Equations of motion:

Notes

1. Forcing of the ocean - general
How do external forces act on the ocean? Gravity creates a pressure gradient that points upward and keeps the water from moving (hydrostatic balance). Others create pressure gradients that cause water to try to flow from high pressure to low pressure. Friction (viscosity) slows the water down. The earth's rotation introduces centrifugal and Coriolis accelerations. (more later).

What equations do physical oceanographers use to describe the ocean?

1. Mass conservation: what goes into an enclosed box must come out
2. ma =F (vector form; F = pressure gradient, body forces and dissipation and a includes time change of velocity, advection and Coriolis acceleration)
   acceleration + Coriolis acceleration = pressure gradient force + forcing + mixing
   This is 3 equations (vertical and 2 horizontal).
3. Equation of state: how density depends on temperature, salinity and pressure
4. Equation for density change: how density changes as a result of heating/cooling and evaporation/precipitation. (Or equivalently, have separate equations describing how salinity and temperature each change and then use the equation of state to get the density changes.)

2. Mass conservation

3. Dynamical concepts:
Advection, flux and diffusion: see DPO Chapter 5.1.
Momentum balance. DPO Chapter 8.2.1, 8.2.2, 8.2.4, 8.2.5. (Skip 8.2.3 on rotation for now.)

html transcript of Hendershott dynamical concepts notes.
Scanned pdf of Myrl Hendershott's original dynamical concepts notes

Additional material:

Mean marine sea surface, compiled from 10 years of altimeter data. Figure source: AVISO (CNES, Toulouse, France).

Text from the AVISO website: "The Mean Sea Surface represents the sea level due to constant phenomena. It can thus be likened to a flat, calm sea. But it would be wrong to think this surface is smooth. Ocean topography is shaped by permanent ocean currents and, mainly, by the gravity field. The geoid is the undulating surface related to this gravity field that reflects differences below the surface of the Earth (for example, variations in magma temperature). These differences can generate sea level variations of over 100 meters between two ocean regions thousands of kilometers apart.
At smaller scales (a few kilometers), we can also observe ridges and valleys in the ocean floor (submarine mountains, ocean trenches, oceanic ridges, etc.) that cause variations of several meters at the ocean surface."

Wikipedia entry for viscosity

Study questions:

1. Suppose a current is flowing through a passage between an island and the mainland. Suppose it is completely uniform (same velocity at all points as it enters the passage), and steady.
(a) If it is flowing at 10 cm/sec as it enters the passage, and 5 cm/sec as it leaves the passage, what can you say about the geometry of the passage? Assume there is no evaporation or precipitation or runoff in the passage.
(b) If the passage is 2 km wide and 50 meters deep at its entry, how large is it at the exit?
(c) Compute the volume transport of the current.

2. Suppose you had a wind carrying some kind of pollutant. What would you measure in order to calculate how much pollutant stays behind in your county as the wind passes over it? (Assume that the wind is pretty simple, i.e. not a lot of turbulence, but it could vary from place to place.) What would you calculate from these measurements?

3. What are the principle driving forces for the ocean?

4. Suppose you mistakenly drop a large, heavy suitcase as you reach 5 km altitude, as you head out of San Diego back to your home town.
(a)How fast would the suitcase be going as it hits the ground if there were no air resistance?
(b) Now you are going out on your first seagoing cruise from Scripps. You are passing out to the deep ocean, over the abyssal plain, where the water is 5 km deep. Why don't you fall? What is the force balance that keeps you from falling?

4. What do oceanographers use pressure measurements for?

5. Calculate the hydrostatic pressure at 5 km depth. Calculate the hydrostatic pressure at 8 km depth. Assume the density of seawater is constant at 1030 kg/m³ and that seawater is incompressible. (You will have seen that this assumption is pretty good, but not exact, when you looked at properties of seawater.)

6. Why do oceanographers have to use indirect methods to infer the pressure gradient force that drives meso-scale and large-scale flows?

7. Think about some ocean flows that we might not have mentioned in class, and consider the force balances in them. For instance, what would be the force balance in a tsunami? What would be the force balance in a surface wave running up and breaking on the beach? What would be the force balance for a flow entering an estuary through a narrow mouth?