A simple demonstration of chaos and unreliability of computer models

A simple demonstration of chaos and unreliability of computer models

December 5, 2015

Guest essay by  Anthony R. E.

Recently, I read a posting by Kip Hansen on Chaos and Climate. (Part 1 and Part 2) I thought it will be easier for the layman to understand the behavior of computer models under chaotic conditions if there is a simple example that he could play. I used the attached file in a course where we have lots of “black box” computer models with advance cinematic features that laymen assume is reality.
Consider a thought experiment of a simple system in a vacuum consisting of a constant energy source per unit area of q/A and a fixed receptor/ emitter with an area A and initial absolute temperature, T₀ . The emitter/receptor has mass m , specific heat C, and Boltzmann constant σ. The enclosure of the system is too far away such that heat emitted by the enclosure has no effect on the behaviour of the heat source and the emitter/receptor.
The energy balance in the fixed receptor/emitter at any time n is:
Energy in { q/A*A= q} + energy out {-2AσT_n ⁴ } + stored/ released energy {- mC( T_n+1 – T_n )} = 0 eq. (1)
If T_n+1 > T_n the fixed body is a heat receptor, that is, it receives more energy than it emits and if T_n > T_n+1 it is an emitter, that is, it emits more energy than it receives. If T_n = T_n+1 the fixed body temperature is at equilibrium.
Eq (1) could be rearranged as :
T_n+1 = T_n -2AC T_n ⁴ /mC +q/mC eq(2)
Since 2AC/mC is a constant, we could call this α, and q/mC is also a constant we could call this β to facilitate calculations. Eq (2) could be written as:
T_n+1 = T_n – αT_n ⁴ + β eq.(3)
The reader will note this equation exhibits chaotic properties as described by Kip Hansen in this previous post at WUWT on November 23, 2015, titled “Chaos & Climate –Part 2 Chaos=Stability”. At equilibrium, T_n=+1 = T_n , and if the equilibrium temperature is T_∞ then from equation (3)
T_∞ ⁴ =β/α or α = β / T_∞⁴ if eq. (4)
And eq (3) could be written as
T_n+1 = T_n – βT_n ⁴ /T_∞⁴ + β or T_n+1 =T_n +β(1-T_n⁴ /T_∞⁴ ) eq (5)
Eq (5) could be easily programmed in Excel. However, there are several ways of writing T⁴ . One programmer could write it as T*T*T*T, another programmer could write it as T ^2* T ^2, another programmer could write it as T*T ^3 and another could write as T^4. From what we learned in basic algebra, it does not matter as all those expressions are the same. The reader could try all the variations of writing T⁴ . For purposes of illustration, let us look at β= 100, T_∞ =243 ( I am using this out of habit that if it were not for greenhouse gases the earth would be -30^o C or 243⁰ K but you could try other temperatures) and initial temperature of 300 K. After the 17^th iteration the temperature has reached its steady state and the difference between coding T⁴ as T^4 and T*T*T*T is zero. This is the non-chaotic case. Extract from the Excel spreadsheet is shown below:

				beta=	100
Iteration	w T^4	w T*T*T*T	% diff.	T ∞=	243
0	300.00	300.00	0.00
1	167.69	167.69	0.00
2	245.01	245.01	0.00
3	241.66	241.66	0.00
4	243.85	243.85	0.00
5	242.44	242.44	0.00
6	243.36	243.36	0.00
7	242.77	242.77	0.00
8	243.15	243.15	0.00
9	242.90	242.90	0.00
10	243.06	243.06	0.00
11	242.96	242.96	0.00
12	243.03	243.03	0.00
13	242.98	242.98	0.00
14	243.01	243.01	0.00
15	242.99	242.99	0.00
16	243.00	243.00	0.00
17	243.00	243.00	0.00
18	243.00	243.00	0.00
19	243.00	243.00	0.00
20	243.00	243.00	0.00

If β is changed to170 with the same initial T and T∞, T does not gradually approach T∞ unlike in the non chaotic case but fluctuates as shown below. While the difference in coding T⁴ as T^4 and T*T*T*T is zero to the fourth decimal place, differences are really building up as shown in the third table.

				beta	170
Iteration	w T^4	w T*T*T*T	% diff	T ∞	243
0	300.0000	300.0000	0.0000
1	75.0803	75.0803	0.0000
2	243.5310	243.5310	0.0000
3	242.0402	242.0402	0.0000
4	244.7102	244.7102	0.0000
5	239.8738	239.8738	0.0000
6	248.4547	248.4547	0.0000
7	232.6689	232.6689	0.0000
8	259.7871	259.7871	0.0000
9	207.7150	207.7150	0.0000
10	286.9548	286.9548	0.0000
11	126.3738	126.3738	0.0000
12	283.9386	283.9386	0.0000

By the 69^th the difference between coding T⁴ as T^4 and T*T*T*T is now apparent at the fourth decimal place as shown below:

69	88.6160	88.6153	0.0008
70	255.6095	255.6088	0.0003
71	217.4810	217.4824	0.0007
72	278.4101	278.4086	0.0005
73	155.4803	155.4850	0.0030
74	296.9881	296.9894	0.0004

The difference between the two codes builds up rapidly that by the 95^th iteration, the difference is 4.5 per cent and by the 109^th iteration is a huge 179 per cent as shown below.

95	126.5672	132.2459	4.4866
96	284.0558	287.3333	1.1538
97	136.6329	125.0047	8.5105
98	289.6409	283.0997	2.2584
99	116.5073	139.9287	20.1029
100	277.5240	291.2369	4.9412
101	158.3056	110.4775	30.2125
102	297.6853	273.2144	8.2204
103	84.8133	171.5465	102.2637
104	252.2905	299.3233	18.6423
105	224.7630	77.9548	65.3169
106	270.3336	246.1543	8.9442
107	179.9439	237.1541	31.7934
108	298.8261	252.9321	15.3581
109	80.0515	223.3877	179.0549

However, the divergence is not monotonically increasing. There are instance such as in the 104^th iteration, the divergence drops from 102 per cent to 18 per cent. One is tempted to conclude T^4 ≠ T*T*T*T.
Conclusion:
Under chaotic conditions, the same one line equation with the same initial conditions and constant but coded differently will have vastly differing results. Under chaotic conditions predictions made by computer models are unreliable.
The calculations are made for purposes of illustrating the effect of instability of simple non-linear dynamic system and may not have any physical relevance to more complex non-linear system such as the earth’s climate.
Note:
For the above discussion, a LENOVO G50 64 bit computer is used. If a 32 bit computer is used the differences would be noticeable at a much earlier iterations. A different computer processor with the same number of bit will also give different results.