Mathematics of Climate Seminar 9//6 uko s Energ alance Model Richard McGehee School of Mathematics Universit of Minnesota Mathematics of Climate Seminar September, 6 Conservation of Energ temperature change ~ energ in energ out short wave energ from the Sun long wave energ from the Earth Everthing else is detail. Dnamical Models Dnamical Models Model Equilibrium Perfectl Thermall Conducting lack o R QT T Q dd Heat Transport T R Qs( ( ( T C( T T t Plus lbedo Switch to Surface Temperature Dependence on Latitude R Q( T T ( Q R Q( ( T T ( Q T(, t R Qs( ( T(, t t T( ( Qs( global mean temperature T( t T(, t Second Law of Thermonamics: Energ travels from hot places to cold places. Equilibrium temperature profile? heat capacit uko s Equation T R Qs( ( ( ( T C( T T t Smmetr assumption: albedo sin(latitude OLR sin(latitude Chlek and Coakle s quadratic approximation: s. 3 T T( heat transport uko s Equilibrium T R Qs( ( ( ( T C( T T t albedo depends on latitude equilibrium solution: T = T( Qs( ( T ( C T T ( Integrate: Qs( ( T ( CT T ( Q s ( Q s ( ( T( C T T( T T T Q T equilibrium global mean temperature T Q Richard McGehee, Universit of Minnesota
Mathematics of Climate Seminar 9//6 Solve for T (. where uko s Equilibrium Qs( ( T ( C T T ( Global mean temperature at equilibrium: T Q ( s ( Qs( ( CT T ( CT ( ( C T ( T ( Qs( ( CT C Equilibrium temperature profile: T ( Qs( ( CT C T Q and ( s ( temperature (Celsius 6 6 uko s Equilibrium Qs( ( T ( C T T ( Equilibrium temperature profile: T ( Qs( ( CT C C = 3. α( =.3: ice free α( =.6: snowball (constant albedo 8...6.8 sin(latitude ice free snowball ice free (C= snowball (C= uko s Equilibrium 6 temperature (Celsius ice free snowball ice free (C= snowball (C= ice melts albedo decreases more sunlight absorbed REPET 6 8 ice won t melt...6.8 (no exit from snowball sin(latitude ice will form (icecap Wh would it stop? http://www.i-fink.com/melting-polar-ice/ ice melts albedo decreases more sunlight absorbed REPET Wh would it stop? M. I. uko, "The effect of solar radiation variations on the climate of the Earth," Tellus XXI, 6-69, 969. Wh would it stop? uko s Equation sin(latitude T T( T R Qs( ( ( ( T C( T T t heat capacit albedo OLR heat transport http://www.inenco.org/index_principals.html Richard McGehee, Universit of Minnesota
Mathematics of Climate Seminar 9//6 What if the albedo is not constant? Ice Line ssumption: There is a single ice line at =η between the equator and the pole. The albedo is α below the ice line and α above it. (, Equilibrium condition: Equilibrium solution: where Qs( (, T ( C T T ( T ( Qs( (, CT C T Q ( ( (, s( equilibrium temperature profile: T ( Qs( (, CT, where T Q ( C.3, albedo: (,.6, global albedo: let: then: ( (, s ( s ( s ( S( s(, S( s(, since s( ( S( ( S( ( S(.6.3 S( 3 S( s(. 3. Chlek & Coakle equilibrium temperature profile: T( Qs( (, CT, where T Q ( C.3, albedo: (,.6, global albedo: ( (, s ( s ( s ( let: S( s(, S( s(, since s( then: ( S( ( S( ( S(.6.3 S( 3 S( s(. 3. Chlek & Coakle IM Mathematics Summer Schol of Climate 7/8/6 9//6 For each fixed η, there is an equilibrium solution for uko s equation. C T ( Qs( (, CT C 3 3 5 3.5 5 66.5...3..5.6.7.8.9 sine(latitude Dnamics T R Qs( ( ( ( T C( T T t Let X be the space of functions where T lives. (e.g. L ([,] Let L : X X : LT CT ( C T, f ( Qs( ( uko s equation can be written as a linear vector field on X. R f LT The operator L has onl point spectrum, with all eigenvalues negative. Therefore, all solutions are stable. True for an albedo function. experts onl For each fixed η, there is a globall stable equilibrium solution for uko s equation. How to pick one? C 3 3 5 T R Qs( ( (, ( T C( T T t 3.5 5 66.5...3..5.6.7.8.9 sine(latitude Richard McGehee, Universit of Minnesota 3
Mathematics of Climate Seminar 9//6 Summar If we artificiall hold the ice line at a fixed latitude, then the will come to an equilibrium. However, if the temperature is high, we would expect ice to melt and the ice line to retreat to higher latitudes. If the temperature is low, we would expect ice to form and the ice line to advance to lower latitudes. How to model this expectation? T( 3 3 5 For each fixed η, there is a stable equilibrium solution for uko s equation. Standard assumption: Permanent ice forms if the annual average temperature is below T c = C and melts if the annual average temperature is above T c. dditional condition: The average temperature across the ice boundar is the critical temperature T c. looks oka T T T c 3 3 3 3 5 5....6.8.....6.8.....6.8. T( ( ( not good T( looks oka T( 3 3 5 ice line condition:.. η..6.8. ( ( T T T c T η (η T η (η Equilibrium: T R Qs( ( (, ( T C( T T t T ( Qs( (, CT C T ( T ( T c.3, (,.6, (,, (, Ice line condition: lbedo: T ( Qs( CT T ( Qs( CT C C Ice line condition: T ( T ( Qs( CT T c C where:.7 T R Qs( ( (, ( T C( T T t Ice line condition: Qs( CT T c C Rewrite: h( Qs( CT T c C Recall equilibrium GMT: Recall average albedo: where: T Q ( ( (, s( ( S(.6.3 S( S s ( ( 3. Q C h s S( Tc C can be written: Two equilibria (zeros of h satisf the additional condition. T R Qs( ( (, ( T C( T T t T( T ( T c The additional condition: Q C h s S( Tc C h(η 6 6 8...3..5.6.7.8.9 η Richard McGehee, Universit of Minnesota
Mathematics of Climate Seminar 9//6 Equilibrium temperature profiles Interesting Solutions: small cap large cap ice free snowball temperature (ºC 3 - - -3 - -5 T ( Qs( (, CT C....6.8. sin(latitude ice free snowball small cap big cap T( 3 3 5 Dnamics of the Ice Line T R Qs( ( (, ( T C( T T t Idea: If the average temperature across the ice line is above the critical temperature, some ice will melt, moving the ice line toward the pole. If it is below the critical temperature, the ice will advance toward the equator. stationar 3 3 3 3 5 5....6.8.....6.8.....6.8. Widiasih s equation: T( ice melts d T( Tc T( stationar Widiasih s Theorem. For sufficientl small ε, the sstem has an attracting invariant curve given b the graph of a function Φ ε : [,] X. On this curve, the namics are approximated b the equation d h( experts onl Dnamics of the Ice Line d T( Tc State space: [,] X T R Qs( ( (, ( T C( T T t unstable stable h(η 6 6 8...3..5.6.7.8.9 η Esther R. Widiasih, Dnamics of the uko Energ alance Model, SIM J. ppl. Dn. Sst., (, 68 9. d h 3 - - -3 - -5 uko Widiasih Model Temperature profiles -6...3..5.6.7.8.9 Summar uko Widiasih Model sin(latitude T T( sin(latitude T T( T R Qs( ( ( ( T C( T T t heat capacit albedo OLR heat transport reduces to d h( d Q C h s S( Tc C heat capacit T R Qs( ( ( ( T C( T T t albedo What about the greenhouse effect? T is the outgoing long wave radiation. This term decreases if the greenhouse gases increase. We view as a parameter. d h, OLR heat transport Richard McGehee, Universit of Minnesota 5
Mathematics of Climate Seminar 9//6 uko Widiasih Model d h, isocline h,.9.8.7.6 Is it possible for Earth to become completel covered in ice? (.5..3 Did it ever happen?.. 75 8 85 9 95 5 5 high CO low CO http://www.astrobio.net/topic/solar-sstem/earth/new-information-aboutsnowball-earth-period/ There is evidence that Snowball Earth has occurred, the last time about 6 million ears ago. http://www.snowballearth.org/when.html The continents were clustered near the equator. Hoffman & Schrag,, SCIENTIFIC MERICN, Januar, 68-75 Large limestone deposits cap carbonates are found immediatel above the glacial debris, indicating a rapid warming period following the snowball. Ice rafted debris occurred in ocean sediments near the equator, indicating large equatorial glaciers calving icebergs. Hoffman & Schrag,, SCIENTIFIC MERICN, Januar, 68-75 Hoffman & Schrag,, SCIENTIFIC MERICN, Januar, 68-75 Richard McGehee, Universit of Minnesota 6
Mathematics of Climate Seminar 9//6 Idea: When the Earth is mostl ice covered, silicate weathering slows down, but volcanic activit stas the same, allowing for a build up of CO in the atmosphere. When the Earth is mostl ice free, silicate weathering speeds up, drawing down the CO in the atmosphere. uko Widiasih Model d h, What if is a namical variable? Simple equation: d c New sstem: d c d h, MCRN Paleocarbon equation (silicate weathering uko Widiasih Paleocarbon Model d d h,, c.9 What if η c were here? uko Widiasih Paleocarbon Model d d h,, c.9.8 c.8.7.6 stable rest point.7.6.5..3.5..3 unstable rest point What if η c were here?.. c.. 75 8 85 9 95 5 5 high CO low CO 75 8 85 9 95 5 5 high CO low CO uko Widiasih Paleocarbon Model Suggested Reading Snowball Hothouse Oscillations.9.8.7.6.5..3. c. 75 8 85 9 95 5 5 high CO low CO Hoffman & Schrag,, SCIENTIFIC MERICN, Januar, 68-75 K.K. Tung, Topics in Mathematical Modeling, PRINCETON UNIVERSITY PRESS, 7, Chapter 8 Richard McGehee, Universit of Minnesota 7