GEOPHYSICAL RESEARCH LETTERS, VOL. 27, NO. 8, PAGES 1179-1182,APRIL 15, 2000
An investigation
thermohaline
John W.
of the link between lead-induced
convection
and
arctic
eddies
M. Bush
Department of Mathematics, Massachusetts Institute of Technology
Andrew
W. Woods
BP Institute for Multiphase Flow, University of Cambridge
Abstract.
A recent laboratory study indicates that a turbulent
buoyant line plume discharging into a rotating stratified fluid provides a natural mechanismfor generating
Hunkins1985]. In the BeaufortSea, sucheddiesaccountfor nearly 30% of the kinetic energyin the uppermost 200 m, and coverapproximately25% of the sea
1974].Whilelead-induced
convection
a seriesof anticyclonicgeostrophic
vortices [Bushand surface[Hunkins
has
been
tentatively
proposed
as
a
possible
sourceof
Woods1999].Hereweexaminethe implicationsof these
experimental results in the context of the thermohaline
convectionwhich developswhen saline water is released
as leadsfreezeover in the polar oceans. Using the experimental results in conjunction with a simple numerical
model of plume dynamics in a non-uniformly stratified
environment, we develop a model which characterizes
the geometry of the eddieswhich would develop owing
to lead-induced convection. The model predicts that, in
the absenceof strong currents, lead-induced thermohaline convection may generate anticyclonic geostrophic
vortices
of characteristic
radius
2-10
km
at the
base
of the mixed layer; however, this mechanism cannot
account
for the
arctic
eddies
observed
at substantial
depths beneath the mixed layer, cyclonic, or strongly
ageostrophiceddies.
Introduction
The production of polar ice from seawater leads to
the rejection of relatively saline water and so to thermohaline convectionin the polar seas. The majority of
suchfreezingeventsoccursalong 'leads', linear fractures
arcticeddies[ManleyandHunkins1985],therehasyet
to be a modelcapableof predictingthe sizeanddepthof
the eddies which would result from lead-induced convec-
tion. Here we developsucha model and so investigate
the plausibilityof lead-inducedconvectionas a source
of the observed arctic eddies.
Model of vortex generation
A seriesof laboratory experiments examining the motion of turbulent line plumes in a stratified rotating environment have recently been reported by the authors
[Bushand Woods1999]. Buoyantdyedwater wasfed
for a finite time ts through a 40 cm long line sourceinto
a rotating stratified saltwater solution, and rose in the
form of a turbulent buoyant line plume. As the plume
rises and entrains ambient fluid, its buoyancy decreases
until it reaches the neutral buoyancy height Z,. The
plume fluid then spreadsout laterally in the form of a
neutral cloud, acrosswhich a strong along-sourceshear
developsas a result of the influence of the Coriolis force.
After several rotation periods, the neutral cloud breaks
into a chain of lenticularanticyclonicvortices(Figure
1). The ratio of the half-height,h, to radius,R, of the
in the ice covertypically 10-100km long [Morisonet
al. 1992]. There havebeenseveralrecentfield studies resulting individual vortices was found to be
of lead-inducedthermohalineconvection[The Leadex
Group1993;MorisonandMcPhee1997],andthesehave
R
N
'
h _ (0.47
+0.12)
f
(1)
motivated the development of a number of numerical
[Smith and Morison 1993; Kantha 1995; Lavelle and where f is twice the rotation rate and N is the BruntVaisala frequency associatedwith the ambient stratifiSmith 1996] and experimental [Fernandoand Ching
1993; Ching et al. 1993]modelsof lead-inducedther- cation. Relation(1) indicatesthat the dynamicsof the
mohaline convection. Field studies of polar oceanshave
identified the prevalenceof lenticular eddies with characteristic
radii
of ten
kilometres
between
the
mixed
layer and 200 m depth [Hunkins1974; Manley and
Copyright
2000bytheAmerican
Geophysical
Union.
individual vorticesis governedby a geostrophicbalance
between the radial pressureforce and the Coriolis force
associatedwith the anticyclonic swirling motion within
the vortex. The size of the vortices may be derived from
conservationof mass. The total volume per unit length
supplied to the neutral cloud is given by
(2)
Papernumber1999GL002314.
0094-8276/00/1999GL002314505.00
where t s is the source dischargetime, and the depen1179
1180
BUSH
& WOODS:
LEAD-INDUCED
CONVECTION
AND
ARCTIC
EDDIES
natural mechanism for generating a series of anticyclonic lenticular vortices. We proceed by examining the
dimensionsand depth of the eddieswhich would be pro-
ducedby lead-inducedconvectionin the polar seas(Figure 3). As a first simplifiedmodel,we couldexaminethe
penetration of the line plume acrossthe density interface between two uniform layers of fluid corresponding
to that aboveand belowthe pycnocline(e.g. [Ching
et al. 1993]). However,in the presentwork, we also
ii)
account
for the
stratification
of the
water
above
and
'!i'-•:.....,-.:•i!•..•;X•:?:
.. :::;.-.:
. ....'•'..:':ii;
':...".15:::'::'iiiiii.:.:•i;:ii•i!".i•!?:::i:•il;;'-:...i•:•:•.•.:..:i::i:-:..::?
......
:.:, below the pycnocline which has an important role in
-•:::.
:.=================================:
...:...:.-...
-:........::
....-..:...:...-.•,
:.::.::......
;..• ...::.:::::::.....
:j--:......
:.•:::
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:•::•iii..:.:•?i:•::;.:;•;•:•.•i•i•!•;•?:•..•.?•?.:•..•`•...`•::•:•:.::...i•i•::•?:!::.•!iiii•!;:::•:•!:.:.•i•...`..•!•.....!!.%:..:.<•.:
.....
-'".•.i
.............
'•:"'
ß":'.':"=================================================================
•.::::::.>.•.:.%•::.:..:•!:i:i:i:`..•i•::•:.::!:!:!:[•i:•i•!:i:•:i:i:!:i:•:i:!•i:!:i:!:!:!:i:i:i:i:!:!•i:i:!•i:•:•.?....•.•
:::•(-'...Xxx..•.:.•:.::::::..
:.?.-"-
arresting the plume. Specifically, we develop a model
of a turbulent line plume descendingthrough a non-
uniformlystratifiedenvironment
( [Mortonet al. 1956],
[Caulfieldand Woods1998]). When the lead is sufficiently narrow to be treated as a line plume, the cross-
plumeaveraged
massfluxQ - f_• udx,momentum
fluxM - f_• u2dxandbuoyancy
fluxB - f_• ug'dx
iii)
per unit length along the lead vary with distance z below the source according to
dQ
dz =e•M
, dM
dz = BQ
M ' dB
dz = _N•.(z)Q
' (4)
where the entrainment coefficient for a line plume is
taken to be e • 0.1 [List 1979]. We adopt a simple model of the ambientstratificationN(z) in a polar
iv)
ocean, based on winter measurementstaken during the
LeadExexpedition.:The profilein Figure4a revealsa
very strong density gradient at the pycnocline, 40-45m
below the surface at the base of the well-mixed upper
layer. Numericalsolutionsof equation(4) relate the
properties of the plume at the neutral buoyancyheight,
Figure 1. The developmentof coherentvorticesfrom Z•, to the buoyancy flux, B0, released by freezing at
the dischargeof a 44cm long line plume in a rotating
stratified fluid as viewed from above.
Here N -
0.82
s-1, B0 - 3.1 cm3/s3, f -- 1.5s-: andt, - 15/f. (i)
t - 11/f: the plumefluidspreads
at its levelof neutral
1.1
buoyancy.The Coriolisforce generatesa strongshear
acrossthe neutral cloud. (ii) t - 17/f: the neutral
clouddevelops
an instability.(iii) t - 23/f' the neutral
cloudbreaksinto six distinctanticyclonicvortices.(iv)
t - 37/f' the persistence
of six lenticularvorticesof
comparable size.
1.0
0.9
0.8
0.7
B:13t•/2
0.6
0.5
0.4
dence of the volume flux per unit length supplied to
the neutral cloud on the sourcebuoyancy flux per unit
length B0 is knownfrom dimensionalconsiderations
to
0.3
beQ(Z,•)• B•/a/N.Substituting
Q(Z,•)and(1)into
0.0
0.2
I
0.1
0.0
1.0
2.0
(2), we predictthe eddyradiusto be
3.0
4.0
5.0
6.0
?t
(a)
where ,Xis a dimensionlessconstant. Figure 2 showsthe
Figure 2. The average size of the emerging eddies.
The data were gathered from a seriesof experiments
in which 1 to 6 eddies were produced through varying the source strength, rotation rate and stratifica-
excellentagreementbetweenthe measurededdy radii tion of the ambient fluid. The dashed line represents
and the scaling(3), and indicatesthat A -- 0.77q-0.15.
R- 0.77B1/at}/2
/ f •/2.
Application to lead-induced convection
The instability of the neutral cloud generatedby a
buoyant line plume in a stratified ambient providesa
1Morrison, J.H., private communicationvia J.S. Wettlaufer,
data collected at BASE302 during LeadEx expedition.
BUSH &: WOODS: LEAD-INDUCED
CONVECTION
Linear source of
brine-rich fluid
lead - longlinear
of surface water at
openingin the ice cover , _
10-100m
wide;
10-100leao
•
Negatively
buoyant
turbulent
lineplume, •i:•:".•
descending
through
•i"i
unit area Q0, the mixed layer depth D, the drag coefficient at the ice-water boundary Ca and the ice-drift
velocityU), is sufficiently
large;for mostleadsL >> 1,
sothat the plume-inducedconvectiondominatesthe influenceof ambientcurrents [Morisonet al. 1992] .
ice-coveron
polar
ocean
•,0-50m
60--
deep
water
t r
mixed
layer
to
neutralcloud
•:"''•'
50'"•
1181
ber,L = QoD/(CaUa) (based
onthebuoyancy
fluxper
formedby freezing
k
AND ARCTIC EDDIES
•'•
weakly
stratified•f
VIEW
neutral
cloud
intrudes
laterally
]rowing
toawidth
of notto
order
1-10• andathickness
10-50
m before
scale
30-
becoming
unstable
andbreaking
upintoa seriesof
upper mixed layer
20-
eddies
with
very
weak
shear is set
non-linear
10 density
gradient
up in the
break-upof the
0.0()01
neutral
cloud owing
to the
Codoils
force
•
cloud into a
instability
trainof eddies
aftera
(seefig2)
time of
order 10/f
0.001
0.01
0.1
stratification (Brunt Vaiasala frequency) (1/s)
60--
55-
TOP VIEW
OF NEUTRAL
CLOUD
50-
Figure 3. Schematic
of lead-induced
eddyformation
beneath the ice cover.
4540-
35-
thelead.In turn,theleadbuoyancy
flux,B0,isrelated
to thefreezing
rate,5, of thesurface
iceaccording
to
3025
10'7
Bo = gl• ( $=-Si ) W 5 ,
(5)
where
g isthegravitational
acceleration,
W isthelead
width,• is the solutalexpansion
coefficient
forseawater,and$•0andSi represent
thesalinity
oftheambient
seawaterand the ice, respectively[Wettlauferet al.
1997].
Using(1), (2) and (5) togetherwith the numerical
solution
of (4) to determine
Q(Z•), wecanestimate
the
radiusand depthof the resultingvorticesas a function
of the rate of freezingat the lead. In figure4, wepresent
predictions
from this modelshowingthe eddy depth
10's
10's
0.0001
freezing rate (m/s)
discontinuties arise from the discontinuous
modelprofileof BruntVaiasalafrequency
(3a)
14
-
12
10
8
6
4
(that of the centroid)andradiusasa functionof the
leadfreezingrate. We notethat the penetrationdepth
is independent
of discharge
time, whilethe eddyradius
10'7
10's
I 0'5
0.0001
freezing rate (m/s)
increases
as•8 . Owing
tothelarge
density
gradient
Figure
4. (a)Model
oftheapproximate
stratification
atthebase
ofthemixed
layer,
there
isa sizeable
rangein a polarocean,
based
ondatafromtheLeadex
Exoffreezing
rates(e.g.7x 10-7 < 5 < 3x 10-5m/s
fora periments.
Thepycnocline,
ata depth
ofabout
40-45m
100mwidelead)for whichplumesaretrappednearthe whichdividesthe uppermixedlayerfromthe deepwa-
pycnocline.
Ourcalculations
suggest
thatin order
for terisa region
ofveryhighstratification;
(b,c)Calcula-
a plume
topenetrate
thepycnocline,
thefreezing
ratetionofthedepth
andradius
ofaneddy
formed
from
a
would
need
tobeunrealistically
high.
line
plume
as
a
function
of
the
freezing
rate
at
the
lead.
The sourceis assumed
to persistfor a time 100/f. OwDiscussion
ing to the largestratificationat the pycnocline,plumes
producedby a widerangeof freezingratesare confined
to depths of 40-45m. Only the most vigorousplumes
In developing
oursimplemodel,wehaveneglectedproduced
by the highest
freezing
ratescanpenetrate
the influenceof ambientcurrents. The influenceof the pycnocline.Curvesare shownfor 10 and 100mwide
background
flowis negligible
provided
the Leadhum- leads.
1182
BUSH & WOODS: LEAD-INDUCED
CONVECTION
Nevertheless, we expect that situations will arise where
vigorousbackgroundflows will dispersethe plume fluid
to such an extent as to preclude the generation of coherent
AND ARCTIC
Acknowledgments.
EDDIES
This work was supported by a
NERC BRIDGE grant. We thank John Wettlaufer for a
number
of valuable
discussions.
vortices.
In order to model the generation of eddies by leads
more accurately, it would be necessaryto considerthe
influence
of the finite
width
W
of the lead source on
the buoyancy flux at the intrusion depth H. In particu-
References
Bush, J.W.M. and Woods,A.W., The generationof vortices
by lineplumesin a rotatingstratifiedfluid, J. Fluid Mech.,
388, 289-313, 1999.
lar, a recentnumericalstudy [SmithandMorison1993] Caulfield, C.-P. and Woods, A.W., Turbulent gravitational
convectionfrom a point sourcein a non-uniformlystratihas suggestedthat for W > H, the convectionmay be
fied environment,J. Fluid Mech., 360, 222-248, 1998.
characterized by the descent of a number of discrete Ching, C.Y., Fernando, H.J.S. and Nob, Y., Interaction of
plumes, in which casethe volume flux impinging on the
a negativelybuoyantline plume with a densityinterface,
Dyn. Atmos. Oceans,19, 367-388, 1993.
pycnoclineis expected to be significantly different from
that beneath a simple line plume. Finally, while our Fernando,H.J.S. and Ching, C.Y., Effectsof backgoundrotation on turbulent line plumes,J. Phys. Oceanogr.,23,
model assumesexplicitly that the buoyancyflux is con2125-2129, 1993.
stant in time, it is straightforward to considera source Hunkins,K., Subsurfaceeddiesin the Arctic Ocean,Deepwith buoyancy flux of the form Bot•. Doing so would
Sea Res., 21, 1017-1033, 1974.
allow one to make more sophisticated predictions for Kantha, L.H., A numericalmodelof arcticleads,J. Geophys.
Res., 100 (C3), 4653-4672, 1995.
the properties of the coherent structures generated by
Lavelle, J.W. and Smith, D.C., Effectsof rotation on convecleads, whose buoyancy flux necessarilydecreaseswith
tive plumesfromline segmentsources,J. Phys. Oceanogr.,
time [ Wettlauferet al. 1997].
Our experiments and theoretical calculations suggest that lead-induced thermohaline convection provides a robust mechanismfor generating anticyclonic
geostrophicvortices at the base of the mixed layer.
However,we concludethat this mechanismis unlikely
to be responsiblefor the generationof the majority of
the large, deep arctic eddies which have been observed
on field expeditions.First, our model suggeststhat the
plumes generated by refreezing leads will not typically
be sufficientlyvigorousto penetrate to great depths beneath the pycnocline, and so to account for the deepest
observededdies. Second,while the observededdieswere
predominantly anticyclonic, two of sevenwere cyclonic
[Manleyand Hunkins1985].It is noteworthythat situations arose in our experiments where the anticyclonic
eddies spun-up a cyclonic pair, thus forming a dipolar
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No.35, p. 393, 396-397, 1993.
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coastalwaters(Ed. H.B. Fischer)Chap. 9, 315-389,Academic, New York, 1979.
Manley, T.O. and Hunkins, K., Mesoscaleeddiesof the arc-
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The oceanographyof leads, J. Geophys.Res., 97 (C7),
11199-11218, 1992.
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with an autonomousunderwatervehicle,J. Geophys.Res.,
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from maintained
and instantaneous
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vortexwhichpropagatedaway from the source [Bush Wettlaufer, J.S., Worster, M.G. and Huppert, H.E., Natural convectionduring solidificationof an alloy from above
and Woods1999]; however,the mechanismby which
with applicationto the evolutionof seaice, J. Fluid Mech.,
a cyclonic eddy might arise in isolation is not clear.
344, 291-316, 1997.
Finally, our model indicates that an alternate generation mechanismmust be soughtfor ageostrophiceddies,
characterisedby relatively small dimensionsand large
internal velocities,for which fluid inertia plays a signif- (ReceivedMarch 5, 1999;revisedSeptember1, 1999;
icantrolein the dynamics[ManleyandHunkins1985].
acceptedJanuary14, 2000.)