fredag 24 augusti 2012

A Discrete Model Atmosphere

(A few additional comments since this post seems to have attracted certain attention. Although it might look odd, in reality it describes an ordinary diffusion process. The corresponding differential equation is described on the post "A new attempt". Here, the thermal conductivity k is normalized to 1 but if you want to vary it all you have to do is to divide the forcing vector F by k. The derivation of this I leave to the reader.)


This post is a bit technical but if you bear with me I hope it might shed some light on certain important topics. The reason why I have constructed a discrete model is that by doing so I have avoided the tricky problem of specifying the boundary conditions, which you are invariably faced with if you formulate the model in terms of differential equations. The model which I am going to present admittedly bear some SUPERFICIAL ressemblance with the greenhouse effect, but one of the questions I am asking is if it ACTUALLY reproduces the greenhouse effect. Take a look at the diagram below:

The atmosphere has been discretized into a finite number of layers each having its own "weight" w1, w2 and so on. The arrows may represent either a "real" or a "formal/mathematical" energy flow. What do I mean by this? There is actually a point in not specifying this at the moment. It could be interpreted as if the "weight" of the atmospheric layer is simply a function of the amount of greenhouse gases present and that the arrows represent upward or downward radiation. On the other hand, if we allow a certain "flexibility" in our thinking, the "weight" could be interpreted as a function of the actual weight (mass) of the layer and the arrows merely illustrate a "formal" energy conserving representation of the obstruction to cooling caused by these layers. Although that may sound like mumbo-jumbo in your ears, somehow we have to account for the eventual energy loss to outer space, and I haven't found any better way to do it than this. Now for the technicalities:

The amount of energy absorbed per unit time by each layer is the amount of incident energy times its wight. (Hence, the wight must be less than 1). Each layer loses energy per unit time proportional to its temperature AND weight. This loss is equally distributed upwards and downwards. Keep in mind that the energy that is not absorbed by the layer is free to pass on to the next and so on. (This is accounted for by inserting a factor (1-w) for each intermediate layer). Below is a Mathematica worksheet where the energy balance equations for the STATIONARY state of a model with four layers are written down and solved. Staionary means that all of the layers have reached energy balance, hence, the amount of energy lost per unit time equals the amount of energy absorbed per unit time. The left hand sides represents the energy lost and the right hand sides contains the absorbed energy.

This model is linear in temperature, unlike mainstream climatology where one is instead obsessed with the SB-law. A linear dependence on temperature may very well be physically motivated, especially if we are dealing with ideal gases, though admittedly, here it is primarily employed for computational efficiency, things would become very much more cumbersome with a T4-dependence. If the wight of the layers are assumed to decline exponentially with altitude, from the analytical and numerical solutions we see not only the formation of a stratosphere but we can also infer with almost certainty (although I have no strict proof for this) that the temperature approaches the finite value F/2 at the top of the atmosphere, where F is the solar forcing at the surface. This can be verified also with a numerical algorithm solving the equations for a model atmosphere discretized in 60 layers as is shown below (The Python-script for this is appended at the end)


What is then the main conclusion to be drawn from this. The most obvious one is that this model does not predict a cooling of the stratosphere upon an increases in the weight-function. On the contrary it either produces a warming of the entire atmosphere upon an increase in the weight or solar forcing, or conversely a cooling of the entire atmosphere upon a decrease in solar forcing or weight. In other words, the temperature gradient does not pivot, and this holds true even if the weight is interpreted as the amount of greenhouse gases present. (Admittedly it also somewhat contradicts some of my previous statements/guesses, which the reader can verify on his/her own.) 

How the cooling of the stratosphere comes about in the mainstream models is very obscure. From the most simple equations I can see that this follows from foolishly applying a Dirichlet boundary condition at the surface rather than at the top of the atmosphere, but how it emerges in their time-stepping algorithms remains a mystery to me. One of the lessons to be learned, however, is that there is much mathematical subtlety involved and that any "wordy" explanations from people like Roy Spencer must now be replaced by reproducible algorithms/equations.

Moving on, one could now ask if the model presented above could actually be useful as a model atmosphere. My answer to that is: Yes maybe, if the weight is interpreted as a function of the TOTAL MASS. The reason why I believe this is that:

1. It predicts an elevation in surface temperature as a function of atmospheric mass

2. It predicts the formation of a stratosphere whose temperature is a function of solar forcing alone

3. It can easily be extended so as to reproduce a thermosphere higher up if you add solar forcing terms to the uppermost layers.


Below is the Python code for your enjoyment:

***

import numpy as np
import matplotlib.pyplot as plt

interval = 6
meshsize = 0.1
N = int (interval/meshsize)

weight = np.zeros(N)
weight[0] = 1


for idx in range(1,N):

    weight[idx] =  np.exp(-idx*meshsize)*meshsize
    


A = np.zeros([N,N])
    
for idx1 in range(N):
    for idx2 in range(N):
        if idx1 == idx2: 
            if idx1 == 0:
                A[idx1,idx2] = -1
            else:
                A[idx1,idx2] = -2
            
        else:
            A[idx1,idx2] = weight[idx2]

            if idx2>idx1:

                for idx3 in range(idx1+1, idx2):
                    A[idx1,idx2] = A[idx1,idx2]*(1-weight[idx3])

            else:
                for idx3 in range(idx2+1, idx1):
                    A[idx1,idx2] = A[idx1,idx2]*(1-weight[idx3])    
            
F = np.zeros(N)
F[0] = 1



temp = np.linalg.solve(A,-F)
print temp

x = np.arange(0,interval,meshsize)

plt.plot(x, temp)
plt.ylabel('Temperature')
plt.xlabel('Position')
plt.show()

***

onsdag 22 augusti 2012

När ska någon punktera Lennart Bengtsson?

När man trodde att det värsta klimattjatet hade upphört på debattsidorna sprutas det tändvätska på brasan igen av ingen mindre än vår egen klimatsvamlarnestor Lennart Bengtsson. Jag noterar dock att det denna gången sker på Newsmill och inte på Svd:s debattsida. Tycker du att det blivit lite för tyst Lennart? Har du inte fått stå i rampljuset på ett tag? 

Lennarts budskap innebär bland annat att de fattiga länderna inte kan utvecklas ekonomiskt utan tillgång till fossilt bränsle men samtidigt att detta hotar att leda till en klimatkatastrof för hela mänskligheten. När detta sedan applåderas av välkända "klimatskeptiker" torde väl debatten ha nått en sådan grad av principlöshet att varje person med en gnutta intellektuell heder i kroppen borde tycka att måttet nu är rågat. 

Hur har vi hamnat i denna situationen? Jag ber er först betänka det faktum att Lennart Bengtssons specialområde är den grundläggande strukturen hos atmosfären, med andra ord den så kallade växthuseffekten. Detta har han gemensamt med en anan välkänd skeptiker-guru Richard Lindzen som råkar vara elev till Richard Goody, en av förgrundsgestalterna i formuleringen av den moderna versionen av teorin. Börjar poletten trilla ner?


När man granskar litteraturen på området framträder en bild som ganska radikalt skiljer sig från den som man matas med i den alldagliga debatten. Detta kommer iochförsig inte som en överraskning. Debatten har hittills handlat om många saker men en sak som aktivt motarbetats är en seriös och öppen debatt om grunderna för hela spektaklet, just det, växthuseffekten. I korta drag går den ut på att några spårgaser i atmosfären värmer upp jordytan samtidigt som den kyler den övre atmosfären, denna värmepump arbetar dessutom med en sådan infernalisk effektivitet att ytan skulle förvandlas till en bastu om inte konvektion inträdde och räddade oss. Detta är naturligtvis så absurt att ledande klimatforskare, inklusive många skeptiker, inser att detta måste hållas hemligt för allmänheten.

Om du ifrågasätter denna hypotes, vilken som sagt hålls hemlig, stämplas du dock som förnekare. 

Jag är övertygad om att det finns vettiga människor där ute som nu tycker att Lennart Bengtsson och hans lakejer har fått härja ostört lite för länge nu. Om du är en av dessa råder jag dig till att noggrant studera relevant material, till exempel det som presenteras på denna blogg och börja ställa relevanta frågor till ledande befattningshavare och forskare. Det kommer att mötas med hårt motstånd, men med lite ihärdighet tror jag att vi kan vinna striden. Om ni har frågor med vilka jag kan vara behjälplig skicka gärna ett email. Jag är säker på att även Claes Johnsson kan vara behjälplig. Lycka till!

söndag 19 augusti 2012

Is Robert G. Brown a denier?

Recently I became aware of a blog post on WUWT written by Robert G. Brown at Duke University dating back to january this year, so I apologize for not having noticed it earlier. The aim of text is to disprove the long lasting conjecture that an isolated column of air maintains a nonzero temperature gradient solely by the influence of a graviational field. I will not comment on the details of the refutation here since there is already plenty of material out there that does, instead I would like to focus on a sentence in the concluding part of the text:

In nature, the dry adiabatic lapse rate of air in the atmosphere is maintained because the system is differentially heated from below causing parcels of air to constantly move up and down.

If it hasn't already become clear to the reader I would like to once again point out the obvious fact that this is not in accordance with the greenhouse gas hypothesis. Hence, that would make Robert G. Brown a denier in the eyes of Fred Singer, wouldn't it. To clarify things a bit further, at present there appears to be three explanations for the lapse rate advocated by various people:

1. The gravity theory

This theory (which dates back at least 150 years) claims that gravity alone induces a lapse rate even in a state of equilibrium. The theory is thus essentially an equilibrium theory.

Keynote: Gravity destabilises the atmosphere

2. The differential heating theory

This theory claims that it is the differential heating of the surface and the atmosphere that causes the lapse rate. The same argument could be used to explain why it s hotter 1cm away from a lit candle than it is 1m away. The lapse rate is however reduced in magnitude by convection, but the convection itself is limited by pressure induced bouyancy. This is essentially a non-equilibrium theory.

Keynote: Sunlight destabilises the atmosphere

3. The greenhouse gas hypothesis

The radiative transfer of greenhouse gases in the atmosphere creates a temperature gradient (see the two previous posts) which is limited in magnitude by convection is the same way as expressed above. This doesn't fall into any of the categories Equilibrium/Non-equilibrium.

Keynote: Greenhouse gases destabilise the atmosphere



To my knowledge, the only person in the world that openly defends the greenhouse gas hypothesis is Roy Spencer, and the only one in the world that points out that he is indeed defending the greenhouse gas hypothesis as it is presented in the literature is myself. The reason for this state of affairs puzzles me a bit. Recently if have become more and more interested in Theory nr 2, simply because it seems to put the pieces together in a more satisfactory way than Theory nr 1, though I welcome all well argued developements of either theory. In any case, the main point of this post is:

Theory nr 2 is not identical to Theory nr 3 

I will elaborate on this in detail in an upcoming post. This is just a preemptive step to make sure that the lukewarmers not yet again take the opportunity to obfuscate things so as to make people believe that the greenhouse gas theory is simply Theory nr 2 (or just any theory that is not Theory nr 1).