ESCI 342 – Atmospheric Dynamics
Selected Answers to Exercises for Lesson 6
1. Perform a scale analysis of the horizontal momentum equations (in component form)
for the whirlpool formed as your bathroom sink drains. Which terms are important in
this case? Water has a density of 1000 kg/m3 and a kinematic viscosity of ν =
1.8×10−6 m2 s−1. The horizontal pressure difference across the whirlpool is ~ 10 Pa.
(Use a reasonable estimate for the horizontal velocity based on your own
experiences.)
Answer:
Name
Horizontal velocity
Vertical velocity
Horizontal distance
Vertical distance
Horizontal pressure change
Density
Time
Kinematic viscosity
Omega
Latitude
Radius of Earth
∂u
∂t
U 2/L
10−2
m/s2
∂v
∂t
U 2/L
10−2
m/s2
u
∂u
∂u
+v
∂x
∂y
U 2/L
10−2
m/s2
u
∂v
∂v
+v
∂x
∂y
U 2/L
10−2
m/s2
w
∂u
∂z
WU/H
10−3
m/s2
w
∂v
∂z
WU/H
10−3
m/s2
−
Symbol
U
W
L
H
δP
Order of magnitude
0.01 m/s
0.01 m/s
0.01 m
0.1 m
0.1 Pa
1000 kg/m3
ρ
τ = L /U 10 s
1.8×10−6 m2/s
ν
7.292×10−5 rad/s
Ω
45°
φ
a
6.378×106 m
uv tan φ
a
uw
a
U 2/a
10−11
m/s2
UW/a
10−11
m/s2
u 2 tan φ
a
vw
a
UV/a
10−11
m/s2
UW/a
10−11
m/s2
−
1 ∂p
ρ ∂x
δP/(ρ L)
−2
10
m/s2
−
1 ∂p
ρ ∂y
δP/(ρ L)
−2
10
m/s2
2Ωv sinφ
−2Ωwcosφ
 ∂ 2u
ν 
 ∂x
2ΩU sin 45
10−6
m/s2
−2Ωu sinφ
2ΩW cos 45
10−6
m/s2
 ∂ 2v
ν 
 ∂x
2ΩU sin 45
10−6
m/s2
2
+
∂ 2v 
∂y 2 
νU/L2
−4
10
m/s2
2
+
∂ 2u 
∂y 2 
νU/L2
−4
10
m/s2
ν
∂ 2v
∂z 2
νU/H2
10−6
m/s2
Pressure gradient and acceleration are dominant terms. Flow is cyclostrophic.
2. What is the Rossby number for a tornado? Does the Coriolis force effect a tornado?
Answer: Using U ~ 100 m/s and L ~ 100 m, Ro ~ 104. Therefore, Coriolis does not
influence tornadoes.
ν
∂ 2u
∂z 2
νU/H2
10−6
m/s2
5. At the four points shown in the picture below, estimate the magnitude of the
geostrophic wind. Assume a density of 1.23 kg/m3 and a latitude of 45°. The isobars
are labeled in mb.
Answer: A - 19 m/s; B - 39 m/s; C - 58 m/s; D – 0 (answers will vary slightly due to
measurement error).
6. Perform a scale analysis of the vertical momentum equation for a thunderstorm to find
out what terms can be ignored.
Answer: Using U ~ W ~ 10 m/s and L ~ H ~ 104 m, the resulting equation is
∂w
∂w
∂w
∂w
1 ∂p
+u
+v
+w
=−
− g.
∂t
∂x
∂y
∂z
ρ ∂z
2