Fundamentals of Ocean Circulation
As most people know, the atmospheric circulation is thermally driven by
the differential heating from the sun. Thermodynamically speaking, it is
a heat engine. However, contrary to what many people think, this is not
so for the ocean circulation, which is mechanically driven.
This was demonstrated about a century ago by the Swedish oceanographer
Sandström, who performed a simple laboratory experiment. In a vessel
filled with water he introduced a heating source and a cooling source, in
the form of metal tubes with runnig hot or cold water inside. He found that
when the heating was at a lower level than the cooling, a vigorous circulation
was excited between the two levels. If, on the other hand, they were at the
same level, the circulation was very weak, and confined to a thin layer.
Thus, if there were only the heating and cooling of the ocean surface
by the sun and the atmosphere, but no mechanical forcing, there would exist
a significant flow only in a thin layer just below the surface, driven by
molecular heat conduction. The rest of the ocean would be filled with stagnant
cold water. But in reality there are strong ocean currents at great depths.
According to the conventinal picture of the overturning circulation of
the ocean, dense and cold water sinks to the bottom at narrow convection
sites in the North Atlantic and near Antarctica, and rises again in a broad
upwelling in the tropics and midlatitudes.
As the water rises it must also be heated, in order to attain the higher
temperature prevailing at smaller depths. This can only be accomplished
by turbulent mixing, since the molecular heat conductivity is much too low.
The turbulence and the mixing are caused by breaking internal waves. These
waves are excited by winds and by tidal flows that encounter rough topography
on the bottom of the ocean. In this view, therefore,
the deep circulation of the ocean is driven by the small-scale mixing, and
ultimately by winds and tides.
My Research
Internal tides
By using linear wave theory, it is possible to compute the radiation of
internal waves that are generated by tides over bottom topography. The figure
shows the result of such a computation for the global ocean, and is taken
from a recent paper
(pdf file).
The color represents radiated power, in units of Watts per square meter,
and the scale is logarithmic; for example, -3 represents 1 mW/m2.
The input data used in the computation were the bottom topography with a
resolution of 2 nautical miles, the tidal velocity field, and the
stratification of the ocean.
About half of the internal wave energy was in this computation generated
in points where the bottom slope is supercritical (i.e. the slope is greater
than the slope of the internal wave rays). At these points the linear theory
on which the computation was based is not valid. Will the nonlinear effects
caused by supercriticality increase or decrease the radiated power compared
to the prediction of linear theory? This is currently an area of very active
research, but the answer in still not known. Personally, I strongly believe
that the nonlinear effects will decrease the radiation. The arguments for
this are presented in a recent paper, which also contains
an analytic solution to a simple test problem that gives some (but not very
strong) support for this hypothesis. Much more remains to do in this field.
Analysis of Ocean Circulation
Oceanic currents mainly flow along isopycnal surfaces (i.e. surfaces
of constant density). However, from many points of view, the small
component of the flow which is perpendicular to these surfaces is the most
interesting one, since it necessarily involves irreversible energy
transformations, connected with mixing. This is, for example, the most
important aspect of the conveyor belt circulation.
In a recent paper
(pdf file)
, we propose calculating a stream function as function of depth and density
as a new way of analysing the thermodynamic character of the
overturning circulation. Since depth is linearly
related to pressure, and density is the inverse of specific volume,
the circulation is in effect displayed in a classical thermodynamic
pV-diagramme. The sign of an overturning cell in this diagramme
directly shows whether it is driven mechanically by large-scale wind
stress, or 'thermally' by heat conduction and small scale mixing.
The figure below shows this stream function for the high-resolution global
ocean model OCCAM, with potential density and depth on the axes.
The color represents the stream function in Sverdrups
(106 m3/s), with clockwise circulation in negative
overturning cells. Three main cells are seen in the figure: 1) a tropical
thermal cell in the warmest and shallowest waters, 2) a mechanically driven
cell (i.e. the sinking water is lighter than the rising water) dominating
depths between 200 m and 2000 m, 3) a thin thermal cell in the very
densest waters. Only the third cell is compatible with the conventional
ideas about the conveyor belt circulation.
Vortex Dynamics
Vortices are common in many parts of the ocean. They can be generated by
instabilities in frontal currents (for example the Gulf Stream rings), by
intrusion of water into a basin where the temperature and salinity is different
(for example the outflow of Meditteranean water into the Atlantic, which creates
Meddies), by flow over topography, or by other mechanisms. I have tried
to understand various properties (such as stability or drift speed) of such
vortices.
In
this paper, a very general
variational principle was used to understand the properties of vortices
attached to seamounts. It was found that there exists a large class of
stationary and stable anticyclonic vortices that are attached to a given
localized seamount of arbitrary shape. If the
seamount is circular there are also stable cyclones, but these are destabilized
by noncircularities in the topographic shape, unlike the anticyclones.
In
this paper,
the same variational
principle is applied to flow in closed basins with sloping boundaries. It is
shown that weak (Ro << 1) cyclonic basin flow with monotonic radial profile of
potential vorticity (PV) is nonlinearly stable, even if the basin has an
irregular shape. By contrast, anticyclonic flow can only be shown to be stable
if the basin is circular. If the basin is non-circular, anticyclonic flows can
therefore be expected to be unstable, even if the PV-profile is monotonic.
This striking difference between cyclonic and anticyclonic flows is confirmed
by both numerical simulations and laboratory experiments
with non-circular basins. They demonstrate that cyclonically forced flows
nicely follow the isobaths, while corresponding anticyclonically forced flows
develop strong cross-slope flows. This agrees with the real ocean, where
cyclonic flows tend to follow isobaths closely, while anticyclonic flows
(such as the Gulf Stream) separate more easily from the topography.
Geophysical fluid dynamics
This course shows how to explain important dynamic phenomena in the ocean and atmosphere by using basic fluid equations, in particular the shallow-water equations. The subjects dealt with include geostrophic adjustment, the separation between fast and slow modes, reduction to quasigeostrophy, various geophysical waves (e.g. Rossby waves and gravity waves), conservation laws and stability theory.
Course plan pdf file
Schedule pdf file
Grading criteria pdf file
Exercises pdf file , and solutions to some exercises:
Exercise12.pdf ,
Exercise13.pdf ,
Exercise15.pdf
Computer lab pdf file
Tutorial
on geostrophic adjustment
Movie with SSH 1993-2009 (thanks to Leon Chafik!) mp4 file
