Math /22/2014. Richard McGehee, University of Minnesota 1. Math Energy Balance

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1 Math 59 Topics in Applied Mathematics: Introduction to the Mathematics of Climate Mondas and Wednesdas :3 3:5 Streaming video is available at Click on the link: "Live Streaming from 35 Lind Hall". Participation: Perfectl Thermall Conducting lack o Plus Albedo Switch to Surface Temperature Dependence on Latitude Energ alance Dnamical Models Model R QT R Q( T R Q( ( AT Equilibrium T Q T ( Q T ( Q A T(, t R Qs( ( A T (, t t T( ( Qs( A Add Heat Transport global mean temperature Energ alance Dnamical Models T R Qs( ( ( A T C( T T t 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( ( ( ( A 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 Energ alance uko s Equilibrium T R Qs( ( ( ( A T C( T T t albedo depends on latitude equilibrium solution: T = T( Qs( ( AT ( C T T ( Integrate: Qs( ( AT ( CT T ( Q s ( Q s ( ( A T( C T T( T Q AT equilibrium global mean temperature T T Q A T Solve for T (. where Energ alance uko s Equilibrium Qs( ( AT ( C T T ( Global mean temperature at equilibrium: T Q A ( s ( Qs( ( A CT T ( CT ( ( C T ( T ( Qs( ( ACT C Equilibrium temperature profile: T ( Qs( ( ACT C T Q A and ( s ( Richard McGehee, Universit of Minnesota

2 temperature (Celsius 6 6 Energ alance uko s Equilibrium Qs( ( AT ( CT T ( Equilibrium temperature profile: T ( Qs( ( ACT C C = 3. α( =.3: ice free α( =.6: snowball (constant albedo sin(latitude ice free snowball ice free (C= snowball (C= temperature (Celsius 6 6 Energ alance uko s Equilibrium 8 ice won t melt (no exit from snowball sin(latitude ice free snowball ice free (C= snowball (C= ice will form (icecap ice melts albedo decreases more sunlight absorbed REPEAT Wh would it stop? ice melts albedo decreases more sunlight absorbed REPEAT Wh would it stop? M. I. uko, "The effect of solar radiation variations on the climate of the Earth," Tellus XXI, 6-69, Wh would it stop? uko s Equation sin(latitude T T( T R Qs( ( ( ( A T C( T T t heat capacit albedo OLR heat transport What if the albedo is not constant? Ice Line Assumption: 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. (, and Equilibrium condition: Equilibrium solution: where Qs( (, AT ( C T T ( T ( Qs( (, ACT C T Q ( A ( (, s( Richard McGehee, Universit of Minnesota

3 equilibrium temperature profile: T ( Qs( (, ACT, where T Q ( A 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( (, ACT, where T Q ( A 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 T ( Qs( (, ACT C Dnamics T R Qs( ( ( ( A T C( T T t For each fixed η, there is an equilibrium solution for uko s equation. C sine(latitude Let X be the space of functions where T lives. (e.g. L ([,] Let L : X X : LT CT ( C T, f ( Qs( ( A 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 T R Qs( ( (, ( A T C( T T t 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? sine(latitude Richard McGehee, Universit of Minnesota 3

4 T( 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. Additional condition: The average temperature across the ice boundar is the critical temperature T c. looks oka T T T c T( ( ( not good T( looks oka T( ice line condition:.. η T T T c ( ( T η (η T η (η Equilibrium: T R Qs( ( (, ( A T C( T T t T ( Qs( (, ACT C T ( T ( T c.3, (,.6, (,, (, Ice line condition: Albedo: T ( Qs( ACT T ( Qs( ACT C C Ice line condition: T ( T ( Qs( ACT T c C where:.7 T R Qs( ( (, ( A T C( T T t Ice line condition: Qs( A CT T c C Rewrite: h( Qs( ACT T c C Recall equilibrium GMT: Recall average albedo: where: T Q ( A ( (, s( ( S(.6.3 S( S s ( ( 3. Q C A h s S( Tc C can be written: Two equilibria (zeros of h satisf the additional condition. T R Qs( ( (, ( A T C( T T t T( T( T c The additional condition: Q C A h s S( Tc C h(η η Equilibrium temperature profiles Interesting Solutions: small cap large cap ice free snowball temperature (ºC T ( Qs( (, ACT C sin(latitude ice free snowball small cap big cap Richard McGehee, Universit of Minnesota

5 T( Dnamics of the Ice Line T R Qs( ( (, ( A 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 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( ( (, ( A T C( T T t unstable stable h(η η Esther R. Widiasih, Dnamics of the uko Energ alance Model, SIAM J. Appl. Dn. Sst., (, d h uko Widiasih Model Temperature profiles heat capacit Summar T R Qs( ( ( ( A T C( T T t albedo reduces to d h( sin(latitude OLR T T( heat transport d Q C A h s S( Tc C uko Widiasih Model uko Widiasih Model sin(latitude T T( d h, A isocline h, A T R Qs( ( ( ( A T C( T T t.9.8 heat capacit albedo OLR heat transport.7.6 What about the greenhouse effect? AT is the outgoing long wave radiation. This term decreases if the greenhouse gases increase. We view A as a parameter. d h, A A high CO low CO Richard McGehee, Universit of Minnesota 5

6 Is it possible for Earth to become completel covered in ice? ( Did it ever happen? There is evidence that Snowball Earth has occurred, the last time about 6 million ears ago. The continents were clustered near the equator. Hoffman & Schrag,, SCIENTIFIC AMERICAN, Januar, Ice rafted debris occurred in ocean sediments near the equator, indicating large equatorial glaciers calving icebergs. Hoffman & Schrag,, SCIENTIFIC AMERICAN, Januar, Large limestone deposits cap carbonates are found immediatel above the glacial debris, indicating a rapid warming period following the snowball. 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. Hoffman & Schrag,, SCIENTIFIC AMERICAN, Januar, Richard McGehee, Universit of Minnesota 6

7 uko Widiasih Model d h, A What if A is a namical variable? Simple equation: da c New sstem: da c d h, A MCRN Paleocarbon equation (silicate weathering uko Widiasih Paleocarbon Model d da h, A, c stable rest point high CO A low CO c What if η c were here? uko Widiasih Paleocarbon Model d da h, A, c uko Widiasih Paleocarbon Model Snowball Hothouse Oscillations unstable rest point What if η c were here? c. c high CO A low CO high CO A low CO Suggested Reading Hoffman & Schrag,, SCIENTIFIC AMERICAN, Januar, K.K. Tung, Topics in Mathematical Modeling, PRINCETON UNIVERSITY PRESS, 7, Chapter 8 Richard McGehee, Universit of Minnesota 7