# Document 17799186 ```METO 658B Boundary Layer Meteorology
Homework I due 9/16/04
1. Prove the following identities using summation (index) notation:
a.
2
u
  u u  u  (u)    2
b.     u 0
c.     0
hint:

u  i u j ;  u  i ui ;  u  ijk j uk ;

2. Derive the Navier-Stokes equation (2.40 in Garratt), starting from

equations 2.3 and 2.39, and making use of the Boussinesq
approximations.
3. Consider the TKE equation:
eu j ' 1  p' ui '
e
e
 u  g 
 uj
 ui ' u j ' i   ui ' v '3i 


t
x j
x j  v 
x j
x i

Derive this equation, starting from the momentum and continuity
equations.
Identify, and explain the physical significance of each term.
4. a. Determine the equilibrium solutions of the equation:
dy
 A  y2
dt

where A is positive. Sketch the equilibrium solutions qualitatively on a y
vs. A diagram.
b. Now consider the stability of the equilibria on a linear basis. Show
which part of the diagram corresponds to a stable, and which to an
unstable equilibrium state.
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