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Physical Oceanography:
Marine Science 451/501
Fall 2004 Lectures
John Wilkin
Oct 7: Friction, viscosity and shear stress
Oct 11: Friction and stress divergence in the equations of motion:
Oct 14: The upper ocean response to winds
Wind stress and drag coefficients
Effects of the Earth’s rotation on oceanic motion
Inertial motion
Oct 18: Ekman currents
Important concepts
Oct 25: Ekman Transport
Important concepts:
Oct 28: Ekman pumping and wind stress curl
Upwelling
Ekman pumping.
Nov 1: Geostrophic currents
Ocean density
Ocean pressure and the geostropic balance
Nov 4: Geostrophic currents from hydrography
Surface geostrophic currents from altimetry
Geostrophic currents from hydrography
Geopotential surfaces in the ocean
Light on the right
Thermal wind
Nov 8: Exam 2
Nov 11: Rossby waves and westward propagation
1½ layer approximation
Variation in f
Rossby waves
Nov 15: Rossby waves and Sverdrup circulation
Nov 18: Sverdrup’s theory of the oceanic circulation
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Orders of magnitude: a simple Sverdrup calculation for typical values
Nov 22: Vorticity
Vorticity
Equatorial currents
Nov 29: El Niño – Southern Oscillation
Oct 7: Friction, viscosity and shear stress
Governing equations: the math and physics of oceanography
So far in class you have covered the concepts of:
· conservation of mass
· conservation of scalar quantities (like temperature and salt)
o the role of advection, mixing and air-sea fluxes
The equations that govern fluid motion describe the influences of different forces that add or remove momentum to a
fluid. The equations of motion are therefore essentially a statement of Newton’s F = ma.
· conservation of momentum
o acceleration and advection – material derivatives “following the fluid”
o pressure force and pressure gradients
o Coriolis (not really a force at all, just a matter of how you look at it)
o gravity and hydrostatic pressure
The final step to a complete description of the governing equations is consideration of friction.
Frictional forces transfer momentum from the wind to the ocean surface, generating currents and waves and causing
mixing.
Friction also explains the drag on a fluid that will eventually bring it to rest if all the driving forces, such as wind, were
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removed.
Friction acts by transferring momentum from one fluid parcel to an adjacent parcel by internal stresses. These stresses
arise because a real fluid is viscous.
Friction is the ultimate sink of energy in fluid flow. Friction dissipates the kinetic energy and momentum of fluids.
If fluids had no viscosity, or were inviscid, the wind blowing over a flat sea surface would have no effect, and fluid flow
could be put into perpetual motion by the application of pressure forces and gravity.
But in real fluids, friction and stress are important.
You’ve already been introduced to the concept of molecular diffusion of heat and salt in the governing equations.
Molecular diffusivity is a property of the fluid.
Mixing and stirring on small scales typically appear to act in the same way as molecular diffusion, but with a larger
“eddy” diffusivity that parameterizes the net effect of small eddies and turbulence in a fluid that mix scalar quantities.
Eddy diffusivity is a property of the flow, not of the fluid itself.
In molecular diffusion, the fluxes of tracer across the faces of a fluid element are given by, e.g.
and it is the divergence of the flux that can cause a net gain or loss of heat in the fluid element:
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Momentum will diffuse though a fluid in much the same manner as heat and salt, only momentum is a vector and this
gives arise to stresses which are a little different to the analogous fluxes of tracers of heat and salt.
The experiments of Hagen (1839) and Poiseuille (1840) of steady flow through a long pipe showed that the discharge
(flow rate in m3/s) is proportional to the pressure difference at the ends of the pipe and the 4th power of the tube
diameter.
This result was consistent with hypotheses concerning two fluid properties that were suggested by observations:
Fluid immediately adjacent to the wall of the pipe was not moving; the so-called “no slip” property or boundary
condition.
u=0 at z=0
The shear stress, per unit mass, within fluids is proportional to the “rate of strain” of the fluid.
in a Newtonian fluid
where n is the kinematic viscosity 1 x 10-6 m2 s-1 for water.
Examples of different velocity (and hence stress) profiles.
Left: flow between two flat plates with the top plate moving a velocity U.
Right: flow with a free surface where no stress is applied, hence
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Oct 11: Friction and stress divergence in the equations of motion:
Friction and the viscosity give rise to important properties of fluid dynamics.
1. Fluid immediately adjacent to a rigid boundary does not move. This is the so-called “no slip” property or boundary
condition
u=0 at e.g. z=-h
2. Friction transmits momentum, via shear stresses, through a fluid.
Without friction, stresses applied at fluid boundaries, e.g. the sea surface or the seafloor, would not get distributed
through the water column.
The shear stress (per unit mass) within fluids is related to the “rate of strain” of the fluid,
This is rather different from a solid, where a stress is sustained by a finite displacement of the solid – an elastic
response.
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A fluid deformed by an applied stress does not snap back to its original position once the stress is removed, so to sustain
a stress the fluid has to keep moving and maintaining the rate of strain.
Stress can be thought of as a flux of momentum, and is analogous to a flux of heat or salt. It can be parameterized in a
similar manner as a Fickian diffusive flux of a scalar such as heat or something dissolved (e.g. salt).
Different fluids have different relationships between stress and rate of strain.
In a Newtonian fluid (e.g. water):
, or
where m is the dynamic viscosity and n=m/r is the kinematic viscosity.
The molecular kinematic viscosity of water is 1 x 10-6 m2 s-1.
Aside: Non-Newtonian fluids:
Shear thickening or dilatant fluids: The apparent viscosity increases with the rate of strain, e.g. wet sand, suspensions of corn
starch, silly putty: the stress increases rapidly with rate of strain – if you hit the silly putty it snaps into solid chunks
Bingham plastic: can sustain a finite “yield” stress like a solid, but then suddenly starts moving like a fluid
Shear thinning or pseudo-plastic fluids: the apparent viscosity decreases with the duration of the stress, e.g. ketchup, paint
Shear stress has units of force per area = N m-2 = Pa (Pascal), the same as pressure. Pressure is simply a “normal”
stress that pushes a fluid element, whereas shear stresses are tangential forces that try to shear or distort the shape of the
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fluid element.
We can derive an equation for the effect of stresses by considering the forces on a fluid element (just for a change).
The stress in the x-direction acting on the z face of the cube is denoted
See Box 5.4 in Knauss, page 100.
Consider the sum of stresses (forces) on opposing faces of the fluid element. If the stress at
is different from at z,
then momentum has been imparted to the fluid and we expect it to either accelerate or balance the stress divergence
through some other term.
The force on each side of the fluid element due to the shear stress is the stress times the elemental area, so we get a net
force due to the vertical gradient in
of:
Divide this by the mass of the fluid element
governing equations.
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to get the acceleration in units m s-1 so we can include it in the
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The shear stresses on a fluid element therefore add terms to the momentum equation of the form:
There are also shear stresses in the x-direction acting on the y and x faces of the element, and they give similar terms
Using the Newtonian fluid result for stress proportional to rate of strain
…
we get governing equation terms of the form:
Molecular viscosity n is constant and could be taken outside the partial derivatives, but here we write it inside to
acknowledge the fact that in turbulent fluids, like the ocean, a viscosity coefficient that varies spatially is often used to
parameterize the mixing effects of eddies.
This was the case in the derivation of governing equations for heat and salt, where eddy diffusivity coefficients Az and Ah
were introduced that had much greater magnitudes that their molecular values. Furthermore, Az << Ah because the
vertical scale of turbulent fluid motions is typically much smaller that the size of horizontal eddies.
The same is true for eddy viscosity which it can be argued should have similar magnitudes to the diffusivities because
both heat and momentum and being mixed and stirred by the same eddies and turbulence.
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Eddy viscosity is a property of the flow, not of the fluid.
The dominance of mixing and stirring of momentum over molecular viscosity is a feature of turbulent flow.
Generally speaking, there is no unambiguously ‘correct’ choice of eddy diffusivity or viscosity in any particular
physical setting, but physical arguments can be developed to argue for certain forms of parameterization of eddy mixing
in some idealized circumstances.
We can get an idea of the range of scales for which viscosity is important by comparing the order of magnitude of terms
in the equations of motion.
The nonlinear advection terms, e.g.
ratio of these two is:
are of order U2/L, while the viscous terms
are of order nU/L2. The
which is termed the Reynolds number.
This expresses the ratio of inertia forces to molecular friction stresses.
Picking a typical oceanic length scale of 1 km, and velocity of 0.1 m s-1, we get
Re = 0.1 x 103/10-6 = 108
Experimentally, we know that a transition from laminar flow (think of slowly drooling oil or honey) to smooth turbulent
flow begins at around Re of 100 to 1000, and full turbulent flow occurs for Re > 106.
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The kinematic viscosity of honey at room temperature is 7.3 x 10-5 m2 s-1, or approximately 10-4.
A 5 cm thick stream of honey pouring from a pitcher at 2 cm s-1 would have
Re = 5 x 10-2 x 10-2 / 10-4 = 10
whereas water would have Re = 1000 and be turbulent.
More reading on viscosity:
http://xtronics.com/reference/viscosity.htm
http://www.engineeringtoolbox.com/21_412.html
Acknowledging that the ocean is always turbulent at smaller space and time scales than we are principally interested in,
we can consider how these seemingly random turbulent motions can act to redistribute momentum.
This brings us to the concept of Reynolds stresses.
The Reynolds stresses arise when if we decompose the flow we are interested in into a slowly varying mean flow and a
turbulent fluctuation.
This is the so-called Reynolds decomposition.
The random turbulent fluctuations are defined such that over the time scales we are interested in they average out to
zero, i.e.
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It is implied therefore that varies slowly on timescale greater than T, and that we consider these to be the scales of
interest, and variability on shorter time scales is classed as “turbulence”.
Consider now the product of velocity components u and v.
Average
Now introduce the Reynolds decomposition into the nonlinear terms of the equations of motion:
Then average:
|
|
slowly varying flow + Reynolds stress
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If we do this for all the nonlinear terms we get:
to which we can add zero in the form of
from continuity (mass conservation) equation
to get
= slowly varying flow +
Reynolds stresses
We treat the spatial gradients of the correlations in the turbulent fluctuations as forcing terms, and take them over to the
right-hand-side of the equations of motion:
where we have included the Coriolis and pressure gradient terms for completeness.
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The correlations in the turbulent fluctuations appear as if they are stresses in the equations of motion, and we term them
the “Reynolds stresses”. We parameterize them as the product of an eddy viscosity Az and the gradient of the mean flow
e.g.
See Box 5.5 in Knauss page 103.
Notice the sign of the velocity perturbation correlation u’w’ is opposite to the sign of the stress…
Consider the case of a boundary layer where the “mean” flow in the x-direction decreases with z as we approach the
seafloor (a typical boundary layer profile).
In this situation, we expect positive fluctuations u’ to be correlated with negative fluctuations w’ because u-momentum
is being removed by bottom friction,
i.e.
which says there is a flux of u-momentum toward the seafloor in this case.
(To re-state this: fluctuations of the flow that carry water downward w’<0 tend to be associated with excess u’>0
momentum, i.e u > , whereas fluctuations of the flow that carry water upward w’>0 tend to be associated with a deficit
of momentum u< . Thus we get a negative correlation
).
Turbulence diminishes as we approach the seafloor because the presence of the bottom restricts turbulent motions, and
the mean flow itself weakens, so the magnitude of
decreases, from which it follows that
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and
This fits with our choice of parameterizing the effect of the Reynolds stresses in terms of the vertical profile of (z) and
a positive eddy viscosity coefficient.
Turbulent boundary layer over a flat plate
In the early 20th century (1915-1930) G.I. Taylor and Theodore von Karman developed an empirical “mixing length”
theory for turbulence in a simple boundary layer.
They assumed that large eddies would be more effective in mixing momentum than small eddies, and therefore Az ought
to increase with distance from the wall.
Von Karman assumed that it had the particular functional form
where
is a dimensionless constant, and the shear velocity is defined as u*=(t/r)1/2
With this assumption, the equation for mean velocity profile becomes
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This result can be verified by substituting in the assumed eddy viscosity profile:
The parameter zo is a roughness length scale related, rather vaguely, to roughness of the seafloor (it depends also on
other sources of turbulence like high-frequency waves, and varies if the bed is moveable e.g. loose sand or silt).
Theodore von Karman’s name is now associated with process of regular eddy or vortex shedding that occurs in the wake
of a solid objects. The so-called Karman Vortex Street occurs over a limited range of Reynolds number in the vicinity of
the transition from smooth turbulent flow to fully turbulent.
For an animation of the vortex street phenomenon see:
http://www.itsc.com/movies/acel.mpg
Observations of Karman vortex streets in the atmosphere:
http://www.galleryoffluidmechanics.com/vortex/selkirkb.htm
http://www.galleryoffluidmechanics.com/vortex/guadb.htm
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Oct 14: The upper ocean response to winds
Wind stress and drag coefficients
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Wind blowing over the sea surface exerts a stress on the ocean that imparts momentum. This drag force at the surface
slows the wind speed and forms a boundary layer in the atmosphere. The precise details of how the atmosphere and
ocean interact to exchange momentum is complicated by the stratification or stability of the atmospheric boundary layer,
the presence of waves, wave breaking, and a host of other processes.
For practical applications in oceanography, it’s enough for us to use an empirical formula to calculate wind stress from
wind speed.
where
U10 = wind speed at 10 m above the sea surface
= 1.22 kg m-3
CD = dimensionless drag coefficient
a typical value might be 0.0013
this gives
in units of N m-2, or Pascals (Pa).
A popular formula for a neutrally stable boundary layer is that of Large and Pond (1981), J. Phys. Oceanog., 11, 324336.
CD = 1.2 x 10-3
for
CD = 10-3 (0.49 + 0.065 U10)
Example: U10 = 8 m s-1
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for
4 < U10 < 11 m s-1
11 < U10 < 25 m s-1
= 1.2 x 10-3 * 1.22 * 64 = 0.0937 Pa
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Typical oceanic values of wind stress are around 0.1 Pa.
Because wind stress is a quadratic function of wind speed, gusty winds produce larger stresses than would steady winds
of the same average speed.
Stormy regions, such as the Southern Ocean, have particularly high mean wind stress.
In practice, we often have observations from instruments located as some height other than 10 m above the sea surface.
However, it is meteorological convention to report wind speeds as equivalent 10 m values by using the log layer theory
(Monin-Obukov theory) to adjust the observed wind speed to that which would have been observed if the anemometer
were at 10 m. This adjustment can be a source of error.
At low wind speeds there is considerable uncertainty about the correct parameterization of drag coefficient, and this is
an active research problem.
Direct estimates of drag coefficients at low wind speed vary wildly, and there appear to be more factors involved that
simply the wind speed profile through the logarithmic atmospheric boundary layer. These include:
·
·
·
·
whether the log layer assumption is valid at low wind speed
the presence of waves and swell
surfactants
meso-scale variability affecting stability of boundary layer
We have now introduced a complete description of the various forces that act on a fluid and govern its motion.
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which include the forces:
·
·
·
·
pressure gradients
gravity
stresses (viscous and turbulent)
Coriolis
The sum of these forces starts or keeps a fluid in motion by producing a net acceleration of the fluid.
The acceleration is comprised of a local time rate of change, but also changes following the fluid. So even in steady
flow, where nothing changes with time, fluid may gain or lose momentum as it flows along.
This fluid motion occurs subject to constraints on the conservation of mass, and since motion transports temperature and
salt which affect density and therefore pressure, transport processes are directly linked to the dynamics of the flow itself.
Understanding fluid motion requires an understanding of the balances of forces.
We already have some intuition for how fluid behaves, and this stems from understanding simple force balances:
· water accelerates when you pour it
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(gravity : acceleration)
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· sticky fluids pour at a steady rate
· water pistol
(gravity : friction)
(pressure : acceleration)
Geophysical fluid dynamics is the study of fluid flow on scales large enough that the fact we are on a rotating planet is
of fundamental importance.
The effects of the Coriolis force are often counter-intuitive – but this is just a matter of us developing a new intuition.
Effects of the Earth’s rotation on oceanic motion
Chapter 6 Knauss
In discussing friction we considered the Reynolds number
Re = ratio of nonlinear momentum advection (or inertia) terms to viscous terms.
The Rossby number we’ve met in previous lectures is
Ro = ratio of inertia terms (a.k.a. centrifugal acceleration) to the Coriolis acceleration:
where
U is a velocity scale
L a length
f is the Coriolis parameter 2W sin (latitude)
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W = 2p/(86164 seconds) = 7.29 x 10-4 s-1
typical f at latitude 45o is 10-4 s-1
Let’s consider the dynamics that result for a very simple force balance for a flow with Ro ~ 1.
Inertial motion
Shipwrecked: In a wide expansive ocean with no coastlines and no gradients in anything in any direction
·
·
·
·
no pressure gradient
no friction
no spatial gradients in velocity
the wind has stopped blowing, but the fluid is in motion
All that is left is the balance between inertia and Coriolis
These equations can be solved by differentiating one with respect to time, and substituting in the other to give a single
equation for v (or u).
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Applied mathematicians, engineers and musicians will recognize this as a wave equation with the simple solution
which is readily verified by substitution into the equation. It follows that
The magnitude of the velocity is
which is constant in time.
The water is always going the exact same speed. This speed is the speed the water had attained when the wind (or
whatever started it going) stopped.
This fits with our knowledge of Newton’s Laws of Motion that in the absence of any force to change the momentum, the
fluid will carry on without accelerating or decelerating.
The peculiar thing about this is that the direction keeps changing.
These velocity components describe motion in a circle of radius
The direction of motion is clockwise for f>0 (northern hemisphere) and counter-clockwise for f<0 (southern
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hemisphere).
(We also describe the direction as anti-cyclonic regardless of the sign of f for reasons that will be become apparent when
we discuss the direction of rotation of cyclones and anticyclones in the pressure field. Anti-cyclonic implies clockwise in
the northern hemisphere AND counter-clockwise in the southern hemisphere).
For example, water in motion with a speed of U = 0.5 m s-1, at latitude 42oN where f = 10-4 s-1, describes a circle of
radius
r = 0.5/10-4 = 5000 m = 5 km.
The period of the motion (once around the circle) is
which we call the “inertial” period.
Notice that it changes with latitude.
· At 60o latitude the inertial period is 13.8 hours.
· At 45o latitude the inertial period is 16.9 hours.
· At 30o latitude the inertial period is 23.9 hours.
Although we assumed no spatial gradients in the flow, which would seem to make specifying a characteristic length
scale in the Rossby number rather ill-posed, we see that a natural length scale (the radius of the inertial circle) arises in
the solution.
The Rossby number for this inertial oscillation phenomenon is:
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Ro =
= 1.
This says the inertia forces are of the same size as Coriolis, which is hardly surprising since this is precisely the simple
balance we assumed at the start.
These so-called “inertial oscillations” are often observed in the ocean in situations such as the response to the passage of
an abrupt storm. They will be apparent in current meter observations of u(t) and v(t) at a fixed point, and also show in
the trajectories of drifting buoys.
Stewart: Figure 9.1 Inertial currents in the North Pacific in October 1987 (days 275–300) measured by holey-sock drifting buoy drogued at a
depth of 15m. Positions were observed 10–12 times per day by the Argos system on NOAA polar-orbiting weather satellites and
interpolated to positions every three hours. The largest currents were generated by a storm on day 277. Note these are not individual eddies.
The entire surface is rotating. A drogue placed anywhere in the region would have the same circular motion. From van Meurs (1998).
Further reading :
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Stewart chapter 9
Knauss chapter 6
Oct 18: Ekman currents
Nansen’s qualitative arguments:
Figure 9.2 The balance of forces acting on an iceberg in a wind on a rotating Earth.
Fridtjof Nansen noticed that wind tended to blow ice at an angle of 20°-40° to the right of the wind in the Arctic, by which he meant that the
track of the iceberg was to the right of the wind looking downwind (See Figure 9.2). He later worked out the balance of forces that must
exist when wind tried to push icebergs downwind on a rotating Earth.
Nansen argued that three forces must be important:
Wind Stress, W;
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Friction F (otherwise the iceberg would move as fast as the wind);
Coriolis Force, C.
Nansen argued further that the forces must have the following attributes:
Drag must be opposite the direction of the ice's velocity;
Coriolis force must be perpendicular to the velocity;
The forces must balance for steady flow.
W+F+C=0
Nansen’s ideas led to the work of Walfrid Ekman.
The Ekman balance is another simple balance between two forces.
Steady surface wind stress, when balanced by the Coriolis force, sets up the so-called Ekman transport in the upper
ocean wind-driven mixed layer.
The depth range over which this dynamical balance can be achieved can be estimated by a simple scale analysis:
If we have only Coriolis forces balancing the frictional mixing of wind momentum input by the wind then we have the
Ekman equations:
Consider the order of magnitude of the terms:
fv is o(fV)
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Avd2u/dz2 is o(AV/de2)
where de is some vertical boundary layer scale over which the momentum from the wind is mixes into the ocean by
vertical turbulent eddies.
The ratio of these two is the Ekman number:
Ek = AV/ de 2 V/f = A/f de 2
A typical eddy viscosity would be 10-2 m2s-1 and f is about 10-4 s-1
For an Ek of O(1) we need de 2 = A/f or de =
The depth range over which the Ekman balance can apply is very limited, of order only tens of meters below the sea
surface.
The Ekman equations can be solved exactly for the case of Av = constant. The solution is a spiraling velocity pattern that
decays with depth.
For the case of a wind stress directed in the positive y-direction the solution is:
where
is termed the Ekman depth
As z becomes negative, the magnitude of the velocity decays exponentially and the direction rotates clockwise (for f>0).
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This is pattern of currents is termed the “Ekman spiral.”
Figure 9.3. Ekman current generated by a 10 m s-1 wind at 35°N (from Stewart)
At z=0
This is a surface current 45o (i.e.
) to the right of the wind.
We can verify that the surface wind stress condition is met by evaluating
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which gives the maximum surface velocity in terms of the wind stress:
For a wind stress of 0.1 Pa and an assumed Av of 10-2 m2s-1 that gave rise to a o(10 m) Ekman depth, we get
Uo ~ (0.1)(10-3 )(10-2.10-4)-1/2 = 0.1 m s-1 or 10 cm s-1
Since a stress of 0.1 Pa is produced by a roughly 10 m/s wind, this calculation suggests surface wind driven Ekman
currents are typically order(100) times smaller than the wind speed.
This result also shows that the same wind produces a different maximum surface current at different latitudes.
From the equation for the idealized Ekman spiral solution, we see that velocity pattern depends on the details of the
eddy viscosity profile.
In reality, a distinct Ekman velocity spiral is seldom observed in the ocean.
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Important concepts
· Ekman number o(1) implies the direct influence of the winds is limited to a relatively shallow depth in the ocean
· a fundamental depth scale arises
which shows how the depth of the Ekman layer scales with latitude
and magnitude of the mixing coefficient (the wind-driven currents decay roughly exponentially with this scale)
· the velocity pattern is predominantly to the right (left) of the wind in the north (south) hemisphere
Progressive vector diagram, using daily averaged currents relative to the flow at 48 m, at a subset of depths from a
moored ADCP at 37.1°N, 127.6°W in the California Current, deployed as part of the Eastern Boundary Currents
experiment. Daily averaged wind vectors are plotted at midnight UT along the 8-m relative to 48-m displacement curve.
Wind velocity scale is shown at bottom left. (From: Chereskin, T. K., 1995: Evidence for an Ekman balance in the
California Current. J. Geophys. Res., 100, 12727-12748.)
Oct 25: Ekman Transport
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The details of the Ekman spiral velocity profile as a function of depth are more of theoretical interest that practical
importance. Current profiles closely resembling the theoretical result are seldom, if ever, observed.
The details of the spiral profile depend on the assumed eddy viscosity, and Av = constant is not a particularly good
assumption. Recall that the size of turbulent eddies tends to scale with distance from the boundary so that Av is generally
proportional to z which leads to the log-layer dependence. In Ekman dynamics, the log-layer structure is modified by
Coriolis.
The fact that the surface current is to the right of the wind (f > 0) is a key result, but the magnitude of the angle will
depend on the details.
In practical applications such as oil-spill tracking and search-and-rescue, empirical values for the angle of motion with
respect to the wind direction are used based on experience and observation.
However, a robust and important result that is independent of these details is obtained if we integrate the equations over
a depth large enough to encompass the whole Ekman layer (in practice, just a few times the Ekman scale depth).
Start with the Ekman equations expressed in terms of the stresses rather than eddy viscosity
u-momentum equation:
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We need not actually integrate to –∞ because the Ekman currents decay exponentially fast.
In practice, it is sufficient to integrate from the surface to some depth z=-D, where D is a few times the Ekman depth, at
which depth
Similarly, for the v-momentum equation:
These components of the Ekman transport describe depth integrated flow (in m2s-1) (velocity times depth) that is 900 to
the right (left) of the wind stress in the northern (southern) hemisphere.
The details of the eddy viscosity profile have no influence on this result, and the calculation is very robust.
This Ekman balance between wind stress and Coriolis is established over several inertial periods, i.e. the balance is not
established instantly when the wind starts blowing.
The ocean response to suddenly imposed winds is a set of inertial oscillations. The inertial oscillations decay over a
period that is several times their natural oscillatory timescale f-1 leaving steady Ekman transports in their wake.
Important concepts:
· Ekman currents are stronger at the surface, and decay approximately exponentially over a depth scale given by
. Typical Ekman depths are of order 10 to 30 meters.
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· Regardless of the details of the eddy viscosity profile, the Ekman transports are:
which are directed perpendicular to the wind stress direction; right (left) in the northern (southern) hemisphere
· Ekman transports are fully established after several inertial periods, i.e. 1 to 2 days
Ekman dynamics is a very practical way to estimate the oceanic response to winds on time scales of a few days.
Objects floating near the surface within the Ekman layer will be transported by Ekman currents, and their drift can be
predicted with considerable skill using these simple equations.
However, Ekman transports have a far more significant impact on the entire upper ocean circulation (over much greater
depths than the Ekman layer) through a rather subtle interaction with the oceanic pressure field.
Where Ekman transports converge and diverge they generate pressure gradients that are in turn balanced by the Coriolis
force, and the resulting geostrophically balanced currents form the upper ocean pattern of gyres and western boundary
currents.
Oct 28: Ekman pumping and wind stress curl
Balance between Coriolis and the vertical friction mixing of momentum, or stress, leads to Ekman currents. Integrating
this balance vertically (in practice over the top few tens of meters) gives us the Ekman transport:
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These components of Ekman transport describe depth integrated flow (in m2s-1) (i.e. velocity times depth) that is 900 to
the right (left) of the wind stress in the northern (southern) hemisphere.
The units of m2 s-1 can be thought of as the total transport (m3 s-1) per meter perpendicular to the current.
Conveniently, the details of the eddy viscosity profile (the vertical rate of momentum mixing) are irrelevant to this
transport result.
In an infinite ocean, uniform winds would generate uniform Ekman transports and the ocean currents would be the same
everywhere.
But the ocean is not infinite, and winds are variable, so Ekman transports are not spatially uniform, which leads to
converge and divergence of the surface currents.
Upwelling
This effect is dramatic at the coast where winds parallel to the coast cannot drive Ekman transports across the coastline.
The details of the ocean circulation response in this case are complicated, but the dominant features of the transport
patterns can be deduced from mass balance concepts.
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Assuming the upwelling pattern is 2-dimensional and uniform alongshore, the Ekman transport offshore must be
balanced by water uplifted from below.
The zone of active upwelling can be seen as a band of cold water adjacent to the coast, and this has a characteristic with
determined by the Rossby radius which depends
In coastal NJ waters the scales are roughly h = 10m, density difference of 2 kg m-3, and f=10-4 s-1. g =9.81 m s-2, so
R ~ (10 x 10 x 2/1025)1/2 104 = 14 km
The vertical transport due to upwelling occurs over this horizontal distance next to the coast, so an average vertical
velocity can be estimated from mass conservation.
Mass conservation also demands that flow feed the upwelling, and this would come from offshore in the 2-dimensional
idealized case, or possibly from a divergence of the along shelf flow.
There must be an alongshelf flow because of the pressure gradient set up by horizontal density pattern (geostrophy)
which will come to later.
Ekman pumping.
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Q: Recalling what you know about the global patterns of winds, what latitudes would you expect to be characterized by
converging Ekman transports and therefore downward Ekman pumping?
A: The region between the Trades and Westerlies
The net influence of converging or diverging horizontal Ekman transports can be quantified by considering the
conservation of mass equation:
Figure 3.24 Ocean Circulation: Ekman pumping (convergence and divergence)
Recall that the Ekman transport components are related to the wind stress:
Where
the Ekman pumping velocity wE is negative, i.e. there is a convergence of Ekman transports that
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pump water downward into the ocean interior.
This downward Ekman pumping between the Trades and Westerlies generates a depressed thermocline in the center of
the subtropical gyres.
This is the case in both hemispheres, because the sign of curl wind stress and f both differ, so we < 0.
The baroclinic pressure gradients associated with this drive the large scale gyre circulations, and conservation of mass
closes the gyre circulations with intense poleward western boundary currents.
Suggested reading:
· Chapters 3.2, 3.3, 3.4 of Ocean Circulation
· Section 9.4 of Pond and Pickard
· Chapter 9 of Stewart
Nov 1: Geostrophic currents
Vertical hydrographic cross-sections of North Pacific
· North-south illustrates deepening of thermocline in center
· East-west illustrates trend of increasing thermocline depth toward west, reversed by abrupt shoaling in a narrow
western boundary current
These gradients in temperature, and hence density, imply the presence of horizontal pressure gradients. At large space
scales in the ocean, these pressure gradients are balanced by Coriolis force and associated geostrophic currents.
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Ocean density
The density of seawater depends on
(a) temperature - thermal expansion and contraction
(b) salinity - concentration of dissolved salts
(c) pressure - compressibility and thermodynamics
An empirical formula – the UNESCO equation of state – gives the density in kg m-3 as a function of temperature,
salinity and pressure.
Density is often expressed in ‘sigma’ units:
- 1000
With an instrument like a CTD, an oceanographer can observe a vertical profile of salinity and in-situ temperature as a
function of depth.
Tomczak, Matthias & J Stuart Godfrey: Regional Oceanography: an Introduction
http://gyre.umeoce.maine.edu/physicalocean/Tomczak/regoc/index.html
[Tomczak and Godfrey, fig. 2.1 - typical T, S profile]
[Tomczak and Godfrey, fig. 2.2 - CTD deployment]
Using empirical formulae, these in-situ observations are converted to potential temperature and potential density
= function (T, S, p)
= function (T or , S, p)
usually relative to p=0
usually relative to p=0
Temperature and salinity themselves can tell us a lot about ocean circulation because water masses in different parts of
the ocean have distinctive -S relationships that are very stable and persistent: they define water mass properties and we
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can actually track particular water masses as they spread away from the region where they acquired their -S properties
... namely, regions at the ocean surface of particularly strong temperature or salt (freshwater) fluxes.
See Knauss Chapter 2
A useful exercise on the information contained in vertical profiles of temperature and salinity is presented on-line by
Matt Tomczak in his Exercise 4 at:
http://gyre.umeoce.maine.edu/physicalocean/Tomczak/IntExerc/basicentry.html
Temperature and salinity observations, and hence density, represent the principal data source upon which our
knowledge of the oceanic general circulation is based.
Ocean pressure and the geostropic balance
For ocean DYNAMICS, the oceanic pressure field is of primary importance.
Within the ocean's interior away from the top and bottom Ekman layers, for horizontal distances exceeding a few tens of
kilometers, and for times exceeding a few days, horizontal pressure gradients in the ocean almost exactly balance the
Coriolis force resulting from horizontal currents.
This balance is known as the geostrophic balance.
Unfortunately, it is NOT straightforward to measure
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Why not?
Who SCUBA dives?
What do you use to tell you how deep you are?
A pressure gauge is the most practical way to tell how deep you are in the ocean, hence a dive computer and a CTD
measure pressure, not depth.
In fact oceanographers often use pressure in decibars and meters interchangeably when discussing depth below the sea
surface because 1 decibar is very close to 1 meter (within a few percent).
But if the pressure at 500 m were always 500 decibars, then
“z = -500 m” would always be zero!
at
Precise knowledge of the density, using the equation of state, is the basis for calculation of the pressure field using the
hydrostatic relation:
This is actually the vertical momentum equation, though is doesn't look much like the horizontal momentum equations.
The order of magnitude of the term
and estimate of the size of w:
U/L
U/L
can be estimated by first considering the mass conservation equation to obtain
W/H
=> W ~ UH/L
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Now compare to the gravity force:
UH/L /T
:
g
(0.1 m/s * 1000 m) / 100 km / 104 s
= 10-1 * 103 * 10-5 * 10-4 = 10-7 compared to 10
which is clearly negligible and to a very good approximation the only plausible vertical momentum balance is between
vertical pressure gradient and gravity.
Then, from knowledge of the in-situ density we can integrate vertically to get pressure:
But we have observations of r at pressure levels, not depths.
This is what we do:
Re-write the hydrostatic relation as:
where
is termed the specific volume.
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Define dynamic height D (a.k.a. or geopotential – a surface where the gravitational potential is constant: If the free
surface coincided with a geopotential surface it would have no slope and there would be no tendency for a particle to
roll along it due to gravity. If dynamic height is constant at two adjacent points in the ocean, there is no pressure
gradient. Dynamic height maps can substitute for pressure maps and chart geostrophic flow.)
An elemental change in dynamic height is
Units are energy per unit mass, J/kg, or m2/s2.
dD is the change in potential energy associated with raising 1 kg through a distance of dz
Dynamic height is referred to as the dynamic height of pressure surface p1 w.r.t. p2:
(See Knauss Chapter 2)
We can use ocean density observations to make estimates of pressure gradients and hence geostrophic ocean currents.
However, before examining the practicalities of estimating pressure gradients from observations, we re-visit the
assumptions underlying geostrophy.
Geostrophic balance between Coriolis and pressure gradients assumes all the other terms in the horizontal momentum
equation are negligible.
The assumptions underlying geostrophy are that:
· viscosity (friction)
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· and nonlinear advection
are negligible, and that
· time scales exceed several days (for Coriolis to be important)
Consider viscosity:
A rowboat weighing 100 kg will coast for maybe 10 m after the rower stops.
A super tanker moving at the speed of a rowboat may coast for kilometers.
A cubic km of water weighing 1012 kg would coast for perhaps a day before slowing to a stop.
An ocean mesoscale eddy, like a Gulf Stream Ring, is about 200 km in diameter and a few 100 m deep so contains
about x 1002 x 0.1 = 3000 km3. Typical currents in a mesoscale eddy are around 10 cm/s.
Intuition suggests the momentum of this mass of water would not be dissipated by friction for at least a few days –
certainly long enough for the Coriolis force to become important (a couple of days).
We showed earlier that for friction to be important the vertical scale must be very small. This vertical scale was the
Ekman depth, typically or order 10 meters, and within a layer of this depth vertical mixing of momentum (friction) will
accommodate all the momentum of the surface wind stress by achieving an Ekman balance (between friction and
Coriolis).
Consider the nonlinear terms
A scale analysis shows:
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V2/L
:
fV
=> V2 / LfV = V/Lf ~ 0.1/(100x103x10-4) = 10-2
which you will recall is the Rossby number.
The geostropic balance is therefore one between Coriolis and pressure force, or pressure gradient.
(The geostrophic theory will begin to fail when the Rossby number gets large due to very strong currents or small space
scales – e.g. intense eddies to very swift boundary currents. In these situations we may need to include the nonlinear
terms, even in steady flow, to get a more accurate measure of current speeds.)
The geostrophic balance is:
An ocean at rest, u=v=0, must have no pressure gradients.
So a level surface in the ocean is one along which the pressure is constant.
Pressure can be computed from the density field through the hydrostatic relation, which we showed was an excellent
approximation good to about 1 in 108.
Before we get deep into the ocean where density variations are significant, let’s first consider a simpler problem at the
sea surface.
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are constant, p is simply p =
Then the geostrophic equations give the velocity near the sea surface as:
where
is the sea surface height.
In fact, if the density were constant this pressure gradient and geostrophic velocity would persist vertically throughout
the entire water column.
But the density of sea water varies, and the density field typically organizes itself in a way to counteract, or cancel out,
the surface pressure gradient with a baroclinic pressure gradient that leads to weak geostrophic currents at depth (~2000
m).
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So the geostrophic velocity at any depth can be viewed as the combination of part due to the sea surface slope, and the
internal pressure gradients associated with horizontally varying vertical density stratification.
If r(z) were the same everywhere, the first term in the equations for u,v above would vanish.
If we could observe the sea surface height directly we could at least get the surface current without having to go
charging around in boats with a CTD.
Geostrophic currents from altimetry
Surface geostrophic currents are proportional to surface slope.
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(Stewart figure 10.2)
To use these equations we need to know the surface height with respect to a level surface. A level surface is one of
constant gravitational geopotential, i.e. moving along a geopotential does no work against gravity.
The surface slope is a quantity that can be measured by satellite altimeters if the geoid is known.
If the ocean were at rest, the sea surface would be a geopotential surface referred to as the geoid.
The ocean is constantly moving so even over a long time average the ocean surface never assumes the shape of the
geoid.
Sea surface height is variations introduced by ocean dynamics, what we refer to as the dynamic topography, are seldom
more that 1 meter.
As we see above, a 1 meter height change over 100 km would imply a rather strong geostrophic velocity of order 1 m/s.
Though you wouldn’t think it, the mean sea surface height varies by several tens of meters globally on length scales of
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100s of km because the geoid is dominated by variations in the gravitational force.
(See mean sea surface topography from TOPEX/Poseidon – note that the long time sea surface includes the geoid +
mean ocean dynamic topography).
Errors in knowing the height of the geoid are larger than the dynamics topography for features with horizontal extent
less than roughly 1600 km. This is because of limitations in our ability to map the gravitational field (though dedicated
satellite gravity missions GRACE and CHAMP are reducing these errors rapidly).
Errors are around +/- 15 cm at scales > 1600 km, but more like +/- 50 cm locally.
To measure sea surface height variations of order a few cm requires very accurate radar altimeter satellites.
Presently there are 4 radar altimeters in orbit:
· Jason-1 (and TOPEX/Poseidon) http://topex-www.jpl.nasa.gov/
· GFO http://gfo.bmpcoe.org/Gfo
· Envisat http://www.aviso.oceanobs.com
o Follow the link to the Live Access Server to experiment with plotting sea level height anomalies observed by satellite. Try
the near-real-time sea level maps, and geostrophic currents to see how the current patterns and surface heights are related.
Because the geoid is imprecisely known, altimeters are usually flown in orbits that exactly repeat their ground-track.
Jason and Topex/Poseidon both fly in an orbit that exactly repeats every 10 days (actually 9.9156 days). The Jason and
T/P ground tracks are 315 km apart at the equator.
Envisat repeats every 35 days so has much more closely spaced ground tracks.
By subtracting the sea level observed in one traverse from the long-term mean over the entire mission (10 years of data
in the case of T/P) the effect of the geoid but also the mean dynamic topography can be removed to show mesoscale
eddies and variable fronts like meanders of the Gulf Stream.
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Near real-time analyses of T/P and ERS altimeter data are accessible on the web
(Stewart Figure 10.3 shows sea level and sea level anomaly across the Gulf Stream)
Nov 4: Geostrophic currents from hydrography
Surface geostrophic currents from altimetry
Sea surface elevation gradients are balanced by geostrophic surface currents given by
Satellite altimeter observations are now precise enough to measure:
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·
·
·
·
·
·
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Changes in mean volume of the ocean (sea level rise and global warming)
Seasonal heating and cooling
Tides
Mean dynamic topography (on long length scales)
Variability in surface geostrophic currents
Variations in the topography of the equatorial current systems associated with El Nino (but geostrophy can’t be
used at the equator)
Figure 10.4 Global distribution of variance of topography from Topex/ Poseidon altimeter data from 10/3/92 to 10/6/94. The height variance
is an indicator of variability of currents. (From Center for Space Research, University of Texas).
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Figure 10.5 Global distribution of time-averaged topography of the ocean from Topex/ Poseidon altimeter data from 10/3/92 to 10/6/99
relative to the JGM-3 geoid. Geostrophic currents at the ocean surface are parallel to the contours. Compare with Figure 2.8 calculated from
hydrographic data. (From Center for Space Research, University of Texas).
From Stewart Chapter 10
The pressure gradient associated with the departure of sea surface height from a geopotential is felt throughout the water
column, and these currents are often referred to as barotropic currents.
They are often said to be the part of the flow that does not vary with depth, but strictly speaking barotropic processes are
associated with pressure surfaces that are parallel to density surfaces.
Baroclinic flows, on the other hand, are the result of pressure surfaces that are not parallel to density surfaces.
Geostrophic currents from hydrography
To use geostrophy to infer currents at depth we need to determine not only the pressure gradient due to the sloping sea
surface, but also the subsurface pressure gradients due to variable density stratification.
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Though we can measure water pressure with a pressure transducer lowered from a ship, we can’t simply use this
observation because we seldom have an independent way of measuring depth.
Even if we could measure depth independently, it would have to be a very precise measurement:
A 10 cm/s current corresponds to a pressure gradient of
Pa m-1 or 1000 Pa in 100 km
From the hydrostatic relation we know that 1000 Pa is equivalent to the pressure change due to 10 cm of water.
We would need to know the depth of the pressure gauge to accuracy much better than 10 cm to make an observation
adequate for calculating geostrophic currents, and we would still need to deal with the issue of the slope of the geoid.
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Geopotential surfaces in the ocean
In practice, what we do in oceanography is to estimate the slope of the geopotential surface at one depth compared to
another, and this tells us the relative strength of the current at the two depths.
This is a complimentary approach to that used in satellite altimetry which calculated the slope of a constant pressure
surface (p = patmosphere).
Stewart Figure 10.7: Sketch of geometry used for calculating geostrophic current from hydrography.
The steps taken are:
1.
2.
3.
4.
Calculate the differences in geopotential between two different pressure surfaces 1 and 2
Calculate the slope of the upper surface relative to the lower from observations at two locations A and B
Calculate current at the upper surface relative to the lower – this is the current shear
Integrate vertically the shear in the current assuming some knowledge of the current at a reference depth
We use a modified form of the hydrostatic equation, which for historical reasons is written:
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so that d is the change in potential energy associated with raising 1 kg through a distance of dz. Units are energy per
unit mass, J/kg, or m2/s2.
The geostrophic balance is written:
where
is the geopotential along a constant pressure surface.
Now consider how hydrographic data can be used to evaluate
At station A, the difference in geopotential between surfaces P1 and P2 is:
where the specific volume anomaly is written as the sum of two parts:
where
is the specific volume of a standard ocean of S=35, T=0 at pressure p.
The term
is the specific volume anomaly, and tables and computer programs exist for easily calculating this for any
observed hydrographic data.
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which is the sum of:
· the standard geopotential distance between the pressure surfaces
and in meters would be approximately
- this is what a SCUBA depth gauge measures
· the geopotential anomaly
- usually about 0.1% of the geopotential distance
The standard geopotential distance is the same at any horizontal location in the ocean because there is no variation in the
vertical profile of T or S, so this is not going to enter into the calculation of pressure gradients.
Consider now the geopotential anomaly between P1 and P2 at two different stations A and B:
For simplicity, assume the lower surface is a level surface i.e. the constant pressure and geopotential surfaces coincide.
The slope of the upper surface is:
slope of constant pressure surface P2
because the standard geopotential distance is the same at stations A and B.
The geostrophic velocity at the upper surface is calculated from:
similar to the way that we calculated surface velocity from altimetry from the slope of the sea surface (also a constant
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pressure surface).
Units: s.(m2/s2)/m = m/s (geopotential anomaly has units of m2/s2)
Geopotential anomaly is often referred to as dynamic height.
Oceanographers also often scale
by 1/g, and call this steric height, h, with units of meters.
Steric height measures variations in the vertical distance between two surfaces of constant pressure, and should be stated
as the steric height of surface p1 relative to po, e.g., the height of the sea surface (p1 = 0) relative to 1000 decibars
(approximately 1000 m).
The velocity v is perpendicular to the plane of the two hydrographic stations and directed into the plane the way the
figure is sketched.
Light on the right
A useful rule of thumb is that the flow is such that lighter (less dense, warmer)
water is on the right looking in the downstream direction in the northern
hemisphere – light on the right.
This only works if the level surface is below.
The notions of geopotential anomaly, steric height, and pressure are somewhat interchangeable. All can be used to
visualize the pressure gradients that give rise to geostrophic currents.
Tomczak and Godfrey, Figure 2.7 – Schematic steric height and pressure section across a cold core eddy:
http://gyre.umeoce.maine.edu/physicalocean/Tomczak/regoc/text/2steric.html
Since the weight of water above z0 must be the same at all locations, the sea surface must be lower over the region of
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higher density.
· pressure surfaces parallel the sea surface, but become flat at depth
· some density surfaces outcrop at the surface, and this feature would have a cold center (possibly visible in
satellite imagery)
The distance between the surface p=0 and reference level p(z0) is the steric height h(0,z0).
If we plot the pressure map at constant depth, z = zr, we get:
[See Tomczak and Godfrey, fig. 2.7b]
If we plot the steric height at constant pressure, p = p1, we get:
[See Tomczak and Godfrey, fig. 2.7c]
At any depth level, contours of steric height coincide with contours of pressure. Since the shape of the sea surface is so
difficult to map accurately, oceanographers instead use maps of the steric height relative to a reference level of no
motion to map the "dynamic topography" of the ocean.
This is the oceanographic equivalent of a meteorologists pressure map, and is an effective way of visualizing
geostrophic currents.
We’ve already seen in the case of satellite altimetry that we can relatively easily measure the variability of geostrophic
currents, but we are left with uncertainty in the mean circulation.
Similarly, the dynamic method used to compute geostrophic velocity of one depth relative to another leaves us uncertain
about the absolute velocity.
Q: Is it possible to find a flat surface in the ocean ... one where horizontal pressure variations vanish?
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Yes ... at depth in most ocean basins the density field is so uniform horizontally that, for example, steric height of the
1500 m relative to 2000 m varies by only a cm or so.
[Tomczak and Godfrey, fig. 2.8 - dyn hgt 1500/2000 db and 0/2000 db]
Steric height of the surface relative to 2000 m shows differences of order 0.5 m in a single basin.
The Southern Ocean is a marked exception to this – here strong geostrophic currents extend to the bottom.
Since maps of dynamic height and pressure are similar, we can sketch the pattern of geostrophic currents on a dynamic
height or steric height map.
This method relies on the assumption that there is little or no flow at, say, 2000 db. Assuming this, we can compute the
dynamic topography at other depths w.r.t. 2000 db (including deeper depths).
This gives ocean currents at any depth we select, so we get u(z), v(z).
You won’t go too far wrong in the gyre centers, but there are regions where this won’t work (notably coastal, and
boundary currents).
There are other observational clues we can use to make a more informed choice of reference level (such as O2
minimum, or breaks in tracer property distributions).
Oceanographers play fast and loose with the terms
· dynamic height/topography
· steric height
Units are your friend
· use them to check how to calculate geostrophic currents (m/s)
· f is 10-4 s-1, g is 10 m/s2
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· think about whether you have a sensible answer …
* 0.1 m/s is a moderate to brisk flow
* 1 m/s/ is hauling
* 10 m/s is ballistic
Thermal wind
The “light on the right” rule can be derived another way by reconsidering the geostrophic balance:
Multiply through by r and differentiate w.r.t. z
Recalling the hydrostatic relation, we can replace the vertical pressure gradient:
to get
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We can simplify this further by approximating the left-hand-side terms.
(the chain rule)
Consider the magnitude of the two terms in the expansion.
3 kg/m3 / 1000 m * 1 m/s : 1000 kg/m3 * 1 m/s / 1000 m
3/1000
:
1
Since varies so little (3 kg/m3) compared to its mean value, this first term is negligible, and we can take
outside the vertical derivative, leaving
(and f)
The slopes of the density field are large (100 to 300 times greater than the surface) and readily measured from data. So
these thermal wind equations give us a straightforward way to compute the velocity shear. If we have direct
observations of the velocity at some depth, such as from current meters, drifters, or by assuming a level of no motion,
we can compute flow at all other depths.
Rule of thumb: “light on the right” (in the northern hemisphere)
Thermal wind: If in the northern (southern) hemisphere isopycnals slope upward to the left (right) across a
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current when looking in the direction of flow, current speed decreases with depth; if they slope downward,
current speed increases with depth.
[Gulf Stream sections along 64oW]
Hydrographic data and the geostrophic method can be used to calculate only the currents relative to the current at
another depth. To convert these to absolute velocities, an oceanographer must measure or assume something about the
current at a chosen reference depth:
1. assume a level of no motion – useful for the deep ocean but not in shallow water such as over the continental
shelf or in a very strong boundary current such as the Gulf Stream
2. use known currents (e.g. from moored or shipboard current meters)
3. use conservation equations
4. geostrophic currents must be computed for stations that are tens of km apart, and only give the part of the flow
that is in geostrophic balance
Nov 8: Exam 2
Nov 11: Rossby waves and westward propagation
Recommended reading:
Tomczak and Godfrey “Introduction to Regional Oceanography” Chapter 3
East-west hydrographic sections in the sub-tropical Pacific showed a thermocline that deepens toward the west. The
dynamic height of the surface therefore increases toward the west.
This implies there is geostrophic flow out of the page all across that section (except for the western boundary current).
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Deepening of the thermocline moving westward is a feature of all the subtropical gyres. The dynamical reason for this is
related to the large scale planetary waves, called Rossby waves, which determine how the subtropical ocean adjusts to
variations in wind and buoyancy forcing.
The essence of the dynamics of Rossby wave propagation (Rossby waves always travel westward) can be captured in a
very simple dynamic model, one consisting of a simple 1½ layer ocean.
1½ layer approximation
To understand the mechanism that gives rise to Rossby waves it is useful to consider an idealized approximation to the
oceanic density structure known as the 1½ layer ocean.
[Tomczak and Godfrey, fig. 3.3 - 1½ layer ocean]
In such a model, the ocean is represented as a very deep layer of density
density .
capped by a much shallower layer of
The lower layer is assumed to be motionless. This is consistent with our observation that the deep ocean is typically
relatively quiescent.
The interface between the two layers is at z=H(x,y,t) and the free surface is at z=h(x,y,t).
The shape of the free surface corresponds to the steric height of the p=0 surface relative to a level of no motion in the
lower layer at znm.
As we have seen previously, znm must coincide with p=constant for there to be no flow and the weight of water between
z = h and z = znm must be constant. This gives us information on the relationship of the thermocline and sea surface
displacements.
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The pressure difference over the depth range associated with the displacement of the layer interface (the “thermocline”)
is less in the eddy center because of the less dense water. The difference in the weight of water inside compared to
outside the eddy is
Constant pressure at depth means this internal pressure difference due to displacement of thermocline is compensated by
the elevated sea level h.
Density varies by perhaps 3 kg/m3 across the thermocline, so
is around 3/1000 = 0.003.
The only way we can get this to balance is to have H and h with opposite sign, and since the factor
is so small,
we see that displacement of the thermocline H(x,y) must be much larger than displacement of the ocean surface h(x,y),
and in the opposite direction.
Sometimes oceanographers will be a bit cavalier about signs here, changing the sign of H for convenience. Then h and H always have the
same sign. In effect this is re-interpreting H as the thickness of the upper layer. I actually prefer to think of H and h this way. My advice is
don’t be dependent on getting the math right – develop intuition for what ought to happen and work from that – this will get you out of sign
troubles.
Where h slopes upward, H slopes steeply downward, and vice versa.
Rule: In most ocean regions (where the 1½ layer model is a good approximation) the thermocline slopes opposite
to the sea surface, and at an angle usually 100-300 times larger than the sea surface.
This simple rule allows a very easy check on the current direction from hydrographic measurements. You may want to
verify this by looking at hydrographic sections across currents for which we know the direction of flow by independent
measure, such as current meters or drifters.
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Return to the Pacific dynamic topography map…
Notice that the dynamic topography of the ocean sea surface has a maximum in the west of every oceanic basin.
To understand how this arises we need to examine how the mass transport of geostrophic flow is modified by the
variation of the Coriolis parameter with latitude.
So first we look at geostrophic currents and mass transport that goes along with the steric height map we have computed
from observations of temperature and salinity.
The mass transport between two stations A and B on a steric height map is
in kg s-1
This is mass x velocity x area kg/m3.m/s.m2 = kg s-1
The volume transport is simply mass transport divided by density
in m3/s
Typical values for a current such as the Gulf Stream would be of the order of millions of m3 s-1.
Recall that oceanographers use a unit called the Sverdrup; 1 Sv = 106 m3/s
Consider a map of steric height of the ocean surface relative to some depth of no motion:
[Tomczak and Godfrey, fig. 3.2 - transport and dynamic topography]
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(define local x,y coordinates on the diagram)
We can use the geostrophic relation to calculate the average velocity between stations A and B.
where h is the steric height (of the surface relative to some depth).
The mass transport per unit depth between A and B is
where H is now the average thickness of the layer (it could be the depth to the reference level znm).
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Note that the total transport does not depend on how far apart the points A and B are.
If the contours of steric height converge:
· the velocity between them increases
· the separation decreases
· the net transport remains fixed
The geostrophic current remains confined between the contours of steric height, so the height contours are effectively
streamlines of the geostrophic flow. This is why mapping steric height is such an effective way of visualizing the flow
field.
Variation in f
An important thing to notice is that the transport M depends on f, and f varies with latitude. This gives rise to waves of
very long wavelength (1000s of km) that we call Rossby waves.
Rossby waves
Equipped with this knowledge about how the pressure field, thermocline displacement, and mass transport interact
through geostrophic dynamics, we are set to consider the properties of Rossby waves.
Consider an eddy in a 1½ layer ocean.
To keep you on your toes, it’s a Southern Hemisphere anticyclonic eddy.
Is the sea level high or low?
Which way do the currents go?
What is the displacement of the thermocline?
[Tomczak and Godfrey, fig. 3.4 – Ekman convergence/divergence for eddy]
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Sketch the layer depth (or thickness, since h<<H)
Isobars are parallel to the interface displacement, so we consider two isobars, H1 and H2, which will be streamlines of
the geostrophic current.
I want to look at the mass transport between these two isobars at different latitudes y1 and y2.
The difference in steric height between the two contours is
Q: The mass transport per unit depth between A and B is?
Think of the surface geostrophic current:
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Slope of the sea surface is
, so the velocity is (don’t worry about signs)
and the transport is this average velocity, times the width of the current, times the average layer thickness, say
½(H1+H2)
where I’ve used
=
with both h and H positive (surface height and layer thickness)
Units are transport in m3/s (velocity time width of current times depth)
Check signs on the basis of our rules:
> 0, f1 < 0, get M < 0 for southward transport – GOOD.
What about at y2?
The equation for MCD at section C-D is almost the same:
·
is the same
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· the distance LCD doesn’t matter
· but f2 is different
So we get
The Coriolis parameter increases in magnitude with distance away from the equator, so
| f2 | > | f1 |
Q: What does this mean for the transport between the two isobars at CD compared to AB?
A:
|MCD| < |MAB|
because of the change in f.
Q: Where does the water go?
A:
It pushes the thermocline down, so H increases.
On the western side of the eddy, the variation of f causes convergence of flow and pushes the thermocline down.
Q: What happens on the eastern side of the anticyclone?
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A: Transport magnitudes are the same because Dh is the same
and f1 and f2 are the same … only the direction is reversed.
This causes divergence.
On the eastern side of the eddy, the variation of f causes divergence of flow and pushes the thermocline? … UP.
Q: What happens to the eddy because of this convergence and divergence?
A:
It goes west
The westward movement of such planetary eddies is known as Rossby wave propagation.
Rossby waves move thermocline displacements westward, though they don’t actually move the water.
Nov 15: Rossby waves and Sverdrup circulation
In a simple 1½ layer ocean for which geostrophic dynamics holds, the layer interface (pycnocline) displacement is
opposite in sign to the sea surface displacement and greater in magnitude by the ratio of the relative density difference in
the layers:
Contours of surface dynamic height, interface displacement and geopotential height are all parallel and represent
streamlines of the geostrophic flow.
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The transport between two points A and B can be calculated a number of ways. One approach is to consider the velocity
implied by the surface height gradient:
and multiply this by the distance between A and B and the average layer thickness to get an effective volume transport:
where the ratio of sea surface and interface displacement is used to write this as an equation in H only.
The dependence of transport on f means that on large “planetary” scales, variation in Coriolis causes convergence and
divergence to the west and east of eddies such that their pattern propagates westward.
This propagation leads to the accumulation of energy in the west of the ocean gyres and produces the intensification
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currents on the western side of the ocean basins.
We can estimate the speed that the Rossby wave moves by considering these convergence and divergence processes and
the rate at which they displace the thermocline.
First we introduce the b-plane approximation which is just a convenient way of representing the variation of f with
latitude.
Between two latitudes y1 and y2, f changes by an amount
:
i.e.
where
= 7.292 x 10-5 s-1 and
Radius of the Earth is 6371 km, so
= 2 x 7.292 x 10-5 / 6371 x 103
= 2.28 x 10-11
and the units are?
… f / length = s-1 m-1
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At 20oN
= 2.15 x 10-11 m-1s-1
At 40oN
= 1.75 x 10-11 m-1s-1
What about southern hemisphere latitudes, 20oS and 40oS?
The net volume convergence between y1 and y2 is
and for small
calculate
we can assume that f1 f2 = f2 where f is the Coriolis parameter at the average latitude (where we
).
This volume convergence must be balanced by a pycnocline that is being driven down at vertical velocity of
box ABCD.
in the
Balancing these we get:
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Check the signs here:
If the thickness of the layer H is increasing with time, the LHS is positive.
Everything on the RHS is also positive because the way I defined
changes sign and this means
was the difference right-left. On the east side,
is negative, consistent with a thinning upper layer.
Divide through by
:
Now, the ratio
is the speed c at which a line of constant H (a wave crest for example) moves eastward.
So the planetary eddy pattern moves westward at a speed of –c, with
(units are m s-1)
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[Indian Ocean Hovmueller diagram of Rossby wave crests]
http://www.soc.soton.ac.uk/JRD/SAT/Rossby/ltplotprod_largerfont.gif
Thermocline displacements have small sea surface displacements associated with them and we can observe these from
space with a satellite altimeter.
Imagine a snapshot in time of a series of wave crests and troughs across the ocean at some latitude.
A short while later, the pattern has moved westward by a fixed amount that is roughly the same at every longitude.
Plot this pattern offset in time, now consider what it looks like if we color in the picture.
In an interval
, the pattern moves west a distance
, so the slope of these lines is
which gives the wave speed.
Let’s check how well our simple theory fits these observations in the Indian Ocean.
c=
/
30o lon * 111 km * cos(25)
60 cycles * 10 days * 86400 sec
= 5.8 x 10-2 m s-1
Indian Ocean thermocline is at around 1000m depth, and the average r in the surface is 1026.6, and at depth is 1027.8,
so we get:
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= 2.1 x 10-11 * (27.8-26.6)/1027.8 * 9.81 * 1000 m / (6.16 x 10-5)2
= 6.3 x 10-2 m s-1
(close enough for such a simple estimate)
Things to notice about the Rossby wave speed:
· gets larger approaching the equator
· always positive (i.e. westward propagation in our sign convention)
This is an approximate equation for very long wavelength, long period (many months) Rossby waves, for the idealized
1½ layer ocean.
Choose some reasonable approximate values:
H = 300 m,
= 3 x 10-3
We find that
c = 1.27 m/s
at 5oS or 5oN
c = 0.08 m/s
at 20oS or 20oN
c = 0.02 m/s
at 40oS or 40oN
(=> 6 months to cross Pacific)
(=> 20 years to cross Pacific)
[Chelton and Schlax - TOPEX Rossby wave propagation across Pacific]
Hovmueller diagrams at different latitudes show different speeds.
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Pacific transit times at 4oN are only a year, compared to many years at higher latitudes.
At the equator, there is no obvious westward propagation. As we will see when we consider ENSO, there is another
class of planetary waves (Kelvin waves) with quite different features that propagate eastward along the equator.
http://www.po.gso.uri.edu/demos/
Suggested reading for next topic:
· Pond and Pickard section 9.5
· Tomczak and Godfrey chapter 4
· Stewart Chapter 11
[Video of Australian region]
Rossby waves are a general phenomenon of planetary scale motion of fluids and gases, including the atmospheres of the
other planets as well as the Earth. In the Earth’s atmosphere, planetary eddies are the atmospheric highs and lows and
play a key role in determining the weather.
Atmospheric highs and lows generally move eastward because they are carried along by the mean flow, such as the jet
stream. Relative to the mean flow of the air however, they are going westward. So the highs and lows move slower than
the jet stream around them.
Current velocities in the ocean gyres are generally much slower than the Rossby wave speed, except at high latitudes, so
oceanic Rossby wave movement is almost always westward and can be seen in the sea surface height displacements
observed by orbiting radar altimeters, and also in sea surface temperature patterns.
An exception to this is the Southern Ocean, where the Antarctic Circumpolar Current is the oceanic analogue of the
atmospheric jet stream. It is able to circumnavigate the planet without interruption, unlike the mid-latitude oceanic
gyres, and eddies and Rossby waves (which are very slow at such high latitudes) riding on the ACC do get swept
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eastward.
If the ocean were purely geostrophic, then the depressions and bulges in thermocline seen, for example, in the transPacific hydrographic sections or in horizontal maps of temperature and salinity, would all move toward the western
boundary at the Rossby wave speed. Within a few years the ocean would come to a state of horizontally uniform
stratification, and no flow.
There must be some process constantly replenishing these bulges, or eddies
Q: What is this process?
A: The winds
A rough equilibrium is established between:
· convergence of wind-driven Ekman transport (creating bulges in the thermocline) via the process of Ekman pumping
· and the westward propagation of Rossby waves
It is the winds that establish the global distribution of steric height that we observe from density patterns – a pattern
characterized by large, slow, circulating gyres, closed by intense western boundary currents.
Momentum is transferred from the winds to the ocean by friction, and we’ve already learned in class that friction is
important within a shallow boundary layer near the surface than we term the Ekman layer.
This balance of forces in this Ekman layer determines the Ekman transport:
This is a volume transport per unit width across the current. It is velocity integrated over the depth of the Ekman layer. It
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is extremely convenient that we do not need to know any details about the actual profile of velocity within the Ekman
layer to get the total transport. All we need to know is the wind stress (and f).
Here, the wind stress
is in SI units of Pascals (Pa), or kg m-1 s-2
Changes in Ekman transport can occur from changes in wind stress and changes in latitude (Coriolis).
[Tomczak and Godfrey fig. 4.1 – Illustration of Ekman transport and Ekman pumping]
· Box A: Between the Trades and the westerlies the Ekman transport is converging
· Box B is the same: the reversal in direction of the Ekman transport in the Southern Hemisphere means this is still
convergence
Q: Convergence of Ekman transport is going to go where?
A: It pumps the thermocline down
· Box C: The stronger westerlies to the north cause a divergence, which will upwell water poleward of the
maximum of the westerlies (same in the northern hemisphere)
· Box D: The Trades blow toward the west, but the change in sign of Coriolis means the Ekman transport is
opposite on either side of the equator
This causes divergence and equatorial upwelling
· Box E: Coastal upwelling
Let’s quantify this net vertical downward Ekman pumping (or upward “suction”).
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Recall: The net divergence/convergence of the Ekman transports gives the Ekman pumping velocity:
we is defined as positive upward.
Therefore, negative curl implies downward we, because negative we is the results of convergence and pumping
downward of the pycnocline.
In the absence of any other processes affecting a 1½ ocean, we would drive a changing layer thickness
Sign check: the layer thickness is increasing with time if water is being pumped downward, i.e. we< 0
Then
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[Tomczak and Godfrey – fig 4.3 map of curl(t/f)]
So now reconsider the question I posed about what maintains the bulges in the thermocline that we see propagating
westward as Rossby waves.
In the 1½ layer ocean, the local variation in the thermocline depth with time was:
e.g. On the trailing edge of the eddy
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: the layer thins as the eddy goes west.
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In the annual mean we have a steady state in the thermocline that sees it slope such that it gets deeper going from east to
west.
So in the long-term average
= 0 (steady state means not changing in time)
We need to consider that in the long term average the convergence of Ekman transport (Ekman pumping) would
perpetually drive down the thermocline, yet we know it reaches equilibrium.
If the vertical velocity of the layer interface expected from the passage of the Rossby wave (due to divergence of the
horizontal geostrophic flow that arises because of ) is held in check by continuous Ekman pumping, then instead of
we get
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[Apel fig 6.50 – Numerical evolution over time of the thermocline, under the influence of wind stress (Rossby waves
deepen thermocline over time)]
If we do this entire analysis more precisely, using continuous stratification we get a very similar result but all the basic
properties are the same.
In the more realistic case of a continuously stratified ocean the Sverdrup relation takes on the form:
where
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is the depth integrated steric height
is the generalization of
The details of the analysis aren’t important: but I think you can see the connection between the simple 1½ layer model
and the continuously stratified case, just as we saw the similarities in the layer model and the more general thermal wind
relation.
The reason to introduce the depth integrated steric height gradient is that this is a quantity we can evaluate from
observations of the oceanic density field.
We can test the Sverdrup balance by comparing maps of
from hydrography to
from winds.
[Tomczak and Godfrey: curl(tau) and steric height]
There is a maximum in P near the western boundary of each ocean basin, and the number of contours across each basin
is roughly correct.
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The poorest agreement occurs at the outflows of the western boundary currents (EAC, Agulhas, Kuroshio, Gulf Stream)
which are regions of very intense flow and vigorous eddies. These are situations where the simplifying assumptions of
the Sverdrup method do not hold well.
But the qualitative pattern of circulation is quite good.
Read Tomczak and Godfrey Chapter 4
Nov 18: Sverdrup’s theory of the oceanic circulation
We can arrive at a very robust version of the key features of the Sverdrup balance without needing to make the 1½ layer
assumption, or consider the details of the vertical stratification.
The approach is similar to the way we derived the Ekman transport relation by integrating the momentum equations
over a large enough depth to cover the entire Ekman layer (the near surface region where vertical mixing of the
momentum imparted by the wind is significant).
It turned out we didn’t need to explicitly know the Ekman layer depth, or indeed any details about the vertical profile of
the turbulent mixing coefficient (the eddy viscosity). All that mattered was that over some several tens of meters (a
depth range estimated from a simple scale analysis) it had to be that all the wind momentum was transferred to the
ocean.
Start with the steady (no time derivative) momentum equations at small Rossby number (advection terms are negligible)
with both friction and Coriolis
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Sverdrup integrated these equations from the surface to a depth at which the horizontal pressure gradient becomes zero
(i.e. our level of no motion)
Notice that if there were no pressure gradient we would just have Ekman transports – because the depth zo is (much)
deeper than the Ekman layer.
Now take
of x equation and add to
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The second term on the left-hand-side is the mass conservation equation integrated over depth from the surface to the
level of no motion. It is therefore zero. This leaves:
where
Notice that
has dimensions of:
density.velocity.depth = kg s-1 m-1
My is the mass transport in the y direction per unit distance in the x direction.
Integrated across the whole width of the ocean basin this will be the total north-south direction mass transport of the
gyre (i.e. not including the western boundary current), and it is driven by the wind stress curl.
At some latitudes
and therefore My = 0, i.e. there is no north-south transport.
= 0 lines are the natural boundaries that divide the ocean up into the subtropical and subpolar gyres.
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With wind stress curl computed from observations, we can integrate My westward from the eastern boundary and map
the streamlines of the depth-integrated flow.
[Apel fig. 6.36 – Schematic of zonal winds and gyres]
As presented here, the Sverdrup balance only describes the north-south component of flow, and doesn’t immediately say
anything about the east-west flow.
Consider the outcome of having Sverdrup transport that changes with latitude.
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This is typically the case, because the westerlies change smoothly to the Trades. (Often the maximum in wind stress curl
is close to the minimum wind speed, but this is not necessarily so).
Between the maximum of the westerlies and the maximum of the wind stress curl, there is increasing equatorward
Sverdrup transport as one goes toward the equator. This has to come from somewhere, and is fed from the west
· consider a box up against the eastern boundary
o there is more flow out the south face than in through the north
o flow must enter from the west to balance mass
· consider the next box to the west
o mass is lost out the eastern face to the eastern box
o so even more flow must enter though the west face
o so the inflow from the west builds are we move westward, implying the streamlines becomes closer
together going west.
This gives the distinctive westward distorted ellipse pattern to the circulation.
Southward of the latitude of the maximum wind stress curl, the equatorward flow is weakening. More must flow out the
west face of each box than in.
The Sverdup balance flow pattern that corresponds to the observed mean zonal (west-east) winds in the Pacific was
computed in 1950 by Walter Munk.
Stewart figure 11.6: Munk’s calculation of the Sverdrup circulation of the North Pacific calculated from wind stress curl
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The streamlines of the flow that show this distorted ellipse pattern are computed using mass conservation to evaluate the
east-west part of the transport that balances the north-south transport given by the Sverdrup relation.
Typically, the north-south component of the wind, , and its variation with longitude,
, are negligible compared
to the zonal winds. In fact, the very large x-scale compared to y-scale means that x-variations are generally negligible in
the equatorial region in almost all terms except the pressure gradient,
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= 0 and
Then the continuity equation
can be used to calculate the zonal (west-east direction) transport Mx from the Sverdrup relation:
These terms depend only on latitude, so integrating with respect to x gives:
Variations in the wind stress dominate over variations in
in this analysis.
The point here is that the zonal transport Mx is roughly linearly proportional to longitude x, recognizing that Mx=0 at the
eastern boundary.
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Key concepts of the Sverdrup solution
The Sverdrup solution was derived without needing to consider any details about how the oceanic density field arranges
itself.
We integrated momentum equations vertically over the whole water column from the surface to the level of no motion.
We kept the Coriolis, pressure gradient, and wind stress terms in the momentum equations.
The assumed dynamics is that there is a steady state geostrophic balance to the net influence of the Ekman pumping.
The general solution for the pattern of streamlines of the Sverdrup flow can be obtained by integrating the wind stress
curl westward starting from the eastern boundary.
The Sverdrup transport is the combination of geostrophic and Ekman transports together. The individual contributions
of geostrophic transport and Ekman can be in different directions.
· The direction of the Ekman flow depends on the sign of the zonal wind stress
· The direction of the total Sverdrup=Ekman+Geostrophic depends on the sign of the wind stress curl
The Sverdrup transport result still holds for a continuously stratified ocean.
What we have lost (by integrating over a large depth range) is any information about the shape of the thermocline, but
we know from the 1½ layer model that net equatorward flow would be balanced by a thermocline deepening toward the
west (to give higher dynamic height or geopotential in the west). This is consistent with thermal wind, which says the
southward flow in the subtropical gyre requires “light water on the right” so density surfaces slope downward toward
the west across the entire basin where the Sverdrup balance holds.
Only in the western boundary current does this slope of the isopycnals and isotherms reverse. In the boundary current
the Sverdrup balance doesn’t hold, but we do know from the principle of mass conservation that the gyre scale Sverdrup
transports tells us the total mass transport of the (equal and opposite) western boundary current.
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Orders of magnitude: a simple Sverdrup calculation for typical values
If we can ignore meridional (north-south direction) winds, then
= 0 and the wind stress curl is simply
Say is a maximum of 0.05 Pa in the maximum of the westerlies, and similarly -0.05 Pa in the center of the Trades, and
the meridional (north south) length scale between these latitudes is 1000 km. Then
= -0.1 Pa/1000 x 103 m = -10-7 N/m3 (or kg m-1 s-1)
The meridional transport per unit distance in the x direction is
= -10-7 / 2 x 10-11 = - 5000 kg m-1 s-1 (southward)
in kg s-1 per meter zonal (west-east) width.
We can compare this to the directly wind-driven Ekman transport:
For
of -0.05 Pa in the center of the Trades, the Ekman mass transport is simply
MEkman = -
= 0.05 / 10-4 = 500 kg m-1 s-1
(northward)
in kg s-1 per meter zonal distance. (This is the volume transport multiplied by density).
We see that the magnitude of the Sverdrup transport is 10 times greater than the Ekman transport itself. This is typical of
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the mid-latitude gyres.
Note than My is the total mass transport in the y-direction per unit distance in x, and is the sum of Ekman and
geostrophic (thermocline) components. We can break My into the separate contributions:
My = MyGeostrophic + MyEkman
In the example above, we get
MyGeostrophic = 5500 kg m-1 s-1
These transports are per unit width in the east-west direction. We can sum (integrate) across all longitudes using the
local values of My to determine the total southward transport.
In the example above, if the wind stress is uniform across an ocean basin 12,000 km wide, we would get a total
southward Sverdrup transport of
MTOT = -5000 kg m-1 s-1 x 12000 x 103 m = 60 x 109 kg s-1
or, dividing by a density of
= 1000 kg m-3
MTOT = -60 x 106 m3 s-1 = -60 Sv
In a closed basin such as the North Pacific, all this southward transport has to be balanced by northward flow
somewhere else; namely, the western boundary current (Kuroshio).
Similarly,
MEkman = 6 x 106 m3 s-1 = 6 Sv
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and geostrophic interior flow, not including the Ekman layer, is 66 Sv southward.
Now, the Pacific is a closed basin with virtually no flow out through the Bering Strait (it’s actually about 1-2 Sv
southward).
To conserve mass, the Sverdrup flow must be balanced by …?
the western boundary current (Kuroshio) with a transport of?
60 Sv northward
Now we can make an approximate heat transport estimate by looking at the temperatures in the hydrographic data.
The interior of the ocean doesn’t fluctuate all that much seasonally, and I going to propose average temperatures:
in the thermocline of TThermocline = 15oC
in the Kuroshio TKuroshio = 18oC (warmer because it is moving equatorial water northward subtropical)
In the Ekman layer, it’s important to remember that there is a strong seasonal cycle, so use a value typical of annual
mean conditions, say
in the Ekman layer TEkman = 22oC
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with units of Watts (= power = energy per second = Joules per second)
cp is Joules C-1 kg-1
so
cp is J C-1 m-3
then multiply by transport m3 s-1 and temperature oC
and we get Joules per second, or Watts, which is heat transport.
This estimate of 0.91 PetaWatts is of about the right magnitude for the annual mean oceanic heat transport across 24oN
in the Pacific.
Nov 22: Vorticity
Related reading:
· Pond and Pickard, Chapter 9
The gyre circulation depicted in Munk’s (1950) solution is plotted as a pattern of streamlines computed by integrating
the Sverdrup balance westward, starting from zero at the eastern boundary.
Streamlines are contours of the mass transport streamfunction which is a useful and intuitive way of depicting 2dimensional flows in fluid dynamics.
Define a mass transport streamfunction
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Two dimensional horizontal flow, such as geostrophically balanced currents, can be defined by a such a streamfunction.
Lines of constant (psi) are streamlines of the depth-integrated flow. Flow is parallel to streamlines, and between a pair
of streamlines the mass transport is constant. For geostrophic flow where f is constant, a surface velocity streamfunction
could be defined:
which gives
Thinking in terms of the streamfunction
(continuity) equation.
and
is particularly useful because
automatically satisfies the mass conservation
The streamfunction of the depth-integrated flow is very closely related to depth integrated steric height.
The Sverdrup balance can be written in terms of streamfunction and rather easily integrated assuming
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eastern boundary.
Once is computed along each latitude line, we can calculate the meridional component also. This will show the
distorted ellipse pattern of Munk’s solution.
The zonal flows feed into, and exit from, the western boundary current, so the western boundary current transport is
strongest at the latitude of maximum meridional transport, which occurs at the latitude of maximum wind stress curl.
This simple theory, however, does not give us any details of the solution within the western boundary current.
If we were to include horizontal friction in the model, we would find that the equation for
would become:
and AH is a horizontal friction coefficient (eddy viscosity).
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This term arises from taking
of the x momentum equation and adding to
friction term is added on the right hand side.
of y equation when a horizontal
…
Where the streamfunction is varying gradually the frictional terms wil be small.
Only in regions where the current is changing very abruptly over a small distance can this frictional term be significant.
The ratio of the friction term to the beta term is
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The region where this becomes of order unity is the western boundary current, where the large value of
zero over a short distance in x across the current.
is brought to
Vorticity
Vorticity is a characteristic of the kinematics of fluid flow which expresses the tendency for portions of the fluid to
rotate.
It is directly associated with the velocity shear – how a current varies in the direction perpendicular to the direction of
flow, u(y) and v(x).
It is the curl of the velocity.
If the current in the u-direction varies with y, then an object floating in the fluid will start to spin as it is transported
along.
Counterclockwise rotation is positive vorticity.
When measured relative to the Earth, in a rotating reference frame, it is the:
(units are s-1)
relative vorticity.
By virtue of the rotation of the Earth in space, the Coriolis effect adds to the vorticity of the fluid the
planetary vorticity which is simply
absolute vorticity is
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.
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The absolute vorticity divided by the Z, a layer of water with that vorticity, is called the
potential vorticity:
An equation for the conservation of potential vorticity can be derived by taking the curl of the momentum equations,
integrating over the fluid layer thickness D, and applying the mass conservation (continuity) equation.
In the absence of friction, potential vorticity is conserved. (Knauss chapter 5, Box 5.6).
This equation is written in terms of the total (or material) derivative following the flow, and retains the non-linearity of
the momentum equations.
You may recall that the steps (i) curl of momentum equation, (ii) vertical integral and (iii) apply continuity, were the
steps we took when deriving the Sverdrup balance.
The Sverdrup balance is a statement of conservation of potential vorticity, with the wind stress curl providing a source
or sink or vorticity that would be added to the right-hand-side of the vorticity conservation equation. The equivalence
between wind stress curl and the rate of Ekman pumping can be thought of as increasing or decreasing the layer
thickness Z and therefore altering the total potential vorticity.
In a northern hemisphere sub-tropical gyre, where the wind stress curl is negative, vorticity is being removed from the
flow. This is balanced by loss of planetary vorticity as the fluid flow south according to the Sverdrup balance.
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very small + negative = negative
In the western boundary current, flow is northward and so the fluid gains planetary vorticity due to increasing f.
large and negative + positive = negative
The vorticity equation (for streamfunction) can’t balance unless we allow a significant contribution from friction in the
boundary current.
Increasing east-west shear in accelerating north-south component of the current represents a large increase in the relative
vorticity of the flow.
i.e.
The term
is dominated by the dv/dx term.
describes the increase in vorticity input from friction in the western boundary current.
Vorticity conservation is therefore one way of explaining the reason for the existence of western boundary currents in
the form of a strongly sheared (in the x direction) meridional (v) flow.
Potential vorticity conservation arguments can also be used to explain some other fundamental properties of ocean flow.
(a) If the column thickness D decreases, as for example when in a 2-layer coastal regime the surface layer thins near the
coast due to upwelling, then z+f must decrease if it is initially positive to keep the ratio
constant. In a limited
extent coastal regime f won’t change much, so the conservation of vorticity is achieved by having
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coastally trapped jet with the current near the coast (x=0) large with respect to weaker flow offshore.
(b) In flows where the current is weak and the relative vorticity makes a small contribution to the potential vorticity, the
ratio
= constant is dominated by
= constant. Flow will tend to follow lines of constant
.
1. Zonal flows approaching a mid-ocean ridge (D decreases) will be deflected toward the equator (f
decreases).
2. Topographic gradients are large on the continental slope. The steepness of the bathymetry across the
continental shelf break tends to constrain along-shelf flows to travel parallel to the isobaths. To cross
isobaths, a process that adds or removes relative vorticity, such as bottom friction, is required.
Equatorial currents
Suggested reading:
· Ocean Circulation sections 5.1, 5.3, 5.4
· Pond and Pickard, Chapter 9
· Tomczak and Godfrey, Chapter 19
Equatorial winds and Ekman divergence/convergence
The notion of convergences and divergences of Ekman transport, and Sverdrup dynamics, can also be applied to the
equatorial regions to understand the pattern of currents there.
Right AT the equator, Ekman dynamics don’t hold and the ocean response is more like we would expect form natural
intuition. Water tends to flow directly downwind as it gains momentum in response to the applied wind stress.
Chapter 9 of Pond and Pickard gives an analysis of this Sverdrup balance in the equatorial region.
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They show that the dominant term is
which emphasizes that quite subtle features of the latitudinal variation of winds drive the major east-west current
systems.
But we can obtain the essence of this equatorial Sverdrup balance qualitatively by considering the pattern of equatorial
winds and the associated Ekman divergence and convergence, and then deduce the effect of these on sea level.
[Equatorial winds and ITCZ: Figure 2.3 in Ocean Circulation]
This equatorial convergence occurs in a zone referred to the Intertropical Convergence Zone (ITCZ)
The ITCZ shifts seasonally toward the summer hemisphere.
This is most pronounced in Indian Ocean. The northern summer is the time of the southwest monsoon, with strong
heating over the land causing low air pressure and moist warm winds approach the Indian subcontinent from the
southwest.
In the Pacific Ocean this seasonal movement is much less pronounced because there are no heating extremes over
surrounding land. Nevertheless, the imbalance in the distribution of the continents globally means that the location of
the ITCZ is biased toward the northern hemisphere.
The ITCZ is a region of low wind speeds – a quiet zone between the relentless, steady, easterly flow of the Trade winds.
These are the Doldrums.
In Moby Dick, Captain Ahab and crew of the Pequod languished for weeks in the doldrums of the Pacific Ocean before
they could continue Ahab’s obsessive quest to find the white whale.
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The doldrums do not fall on the equator, so there is a modest easterly Trade wind along the equator in the Pacific. This
has important consequences for the oceanic circulation at the equator.
North-south vs. depth schematic cross-section across the equatorial current systems showing convergences and
divergences:
[Pond and Pickard fig 9.6] [also adapted in Ocean Circulation figure 5.1(b)]
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At the equator – upwelling and westward downwind flow
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At the equator, there is no Coriolis force, and the easterly wind blows water toward the west. The accumulation of warm
equatorial water in the western Pacific is the source of the West Pacific Warm Pool.
Off equator currents
Just off the equator, easterly flow generates Ekman currents in the two hemispheres that diverge, causing upwelling and
locally decreased sea level.
Low sea level at the equator leads to geostrophic westward currents off the equator, i.e. flowing in the same direction as
the directly wind driven surface at the equator.
In the center of the doldrums the wind stress is a minimum, so
with longitude, then
= 0 means
.
= 0 and if we ignore the variation of wind stress
The meridional transport is therefore zero, and we expect this latitude to form a natural boundary between current
systems where north-south
What are the zonal currents doing?
Think first of the Ekman transports:
South of the equator:
· sea level is lowered at the equator due to the Ekman divergence and the upwelling of cold water
· from south to north the sea level falls, which in the southern hemisphere directs geostrophic flow to the left and
westward along the equator
· this is the South Equatorial Current (SEC)
North of the equator but south of the doldrums:
· Northward Ekman transport is greater to the south, so there is a convergence that locally elevates the sea level
· The convergence is greatest where the winds are changing most rapidly.
· South of the locally high sea level, the geostrophic flow is westward, and this is still considered part of the SEC
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So ON the equator there is westward flow that is the SOUTH Equatorial Current
North of the locally high sea level, flow is eastward.
· This is the North Equatorial Countercurrent (NECC)
· It is termed a countercurrent because it flows directly opposite the local winds, which are the easterly Trades.
North of the doldrums:
· the northeast Trades build in strength going northward
· the Ekman transport again diverges, causing locally lowered sea level
· North of this divergence the sea level rises heading toward the gyre center, so we again have westward flow
· this is the North Equatorial Current (NEC).
At the latitude where the wind stress curl given by
does not change with latitude, i.e.
the zonal flow is zero because
and this is the latitude defining the boundary between westward NEC and eastward NECC.
Remember this pattern of sea level because we will see it shortly in satellite observations of equatorial sea level from the
TOPEX altimeter.
The pattern of equatorial currents has a dramatic influence on biological productivity. Notably in the east Pacific, where
the thermocline is shallow, the divergence of equatorial Ekman transport causes upwelling that brings deep nutrients to
the surface.
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Upwelled water is rich in nutrients but not plankton.
[Mann and Lazier fig 3.06 equatorial distribution of biological zones]
Equatorial biological oceanography
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Ocean primary productivity is kicked off by the new nutrient source, causing a local maximum in phytoplankton
biomass close to the equator (0.5 – 1.0o latitude)
Herbivorous zooplankton increase with the available food supply, and subsequently carnivorous zooplankton increase.
Predatory fish tend to congregate near the convergence at the boundary between the SEC and NECC, because prey
species in the plankton accumulate there.
Zonal patterns
While all this is going on north-south, what east-west structure is brought about by the east-west currents?
Right at the equator we forget about the whole Coriolis business.
Water flows downwind just like in ought to.
Steady easterly winds blow water westward, and these waters continually warm the whole way. This leads to an
accumulation of very warm water in the western Pacific:
The West Pacific Warm Pool is a huge mass of very warm water much of it greater than 28oC.
The accumulated warm pool depresses the thermocline in the west, and accordingly sea level is elevated in the WPWP
compared to the east Pacific.
[Ocean Circulation figure 5.4]
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So there is a pressure force directed from west to east. This is obviously not sufficient to overcome the westward flow of
warm water at the surface (otherwise we’d never get the Warm Pool).
But below the upper ocean mixed layer, where the direct influence of the winds is lost, the pressure gradient remains
and a west to east undercurrent forms at the bottom of the mixed layer.
This Equatorial Undercurrent (EUC) is a remarkable flow. It is effectively a ribbon of intense flow with speeds
approaching 1.5 m s-1 only 200 m deep yet 300 km wide. It is deepest in the west and gradually shoals eastward along
with the thermocline itself.
[Ocean Circulation figure 5.5]
A down pressure gradient flow like this can only occur at the equator because otherwise geostrophic balance would
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require the current to flow perpendicular to the pressure gradient.
However, Coriolis still plays a role in stabilizing the EUC.
Any strong current is prone to instabilities that cause it to meander. We’ve seen this in the meandering path of the Gulf
Stream and its tendency to spawn cyclonic and anticyclonic eddies.
If the EUC strays from its zonal path, say into the Southern hemisphere, the Coriolis force starts to come into play
deflecting the current left and back toward the equator. Similarly, northward meanders are deflected southward, and the
EUC is trapped.
Isotherms and other property distributions are deflected around the core of the EUC.
The waters of the EUC originate from the southern hemisphere, at the western boundary, from the New Guinea Coastal
Current, though there is some exchange with surrounding waters during the course of the EUC transit across the basin.
The NECC is supplied primarily by northern hemisphere waters, also originating in the west from the Mindanao Eddy.
[Mann and Lazier – fig. 9.01 zonal cross section of equatorial thermocline]
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Nov 29: El Niño – Southern Oscillation
Suggested reading:
· Ocean Circulation sections 5.1, 5.3, 5.4
· Pond and Pickard, Chapter 9
· Tomczak and Godfrey, Chapter 19
· Stewart, chapter 14
· El Nino Theme page
http://www.pmel.noaa.gov/tao/elnino/nino-home.html
[Godfrey and Wilkin, JPO
western equatorial current systems
Tomczak&Godfrey figs 8.7 and 8.8]
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The West Pacific Warm Pool (WPWP):
· drives strong atmospheric convection in the west
· the rising moist air causes heavy precipitation over the islands of the western Pacific
This air returns eastward aloft in a circulation named for Sir Gilbert Walker, who in 1904 was appointed Director
General of Observatories in the British colonial service in India. Walker compiled extensive observations throughout the
region and identified and named the Southern Oscillation which we now recognize as a pattern of interannual variability
in the tropical ocean and atmosphere that causes El Niño. The subsiding air of the Walker circulation occurs in the
eastern Pacific, and feeds into the Trade winds.
The eastward traveling air aloft subsides in the east, and this so-called Walker cell in the atmosphere maintains the east
to west pressure gradient along the equator that drives the Trades … (that produce the WPWP, that drive convection,
that sustains the Walker cell, that…)
[Summary plot from web of the Equatorial Atmosphere-Ocean Circulation]
A tight coupling of atmosphere and ocean dynamics underlies the mean equatorial patterns of winds, currents, sea
temperatures and precipitation.
Seasonal cycle of the Trades
The coupling of winds and SST is apparent if we consider the seasonal cycle at the equator.
[Monthly Equatorial Pacific SST and zonal wind anomalies 1999-present]
Weaker southeast Trades in the austral late summer cause February-April to be the equatorial warm season.
· upwelling is weaker and sea surface temperature warms
· upwelling is strongest at the end of the austral winter in Sept-Oct when the southeast Trades blow at a steady 6 m
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s-1
Some years, this seasonal cycle seems to get amplified and the warming of the eastern Pacific early in the year becomes
dramatic and persistent:
[Equatorial SST and anomalies 1986-present]
This is an El Niño, or an ENSO warm event.
ENSO
What has become apparent through research over the past decade is that the phenomenon we call ENSO is an instability
of the tightly coupled tropical atmosphere-ocean system.
The term ENSO is a combination of
· El Niño: the episodic warming of waters along the Peruvian coast
· Southern Oscillation: pattern of sea level pressure variability that is coherent over the Pacific and much of the
Indian Ocean region
SOI
The Southern Oscillation index (SOI) is classically defined as the anomaly in the difference in sea level pressure
between Tahiti and Darwin.
· When SOI is negative, the air pressure gradient along the equator is less than usual, and we would expect the
Trade winds to be weaker than usual
· Conversely, positive SOI would indicate stronger Trades.
But there is more to ENSO than just variability in the equatorial winds. It has become clear that in an ENSO event the
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entire Pacific air-sea system …
·
·
·
·
rain bands
associated winds
wind-driven currents
and SST patterns
… all move eastward together.
It was only in the late 1960s that it was recognized that these two processes (El Niño, and SO) were linked.
Now we know that just about any upset to a component of the coupled ocean-atmosphere circulation in the tropics will
cause a feedback that links all the components. (Chicken and Egg)
Understanding an ENSO event begins with understanding the evolution of the SST field.
Many factors influence SST
· Change in wind speed
1. evaporative (latent) and sensible heat loss
2. Ekman transports
§ advect heat laterally
§ produce Ekman pumping which changes deeper density field and affects temperature of water
available for upwelling
§ alters geostrophic flow
3. alter the depth of directly wind mixed layer
· SST itself alters cloud cover, and incoming solar radiation
This plethora of processes affecting SST has made it difficult to understand the details of the ENSO cycle, and it is not
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clear that there is one single mechanism that is the trigger for an ENSO warm event
Westerly wind bursts
However, there is general agreement, that a necessary ingredient in the commencement of an ENSO event is a reversal
of the Trade wind pattern in the western Pacific.
Winds are typically light in the west Pacific, but occasionally an outbreak of westerly winds occur that persist for
perhaps a week or more, coherent over many 1000s km (sometimes from Indonesia to the Dateline).
[Tomczak and Godfrey fig. 19.8 tropical cyclone pair from severe westerly wind burst]
A wind burst such as this sets in train wave motions that are characteristic of the equatorial region.
[Sketch schematic of Kelvin wave dynamics]
The westerly wind burst causes:
1.
2.
3.
4.
5.
converging Ekman transports (off equator) that increase sea level
and depress the thermocline
eastward geostrophic flow converges to the east
and diverges to the west
the pattern moves eastward
Note that the same happens for an easterly wind burst: the equatorial Kelvin waves only go east so the anomalous
pattern cannot easily reset itself even if the WWB is followed by easterly wind anomalies.
6. the equatorial Kelvin wave speed
is about 2.5 m/s
The observed speed is about 10 – 20% faster than this due to advection by the EUC
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The cyclonic wind patterns off equator cause local doming of the thermocline - wind stress curl generates a pair of
Rossby waves.
[Tomczak and Godfrey fig. 19.9 wave propagation during ENSO event]
At 5-7o latitude the simple 1½ layer model we used to consider Rossby wave propagation speed doesn’t work all that
well, and the propagation speed is (for H=150 m and
= 0.004) is about 0.3 m/s.
These Rossby waves take a few months to reach the western boundary. At the west, they reflect as Kelvin waves but
with a thermocline elevation (that partly resets the thermocline deepening of the original Kelvin wave).
Meantime, the Kelvin wave has reached the eastern boundary of the Pacific.
It lowers the thermocline …
… raises sea level
… and warms SST
At the coast, it generates elevated sea level (and depressed thermocline) that propagates poleward on both coasts.
These in turn radiate Rossby waves westward back across the basin.
As the Equatorial Kelvin wave propagates eastward it takes thermocline waters with it, expanding the West Pacific
Warm Pool toward the east.
In doing so, it moves the region of warm SST and strong atmospheric convection eastward.
This movement in the convection effectively short-circuits the Trades. The local winds become westerly (or at least their
anomaly from the mean does) so that the Kelvin wave generation mechanism continues.
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By generating further westerly wind anomalies as it propagates, the Kelvin wave generation mechanism remains active
further fostering downwelling of the thermocline
· more warm water is drawn eastward
· this further translates the convection center
· and further weakens the Trades
This positive feedback mechanism is the essence of the ENSO phenomenon.
The deepening of the thermocline in the east Pacific propagates along the Peruvian coast, so that the water that is
upwelled (along the equator, and the coast) due to divergent Ekman flow, is no longer the cooler nutrient rich water
below the thermocline, but warmer already nutrient depleted near surface water.
This leads to a collapse of the primary production that sustains the equatorial and coastal fisheries, and is the reason that
ENSO warm events are such a catastrophe in this region.
In the west Pacific, the loss of thermocline waters causes a shrinking and shallowing of the West Pacific Warm Pool.
Eastward displacement of the major convection center causes negative rainfall anomalies in Australia and Indonesia.
Drought has severe negative impacts on the economy and biosphere in those regions.
The ENSO process actually seems to be more sensitive to the modest SST anomalies in the west Pacific, than to the
dramatic SST anomalies in the east.
The reasons for this aren’t all that clear, but efforts to improve predictability of ENSO focus more on understanding the
coupled west Pacific atmosphere-ocean system, and especially the trigger mechanism related to the WWBs.
[Video: Evolution of the 1997-1998 El Nino: A view from space (animations are also available on the web at the NOAA
PMEL El Nino Theme Page]
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Observational networks in the equatorial ocean
Important advances in ENSO predictability have been achieved through the establishment of an observational network
in the Pacific (and Atlantic):
The TAO array gives real-time data on:
·
·
·
·
subsurface ocean temperature
velocity
surface winds
air temperature and humidity
[Schematic summary of Normal and ENSO coupled atmosphere-ocean]
Cold episodes: La Nina
The 1997 El Nino was followed by a dramatic cold episode, or La Nina.
However, a quick look at the SOI shows that warm episodes do not always transition to cold episodes. Often the SOI
stays near zero and then returns to a negative anomaly.
Researchers haven’t actually been able to determine what causes the transition to La Nina following El Nino, or not.
To reset the ocean back to “normal,” or more correctly “non-ENSO,” conditions, requires transport of heat to recharge
the Warm Pool.
Planetary wave dynamics (Kelvin and Rossby) mean that temperatures can’t be reset just by running everything in
reverse. The delay in resetting the SST patterns that allows anomalous patterns to persist for many seasons, and even
years, is probably related to the slow westward speed of the Rossby waves.
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The strong dependence of Rossby wave speed on latitude only serves to complicate this resetting process.
What is clear is that significant global scale changes in the ocean-atmosphere heat budget result from ENSO.
During El Nino, heat is transported to higher latitudes:
· the warming of the eastern equatorial Pacific reduces the air-sea heat exchange there that usually draws a lot of
heat out of the atmosphere
· east Pacific positive SST anomalies propagate poleward along the coast
· additional heat goes to the atmosphere through evaporation
· global average air temperatures rise by as much as 0.3oC in the months after a strong El Nino
After La Nina
· increased solar input occurs due to lower than normal cloud cover in the West Pacific Warm Pool
The tropical Pacific loses heat during El Nino, and gains it during La Nina.
Read the current ENSO climate advisory from NOAA at:
http://www.cpc.ncep.noaa.gov/products/analysis_monitoring/enso_advisory/index.html
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