IMA. Budyko s Model as a Dynamical System. Math and Climate Seminar
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1 Math and Climate Seminar IMA as a Dnamical Sstem Richard McGehee Joint MCRN/IMA Math and Climate Seminar Tuesdas :5 :5 streaming video available at Seminar on the Mathematics of Climate IMA, MCRN, School of Mathematics November, 3 Budko s Equation Budko s Equation T T( d T R Qs( ( ( ( A BT C( T T t heat capacit T R Qs( ( ( ( ABT C( T T t Smmetr assumption: Chlek and Coakle s quadratic approximation:.43 s equilibrium solution: T = T( Qs( ( A BT ( C T T ( Integrate: Q ABT Qs( ( A BT ( C T T ( d where ( sd ( Global mean temperature at equilibrium T Q A B Equilibrium temperature profile: T ( Qs( ( ACT B C Example: C = 3.4 α =.3: ice free α =.6: snowball Budko s Equilibrium Qs( ( A BT ( C T T ( Global mean temperature at equilibrium: T Q A ( sd ( B temperature (Celsius ice free snowball What if the is not constant? Ice Line Assumption: There is a single ice line at =η between the equator and the pole. The is α below the ice line and α above it. (, Equilibrium condition: Equilibrium solution: where Qs( (, A BT ( C T T ( T ( Qs( (, ACT B C T Q B ( A ( (, s( d
2 Dnamics T ( Qs( (, ACT B C T T( d For each fixed η, there is an equilibrium solution for Budko s equation. C sine(latitude T c heat capacit T R Qs( ( ( ( ABT C( T T t Let X be the space of functions where T lives. (e.g. L ([,] Let L : X X : LT CT ( BC T, f ( Qs( ( A Budko s equation can be written as a linear vector field on X. dt R f LT Budko s Equation dt R f LT LT( C T( d ( BC T( If X is a decent space (e.g. L ([,], then L is a continuous linear operator on X. Equilibrium solution: T L f if L is invertible. Stabilit depends on the spectrum of L. Linear Operator LT( C T( d ( BC T( Let : (, : is constant X T X T d X T X T Then X X X LT ( BC T, for T X LT BT, for T X L is invertible. Spectrum B is an eigenvalue, with eigenspace X (dimension ( BC is an eigenvalue, with eigenspace X (codimension The equilibrium solution is stable. For each fixed η, there is a stable equilibrium solution for Budko s equation. How to pick one? C T R Qs( ( (, ( A BT C( T T t sine(latitude T c T( For each fixed η, there is a stable equilibrium solution for Budko 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
3 T R Qs( ( (, ( A BT C( T T t T ( T ( T c can be written: The additional condition: Q C A h s sd Tc B C B B Two equilibria (zeros of h satisf the additional condition. h(η η Interesting Solutions: small cap large cap ice free snowball T R Qs( ( (, ( A BT C( T T t temperature (ºC ice free snowball small cap big cap T( Dnamics of the Ice Line T R Qs( ( (, ( A BT 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 dnamics are approximated b the equation d h( Dnamics of the Ice Line d T( Tc T R Qs( ( (, ( A BT C( T T t State space: [,] X h(η unstable η stable 3 Budko Widiasih Model d h Temperature profiles Summar T T( d T R Qs( ( ( ( A BT C( T T t heat capacit reduces to d h( -4-5 d Q C A h s sd T c B C B B
4 Budko Widiasih Model Budko Widiasih Model T T( d d h, A isocline h, A heat capacit T R Qs( ( ( ( A BT C( T T t What about the greenhouse effect? ABT 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 Budko Widiasih Model d h, A What if A is a dnamical variable? Simple equation: c MCRN Paleocarbon equation Budko Widiasih Paleocarbon Model d h, A, c stable rest point.6 c New sstem: c d h, A high CO A low CO Budko Widiasih Paleocarbon Model d h, A, c Budko Widiasih Paleocarbon Model Snowball Hothouse Oscillations unstable rest point.4.3. c. c A high CO A low CO 4
5 The snowball Earth hpothesis is not universall accepted. Complex ocean dwelling photosnthetic organisms appear to have survived through the time when the snowball would have occurred. Perhaps the equator never full froze over, leaving a belt of open water covering the equator. Hence water belt planet, aka Jormungand. d h, A, c h is modified, so the isocline looks like this:.3 D.S. Abbot, A. Voigt, and D. Koll (, The Jormungand global climate state and implications for Neoproterozoic glaciations, J. Geophs. Res d h, A, c Stable Small Icecap stable rest point c d h, A, c Stable Water Belt stable rest point c Large Ice Ages Banded Iron Formations?.9.8 unstable rest point.9.8 d h, A, c c d h, A, c c
6 Budko s Equation T T( d T R Qs( ( ( ( A BT C( T T t heat capacit Next Time Glacial Ccles 6
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