AMATH/ATMOS 505, OCEAN 511—Autumn 06
Midterm/Homework 3
1. Buoyancy and alternative expressions for the pressure-gradient force: Define the
Exner function as π= (p/p00)R/cp, where pis pressure, cpis the specific heat of
air at constant pressure, Ris the gas constant for air, and p00 = 1000 mb. The
potential temperature θsatisfies πθ =T, where Tis temperature.
(a) Show that
cpθ∇π=∇p
ρ,
which provides an alternative expression for the pressure gradient force.
(b) Express the hydrostatic relation in terms of πand θ.
(c) Divide the total Exner function and potential temperature into a vertically
varying hydrostatically-balanced piece (denoted by an overbar) and a remain-
der, such that
π(x, y, z, t) = π(z) + π0(x, y , z, t),
θ(x, y, z, t) = θ(z) + θ0(x, y , z, t).
Show the the unapproximated momentum equation for inviscid flow (neglecting
Coriolis forces) may be expressed as
Du
Dt +cpθ∇π0=gθ0
θk
Note that this approach gives a simple exact way to relate potential temperature
perturbations to buoyancy forces.
2. As discussed in class, the Boussinesq equations
Du
Dt +∇P=bk,Db
Dt +N2w= 0,∇ · u= 0,(1)
may be expressed in terms of pressure potential, buoyancy, and Brunt-V¨ais¨al¨a fre-
quency, where these quantities are defined either as
P=p−p
ρ0
b=−gρ−ρ
ρ0
N2=−g
ρ0
dρ
dz ,(2)
or
P=cpθ0(π−π)b=gθ−θ
θ0
N2=g
θ0
dθ
dz .(3)
“Derive” the Boussinesq system from the appropriate π-θform of the unapproxi-
mated governing equations. Do not do a scaling analysis as part of your derivation,
simply note what approximations are required to arrive at (1) with variables defined
by (3).
