söndag 16 oktober 2011

Stability of True-SB vs False-SB

The different stability properties of the two versions discussed in recent posts of Stefan-Boltzmann's radiation law, True-SB and False-SB, can be exhibited by comparing the stability of the following simplified SBs:
  1. $R = T_1 - T_2$
  2. $R = T_3$ where $T_3 = T_1 - T_2$,
where $R$ is the heat flow between two bodies of temperature $T_1$ and $T_2$, expressed in two different forms: 1. as the difference of two-way gross flows and 2. as one-way net flow.

Stability concerns the effect on the output $R$ from perturbations of input ($T_1$, $T_2$ and $T_3$). Denoting the perturbation of $T_i$ by $dT_i$ for $i=1,2,3$, it is natural to assume a bound $E$ on relative perturbation of the form
  • $\frac{\vert dT_i\vert}{T_i}\le E$ for $i=1,2,3$.
Estimating the effect $dR$ on the output $R$ from the $dT_i$, we have

$\vert dR\vert \le \vert dT_1\vert +\vert dT_2\vert = \frac{\vert dT_1\vert}{T_1} T_1 + \frac{\vert dT_2\vert}{T_2} T_2 \le ET_1 + ET_2=E(T_1+T_2)$,

which is to be compared with

$\vert dR\vert \le \frac{\vert dT_3\vert}{T_3}T_3 \le E T_3$.

If $T_3$ is much smaller than $T_1+ T_2$, then the second bound is much smaller which expresses that 2. is more stable than 1. or that 1. is more unstable than 2.

A particular case is given by $T_1=T_2$ with $T_3 =0$. This is the case of two blackbodies of equal temperature which according to 2. is very stable, but according to 1. much less stable.

The different stability gets expressed in different assessments of climate sensitivity, from 1. unstable with alarm to 2. stable without alarm.

12 kommentarer:

  1. "it is natural to assume a bound $E$ on relative perturbation"

    No it isn't. This statement is meaningless.

    "If $T_3$ is much smaller than $T_1+ T_2$, then the second bound is much smaller which expresses that 2. is more stable than 1. or that 1. is more unstable than 2."

    Refer to your own equation 2 which says:

    $T_3 = T_1 - T_2$

    It is not even clear what you are attempting to prove here, but it is clear that you are doing it extremely ineptly.

  2. I explain why one-way net flow is more stable than two-way gross flow,
    which is not difficult to understand.

  3. You don't explain anything. You waffle; you use words like "stable" without any rigorous definition, and you keep on attempting to find some difference between mathematically identical statements, for reasons you seem unable to elucidate.

  4. Stable means small effect of small cause. Climate sensitivity is about stability and since climate sensitivity is the basic question of climate science, stability is
    very important.

  5. I kind of agree. It would be very helpful if you (Claes) would write a post where you give a rigorous explanation of the stability concept that you are using.

  6. "Stable means small effect of small cause" - define small. You are not even making a token effort here. Your insincerity is very immature.

  7. In climate science a small effect would be doubled CO2 corresponding to a radiative forcing of less than 3 W/m2 about 1% of total forcing, and a small effect (stable) would be 1% of total warming by the atmosphere of 33 C , that is 0.3 C. A big effect (unstable) would be the 10% of IPPC = 3 C.

    If an input perturbation of 1% gives an output effect of 1%, then the problem is stable.

    If 1% gives an effect of 10% then the problem is unstable.

    Do you get the message?

  8. As a control engineer I find it very weird to used the word unstable here. What's wrong with "high gain" (compared to low gain).

  9. Hi gain from a steady non-oscillating force may be stable, but an oscillating force mat trigger instability. The later is like driving your car with only full throttle or full brake, which will quickly make you lose your drivers licence.

  10. An unstable system (a car for example) is unstable independently of the input. Haven't you studied Control Engineering (that is feedback control) at the university?

  11. There is something to what you are saying, but it misses the point here which is instability from cancellation of opposing forces. A car is stable under normal driving, otherwise we would all be dead by now, but unstable under bang- bang driving. May this is not included in basic courses at the university.