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 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:
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 
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.
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 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 
to get the
acceleration in units m s-1 so we can include it in
the governing equations.
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.
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. 
are of order U2/L,
while the viscous terms 
are of order nU/L2. The ratio of these two is:
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.
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.
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
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.
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
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
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
Oct 14: The upper ocean response to winds
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, 324-336.
CD
= 1.2 x 10-3 for 4
< U10 < 11 m s-1
CD
= 10-3 (0.49 + 0.065 U10) for 11
< U10 < 25 m s-1
Example: U10 = 8 m s-1 
= 1.2 x 10-3
* 1.22 * 64 = 0.0937 Pa
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.
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 (gravity : acceleration)
·
sticky fluids pour at a steady rate (gravity
: friction)
·
water pistol (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.
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)
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.
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).
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 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:
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 :
Stewart
chapter 9
Knauss chapter 6
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;
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)
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).
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
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.
·
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.)
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:
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.
·
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.
·
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.
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:
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.
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.
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.
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 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
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.
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) usually
relative to p=0
= function (T or , S, p) 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 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.
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 
at any chosen depth
(z) directly.
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 at
“z = -500 m” would always be zero!
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 
can be estimated by
first considering the mass conservation equation to obtain and estimate of the
size of w:
U/L U/L W/H
=> W ~ UH/L
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.
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)
·
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:
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.
If g and 
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).
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.
(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 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.
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)
Sea surface
elevation gradients are balanced by geostrophic surface currents given by
Satellite altimeter
observations are now precise enough to measure:
·
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).
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.
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.
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.
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. Calculate the
differences in geopotential 
between two different pressure surfaces 1 and 2
2. Calculate the slope
of the upper surface relative to the lower from observations at two locations A
and B
3. Calculate current
at the upper surface relative to the lower – this is the current shear
4. 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:
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.
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 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.
|
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 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?
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
·
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
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
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 (and f) outside the vertical derivative,
leaving
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 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 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).
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 1˝ layer
ocean.
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 1˝ layer ocean.
[Tomczak and
Godfrey, fig. 3.3 - 1˝ layer ocean]
In such a model,
the ocean is represented as a very deep layer of density 
capped by a much
shallower layer of density
.
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.
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 1˝ 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.
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]
(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).
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.
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.
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 1˝ 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]
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:
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 
= ˝(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
·
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?
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.
In a simple 1˝
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.
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 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
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 
we can assume that f1 f2 = f2 where f is the Coriolis parameter at the average
latitude (where we calculate ).
This volume convergence must be balanced by a pycnocline
that is being driven down at vertical velocity of 
in the box ABCD.
Balancing these we
get:
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 
was the difference right-left. On the east
side, 
changes sign and this means 
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)
[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:
=
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 1˝ 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 (=> 6 months to cross Pacific)
c = 0.08
m/s at 20oS or 20oN
c = 0.02
m/s at 40oS or 40oN (=> 20 years to cross Pacific)
[Chelton
and Schlax - TOPEX Rossby wave propagation across Pacific]
Hovmueller diagrams
at different latitudes show different speeds.
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 eastward.
If
the ocean were purely geostrophic, then the depressions and bulges in
thermocline seen, for example, in the trans-Pacific 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 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”).
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 1˝ 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
[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 1˝ layer ocean, the local variation in the thermocline depth with time was:
e.g. On the trailing edge of the eddy 
and 
: the layer thins as the eddy goes west.
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
[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
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 1˝ 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.
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
We
can arrive at a very robust version of the key features of the Sverdrup balance
without needing to make the 1˝ 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
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
of y equation
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.
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.
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
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, 
If we ignore
meridional winds, then = 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.
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 1˝ 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.
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 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
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
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.
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 2-dimensional flows in fluid dynamics.
Define
a mass transport streamfunction 
:
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 
and 
Thinking
in terms of the streamfunction is particularly useful because automatically satisfies the mass conservation (continuity)
equation.
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 
at x = the 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).
This
term arises from taking of the x momentum equation and adding toof y equation when
a horizontal friction term is added on the right hand side.
…
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 
The
region where this becomes of order unity is the western boundary current, where
the large value of is brought to zero over a short distance in x across
the current.
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:
relative
vorticity. 
(units are s-1)
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 f.
absolute
vorticity is 
.
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.
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. 
is dominated by the
dv/dx term.
The 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 
decrease, i.e. a
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.
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.
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.
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)]
At the equator – upwelling
and westward downwind flow
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 
= 0 and if we ignore the variation of wind stress with
longitude, then = 0 means .
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
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.
Upwelled water is
rich in nutrients but not plankton.
[Mann
and Lazier fig 3.06 equatorial distribution of biological zones]
Equatorial biological
oceanography
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]
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 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]
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]
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 Nińo. 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 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 Nińo, 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 Nińo:
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 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 Nińo, 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 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. converging Ekman
transports (off equator) that increase sea level
2. and depress the
thermocline
3. eastward
geostrophic flow converges to the east
4. and diverges to the
west
5. 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
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 1˝ 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.
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]
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.
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
![[Monthly Equatorial Pacific SST and zonal wind anomalies 1999-present]](../../pics/time_lon_EQ_lf_sst_uwnd_mean_mean_199901_200110.gif){kind=link}
![[Equatorial SST and anomalies 1986-present]](../../pics/EQSST_xt.gif){kind=link}