Mathematics of Our Ice Dependent World

Transcription

1 Mathematics of Our Ice Dependent World Ivan Sudakov

2 Predictions from the Bulgarian prophet Baba Vanga all of the polar ice caps melt and the world ocean levels rise a new kind of weapon - the climate The sharp cooling (instant freezing) Climate Change 2013: The Phsical Science Basis IPCC Jeffries M.O. et al. The Arctic shifts to a new normal. Phs. Toda (2013) 2

3 Map of potential polic-relevant tipping elements in the climate sstem Tipping points ~ Rapid change in climate sstem or sub sstem Irreversibilit Significant impact 3

4 A phase transition is the transformation of thermodnamic sstem from one phase or state of matter to another. A phase of a thermodnamic sstem and the states of matter have uniform phsical properties The phenomena associated with continuous phase transitions are called critical phenomena due to their association with critical points 4

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6 reflected sunlight incident sunlight 6

7 A simple classical approimation to an electronic magnetic moment is provided b an Ising spin which can take two values s i 1"spin up" 1"spin down" the Hamiltonian is the operator corresponding to the total energ of the sstem In general domains do not line up no macroscopic magnetization Can be forced to line up in one direction The interaction energ can be approimated b E J i j s i s j H i s i If the interaction strength J > 0 the sstem is ferromagnetic: the energ is minimized if the spin point in the same direction s i s j = +1. If J < 0 the sstem is antiferromagnetic. H represents an eternal magnetic field which couples to the magnetization of the sstem There is a critical value of the temperature T c called the Curie temperature at which a phase transition between the ferromagnetic (permanentl magnetized) and paramagnetic phases occurs. Iron: 1043 K Critical point phase transition 7

8 In the Random Field Ising Model (the magnetic field ehibits quenched disorder) the interaction energ can be approimated b s i 1"melt water" 1"ice" E Jijsis j i j i H i s i H represents solar radiation field J relates to kink in the melting front J 0 J 1 8

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10 V The permafrost lake methane emission rate: ( BR 1) e p ( b 0 / u ( t )) av V the total rate of the methane emission; R av the averaged lake radius; u av the average temperature; β b 0 B positive constants av 10

11 Let us consider a clindrical domain D [ 0 h] where Ω is a sub-domain of 2D sphere corresponding to the permafrost surface; ( ) are coordinates on z [0 h] The unknown function is temperature u u( z t) We assume that h L where L diam() and that the boundar of Ω is a smooth curve; l( t) - the thawing front position the front surface is defined b z l The classical Stefan Problem: u t K 1 u 0 z l( t) u~ t K 2 u~ l( t) z h 11

12 12 Boundar conditions on the depth: ) ( ) ) ~ ( ( ) ( ~ t V t h u b t h u z on surface: ) ( ) 0 ( t U t u at the thawing transition front: Qv t z u t z u z ) )) ~ ( ) ( ( ( ) ( n where n is the unit normal vector with respect to the thawing front v is the normal front velocit Q is a dimensionless latent heat. We define a separating surface Γ b the following equation ) ) ( ( t t l u

13 Schematic plot of the free energ difference between the normal and superconducting states as a function of the order parameter Free energ is a thermodnamic potential that measures the "usefulness" or process-initiating work obtainable from a thermodnamic sstem Landau theor of phase transitions The free energ of an sstem should obe two conditions: that the free energ is analtic and that it obes the smmetr of the Hamiltonian The free energ F of a superconductor near the superconducting transition can be epressed in terms of a comple order parameter fieldφ which is nonzero below a phase transition into a superconducting state F s F n 2 2 F n F s free energ at the normal and superconducting states; α parameter which depends on temperatures at critical point; β a parameter 4 The superconducting phase transition 3 g 05( ) F Meican hat potential for superconductor 13

14 The two main functions of interest are the order parameter ( t) and the temperature u u t) We have two etreme states iced ground and soil. The frozen state is characterized b 1 and the melted state b 1 The Stefan Problem can be written as (Caginalp 1989) 2 2 Let us assume then equation for order parameter has the solution 1/ 2 1 a is a parameter that defines the front width g( ) v is the normal thawing front velocit at the point ( z) k is the mean front curvature at this point μ is a positive coefficient δ is function of the microscopic parameters of the model: v( z t) k( z t) u t t Ku zz b 2 t a 1 a b K dimensionless parameters tanh(( z) / ) Asmptotical relation for averaged lake growth: dr dt R 1 14

15 mass mg mass mg The positive feedback in to climate model u ( t ) u av X u c the climate sstem temperature; u av a mean temperature for the summer season without warming effect; X the total methane mass in atmosphere; γ the feedback coefficient. time r time r Numerical eperiment 1: A linear growth of methane mass in case of a small feedback coefficient Numerical eperiment 2: A sharp increasing of methane mass in case of a large feedback coefficient 15

16 Credit: Imbrie Models suggest nuclear winter could have a severe effect on the climate causing cold weather and reduced sunlight for a period of months or even ears. This is caused b large amounts of smoke and soot being ejected into the Earth's stratosphere. T T w s 1 4 Credit: A. Robock A simple two - laer energ balance model of the anti greenhouse effect of high-altitude soot. The surface temperature T w depends on smoke-laer emissivit ε s and must be less than equilibrium temperature T 0. 16

17 The conceptual phsical models of the Crosphere can be used in the Earth Sstem Models for developing of the phsical basis and numerical techniques Mathematical methods of theoretical phsics are universal for all sstem: Ferromagnets - Melt Ponds Superconductors - Permafrost Lakes The phase transition theor improves our knowledge about climate tipping points 17

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