Miskolczi [2007] proposes that feedbacks constrain the gray infrared optical depth of Earth's atmosphere to a value close to 1.841. If true this would imply a very low value for climate sensitivity to a doubling of carbon dioxide, contrary to most findings (see e.g. Boer and Yu 2003; Boer et al. 2000; Dai et al. 2001; Delworth et al. 1999; Goosse et al. 2006; Hegerl et al. 2006; Roeckner et al. 1999; Sumi 2005; Washington et al. 2000; Wetherald et al. 2001). However, this conclusion depends on some doubtful assumptions.

1. "According to the Kirchhoff law, two systems in thermal equilibrium exchange energy by absorption and emission in equal amounts..." [Miskolczi 2007]. In fact, Kirchhoff's Law states that for a body in local thermodynamic equilibrium (LTE), emissivity and absorptivity must be equal at a given wavelength. Miskolczi confuses emission with emissivity. This can lead to large numerical errors, since emissivity is of course constrained to the range 0 - 1 by definition, but emission can have any nonnegative value, and is typically in the hundreds of watts per square meter for low levels of atmosphere.

2. Miskolczi proposes that when greenhouse gases increase, water vapor decreases. This would seem to violate the Clausius-Clapeyron law, but of course the complexity of atmospheric processes might, in theory, lead to some net feedback of the type described. As an empirical matter, however, “[t]he global trend in precipitable water vapor is found to be 0.9 ± 0.06 mm/decade” [Brown et al. 2007]. With temperature rising [NASA GISS 2008], and water vapor rising as well, and at a rate consistent with the Clausius-Clapeyron law and a fixed relation of relative humidity to altitude [Manabe and Wetherald 1967; Inamdar and Ramanathan 1998; Held and Soden 2000], the proposed feedback seems unlikely to exist.

The consequences of this observation alone mitigate strongly against the Miskolczi paper's conclusions. Note, too, that if those conclusions held, it is difficult to see how Earth could have undergone ice ages, or why Venus is as hot as it is.

Miskolczi proposes that the gray infrared optical thickness of Earth's atmosphere is constrained to stay near τ = 1.841. This is noticeably lower than most other estimates; e.g. Hart [1978] estimates this value as τ = 2.49. Taking Earth's radiative equilibrium temperature as 254 K, its surface temperature as 288 K, and assuming the known surface heat loss mechanisms enumerated by Kiehl and Trenberth [1997] (atmospheric absorption 67 watts per square meter, sensible heat 24 W/m^{2}, latent heat 78 W/m^{2}, window radiation 40 W/m^{2}) hold, τ = 2.07 is implied. This is a minor point, but once again points out that Miskolczi's findings are outside the usual consensus.

3. Miskolczi states: "The atmosphere is a gravitationally bounded system and constrained by the virial theorem: the total kinetic energy of the system must be half of the total gravitational potential energy. The surface air temperature t_{A} is linked to the total gravitational potential energy through the surface pressure and air density. The temperature, pressure, and air density obey the gas law, therefore, in terms of radiative flux 4 S_{A} = σ t_{A}^{4} represents also the total gravitational potential energy."

This would be correct if the Earth's atmosphere were in orbit around the Earth. But the atmosphere rotates with the Earth.

A simple check of this proposition would be to find the gravitational potential and kinetic energies (U and K, respectively) of the Earths atmosphere and determine if the ratio between them is actually 2:1.

The Earth's atmosphere has a total mass of about 5.136 x 10^{18} kg [Walker 1977, 20]. 99% or so of its mass is in the troposphere and stratosphere, the two respectively constituting 80% and 20%, roughly. The US Standard Atmosphere [NOAA 1976] uses a surface temperature of 288.15 K, a tropospheric lapse rate of 6.5 K km^{-1}, and an isothermal stratosphere at 216.67 K. The present author divided the atmosphere into 20 layers, four of stratosphere on top and 16 of troposphere, and used a simple iterative model to fit the facts listed above:

Table I. Simple Model Earth Atmosphere.

Layer | Pressure (Pa) | Temp. (K) | Altitude (m) | |
---|---|---|---|---|

1 | 2468.6 | 216.67 | 26598.2 | |

2 | 7405.9 | 216.67 | 18141.8 | |

3 | 12343.1 | 216.67 | 14759.2 | |

4 | 17280.4 | 216.67 | 12584.7 | |

5 | 22217.7 | 216.82 | 10973.4 | |

6 | 27154.9 | 225.30 | 9668.7 | |

7 | 32092.2 | 232.60 | 8545.4 | |

8 | 37029.4 | 239.04 | 7555.2 | |

9 | 41966.7 | 244.81 | 6667.2 | |

10 | 46904.0 | 250.06 | 5860.4 | |

11 | 51841.2 | 254.87 | 5119.9 | |

12 | 56778.5 | 259.33 | 4434.6 | |

13 | 61715.7 | 263.48 | 3796.0 | |

14 | 66653.0 | 267.37 | 3197.7 | |

15 | 71590.3 | 271.03 | 2634.2 | |

16 | 76527.5 | 274.49 | 2101.5 | |

17 | 81464.8 | 277.78 | 1595.9 | |

18 | 86402.0 | 280.91 | 1114.6 | |

19 | 91339.3 | 283.89 | 655.0 | |

20 | 96276.6 | 286.75 | 215.2 |

The gravitational potential energy of an object is

U = G M m r^{-1} | (1) |

where G is the Newton-Cavendish constant (6.67428 x 10^{-11} m^{3}/kg/s^{2} in the SI), M and m the masses of the reference and target bodies, respectively, and r the radial distance from M's center of gravity. For the Earth, M = 5.9736 x 10^{24} kg. Taking the Earth's volumetric mean radius as 6,371,010 m [Lodders and Fegley 1998, 128), the mean altitudes of the atmosphere layers in the Table I model and their individual masses (5.136 x 10^{18} kg / 20 = 2.568 x 10^{17} kg) lead to a total gravitational potential energy of U = 3.210 x 10^{26} J for Earth's atmosphere.

The mean molecular velocity for a gas is

v = (8 R T / (p MW))^{0.5} | (2) |

where v is the velocity, R the universal gas constant (8,314.472 J/K/kmol in the SI), T the absolute temperature, p the circle constant and MW the gas's mean molecular weight. The mean temperature of the model atmosphere in Table I is 249.76 K, which leads to v = 427.6 m s^{-1} and a total kinetic energy for the atmosphere of K = 4.696 x 10^{23} J.

The observed ratio of U to K, therefore, is (3.210 x 10^{26}) / (4.696 x 10^{23}) or about 684. This is substantially different from 2. Increasing the resolution of the atmosphere model or using local saturated lapse rates for each level makes no substantial difference to this conclusion.

It is therefore impossible not to conclude that the model proposed in Miskolczi [2007] is fatally flawed, and thus so is its conclusion of startlingly low climate sensitivity to a doubling of atmospheric CO2.

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Note added 2/02/2010: Certain smart-ass deniers insist that the potential energy of the atmosphere should be measured relative to the ground, not to the center of the Earth, and that if you do that, the ratio is indeed two to one. Noting that

Ep = m g h | (3) |

where m is the mass of the object in question (for an atmosphere layer with 5% of the atmosphere's mass, 2.568 x 10^{17} kg), g local gravity (9.80665 m s^{-2}), and h height above the reference level in question (same as altitude z, meters).

We can see from the table that the total potential energy measured that way is about 3.682 x 10^{23} Joules. Wow! Much closer! But the ratio Ep/Ek is then 0.784, which again, is not 2. Correct for the change in gravity with height and the disparity widens. So once again, the Earth's atmosphere is not in free orbit around the Earth, the virial theorem does not apply, and Ferencz M. Miskolczi is still **Wrong**.

Page created: | 08/06/2008 |

Last modified: | 02/09/2011 |

Author: | BPL |