Fall, 2018

Descriptive Physical Oceanography 6th edition (Talley, Pickard, Emery, Swift): Chapter 1.2 and Chapter 2.1.

Note that figures can be viewed in full color from the Elsevier website image gallery (left menu bar).

What phenomena do physical oceanographers study? (e.g. surface and internal waves, air-sea exchanges, turbulence and mixing, acoustics, heating and cooling, wave and wind-induced currents, tides, tsunamis, storm surges, large-scale waves affected by earth's rotation, large-scale eddies, general circulation and its changes, coupled ocean-atmosphere dynamics for weather and climate).

What external forces act on the ocean? (e.g. wind [waves, turbulence, large scale waves, circulation], heating due to the sun and geothermal energy, cooling, evaporation due to sun and wind, precipitation, tidal potential [the moon and sun], earthquakes, gravity, friction)

What internal forces act on the ocean? (pressure gradients, viscosity or friction)

**
2. Geographical setting
**

(zonal = east-west and meridional = north-south).

Earth is 70.96% water-covered. In the zonal direction, there is no land between 85-90 degrees N and between 55-60 degrees S. At latitudes 45-70N, there is more land than water. At latitudes 70-90S there is only land (Antarctica).

The areas of the oceans are: Pacific (179 x1e6 km2), Atlantic (106x1e6 km2), Indian (75 x1e6 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 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 (Mindanao Trench).

We divide the ocean regions into:

Oceans - Atlantic, Pacific, Indian. We often call the
region south of about 40S or 30S the "Southern Ocean".

Mediterranean seas - Mediterranean, Arctic, Gulf of Mexico, Red Sea,
Persian Gulf.

Marginal seas - many. Examples: Pacific (Bering Sea),
Atlantic (Caribbean), Indian (Andaman).

Inland seas - Black, Caspian Seas, Lake Baikal, Great Lakes.

Areas of open oceans are sometimes referred to as "seas", mainly for
historical reasons and geographical convenience. There are
no rules about whether some areas should have names and others
should not. Examples are: Arctic (Greenland, Norwegian,
Iceland, Kara, Barents, Chukchi Seas), N. Atlantic (Labrador, Irminger,
Sargasso Seas), Southern Ocean (Weddell, Ross Seas), Indian (Arabian
Sea, Bay of Bengal), South Pacific (Tasman Sea), North Pacific (Gulf of Alaska).

**
3. Methods of study
**
There are several complementary approaches to studying the ocean: (1)
observations, (2) process models, including pure theory and simplified
numerical models, (3) simulations of the flow using complex numerical
models, (4) combined observational/numerical modeling simulations (data
assimilation). The boundaries between these are not firm -
observational analysis often includes modeling, process models are
usually based on observations, complex numerical models are often
exploited to understand general processes.
This course concentrates on simple physical processes and
observations, but some examples will be drawn from numerical modeling.
The powerpoint figures shown above include examples from both
observations and modeling.

**
4. Scale analysis
**
A very small set of equations governs all the motions of fluids (5
equations for the ocean, to be introduced in the next lecture).
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 compuatation 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.

**
5. Example of scales: the general circulation
**

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?

4. Using a radius of the Earth of 6371 km, calculate the circumference, the surface area, and the volume.

5. Take this diagram from the lecture (Fig. 1.2 from DPO).

Link to the figure

Replace the "time" axis in the figure with a vertical length scale axis.
Place the same phenomena (bubbles to Milankovitch variations) on the
new graph, and give the approximate vertical length scales.

6. What are typical horizontal velocities for the ocean? What are typical vertical velocities? (Choose, say, two different phenomena with wildly different length scales, for instance from the figure in #4.)

7. What are a typical aspect ratio and a typical Rossby number for the different phenomena you selected in #5? Think about what non-dimensional parameters such as these actually tell you.

J. Pedlosky text (Geophysical Fluid Dynamics): sections 1.1 and 1.2

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