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.