Amath-Atm. Sci. 505/Ocean 511
Due Thursday, October 9, 2014
Homework 1
1. Consider the hollow circular glass pipe shown at right. The circle into
which the pipe is bent is 1 m in radius and the inner radius of the pipe is
1 cm. The upper half of the pipe contains a vacuum and its lower half
contains water at rest of density ρ = 1000 kg m-3.
(a) Roughly what is the water pressure at the bottom of the circular
pipe?
(b) The air surrounding the pipe is at room temperature of 20 C and pressure 1 bar. If a small
hole is drilled in the bottom of the circular pipe, will water flow out of the pipe, will air
flow into the pipe, or will nothing happen? (neglect surface tension of the water surface,
and consider the force balance at a water surface extending horizontally across the hole).
2. Consider two rigid 1 m3 containers of air. The air temperature is 0
C in container 1 and 40 C in container 2. The air in each container
is at a pressure of 1 bar.
(a) Calculate the mass M1 of air in container 1 and M2 of air in
container 2.
(b) Calculate the total entropy S = M1s1 + M2s2 of the system of
two containers (using a reference pressure of 1 bar and temperature of 0 C = 273 K).
Here s1 and s2 are the specific entropy of the air in containers 1 and 2, respectively.
(c) The air is suddenly allowed to mix between the two containers, conserving total mass and
total internal energy in the process. Does the entropy of the system change? If so, by
how much? If not, why not?
3. At the top of Mt. Rainier, the summertime air temperature and pressure average about 270 K
and 0.6 bar. What are the potential temperature and potential density of this air, assuming a
reference pressure pref = 1 bar? Compare this to the average potential temperature and
potential density of air at sea level, where the temperature and pressure are 295 K and 1 bar.
4. (a) Using the Cartesian representation of the curl vector, prove vector identity (7) on the
vector operator class handout sheet.
(b) Using indicial notation, prove vector identity (8) on the handout.
(c) Using the vector identities given in the handout, show that if u is an arbitrary vector field
and ω = ∇ × u ,
⎛1
⎞
∇ ⎜ u ⋅ u⎟ = ( u ⋅ ∇) u + u × ω
⎝2
⎠
(d) Derive this result directly using indicial notation.