Problem 4 - Solutions
These solutions are based on the J's given in the first array of PS3-Solutions.
Analytical Solution
z O3P O3 O1D
20 6.547826494089E+06 1.511558689720E+13 4.556588652574E+00
30 2.348009143518E+08 1.337130829941E+13 2.001469166125E+02
40 2.463123236634E+09 1.318923612015E+12 1.027247982852E+03
50 9.864428434788E+09 9.406740016156E+10 1.013681369012E+03
Family Approximation [The family approximation looks too good.
I have to check with Andrzej]
z O3P O3 O1D
20 6.547793790082E+06 1.511566239430E+13 4.556611411151E+00
30 2.347594274285E+08 1.337367129052E+13 2.001822867779E+02
40 2.460468749599E+09 1.320346538288E+12 1.028356233648E+03
50 9.836918249376E+09 9.433047153758E+10 1.016516257104E+03
Integration (using the exponential Euler solution) dt=1000sec
z O3P O3 O1D
20 6.547826494023E+06 1.511558689705E+13 4.556588652529E+00
30 2.348009143517E+08 1.337130829940E+13 2.001469166124E+02
40 2.463123236634E+09 1.318923612014E+12 1.027247982851E+03
50 9.864428434787E+09 9.406740016155E+10 1.013681369012E+03
At 20km 2644050 time-steps required, 30km - 71766, 40km -10126,
50km -4044.
With the linear Euler method similar results, but a much smaller
time-step is required.
Lifetime of O3P at:
50km - 15sec; 40km - .77sec; 30km - .029sec; 20km .001sec.
Lifetime of ozone at:
50km - 141sec; 40km - 414sec; 30km - 1642sec; 20km - 2530sec.
Lifetime of O1D at:
50km - 1.6e-6sec; at 40km - 4.0e-7sec; at 30km - 8.2e-8;
at 20km - 1.66e-8sec.
SAME SOLUTION AS THE ANALYTICAL SOLUTION ABOVE, but in PPM
20 3.53142217054340968E-6, 8.1522500233814803, 2.45749306342695525E-12
30 6.16881364949643102E-4, 35.129807469750119, 5.25836551579102009E-10
40 2.99925566721665066E-2, 16.06005359019785, 1.25084254347536859E-8
50 0.47899361696078502, 4.5676933579426606, 4.92220009095017816E-8
_________________________________________________________________________
VALUES USING J's CALCULATED WITH NO O3 ABOVE 50km
20 6.585953034144E+06 1.519394541920E+13 4.611053552082E+00
30 2.373298258442E+08 1.344575844667E+13 2.044560671470E+02
40 2.633277891042E+09 1.286197888340E+12 1.120012005719E+03
50 1.109646432812E+10 8.803362780613E+10 1.153275879640E+03
_________________________________________________________________________
VALUES USING J's CALCULATED WITH O3 ABOVE 50km
but with 'the average number density' approach for O3
20 6.847039276488E+06 1.572832921709E+13 4.992879264566E+00
30 2.186711190216E+08 1.287220226158E+13 1.735448471797E+02
40 2.321146271924E+09 1.346769827867E+12 9.497835542096E+02
50 9.864428434788E+09 9.406740016156E+10 1.013681369012E+03
__________________________________________________________________________