lördag 10 maj 2014

Basic Atmospheric Thermodynamics as 2nd Law

The debate on the temperature distribution in the atmosphere is going around in never-ending circulation just like the air in the atmosphere. Let us here recall the basic statements of my chapter Climate Thermodynamics in a famous book, which is condensed as the 2nd law of thermodynamics expressed in the following form with the dot signifying time differentiation:
  • $\dot K+\dot P = W-D$
  • $\dot E = -W + D$,  
where $K$ is kinetic energy, $P$ potential energy, $W$ work, $E$ heat energy and $D\ge 0$ is turbulent dissipation with $W > 0$ under expansion and $W < 0$ under compression. The sign of $D$ sets the direction of time with always transfer of energy from $K+P$ to $E$.

There are two basic temperature distributions with linear decrease with height as lapse rate (assuming zero heat conductivity): 
  • Isothermal atmosphere with zero lapse rate: $D$ maximal with $W=D$.
  • Maximal (dry adiabatic) lapse rate $=9.8\, C/km$ with $D=0$ minimal.
The observed lapse rate (of about 6.5 C/km) is somewhere between maximal and minimal. We note:
  1. Lapse rate may increase by slow laminar vertical circulation with ascending air cooling and descending air warming with $D=0$.
  2. Lapse rate may decrease by turbulent dissipation $D>0$ heating upper layers.
  3. A (partially) transparent atmosphere (like on Earth) heated from below may naturally develop a positive lapse rate by 1. 
  4. An opaque atmosphere (like on Venus) heated from above may become isothermal by heat conduction and may then develop a positive lapse rate by 1.  
The lapse rate is basic to planetary climate since it determines the surface temperature from the temperature at the top of the troposphere, and its dependence on the radiative properties of the atmosphere is a key question in global climate science.

Compare with the previous post Lapse Rate by Gravitation: Loschmidt or Boltzmann/Maxwell?

2 kommentarer:

  1. In your related post "Lapse Rate by Gravitation: Loschmidt or Boltzmann/Maxwell?" you said "Maxwell claimed that the atmosphere would be isothermal."

    However, Maxwell actually said the atmosphere would not be isothermal in his book Theory of Heat, 1888, pp. 330-331:

    ”This result is by no means applicable to the case of our atmosphere. Setting aside the enormous direct effect of the sun’s radiation in disturbing thermal equilibrium, the effect of winds in carrying large masses of air from one height to another tends to produce a distribution of temperature of a quite different kind, the temperature at any height being such that a mass of air, brought from one height to another without gaining or losing heat, would always find itself at the temperature of the surrounding air. In this condition of what Sir William Thomson has called the convective equilibrium of heat, it is not the temperature which is constant, but the quantity ϕ [entropy], which determines the adiabatic curves.
    In the convective equilibrium of temperature, the absolute temperature is proportional to the pressure raised to the power (γ-1)/γ, or 0,29.
    The extreme slowness of the conduction of heat in air, compared with the rapidity with which large masses of air are carried from one height to another by the winds, causes the temperature of the different strata of the atmosphere to depend far more on this condition of convective equilibrium than on true thermal equilibrium.”


  2. The study of the thermodynamics of the atmosphere lies wholly within the discipline of physics. Correct me if I’m wrong. but my understanding is that climatology courses do not include anywhere near as much physics as is covered in a degree in science with a major in physics, such as I have.

    More importantly, physics is about understanding the real world. If you use equations and “laws” of physics, you need to know about the prerequisites for these to apply. Many of my students over the years have not understood such.

    For example, the Stefan-Boltzmann equation only applies to true black and gray bodies, and such bodies have to be perfectly insulated against any loss of thermal energy by conduction and other sensible heat transfers. And of course a black body is not transparent like the thin surface layer of the oceans. So all James Hansen’s fiddling with back radiation in order to get the magical 390W/m^2 to “explain” the surface temperature of 288K is garbage, because radiation is not the primary determinant of that temperature, for the simple reason that it is not a black or gray body and only the solar radiation should be counted anyway.

    You have to understand that you can only claim that sensible heat transfers occur only from warmer to cooler regions in a horizontal plane, because, as we saw above, the equations for thermodynamic potentials are derived with the assumption that gravitational potential energy does not vary.

    You have to understand that the Second Law is all-pervasive and plays a part in determining what happens in all heat transfers and other energy transfers also. That’s why the density gradient is a result of the Second Law. To claim that the state of thermodynamic equilibrium has no density gradient in a gravitational field would be ludicrous. It is just as ludicrous to claim that it has no temperature gradient. There must be no unbalanced energy potentials, and that sure ain’t the case for the assumed isothermal troposphere.

    It is not greenhouse gases which establish the temperature gradient: it is gravity. Until you understand that fact, you will never understand the downward convective heat transfers which explain why all planets’ surfaces are hotter than their tropospheres.