Nov 15: Rossby waves and
Sverdrup circulation
In a simple 1½
layer ocean for which geostrophic dynamics holds, the layer interface
(pycnocline) displacement is opposite in sign to the sea surface displacement
and greater in magnitude by the ratio of the relative density difference in the
layers:
Contours of surface
dynamic height, interface displacement and geopotential height are all parallel
and represent streamlines of the geostrophic flow.
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
c = /
30o lon *
111 km * cos(25)
60 cycles * 10 days
* 86400 sec
= 5.8 x 10-2 m s-1
=
2.1 x 10-11 *
(27.8-26.6)/1027.8 * 9.81 * 1000
m / (6.16 x 10-5)2
=
6.3 x 10-2 m s-1
(close enough for such a simple estimate)
Things to notice
about the Rossby wave speed:
·
gets larger approaching the equator
·
always positive (i.e. westward propagation in our sign
convention)
This is an
approximate equation for very long wavelength, long period (many months) Rossby
waves, for the idealized 1½ layer ocean.
Choose some
reasonable approximate values:
H = 300 m, 
= 3 x 10-3
We find that
c = 1.27
m/s at 5oS or 5oN (=> 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 1½ ocean, we
would drive a changing layer thickness 
Sign
check: the layer thickness is increasing with time if water is being pumped
downward, i.e. we< 0
Then
[Tomczak and Godfrey – fig 4.3 map of curl(t/f)]
So
now reconsider the question I posed about what maintains the bulges in the
thermocline that we see propagating westward as Rossby waves.
In
the 1½ layer ocean, the local variation in the thermocline depth with time was:
e.g. On the trailing edge of the eddy 
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
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 1½ layer model and the continuously
stratified case, just as we saw the similarities in the layer model and the
more general thermal wind relation.
The
reason to introduce the depth integrated steric height gradient is that this is
a quantity we can evaluate from observations of the oceanic density field.
We
can test the Sverdrup balance by comparing maps of 
from hydrography to 
from winds.
[Tomczak and Godfrey: curl(tau) and steric height]
There
is a maximum in P near the western
boundary of each ocean basin, and the number of contours across each basin is
roughly correct.
The
poorest agreement occurs at the outflows of the western boundary currents (EAC,
Agulhas, Kuroshio,
But
the qualitative pattern of circulation is quite good.
Read Tomczak and Godfrey Chapter 4
Sverdrup’s
theory of the oceanic circulation
We
can arrive at a very robust version of the key features of the Sverdrup balance
without needing to make the 1½ layer assumption, or consider the details of the
vertical stratification.
The
approach is similar to the way we derived the Ekman transport relation by
integrating the momentum equations over a large enough depth to cover the
entire Ekman layer (the near surface region where vertical mixing of the
momentum imparted by the wind is significant).
It
turned out we didn’t need to explicitly know the Ekman layer depth, or indeed
any details about the vertical profile of the rate of turbulent mixing. 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
passed 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 equal to or greater than
the depth at the which the horizontal pressure gradient becomes zero (i.e. our
level of no motion)
Notiver
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:
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 
= 0 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.
[Apel fig. 6.36 – Schematic of zonal
winds and gyres]
With
wind stress curl computed from observations, we can integrate westward from the
eastern boundary and map the streamlines of the depth-integrated flow.
Define
a mass transport streamfunction:
Lines
of constant 
(psi) are streamlines of the depth-integrated flow. Flow is
parallel to streamlines, and the between a pair of streamlines the mass
transport is constant.
The
streamfunction of the depth-integrated flow is very closely related to depth
integrated steric height.
