GFD-1 Problem Set 1

out: 10 Jan 2001

back: 17 Jan 2001

1. Write an essay on the similarities and differences (‘compare and contrast’) between Earth’s atmosphere (A) and ocean (O). Use ideas in Gill chaps. 1 and 2 as a start. Other standard texts may be consulted. This can involve the nature of the boundaries of the respective fluids, and also touch on the interaction between O and A. It could extend as far as the chemical makeup of the fluids. Length: about 1 page single spaced. Examples: ‘the eddying motions dominating the kinetic energy have lateral scale of about 100 km in the ocean, and about 1000 km in the atmosphere, yet they have similar dynamic properties’.

2. Show that the centripetal acceleration can be written where W is the magnitude of the rotation vector and is the cylindrical radius ( is the full radius vector).

2. A pendulum hangs from a support, at the center of a rotating platform in the laboratory, where the gravitational potential is gz (z being the vertical coordinate).

(a) Describe its equilibrium position (that is, where it can hang without any movement, when viewed in the rotating frame. This involves graphical and analytical discussion.

(b) Suppose the pendulum is rigidly fixed to an axle held by two supports so that it can swing only in the plane perpendicular to the axle. what will its natural frequency be (assuming very small amplitude swings)?

(c) If the pendulum is not so constrained (say, it is a simple string with a bob at the end), describe the oscillation when the pendulum is held by a non-rotating observer and released (there is a swivel mount so that the string does not twist up). How does the oscillation appear to a rotating observer?

(d) Again, for the unconstrained pendulum, think about the oscillation if a rotating observer riding on the rotating table releases the pendulum bob. This can lead to some quite elaborate math, which is not necessary here, but take it as far as you can.