The Global Atmospheric Circulation

Exercise sheet 1

Due by 07.03.2019

Tutors:

Jake Casselman (jake.casselman@env.ethz.ch)

Bernat Jimenez (bernat.jimenez@env.ethz.ch)

Q1. Thermal wind balance

(a) Derive the thermal wind equation (the vertical variation of the geostrophic wind ug in

hydrostatic equilibrium) for a zonal mean flow in isobaric coordinates.

• The geostrophic wind:

ug = − 1

f

∂Φ

∂y

• Hydrostatic balance:∂Φ

∂p= −rdT

p

where Φ is the geopotential, p is the vertical pressure coordinate, f is the Coriolis parameter,

rd is the dry air constant and T is the temperature.

Hint: use the following relationship dpp = d ln p

(b) Use the derived thermal wind equation and zonal mean temperature climatology in boreal

winter (DJF) and summer (JJA) to qualitatively determine the location of easterly and

westerly jets, both in the troposphere and stratosphere. Indicate it on top of the temperature

plot. Justify your answers.

DJF JJA

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Q2. Rainfall climatology in the tropics

Use the link below to access the IRI data library. For each of the following coordinates listed

below, click to generate a grid point average monthly climatology. Copy and paste the precipitation

graphs into your answer (in the website, the coordinates can be seen above the precipitation chart).

Briefly describe the rainfall distribution and explain why the three climatologies vary.

Approximate locations:

• Location 1: 0.25-0.75◦S, 13-13.5◦E

• Location 2: 20.25-20.75◦S, 16-16.5◦E

• Location 3: 10.25-10.75◦N, 10-10.5◦E

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Website: http://iridl.ldeo.columbia.edu/maproom/Global/Climatologies/Select_a_Point.

html

Note: It is easier if you go to the region selection tool at the top center of the page, and choose

Africa. When gathering the point averages, avoid dragging the mouse as this will generate an

average over the area. Also, if you click on an area and receive a red box with a question mark for

temperature, you are outside the data boundaries and must click in a different area.

Q3. Coriolis effect

George Philander, a pioneer in tropical dynamics, once told the following story: At the equator, a

man demonstrated the Coriolis effect to a few visitors by using a bucket with a plugged hole at the

bottom. First, he filled the bucket with water, stepped north of the equator and pulled the plug at

the bottom of the bucket. The water circulated to the right when exiting the bucket. Then, he

re-filled the bucket, stepped south of the equator and pulled the plug again. The water circulated

to the left when exiting the bucket.

Explain: Was this (a) physics or (b) a tourist trap? Give a reason for your answer.

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Q4. Rossby wave dispersion relation, phase speed and group velocity

(a) The Rossby wave dispersion relation (the relationship between the frequency ω and the zonal

and meridional wave numbers (k and l) of the wave) can be written as:

ω = Uk − βk

k2 + l2

where β = dfdy at φ = φ0. For which background wind U will the wave be stationary?

Remember that the zonal phase speed of a wave is:

c = ω/k

Assume a Rossby wave with the following characteristics:

l = 0, k =2π

λ=

2π

5000km

use β = 1.6 · 10−11m−1s−1 at φ0 = 45◦

(b) The group velocity of a wave packet can be calculated using the following expression:

cg =∂ω

∂k

For the previous wave, given in (a), is the energy propagating eastward or westward?

(c) For an easterly background flow of U = −5m/s, what would be the minimum wavelength λ

that would allow the energy to propagate eastwards (cg > 0)? What is the phase speed of

this wave?

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