## Transcribed Text

Atmospheric Dynamics
1. Consider a form of the QG vorticity equation,
(1)
with B2 III (3)+V.V) v3 the geostrophic velocity, 53 the vorticity associated with the
geostrophic velocity, Vo = v - v2 the ageostrophic component of the velocity, and the beta-
plane approximation f = ++By
(a) Assuming purely geostrophic flow, consider how a small amplitude initial condition
that is wavy in x and y will evolve: find the dispersion relation. (Hint: Linearize the
equation, then introduce a streamfunction [or equivalently the geopotential] to write it
in terms of a single quantity; note that geostrophic flow is nondivergent.)
(b) Consider a small amplitude initial condition in eq. (1) that is wavy in the x direction
only (uniform in the y and Z directions). Assume the flow is nearly geostrophic and is
in a homogenous incompressible ocean basin (i.e., constant g) with zero background
motion. Here variability in x has to lead to divergent (hence ageostrophic) flow and
changes in depth (h), so you'll need the continuity equation in the shallow water
approximation, as we used for surface gravity waves. Let u,v,h) =
and
linearize (here H is the constant depth of the unperturbed basin). Write the governing
equation as a PDE in h and derive the dispersion relation. Compare with the answer to
(la). What kind of wave is this?
2. Consider conservation of Ertel PV for nearly geostrophic flow. Consider small perturbations
about the mean potential temperature, 0 = Bote', with 00/0p being constant, and show
that the Ertel PV can be written as the sum of relative vorticity, planetary vorticity, and a
stretching vorticity (note that constant factors scaling the Ertel PV are irrelevant and can be
dropped, since the relevant feature of Ertel PV is that it is conserved). Compare your result
with the QGPV equation, which we'll discuss in lecture:
O.
9 III fo 1 / + a o
,
with 9 the QGPV. You should be able to readily show that the first two terms in your
expression of the Ertel PV are identical to the first two terms in the QGPV. For the third
term in your expression of the Ertel PV, it is sufficient to just argue that it represents the
stretching of columns, even if you can't get it into a form identical to the third term in the
QGPV.
3. Optional: Consider the shallow water QGPV equation derived in class on Tuesday, which
is a single equation for H1. Linearize and insert a wavy ansatz, and show that it gives a
dispersion relation that is similar to the free barotropic Rossby waves (which were for a fluid
of constant beight) that we discussed in class except for an additional term involving H in
the denomenator.
1

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