The Budyko Energy Balance Models of the Earth s Climate

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1 The Budyko Energy Balance Models of the Earth s Climate Esther Widiasih Summer Seminar July

2 Where we were left off!"#$%&'()*)"+#',-.#*/ (0.&1-23#**#$/'!"#$%&'()*)"+#',-.#*'4/5 %" ' %& $ "#! "#!"! 6478'#904*4:$40;'/-*074-" $ "#! " ()! # <84/'#904*4:$40;'/-*074-"'4/'/7):*# =478'#4%#">)*0# (?''@A#+)**'(BC?D E0#/74-"5 )*:#.- =)/'"-7'FCGH.+( *+(,- '&(% '&$% from Samantha s talk

3 Where we were left off!"#$%&'()*)"+#',-.#*/ (0.&1-23#**#$/'!"#$%&'()*)"+#',-.#*'4/5 %" ' %& $ "#! "#!"! Global EBM T = global annual average 6478'#904*4:$40;'/-*074-" $ "#! " ()! # <84/'#904*4:$40;'/-*074-"'4/'/7):*# =478'#4%#">)*0# (?''@A#+)**'(BC?D E0#/74-"5 )*:#.- =)/'"-7'FCGH.+( *+(,- '&(% '&$% from Samantha s talk

4 Where we were left off!"#$%&'()*)"+#',-.#*/ (0.&1-23#**#$/'!"#$%&'()*)"+#',-.#*'4/5 %" ' %& $ "#! "#!"! Global EBM T = global annual average 6478'#904*4:$40;'/-*074-" $ "#! " ()! # <84/'#904*4:$40;'/-*074-"'4/'/7):*# =478'#4%#">)*0# (?''@A#+)**'(BC?D E0#/74-"5 )*:#.- =)/'"-7'FCGH.+( *+(,- '&(% '&$% Next: zonal EBM with ice albedo feedback from Samantha s talk

5 Recall: Earth s Energy Budget energy received = energy re-emitted temperature change = (heat capacity) (heat imbalance)

6 Recall: Earth s Energy Budget energy received = energy re-emitted temperature change = (heat capacity) (heat imbalance) to use in a course: explain the numbers?

7 Global Energy Balance Models heat imbalance = insolation - reflection - reemission incoming solar radiation short wave long wave The Earth s climate is represented as a temperature over one point R dt dt = Q(1 α) (A + BT) all variables and parameters are global averages.

8 Zonal Energy Balance Models "! T(y)!"!%&#$%&" 0 1 y # ' $ equator pole The climate on Earth is represented by a function of temperature over an interval (temperature profile).

9 Zonal Energy Balance Models Why might we want to consider this type of model? What is the advantage of the spatial dimension?

10 Zonal Energy Balance Models heat imbalance = insolation - reflection - reemission - transport 0 y 1 y equator pole transport processes eg. wind, storm, the gulf stream modeling transport eg. diffusion, relaxation to global average

11 Zonal Energy Balance Models heat imbalance = insolation - reflection - reemission - transport 0 y 1 y equator pole reemission processes include the effects of green house gasses modeling reemission is often based on the Stefan-Boltzman s law of black body radiation

Zonal Energy Balance Models heat imbalance = insolation - reflection - reemission - transport Scatter plot for OLR vs T (10 years) 300 ß ot4._o 0.0% : 250 > - o 200 --.N 0 - o 150 --... i!ig.

12 Zonal Energy Balance Models heat imbalance = insolation - reflection - reemission - transport Scatter plot for OLR vs T (10 years) 300 ß ot4._o 0.0% : 250 > - o N 0 - o i!ig... ' ---- :-:-:b-...:.: ::::::::::.....:....,,... [ i ';':[ :-: :.. // : =:':: ':' " :: =:si::. ;'... ß E:>;. :::::-::: ::; F.. 1.0% I ' I ' I ' I ' I ' I ' I ' I Surface Temperature (in C) Outgoing Longwave Radiation = A+BT

13 Zonal Energy Balance Models heat imbalance = insolation - reflection - reemission - transport ocean snow/ ice Satellite data from NASA NASA Observation surface dependent albedo

14 Zonal Energy Balance Models McGehee & Lehman "#$%&'(')!$*!$+#!#,-(.$(,!,**&/('0$#1! M2 3 J2!0'/!'*$(')!$+0$!,*1 M! M! J Ice-Albedo Feedba,*1 M2! M2! J2 3!4#!+05#! " S 6 6S,*1 M2,*1 M2! J2!M2 6 ³ 7 8S $: S 6! S 6 " " S " M M,*1! 3 ³ S 6 6S $: 6S $: 6 8 $: 6#! heat imbalance = insolation - reflection - reemission - transport 1*!! # "! 9 $: 0'/! 6S! % & 6 µ S 6 7 : 6 : & 1(' E,*1 J &,*1 E! J ;! 6 <+#!)&0.+!*=!$+(1!=%',$(*'!(1!1+*4'!('!>()%&#!9!=*&!$+#!,%&&#'$!50-%#!*=!*?-(@%($A! E ' 0 equator y 1 pole y '"$ '"# '!"&!"%!"$!"#!!"!!"'!"#!"(!"$!")!"%!"*!"&!"+ '"! ' ()*+,-'./''01%2345)21'!)%5,)6+5)21'4%'4'7+185)21'27'&/' D#.-#&E1!$+(&/!-04!$#--1!%1!$+0$! McGehee and Lehman, B;CD!

15 Zonal Energy Balance Models: heat imbalance = insolation - reflection - reemission - transport What affects insolation

16 The Budyko Zonal Energy Balance Models heat imbalance = insolation - reflection - reemission - transport R T t = Qs(y) insolation Qs(y)(α(η,y) reflection (A + BT(y)) reemission C T (y) T transport R = planetary heat capacity T = T (y) =T (y, t) s(y) = a distribution function α(η,y) = the albedo at y given that the ice line is at η T = 1 0 T (ξ)dξ A, B, C are parameters Q and s(y) contain Earth orbital elements Budyko, KK Tung, 2007.

17 The Budyko Zonal EBM T(y, t) forward in time Water covered Ice covered Ice cap boundary

18 The Budyko Zonal EBM: Steady State R T t = Qs(y)(1 α(η,y)) (A + BT(y)) C(T (y) T ) if the ice cap boundary is fixed at η, the steady state is T (η,y)= Q s(y) (1 a(η)(y)) + C 1 0 T (η, ξ)dξ A B + C η

19 The Budyko Zonal EBM with Ice Albedo Feedback!"#$%#& '()$"*% '2(0%#& '()$"*% +%,,&-.%& "/0&1/23 42#%&-.%& "/0&1/23 42#%&15/()67* 89,2#9%0&9: +"/0&"/0&1%" +%,,&15/()67* 89,2#9%0&9: +"/0&"/0&1%"!"#$%&#$'()#*+$,##*)-&.

20 The Budyko Zonal EBM with Ice Albedo Feedback!"#$%#& '()$"*% '2(0%#& '()$"*% +%,,&-.%& "/0&1/23 42#%&-.%& "/0&1/23 42#%&15/()67* 89,2#9%0&9: +"/0&"/0&1%" +%,,&15/()67* 89,2#9%0&9: +"/0&"/0&1%"!"#$%&#$'()#*+$,##*)-&.

21 The Budyko Zonal EBM with Ice Albedo Feedback!"#$%#& '()$"*% '2(0%#& '()$"*% +%,,&-.%& "/0&1/23 42#%&-.%& "/0&1/23 42#%&15/()67* 89,2#9%0&9: +"/0&"/0&1%" +%,,&15/()67* 89,2#9%0&9: +"/0&"/0&1%"!"#$%&#$'()#*+$,##*)-&.

22 The Budyko Zonal EBM with Ice Albedo Feedback!"#$%#& '()$"*% '2(0%#& '()$"*% +%,,&-.%& "/0&1/23 42#%&-.%& "/0&1/23 42#%&15/()67* 89,2#9%0&9: +"/0&"/0&1%" +%,,&15/()67* 89,2#9%0&9: +"/0&"/0&1%"!"#$%&#$'()#*+$,##*)-&. Could the Earth s climate reach extreme states? What makes the current small ice cap possible?

23 The Budyko Zonal EBM with Ice Albedo Feedback!"#$%#& '()$"*% '2(0%#& '()$"*% +%,,&-.%& "/0&1/23 42#%&-.%& "/0&1/23 42#%&15/()67* 89,2#9%0&9: +"/0&"/0&1%" +%,,&15/()67* 89,2#9%0&9: +"/0&"/0&1%"!"#$%&#$'()#*+$,##*)-&. Could the Earth s climate reach extreme states? What makes the current small ice cap possible? What do we need to do to the Budyko EBM?

24 The Budyko Zonal EBM with Ice Albedo Feedback!"#$%#& '()$"*% '2(0%#& '()$"*% +%,,&-.%& "/0&1/23 42#%&-.%& "/0&1/23 42#%&15/()67* 89,2#9%0&9: +"/0&"/0&1%" +%,,&15/()67* 89,2#9%0&9: +"/0&"/0&1%"!"#$%&#$'()#*+$,##*)-&. Could the Earth s climate reach extreme states? What makes the current small ice cap possible? What do we need to do to the Budyko EBM? Need to evolve the ice (cap) boundary.

25 The Budyko Zonal EBM with Evolving Ice Boundary R T t = Qs(y)(1 α(η,y)) (A + BT(y)) C(T (y) T ) η

26 The Budyko Zonal EBM with Evolving Ice Boundary R T t = Qs(y)(1 α(η,y)) (A + BT(y)) C(T (y) T ) Assumptions to evolve η 1. Ice forms or melts at a much slower rate 2. At the ice line, ice forms above and melts below a critical temperature η

27 The Budyko Zonal EBM with Evolving Ice Boundary R T t = Qs(y)(1 α(η,y)) (A + BT(y)) C(T (y) T ) Assumptions to evolve η 1. Ice forms or melts at a much slower rate 2. At the ice line, ice forms above and melts below a critical temperature η dη dt =(T (η) T c)

28 The Budyko Zonal EBM and a Slow Moving Ice Line T(y, t) forward in time Water covered Ice covered Ice cap boundary

29 The Budyko Zonal EBM and a Slow Moving Ice Line T(y, t) forward in time Water covered Ice covered Ice cap boundary Where would the ice line (the ice cap boundary) end up?

30 The Budyko EBM with a Slow Moving Ice Line: Some Results

31 The Budyko EBM with a Slow Moving Ice Line: Some Results Budyko EBM Slow moving ice line Temperature profile steady state R T t = Qs(y)(1 α(η,y)) (A + BT(y)) C(T (y) T ) dη dt = (T (η) T c) T (y, η) = Q s(y) (1 a(η,y)) + C 1 0 T (η)(ξ)dξ A B + C

32 The Budyko EBM with a Slow Moving Ice Line: Some Results Budyko EBM Slow moving ice line Temperature profile steady state R T t = Qs(y)(1 α(η,y)) (A + BT(y)) C(T (y) T ) dη dt = (T (η) T c) T (y, η) = Q s(y) (1 a(η,y)) + C 1 0 T (η)(ξ)dξ A B + C 1. The temperature profile T(y, t) quickly approaches T*(y, η), while the ice line η moves slowly. 2. When the temperature profile is near T*(y, η), the dynamics of the ice line η is approximately the following

33 The Budyko EBM with a Slow Moving Ice Line: Some Results Budyko EBM Slow moving ice line Temperature profile steady state R T t = Qs(y)(1 α(η,y)) (A + BT(y)) C(T (y) T ) dη dt = (T (η) T c) T (y, η) = Q s(y) (1 a(η,y)) + C 1 0 T (η)(ξ)dξ A B + C 1. The temperature profile T(y, t) quickly approaches T*(y, η), while the ice line η moves slowly. 2. When the temperature profile is near T*(y, η), the dynamics of the ice line η is approximately the following dη dt = (T (η, η) T c )

34 The Budyko EBM with a Slow Moving Ice Line: Some Results dimension: R T = Qs(y)(1 α(η,y)) (A + BT(y)) C(T (y) T ) t dη dt = (T (η) T c) T*(η, η ) dimension: one dη dt = (T (η, η) T c ) Tc

35 The Budyko EBM with a Slow Moving Ice Line: Some Results dimension: R T = Qs(y)(1 α(η,y)) (A + BT(y)) C(T (y) T ) t dη dt = (T (η) T c) T*(η, η ) dimension: one dη dt = (T (η, η) T c ) Tc Quiz: What is (are) the stable ice line state(s)? The unstable one(s)?

36 The Budyko EBM with a Slow Moving Ice Line: Preliminary Bifurcation Analysis dη dt = (T (η, η) T c ) T (y, η) = Q s(y) (1 a(η,y)) + C 1 0 T (η)(ξ)dξ A B + C T*(η, η ) Tc

37 The Budyko EBM with a Slow Moving Ice Line: Preliminary Bifurcation Analysis dη dt = (T (η, η) T c ) T (y, η) = Q s(y) (1 a(η,y)) + C 1 0 T (η)(ξ)dξ A B + C T*(η, η ) Some Interesting questions: 1. What are the effects of increasing green house gas? --see Jim s session for more details Tc 2. What are the effects of solar insolation variability ie. the Milankovitch cycle? -- see Dick s session. 3. What climate states can this simple model predict? -- see Anna s session

38 Suggested Exercises Software oriented -page 10, Section 1.2 : modeling, stability, green house effect (walking through the details of this lecture) Minimum software -page 10, Section 1.2 -page 21, Section 2.1: stability, bifurcation -pagesection 2.2: dimension reduction of the Budyko EBM -page 30, Section 2.5: Budyko EBM with contraction -page 33, Section 2.6: Snowball Earth, Jormungand state

39 Thank you for your attention!

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