måndag 1 augusti 2011

The heat equation revisited

Let's recall the heat equation in one spatial dimension:

dU/dt = D*d^2U/dx^2

Here U is the heat-content or temperature at a given position and time, and D is the diffussivity constant. In order to completely solve the heat equation one needs to specify boundary conditions which might depend on various energy sources et.c., but we will not be so much concerned with that here. Often one is interested in finding a stationary solution that does not change in time, and for such a solution we have that dU/dt = 0 everywhere. The stationary one-dimensional heat equation is thus D*d^2U/dx^2 = 0.

Now let's try to construct a simple radiation model for an almost ideal gas. Many of the thermodynamic properties of an ideal gas, such as the energy content and pressure, are proportional to temperature so let's assume that we can interchangeably speak about the heat content and temperature. Suppose that the radiation is proportional to the temperature U and also proportional to a parameter A which is a measure of the emmisivity. Let's discretize the position variable with an index n. To start with, also suppose that every layer absorbs all incoming radiation (an unphysical assumption that we will later relax). If we now try to find a stationary solution, that is a solution that does not change in time, then there is no build-up or loss of energy and hence we must have balance between incoming and outgoing radiation. Now take the perspective of the layer at position n

In words:

Heat absorbed from the adjacent layers - Heat lost by radiation = 0

In numbers:

AU(n-1) + AU(n+1) - 2AU(n) = 0

Notice in particular the "back-radiation" term AU(n+1). However, what we have just written is simply the discrete form of the stationary heat equation:

A*d^2U/dx^2 = 0,

now with the diffusivity constant simply given by the emissivity parameter A. Relaxing the assumption that all incoming radiation is absorbed leads in its simplest form to the model I described in the post "A simple radiation model". One can of course take into account all sorts of other circumstances, for example a position dependent thickness of the gas and so on, but very quicly such more elaborate models become analytically intractable and one would probably need to use computers. 

However, the main message is that a model which incorporates "back-radiation" does not necessarily lead to a greenhouse effect.

3 kommentarer:

  1. Hi Anders,

    It's pleasing to see a quasi-familiar formulations at the top of your post. However, I'm not sure that the way you use them illuminates the problem you have set yourself.

    As an aside, of course heat diffusion plays almost no role in atmospheric thermodynamics, so your approach will not in any case come near to representing the real processes at work. But you know this, I believe. So we won't worry about that, and just look at your maths :-)

    Perhaps you do not realise it, but you have implicitly assumed your conclusion in your starting conditions. In other words, you have discovered nothing but the situation you began with. I will add some quick notes that might help you see where your confusion comes from.

    You give a form of the stationary heat equation in one dimension as D*d^2U/dx^2 = 0. On reflection, you must realise that, by simple algebra, the factor D divides out.

    So you see that your factor A, which you use for emissivity by analogy to D, likewise vanishes in the stationary case.

    Remember that by using the homogeneous stationary heat equation, you have assumed that the system is in equilibrium, or in other words that its temperature is constant throughout.

    So, from the beginning you have assumed an imaginary static, homogeneous system at constant temperature. There cannot possibly be a 'greenhouse effect', or any other effect, in this system.

    Still you manage to 'discover' that

    AU(n-1) + AU(n+1) - 2AU(n) = 0

    But you fail to notice that, because the system is stationary, we can divide both sides by A and hey presto, A vanishes from the equation.

    So what you have 'discovered' is that U(n)=(U(n-1)+U(n+1))/2, or in fact that, in a stationary system in thermal equilibrium, the temperature at one point is the same as the average temperature of the surrounding point.

    Which is the assumption you began with.

  2. And that should be 'formulation', singular, of course...

  3. Hi AT,

    I shall confess that my experience in thermodynamic modelling is limited, to say the least, so I'm quite open to the possibility that I will make some major and minor mistakes here and there. Nevertheless,

    I do not assume a priori that the process is diffussive, I merely try to construct a model, including backradiation as Roy and others advice us to do, and I end up with something that looks quite familiar to a diffussive heat equation.

    You are right that the constants divide out, but that changes quickly as one adds some further complications and inhomogeneties, as I do in the slightly more complicated model with variable absorption.

    I do not agree that "stationary" is the same as "equilibrium". A house can be heated by radiators and reach a stationary state without it beeing in equilibrium. The general solution to the equation d^2U/dx^2 = 0 is U = Bx + C which is not necessarily constant but can be linear.

    I am not so sure that diffusive heat transport is so insignificant as you state, especially in the thin atmosphere. But perhaps it is more a matter of definition, I argue that the raditive heat transfer processes that you think are so dominant in the atmosphere might actually be (formally) diffusive radiative processes.