Dynamics of Energy Balance Models for Planetary Climate

Transcription

1 Dynamics of Energy Balance Models for Planetary Climate Alice Nadeau University of Minnesota April 13, 2016

2 Motivation Low dimensional climate models are important for understanding the predominant forces affecting the climate of Earth Fly-bys of Pluto and other rocky celestial bodies in our solar system have raised interest in other climates Photos from nasa.gov

3 Energy Balance Models in 1969 Budyko and Sellers (independently) proposed energy balance models for the Earth (1, 14) Wanted to study if another glacial age was possible Both models had the same major components: incoming solar radiation, outgoing radiation, and energy transfer: R T = Q(y)(1 α(y)) (A + BT ) + Γ(T )

4 Energy Balance Models Today where R T t = Qs(y)(1 α(y, η)) (A + BT ) + Γ(T ) Γ(T (y, η)) = C ( T (y, η) 1 0 ) T (γ, η)dγ. Widiasih introduced an equation for the dynamics of the ice line, η, in 2012 (18) d dt η = ɛ(t (y, η) T c)

5 Quadratic Approximation In (9) McGehee and Widiasih consider the Budyko-type equation for y, η [0, 1] t T = 1 ( ( )) Qs(y)(1 α(η, y)) (A + BT (y, η)) C T (y, η) T R with dynamic ice line η = ρ(t (y, η) T c ) and piecewise constant albedo function α w y < η α(y, η) = α 0 y = η y > η α i where α 0 = α w + α i. 2

6 Piecewise Function Space The equilibrium temperature profile can be written 1 Tη B+C (Qs(y)(1 α w ) A + CTη ) (y) = 1 B+C (Qs(η)(1 α 0) A + CTη ) 1 B+C (Qs(y)(1 α i) A + CTη ) y < η y = η y > η which motivates the four-dimensional function space X whose elements are of the form w z 0 + ( w z 2) p2 (y) y < η T (y) = w 0 + w 2 p 2 (η) y = η w z 0 + ( w z 2) p2 (y) y > η

7 New Budyko s Equation Reformulating the t T equation in this function space gives Rẇ 0 = Q(1 α 0 ) A Bw 0 + C ((η 12 η ) )z 0 + z 2 p 2 (y)dy Rż 0 = Q(α i α w ) (B + C)z 0 Rẇ 2 = Qs 2 (1 α 0 ) (B + C)w 2 Rż 2 = Qs 2 (α i α w ) (B + C)z 2 and the ice line equation becomes ( R η = ρ Write w 0 Q(1 α 0) B + C s 2p 2 (η) + T c Rẇ 0 = B(w 0 F (η)) R η = ρ(w 0 G(η)) 0 ).

8 (η, w 0 ) Phase Space Rẇ 0 = B(w 0 F (η)) R η = ρ(w 0 G(η))

9 The Jormungand Model Define the function δ(η) = { η +.35, η <.35 0, η.35 which represents the extent of the bare ice and the Jormungand albedo function α J (y, η) = α s + α w 2 + α s α i 2 + α i α w 2 tanh(m(y η)) tanh(m(y (η + δ(η))))

10 Jormungand albedo function and Temperature Profile Widiasih s Theorem still applies with this albedo function There is a locally attracting invariant manifold PJ = {(Φ J (η), η) : η R} within O(ɛ) of the manifold of fixed points TJ = {(T J (y, η), η) : η R}

11 Jormungand Bifurcation in the Greenhouse Gas Parameter

12 Insolation Distribution on Rapidly Spinning Planets Actual distribution can be found using orbital parameters (as seen in (5, 8, 17)): s(y, β) = 2 2π ( 2 π y 2 sin β sin γ y cos β) dγ 0 We use Legendre approximations in EBMs because the above integral doesn t have a closed form expression. Instead use s(y, β) = s 2n (β)p 2n (y) n=0

13 Insolation Distribution on Rapidly Spinning Planets Write the 2n-th degree Legendre polynomial as and where c 2k (β) = k ( 2k 2j ( π/2 j=0 p 2n (y) = s 2n (β) = P 2n n a 2k y 2k k=0 n a 2k c 2k (β). k=0 ) (sin β) 2(k j) (cos β) 2j π 2 (cos ˆφ) 2(k+1 j) (sin ˆφ) 2j d ˆφ π/2 ) ( 2π 0 ) (cos ˆθ) 2(k j) d ˆθ

14 Insolation Distribution on Rapidly Spinning Planets Earth (a) (a) Pluto (a) sin(ϕ) (b) (b) (b) sin(ϕ) (c) 1.2 (c) (c) Actual Approximation sin(ϕ) sin(ϕ) sine of latitude sine of latitude

15 Small Integer Spin-Orbit Resonances Mean annual insolation distributions for obliquity β = 60 and eccentricity e =.2: -1:1 0:1 1:1 2:1 3:1 4:1 1:2 3:2 5:2 A. Dobrovolskis, Insolation on exoplanets with eccentricity and obliquity.

16 Open Questions about Insolation Can we quantify when we can use the rapidly spinning planet method/formula and still have error less than τ in our approximation? Can we find a closed form expression for insolation on planets with small integer resonances?

17 Open Questions about Insolation For a reasonable range of parameter values, the Budyko map for Pluto doesn t have any nontrivial stable fixed points. Why? Could Pluto s upside down insolation be playing a factor? Is it because Pluto s insolation is relatively flat? (a) (b) (c) Actual Approximation sin(ϕ) sine of latitude

18 Open Questions about Ice Lines Reformulate model to accommodate ice planets Is more than one ice present and how do we account for different albedos? How are the ices situated on the surface? Are ices mixed? Are they layered? From nasa.gov

19 Open Questions about Ice Lines Remove symmetry assumptions? Investigate four ice lines (η SP, η SE, η NE, η NP ) with the properties (i) η SP, η SE, η NE, η NP [ 1, 1]. (ii) 1 η SP η SE η NE η NP 1. (iii) Ice is located between η SP and η SE and between η NE and η NP. Initial investigations into this case show potential oscillations in the ice line dynamics

20 Other Open Questions Concerning EBMs How do we accurately model a planet whose atmosphere freezes out (as Pluto s might)? Is the diffusion model more accurate for a planet with no oceans? For a planet without an atmosphere? In which cases do we get the same results with both models? In which cases do we get different results?

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