Budyko’s Energy Balance Model Gareth E. Roberts Department of Mathematics and Computer Science College of the Holy Cross Worcester, MA, USA Mathematical Models MATH 303 Fall 2018 November 19, 26, and 28, 2018 Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 1 / 14
FAQ 1.1, Figure 1. Estimate of the Earth’s annual and global mean energy balance. Over the long term, the amount of incoming solar radiation absorbed by the Earth and atmosphere is balanced by the Earth and atmosphere releasing the same amount of outgoing longwave radiation. About half of the incoming solar radiation is absorbed by the Earth’s surface. This energy is transferred to the atmosphere by warming the air in contact with the surface (thermals), by evapotranspiration and by longwave radiation that is absorbed by clouds and greenhouse gases. The atmosphere in turn radiates longwave energy back to Earth as well as out to space. Source: Kiehl and Trenberth (1997). Figure: Heat Balance. Recall: Q = S / 4 = 342 W / m 2 . Source: “Historical Overview of Climate Change Science,” IPCC AR4, (2007) p. 96. Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 2 / 14
Tilt of the Earth Figure: The Earth is tilted (obliquity) 23 . 5 ◦ from the normal to the plane of the ecliptic (the plane the planets travel in around the sun). The obliquity changes on a 40,000 year cycle. Source: http://www.rsd17.org/TeacherWebPage/ HighSchool/JAnderson/A/introduction/earthinspace/earthsTilt.jpg Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 3 / 14
Insolation Distribution � � � � � � � � � 1.3 approx today 1.2 green = quadratic � � � approximation 1.1 � (Chylek & Coakley) relative insolation 1 0.9 fuchsia = formula α 0.8 using obliquity of α 23.5° 0.7 ‐ ‐ 0.6 ‐ 0.5 ‐ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sine(latitude) Figure: The quadratic approximation to the insolation distribution s ( y ) is quite good. Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 4 / 14 � � � � � � � � � ‐ ‐ ‐ ‐
Figure: Archimedes’ Hat-Box Theorem: S 1 = S 2 = 2 π Rh . The cylinder and sphere have the same radius ( a = R ). Think of the sphere being circumscribed by the cylinder. Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 5 / 14
� � � � � � � � � � � θ θ 1 � θ � � � � � � � � � � 0.9 � � � � 0.8 surface area proportion θ 0.7 Arctic Circle θ 0.6 � � � � � � � � � � � � � � � � � 0.5 θ θ � � 0.4 Minneapolis � 0.3 0.2 � � Tropic of Cancer 0.1 � 0 � � 0 10 20 30 40 50 60 70 80 90 latitude Figure: A plot of y = sin θ along with some key latitudes. Due to Archimedes’ Hat-Box Theorem, the proportion of the Earth’s surface area from the equator to a given latitude θ is simply y / 2, and between − θ and θ it is just y . Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 6 / 14 � � � � � � � � � � � � � � � � � � � � �
Equlibrium temperature profiles 60 Ice free Ice free Worcester 40 Snowball Snowball 20 T (celsius) 0 -20 -40 -60 -80 0 0.2 0.4 0.6 0.8 1 y = sin Figure: Graphs of equilibrium temperatures with (solid) and without (dashed) latitude dependence. Ice free is α = 0 . 32 (red); snowball Earth is α = 0 . 62 (blue). Note that incorporating latitude allows ice caps to form in the ice free case. Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 7 / 14
Equilibrium temp. profiles with and without heat transport 60 ice free (C = 0) ice free snowball (c = 0) 40 snowball 20 T (celsius) 0 -20 -40 -60 -80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y = sin (latitude) Figure: Graphs of equilibrium temperatures with ( C = 3 . 04; solid) and without ( C = 0; dashed) heat transport. Ice free is α = 0 . 32 (red); snowball Earth is α = 0 . 62 (blue). Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 8 / 14
Equilibrium temperature profiles for different ice lines 30 20 10 = 1 0 T (celsius) -10 -20 -30 = 0 -40 -50 -60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y = sin Figure: Graphs of equilibrium temperatures with two-step albedo function for different ice lines: η = 1 (red; ice free), η = sin( 70 ◦ ) (orange; current), η = sin( 42 . 3 ◦ ) (green; Worcester), η = sin( 23 . 5 ◦ ) (light blue; Tropic of Cancer), η = 0 (blue; snowball). Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 9 / 14
Plot of h( ) 5 4 2 0 1 2 -2 d /dt -4 -6 -8 -10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure: Plot of h ( η ) for the Widiasih ice-line equation d η/ dt = ǫ h ( η ) showing two equilibria ice line positions at η 1 ≈ 0 . 2562 (unstable) and η 2 ≈ 0 . 9394 (stable). Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 10 / 14
Equilibrium temperature profiles for 1 and 2 30 20 = 2 10 0 = 1 T (Celsius) -10 -20 -30 -40 -50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y = sin Figure: Equilibrium temperature profiles for the two ice line equilibrium points η = η 1 ≈ 0 . 2562 (blue) and η = η 2 ≈ 0 . 9394 (red). The red curve is very close to our current climate. Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 11 / 14
Figure 3: Figure: Bifurcation diagram showing the location of the ice line equilibria (roots of h ( η ) ) as the albedo parameter α s is varied. Note the tipping point at α s ≈ 0 . 69557. Figure by Cara Donovan. Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 12 / 14
Plot of h( ) at bifurcation 0 -5 -10 d /dt -15 -20 -25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure: Plot of h ( η ) for α s = α ∗ s showing saddle-node (tangent) bifurcation. Once α s increases above α ∗ s , there are no equilibria and the ice line decreases toward the equator (Snowball Earth scenario). Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 13 / 14
Less CO 2 ★ More CO 2 Figure 4: Figure: Two-dimensional bifurcation diagram indicating the number of ice line equilibria as A and α S are varied. Red means two equilibria (one stable, one unstable); green means one equilibrium (the other root is less than 0 or greater than 1); blue indicates no equilibria. Figure by Cara Donovan. Roberts (Holy Cross) Budyko’s Energy Balance Model Mathematical Models 14 / 14
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