Planetary Heat Engine Theory

Taras K               Revised  15 May 2004
                               
(First Edition- 26 August 2000)

 

Abstract:

      This simple theoretical heat engine model uses an ice-air cycle.   The “engine” converts radiant solar energy into planetary mechanical energy.   The mechanical energy produced is in the form of increased planetary wobble.   Wobble or ‘free Eulerian precession’ is like a pendulum oscillation within a rotating frame, it increases a planet's rotational kinetic energy, while keeping its angular momentum constant.   On Earth, wobble energy could possibly cause geological deformation as well as dissipate into internal heat by friction.   Calculations are included for several cases, showing the theoretical power output of this heat engine process on an earth-like planet.   The predicted energy output of this heat engine cycle is big enough to drive many geophysical processes that are now generally considered to be powered by other sources.   Friction heating of a planetary interior can provide a governor mechanism for heat engine output and regulate internal planet temperature.


A Planetary Heat Engine Theory:

      For many years people have known that wind, ocean currents, and rain are powered by the sun.(1)   Geological processes like wind and water erosion are thus solar powered.   The solar heat engine idea can be expanded to explain some other planetary phenomena, including:  daily rotation and wobble, plate tectonics, glaciation, and the melting of subterranean lava.     This may seem like an outrageous assertion.    How could solar energy spin the Earth?   power tectonic motion?   cause an ice age?   or heat a planet’s interior?   Answers to these questions may be found by studying the behavior of a freely turning inertial body, like a planet, as it is perturbed by the thermodynamic mass transport of a multi-phase substance, like water, on the body and in its atmosphere.  

      First, a brief discussion of solid body rotation . . .   the bottom line is— Extra energy often causes spinning things to wobble.(2,3)     Rotary motion can be simulated or calculated knowing a body’s inertia and its initial spin conditions.   The inertia (a tensor) is generally written as a 3 x 3 matrix, which has diagonal form when expressed in the principal axis basis.(4)   The three diagonal elements are then the three principal moments of inertia.(5)   If not equal, the principal moments can be ordered: maximum, intermediate, and minimum moments of inertia.   The maximum and minimum axes have stable spin.(6)   The intermediate axis sits in an energy saddle and is a chaotic neutral point for the spin axis.   When turning about one of these principal axes the body’s angular velocity remains constant over time (simple spin).   When turning about any other axes the angular momentum remains constant, but not the angular velocity vector.   This kind of  “complex spin” motion is demonstrated by a wobbly USA football pass.  200 years ago, Leonhard Euler analyzed this motion and predicted that the Earth would wobble slightly, with a ten-month period. 

       In energy terms, a free body with three different principal moments of inertia and a fixed net angular momentum has simple stable spin at its lowest and highest kinetic energy levels.   All other energy levels imply a wobble.   There are energy saddles associated with the intermediate inertial axis, which can produce interesting quasi-stable and variable-frequency polarity reversing behavior.   

Note ~  Spin, rotation, and turning, as used in this paper, all mean essentially the same thing.  

 

      Planets and other objects frequently turn nearly about their maximum inertial axis, as this is the lowest kinetic energy state for a given amount of angular momentum.   If a body is free of external torques, yet has a way to dissipate internal mechanical energy, it will approach the maximum inertial axis spin state over time, as its rotational kinetic energy decreases.   A classic example of this is the spinning, bullet-like, Explorer 1 satellite with its energy-absorbing whip antennas; within a few days it was turning end over end. . .   its lowest energy state.(7)

      The Earth now turns very close to its lowest-energy, maximum inertial axis, but with a slight wobble.  Our planet oscillates a bit, causing the geographical positions of the North and South Poles to describe various approximately circular paths, up to about 20 meters in diameter.   The frequency spectrum of the solid earth’s oscillation has two major peaks.   There is a 12 month period ‘annual’ peak, with harmonics.   An internal oscillation within the {solid earth, atmosphere, and oceans}  system causes the annual wobble.   Seasonal changes in weather and solar flux are synchronized with this annual component.   With reference to the solid earth, the annual wobble is called a ‘forced oscillation’.  In addition to this annual term, there is 14 month period ‘Chandler Wobble’ spectral component due to ‘free oscillation’ of the above system as a whole; this is akin to the solid body wobbles or 'Free Eulerian Precession' discussed earlier.   These wobble components combine so that the astronomical latitude of a point on the ground changes by up to 20 meters (~ 0.7 arcseconds or 1/5000 th of a degree) in a combined semi-period of six and a half months.   The International Earth Rotation Service has observations of the wobble going back to 1846.(8)   The energy source of the Chandler Wobble continues to be a topic of considerable discussion in geophysics.(9)  

      When a surface area's astronomical latitude or "spin latitude" changes as a result of wobble, its perpendicular distance from the rotation axis changes (except right near the Equator).   This variation in distance causes variation in centripetal potential.   When a planet is solid and rigid the changes in potential can be quite significant.   For a totally rigid, earth-size planet, turning once a day, the change in centripetal potential, from equator to pole, is about 105 joules / kg, which is equivalent to 11 km of height change at one gee.   This is at a constant radial distance from the planet's center of mass and is separate from gravitational potential changes due to non-spherically symmetric mass distribution.

      The next figure shows how wobble induces rigid surface potential changes during oscillation of the surface relative to its spin axis. 

Color, digital versions of some figures may be found on the web at:  www.icarusengineering.com

::

       In the figures, the energy reference frame turns around the planet's fixed angular momentum vector at the rate of daily rotation, hence the ellipsoidal equipotentials (shown exaggerated) and almost-fixed position of the spin axis.   On Earth, near 45 degrees spin latitude, the slope of an equipotential relative to a constant radius arc is about   1:580   (  pi / 2 times the average slope from zero spin latitude to a pole).   Thus, near 45 degrees, a 17 meter change in spin latitude produces an equivalent height change of    17 meters / 580    or about 3 cm.   (17 meters along the surface is about 0.6 arcsecond of latitude.)

     On Earth, even in the case of large tectonic oscillations, the shifting of the surface relative to its spin axis would be very slow compared to the rate of daily rotation.   This is because Earth's principal moments of inertia are nearly equal (less than 1/2 percent different).   This near-symmetry also means that the rotation vector is always within 1/10 degree of the angular momentum vector.

      Real planets are not totally rigid; they flex and flow somewhat in response to changes in centripetal potential.(10)   A "Rigidity Index" can be used to compare the actual situation to the idealized rigid case.   The rigidity index is defined as the ratio of the actual surface potential change to a theoretical rigid case.   This is a simplified way of expressing what is traditionally described with Love numbers.(11)   The rigidity index can be a function of location as well  as  other  variables.   The average  planet-wide  index  effects  the  period  of  any      "Tectonic Oscillations"  (Chandler-type wobbles).    This average should be weighted to emphasize the effects of 45-degree latitude areas.   On the Earth we can compare the actual observed Chandler  'free oscillation'  period of 14 months with Euler’s calculated period of 10 months for a rigid, 0.33 % oblate Earth; this gives an average rigidity index of about 0.7

      Multiplying this rigidity index by the 3 cm rigid body height change, calculated previously, gives about 2 cm.   In theory, this means that often much of the mid-latitude surface land mass of the Earth goes energetically  "up and down"  by 2 cm. in little over a year.   This motion is rather similar to that of a piston.   And this movement could, perhaps, become the stroke of a huge planetary heat engine.

      The fluctuations in potential of surface areas produce no power by themselves, if there are no changes in mass distribution.   The power comes from varying the surface loading with movable water mass.   In this simple engine analogy, mass is added to the "surface pistons" in the form of snow and ice at the tops of their strokes and removed by melting and evaporation at the bottoms.   The mechanical energy from one cycle is equal to the difference in surface loading times the stroke length.   If the surface is loaded with an ice mass  1000 km x 1000 km x 0.1 meter  on the "down" stroke and empty on the "up" stroke the variation in mass loading is 1014 kilograms, producing a weight force variation of 1015 newtons.   With a potential height ‘stroke’ of 2 cm, this force difference yields 2 x 1013 joules of energy.   Over a period of 14 months this correspond to a power output of about 600 kW or 800 HP.   The energy from one cycle above is equal to approximately 2 % of the kinetic energy in the original  17 meter wobble.   This corresponds to an energy 'Q' of about minus 350.   For small wobbles, Energy is proportional to Amplitude squared; so a 2 percent increase in energy results in a 1 percent increase in amplitude.   The author does not currently know whether ice mass ‘forcing’ has actually been historically significant on Earth, and more references on the subject are sought.(12)

      To model the efficiency of this heat engine cycle, the thermal energy required to return the mass of ice to the top of a stroke is compared with the mechanical energy output.   To return the ice mass, it is melted and evaporated.   Once the H2O is vaporized and in the atmosphere, wind can move it around; allowing it to re-freeze and fall on ground that is at the top of a stroke.   How much energy is required?   The combined heat of fusion and vaporization for water is about 2.5 x 106 joules / kg, plus there's the pressure-volume work of 105 joules needed to push 1 kg (~1 cubic meter) of vapor into atmospheric pressure.   In the case above, with 1014 kg of ice, it means that at least 2.6 x 1020 joules of solar energy go into each cycle.   Dividing the output of 2 x 1013 joules by the solar input gives a thermodynamic efficiency of 8 x 10-8 or eight millionths of a percent—   not very much!   However the energy output and efficiency of the cycle are proportional to the length of the stroke, which could, perhaps at times, increase dramatically.   Another way to look at the cycle efficiency is to convert the heat of fusion + vaporization + P-V work into an equivalent (1 gee) height change—   2.6 x 106 joules / kg corresponds to 265 km of height change at 9.8 m/s2, thus the maximum efficiency for a rigid-earth, ice-air cycle (90 degree amplitude oscillation) would be the ratio of the centripetal equipotential oblateness to this height---   11 km /  265 km = about 4%.    At the current rigidity index of 0.7 the maximum efficiency would be about 3%.   These numbers are presented for comparison only, rigidity indices will most likely lessen as wobble amplitudes increase and also rigidity may become longitudinally polarized,   i.e., the wobble pole path could become very elliptical or otherwise elongated.

      What is needed for this type of heat engine to work?


    

 Planetary Heat Engine Checklist . . .

  • Spin Rate

A planetary spin rate fast enough to produce substantially oblate equipotentials (imaginary "sea level" surfaces) in the planet’s combined gravitational and centripetal acceleration field

  • Rigidity

A rigid or semi-rigid planet that resists deformation by changing relationship to equipotentials

  • Complex spin

A motion producing changes in "Spin Latitude" of surface areas.   In various forms this motion is known as: ‘Wobble', 'Free Nutation’, ‘Free Eulerian Precession’, ‘True Polar Wander’, ‘Pole Shifts’, or “Tectonic Oscillation”. 

  • Heat

A source of radiant heat, like a very close-by star, that is located approximately perpendicular to the planetary spin axis

  • Heat Sink

A heat sink to low temperature, like space.
This allows condensation and freezing in cool areas of the planet and its atmosphere (particularly at night or during dark winter months).

  • Working Fluid

An atmosphere containing a compound, such as water  or  CO2 , which can freeze, precipitate and remain frozen on the ground for a time.

 

 

 

     The following calculations are for a variety of hypothetical heat engine cycles on an earth-like planet.   The first case is similar to the one worked through in the text.   The other cases show how much power could be produced if wobble amplitude were increased.

:
Heat Engine Calculations for an Earth-Like Planet
Wobble Oscillation Amplitude   
Compared to Typical - 1x typ. 500x 30,000x 300,000x
      20 th Century Values
Meters at Surface 17 7,500 450,000 4,500,000
degrees, minutes, secconds 0.6" 4'  4 deg. 40 deg.
Oscillation Amplitude, rad 2.7E-06 1.2E-03 7.0E-02 7.0E-01  
Oscillation Period, days 430 430 602 1003  
Mean Latitude, degrees 45 45 45 45  
Rigidity Index, ratio 0.7 0.7 0.5 0.3  
Centripetal Equipotential    
           Oblateness, m 11000 11000 11000 11000  
Potential "Height" Change, m 0.021 9 389 1500  
Surface Area m ^2 1E+12 5E+12 1E+13 2E+13  
Variation in Ice Thickness, m 0.1 1 10 100  
Ice Mass per Cycle, kg 1.0E+14 5.0E+15 1.0E+17 2.0E+18  
Heat per Cycle, joules 2.5E+20 1.3E+22 2.5E+23 5.0E+24
Srface Load Change, N 9.8E+14 4.9E+16 9.8E+17 1.96E+19  
Energy per Cycle, joules 2.0E+13 4.4E+17 3.8E+20 2.9E+22  
Power Output, watts 5.4.E+05 1.2E+10 7.3E+12 3.4E+14  
Horse Power 723 1.6E+07 9.8E+09 4.5E+11  
Energy in Wobble, joules 1.1E+15 2.2E+20 5.6E+23 3.2E+25
Cycle Output / EiW(above) 1.8E-02 2.1E-03 6.9E-04 9.2E-04
System "-Q" 3.5E+02 3.1E+03 9.2E+03 6.9E+03
Energy Flux Ratio* 5.5E+01 4.9E+02 1.5E+03 1.1E+03
Heat Eng. efficiency, (ratio) 8.1E-08 3.6E-05 1.5E-03 5.9E-03
Power / Planet's Solar Flux 3.2E-12 7.0E-08 4.3E-05 2.0E-03  
Cycle / Earth Spin Energy 6.7E-17 1.6E-12 1.4E-09 1.1E-07  
Yearly Output / ESE 5.7E-17 1.4E-12 8.6E-10 4.0E-08
Water, Heat of Fusion & Vap. 2.5E+06
( joules / kg )   & Mars-Like ?
Total Earth Solar Flux, W 1.7E+17 Revised
Earth (mantle) Spin Energy, J 3E+29 Apr-02
 
:

      The energy flux ratio, above*  is an interesting parameter.   It shows the number of cycles whose energy output equals the total kinetic energy in the wobble oscillation.  It is like Q / 2 pi .   When  the ratio is multiplied by the cycle period, the resulting time scales are roughly seventy to three thousand years, in the different cases.   This gives an indication of how long oscillations would take to build up, without any damping.  The calculations assume uniform planet-wide rigidity, if continental crust (surface piston area) stays more rigid than the planet-wide average, then the cycle may self-excite more easily.  This is because a smaller average rigidity index means that less kinetic energy is needed for a given size wobble.


      A key factor governing cycle power output is the timing between the surface potential changes and the variations in surface loading.  The previous calculations assume the most powerful timing relationship.  Any actual feedback mechanisms that connect Earth's wobble to global snowfall distribution are not yet known.   Theoretically, in large oscillations the variations of an area's spin latitude could produce self-energizing feedback via local changes in sun angle and atmospheric pressure-altitude.  This process is shown in the graphs of the preceding figure.   In smaller amplitude oscillations the strongest feedback may come from variations in ocean currents.   These variations are caused by the movements of the oceans relative to the planet's spin axis.   Changes in ocean currents could likely produce rapid changes in worldwide snowfall distribution and stronger currents could perhaps generate much bigger snowstorms than we now experience.

      Wobble induced Coriolis accelerations produce ocean currents by making the waters turn in their basins.   With a wobble of 20 meters (3 x 10-6 radians) the net swirl induced over a 7 month period is about 3 x 10-6 revolutions per day, which corresponds to a flow rate of about 100 meters / day around the periphery of a 10,000 km diameter basin.   This is small compared to typical ocean currents, but it extends to all ocean depths.   For wobbles less than 1 radian, or so, the undamped swirl is roughly proportional to the wobble amplitude.  

      The constant angular momentum heat engine modeled here, if not limited by damping, would continue to pump kinetic energy into the planet until it was spinning close to its minimum inertial axis, where rotation has the maximum kinetic energy for a given angular momentum.   As noted before, damping tends to do the opposite, inducing maximum inertial axis spin.   By way of a side note~   There are heat engine models that rely on tidal transfer of angular momentum between a planet and its parent star, such cycles can increase planetary spin rate without inducing wobble.    These angular momentum exchanging cycles will hopefully be discussed in future papers. 

      On oscillating planets damping can come from many processes.   Energy is dissipated by ocean tides and currents; they can damp wobble, as can subterranean friction.  Underground friction heating may be particularly important to the cycle, because in addition to providing damping, it could regulate the heat engine’s wobble production.  

      When mechanical energy is dissipated inside an earth-like planet it generates heat at high temperatures.   In this way, a planetary phase-change heat engine could pump a small fraction of Earth's incident solar heat flux down into the planet’s incandescent interior.   We do not normally consider that heat might “naturally” get from a relatively cool place (Earth’s surface and atmosphere) to a much hotter place  (Earth’s interior).   Man-made machines that move heat “uphill” (against a temperature gradient) are called ‘heat pumps’.

      Friction heating of the Earth's crust and mantle would likely reduce rigidity.   This reduces the fluctuations of potential at a given wobble amplitude.  Smaller potential fluctuations produce less heat engine power output for the same variations in surface loading.   So wobble production can depend on internal heat content.   The next figure shows how surface potential changes are related to planetary rigidity.

::

      Note that the rigid planet need not be spherical, though it makes the diagram simpler.     It's the rigid, constant radius motion (relative to the spin axis) that produces the upper circular arc C1-C2-C3.

      There are many factors governing planetary heat engines.   The dynamics of a spinning, visco-elastic, cosmic body that is perturbed by heat-driven shifting of surface mass could be quite intricate.  The motion could be seen as a sort of   "planetary dance".   And this kind of heat engine “dancing” activity would surely leave some significant marks . . .   such as evidence of fluctuating ice deposits or indications of varying Coriolis currents in deep sea sedimentation.  

     The "unrigid" part of planetary behavior during tectonic oscillations could cause a complicated pattern of changing strain within a planet.   These cyclic strains might be drivers of plate tectonics, particularly during orogenies and glacial epochs.   Also, strains tend to release heat in localized areas underground, creating 'hot spots'.   This is because movement, and its associated friction heating, generally occur at the weakest (often hottest) place.  Comparing the Earth’s tectonic motion patterns with those produce by various types of theoretical tectonic oscillations could prove quite interesting.

     Possibly some of the heat-flow and isotope-ratio data fit problems experienced with current geophysical paradigms(13) may be resolved by the energy production of planetary heat engine cycles.  As can be seen in the power output chart the theoretical mechanical output of an earth-like engine could become geologically significant at fairly small wobble amplitudes.  The second column case in the chart produces 1.5 x 1010 watts with less than one tenth of a degree of wobble.   What might that mean in geological terms?   To get a rough idea of the energy scale we can compare this heat engine output to the potential energy in a large non-isostatically compensated mountain range.    If the mountain range is 10,000 km long,   200 km wide, and  6 km high, then the volume is about 6 x 1015 cubic meters.  At a density of 3000 kg/m3, it has a mass of 1.8 x 1019 kg, with a weight of about 1.8 x 1020 newtons.  If the range is triangular in cross section then the center of gravity is  2000 meters  above the base.  2000 meters times  1.8 x 1020 newtons  equals   3.6 x 1023 joules of energy.  This energy divided by the  1.5 x 1010 watt  engine output equals  2.4 x 1013 seconds, or about  760,000 years, not very long in the geological time frame. 

      The friction heat produced by the engine cycles may also be significant.   The theoretical power output of the engine in the fourth column case --   3 x 1014 watts   is an order of magnitude greater than Earth’s current outward subterranean heat flux of approximately 3 x 1013 watts.(14)   So heat engine activity could possibly account for the heat production now ascribed to ‘radioactive decay’ and ‘primordial heat’.   We now know of little direct evidence showing ancient wobble amplitudes, but do have some evidence of fluctuating ice age glacial deposits, these might be indicative of  “ice-engine” activity.

     It is an intriguing possibility that there could be a feedback relationship between the output of a phase-change heat engine and planetary rigidity---    As a planet cools and hardens, the engine modeled here would produce more power and “rev up” (more wobble).   This would produce more friction and heat the planet until it became fluid or elastic enough inside to slow or shut-down the engine cycle.   This feedback process provides an internal thermostatic action for spinning planetary bodies.   Perhaps it’s an aspect of Gaia theory.(15)  Animals that thermoregulate are called warm blooded. . .   but who ever heard of a "warm blooded planet"?

Acknowledgments:

      Thanks to Caltech geology professors James Westphal and Gene Shoemaker for conveying the vision of rocks moving and forming a dynamic system.   Thanks to Bob Leighton, CIT professor and family friend, for teaching how to analyze situations and make quick mental calculations.   Thanks to Caltech professor E. C. Stone for a wonderful    Feynman I  physics course, which encouraged the author’s interest in dynamics.   Thanks to math professors Bohnenblust and Dean for a great course in numerical modeling.  Thanks to classmate Mike Purucker for pointing out a possible correlation between episodes of geomagnetic reversals and ice ages (circa 1975).   Thanks to Jacques Labeyrie for fascinating discussions, including many on geophysics (c 1991).   Thanks to John Dickenson for suggesting that the cause-effect relationship between glaciation and planetary shifting could work two ways (c1998).   Thanks to CIT professor Tom Caughey for interesting rotational dynamics discussions.   Thanks to Joe Kirschvink, CIT professor and former Lloyd House-mate for valuable discussions and references.   Thanks to Professor David Stevenson for valuable discussions and comments.    Thanks to Professor  Bill Iwan,  Professor Tom Caughey and the Caltech Civil Engineering Department for hosting the author's seminar talk on Planetary Heat Engines (5-11-2000).   Thanks to Dr. Dan Gezari for organizing an informal lunch seminar at NASA Goddard (2-11-2000).   Thanks to Cari and Bob Leidig for reading over early manuscripts and giving many helpful suggestions.  And particular thanks to James Lovelock for the Gaia Hypothesis, which stimulated a lot of thought on self-regulation and Earth's dynamics.

 Taras K,   Copyright   2004

 

References:

(1)   Gamow, ch 6
(2)   Feynman, vol 1, ch 20-4
(3)   Munk and MacDonald, ch 2
(4)   Apostol, vol II, chs 2, 3 & 4
(5)   Goldstein, ch 5
(6)   Webster, ch VII, arts. 84-88
(7)   Personal communication with Professor Tom Caughey, Caltech, 1999;   also Leighton and Vogt, B-17, p. 91
(8)  IERS Website- http://hpiers.obspm.fr/
(9)  Anderson- TotE;  Cannon;  Cazenave;  Chao, et. al.;  Gross;   Lambeck- TCAR, TEVR ch 5;   Munk and MacDonald, ch 10-6;   White
(10) Lambeck- GG
(11) Munk and MacDonald, ch 5
(12) Chao, et. al.
(13) Anderson- AtotE:HaHDaH, TIoE:DESFtTD
(14)  Skinner and Porter,  Figure CI.1, p. 15
(15) Lovelock
 

 

Bibliography, [with some notes]:

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Anderson, Don L.   Theory of the Earth, Blackwell Scientific Publications, 1989.

Apostol, Tom M.   Calculus vol. II, Blaisdell Publishing Co., Waltham, Mass., 1969.   [linear algebra can be very powerful-- a great book]

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Cazenave, Anny (Editor)   Earth Rotation: Solved and Unsolved Problems, D. Reidel Publishing Co., Holland, 1986.

Chao; O'Connor; Chang; Hall and Foster   "Snow Load Effect on the Earth's Rotation and Gravitational Field, 1979-1985",  Journal of Geophysical Research, vol. 92,  pp. 9415-9422, 1987.

Decker, R. and B. (introduction by)   Volcanoes and the Earth's Interior, Scientific American, W. H. Freeman and Company.

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Gamow, George   A Planet Called Earth, The Viking Press, 1963.   [The figures on pages 145,146 highlight the idea of Earth as a heat engine, complete with bolt-up pipe flange fittings]

Gold, T.   "Instability of the Earth's Axis of Rotation"  Nature, vol. 175,  pp. 526-529, 1955.   [key early reference on ‘True Polar Wander’]

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International Earth Rotation Service Website ---  http://hpiers.obspm.fr/

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Lovelock, James   Gaia: A new look at life on Earth, Oxford University Press, 1995.   [recent edition of an important book, which many years ago, prompted my interest in the Earth's energy balance]

Miller, Russel   Continents in Collision,  Time Life Books, 1983.

Mound; Mitrovica; Evans; and Kirschvink   "A sea level test for inertial interchange true polar wander events"  Geophys. J. Int.,  vol. 136,  pp. F5-F10,  1999

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Skinner and Porter    The Blue Planet,   Wiley, John & Sons Inc., 1994.

Steinberger and O'Connell   "Changes of the Earth's Rotation Axis Owing to Advection of Mantle Density Heterogeneities", Nature, 8 May 1997, pp 169-173.

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White, John   Pole Shift,   Doubleday and Company, 1980.   [lots of wild ideas collected here]

Wilson, J. Tuzo   Continents Adrift, Scientific American, Inc., W.H. Freeman and Co., 1972.           [a very influential and popular classic]

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