Basic course of tropospheric composition modelling
Exam assignment
Task 1. Two identical NO2 monitoring instruments settled at a car and at a stationary
platform. The car drives along the road westwards 40 km hr-1 and its NO2 monitor shows an
increase of 5 g N m-3 every 30 minutes. The time trend of the NO2 concentration recorded by
the monitor at the fixed site is 0.5 g NO2 per hour. What is the spatial gradient of the field
(assuming the trends and spatial gradients are all homogeneous)?
Task 2. Derive the pressure dependence on altitude for the hydrostatic atmosphere with the
following temperature dependence on altitude:
300  4 * z, z  0, 20km
. Here temperature T is in Kelvin, altitude z is in km.
T ( z)  
z  20km
 220,
Cork
Task 3. Molecular diffusion processes
At the lectures (section “Viscosity and heat conductivity”), it was
Low-pressure
shown that neither molecular viscosity nor heat conductivity depend
air
Insulated
on the gas density. However, this is in contradiction with the daily
volume
practice of heat-preserving thermos-type devices where a thin layer of
low-pressure air contained in-between the hermetic thermos walls (see
figure) serves as a very good insulation material, so that most of the
heat loss takes place through the cork. Explain the contradiction and derive a formula
(possibly, crude) estimating the molecular viscosity and heat conductivity for this case.
Task 4. Programming practice
Implement the one-dimensional vertical diffusion using Crank-Nicolson time-implicit
scheme. Define a semi-uniform 1-D grid with 40 layers, first 10 of which are with 50m
thickness, while the other 30 are all 250m thick. Let the grid meshes be located in the middle
of the corresponding layers.
Test the scheme:
- a) Let the diffusion coefficient be constant, Kz= 10 m2 sec-1; let initially the
concentrations be all-zero, except for C(k=20)=1g m-3, where k is the vertical grid
index. Compute the pollution profile after 10sec, 100sec, 1000sec, 10000sec.
- b) For the same initial mass distribution, let the diffusion coefficient growing linearly
with height: K=0.01 z until z=1000m and then stays constant. Here z is an absolute
vertical coordinate in m. Compute profile for the above times.
0.02 z [m 2 sec 1 ], z  1000m

- c) Let K z ( z )   20 m 2 sec 1 , 1000m  z  2000m . Compute the concentration
 1 m 2 sec 1 , z  2000m

profiles for the above time steps for the following initial mass distributions
o C(k)=1g m-3(k-5),
o C(k)=1g m-3(k-150),
Here  is the Kroneker’s symbol meaning, for example, that: (k-5)=1 for k=5 and 0 for all
other k.
Task 5. Programming practice.
Compute the 24hrs average and max over the same period of NO / NOx ratio assuming the
photostationary equation is fulfilled. Use ozone level of 50 g O3 m-3. Assume that maximal
photolysis rate is jNO2 = 0.01 sec-1; it varies parabolically over a day starting from zero at
6:00, raising till 12:00 and fading out by 18:00. Should you need the reaction rate, let the
coefficient of NO + O3 reaction be 1.9*10-14 molec cm2 sec-1.