WEATHER & THE ENVIRONMENT - WEEKLY PROBLEMS
WEEK 1
1. Estimate the total mass of the atmosphere. The radius of the earth may be taken as 6370 km
and atmospheric pressure as 105Pa.
2. (a) A body at temperature, T , radiates energy at a rateσT 4 per unit area, withσ == 5.67 x
10-8 Wm-2 K4. The measured flux at the earth’s distance from the sun’s centre is about
1376 Wm-2. If the radius of the sun is taken to be 6.95 x 105km and the mean sun-earth
distance to be 1.49 x 108km calculate the (effective) temperature of the sun’s surface.
(b) Compute the temperature, which the earth would have, in the absence of the atmosphere,
were it to radiate the energy received on average from the sun.
(c) The albedo is defined as the fraction of the energy incident on a planet, which is reflected.
Repeat the calculation in part b assuming the earth has no atmosphere and the current albedo of 0.30.
3. Estimate the difference in pressure between sea level and 10m above sea level. Assume P0
= 101325 Pa, R = 287 J kg-1 K-1, T = 288 K, g = 9.8 ms -2 Estimate the pressure at 50 km
above sea level, i.e. at the top of the stratosphere, (assume
an average temperature of
260K). [The measured pressure at this altitude is approx. 0.8 mbar]
4. Outline a positive (destabilising) feedback mechanism associated with water
as a greenhouse gas, that could lead to increasing temperatures. Can y ou think of a
possible negative (stabilising) feedback mechanism associated with water
vapour that
might counteract this?
WEEK 2
1. A saturated parcel of air is rising within a cloud. When will it stop rising?
2. Smoke from smoke stacks disperses in different ways depending on the weather conditions.
Sometimes it disperses in uniform horizontal plumes, whilst at others it mixes rapidly with
the air in almost vertical disorganised puffs. Explain this. (Hint: In one case dt dz is close
to the DALR, in the other it is in a region of stability.)
3. Suppose that at sunrise the temperature is 10oC and the atmosphere is saturated with water
vapour. During the day the air is heated, but the total water vapour content does
not
change. W hat will the relative humidity be when the temperature reaches 20oC?
(Hint: You will need to use the SVP vs T curve in the notes.)
WEEK 3
1. Consider a droplet of pure water in air containing water vapour at the saturation value for
the droplet, er , the system therefore being in equilibrium. I s this a stable or unstable
equilibrium? (The question you need to ask is whether, if the droplet grows a little will it
continue to grow, or will it shrink back.)
2. A sphere moving in viscous flow through a fluid experiences a drag force given by Stokes’
law, f drag = 6πηrv , where η is the viscosity of the fluid, r is the radius of the sphere and v is the
velocity. Derive an ex pression of the form, v = v(r ) , for the fall in the ai r of droplets not
exceeding about 30 µm in radius (for r ≤ 30 µm the air flow over the droplet may be taken as
viscous).
3. A drop g rows by accretion as it falls throug h a cloud of uniform droplets of radius r = 10
µm. A set of estimated collection efficiencies , E, and the corresponding terminal velocities V
of the growing drop are given in the table below as a function of the drop radius, R .
R (µm)
30
40
50
80
100
E
.17
.45
.56
.62
.69
V (cm/s)
9.0
18.7
36.1
56.0
74.7
The terminal velocity of the small cloud drops, v , is 1.3 cm s-1 and the water content of the
cloud, w, is 2 x 10-3 kg m3.
(a) Compute the growth rate dR dt between R = 30 µm and 100 µm, and plot it as a
function of R .
(b) Plot dt dR on the same graph and estimate (from your graph) the time taken for the
drop to grow from 30 to 100 µm.
WEEK 4
1. Compute the magnitude of the centrifugal force as a function of the latitude and compare it
with (a) gravity; (b) the Coriolis force for a wind speed of 5 ms-1.
2. At the equator a wind is blowing at a speed of 10 ms -1. What is the magnitude of the
Coriolis force there? Can the wind be g eostrophic? What does this imply about the direction
of the wind relative to the isobars there?
WEEK 5
1. Calculate the geostropic wind speed for a pressure gradient of 0.03 mbar km-1 assuming an
air density of 1 kg m-3 and a latitude of 45o.
2. Why do thunderstorms usually only occur at cold fronts and not at warm ones?
3. In lectures we mainly considered the wind patterns in northern hemisphere cyclones and
anticyclones. Make drawings of the wind patterns around low and high-pressure centres in the
southern hemisphere.