Math 5490 November 12, 2014 Topics in Applied Mathematics: Introduction to the Mathematics of Climate Mondays and Wednesdays 2:30 3:45 http://www.math.umn.edu/~mcgehee/teaching/math5490-2014-2fall/ Streaming video is available at http://www.ima.umn.edu/videos/ Click on the link: "Live Streaming from 305 Lind Hall". Participation: https://umconnect.umn.edu/mathclimate
Henry Stommel, Thermohaline Convection with Two Stable Regimes of Flow, TELLUS XII (1961), 224-230.
dt ct ( T) 2qT dt ds d( S S) 2q S dt kq ( 2T 2 S) 1 2 0 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
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.
Equilibria R 1 f ( f; R, ) f 1 f Graphical Interpretation ( f ) 1.2 1 0.8 f 16 R 2 15 density 0.6 0.4 0.2 0 0.2 0.4 0.6 2 1.5 1 0.5 0 0.5 1 1.5 2 f (flow rate)
Equilibria Graphical Interpretation ( f ) 16 R 2 15 density 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 f 0.6 2 1.5 1 0.5 0 0.5 1 1.5 2 Temperature dominates. capillary flow: cold to warm f (flow rate) Salinity dominates. capillary flow: warm to cold
Temperature dominates. capillary flow: cold to warm Salinity dominates. capillary flow: warm to cold 16 R 2 15 Stommel, TELLUS XII (1961)
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
3 Equilibrium Conditions: Solving for f f (1 ) f f f (1 R) f ( R1) 0 Parameters: 1 6 R 2 1 5 1 3 f 1 1 f f 1 1 f 1 f 1 5 5 6 5 6 6 6 1 3 f 7 f f 1 f 2 f 1 0 5 30 30 3 6 (1 ) (1 2 ) (2 1) 0 Case 1: f 0 1 3 7 2 f f 21 f 1 5 30 30 6 0 Solve numerically. Only one positive root: f 0.21909 Case 2: f 0 1 3 7 2 f f 19 f 1 5 30 30 6 0 Solve numerically. Two negative roots: f -1.06791, -0.30703
Graphical Interpretation ( f ) 16 R 2 15 density 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 f 0.6 2 1.5 1 0.5 0 0.5 1 1.5 2 f (flow rate) -1.068-0.307 0.219
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 -1.06791: 1 1 0.13500, ye 0.48358 16-1.06791 1-1.06791 point b: f -0.30703: 1 1 0.35184, ye 0.76510 16-1.06791 1-1.06791 point c: f 0.21909: 1 1 0.43205, ye 0.82028 16-1.06791 1-1.06791
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
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
Rest Point c f 0.21909 0, x 0.43205, y 0.82028 e e 16 R 2 15 Jacobian matrix R 1 1 2 1 f x x 0.21909 0.43205 0.43205 6 15 15 R 1 2 1 y 1 f y 0.82028 1 0.21909 0.82028 15 15 4.70627 2.16025 8.20284 2.88233 determinant 4.15521 0 trace 1.82394 0 discriminant 13.29410 0 stable spiral
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
Rest Point b f 0.30703 0, x 0.35184, y 0.76510 e e 16 R 2 15 Jacobian matrix R 1 1 2 1 f x x 0.30703 0.35184 0.35184 6 15 15 R 1 2 1 y 1 f y 0.76510 1 0.30703 0.76510 15 15 3.04476 1.75922 7.65095 5.13250 eigenvalue 7.60883 0.61023 eigenvector 0.79222 eigenvalue 0.28486 0.28604 eigenvector 0.95822 saddle
f 0 stable vector unstable vector f 0 b c stable spiral 16 R 2 15 a saddle f 0
Rest Point a f -1.06791 0, x 0.13500, y 0.48358 e Jacobian matrix R 1 1 2 1 f x x -1.06791 0.13500 0.13500 6 15 15 R 1 2 1 y 1 f y 0.48358 1-1.06791 0.48358 15 15 0.11541-0.67500 0.48358-4.48581 e 16 R 2 15 eigenvalue 0.76088 0.61023 eigenvector 0.79222 eigenvalue 3.60951 0.17831 eigenvector 0.98398 stable node
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
stable manifold 16 R 2 unstable manifold b c 15 stable spiral slow vector a saddle fast vector stable node
16 R 2 15 stable manifold b c stable spiral saddle a stable node
stable manifold 16 R 2 unstable manifold b c 15 stable spiral slow vector a saddle fast vector stable node
16 R 2 15 stable manifold b c stable spiral saddle a stable node
16 R 2 15 stable manifold b c stable spiral saddle
stable manifold 16 R 2 unstable manifold b c 15 stable spiral slow vector a saddle fast vector stable node
Vague Analogy to Atlantic Overturning Circulation Gulf Stream reversed. Gulf Stream flowing North Stommel, TELLUS XII (1961) Math 5490 11/10/2014
- - 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
stable manifold 16 R 2 unstable manifold b c 0.3 a stable spiral Increase the flow resistance. saddle stable node
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
stable manifold 16 R 2 0.33 a b c stable spiral Increase the flow resistance. The saddle and the stable node start to merge. stable node saddle
stable manifold 16 R 2 0.33 a b c stable spiral Increase the flow resistance. The saddle and the stable node start to merge. stable node saddle
stable spiral 16 R 2 0.4 c Increase the flow resistance. The saddle and the stable node have disappeared. The Gulf Stream will eventually reverse.
- - 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
stable manifold 16 R 2 0.33 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
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
stable manifold 16 R 2 0.1 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
Henry Stommel, Thermohaline Convection with Two Stable Regimes of Flow, TELLUS XII (1961), 224-230.