constant during an experiment to
a few hundredths of a degree.
All temperature measurements were made with a bulb
thermometer.
Densities of the tank fluid and the injected fluid were determined in one of two ways:
1. Specific gravity was measured with a precision hydrometer and converted to density:
standard error = 0.3 kg m- 3 ;
2. The solution was cooled to 20* C, then measured into a volumetric or pycnometric
flask and weighed; the flask weight was then subtracted and the density at 20* C
calculated. The formulae of Ruddick and Shirtcliffe (1979) were used to correct the
density to room temperature: standard error = 0.5 kg m~ 3 .
A linear equation of state p = po(1 + aAT +PAS) was assumed, where po is the density
of fresh water at the measured room temperature, calculated using Ruddick and Shirtcliffe
(1979). Here T denotes salt and S denotes sugar. The departures of density from this value
gave the initial aAT and
#AS.
The initial density difference is 2
Ap = poIaAT -
PASI,
and it is about 2.2 kg m- 3 in all the experiments. The density ratio is defined as
R
=AT
PAS
in the fingering experiments, and
=lAS
_
aAT
in the diffusive experiments.
Top and side view pictures were taken simultaneously using 35mm cameras rotating
with the tank. Photographs were taken about every three rotation periods for the first
seven minutes of the experiment, and less frequently thereafter. Side views were taken with
a 24mm lens on a camera mounted above the mirror. The side view showed the shape of
the interface and the presence of double-diffusive convection. The top view was taken using
a camera with a 36mm lens mounted above the center of the tank. Streaks were made
in the injection experiments by floating paper punches or particles of fluorescent dye on
the surface. Exposure times were a few seconds; the streaks were a few centimeters long.
The side view photos had exposure times of 1/15" or 1/30". The side view was back-lit
and the top view was illuminated by lights mounted just above the edge of the tank. This
arrangement allowed simultaneous measurements of interface shape and azirmuthal velocity.
6.2
Choice of parameters
Similar parameters were used in all the experiments. These parameters are listed in Table 2.
The inviscid theory of Griffiths and Linden (1981) was used to indicate which choice of
2
In the one-component experiments, Ap = po jaATi - aAT |.
2
parameters would avoid the occurrence of vortex breakdown by baroclinic instability, in
order to isolate the effects of double-diffusion. In the notation used in Section 5.4, the value
of 8 was kept large-so that the interface depth could easily be measured from the side
view photographs-while keeping e larger than the critical value 0.02. The inviscid theory
gives (cf Equation (8) )
8
Let Co = R/Ro, and note that
Io(R/Ro) - 1
-
2Io(R/RZo)
(R/R)2 =
e = (-2[6(1
-
6(Ro/R) 2 . Then (see Section 5.4)
)11/2 =
V'Io(o2
_
1
The vortex should be stable when Co < 5.0.
In addition, we have the following constraints:
" R < 0.30 m-i.e. vortex is much smaller than the tank
" H
0.27 m
* 9'
0.002ms- 2, i.e. Ap 5 2.kg m- 3 . This is the smallest practicable Ap because of
errors in the measurement of p. With Ap = 2.2 kg m- 3 , R, can be made as small as
1.02.
To maximize E within these constraints, Co = 1.7 (R 5 6.2 cm) was chosen. This determines
the vortex volume
Qt =
1rR 2 H
?
2
1-
1
2I1(_0)
C (I
I m4 x
o)I
10~ 4m3 .
Choosing t = 5 minutes gives Q s 80 ml/min. This flow rate is slow, so that the vertical
velocity of the incoming jet was small; yet it is fast enough that a vortex deep enough to be
stable to baroclinic instability was formed in five minutes. A non-double-diffusive vortex
with these parameters lasts ten to fifteen minutes after turning off the flow.
6.3
Experimental procedure
The fluids were spun up for about ten times the spin-up time scale rEk =
minutes. The period of rotation was measured with a stopwatch.
H/IV/2
s
2
The water depth H was
measured with a ruler on the side of the tank before spin-up. Since only about 400 cm 3 of
fluid was injected, H does not change appreciably during an experiment.
R,
g' (m s-2)
exp't
type
1
0
.01933
2
0
3
0
4
0
5
D
6
f
(s-')
Q
(10- 6m 3 s- 1)
H (m)
RZo (M)
2.083
.270
.0347
1.342
.02095
2.083
.270
.0361
1.374
-
.02124
2.087
.270
.0363
1.333
-
.02086
2.085
.268
.0359
1.368
1.090
.02086
2.083
.270
.0360
1.341
D
1.045
.02102
2.069
.270
.0364
1.342
7
D
1.025
.02400
2.127
.270
.0378
1.467
8
F
1.093
.02144
2.076
.270
.0366
1.302
9
F
1.046
.02129
2.097
.270
.0362
1.255
10
F
1.022
.02117
2.086
.268
.0361
1.446
Table 2: Experiment parameters. 0-one component experiment; D--diffusive interface;
F-fingering interface
The fluid was injected as close to the axis of rotation as possible (within a mm or so).
A diffuser made of cheesecloth covered the end of the 0.6 cm diameter glass tube, and the
end of the tube was about 0.5 cm below the surface of the water. The lighter fluid was
injected for five minutes; during that time the flow rate was held constant, although it
varied somewhat more during the first few rotations as the valve was adjusted.
The experiment was continued until the vortex had spread over most of the water
surface. This was about fifteen minutes after the flow was turned off in the one-component
experiments, and less in the double-diffusive ones, depending on the strength of the doublediffusion.
6.4
Observations
As fluid was added, the vortex grew in radius and depth, then flattened and broadened
after the flow was turned off. An exception to this occurred when the double-diffusion was
very strong; in those experiments the vortex slumped so quickly that the interface depth
started to decrease before the flow was turned off.
6.4.1
Instabilities
In these experiments, vortex wandering is common, although the rotation rate is not as
great as that in the Griffiths and Linden (1981) experiments which showed the n = 1
instability. I seldom saw the "spiral arms" which are the first manifestation of the n = 2
instability in Griffiths and Linden's experiments. A small-scale barotropic instability was
sometimes observed, especially in the double-diffusive experiments. This was indicated by
columns of fluid rotating cyclonically and extending, eventually, to the bottom of the tank.
These could be distinguished from salt fingers in the fingering experiments because the dye
could often be seen spiraling down inside them. The cyclones had a radius at the surface of
about 1.5 cm. They formed not just at the edge of the vortex (where they were indicated
by the dye), but anywhere in the fluid; this can be seen in some of the streak photographs.
These cyclones are probably due to convection caused by evaporative cooling at the surface.
The water in the tank was at room temperature (dry-bulb temperature), so there were no
thermal effects at the side walls, but the tank was not covered and evaporation could occur.
The cyclones resemble those observed by Nakagawa and Frenzen (1955) in their study of
convection in rotating fluids.
6.4.2
General character of interface
The interface was clearly different in the three cases. Typical side-view photographs are
shown in Figure 7. In the one-component case, the interface is fairly sharp and smooth.
There is a fuzzy patch at the tip of the vortex.
This was also noted by Griffiths and
Linden (1981), and is caused by mixing due to Ekman convergence in the lower layer. The
shape of the interface is more like a Gaussian than the Bessel function predicted by the
inviscid theory. This is evidence of the importance of friction in the vortex dynamics. The
inviscid theoretical interface shape for the two layer and 1} layer models, and a Gaussian
fit, are superimposed on the non-double-diffusive interface in Figure 8. The two inviscid
models are similar, indicating that the lower layer velocity is small. A Gaussian curve
(cf. Equation (27) ) is a better approximation of the interface shape. The fit was made
by measuring the maximum interface depth and the half-width.
If the interface shape
were truly Gaussian, it would extend to infinity in radius; however, we can expect that
the similarity solution of the linear friction model breaks down when the interface depth
(a)
(b)
(b
(e)
(f)
Figure 7: Side view photographs, taken at t = 149 s and t = 386 s: (a,b) one-component
experiment; (c,d) diffusive interface, R, = 1.02; (e,f) fingering interface, R, = 1.02. The
flow was turned off at 300 seconds.
approaches the Ekman layer thickness (Zhurbas and Kus'mina, 1981).
The effect of the double-diffusive convection can clearly be seen in the photographs. In
the salt-fingering case, the interface is most diffuse and the fingers have carried the dye
into the lower layer. The diffusive interface is sharp, but the convection in the layers has
distorted it. Again, dye is seen in the lower layer. The wavy diffusive interface and the
diffuse fingering interface were also noted by Yoshida et al. (unpublished).
"Sections" of the interface depth have been drawn in Figure 9 at various times. These are
not scaled, but as the parameters are very similar for each experiment, direct comparison is
possible. The depth of both the diffusive and fingering interfaces decays more quickly than
that of the one-component case. This indicates that the frictional spin-down is nore efficient
in the double-diffusive experiments. In the next section, I will try to quantify this result.
Note that the differences from the one-component case are greater for the experiments in
which R, is close to one. This is as expected, since the double-diffusion increases in intensity
as the density ratio approaches one.
6.4.3
Quantitative measurements
Maximum interface depth and radial extent were measured from (projected) slides. The
mixed patch at the tip of the vortex in the one-component case and the radial spreading in a
thin surface Ekman layer were not included in the measurements. In Figures 10, 11, 12 and
13, 14, 15, the unscaled maximum interface depth h and radius R are plotted as functions of
time for the one component, diffusive, and fingering experiments; the corresponding aspect
ratios h/R are plotted in Figures 16, 17, and 18. The scatter in the data from the different
one-component experiments gives an estimate of the uncertainty. Because of the parallax
due to the wide-angle lens, the vortex is not viewed exactly edge-on (see Figure 7), so h can
not be measured when it is less than about one centimeter. Thus, there are unfortunately
not many measurements after the flow is shut off in the double-diffusive experiments.
The top view streak photographs gave (surface) azimuthal velocities as a function of
radius and time. Velocity was measured from the photographs as follows: The length of
the chord and the distance from the center are measured with calipers on the print; the arc
length is calculated from this information assuming that all the particles move in a circle
around the center. The center is the point about which all the particles seem to be moving;
Figure 8: Theoretical interface shapes superimposed on a non-double-diffusive vortex side
view. inviscid two-layer solution,.
inviscid 1I layer solution, - - - Gaussian fit
33
( (b)
(d
(c)
(d)
(a)
t=88 s
146 s
~7Z7~
270 s
~-
378 s
455 s
(e)
t
(g)
'77_
= 88 s
7II7~~
~~7Z7~
146 s
270 s
378 s
(h)
7II7~'
455s
Figure 8: "Sections" of interfaces at various times. (a,e) one component
experiments, (b) diffusive experiment, R, = 1.09, (c) diffusive experiment,
R, = 1.04, (d) diffusive experiment, R, = 1.02, (f) fingering experiment,
R, = 1.09, (g) fingering experiment, R, =1.04, (h) fingering experiment,
R, = 1.02.
-+ turn-off time = 300 seconds.
it may change with time as the vortex wanders off center. The arc length is then divided by
the exposure time to give the velocity. I did not attempt to measure radial velocities, which
are much smaller than the azimuthal ones (until the vortex breaks up and goes unstable).
A typical streak photograph is shown in Figure 19.
6.5
6.5.1
Results
Interface shape
Because the experimental parameters were similar for the various runs, the interface depth
data is presented without scaling. The flow was turned off in all experiments at 300 seconds. The uncertainty in the measurements can be estimated from the scatter for each
experiment and between the four one-component experiments. Scaling does not reduce the
scatter significantly. For comparison to the double-diffusive runs, h and R from the onecomponent runs were averaged. The solution to the inviscid two-layer model is presented
for comparison. Of course, in that solution the interface depth and radius do not change
after the injection of fluid is stopped.
The one-component interface depth data, in Figure 10, show that the interface depth
increases until the flow is turned off, then decreases. The depth is less than that predicted by
the inviscid model. In the double-diffusive runs, shown in Figures 11 and 12, the vortices are
shallower than in the one-component case, and the interface depth decreases more rapidly.
The rate at which the vortex slumps increases as the density ratio approaches one. The
results are similar for the diffusive and fingering cases. At the density ratios closest to one,
the vortex begins to become shallower even before the flow is turned off.
The radius of the vortex is plotted in Figures 13, 14 and 15. While filling, the vortex is
broader than the inviscid theory, as noted by Griffiths and Linden. It continues to spread
out after the inflow is stopped; the double-diffusive vortex spreads out faster than the
one-component case when R, = 1.04 or 1.02.
It is interesting to note the difference between the measurements and the inviscid theory
in the plot of aspect ratio h/R, Figure 16. The vortex continually becomes broader and
shallower; the inviscid model predicts that it will have an aspect ratio of about one at the
time the flow is turned off. In the double-diffusive experiments, the aspect ratio plots again
show that the vortex is shallower and broader when acted upon by double-diffusion.
o 0 o 7 -
one component salt)
one component sugar)
one component (salt)
inviscid two-layer solution
-
o
0
E
U
one component salt)
V
0
-U
Q-o
(4-.
0
070
U
0
0
00
0
0
0
000
V
L
Q)
00
C
0
0
0
0
300.0
600.0
900.0
0
0
03
(M 0
1
1200.0
tUme (sec)
Figure 10: Interface depth h versus time from the side view photographs, for four
one-component experiments. The inviscid two-layer solution is shown for comparison. The
flow was turned off at t = 300 s.
A
one component exp'ts
+ t z -
diffusive, R,=1.09
diffusive, R,=1.04
diffusive, R,=1.02
-
inviscid two-layer solution
A6
0
Q).
Q)
AA
A
*
p
N
C.)
3
41
I
-
a
A
a
3 A
A
,
300.0
A
,
600.0
1200.0
tLme (sec)
Figure 11: Interface depth h versus time for three diffusive experiments, with the average
of the one-component experiments and the inviscid solution for comparison.
37
- one component exp'ts
+ - fingering, R,=1.09
x - fingering, R,=1.04
x . fingering, R,=1.02
A
-
inviscid two-layer solution
E
A+~
++
Xx
A
L
)
A
A
C
A
A
+
+
+
+
A
A
x
04
0.0
300.0
600.0
tLme
900.0
1200 0
(sec)
Figure 12: Interface depth h versus time for three fingering experiments, with the average
of the one-component experiments and the inviscid solution for comparison.
38
o
- one component
o - one component
o - one component
v - one component
-
(salt)
salt
sugar)
salt)
inviscid two-layer solution
C,
0
N
0
VO
M
0
E
0
Oj
0C
0
0 0
0
13
U,
ao
0
0
L
0
0
7
V
7
V
.0
V0
0
00
40o
300.0
600.0
900.0
1200.0
t Lme (sec)
Figure 13: Vortex radius R versus time for four one-component experiments. The flow was
turned off at t = 300 s.
one component exp'ts
diffusive, R,=1.09
- diffusive, R,=1.04
- diffusive, R,=1.02
inviscid two-layer solution
=
+
*
*
-
U
*
A aA
A
AA
A
U
~*
0.0
A
A~ A
300.0
600.0
900.0
1200
0
tLme (sec)
Figure 14: Vortex radius R versus time for three diffusive experiments, with the average of
the one-component experiments and the inviscid solution for comparison.
a - one component exp'ts
+ x -
-
fingering, R,=1.09
fingering, R,=1.04
fingering, R,=1.02
inviscid two-layer solution
A
A
E
0
x
+
Ln
L
Q)
L
A
+
0
A
+
X
~<A
900.0
300.0
1200 0
tLme (sec)
Figure 15: Vortex radius R versus time for three fingering experiments, with the average of
the one-component experiments and the inviscid solution for comparison.
41
o = one component (salt)
o - one component salt)
o - one component sugar)
, - one component salt)
inviscid two-layer solution
-
0
0
0
D
J
0
0
0o
970
0
L
Q)
V(
CL 0
07)
0
V
0V
7
09P0
0
oVV
~Oo
0
00
0
0
0n
0
0
0
00
0
C0
300.0
600.0
tLme (sec)
Figure 16: Aspect ratio h/R versus time for four one-component experiments, with the
inviscid solution for comparison. The flow was turned off at t = 300 s.
A
one component exp'ts
diffusive, R,=1.09
+
-
H
-
diffusive, R,=1.04
diffusive, R,=1.02
U/)
0
0
0
L
+
-+j
U
Q)
A
*C
0
AA
A
t-n
0
*
N
H
A
A
a
,
N
N::
0
a
A
N
A
H
0-
300.0
600.0
900.0
1200 0
tLme (sec)
Figure 17: Aspect ratio h/R versus time for three diffusive experiments, with the average
of the one-component experiments for comparison.
a
+ x x -
one component exp'ts
fingering, R,=1.09
fingering, R,=1.04
fingering, R,=1.02
+x
U
+
++a6++
ULr
+
+
+
+
+
+A
X( )<XM
4 +
+ +
+A
+A
A
A
+
xN
-~
I
300.0
900.0
600.0
tLme
1200 0
(sec)
Figure 18: Aspect ratio h/R versus time for three fingering experiments, with the average
of the one-component experiments for comparison.
Figure 19: A typical streak photograph. The streaks were made by paper punches or
particles of fluorescent dye floating on the surface.
6.5.2
Azimuthal velocity
There is considerable scatter in the velocity data. To make the results more clear, the data
have been smoothed by fitting Equation (30), page 23, to the data, using the method of
least squares. The Gaussian shape of the interface and the velocity data support the choice
of this form for the smoothing. Typical fits are plotted with the corresponding data in
Figures 20 and 21. The parameters of the fits to unscaled velocity are given in Table 3.
Scaled azimuthal velocity as a function of radius at various times is shown in Figures 22, 23, and 24 as a function of radius. The radius r is scaled by the frictional radius
Ry = Q'/ 2(fV)-1/ 4 (Gill et al., 1979), discussed on page 22. The associated azimuthal velocity scale is V, = -f R,/2. After the flow is turned off, this scaling may no longer be appropriate, but all the plots use the same scaling for ease of comparison. The double-diffusive
experiments are again compared with an average of the one-component experiments.
The data show that the velocities are generally slower in the double-diffusive cases, for
density ratios less than 1.09. As discussed in Section 5.2, the change in density difference
would not change the velocity; the reduction in the speed therefore indicates the effect of
friction. The differences between the double-diffusive and non-double-diffusive vortices tend
to increase with the strength of the double-diffusion as measured by the density ratio.
7
Discussion
The effect of buoyancy flux can be estimated from the data by calculating the reduced gravity 9 '(t). At large r the balance is approximately geostrophic; that is, for v = -wore
v 2/r < fv when r > a. Substituting q = he-' 2 /2a2 into fv = g',
,
fwo(t)a 2 (t)
h(t)
gives
(31)
Table 3 shows the measured values of h and the calculated values of g' for the various experiments at several times. This calculation gives only a rough estimate of g'(t), but indicates
the time scale for changes in the density difference. The reduced gravity changes significantly over the length of the experiment only in the double-diffusive runs with R,, = 1.02,
38. seconds
to
124. seconds
88. seconds
to
O
0-i
C)
x
x
-
0.0 2.0
4.0
6.0
I
0.0 2.0
-A
4.0
I
6.0
r/R,9
r/Ry
4.0
0.0 2.0
6.0
r/R,
234. seconds
180. seconds
273. seconds
x
T--A
0.0 2.0
I
I
4.0
6.0
4.0
r/R,
4.0
2.0
6.0
0.0 2.0
1.0
6.0
455. seconds
396. seconds
6.0
4.0
0.0 2.0
r/R,.
r/R ,
r/R ,
330. seconds
0.0 2.0
.0
6.0
0.0 2.0
r/R,
Figure 20: Representative plot showing azimuthal velocity data and fitting function
4.0
6.0
88
38, seconds
124
seconds
seccnJs
162.
sec'-ds
COJ
C)
0
0
0.0 2.0
4.0
6.0
0 0 2.0
4.0
180
0~
6.0
0.0 2.0
r/R
r/Rv
CZ)
215.
seonds
4.0
6.0
0
0
r/R
234. seconds
seconds
2.0
4 0
6 0
r/R3
273. se-conds
0
oo
0
M'
0 0
0-
0.0 2.0
4.0
6.0
0 0 2.0
330
0
0
2.0
378
seconds
4.0
/Rev
4
0
& 0
r / R9
r/R 9
6.0
02
C.0
2.0
4.0
6.0
0.0 2
r/R,
seconds
r0
4
6
r/R.
Figure 21: Representative plot showing azimuthal velocity data and fitting function
0
40
r/RY
6 C
38. seconds
1.0 2.0
4.0
88, seconds
6.0
0.0
2.0
4.0
r/R,0
r/R,
180. seconds
215. seconds
1.0 2.0
4.0
6.0
r/R0
330. seconds
0.0 2.0
4.0
r/R,
6.0
0.0 2.0
4.0
6.0
4.0
r/Rv
0.0 2.0
4,0
6.0
162. seconds
0.0
6.0
0.0 2.0
1.0
6.0
2.0
0.0
2.0
r/R,
0.0 2.0
4.0
r/Rv
6.0
4.0
6.0
r/R,
396. seconds
6.0
4.0
r/R,
273. seconds
r/R,
234. seconds
r/R,
378. seconds
0.0 2.0
124. seconds
455.
I
6.0
0.0 2.0
seconds
1.0
r/R,
Figure 22: Smoothed azimuthal velocities as a function of radius at several times for three
one-component experiments. Flow was turned off at 300 seconds.
6.0
124.
88. seconds
38, seconds
162. seconds
seconds
O
to
("
0
0
.0
4.0
2.0
180.
0.0 2.0
6.0
r/R8
seconds
0 2.0
4.0
4.0
0.0 2.0
6.0
4.0
6.0
r/R,
r/R,
r/R,
215. seconds
234. seconds
273. seconds
0
0~
--
0~
0.0 2.0
1.0
0.0 2.0
6.0
4.0
0.0 2.0
6.0
r/R,
r/R,
330. seconds
378. seconds
0.0 2.0
6.0
4.0
6.0
r/R,
r/R y
Co
4.0
455, seconds
396, seconds
0
to
to
0
0
>0>0
0
.-
V5
r/R
0
0
0
0.0 2.0
1.0
0.0 2.0
6.0
4.0
r/R,
r/Ry
6.0
0.0 2.0
4.0
6,0
0.0 2.0
1.0
r/R,
Figure 23: Smoothed azimuthal velocities as a function of radius at several times for diffusive
experiments.
-
one component, - - - R, = 1.09, - - - R,
= 1.04, -..--
R, = 1.02
6.0
38. seconds
88, seconds
Co
124. seconds
162. seconds
0ED
0-
> CD
0.0 2.0
6.0
4.0
0.0 2.0
4.0
6.0
r/R,
r/R,
180. seconds
215. seconds
0.0 2.0
CO
4.0
6,0
0.0 2.0
6.0
4.0
r/R,
r/R.
234. seconds
273. seconds
0(0
0.0 2.0
6,0
4.0
0.0 2.0
r/R,
330. seconds
4.0
6.0
r/R,
r/R,
378. seconds
396. seconds
0.0 2.0
4.0
6.0
r/R,
455.
seconds
(CI
>
0.0 2.0
1.0
r/R,
6.0
0
0.0 2.0
4.0
r/R,
6.0
0.0 2.0
4.0
6,0
0.0 2.0
4.0
6.0
r/R,
Figure 24: Smoothed azimuthal velocities as a function of radius at several times for fingering experiments.
one component, - - - R, = 1.09, - - - R, = 1.04 .... R, = 1.02
exp't
0
0
0
5
CA
ba
0o
8
time
wo (cm/s)
a (cm)
h (cm)
38.
88.
124.
162.
180.
215.
234.
273.
330.
378.
396.
455.
38.
88.
124.
162.
180.
215.
234.
.467
1.054
.515
.657
.501
.573
.698
.548
.351
.151
3.39
2.41
3.77
3.33
3.83
3.96
3.73
4.15
4.15
4.99
3.35
4.28
4.56
5.24
5.40
5.37
5.15
5.34
4.93
4.24
1.164
1.073
.777
.528
.511
.447
.484
2.06
2.45
3.11
3.84
3.97
4.16
4.21
2.85
3.33
3.77
3.81
4.04
4.16
4.09
3.60
4.05
4.15
4.25
4.15
3.87
4.36
273.
330.
378.
396.
455.
.282
.212
.183
.117
5.33
5.70
5.87
6.88
4.39
3.86
3.39
2.93
2.58
4.33
4.22
4.48
4.49
38.
88.
124.
.527
.719
.643
3.07
2.93
3.41
2.77
3.04
3.22
3.71
4.23
4.82
162.
.609
3.59
3.84
180.
.557
3.94
3.81
215.
.571
3.98
234.
273.
330.
.618
.632
.532
3.91
4.15
4.17
378.
396.
.404
-
455.
.271
g' (cm a 2)1 exp't
3.34
2.97
3.34
2.89
2.84
3.49
3.92
3.69
2.55
1.84
2
time
38.
88.
124.
162.
wo (cm/a)
.655
1.133
.606
.835
a (cm)
2.31
2.29
2.89
2.90
h (cm)
3.39
3.73
4.38
4.52
2
g' (Cm a~ )
2.14
3.33
2.40
3.23
1exp't
1
4
-
time
wo (cm/s)
a (cm)
38.
88.
124.
162.
180.
215.
234.
273.
330.
378.
396.
455.
38.
88.
124.
162.
180.
215.
234.
2.281
.753
.632
.518
.534
.518
.385
.337
.289
.243
.215
.202
.244
.467
.209
.348
.247
1.78
3.08
3.67
4.07
4.18
4.17
5.02
5.44
5.63
5.45
6.11
5.59
5.68
3.34
6.31
4.27
5.04
h (cm)
2.75
3.41
3.80
3.96
4.17
4.18
3.96
4.55
4.37
4.37
4.24
4.32
2.51
3.38
3.48
3.09
2.65
3.25
3.17
273.
330.
378.
396.
455.
.139
.113
.079
-
8.30
8.15
11.61
-
2.51
1.87
1.01
1.12
-
8.15
8.53
22.42
-
38.
88.
124.
.561
.334
.485
3.01
4.00
3.63
2.33
2.84
2.50
4.56
3.93
5.33
g' (Cm
5.50
4.37
4.68
4.52
4.66
4.48
5.10
4.56
4.38
3.45
3.95
180.
-
-
-
215.
-
-
-
-
234.
273.
.716
.618
3.77
4.07
4.81
5.27
4.42
4.06
330.
-
-
-
-
378.
-
-
-
396.
-
-
-
-
455.
-
-
-
-
38.
88.
124.
162.
180.
215.
234.
.603
.503
.339
.342
.360
.269
.281
2.82
3.07
3.89
4.04
3.31
4.68
4.33
2.37
3.94
3.71
4.25
4.26
3.92
3.34
4.18
2.48
2.86
2.72
1.92
3.10
3.26
273.
330.
378.
396.
455.
-
-
1.36
-
38.
88.
124.
.304
.584
.316
3.11
2.95
4.33
2.87
3.51
3.30
2.15
3.03
3.78
4.25
162.
-
-
-
-
162.
-
-
2.62
-
4.70
180.
.315
3.67
3.78
2.35
180.
.519
3.84
2.56
6.23
4.24
4.43
215.
-
-
-
-
215.
-
-
2.70
-
4.23
4.79
4.23
4.65
4.71
4.54
234.
273.
330.
.450
.324
.253
3.84
4.46
5.48
3.84
3.46
3.23
3.62
3.90
4.94
234.
273.
330.
.411
.282
.221
4.07
5.07
5.43
2.43
2.32
1.60
5.82
6.54
8.49
4.58
-
3.90
3.71
4.51
-
378.
396.
.141
7.12
2.33
6.45
378.
396.
.152
-
6.75
-
1.47
1.47
9.80
-
5.04
2.89
4.96
455.
.178
4.12
1.94
3.27
455.
.176
5.73
0.86
14.03
6
9
7
10
Table 3: Velocity fit parameters and estimates of reduced gravity. The azimuthal velocity
has the form v = -w 0 re
. Reduced gravity is calculated using g'= fwoa 2 /h.
3.05
4.95
3.19
5.72
5.10
4.22
-
-2)
i.e. experiments 7 and 10. From the discussion at the end of Section 5.2, R is proportional to
the fourth root of g', while h varies as the inverse square root of g'. The increase in g', in the
strongly double-diffusive cases, of a factor of about four corresponds to a decrease in h of a
factor of two and an increase in R of about a factor of 1.5. This is not enough to account for
the rapid decrease in h with time as compared with the non-double-diffusive experiments
(see Figures 11 and 12.) Therefore, there must be enhanced momentum transport in the
double-diffusive experiments.
The vortex broke up more quickly in the double-diffusive cases; that is, it was more
unstable. The criterion for stability when h < H is that R < 4Ro. Spreading due simply
to the changing density difference would not make the vortex unstable, since Ro would be
increasing at the same rate as R. Provided the time scale for changes to g' is longer than
a rotation period, the vortex will be able to continually readjust itself to cyclogeostrophic
equilibrium. The increased instability of the double-diffusive vortex, also observed by Keeling (1978) and Rudels (1979), is thus an indication of the importance of friction in making
the vortex spread.
The velocity measurements show that the vortex spins down after the flow is shut off.
The initial relative vorticity of the injected fluid is dissipated. If there were no possibility
of instability and breakup of the vortex, it would eventually spin down until all the fluid
was in solid body rotation.
A rough estimate of the friction parameter A can be obtained from the evolution of the
interface shape after the flow is turned off at 300 seconds. Interface depth is used instead of
the change in, say, the velocity maximum with time because there are more measurements
and the dependence on R, is more clear. From Equation (28), the maximum interface depth
h is expected to be inversely proportional to time. A function of the form
h
[1+t - toff-
hoff
rM
was fit to the h(t) data for t > 300 seconds; here hoff is the value of h when the flow was
shut off, and rm is a time scale for the spin down
V
nme b4rhffi cH
For simplicity, the friction is assumed to begin acting at t = toff = 300 seconds. Setting the
vortex volume V =
Q - tog,
the friction parameter A can be estimated from Tm and hff;
A
Qtff
_
Pf
g'
4
1rg'hoffgr
The calculated values of rm, hoff, and A are listed in Table 4.
If the interfacial stress is parameterized as quadratic rather than linear in velocity,
i.e. -CD
vIV, the rate of spin-down changes with time (Garrett and Loder, 1981).
The
Garrett and Loder solution for the Ekman model with quadratic friction has
t7(0,t) = h(t) =
'2
45
V(,'Vt)-2/
where
ICH -
SCD;
here CD is the drag coefficient. The data were fit with the function
t-tg-2/6
h
hoffr
where the time scale is now given by
=( 0 .3 0 1 )hV/2
hoes/ 2H
and so
CD = (0.301)
(Qtoff) 3/ 2 f 3
hoffs/2 (g)
2
m'
The values of hoff, r', and CD are also listed in Table 4.
The effect of the buoyancy
flux in changing g' has been neglected in these calculations; this will cause A and CD to
be overestimated somewhat in the double-diffusive experiments. On the other hand, the
interior flow described by Flierl and Mied (1985) may take up some of the divergence of the
Ekman transport as discussed in Section 5.3. This would cause the momentum transport
calculated from the spreading of the vortex to be underestimated.
Table 4 shows that the momentum transport is increased in the double-diffusive experiments. The friction parameter is increased by a factor of 4-14 in the diffusive case, and
1.4-8 in the fingering case. If the quadratic drag law is used, the drag coefficient increases
by a factor of 4-25 at a diffusive interface, and a factor of 2-50 at a fingering one.
The Ekman layer model is admittedly a simplistic view of the actual dynamics. The
real density stratification is continuous. Stratification allows forces to be redistributed by
exp't
type
linear
R,
quadratic
hoff (cm)
rm (s)
A (m s-1)
hoff (cm)
rm' (s)
4.56
946.
1.6 x 10-4
4.58
262.
0.45
CD
*
0
5
D
1.09
4.15
283.
5.7 x 10-4
4.12
73.
2.00
6
D
1.045
2.96
218.
10.1 x 10-4
2.81
93.
3.96
7
D
1.02
2.36
125.
22.3 x 10-4
2.24
55.
11.24
8
F
1.09
4.27
641.
2.3 x 10~4
4.56
116.
0.88
9
F
1.045
3.17
435.
4.5 x 10-4
3.18
121.
2.05
10
F
1.02
1.89
145.
12.4 x 10-4
1.80
51.
24.85
Table 4: Estimates of friction parameters A and CD from interface depth data. *
of one component experiments.
average
pressure gradients; the effect of interfacial Ekman layers will be weakened. Gill (1981)
and Garrett and Loder (1981) show that with continuous stratification the horizontal diffusivity
If
is replaced by
where N 2 =
9AH,
gp,/p
is the buoyancy frequency. Weaker
stratification therefore leads to a slower spread across the front for the same friction A.
These experiments do not determine what form the friction at the boundary takes. One
possibility is form drag on the interface, which is distorted by the double-diffusive convection
on either side. The asymmetric shape of the diffusive interface, in particular, indicates this.
In the fingering case, momentum could be transported by Reynolds stresses. Ruddick (1980,
1985) suggests another possibility, that stress is transferred by internal waves generated at
the interface by the convection on either side. Whatever the exact mechanism, comparison
of the time scales in Table 4 shows that spin-down of the baroclinic vortex occurs more
rapidly when the interface is double-diffusive. The effect becomes greater as the density
ratio is reduced towards one. The results of the experiments are not directly applicable to
the ocean, but they suggest that the double-diffusively driven momentum flux in oceanic
intrusions could be large enough to be an important element in their dynamics.
8
Conclusions
The results of a series of laboratory experiments on a double-diffusive baroclinic vortex
have been described, with the expectation that they have relevance to the dynamics of
oceanic intrusions. In particular, they indicate the importance of momentum transport
across the double-diffusive interface. A similar problem was studied in a nonrotating frame
by Maxworthy (1983) and Yoshida et al. (unpublished). The earth's rotation affects the
momentum balance in intrusions (Stern, 1967; Toole and Georgi, 1981). In Stern's model,
the divergence in the upgradient buoyancy flux results in an alongfront pressure gradient;
the intrusion grows across the front in geostrophic balance. There is in this inviscid model no
mechanism to set a vertical scale for the intrusions. The fastest-growing intrusion has infinite
vertical wavenumber. Toole and Georgi (1981) add friction to Stern's model. Friction damps
the smallest-scale motions; thus, the addition of viscous dissipation results in a preferred
vertical scale for the fastest growing intrusion.
The two-layer baroclinic vortex was chosen as a simple rotating system in which to study
the effects of double-diffusion. In a rotating system, the efficiency of momentum transport
is enhanced by Ekman layers; through the divergence of the transport in such a layer, the
boundary effects can be felt throughout the fluid. Interfacial Ekman layers exist in the
baroclinic vortex experiments (Griffiths and Linden, 1981) and affect the evolution of the
vortex circulation. In the double-diffusive case, the Ekman layers are probably turbulent.
The thicker Ekman layer is more efficient in causing the observed "diffusion" of the interface
horizontally.
The vortex spreads out with time as friction acts. Double-diffusive vortices spread out
noticeably more quickly than one-component ones. This effect increases as the density ratio
approaches one, that is, as the double-diffusion becomes stronger. From the shape of the
vortex, a crude estimate of the difference in effective viscosity can be made. It can be several
times greater in the double-diffusive experiments, depending on the density ratio. In addition, no clear evidence of the along-front perturbations required by the Stern (1967) model
of intrusions was seen. These results indicate that friction must be included in intrusion
models, as it is in Toole and Georgi's (1981) paper, and the choice of a parameterization
of friction should be carefully made. Double-diffusive momentum transport must be considered along with the buoyancy transport when discussing the physics of double-diffusive
frontal intrusions.
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