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5. THERMOHALINE CIRCULATION
We now turn our attention to the “thermohaline
circulation” — the circulation driven by changes to the
temperature or salinity in some part of the ocean.
The lecture is organised around the following topics:
• Water masses and their formation.
• The role of diapycnal mixing.
• The Stommel-Arons model of the abyssal circulation and
Deep Western Boundary Currents.
• Multiple equilibria and abrupt climate change.
WATER MASSES AND THEIR FORMATION
Sparse observations ⇒ deep circulation often inferred from
large-scale “water mass” properties.
Within the oceanic interior, many properties such as
potential temperature, salinity, and other tracers are
quasi-conserved. The properties distributions therefore give
a zero order indication of the circulation pathways.
Water masses are crudely named in terms of:
• their formation site (e.g., NA = North Atlantic, AA =
Antarctic, M = Mediterranean, etc);
• the depth at which water mass settles (e.g., IW =
Intermediate Water; DW = Deep Water; BW = Bottom
Water).
(a) Atlantic
GEOSECS section along the western trough
(figure from Pickard and Emery 1990)
(b) Pacific
GEOSECS section along 160◦E
(figure from Pickard and Emery 1990)
There are three dominant water masses:
• AAIW that forms in the Southern Ocean and spreads
northwards into both the Atlantic and Pacific;
• NADW that forms at high latitudes in the North
Atlantic and spreads southwards;
• AABW that forms adjacent to Antarctica and spreads
northwards into the Atlantic and Pacific.
However no deep water is formed in the North Pacific.
This is the fundamental reason for the asymmetry in the
heat transports between the Atlantic and the Pacific
(see lecture 1).
Generalisation of the thermohaline conveyorbelt including
the additional water masses (Schmitz 1996):
WATER MASS FORMATION
Surface waters are made dense in three different ways:
• surface cooling;
• evaporation (leaves salt behind ⇒ S increases);
• sea-ice formation (sea ice can hold only 4o/oo salt ⇒excess salt released into underlying water).
The densest water masses are formed in semi-enclosed or
marginal seas, where relatively small volumes of water are
trapped and exposed to intense buoyancy loss for a
prolonged period.
The dense waters subsequently overflow from marginal sea
into the abyssal ocean:
(Price 1994)
Note, the densities of the water masses change dramatically
in the overflows!
IS THE THERMOHALINE CIRCULATION PUSHED OR
PULLED?
If there is a localised source, S ∼ 20 Sv, of NADW at high
latitudes in the North Atlantic, it is necessary for this water
to return to the surface somewhere (i.e., to be converted
back into a lighter water mass).
Suppose that the NADW upwells uniformly over the abyssal
ocean. How large is the required upwelling?
w∗A = S (5.1)
where A ∼ 3 × 1014m2 is the surface area of the oceans.
Thus
w∗ ∼ 0.7 × 10−7m s−1 ∼ 2m yr−1.
However to maintain thermodynamic equilibrium, we
require that the upwelling of cold water is balanced by a
downward flux of heat. Classically, it has been assumed
that this is provided by internal wave breaking.
1-d heat budget (Munk 1966):
w∗∂T
∂z∼ ∂
∂z
κ∂T
∂z
(5.2)
⇒ w∗ ∼ κ
D,
where κ is the coefficient of diapycnal mixing and D is a
typical vertical scale.
Setting D ∼ 103m gives a required mixing coefficient of
κ ∼ 0.7 × 10−4m2s−1
.
However estimates from microstructure measurements and
tracer release experiments (e.g., Ledwell et al.1993) suggest
mixing rates in the ocean interior are an order of magnitude
smaller ...
this has led to a debate over the so-called “missing mixing”.
The leading contenders are:
• Enhanced mixing over rough bottom topography, e.g.,
Polzin et al. (1997) have greatly enhanced mixing over
the mid-Atlantic ridge in the Brazil Basin.
• NADW returns to the surface in the Southern Ocean,
where it is converted to lighter waters as part of the
northward Ekman flux (see figure in lecture 3).
Moreover it has been suggested that these processes might
actually be rate limiting, i.e., they control the overall
strength of the thermohaline circulation (Munk and Wunsch
1998):
STOMMEL AND ARONS MODEL
Why is the abyssal circulation intensified at the western
margin of basins?
... addressed in a famous series of papers by Stommel and
Arons (1958-1960).
Assumptions:
• abyssal ocean represented by a single layer of uniform
thickness;
• no variations in bottom topography;
• localised sources of deep water at high latitudes, balanced
by slow upwelling, w∗, over the remainder of the ocean.
For convenience w∗ is usually assumed uniform.
a. Interior circulation
Away from boundaries, circulation will be geostrophic:
u = − 1
ρ0f
∂p
∂y, v =
1
ρ0f
∂p
∂x. (5.3)
Substituting above into continuity equation,
∂u
∂x+∂v
∂y+∂w
∂z= 0, (5.4)
gives large-scale vorticity balance:
βv = f∂w
∂z. (5.5)
Integrating from sea floor (where w = 0) to the top of the
abyssal layer (where w = w∗) gives:
β∫
v dz = fw∗. (5.6)
⇒ poleward flow in each basin, i.e., towards the sources of
deep water!
b. The Deep Western Boundary Current
The deep flow cannot be poleward everywhere, e.g., we
know there is a net equatorward flow of NADW in the
North Atlantic.
To resolve this paradox, Stommel predicted the existence of
deep western boundary currents. (Can solve for these
mathematically by adding linear friction to the momentum
balance.)
This leads to the following circulation pattern:
The prediction of the Deep Western Boundary Current in
the North Atlantic is perhaps the only example of major
ocean current having been predicted using a theoretical
model before it was actually observed.
Swallow and Worthington (1961) dropped neutrally-buoyant
floats into the predicted DWBC and saw them move rapidly
southward, thus confirming the theoretical prediction.
However, with hindsight, there is a high probability they
could have gone the other way! (due to the eddy field)
Nevertheless the existence of a DWBC is now clearly
established, e.g., from tracer observations (see lecture 1).
MULTIPLE EQUILIBRIA
Can the thermohaline circulation possess more than one
stable mode of operation? (e.g., can deep water be formed
in the Pacific rather than the Atlantic?)
First addressed in a remarkable paper by Stommel (1961).
Consider the following (highly-idealised!) two-box ocean:
Within each box, T and S are assumed well mixed.
A thermohaline circulation, strength q, flows through two
pipes connecting the boxes. We will assume that the
thermohaline circulation acts essentially as non-rotating
density current and write
q = k(α∆T − β∆S). (5.7)
What are the simplest, physically motivated boundary
conditions for T and S?
• Air-sea heat exchange tends to restore the ocean
temperature to equilibrium values over relatively short
time-scales.
• However the evaporation and precipitation rates do not
depend on the salinity of the ocean.
Therefore want restoring boundary conditions on T and
fixed flux boundary conditions on S (known as mixed
boundary conditions).
We can simplify further if we assume that the temperatures
are effectively prescribed (maintained by air-sea fluxes).
Salt budget for either box gives:
|q|S1 = (|q| + E)S2
⇒ |q|∆S ≈ ES0. (5.8)
(The modulus sign is present because the result is
independent of flow direction.)
Eliminating ∆S between (5.7) and (5.8) gives
|q|q − kα∆T |q| + kβES0 ≈ 0. (5.9)
This is a quadratic equation in q, with the added
complication of the modulus signs!
Graph showing equilibrium solutions for q for different
values of E:
The solid lines represent stable equilibria and the dashed
line unstable equilibria.
(values used: α = 2 × 10−4K−1, β = 0.8 × 10−3(o/oo)−1,
k = 0.5 × 1010m3s−1, ∆T = 20 K, S0 = 35 o/oo)
For the present-day North Atlantic, E ≈ 0.5 Sv, and there
are two stable equilibria:
• a fast or thermally-direct equilibrium, with sinking at
high latitudes (q ∼ 15 Sv):
• a slow or thermally-indirect equilibrium, with sinking at
low latitudes (q ∼ −2 Sv):
Global warming scenario:
increased atmospheric CO2 ⇒ warmer air
⇒ increased moisture capacity ⇒ larger E
According to the graph, as E increases the thermohaline
circulation will initially weaken.
However once E exceeds 0.7 Sv, the high-latitude sinking
equilibrium no longer exists, and the circulation collapses
into the low-latitude sinking state.
Note that if atmospheric levels of CO2 subsequently
decrease, the circulation may remain in the low-latitude
sinking state.
Example of the collapse of the thermohaline circulation in a
similar 3-box model. A high-latitude fresh water anomaly of
strength 0.5, 0.558 and 0.6 o/oo is applied impulsively:
(figure courtesy of Helen Johnson)
Stommel’s box model is highly idealised. However very
similar behaviour is observed in more complete models:
Marotzke and Willebrand (1991) looked for these different
equilibria in an idealised OGCM (with identical surface
boundary conditions):
4 hemispheric basins
⇒ 2 × 2 × 2 × 2 = 16
potential equilibria
They found 4 (panels show meridional overturning, Sv):
a. northern sinking b. southern sinking
c. conveyorbelt d. inverse conveyorbelt
Coupled OAGCMs show thermohaline circulation can
shutdown due to anthropogenic climate change, e.g.,
Manabe and Stouffer (1994):
Abrupt changes in thermohaline circulation are also
suggested in paleorecords, such as ice-cores and sediments,
e.g., Broecker (1987):
SUMMARY OF MAIN POINTS:
• Deep water is formed in the Atlantic, but not in the
Pacific.
• The densest water masses are formed in marginal and
semi-enclosed seas. The water mass properties are
modified substantially in the dense water overflows.
• The abyssal circulation is concentrated in Deep Western
Boundary Currents.
• The thermohaline circulation may possess more than one
stable mode of operation. Increased high-latitude
precipitation in a warmer climate may lead to a
reduction, or shutdown, in the North Atlantic
thermohaline circulation.
• However the thermohaline circulation is sensitive to a
number of processes that are currently poorly represented
in ocean general circulation models. Thus the results of
such models should be regarded as tentative in nature.
REFERENCES FOR LECTURE 5
General reading
Siedler, G., J. Church, and J. Gould, 2001: Ocean Circulation and Climate.
Academic Press.
Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cambridge
University Press. (Chapters 1 and 2).
Specific references
Ledwell, J., A. Watson, and C. Law, 1993: Evidence for slow mixing across the
pycnocline from a tracer-release experiment. Nature, 364, 701-703.
Manabe, S., and R. J. Stouffer, 1994: Multiple century response of a coupled
ocean-atmosphere model to an increase of atmospheric carbon dioxide. J.
Climate, 7, 5-23.
Marotzke, J., and J. Willebrand, 1991: Multiple equilibria of the global
thermohaline circulation. J. Phys. Oceanogr., 21, 1372-1385.
Munk, W., 1966: Abyssal recipes. Deep Sea Res., 13, 707-730.
Munk, W., and C. Wunsch, 1998: Abyssal recipes II: energetics of tidal and wind
mixing. Deep Sea Res., 45, 1977-2010.
Pickard, G. L., and W. J. Emery, 1990: Descriptive physical oceanography. An
Introduction. Butterworth-Heinemann.
Polzin, K. L., J. M. Toole, J. R. Ledwell, and R. W. Schmitt, 1997: Spatial
variability of turbulent mixing in the abyssal ocean. Science, 276, 93-96.
Price, J. F., 1994: Dynamics and modelling of marginal sea outflows. Oceanus,
37, no. 1, 9-11.
Schmitz, W. J., 1996: On the World Ocean Ciculation, Vol. II: The Pacific and
Indian Oceans/A Global Update. Woods Hole Oceanographic Institution.
Stommel, H., 1958: The abyssal circulation. Deep Sea Res., 5, 80-82.
Stommel, H, 1961: Thermohaline convection with two stable regimes of flow.
Tellus, 13, 224-230.
Stommel, H., and A. B. Arons, 1960a: On the abyssal circulation of the world
ocean. I. Stationary planetary flow patterns on a sphere. Deep Sea Res., 6,
140-154.
Stommel, H., and A. B. Arons, 1960b: On the abyssal circulation of the world
ocean. II. An idealized model of the circulation patterns and amplitude in
oceanic basins. Deep Sea Res., 6, 217-233.
Swallow, J. C., and L. V. Worthington, 1961: An observation of a deep
countercurrent in the western North Atlantic. Deep Sea Res., 8, 1-19.
Togweiller, J. R., and B. Samuels, 1998: On the ocean’s large scale circulation in
the limit of no vertical mixing. J. Phys. Oceanogr., 28, 1832-1852.
OCEAN CIRCULATION
David Marshall
University of Reading ([email protected])
LECTURES
1. Introduction to the oceans
2. Homogeneous model of the wind-driven circulation
3. Vertical structure of the wind-driven circulation
4. Rossby waves, Kelvin waves and El Nino
5. Thermohaline circulation
6. Dynamics of thermohaline circulation variability
REFERENCES
Both general references for further reading, and specific
references cited in the lecture notes, are listed at the end of
each lecture.
1. INTRODUCTION TO THE OCEANS
Aims for today:
• Why study the oceans?
• Air-sea interaction
• Observation methods and challenges
• Overview of large-scale circulation
WHY STUDY THE OCEANS?
• 71% of the Earth’s surface is covered by water.
• The heat capacity of the upper 3m of the oceans is
equivalent to the entire heat capacity of the atmosphere.
• The oceans transport a similar amount of heat polewards
as the atmosphere.
• Changes in SST can affect the atmospheric circulation
(e.g., El Nino, formation of Hurricanes)
• Long memory of oceans ⇒ potential for seasonal climate
prediction.
• The oceans store about 50 times more carbon than the
atmosphere.
• The oceans take up roughly 1/3 of the carbon released
into the atmosphere through human activity.
• Fisheries.
• Military.
• Mineral deposits.
• Waste disposal?
• Intellectual curiosity.
AIR-SEA INTERACTION
The dominant source of energy for the circulations of the
atmosphere and oceans is the sun.
Excess incoming-outgoing radiation at low latitudes, and
vice-versa at high latitudes ⇒ the atmosphere and oceans
must transport heat polewards.
The the oceanic circulation itself is driven by
• surface wind stresses;
• surface heat fluxes;
• freshwater fluxes (evaporation, precipitation, sea-ice
formation, river discharge);
• body forces (tides).
a. Wind stress
Measured from ship observations (e.g., Josey et al. 2000),
from operational atmospheric analyses (e.g., Trenberth et al.
1990), or from remote sensing (e.g., Liu and Katsaros 2001).
Bulk parameterisation:
τs = ρaCDU210, (1.1)
where ρa is the density of air and U10 is the wind speed at
10 m; the drag coefficient, CD, is a function of wind-speed,
atmospheric stability and sea-state. Typically, use:
103CD = 1.15 (|U10| < 11m s−1)
= 0.49 + 0.065|U10| (|U10| > 11m s−1) (1.2)
(Large and Pond 1981)
Mean wind-stress (from Josey et al. 2000):
Note:
• wind stresses are highly variable;
• there are still significant uncertainties.
b. Heat flux
Four components:
• sensible heat flux from air-sea temperature difference;
• latent heat flux associated with evaporation;
• incoming short-wave radiation from the sun;
• long-wave radiation from the atmosphere and ocean.
Again parameterised using bulk formulae (e.g., Reed 1977).
Air-sea heat flux (W m−2) over the North Atlantic (Isemer
et al. 1989)
Mean sea surface temperature
(from Peixoto and Oort 1992; based on Levitus 1982).
How is this related to the neat heat flux? Chicken or egg?
Freshwater flux
Evaporation and Precipitation
(from Wijffels 2001; based on 13 datasets):
(Evaporation ∝ Latent heat flux;
1.27m yr−1 ⇔ 100W m−2.)
Net E-P flux and standard deviation amongst the 13
contributing datasets.
Surface salinity — how is this related to E-P?
(from Peixoto and Oort 1992)
OBSERVATIONS OF LARGE-SCALE CIRCULATION
The oceans present several major observational difficulties:
• remoteness and size,
• high pressures (e.g., 500 atmos. at 5km),
• highly corrosive,
• opacity to electromagnetic radiation
(⇒ cannot “see” beneath surface),
• turbulence.
The latter is highly problematic: e.g., need ∼ 2 weeks of
continuous data to filter internal waves and measure a
geostrophic current (Wunsch 1996).
The consequence is that the ocean is grossly undersampled
in both space and time. However this situation is
improving rapidly, both due to remote sensing of surface
properties, and intensive observational programmes since
the 1990s, associated with the World Ocean Circulation
Experiment (WOCE).
WOCE one-time hydrographic sections
Southern Ocean hydrographic inventory prior to early 1990s
(http://www.awi-bremerhaven.de/Atlas/SO/)
3 summer months 3 winter months
Approaches:
• In-situ current meters
• Acoustic Doppler Current Profilers (ADCP)
• Hydrographic measurements
Measure T and S (⇒ ρ) along hydrographic sections,
and use thermal wind balance,∂u
∂z=
g
ρ0f
∂ρ
∂y,∂v
∂z= − g
ρ0f
∂ρ
∂x, (1.3)
to infer geostrophic flow field subject to an assumed level
of no motion.
• Floats
• Tracers
• Satellite altimetry
Measure shape of sea surface from space ⇒ surface
geostrophic circulation.
u = −gf
∂η
∂y, v =
g
f
∂η
∂x, (1.4)
Global data every 10 days (TOPEX-POSEIDON) (but
poor knowledge of geoid limits accuracy of mean data).
• Acoustic tomography
Transit time of sound waves ⇒ temperature.
OVERVIEW OF LARGE-SCALE CIRCULATION
Cartoon from Schmitz (1996):
Geostrophic streamlines at surface with assumed level of no
motion at 1.5 km:
Main features:
• Subtropical gyres in all major basins;
anticyclonic,
typical transports T ∼ 30 Sv (1 Sv ≡ 106m s−1)
typical velocities U ∼ 1 cm s−1.
• Subpolar gyres in northern hemisphere basins;
cyclonic.
• Gyres closed by intense western boundary
currents—e.g., Gulf Stream, Kuroshio;
L ∼ 50 km, U ∼ 1 m s−1.
Transports can be enhanced by local recirculations, e.g.,
Gulf Stream transport ∼ 85 Sv at Cape Hatteras, and
∼ 150 Sv after separation.
• Antarctic Circumpolar Current (ACC) in Southern
Ocean; T ∼ 130 − 200 Sv, c.f. atmospheric jet.
• Equatorial jets.
• At depth, find Deep Western Boundary Currents, e.g.,
southward in Atlantic, transport of order 10 − 20 Sv.
Tritium in North Atlantic in 1971 (Ostlund and Rooth
1990):
CFC-11 on σ1.5 = 34.63
(∼ 1.5 − 2 km depth)
(from Weiss et al.,
1985, 1993):
The Deep Western Boundary Current in the North
Atlantic is part of the global thermohaline conveyobelt,
often depicted by Broeker’s “cartoon”:
carries ∼ 1 PW of heat northward in North Atlantic, and
may warm western Europe by several degrees:
Rahmstorf and Ganapolski (1999)
Global salinity section from GEOSECS
• Superimposed on the mean circulation is an intense
transient eddy field, with a dominant energy-containing
scale of order 100 km. Analogue of weather systems in
the atmosphere.
(Richards and Gould 1996):
Visible reflectance ⇒ phytoplankton abundance (Richards
and Gould 1996):
Sea surface height variability from TOPEX-POSIEDON
altimeter
(http://topex-www.jpl.nasa.gov)
Surface velocity snapshot in Indian Ocean from 1/4◦
Semner and Chervin model (Wunsch 1996):
SUMMARY OF KEY POINTS
• Ocean transports a similar amount of heat polewards as
the atmosphere.
• Air-sea fluxes remain highly uncertain.
• Large-scale surface circulation dominated by subtropical
gyres and western boundary currents, and ACC in the
Southern Ocean.
• Thermohaline conveyorbelt carries about 1 PW of heat
northward in the North Atlantic and may warm western
Europe by several degrees.
• Superimposed on the “mean” circulation is an intense,
small-scale eddy field ⇒ major challenges for observing
and modelling the ocean.
POSTSCRIPT: DIFFERENT VIEWS OF THE OCEAN
Wunsch (2001) suggests that the oceanographic literature
suffers from a kind of mulitple personality disorder:
• The descriptive oceanographers’ classical ocean
large-scale, steady, laminar, aims to depict “the” global
circulation
• The analytical theorists’ ocean
quasi-steady, branch of GFD, aims to use simple models
to deepen understanding
• The observers’ highly variable ocean
high temporal and spatial variability, regional focus
• The high-resolution numerical modellers’ ocean
relative newcomer, some elements of each of above, but
differs from all of them
“Little communication between the apostles of these
different personalities appears to exist; nearly disjoint
literatures continue to flourish.”
REFERENCES FOR LECTURE 1
General reading
Open University Course Team, 1989: Ocean Circulation. The Open
University/Pergamon Press.
Peixoto, and Oort, 1992: Physics of Climate. American Institute of Physics.
Siedler, G., J. Church, and J. Gould, 2001: Ocean Circulation and Climate.
Academic Press.
Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cambridge
University Press. (Chapters 1 and 2).
Specific references
Carissimo, B. C., A. H. Oort, and T. H. Vonder Haar, 1985: Estimating the
meridional energy transports in the atmosphere and oceans. J. Phys. Oceanogr.,
15, 82-91.
Hastenrath, S., 1982: On meridional heat transports in the world ocean. J.
Phys. Oceanogr., 12, 922-927.
Isemer, H.-J., J. Willebrand, and L Hasse, 1989: Fine adjustment of large-scale
air-sea energy flux parameterizations by direct estimates of ocean heat transport.
J. Clim., 2, 1173-1184.
Josey, S. A., E. C. Kent, and P. K. Taylor, 2000: On the wind-stress forcing of
the ocean in the SOC and Hellerman and Rosenstein climatologies. J. Phys.
Oceanogr., submitted.
Large W. G., and S. Pond, 1981: Open ocean momentum flux measurements in
moderate-to-strong winds. J. Phys. Oceanogr., 11, 324-336.
Levitus, S., 1982: Climatological Atlas of the World Ocean. NOAA Professional
Paper No. 13, U.S. Government Printing Office, Washington, D.C..
Liu and Katsaros, 2001: Air-sea fluxes from satelite data. In: Ocean Circulation
and Climate, G. Siedler, J. Chruch, and J. Gould, Eds., Academic Press, 173-180.
Ostlund, H. G., and C. G. H. Rooth, 1990: The North Atlantic tritium and
radiocarbon transients 1972-1983. J. Geophys. Res., 95, 20147-20165.
Rahmstorf, S., and A. Ganapolski, 1999: Long-term global warming scenarios
computed with an efficient coupled climate model. Climatic Change, 43, 787-805.
Reed, R. K., 1977: On estimating isolation over the ocean. J. Phys. Oceanogr.,
7, 482-485.
Richards, K. J., and W. J. Gould, 1996: Ocean weather - eddies in the sea. In:
Oceanography, An Illustrated Guide, Eds, C. P. Summerhayes and S. A. Thorpe.
Manson Publishing.
Schmitz, W. J., 1996: On the World Ocean Ciculation, Vol. II: The Pacific and
Indian Oceans/A Global Update. Woods Hole Oceanographic Institution.
Trenberth, K. E., W. G. Large, and J. G. Olson, 1990: The mean annual cycle in
global ocean wind stress. J. Phys. Oceanogr., 20, 1742-1760.
Weiss, R. E., J. C. Bullister, R. H. Gammon, and M. J. Warner, 1985:
Atmospheric chlorofluoromethanes in the deep equatorial Atlantic. Nature, 314,
608-610.
Weiss, R. E., M. J. Warner, P. K. Salameh, F. A. Van Woy, and K. G. Harrison,
1993: South Atlantic Ventilation Experiment: SIO Chlorofluorocarbon
Measurements. Scripps Institute of Oceanography.
Wijffels, S. E., 2001: Ocean transport of fresh water. In Ocean Circulation and
Climate, G. Siedler, J. Chruch, and J. Gould, Eds., Academic Press, 173-180.
Wunsch, C., 2001: Global problems and global observations. In: Ocean
Circulation and Climate, G. Siedler, J. Chruch, and J. Gould, Eds., Academic
Press, 47-58.
2. HOMOGENEOUS MODEL OF THE
WIND-DRIVEN CIRCULATION
The circulation in the different ocean basins contains many
common elements, including:
• subtropical and subpolar gyres,
• western boundary currents,
• inertial recirculation,
• separated meandering jets.
Is there a common dynamical cause of these phenomena,
independent of basin geometry?
Geostrophic streamlines at 100m with assumed level of no
motion at 1500m: (Stommel et al., 1978)
THE HOMOGENEOUS MODEL
The “classical” model of the wind-driven circulation, and
one of the great successes of GFD.
While highly idealised, many underlying ideas carry over to
more complete descriptions of the ocean circulation.
Assume:
• uniform density,
• circulation independent of depth,
• ocean of uniform depth,
• dissipation through linear friction,
• β-plane, i.e., f = f0 + βy.
The final four assumptions are stronger than strictly
necessary, but allow us to considerably simplify the
mathematics.
Equations of motion:
∂u
∂t+ u.∇u− fv +
1
ρ0
∂p
∂x=
τ (x)s
ρ0H− ru, (2.1)
∂v
∂t+ u.∇v + fu +
1
ρ0
∂p
∂y=
τ (y)s
ρ0H− rv, (2.2)
∂u
∂x+∂v
∂y= 0, (2.3)
where τs is ths surface wind stress and r is the coefficient of
linear friction.
Three equations in three unknowns: u, v and p.
We can eliminate p by forming a vorticity equation,
∂(2.2)/∂x− ∂(2.1)/∂y, to give:
∂
∂t+ u.∇
q =1
ρ0H
∂τ (y)s
∂x− ∂τ (x)
s
∂y
− r
∂v
∂x− ∂u
∂y
.
(2.4)
Here
q = f(y) +∂v
∂x− ∂u
∂y. (2.5)
is the absolute vorticity.
Equation (2.4) contains the three essential ingredients of
any ocean gyre:
• a vorticity source (wind stress curl),
• a vorticity redistribution (advection),
• a vorticity sink (friction).
Finally we can use (2.3) to define a streamfunction, ψ, such
that
u = −∂ψ∂y, v =
∂ψ
∂x. (2.6)
Substituting for u and v in (2.4) gives a single equation in
one unknown, ψ.
SVERDRUP BALANCE
First consider the ocean interior.
Estimate magnitude of relative vorticity and planetary
vorticity:∣
∣
∣
∣
∣
∂v∂x
− ∂u∂y
∣
∣
∣
∣
∣
f=
U
fL= Ro
where Ro is the Rossby number.
Typical values: U ∼ 10−2m s−1, L ∼ 106m, f ∼ 10−4s−1
⇒ Ro ∼ 10−4 � 1.
Thus q ≈ f = f0 + βy.
On the advective time-scale (T ∼ L/U ), the
time-dependent term is also small, and friction is unlikely to
important away from the boundaries.
⇒ in the ocean interior, (2.4) simplifies to:
β∂ψ
∂x=
1
ρ0H
∂τ (y)s
∂x− ∂τ (x)
s
∂y
(2.7)
“Sverdrup balance” for a homogeneous ocean.
Local balance between advection of planetary vorticity and
source of vorticity by wind-stress curl.
Boundary conditions?
Would like to set ψ = 0 on both the western and eastern
boundaries. However (2.7) is a 1st-order p.d.e. in x
⇒ can satisfy only 1 b.c. in x.
Sverdrup noted that boundary currents tend to form on the
western margins of ocean basins and applied the eastern
boundary condition.
Example from North Atlantic (Boning et al. 1991):
Predicts both subtropical and subpolar gyres.
Global calculation from Welander (1959):
WESTERN INTENSIFICATION
To close the circulation at western boundary requires
additional physics.
Following Stommel (1948), introduce linear friction ⇒
β∂ψ
∂x=
1
ρ0H
∂τ (y)s
∂x− ∂τ (x)
s
∂y
− r∇2ψ. (2.8)
Now a 2nd-order p.d.e. in x, allowing both western and
eastern boundary conditions.
Solution in a rectangular basin with uniform zonal winds:
β∂ψ
∂x∼ −r∂
2ψ
∂x2
⇒ width δ ∼ r/β
Sverdrup interior
(figure from Cushman-Roisin 1994)
Why does the boundary current form on the western, and
not the eastern, margin of the basin?
Consider sources and sinks of vorticity:
a. western boundary current:
no net source of vorticity
b. eastern boundary current:
net source of anticyclonic vorticity
NONLINEAR EFFECTS
In practice, relative vorticity is not negligible within the
western boundary current.
To include both friction and relative vorticity, it is necessary
to resort to numerical solutions (adapted from Veronis 1966;
also see Bryan 1963 for equivalent with lateral friction).
As r decreases:
• N/S asymmetry develops,
• eastward jet forms along northern edge of gyre,
• gyre transport increases — “inertial recirculation”
To obtain a physical understanding, consider sources and
sinks of vorticity acting on a parcel of fluid as it travels
around a closed streamline:
(i) vorticity source:
u.∇q ≈ 1
ρ0Hcurlτs
(ii) vorticity redistribution:
u.∇q ≈ 0
(iii) vorticity sink:
u.∇q ≈ −r∇2ψ
Nonlinear generalisation of Stommel gyre, but same
underlying principle:
over a closed gyre circuit, net sources and sinks of vorticity
must balance.
Can formalise by integrating vorticity equation [(2.4), with
∂/∂t = 0] over area enclosed by a streamline, to give:
1
ρ0H
∮
ψτs.dl− r
∮
ψu.dl = 0 (2.9)
(Niiler 1966).
What happens as r → 0?
The only way (2.9) can be satisfied is if∮
u.dl increases.
Either:
• the boundary current increases its length,
• or the velocities increase ⇒ inertial recirculation.
c.f. a bicycle rolling down a gentle hill with flat tyres (large
friction) and fully inflated tyres (weak friction)
ROLE OF TRANSIENT EDDIES
So far we have considered only one (subtropical) gyre. Now
consider a more “complete” model in which we have both a
subtropical and subpolar gyre.
Initially, let’s place an imaginary wall between the 2 gyres:
What happens if we remove the wall?
Numerical calculation (J. Marshall 1984):
Can can split the variables into mean and transient
components:
u = u + u′,
q = q + q′, ....
The time-mean vorticity equation is then
u.∇q =1
ρ0H
∂τ (y)s
∂x− ∂τ (x)
s
∂y
− r
∂v
∂x− ∂u
∂y
−∇.u′q′.
(2.10)
Finally, integrating this over the area enclosed by a
time-mean streamline, we obtain:
1
ρ0H
∮
ψ
τs.dl− r∮
ψ
u.dl−∮
ψ
u′q′.dn = 0. (2.11)
Additional “sink” of vorticity in the time-mean vorticity
equation due associated with eddy vorticity fluxes.
Vorticity budget along a time-mean streamline:
(i) vorticity source:
u.∇q ≈ 1
ρ0Hcurlτ s
(ii) vorticity redistribution:
u.∇q ≈ 0
(iii) vorticity sink:
u.∇q ≈ −∇.u′q′
SUMMARY OF MAIN POINTS
• Have developed a simple homogeneous model of
wind-driven gyres, with no vertical structure.
• Model is able to reproduce many features of the observed
circulation, including:
— subtropical and subpolar gyres,
— western boundary currents,
— inertial recirculation,
— separated jets than meander and form rings.
• One reason for the success of the homogeneous model is
that it captures the three essential ingredients of any
ocean gyre:
— a vorticity source,
— a vorticity redistribution,
— a vorticity sink.
In the next lecture, we will see that these ideas carry
over to a stratified ocean if one reinterprets q as the
potential vorticity.
REFERENCES FOR LECTURE 2
General reading
Cushman-Roisin, B., 1994: Introduction to Geophysical Fluid Dynamics.
Prentice-Hall.
Pedlosky, J. 1987: Geophysical Fluid Dynamics, Chapter 5, Springer Verlag.
Pedlosky, J., 1996: Ocean Circulation Theory. Springer-Verlag.
Specific references
Boning, C. W., R. Doscher, and H.-J. Isemer, 1991: Monthly mean wind stress
and Sverdrup transports in the North Atlantic: A comparison of the
Hellerman-Rosenstein and Isemer-Hasse climatologies. J. Phys. Oceanogr., 21,
221-235.
Bryan, K., 1963: A numerical investigation of a nonlinear model of a wind-driven
ocean. J. Atmos. Sci., 20, 594-606.
Marshall, J. C., 1984: Eddy-mean flow interaction in a barotropic ocean model.
Q. J. R. Met. Soc., 110, 573-590.
Niiler, P. P., 1966: On the theory of the wind-driven ocean circulation. Deep Sea
Res., 13, 597-606.
Stommel, H., 1948: The westward intensification of wind-driven ocean currents.
Trans. Amer. Geophys. Union, 29, 202-206.
Stommel, H., P. Niiler, and D. Anati, 1978: Dynamic topography and
recirculation of the North Atlantic. J. Mar. Res., 36, 449-468.
Sverdrup, H. U., 1947: Wind-driven currents in a baroclinic ocean: with
application to the equatorial currents of the eastern Pacific. Proc. Nat. Acad.
Sci., 33, 318-326.
Veronis, G., 1966: Wind-driven ocean circulation, Part II. Deep Sea Res., 13,
30-55.
Welander, P., 1959: On the vertically integrated mass transport in the oceans.
In The Atmosphere and Sea in Motion, B. Bolin, Ed., Rockfeller Institute Press,
75-100.
3. VERTICAL STRUCTURE OF THE
WIND-DRIVEN CIRCULATION
In the last lecture we considered a model of the wind-driven
circulation with no vertical structure. However observations
show that the strongest flows (and strongest density
variations) are concentrated in the upper few hundered
metres of the ocean.
The aim of this lecture is describe the dynamics that sets
the vertical structure of the wind-driven circulation,
specifically:
• the surface Ekman layer,
• the role of the potential vorticity field and eddy mixing,
• the ventilated thermocline.
This lecture will contain no discussion of the role of western
boundary currents. Some of the results are therefore of a
tentative nature in that they rely on an assumption that the
boundary currents merely close the circulation without
feeding back onto the structure of the gyre interior.
THE EKMAN LAYER
The direct effect of the wind stress is only felt within the
upper 30-100m of the ocean, known as the “Ekman Layer”.
Within the Ekman layer, the equations of motion are to
leading order:
−fv +∂p
∂x=
1
ρ0H
∂τ (x)
∂z, (3.1)
fu +∂p
∂y=
1
ρ0H
∂τ (y)
∂z, (3.2)
∂u
∂x+∂v
∂y+∂w
∂z= 0, (3.3)
with τ = τs at the sea surface and τ = 0 at the base of the
Ekman layer.
It is convenient to split the velocity:
u = uEk + ug, (3.4)
such that ug is the geostrophic part of the velocity, and
uEk =1
ρ0f
∂τ (y)
∂z, vEk = − 1
ρ0f
∂τ (x)
∂z, (3.5)
is the wind-driven or “Ekman” part of the velocity.
Integrating over the depth of the Ekman layer, the total
“Ekman transport” is:
UEk =τ (y)s
ρ0f, VEk = −τ
(x)s
ρ0f. (3.6)
The Ekman transport is to the right of the wind-stress in
the Northern Hemisphere and to the left of the wind-stress
in the Southern Hemisphere.
Divergent/convergent Ekman transports ⇒ “Ekman
upwelling/downwelling”, wEk, through the base of the
Ekman layer.
Integrating (3.3) over the depth of the Ekman layer:
wEk =∂
∂x
τ (y)s
ρ0f
− ∂
∂y
τ (x)s
ρ0f
. (3.7)
Generally, wEk < 0 in the subtropical ocean and wEk > 0 in
subpolar ocean — see wind-stress data from lecture 1.
Applications:
a. Coastal upwelling
Generally find equatorward winds along eastern margins of
ocean basins
⇒ off-shore Ekman transport
⇒ coastal upwelling.
SST off coast of South Africa (from Gill 1982).
Upwelling brings cold, nutrient-rich waters to the surface ⇒major fisheries found at eastern margins of basins.
b. Equatorial upwelling
Find easterly trade winds over equatorial Pacific.
Since f changes sign across the equator
⇒ VEk > 0 north of equator and VEk < 0 south of equator
⇒ equatorial upwelling.
wEk evaluated for the tropical Pacific in July
(units: 10−7m s−1; from Gill 1982).
c. Antarctic Circumpolar Current
Can interpret ACC as a huge coastal upwelling current.
Westerly winds drive and equatorward Ekman transport,
and hence upwelling in the Southern Ocean.
Upwelling of dense water ⇒ N-S density gradient
⇒ zonal geostrophic flow through thermal wind balance.
(from Rintoul et al. 2001)
SVERDRUP BALANCE
We now turn to the flow beneath the Ekman layer.
It is straightforward to show that Sverdrup balance carries
over to a stratified ocean:
β0
∫
−Hv dz =
1
ρ0
∂τ (y)s
∂x− ∂τ (x)
s
∂y
, (3.8)
provided the flow vanishes at depth. The integral here is
over the entire depth of the ocean, including the Ekman
layer.
However, more useful for this lecture is a related form of
Sverdrup balance for the depth-integrated flow beneath the
Ekman layer.
Beneath the Ekman layer, the flow is geostrophic:
u = − 1
ρ0f
∂p
∂y, v =
1
ρ0f
∂p
∂x. (3.9)
Substituting the above into the continuity equation,
∂u
∂x+∂v
∂y+∂w
∂z= 0, (3.10)
we obtain the large-scale vorticity balance:
βv = f∂w
∂z. (3.11)
Finally integrating from the sea floor (z = −H) where we
assume w is small, to the base of the Ekman layer
(z = zEk) where w = wEk, gives:
βzEk∫
−Hv dz = fwEk. (3.12)
Physically:
stretching fluid column ⇒ must increase its vorticity
⇒ the column must move poleward to increase f .
(Q: why not increase its relative vorticity?)
Sverdrup balance still tells us nothing about how the
circulation is partitioned over the fluid column.
To solve this problem in a 3-d stratified ocean is extremely
challenging
⇒ try to simplify problem by using a simpler model ...
THE LAYERED MODEL
Approximate the ocean as a series of layers (n = 1, 2, ...),
each of constant but different density, ρn.
Key dynamical results:
• layered form of Sverdrup balance:
β∑
vnhn = fwEk. (3.13)
• layered form of thermal wind balance:
un = un+1 +g′nf
k ×∇ηn, (3.14)
where g′n = g(ρn+1 − ρn)/ρ0.
• conservation of potential vorticity in absence of forcing:
un.∇f
hn= 0, (3.15)
where qn = f/hn is the potential vorticity.
RHINES AND YOUNG (1982A, B)
Will omit mathematical details (see Pedlosky 1996).
Consider a subtropical gyre, and initially assume flow is
confined to layer 1 which is directly forced by wEk:
Thermal wind balance
⇒ interface between layers 1 and 2 must deform.
⇒ potential vorticity field in layer 2 modified:
Flow in layer 2 must conserve its potential vorticity.
If we assume that boundary currents can form at the
western margins of basins, but not at the eastern margins,
then flow is only possible along q2 contours that do not
intersect the eastern boundary. ⇒ flow possible in only the
NW corner of layer 2.
Finally, we need to determine the strength of this flow.
Rhines and Young argued that eddies would homogenize q2
in this region:
Instantaneous q2 from a numerical calculation with both
subtropical and subpolar gyres and resolved eddies (Rhines
and Young 1982b).
Solution including flow in layer 2:
NB: flow in layer 2 ⇒ q3 contours deformed
⇒ flow in layer 3? .... etc
VENTILATION
The SST is not uniform, but decreases with increasing
latitude
⇒ some density layers will “outcrop” at the sea surface.
There is now the additional possibility of a fluid parcel
starting at the sea surface and being “subducted” onto a
subsurface layer. Once shielded from surface forcing, this
parcel will conserve its potential vorticity.
This is the basic idea behind the “ventilated thermocline”
model of Luyten et al. (1983), in which surface density
variations are mapped onto the vertical through via fluid
parcels advecting their potential vorticities into the interior.
There are now 3 types of potential vorticity contours:
• unblocked contours that recirculate through the western
boundary current (the “homogenised pool”);
• ventilated contours that thread down from the sea
surface (the “ventilated zone”);
• blocked contours that intersect the eastern boundary
(the “shadow zone”).
Flow is possible on the first two of these, but not in the
shadow zone.
Find that ventilated zone dominates near the surface.
Deeper down the solution resembles that of the Rhines and
Young model.
Potential vorticity in the North Atlantic:
σθ = 26.3 − 26.5
σθ = 26.5 − 27.0
(McDowell et al. 1982)
“Ventilation age” (from Tr/3He ratio):
σθ = 26.5
σθ = 26.75
(Jenkins 1988)
STOMMEL’S MIXED LAYER DEMON
Iselin (1939) first noted the
properties of the ocean interior
match those of the winter sur-
face mixed layer (not the an-
nual mean conditions).
Explained by Stommel (1979):
(Williams et al. 1995)
Over one year, a fluid parcel in the subtropical gyres moves
a distance
l ∼ 10−2m s−1.3 × 107s ∼ 300 km.
However the annual migration of the surface density
outcrops is an order of magnitude greater:
(Woods 1987)
⇒ Only fluid parcels subducted from the mixed layer in late
winter are able to escape irreversibly into the ocean interior.
Numerical calculation (Williams et al. 1995):
SUMMARY OF MAIN POINTS
• Surface winds drive Ekman transports to the right of the
wind stress in the Northern Hemisphere and to the left of
the wind-stress in the Southern Hemisphere.
• Divergence/convergence in the lateral Ekman transports
⇒ Ekman upwelling/downwelling.
• The vertical structure of the wind-driven circulation is
controlled by the geometry of the potential vorticity field.
• The properties of the ocean interior match those of the
winter mixed layer.
REFERENCES FOR LECTURE 3
General reading
Pedlosky, J., 1996: Ocean Circulation Theory, Springer-Verlag.
Specific references
Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press.
Iselin, C. O’D., 1939: The influence of vertical and lateral turbulence on the
characteristics of waters at mid-depths. Trans. Amer. Geophys. Union, 20,
414-417.
Jenkins, W. J., 1988: The use of anthropogenic tritium and helium-3 to study
subtropical gyre ventilation and circulation. Phil. Trans. R. Soc. London,
A325, 43-61.
Luyten, J. R., J. Pedlosky, and H. Stommel, 1983: The ventilated thermocline.
J. Phys. Oceanogr., 13, 292-309.
McDowell, S., P. Rhines, and T. Keffer, 1982: North Atlantic potential vorticity
and its relation to the general circulation. J. Phys. Oceanogr., 12, 1417-1436.
Rhines, P. B., and W. R. Young, 1982a: A theory of the wind-driven circulation.
I. Mid-ocean gyres. J. Mar. Res., 40 (Suppl.), 559-596.
Rhines, P. B., and W. R. Young, 1982b: Homogenization of potential vorticity in
planetary gyres. J. Fluid Mech., 122, 347-367.
Rintoul, S. R., C. W. Hughes, and D. Olbers, 2001: The Antarctic Circumpolar
Current system. In Ocean Circulation and Climate, G. Siedler, J. Chruch, and J.
Gould, Eds., Academic Press, 271-302.
Stommel, H., 1979: Determination of water mass properties of water pumped
down from the Ekman layer to the geostrophic flow below. Proc. Natl. Acad.
Sci. USA, 76, 3051-3055.
Williams, R. G., M. A. Spall, and J. C. Marshall, 1995: Does Stommel’s mixed
layer demon work? J. Phys. Oceanogr., 25, 3080-3102.
6. DYNAMICS OF THERMOHALINE
CIRCULATION VARIABILITY
This final lecture has more of the flavour of a seminar. I will
discuss some recent work in collaboration with Helen
Johnson (a PhD student at Reading and a recent GEFD
student). The work draws on, and further develops, many of
the ideas we have been discussing over the past five lectures.
We have a reasonable understanding of the dynamics of the
steady-state thermohaline circulation, given the strength
and location of the sources and sinks of each water mass.
The dynamics and thermodynamics that controls the
magnitude of these sources and sinks remains a difficult
problem and is a topic of much ongoing research.
However a separate, but no less important, problem is the
dynamics of the thermohaline circulation on centennial and
shorter time-scales — the relevant time-scales for abrupt
climate change.
PREVIOUS WORK ON THIS TOPIC
• Kawase (1987) and Cane (1989) showed that the abyssal
ocean adjusts to changes in high-latitude conditions
through the propagation of Kelvin waves and Rossby
waves.
• Yang (1999) and Huang et al. (2000) found similar
results for the upper limb of the thermohaline circulation.
• Marotzke and Klinger (2000) suggested that deep
adjustment is dominated by advection within the Deep
Western Boundary Current.
• Goodman (2001) identified fast Kelvin waves, but
adjustment occurs over decades to centuries.
ISSUES
• Over what time-scales does the ocean respond to changes
in deep-water formation at high latitudes?
• How localised is variability on different time-scales?
• Can the adjustment be described using a simple model?
• What are the implications for monitoring and modelling
abrupt climate change?
A secondary issue that we will revisit at the end of this
lecture is:
• On short time-scales, is the thermohaline circulation
“pushed” or “pulled”?
Shallow-water model:
• Dynamic upper layer (initially h = 500 m)
overlying motionless abyss
• Domain from 45◦S to 65◦N, and 50◦ wide
• Prescribed outflow at northern boundary
• Thermocline relaxed to uniform value (500 m)
at southern margin (45◦S - 35◦S)
Shallow-water equations:
∂u
∂t+ u.∇u− fv + g′
∂h
∂x= A∇2u, (6.1)
∂v
∂t+ u.∇v + fu + g′
∂h
∂y= A∇2v, (6.2)
∂h
∂t+
∂
∂x(hu) +
∂
∂y(hv) = 0. (6.3)
Solve on a C-grid at 0.25◦ resolution.
STRATEGY
1. Implusive change in deep water formation
While motivation is the possibility of a shutdown in the
thermohaline circulation, here we will consider the
opposite problem in which the northern outflow is
“turned on” at time t = 0.
(Cleaner, because ocean initially at rest.)
2. Sinusoidal forcing
Imagine a Fourier decomposition of the outflow. Allows
us to investigate the how the spectrum of thermohaline
circulation variability changes with latitude.
THEORETICAL MODEL
Assumptions:
• Kelvin waves infinitely fast;
• thermocline depth uniform on eastern boundary;
• width of western boundary current small;
• interior circulation in geostrophic balance;
• linear.
(i) Overall mass balance:
∂
∂t
∫ ∫
basinh dx dy = TS − TN . (6.4)
(ii) Interior mass budget:
Rossby wave equation: (see lecture 4)
∂h
∂t− c(y)
∂h
∂x= 0, (6.5)
where c(y) is the Rossby wave speed given in (4.6).
Integrating (6.5) over the interior of the basin
(i.e., excluding the western boundary current) gives:
∂
∂t
∫ ∫
interiorh dx dy =
yN∫
ySc(y) (he − hb) dy, (6.6)
where
hb(y) = he
t− L
c(y)
(6.7)
is the layer thickness just outside the western boundary
current.
Delay equation
Finally equating (6.4) and (6.6) gives:
he(t) =1
yN∫
ySc(y) dy
yN∫
ySc(y)he
t− L
c(y)
dy − TN + TS
.
(6.8)
Only one unknown: he(t)
From he(t), the layer thickness in the interior follows from
Rossby wave propagation:
h(x, y, t) = he
t− xe − x
c(y)
. (6.9)
Can also deduce transport as a function of latitude by
equating (6.4) and (6.6) where the integration is carried out
between a latitude y and the northern boundary, to give:
T (y, t) = TN +yN∫
yc(y)
he − he
t− L
c(y)
(6.10)
IMPLICATIONS FOR THERMOHALINE
VARIABILITY?
Equatorial buffer limits magnitude of response in the South
Atlantic.
⇒ the equator acts as a low pass filter to thermohaline
variability.
• Series of calculations with prescribed high-latitude
transport:
TN = A sin (ωt). (6.11)
Allows us to address how the spectrum of variability in the
thermohaline circulation changes with latitude.
Substituting
TN = T0eiωt
into the delay equation (6.8)
⇒ a formula for variations in magnitude and phase of
anomalies as a function of frequency and latitude:
T (ω, y) = TN
g′HfS
− yN∫
yc
(
1 − e−iωL/c)
dy
g′HfS
− yN∫
ySc
(
1 − e−iωL/c)
dy
. (6.12)
IMPLICATIONS FOR MONITORING
Model:
Able to reconstruct the full circulation from only the
eastern boundary thermocline depth and high-latitude
boundary conditions.
Ocean:
Undoubtedly more complicated (e.g., coastal upwelling,
Mediterranean outflow, variable bottom topography), but
suggests that the circulation may be strongly constrained
by relatively few observations at the basin margins.
ATMOSPHERIC RESPONSE TO A SHUTDOWN IN
THE THERMOHALINE CIRCULATION?
Model suggests a rapid response in northward heat
transport in the North Atlantic, but not in the South
Atlantic.
⇒ convergence of hear in the tropics?
In our model this is balanced (by construction) by changes
in ocean heat storage.
However modified advection of SST (in particular in the
subtropics/tropics) may alter air-sea heat fluxes, and thus
modify atmospheric circulation,
e.g., Yang (1999) finds a lagged correlation between the SST
dipole across the equator and the thickness of Labrador Sea
Water (a proxy for deep water formation):
FURTHER ISSUES
• Multiple basins — straightforward, results similar.
• Continuous stratification, variable bottom topography,
other eastern boundary processes.
• Validation using GCMs and observations.
• Other applications (e.g., to assimilation of data into
models, sea-level adjustment, ...)
SUMMARY OF KEY POINTS
• North Atlantic responds rapidly (∼ months) to changes
in deep water formation.
• South Atlantic responds more slowly (∼ decades) due to
the “equatorial buffer” mechanism.
• The equator therefore acts as a low-pass filter to
thermohaline variability.
• On short time-scales, variability is confined to the
hemispheric basin in which it is generated.
• The adjustment is essentially reproduced by a simple
dynamical model.
REFERENCES FOR LECTURE 6
General reading
Johnson, H. L., and D. P. Marshall: On the response of the Atlantic to
thermohaline variability. J. Phys. Oceanogr., in press. (Available from
http://www.met.rdg.ac.uk/∼ocean/pub/thc.html).
Specific references
Cane, M. A., 1989: A mathematical note on Kawase’s study of the deep-ocean
circulation. J. Phys. Oceanogr., 19, 548-550.
Huang, R. X., M. A. Cane, N. Naik, and P. Goodman, 2000: Global adjustment
of the thermocline in response to deep water formation. Geophys. Res. Let., 27,
759-762.
Johnson, H. L., and D. P. Marshall: Localization of abrupt change in the North
Atlantic thermohaline circulation. Geophys. Res. Let., to be submitted.
(Available shortly from http://www.met.rdg.ac.uk/∼ocean/pub/abrupt.html).
Kawase, M., 1987: Establishment of deep ocean circulation dirven by deep-water
production. J. Phys. Oceanogr., 17, 2294-2317.
Marotzke, J., and B. A. Klinger, 2000: The dynamics of equatorially asymmetric
thermohaline circulations. J. Phys. Oceanogr., 30, 955-970.
Yang, J., 1999: A linkage between decadal climate variability in the Labrador
Sea and the tropical Atlantic Ocean. Geophys. Res. Let., 26, 1023-1026.
4. ROSSBY WAVES, KELVIN WAVES
AND EL NINO
The focus of the previous lectures has been on steady-state
circulations. In this lecture we turn our attention to the
adjustment of the ocean through wave propagation.
In particular we will address the following issues:
• westward propagation of long Rossby waves,
• coastal Kelvin waves,
• equatorial Kelvin and Rossby waves.
• the role of Kelvin waves and Rossby waves in the
development of El Nino.
SHALLOW-WATER MODEL
In this lecture, we will restrict our attention to wave
motions in a single shallow-water layer.
The equations of motion are:∂u
∂t+ u
∂u
∂x+ v
∂u
∂y− fv + g′
∂h
∂x= 0, (4.1)
∂v
∂t+ u
∂v
∂x+ v
∂v
∂y+ fu + g′
∂h
∂y= 0, (4.2)
∂h
∂t+
∂
∂x(hu) +
∂
∂y(hv) = 0. (4.3)
Here g′ = g∆ρ/ρ0 is the reduced gravity.
Many of the results we will obtain generalise readily to a
continuously-stratified ocean by projecting onto a series of
vertical modes (g′ and h can then be interpreted as the
“equivalent” reduced-gravities and depths.)
WESTWARD PROPAGATION
First let us restrict our attention to the large-scale interior
of an ocean basin, where Ro � 1, and the momentum
equations can be approximated by geostrophic balance:
u = −g′
f
∂h
∂y, v =
g′
f
∂h
∂x. (4.4)
Substituting the above into the continuity equation (4.3)
gives:∂h
∂t− c(y)
∂h
∂x= 0. (4.5)
where
c(y) =βg′h
f 2
= βL2D (4.6)
is the Rossby wave speed, and
LD =
√g′h
f(4.7)
is the Rossby deformation radius.
Thus all large-scale anomalies propagate westward at the
Rossby wave speed.
[Note (4.6) is the long-wave limit (λ� LD) of the more
general Rossby wave speed; see PPH lecture 3.]
Physical mechanism: (c.f. “traditional” explanation from
atmospheric literature)
Estimated speed of westward propagating anomalies from
altimeter data (• = Pacific; ◦ = Atlantic). Solid line is the
theoretical prediction from (4.6).
(Chelton and Schlax 1986)
KELVIN WAVES
Geostrophic balance is a extremely good approximation over
much of the ocean. However consider a pressure gradient
(here a gradient in h) along a N-S coastline. This pressure
gradient cannot be balanced by a Coriolis force since there
can be no normal flow through the solid boundary.
The vanishing of u at the wall suggests the possibility of a
solution in which u = 0 everywhere.
Linearising the remaining terms in (4.1-4.3) about a state of
rest gives:
−fv + g′∂h
∂x= 0, (4.8)
∂v
∂t+ g′
∂h
∂y= 0, (4.9)
∂h
∂t+H
∂v
∂y= 0, (4.10)
where H is the mean layer thickness.
Elimination of h between (4.9) and (4.10) gives a wave
equation for the along-shore velocity:
∂2v
∂t2− g′H
∂2v
∂y2= 0. (4.11)
This admits two waves propagating at speeds c = ±√g′H :
v = A(x)F (y − ct) +B(x)G (y + ct) . (4.12)
To determine the zonal structure of the wave, eliminate h
between (4.8) and (4.9) to give:
∂2h
∂x∂t+ f
∂h
∂y= 0. (4.13)
Substituting the above solution gives:
dA
dx=
A
LD,dB
dx= − B
LD, (4.14)
and thus:
A = A0ex/LD, B = B0e
−x/LD. (4.15)
Only the wave that decays away from the boundary is
physical
⇒ the wave travels with the coast to its right in the
Northern Hemisphere, and with the coast to its left in the
Southern Hemisphere.
(from Cushman Roisin 1994)
Note: essentially internal gravity waves in direction || to
coast, but in geostrophic balance in direction ⊥ to coast.
Typical numbers:
g′ ∼ 10−2m s−2, H ∼ 400 m (typical thermocline depth)
⇒ c ∼ 2 m s−1
⇒ Kelvin waves are fast
(few months to propagate from high to low latitudes)
At mid-latitudes, f ∼ 0.7 × 10−4s−1
⇒ LD ∼ 30 km .
EQUATORIAL WAVES
The vanishing of Coriolis parameter along the equator
endows the tropics with their own special dynamics.
We start with the linearised shallow-water equations on an
equatorial β-plane (with the equator at y = 0),
∂u
∂t− βyv + g′
∂h
∂x= 0, (4.16)
∂v
∂t+ βyu + g′
∂h
∂y= 0, (4.17)
∂h
∂t+H
∂u
∂x+∂v
∂y
= 0, (4.18)
and again seek wave solutions.
Equatorial Kelvin waves
At midlatitudes, the vanishing of the Coriolis force parallel
to coastlines leads to coastal Kelvin waves.
Likewise, the vanishing of the Coriolis force along the
equator leads to an “equatorial Kelvin wave”
(actually discovered by Wallace and Kousky 1968).
The mathematics exactly mirrors the coastal problem,
except the meridional structure takes the form of a
Gaussian:
u = u0F (x− ct) e−βy2/2c, (4.19)
where c =√g′H is again the wave speed.
By analogy with the coastal problem, (4.19) suggests the
definition of the “equatorial deformation radius”:
LEq =
√
√
√
√
√
√
c
β(4.20)
Typical values:
β ∼ 2.3 × 10−11m−1s−1
g′ ∼ 0.02 m s−1, H ∼ 100 m (see data below)
⇒ c ∼ √g′H ∼ 1.4m s−1 , LEq ∼ 250km.
T (◦ C) along the equatorial Pacific (Colin et al. 1971).
b. Equatorial Rossby waves
More generally, we can seek solutions to (4.16-4.18) of the
form:
u = U (y) cos (kx− ωt), (4.21)
v = V (y) sin (kx− ωt), (4.22)
h′ = A(y) cos (kx− ωt). (4.23)
Eliminating U (y) and A(y) gives:
d2
dy2V (y) +
ω2 − β2y2
g′H− βk
ω− k2
V (y) = 0. (4.24)
Solutions take the form
V (y) = Hn
y
LEq
e−y
2/2L2Eq, (4.25)
where Hn is a “Hermite polynomial” of order n
[H0(λ) = 1, H1(λ) = 2λ, H2(λ) = 4λ2 − 2, .... ]
and the solutions must satisfy the dispersion equation:
ω2
g′H− k2 − βk
ω=
2n + 1
L2Eq
. (4.26)
Dispersion diagram (from Cushman-Roisin 1994):
For each mode (n = 0, 1, 2, ...) there are three wave
solutions:
• two high frequency inertia-gravity waves (which we will
not discuss further here);
• One low frequency Rossby wave
The n = 0 mode is a mixed Rossby/inertia-gravity wave,
and the n = −1 mode is the Kelvin wave already discussed.
Even modes (n = 0, 2, 4, ... ) are antisymmetric about the
equator, and odd modes (n = −1, 1, 3, ... ) are symmetric.
When a symmetic forcing is applied to the equatorial ocean
at low frequencies (ω � f), the dominant modes excited
are the Kelvin wave and the n = 1 Rossby wave.
At low frequencies, the latter propagates westward at a
speed,
c =
√g′H
3, (4.27)
i.e., a third of the Kelvin wave speed.
E.g., response to wind easterly anomaly (from Gill 1982):
EL NINO
“El Nino” is a climate fluctuation, centred in the tropical
Pacific, that occurs every 2-10 years.
SST:
(a) Normal conditions
(b) El Nino conditions
(Philander 1990)
SST difference: December 1982 - climatological mean
(Bigg 1990)
El Nino is intimately connected with the “Southern
Oscillation”:
(Philander 1990)
(“El Nino” + “Southern Oscillation” = ENSO)
El Nino is a coupled phenomenon
⇒ need to consider both the atmosphere and ocean.
The tropical circulation is extremely sensitive to the
distribution of SST over the tropical Pacific:
(a) Normal conditions
(b) El Nino conditions
(Philander 1992)
Suppose we introduce a warm SST anomaly over the
eastern Pacific
⇒ westerly wind anomaly over the central Pacific.
How does the ocean respond?
- Excites Rossby and Kelvin waves (see figure on 4.12)
• A downwelling Kelvin wave propagates eastward. This
wave reinforces the initial warm SST anomaly in the E.
Pacific ⇒ positive feedback.
• An upwelling Rossby wave propagates westward and is
reflected as an upwelling Kelvin wave. This wave reduces
the initial SST anomaly ⇒ negative feedback.
Propagation times ⇒ feedbacks are delayed
⇒ ocean never catches up with its current state
⇒ oscillations.
Sea surface height (proxy for thermocline depth) during the
development of the 1997 El Nino (from
TOPEX-POSEIDON; http://):
Longitude-time sections of projections of
TOPEX-POSEIDON sea-level anomlies into Kelvin (left
and right panels) and n = 1 Rossby waves (middle panel)
(from Boulanger and Menkes 1999):
DELAY-OSCILLATOR MODEL
A simple heuristic model of El Nino including the
time-delayed positive and negative feedbacks.
Many variants (following based on Tziperman et al. 1994):
dh
dt= aF
[
h(t− τ1)]
− bF[
h(t− τ2)]
(4.28)
+ feedback - feedback
where:
• h is thermocline depth in the E. Pacific,
• F [h] = 2.0 tanh
kh
2
(h > 0),
= 0.4 tanh
kh
0.4
(h < 0),
is a nonlinear function that limits the maximum and
minimum values of h (i.e., allows h to saturate),
• k represents the strength of atmosphere-ocean coupling,
• τ1 and τ2 are the two time-delays set by the
Kelvin/Rossby wave transit times,
• a and b control the growth/decay rates.
Parameters are highly tunable, but the model can produce
many realistic features, including:
• oscillations with period of ∼ 2 − 10 yrs (� τ1, τ2),
(from Wan 1996)
• amplitude dependent on basin width ⇒ no/weak
oscillation in the Atlantic.
(from Perella 1999)
SUMMARY OF MAIN POINTS:
• Large-scale anomalies propagate westward in the ocean
interior, with the Rossby propagation speed decreasing
with increasing latitude.
• The vanishing of Coriolis force parallel to coastlines and
along the equator results in (fast) coastal and equatorial
Kelvin waves.
• Equatorial Kelvin and Rossby waves play an important
role in the growth and decay of El Nino.
REFERENCES FOR LECTURE 4
General reading
Cushman-Roisin, B., 1994: Introduction to Geophysical Fluid Dynamics.
Prentice-Hall.
Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press.
Philander, S. G. H., 1990: El Nino, La Nina and the Southern Oscillation.
Academic Press.
Siedler, G., J. Church, and J. Gould, 2001: Ocean Circulation and Climate.
Academic Press.
Specific references
Bigg, G. R., 1990: El Nino and the Southern Oscillation. Weather, 45, 2-8.
Boulanger, J.-P., and C. Menkes, 1999: Long equatorial wave reflection in the
Pacific Ocean from TOPEX-POSEIDON data during the 1992-1998 period.
Climate Dyn., 15, 205-225.
Chelton, D. B., and M. G. Schlax, 1986: Global observations of oceanic Rossby
waves. Science, 272, 234-238.
Colin, C., C. Henin, P. Hisard, and C. Oudot, 1971: Le Couran de Cromwell dans
le Pacifique central en fevrier. Cahiers ORSTROM, Ser. Oceanogr., 9, 167-186.
Perella, R., 1999: The dynamics of El Nino. MSc dissertation, Department of
Meteorology, University of Reading.
Philander, S. G., 1992: El Nino. Oceanus, 33, no. 2, 56-61.
Tziperman, E., L. Stone, M. A. Cane, and H. Jarosh, 1994: El Nino Chaos:
overlapping of resonances between the seasonal cycle and the Pacific
ocean-atmosphere oscillator. Science, 264, 72-74.
Wallace, J. M., and V. E. Kousky, 1968: Observational evidence of Kelvin waves
in the tropical stratosphere. J. Atmos. Sci, 25, 900-907.
Wan, T. C., 1996: El Nino chaos in the delay oscillator. MSc dissertation,
Department of Meteorology, University of Reading.
