Model of Atmosphere with CO2 shows Small Emissivity

Here is a another argument indicating that the effect of the atmospheric trace gas CO2 on the radiation balance of the Earth is small.

Our model of blackbody radiation consists of collection of oscillators with small damping with equal oscillator internal energy T representing temperature, with oscillator resonance frequencies n varying from 1 to a cut-off set at T and each oscillator radiating

  • E_n = \gamma T n^2

where \gamma is a universal constant, which is Planck’s Law.  Summing over n from 1 to T, we obtain the total radiance

  • E =\sum_n \gamma T n^2 = \sigma T^4

which is Stefan-Boltzmann’s law with \sigma = \gamma /3.  In the case of only one resonance frequency n = T, the radiance would be reduced to

  • e = \gamma T T^2 =\gamma T^3 = 3 E/T

with the reduction factor 1/T.

The radiance of an atmosphere which is fully opaque over the entire spectrum would radiate E, while an atmosphere opaque only for a specific frequency near cut-off T, would radiate e\approx 3E/T with a reduction factor  1/T.

We conclude that the emissivity of transparent atmosphere with a trace gas like CO2 with only a few isolated resonances, would scale like 1/T and thus be small as soon as T is bigger than say 100 K.

We thus find theoretical evidence from a basic model that the emissivity of the Earth’s atmosphere with the trace gas CO2 would be small, and thus that CO2 would have little effect on the Earth’s radiation balance.

We thus find theoretical evidence from a basic model that the emissivity of the Earth’s atmosphere with the trace gas CO2 would be small, and thus that CO2 would have little effect on the Earth’s radiation balance.

Blackbody as Transformer of Radiation

Our blackbody model, which has a high frequency cut-off scaling with temperature, absorbs all incoming radiation (forcing), re-emits frequencies below cut-off and stores frequencies above cut-off as internal energy, which is assumed to be equally distributed over frequencies below cut-off.

The result is that a blackbody subject to forcing with frequencies above cut-off, will heat up with a corresponding increase of cut-off until the highest frequency of the forcing is reached, assuming the blackbody is not somehow cooled. The effect is that the blackbody acts as a transformer of radiation from high to low frequencies by absorbing frequencies above current cut-off while emitting frequencies below cut-off.

The Earth subject to forcing from the Sun acts as such a transformer, see Blackbody: Transformer of Radiation.

Kirchhoff’s Law 2

Kirchhoff’s Law states that for a blackbody emissivity = absorptivity. In our wave equation with small damping as a model of blackbody in radiative equilibrium with an exterior forcing, this is built into the model for frequencies below high-frequency cut-off.

This is because for these frequencies the outgoing radiation simply reproduces the part of the incoming radiation which is not reflected. Or put differently: In radiative equilibrium there is below cut-off no distinction between incoming radiation and outgoing radiation as explained in the previous post.

However, if the incoming waves (the forcing) contain frequencies above the present cut-off  of the absorbing body, then the game changes, as these frequencies contribute to the internal energy of the body increasing its temperature and are not (re-)emitted. This is a case with positive absorptivity and zero emissivity of the frequencies above cut-off.

This leads to coefficients of absorptivity depending on temperature, which complicates the picture.

The effect is that different bodies with different cut-off at the same distance to the Sun, can take on different equilibrium temperatures, as evidenced in e.g. Satellite Thermal Control Engineering and What Is the Temperature of Space?

String and Soundboard

Our model of blackbody radiation has an acoustic analog in the form of a string interacting with a soundboard, with the following key components (as total or for each frequency in a spectral decomposition):

  • internal string energy
  • string amplitude
  • soundboard amplitude
  • force balance between string and soundboard = wave equation
  • small damping in the interaction expressed by the wave equation.

The small damping is an important feature of the model since it creates an “optimal” interaction between the string and the soundboard in the sense that the tone generated by the soundboard will be both loud and have long sustain. With zero damping the interaction will small and the tone weak, and with too big damping the sustain will be short. In radiation the damping is small.

The effect of the small damping is that the string velocity is out-of-phase with the forcing, which is the key to optimal interaction between string and soundboard.

Radiative equilibrium corresponds to acoustic equilibrium, which can be interpreted in two different ways:

  1. The vibration of the soundboard is sustained by the vibration of the string = outgoing acoustic waves.
  2. The vibration of the string is sustained by the vibration of the soundboard = incoming acoustic waves.

The same equilibrium can thus be interpreted as incoming or outgoing, which can be expressed as absorptivity = emissivity, which thus is built into the model.

This argument applies to frequencies below cut-off with only radiative damping being active. For frequencies above cut-off the interaction between string and soundboard is different with internal heating of the string, and emissivity in general different from absorptivity.

There is also non-equilibrium dynamics below cut-off with e.g. the string energy being transferred to the soundboard as the string amplitude decreases.

Equidistribution 1

Equidistribution in our wave model of blackbody radiation means that the internal energy of different frequencies is the same, that is, all frequencies have the same temperature. More specifically, the internal energy IE_\nu of the wave component of frequency \nu with amplitude U_\nu is measured by (with subindices indicating differentiation with respect to time t and space x and integration in space and time over a period)

  • IE_\nu = \frac{1}{2}\int (U_{\nu ,t}^2+U_{\nu ,x}^2)dxdt.

Equidistribution means that IE_\nu is the same for all frequencies and then defines a common temperature T =IE_\nu.

In the piano model equidistribution corresponds to pressing all the keys with the same force to give all strings of the piano the same string energy.

Planck’s Law E(\nu ,T) =\gamma T\nu^2  assumes equidistribution with all frequencies having the same internal energy measured by temperature T.

With equidistribution as a characteristic of blackbody radiation the radiation intensity spectrum shows the characteristic quadratic increase with frequency.

Without equidistribution the spectrum can show different dependence on frequency.

Kirchhoff’s Law 1

Kirchhoff’s Law of Thermal Radiation: 150 Years starts off by:

  • Kirchhoff’s law is one of the simplest and most misunderstood in thermodynamics.

Let us see what we can say about Kirchhoff’s Radiation Law stating that the emissivity and absorptivity of a radiating body are equal, in the setting of the wave model with damping presented in Computational Blackbody Radiation and Mathematical Physics of Blackbody Radiation:

  • U_{tt} - U_{xx} - \gamma U_{ttt} - \delta^2U_{xxt} = f,

where the subindices indicate differentiation with respect to space x and time t, and

  1. U_{tt} - U_{xx} models a vibrating material string with U displacement
  2. - \gamma U_{ttt} is a dissipative term modeling outgoing radiation
  3. - \delta^2U_{xxt} is a dissipative term modeling internal heating by friction
  4. f is the amplitude of the incoming forcing,
  5. T is temperature with T^2=\int\frac{1}{2}(\vert U_t^2+U_x^2)dxdt,
  6. the wave equation expresses a balance of forces,

where \gamma and \delta^2 are certain small damping coefficients defined by spectral decomposition as follows in a model case:

  • \gamma = 0 if the frequency \nu > \frac{1}{\delta}
  • \delta = 0 if the frequency \nu < \frac{1}{\delta},

where \delta = \frac{h}{T} represents a “smallest coordination length” depending on temperature T and h is a fixed smallest mesh size (representing some atomic dimension).

This represents a switch from outgoing radiation to internal heating as the frequency \nu passes the threshold \frac{T}{h}, with the threshold increasing linearly with T.

The idea is that a hotter vibrating string is capable of radiating higher frequencies as coherent outgoing radiation. The switch acts as a band filter with frequencies outside the band being stored as internal heat instead of being radiated: The radiator is then muted and heats up internally instead of delivering outgoing radiating.

A spectral analysis, assuming that all frequencies share a common temperature, shows an energy balance between incoming forcing f measured as

  • F = \int f^2(x,t)\, dxdt

assuming periodicity in space and time and integrating over periods, and (rate of) outgoing radiation $R$ measured by

  • R = \int \gamma U_{tt}^2\, dxdt,

and (rate of) internal energy measured by

  • IE = \int \delta^2U_{xt}^2\, dxdt,

together with the oscillator energy

  • OE =T^2 = \frac{1}{2}\int (U_t^2 + U_x^2)\, dxdt

with the energy balance in stationary state with OE constant taking the form

  • F = \kappa (R + IE)

with \kappa\lessapprox 1 is a constant independent of T, \gamma, \delta and \nu. In other words,

  • incoming energy = \kappa\times outgoing radiation energy for \nu <\frac{1}{\delta}
  • incoming energy = \kappa\times stored internal energy for \nu >\frac{1}{\delta},

which can be viewed as an expression of Kirchhoffs’ law that emissivity equals absorptivity.

The equality results from the independence of the coefficient \kappa of the damping coefficients \gamma and \delta^2, and frequency.

Summary: The energy of damping from outgoing radiation or internal heating is the same even if the damping terms represent different physics (emission and absorption) and have different coefficients (\gamma and \delta^2).

PS: Note that internal heat energy accumulating under (high-frequency) forcing above cut-off eventually will be transformed into low-frequency outgoing radiation, but this transformation is not part of the above model.