Fluid dynamics

1.Consider the phase speed of free barotropic Rossby waves.

(a) We often assume earth is a sphere in our calculations, despite it being slightly bulgedat the equator. Compared to a spherical earth, does this bulge cause the crests of freebarotropic Rossby waves at 60N to propagate faster or slower? Assume the bulgedearth has the shape shown in Fig. 1 (below), such that both the bulged earth and thespherical earth have the same radius at 45N (as in Fig. 1). Explain your answer.

Figure 1: Sketch of bulged (orange) and spherical (blue) representations of earth for problem 1a.

(b) How does the height of the tropopause influence the phase speed of Rossby waves?

Here we will consider Rossby waves as barotropic disturbances in a constant density

troposphere (the shallow water model). Begin with the conservation of Rossbypotential vorticity. You may assume that all velocities in this equation are geostrophic,make the b-plane approximation (f = f0+by with f0>>by), and make the hydrostaticapproximation. Assume that the flow has a motionless mean state (i.e., u = u+u0with u = 0 and similar for v) and is confined between a horizontal land surface and atropopause that is free to evolve but has a constant mean height (i.e., h = h+h0 withconstant h). Derive the phase speed for small-amplitude disturbances and determinewhether it is faster or slower when the mean-state tropopause is higher(larger h).

 

2.Determine whether the absolute vorticity is positive or negative for each of the 4 gradientwind cases (regular low, regular high, anomalous low, anomalous high) in the NorthernHemisphere. Also find a constraint on the local Rossby number for the anomalous cases.(Hint: For each case, only one sign of the absolute vorticity is possible. Calculate thecirculation about a circular path and then use this to get the vorticity. Recall the definition ofthe local Rossby number.)

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