Wednesday, May 09, 2007

Cute climate analogies: Introducing opacity

And - unlike the "greenhouse" - these are even accurate.

My favorite is the analogy between temperature and heat flow on the one hand, and pressure and fluid flow on the other. The pressure one is easy to visualize: think of a vertical hose with water flowing in and out. The water pressure at any point in the hose is the result of:
  • The mass of water in the hose at any instant
  • The force of gravity downward
  • The hose cross-sectional area available to the water
  • The rate at which water is flowing through the hose
Assume a steady flow of water and incompressibility - that is, the water density at every point in the hose is the same constant. Then there's a water flow balance: kg of water flowing in per second = kg of water flowing out per second.

Back up for a moment and just have a mass of water in the hose, with no flow. If you squeeze the cross-sectional area down, the height of the top of the water must go up (to preserve the volume and thus the mass). But then the gravitational potential energy difference between the top of the water and the bottom must increase and the pressure difference between top and bottom decrease. If the water top is open to its environment, it must have the same pressure as the ambient air. The pressure difference becomes smaller by the pressure at the bottom of the hose increasing.*

This linear pressure-height relationship is analogous to the linear temperature-altitude relationship in an adiabatic atmosphere. The pressure falls linearly with height as the water does work against gravity, just as temperature falls linearly with height as the air's heat is converted to work against gravity. The analogue to the pressure boundary condition (pressure at the top = ambient air pressure) is the temperature boundary condition (heat flowing out the top of the atmosphere = heat that flowed in - not reflected by clouds - then supplemented by latent heat released in the water cycle).

Now switch on the flow and create an analogue to the pseudoadiabatic atmosphere, one with local thermodynamic equilibrium, but supplemented by steady flows in and out. Hold the flow of water (kg/sec) fixed. There's a new kind of energy in the system, namely the kinetic energy of the flowing water. Squeezing the cross-sectional area down again pushes more liquid up and increases the pressure at the bottom of the hose. But it also increases the fluid's velocity and kinetic energy. Again, the pressure at the top has to be equal to the ambient air pressure. It's the pressure at the bottom of the hose that rises.

If you decrease the cross-sectional area of the hose, but insist on keeping the flow rate unchanged, then something has to give, and that something is the pressure: it goes up. If you increase the cross-sectional area of the hose and leave everything else unchanged, the pressure goes down. OTOH, if you change the cross-sectional area and insist on the pressure remaining the same, then what gives is the flow rate.

The analogy with temperature and heat flow goes like this:
  • Pressure > temperature
  • Mass flow > heat flow
  • Incompressibility > temperatures are steady
  • Cross-sectional area > radiative opacity or thermal conductivity
A source of steady work (a pump) is needed to keep a steady water flow upward against gravity - just as steady heating from the Earth's surface is needed to keep heat flowing up in the air.

The analogy is misleading in one respect. In the Earth's atmosphere, the effect of gravity is dominant. The effect of radiative heat flow (as well as convective heat flow, for that matter) in the atmosphere's temperature distribution is small.** (In the earlier discussion of the lapse rate, those are the rad and conv terms.) That's where the radiative opacity comes in. (If we were talking about heat flow in a solid or liquid, we'd be talking about the thermal conductivity instead.) Upping the opacity - making the atmosphere more opaque to infrared radiation - is like squeezing the hose: with a constant heat or water flow, it raises the temperature or pressure at the bottom, respectively.

The Earth's atmosphere is like a very wide, vertical hose with a strong pump: gravity and heating from below dominate the picture. Changing the hose cross-section a bit only has a small effect on the pressure; we'll see soon that changing the opacity a bit in the air has a real but small effect on the temperature. Soon after that, we'll meet a different planet - Venus - whose atmospheric temperature profile is dominated by opacity, not by gravity, and whose heat flow is like a narrow hose at high pressure.

Further analogies of this type are possible. Opacity is also analogous to electrical resistance. Recall Ohm's law: voltage difference across two points in a circuit = resistance times electrical current flowing between the points. The analogy is:
  • voltage difference > temperature difference
  • electrical current > heat flow
  • resistance > opacity (or thermal conductivity if the medium is a solid or liquid)
The one thing missing is an analogy to gravity. It would have to be a strong electric field parallel to the current wire, although that's not something typical in electrical circuits.**
---
* If you drop the analogy to temperature for a moment and consider real temperature effects on the liquid, heating the liquid causes its pressure to increase and thus the height of the top of the fluid to rise. If the relation of pressure to temperature is proportionate (linear), then the linear relation between the pressure difference and the height yields: temperature change proportional to change in height of fluid column. That's a how a thermometer works.

** And unfortunately, there's no obvious analogy in these examples to evaporation, condensation, and the release of latent heat.

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2 Comments:

At 9:32 PM, Blogger Biby Cletus said...

Nice post, its a really cool blog that you have here, keep up the good work, will be back.

Warm Regards

Biby Cletus - Blog

 
At 8:27 PM, Blogger Binah said...

Thanks for looking and thanks for your comments. Hope you make it back.

 

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