Arctic Armageddon, More Mathematics

Transcription

1 Mathematics Undergraduate Colloquium University of Utah 11/14/2012 Arctic Armageddon, More Mathematics Ivan Sudakov

2 Doomsday

3 Climate Change Past Future Credit: Barnosky, et al., 2011 Credit: IPCC, 2007 Credit: Sun, et al., 2010 Credit: Eleseev, et al.,

4 Mean annual temperature in central Europe for the last 50 million years Credit: Korobeinikov and McNabb,

5 Climate is what you expect (Edward Lorenz, 1997) Climate Science Nonlinear Science 5

6 Developing of Nonlinear Approaches in Climate Science 6

7 A Bifurcation and the Butterfly Effect Abifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior The butterfly effect is the sensitive dependence on initial conditions, where a small change at one place in a deterministic nonlinear system can result in large differences to a later state The name of the effect, coined by Edward Lorenz, is derived from the theoretical example of a hurricane's formation being contingent on whether or not a distant butterfly had flapped its wings several weeks before 7

8 Simple Example Credit: Marsden and McCracken., 1976 Example: A hoop and a small ball Def. Let F t be a C 0 flow on a topological space M and let A be an invariant set; i.e., F t ( A) A for all t. We say A is stable (resp. asymptotically stable or an attractor) if for any neighborhood U of A there is a neighborhood V of A such that the flow lines (integral curves) x( x0, t) Ft ( x0) belong to U if x0 V (resp. Ft (V) A). t0 A is stable (resp. attracting) when an initial condition slightly perturbed from A remains near A (resp. tends towards A). 8

9 The Hopf (Poincaré Andronov Hopf) Bifurcation Another type of behavior that may occur is bifurcation to periodic orbits. This means that there are curves of the form : IP such that ( 0 ) p( 0) and ( 0 ) is on a closed orbit of the flow of x. The Hopf bifurcation is of this type. The appearance of the stable closed orbits (= periodic solutions) is interpreted as a "shift of stability" from the original stationary solution to the periodic one, i.e., a point near the original fixed point now is attracted to and becomes indistinguishable from the closed orbit. Credit: Marsden and McCracken.,

10 Mean annual temperature in central Europe for the last 50 million years Credit: Korobeinikov and McNabb,

11 Credit: Korobeinikov and McNabb, 2001 u v V const the solar radiation intensity k have been chosen as a bifurcation parameter 11

12 p(t) is the total precipitation (snowfall s(t) plus rainfall r(t)) rate); γ v, γ m are the latent heats of vaporization and melting for water; W(p) is the heat spent on work to transport the water of mass p from the ocean onto the land. dq dt O dv( t) s m dt s (t) is the rate of snowfall on the land; m (t) is the rate at which the ice and snow reserves melt; L L E p W ( p) l O q O, q L are the heat contents of the reservoirs O and L; E O, E L the rates of change of thermal energy of the reservoirs O and L; l O, l L the rates of change of radiant energy (to atmosphere); dq dt E v p v m s m O m l L 12

13 Korobeinikov McNabb Theorem If b( a v c)( C C O O C L ) t b a c v t a v 0, then the climate system admits a supercritical Hopf bifurcation of the equilibrium Q 1 to a stable closed orbit, with the solar radiation intensity k as the bifurcation parameter. If, then the equilibrium Q 1 is stable for all k > 0. The equilibrium Q 2 is unstable for all k > 0. 0 Here Q 1 and Q 1 two equilibria of the climate system; С O is the total mean heat capacity of the oceans water and С L is the total mean heat capacity of the land * reservoir;,, a, b, c, t - the physical parameters; γ v - the latent heats of vaporization. 13

14 The Climate System as a System of Reservoirs A basic feature of this system is it has possibilities for the occurrence of a supercritical Hopf bifurcation In this model the bifurcation is triggered by an increase of the solar radiation intensity This research demonstrate that a Hopf bifurcation in a model of the global climate system could (or rather must) cause dramatic changes in the global climate similar to those observed throughout the Pliocene and the Pleistocene 14

15 The Basic Researches in Tipping Points Lenton, T.M. Tipping elements in the Earth's climate system / T.M. Lenton, H. Held, E. Kriegler, J.W. Hall, W. Lucht, S. Rahmstorf, H.J. Schellnhuber // Proceedings of the National Academy of Science of USA V.105 (6). P Thompson, J.M.T. Predicting climate tipping as a noisy bifurcation: a review / J.M.T. Thompson, J. Sieber // Int J. Bif. Chaos V. 21(2). P

16 Simple Tipping Point Definitions A tipping point is a critical threshold at which the future state of a system can be qualitatively altered by a small change in forcing A tipping element is a part of the Earth system (at least sub-continental in scale) that has a tipping point Abrupt climate change is the subset of tipping point change which occurs faster than its cause Tipping point change also includes transitions that are slower than their cause (in both cases the rate is determined by the system itself). In either case the change in state may be reversible or irreversible 16

17 Formal Tipping Point Definitions sub-system Ʃ is a tipping element if there exists a control parameter ρ with a critical value ρ crit, at which a small parameter variation (δρ> 0) leads to a qualitative change in a system feature (F), i.e., F eq ( ) F ( ) crit eq crit F abrupt climate change: when the climate system is forced to cross some threshold, triggering a transition to a new state at a rate determined by the climate system itself and faster than the cause, which is a case of bifurcation (~tipping point) 17

18 Credit: Thompson, et al., 2010 Using Lenton s definitions, Thompson & Sieber were adapted the bifurcation theory to the climate tipping point concept 18

19 Map of potential policy-relevant tipping elements in the climate system 19

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21 Anthropogenic Impact on the Permafrost Regions: Gas and Reindeer Husbandry 21

22 Permafrost as a Storage Photos by Brouchkov 22

23 Problems for the Infrastructure Yakutsk, Russia, 2006 Cherski, Russia,

24 CH 4 Emission from Permafrost Photo by K.Walker Arctic Armageddon Needs More Science Less Hype (R. Kerr; Science, 2010) Photo by K.Walker 24

25 Climate System Feedback Credit: 25

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27 Extended Goody s Radiative-Convective Atmospheric Model v t ( v ) v v P 1( 0) z, ( v ) 3 t Rayleigh Bénard convection in a gravity field v 0 2 C t (v ) C dc b C 0 Boundary conditions for gas velocity v( x, y, z, t) v( x, y, z, t) 0 ( x, y, z, t) ( x, y, z, t) 0 z C C z z z z ( x, y, z, t) zh z 0 zh ( x, y, z, t) ( x, y, ( x, y,0, t)) 0 z zh Boundary conditions for temperature field Boundary conditions for GHG concentration 0 0 ( C) C 1 0 v =(u, v, w) - vector gas velocity, θ temperature, P pressure. We add an equation for greenhouse gas concentration C; α - radiation absorption coefficient 27

28 New Bifurcations in Goody s Model We are linearized the main equations (for small deviations θ, C) at zero state T z T t C t K CT0, d C b C, 0 Boundary conditions ( x, y, z) r0t ( x, y,0), 0 z C z ( x, y, z) z C zh 0, ( x, y, z) z zh v 0, T0 ( z), C C 0, ( x, y, z) T( x, y, z). 0 z. 0 We seek solutions T ( x, y, z) exp( t), C ( x, y, z) exp( t), where λ is complex spectral parameter. So, the spectral problem is: T KT 3 0T 3 1T 0, d b0. h kt sinh( kt h) 3 K fk ( z, )cosh( kt ( z h)) dz This is nonlinear equation should be resolved for each k and have roots depending (k) 0 k c Re( k; ) 0 0 We consider as a bifurcation parameter. For each we can find such that Re ( k; ) 0 for and for c c 28

29 Critical Value c T b d 0 It is the formula for a critical level of GHG emission T TC 1 It is introduced a new parameter of extended Goody s model. This defines a dependence of radiation absorption coefficient on GHG concentration Credit: Sudakov and Vakulenko, 2012 Dependence of c / on GHG diffusion coefficient Dependence potential / c on Global warming 29

30 Methane Hydrate Gun Elementary computation with real parameters implies that β c should be very great (that we observed now) to create a catastrophic bifurcation. It should be increased minimum in times to attain the critical level Only a sharp change of emission under temperature (e.g., methane hydrate gun) leads to a catastrophe 30

31 Bistable Regime S() For a sufficiently large greenhouse gas emission from the soil this system becomes bistable Credit: Sudakov and Vakulenko, 2012 T s,k (surface temperature) This toy planet is sinking slowly into a homogeneous greenhouse gas fog 31

32 The soviet cartoon: Hedgehog in the Fog 32