SIO 217B Atmospheric and Climate Sciences II
Winter 2012
Homework #5
Due March 13 at the beginning of class
1. A simple model for the annual mean Hadley circulation is that air parcels rise to the upper
troposphere at the equator and then move to higher latitudes in both hemispheres.
a) Assuming that air parcels conserve angular momemtum in the absolute reference frame,
derive an expression for zonal wind in the rotating reference frame as a function of
latitude. For simplicity assume that parcels at the equator are at rest with respect to the
rotating Earth.
U = Ω a sin2(φ) / cos(φ)
b) Using the above expression, calculate values of zonal wind at 0°, 10°N, 20°N, 30°N,
40°N, and 50°N.
0 m s-1, 14 m s-1, 58 m s-1, 134 m s-1 251 m s-1, 424 m s-1
c) Assuming surface wind is zero and upper-level wind is at 200 hPa, calculate the layer
mean temperature gradient in thermal wind balance with momentum-conserving wind.
Express your answer in terms of temperature change per 1000 km at 0°, 10°N, 20°N,
30°N, 40°N, and 50°N.
∂<T>/∂y = −f UT / (R log[1000/200])
0 K, 1 K, 6K, 21 K, 51 K, and 103 K per 1000 km decreasing towards the north
2. Momentum-conserving flow cannot extend far into higher latitudes because the vertical wind
shear and the meridional temperature gradient eventually become so large that baroclinic
instability occurs. The waves generated by baroclinic instability then become the primary
process for transporting momentum, and the mean flow no longer conserves angular
momentum. One theory for determining the latitude at which the Hadley circulation ends is
the latitude at which baroclinic instability first occurs.
a) The two-layer model indicates that the onset of baroclinic instability for the most
unstable wave occurs when 2 λ2 UT / β = 1. Change this equation into a form where f0 = a
function of β, ∂T/∂y, σ, and δp. Hint: R T / δp ≈ f0 (ψ1 – ψ3) / δp
f0 = −β σ (δp)2 / (R ∂T/∂y)
b) All else the same, if the stratification of the troposphere increases, will the latitude of the
onset of baroclinic instability shift poleward, equatorward, or stay the same?
f0 must be larger to balance, so shift poleward
c) All else the same, if the depth of the troposphere (represented by δp) increases, will the
latitude of the onset of baroclinic instability shift poleward, equatorward, or stay the
same?
f0 must be larger to balance, so shift poleward
d) All else the same, if the meridional temperature gradient in the troposphere increases,
will the latitude of the onset of baroclinic instability shift poleward, equatorward, or stay
the same?
f0 must be smaller to balance, so shift equatorward
3. During Ice Ages, there is less CO2 in the atmosphere and greater surface albedo at higher
latitudes from more snow and ice cover.
a) Do you think the equator-to-pole temperature gradient is larger or smaller during Ice
Ages than during interglacial times? Why?
Greater albedo at higher latitudes during Ice Ages means less solar absorption and more radiative
imbalance, so the temperature gradient will be larger
b) Colder temperature during Ice Ages implies less water vapor in the atmosphere available
for condensation in the upper troposphere. Do you think that tropospheric stratification is
larger or smaller during Ice Ages than during interglacial times? Why?
Less condensation during Ice Ages means less latent heat release in the upper troposphere so
weaker stratification.
c) Less CO2 during Ice Ages means less radiative cooling in the stratosphere to offset solar
absorption by ozone. Less CO2 and water vapor also mean less greenhouse warming in
the troposphere. Less water vapor available for condensation means less buoyancy for
rising parcels. Do you think the tropopause, defined as the elevation of the coldest
temperature, is higher or lower during Ice Ages than during interglacial times?
Less radiative cooling means the stratosphere will be warmer and less greenhouse warming
means the troposphere will be cooler. Less buoyant parcels will not rise as far. These will
depress the height of the tropopause.
4. Do the following in context of your answers to the previous questions.
a) The plot below schematically indicates positions of the vertical and meridional
components of the annual mean Hadley circulation during interglacial times. Draw
arrows schematically indicating the positions during Ice Ages, if different from
interglacial times.
surface
SP
60°S
30°S
EQ
30°N
60°N
NP
b) Qualitatively draw how annual mean zonal mean average tropospheric temperature varies
with latitude during interglacial times. Qualitatively draw how temperature varies with
latitude during Ice Ages to qualitatively show differences between the interglacial and Ice
Age conditions.
warmer
T
colder
SP
60°S
30°S
EQ
30°N
60°N
NP
c) Qualitatively draw how annual mean zonal mean upper tropospheric zonal wind varies
with latitude during interglacial times. Qualitatively draw how zonal wind varies with
latitude during Ice Ages to qualitatively show differences between interglacial and Ice
Age conditions. Make sure changes in your plot are consistent with zero global average
momentum flux into the atmosphere.
westerly
u = 0
easterly
SP
60°S
30°S
EQ
30°N
60°N
NP