Notes
1.
Geographical setting:
Earth
is 70.8% water-covered. In the zonal direction, there is no
land between 85-90°N and between 55-60°S. At latitudes 45-70°N
there is more land than water. At latitudes 70-90°S there is
only land (Antarctica).
The
areas of the oceans are: Pacific (179 × 1e6 km2),
Atlantic (106 × 1e6 km2), Indian (75 × 1e6 km2).
The order of magnitude of the horizontal length scales that we
associate with these oceans are: Pacific (15,000 km), Atlantic
(5,000 km), Indian (5,000 km).
The
Earth's radius (hereafter Re)
is approximately 6371 km. (The Earth is actually not a sphere,
but this is close enough for us). The average depth of the
ocean is about 4000 m (actually 3795 m). Thus the ocean is a
thin skin on the outside of the Earth. The average height of
land is 245 m. The maximum elevation is about 9,000 m (Mt.
Everest) and the maximum ocean depth is about 11,500 m
(Marianas Trench).
1°
latitude = (2 pi
Re / 360°) km
1°
longitude = (2 pi Re
cos(lat) / 360°) km
1
nautical mile = 1 minute of a great circle of the globe = (2
pi Re)/(360°
× 60) = 1.852 km
2.
Scale analysis:
A
very small set of equations governs all the motions of
fluids (5 equations for the ocean). This small set governs
all scales from nearly-molecular scale to capillary waves to
global circulation. As we will see, this set of equations is
nonlinear,
which means that many terms are products of two things that
vary (i.e. velocity times velocity, or velocity times
density, etc). We can't possibly solve for all ranges of
motion at the same time - direct solutions of the complete
governing equations (pencil and paper) are impossible and
there simply isn't enough computation power to cover all
possible motions. The nonlinearities make theoretical
solutions difficult.
Therefore,
fluid mechanics is a science of approximation.
This differs from the traditional physics and math that you
might have studied before, with precise answers and proofs.
Advanced applied mathematics includes the rigorous,
justified approximation methods used in fluid mechanics.
Approximations must be justified. Much of the debate about
validity of a particular numerical model of this or that
centers on justifying either the physical approximation or
the numerical resolution (time and space scales that are
resolved). Almost all fluids models include scales of motion
that are smaller than the modeled scales. These unresolved
processes might impact the outcome of the resolved
processes, and so a lot of work goes into "parameterizing"
the "sub-grid" scale processes. In theoretical analyses,
this step is called "closure".
As
a first and necessary step, we evaluate the scales
(approximate sizes) of the motion that we wish to resolve.
We then perform a rigorous analysis of the governing
equations, based on this scale
analysis.
All properties that characterize the fluid can be scaled -
for instance, distance, height, time, velocity (horizontal
and vertical), density (variations), temperature, salinity,
pressure.
When
we want to compare the sizes of two terms that have the same
units (i.e. same units in terms of meters, seconds,
kilograms, degrees C), we first figure out the approximate
sizes (scales) of the terms. The scale of a particular term
will have exactly the same units as the term itself. For
instance, the time derivative of velocity, du/dt, where u is
velocity and t is time, has the scale U/T, where U is the
typical size of the velocity, and T is the typical time
scale.
Then
to compare the two terms, we look at the ratio of the two
terms' scalings. In order to compare the two terms, they
must have the same units to start with. Therefore the ratio
of the two scalings will have no units (units = 1), that is,
no length, time, etc. This ratio is called a non-dimensional
parameter
since it is dimensionless. For
example, if we want to compare dw/dt and du/dt, where w is
vertical velocity and u is horizontal velocity, the scales
of the two terms are W/T and U/T. Each of these dimensional
scales has units length/time^2. The non-dimensional ratio of
the two is W/U, i.e. no units. If the horizontal and
vertical velocities in a particular motion are of about the
same size, then this ratio is order (1). If vertical
velocity is much smaller than the horizontal velocity, then
this ratio is small.
After
evaluating the relative size of all of the terms in our
equations, using these scalings, which are based entirely on
the physical system we are looking at (i.e. is it a surface
wave or the whole Pacific general circulation), we then start
our modeling. We carefully take into account which terms are
small, and ignore them "at first order", meaning in the first
step of our analysis. We might look at how they correct the
flow later in the same analysis.
A
very basic non-dimensional parameter that comes from our
discussion of the geographic setting above is the aspect
ratio,
which is the ratio height/length. For some fluid motions in
the ocean, such as surface waves in deep water or convection
cells, the aspect ratio is order (1). (That is, the height
and length are the same order of magnitude, although they
might not be precisely the same number.) In this case, the
horizontal and vertical flows for instance could be about
the same size. For other oceanic flows, such as surface
waves in shallow water or the general circulation, the
aspect ratio might be small (order 0.1 or 0.01 or smaller).
In this case, motion in the vertical direction will be very
different (smaller) than motion in the horizontal direction.
Another
very useful ratio is that of the Earth's rotation time scale
to the time scale of the motion (~1 day/T). For surface waves,
this is a large number, and Earth's rotation is not important.
For internal waves, tides, this is order 1, and rotation and
changing motion are both important. For the ocean circulation,
this ratio is very small, and time dependence is much less
important than rotation.
This
is all rather abstract, and we will not belabor it. We will
also not spend any time in this course developing the
equations of motion and scaling them - this is the task of
fluid mechanics and geophysical fluid dynamics courses.
However, it is important to have a feel for how physical
oceanographers justify simplifying the governing equations to
study each particular process.
4.
Major ocean current system:
Wind driven:
Buoyancy
driven (mixing):
Further
reading:
Study
questions:
1.
How much of the Earth is covered with water? In what latitude
ranges is the Earth covered by water all the way around? What
effect does the presence of continental boundaries at most
latitudes have on the circulation? What happens where there is
no continental boundary?
2.
How deep is the ocean on average? How does this compare with
the average zonal dimension of the three major oceans? What
effect might this difference between the horizontal and
vertical dimensions have on the relative magnitude of the
horizontal and vertical velocities? (Another factor affecting
the vertical velocities is vertical stratification.)
3.
What are the forcing mechanisms for the ocean?
5.
Replace the "time" axis in (4) with a vertical length scale
axis. Now place the same phenomena (bubbles to Milankovitch
variations) on the new graph, and give the approximate
vertical length scales.
Last
modified: Sep 2016