Math 5490 November 12, 2014

Transcription

1 Math 5490 November 12, 2014 Topics in Applied Mathematics: Introduction to the Mathematics of Climate Mondays and Wednesdays 2:30 3:45 Streaming video is available at Click on the link: "Live Streaming from 305 Lind Hall". Participation:

2 Henry Stommel, Thermohaline Convection with Two Stable Regimes of Flow, TELLUS XII (1961),

3 dt ct ( T) 2qT dt ds d( S S) 2q S dt kq ( 2T 2 S) T S y x T S d cdt d S c R k c T 40T flow rate 2q f c flow resistance - - salinity temperature dx (1 x) f x d dy 1 y f y d f yrx

4 dx (1 x) f x d dy 1 y f y d f yrx (1 xe) f xe 0 xe Look for equilibria: f 1 1 ye f ye 0 ye 1 f 1 R f ye Rxe ( f; R, ) 1 f f f ( f; R, ) Solve for f, then solve for equilibrium point.

5 Equilibria R 1 f ( f; R, ) f 1 f Graphical Interpretation ( f ) f 16 R 2 15 density f (flow rate)

6 Equilibria Graphical Interpretation ( f ) 16 R 2 15 density f Temperature dominates. capillary flow: cold to warm f (flow rate) Salinity dominates. capillary flow: warm to cold

7 Temperature dominates. capillary flow: cold to warm Salinity dominates. capillary flow: warm to cold 16 R 2 15 Stommel, TELLUS XII (1961)

8 3 Equilibrium Conditions (1 xe) f xe 0 xe f 1 1 ye f ye 0 ye 1 f 1 R f ye Rxe 1 f f Solve for f, then solve for equilibrium point. 1 1 f f f f R f 2 f (1 ) f f ( R1) (1 R) f f (1 ) f f f (1 R) f ( R1) 0

9 3 Equilibrium Conditions: Solving for f f (1 ) f f f (1 R) f ( R1) 0 Parameters: 1 6 R f 1 1 f f 1 1 f 1 f f 7 f f 1 f 2 f (1 ) (1 2 ) (2 1) 0 Case 1: f f f 21 f Solve numerically. Only one positive root: f Case 2: f f f 19 f Solve numerically. Two negative roots: f ,

10 Graphical Interpretation ( f ) 16 R 2 15 density f f (flow rate)

11 16 R 2 15 x e Rest Points 1 1, ye f 16 f 1 f x x x e e e point a: f : , ye point b: f : , ye point c: f : , ye

12 f 0 Temperature dominates. capillary flow: cold to warm f 0 b c 16 R 2 15 a f 0 Salinity dominates. capillary flow: warm to cold

13 Structure of Rest Points x (1 x) f x y 1 y f y f yrx Jacobian matrix f f f x x x y f y 1 f y x f y f 0: f 1 R f R f 1 y x,, x y f 0: f 1 R f R f 1 y x,, x y f 0 f 0 R 1 f x x R 1 y 1 f y R 1 f x x R 1 y 1 f y

14 Rest Point c f , x , y e e 16 R 2 15 Jacobian matrix R f x x R y 1 f y determinant trace discriminant stable spiral

15 f 0 Temperature dominates. capillary flow: cold to warm f 0 b c stable spiral 16 R 2 15 a f 0 Salinity dominates. capillary flow: warm to cold

16 Rest Point b f , x , y e e 16 R 2 15 Jacobian matrix R f x x R y 1 f y eigenvalue eigenvector eigenvalue eigenvector saddle

17 f 0 stable vector unstable vector f 0 b c stable spiral 16 R 2 15 a saddle f 0

18 Rest Point a f , x , y e Jacobian matrix R f x x R y 1 f y e 16 R 2 15 eigenvalue eigenvector eigenvalue eigenvector stable node

19 f 0 stable vector 16 R 2 unstable vector f 0 b c 15 stable spiral slow vector a saddle fast vector stable node f 0

20 stable manifold 16 R 2 unstable manifold b c 15 stable spiral slow vector a saddle fast vector stable node

21 16 R 2 15 stable manifold b c stable spiral saddle a stable node

22 stable manifold 16 R 2 unstable manifold b c 15 stable spiral slow vector a saddle fast vector stable node

23 16 R 2 15 stable manifold b c stable spiral saddle a stable node

24 16 R 2 15 stable manifold b c stable spiral saddle

25 stable manifold 16 R 2 unstable manifold b c 15 stable spiral slow vector a saddle fast vector stable node

26 Vague Analogy to Atlantic Overturning Circulation Gulf Stream reversed. Gulf Stream flowing North Stommel, TELLUS XII (1961) Math /10/2014

27 - - T S y x T S d cdt d S c R k c T 40T flow rate 2q f c flow resistance Let s increase the resistance in the capillary, so that it is harder for the water to flow between the vessels. salinity temperature dx (1 x) f x d dy 1 y f y d f yrx

28 stable manifold 16 R 2 unstable manifold b c 0.3 a stable spiral Increase the flow resistance. saddle stable node

29 stable manifold 16 R 2 unstable manifold b c 0.3 a stable spiral Increase the flow resistance. Not much different, but it is easier to get to c. stable node saddle

30 stable manifold 16 R a b c stable spiral Increase the flow resistance. The saddle and the stable node start to merge. stable node saddle

31 stable manifold 16 R a b c stable spiral Increase the flow resistance. The saddle and the stable node start to merge. stable node saddle

32 stable spiral 16 R c Increase the flow resistance. The saddle and the stable node have disappeared. The Gulf Stream will eventually reverse.

33 - - T S y x T S d cdt d S c R k c T 40T flow rate 2q f c flow resistance Now decrease the resistance in the capillary, so that it is easier for the water to flow between the vessels. salinity temperature dx (1 x) f x d dy 1 y f y d f yrx

34 stable manifold 16 R a b c stable spiral The saddle and the stable node have reemerged, but it is difficult to get to a. The Gulf Stream is still reversed. stable node saddle

35 stable manifold 16 R 2 15 b c stable spiral We are back to our original parameters, but the Gulf Stream is still reversed. a stable node saddle

36 stable manifold 16 R b c stable spiral The flow resistance is below the original value. Point a is the dominant attractor. Perhaps the Gulf Stream will find a way to return to normal. a stable node saddle

37 Henry Stommel, Thermohaline Convection with Two Stable Regimes of Flow, TELLUS XII (1961),