Math 5490 9/10/2014 Energy Balance Math 5490 Conservation of - PDF document

Math 5490 9/10/2014 Energy Balance Math 5490 Conservation of Energy Topics in Applied Mathematics: Introduction to the Mathematics of Climate temperature change ~ energy in – energy out Mondays and Wednesdays 2:30 – 3:45 http://www.math.umn.edu/~mcgehee/teaching/Math5490-2014-2Fall/ Streaming video is available at short wave energy long wave energy http://www.ima.umn.edu/videos/ from the Sun from the Earth Click on the link: "Live Streaming from 305 Lind Hall". Participation: https://umconnect.umn.edu/mathclimate Everything else is detail. Math 5490 9/10/2014 Energy Balance Energy Balance Stefan ‐ Boltzmann Law Stefan ‐ Boltzmann Law     F T 4 F T 4 power flux (W/m 2 ) temperature (K) power flux (W/m 2 ) temperature (K) Stefan-Boltzmann constant Stefan-Boltzmann constant     8 2 4     8 2 4 5.67 10 W/m K 5.67 10 W/m K Example surface temperature of the Sun: 5780K Reasonable approximation: power flux: 5.67x10 -8 x (5780) 4 = Every body in the solar system radiates energy 6.33x10 7 W/m 2 according to this law. total solar power output: 6.33x10 7 x 4 π ( r S ) 2 , where r S = radius of the sun = 6.96x10 8 m total solar output: 3.85x10 26 W Math 5490 9/10/2014 Math 5490 9/10/2014 Energy Balance Energy Balance Insolation Insolation Global Average Insolation Solar flux at a distance r from the sun: (In coming so lar radi ation) 2 6.33 10 4  7  r 2  r  intercepted flux: F = 1368 W/m 2    7 2 F S 6.33 10 S W/m    Earth cross-section: π r E 2 4 r 2  r  r S = 6.96x10 8 m surface area: 4 π r E 2 average flux: 1368/4 = 342 W/m 2 = Q r = 1.5x10 11 m Simple Model F  1368 W/m 2 solar flux at Earth’s orbit Assume that Earth is a perfectly thermally conducting black body.   Q T 4 F   r 2 W Power intercepted by the Earth: E     1 4 1 4  T  Q   342 5.67 10   8 4 r 2 m 2 Earth’s surface area: E  279K  6 C   43 F    2 F r F   E 342 W/m 2 Average surface flux: 4  r 2 4 Dynamics E dT    R Q T 4 dt heat capacity stable equilibrium Math 5490 9/10/2014 Math 5490 9/10/2014 Richard McGehee, University of Minnesota 1

Math 5490 9/10/2014 Energy Balance Energy Balance Goldilocks Zone Goldilocks Zone Solar flux at a distance r from the Sun: 250   2 6.33 10 4 7 r 2  r  200 F  S  6.33 10  7 S W/m 2   4  r 2  r  150 temperature (Celsius) r S = 6.96x10 8 m 100 50  25 3.07 10  F W/m 2 0 r 2 3.07 10  25 7.67 10  24 ‐ 50 2 Average surface flux: = W/m 2 2 4 r r ‐ 100  24 7.67 10 ‐ 150  Black body temperature: T 4 = W/m 2 r 2 ‐ 200 1 4   ‐ 250 7.67 10  24 1.078 10  8   T   0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60  2  r  r distance from Sun (Tm) T(r) Mercury Earth Saturn Math 5490 9/10/2014 Math 5490 9/10/2014 Energy Balance Energy Balance Goldilocks Zone albedo insolation OLR 150 100 temperature (Celsius) 50 0 ‐ 50 ‐ 100 0.00 0.05 0.10 0.15 0.20 0.25 0.30 distance from Sun (Tm) T(r) Venus Earth Mars Historical Overview of Climate Change Science , IPCC AR4, p.96 http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_CH01.pdf Math 5490 9/10/2014 Math 5490 9/10/2014 Energy Balance Energy Balance Insolation albedo insolation OLR Global Average Insolation (In coming so lar radi ation) intercepted flux: F = 1368 W/m 2 Earth cross-section: π r E 2 surface area: 4 π r E 2 average flux: 1368/4 = 342 W/m 2 = Q Simple Model Assume that Earth is a perfectly thermally conducting black body.   Q T 4     1 4 1 4 T  Q   342 5.67 10   8  279K  6 C   43 F  Dynamics dT    R Q T 4 dt Historical Overview of Climate Change Science , IPCC AR4, p.96 heat capacity stable equilibrium http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_CH01.pdf Math 5490 9/10/2014 Math 5490 9/10/2014 Richard McGehee, University of Minnesota 2

Math 5490 9/10/2014 Energy Balance Energy Balance Albedo albedo insolation OLR Not all the insolation reaches the surface. Some is reflected back into space. The proportion reflected is called the albedo, denoted α . For Earth, α ≈ 0.3 . Simple Model Assume that Earth is a perfectly thermally conducting black body, but only 70% of the insolation is absorbed.     1 4 1 4  T  0.7  F   0.7 342 5.67 10   8  255K   18 C   0 F  Dynamics dT R  Q (1   )   T 4 stable equilibrium dt Historical Overview of Climate Change Science , IPCC AR4, p.96 http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_CH01.pdf Math 5490 9/10/2014 Math 5490 9/10/2014 Energy Balance Energy Balance OLR as a Function of Surface Temperature OLR as a Function of Surface Temperature ( O utgoing L ongwave R adiation)   OLR A BT OLR  A  BT Important: A and B are determined from satellite observations. A+BT is not a linear approximation to the Stefan- T is surface temperature ( in Celsius ). Boltzmann equation.  2 A 202 W/m B  1.90 W/m K 2 different Dynamics Dynamics Kelvin Kelvin dT dT photosphere photosphere      4      4 R Q (1 ) T R Q (1 ) T temperature temperature dt dt becomes becomes Celsius Celsius dT global mean dT global mean R  Q (1   )  ( A  BT ) R  Q (1   )  ( A  BT ) surface temperature surface temperature dt dt Math 5490 9/10/2014 Math 5490 9/10/2014 Energy Balance Energy Balance Homogeneous Earth Latitude Dependence dT Make T depend on y = sin(latitude) R  Q (1   )  ( A  BT ) dt  T y t ( , )        R Qs y ( )(1 ) A BT y t ( , )        Q 1 A BT 0  Equilibrium Temperature: t eq insolation distribution      Q 1 A Stable, since B > 0 . T    Q global annual average insolation 342W/m 2 eq B    1   s y ( ) distribution across latitudes s y dy ( ) 1 Ice-free planet: α = 0.32, T eq = 16 ° C 0 Snowball planet: α = 0.62, T eq = -38 ° C One can show that No glacier would form on an ice-free Earth. 2    2  2    2      s y ( )  1 1 y sin cos y cos d No glacier would melt on a snowball Earth.  2  0 β = obliquity = 23.5° Easy question: Why do we have ice caps? Chylek and Coakley’s quadratic approximation: Hard question:       2  s y 1 0.241 3 y 1 If Earth was ever a snowball, how did we get out? Math 5490 9/10/2014 Math 5490 9/10/2014 Richard McGehee, University of Minnesota 3

Math 5490 9/10/2014 Energy Balance Energy Balance Insolation Distribution Latitude Dependence  T y t ( , )   R  Qs y ( )(1   )  A  BT y t ( , )  t 1.3 Note that y is just a parameter. approx today 1.2 Equilibrium Temperature Profile green = quadratic    Qs y ( )(1 ) A approximation 1.1 T ( ) y  eq B (Chylek & Coakley) relative insolation 1 60 40 0.9 fuchsia = formula temperature (Celsius) 20 α = 0.32: ice free 0.8 using obliquity of 0 α = 0.62: snowball 23.5° ice free 0.7 ‐ 20 snowball ‐ 40 0.6 ‐ 60 0.5 ‐ 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 sine(latitude) sin(latitude) Math 5490 9/10/2014 Math 5490 9/10/2014 Energy Balance Energy Balance Latitude Dependence What’s Missing?  T y t ( , )   R  Qs y ( )(1   )  A  BT y t ( , )  t 60 40 temperature (Celsius) 20 0 ice free ‐ 20 snowball ‐ 40 ‐ 60 ‐ 80 0 0.2 0.4 0.6 0.8 1 sin(latitude) ice won’t melt ice will form (no exit from snowball) (icecap) Math 5490 9/10/2014 Math 5490 9/10/2014 Energy Balance Energy Balance What’s Missing? What’s Missing? Math 5490 9/10/2014 Math 5490 9/10/2014 Richard McGehee, University of Minnesota 4

Math 5490 9/10/2014 Energy Balance Energy Balance What’s Missing? What’s Missing? Math 5490 9/10/2014 Math 5490 9/10/2014 Energy Balance Energy Balance What’s Missing? What’s Missing? Weather! Math 5490 9/10/2014 Math 5490 9/10/2014 Energy Balance Budyko’s Model Why y ? Budyko’s Model  T        R Qs y ( )(1 ) ( A BT ) C T ( T )  t  T y t ( , )        R Qs y ( )(1 ) ( A BT y t ( , )) C T t ( ( ) T y t ( , ))   Weather  1 t T t ( ) T y t dt ( , ) global mean temperature 0   1 global mean temperature: T t ( ) T y t dt ( , ) 0 Second Law of Thermodynamics: Why do we use y = sine(latitude) instead of just latitude? Energy travels from hot places to cold places. Budyko’s equation as a dynamical system: T lives in a function space (temperature as a function of latitude). Math 5490 9/10/2014 Math 5490 9/10/2014 Richard McGehee, University of Minnesota 5