Mathematics of Climate Seminar 9/19/2017

Transcription

1 Mathematics of Climate Seminar 9/9/7 n Introduction to Richard McGehee School of Mathematics Universit of Minnesota Mathematics of Climate Seminar September 9, 7 Dnamical Models T dd Heat Transport R Qs( ( ( T C( T T t global mean temperature T( t T(, t d Second Law of Thermodnamics: Energ travels from hot places to cold places. Equilibrium temperature profile? Conservation of Energ temperature change ~ energ in energ out short wave energ from the Sun Everthing else is detail. long wave energ from the Earth heat capacit udko 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( d heat transport Perfectl Thermall Conducting lack od Plus lbedo Switch to Surface Temperature Dependence on Latitude Dnamical Models Model R QT R Q( T R Q( ( T Equilibrium T Q T ( Q T ( Q T(, t R Qs( ( T(, t t T( ( Qs( udko 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 ( d Q sd ( Q s ( ( d d T( d C Td T( d T T T Q T equilibrium global mean temperature T Q Richard McGehee, Universit of Minnesota

2 Mathematics of Climate Seminar 9/9/7 Solve for T (. udko s Equilibrium Qs( ( T ( C T T ( Global mean temperature at equilibrium: T Q ( sd ( Qs( ( CT T ( CT ( ( C T ( T ( Qs( ( CT C Equilibrium temperature profile: T ( Qs( ( CT C ice melts albedo decreases more sunlight absorbed REPET Wh would it stop? where T Q and ( sd ( temperature (Celsius 6 6 udko s Equilibrium Qs( ( T ( CT T ( Equilibrium temperature profile: T ( Qs( ( CT C C = 3. α( =.3: ice free α( =.6: snowball (constant albedo sin(latitude ice free snowball ice free (C= snowball (C= ice melts albedo decreases more sunlight absorbed REPET Wh would it stop? M. I. udko, "The effect of solar radiation variations on the climate of the Earth," Tellus XXI, 6-69, udko s Equilibrium temperature (Celsius ice won t melt (no exit from snowball sin(latitude ice free snowball ice free (C= snowball (C= ice will form (icecap Wh would it stop? udko s Equation sin(latitude T T( d T R Qs( ( ( ( T C( T T t heat capacit albedo OLR heat transport Richard McGehee, Universit of Minnesota

3 Mathematics of Climate Seminar 9/9/7 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( d For each fixed η, there is an equilibrium solution for udko s equation. C T ( Qs( (, CT C sine(latitude equilibrium temperature profile: T ( Qs( (, CT, where T Q ( C.3, albedo: (,.6, global albedo: let: then: ( (, sd ( sd ( sd ( S( s( d, S( s( d, since s( d ( S( ( S( ( S(.6.3 S( 3 S( s( d. 3 d. Chlek & Coakle 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( ( udko 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 equilibrium temperature profile: T( Qs( (, CT, where T Q ( C.3, albedo: (,.6, global albedo: ( (, sd ( sd ( sd ( let: S( s( d, S( s( d, since s( d then: ( S( ( S( ( S(.6.3 S( 3 S( s( d. 3 d. Chlek & Coakle IM Mathematics Summer Schol of Climate 7/8/6 9/9/7 For each fixed η, there is a globall stable equilibrium solution for udko s equation. How to pick one? C T R Qs( ( (, ( T C( T T t sine(latitude Richard McGehee, Universit of Minnesota 3

4 Mathematics of Climate Seminar 9/9/7 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? 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( For each fixed η, there is a stable equilibrium solution for udko 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 T( ( ( not good T( looks oka 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( d ( S(.6.3 S( S s d ( ( 3. Q C h s S( Tc C T( ice line condition:.. η ( ( T T T c T η (η T η (η 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 η Richard McGehee, Universit of Minnesota

5 Mathematics of Climate Seminar 9/9/7 udko Widiasih Model Equilibrium temperature profiles Interesting Solutions: small cap large cap ice free snowball temperature (ºC T ( Qs( (, CT C d h Temperature profiles sin(latitude ice free snowball small cap big cap T( 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 Widiasih s equation: T( ice melts d T( Tc T( stationar Summar sin(latitude T T( d 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 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 dnamics 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 η Esther R. Widiasih, Dnamics of the udko Energ alance Model, SIM J. ppl. Dn. Sst., (, heat capacit udko Widiasih Model 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, sin(latitude OLR T T( d heat transport Richard McGehee, Universit of Minnesota 5

6 Mathematics of Climate Seminar 9/9/7 udko Widiasih Model d h, isocline h, high CO low CO The continents were clustered near the equator. Hoffman & Schrag,, SCIENTIFIC MERICN, Januar, Is it possible for Earth to become completel covered in ice? ( Did it ever happen? Ice rafted debris occurred in ocean sediments near the equator, indicating large equatorial glaciers calving icebergs. Hoffman & Schrag,, SCIENTIFIC MERICN, Januar, There is evidence that Snowball Earth has occurred, the last time about 6 million ears ago. Large limestone deposits cap carbonates are found immediatel above the glacial debris, indicating a rapid warming period following the snowball. Hoffman & Schrag,, SCIENTIFIC MERICN, Januar, Richard McGehee, Universit of Minnesota 6

7 Mathematics of Climate Seminar 9/9/7 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. udko Widiasih Paleocarbon Model d d h,, c unstable rest point.3 What if η c were here?. c high CO low CO udko Widiasih Model d h, What if is a dnamical variable? Simple equation: d c New sstem: d c d h, MCRN Paleocarbon equation (silicate weathering udko Widiasih Paleocarbon Model Snowball Hothouse Oscillations high CO low CO c udko Widiasih Paleocarbon Model d d h,, c What if η c were here? Suggested Reading.9.8 c.7.6 stable rest point high CO low CO Hoffman & Schrag,, SCIENTIFIC MERICN, Januar, K.K. Tung, Topics in Mathematical Modeling, PRINCETON UNIVERSITY PRESS, 7, Chapter 8 Richard McGehee, Universit of Minnesota 7