An Energy Balance Model for Arctic Sea Ice

Transcription

1 An Energy Balance Model for Arctic Sea Ice Kaitlin Hill October 1, 217

2 Outline Motivation: Arctic sea ice decline Energy balance model Bifurcation analysis Results

3 Arctic sea ice age Sept 1986 Sept 216 (NASA Scientific Visualization Studio)

4 Extent (Millions of km 2 ) Recent decline in Arctic sea ice extent Arctic Sea Ice Extent (since 1979) Average Arctic sea ice extent (Sept) Year 21 1 Mar 1 Sep Date (nsidc.org)

5 Extent (Millions of km 2 ) Recent decline in Arctic sea ice extent Arctic Sea Ice Extent (since 1979) Average Arctic sea ice extent (Sept) Year 21 1 Mar 1 Sep Date (nsidc.org)

6 Extent (Millions of km 2 ) Recent decline in Arctic sea ice extent Arctic Sea Ice Extent (since 1979) Average Arctic sea ice extent (Sept) Year 21 1 Mar 1 Sep Date (nsidc.org)

7 Extent (Millions of km 2 ) Recent decline in Arctic sea ice extent Arctic Sea Ice Extent (since 1979) Average Arctic sea ice extent (Sept) Year 21 1 Mar 1 Sep Date (nsidc.org)

8 Extent (Millions of km 2 ) Recent decline in Arctic sea ice extent Arctic Sea Ice Extent (since 1979) Average Arctic sea ice extent (Sept) Year 21 1 Mar 1 Sep Date (nsidc.org)

9 Arctic energy balance model de dt = 1 α E F s t (F l t + BT(E, t)) incoming outgoing (Eisenman and Wettlaufer 29)

10 Arctic energy balance model de dt = 1 α E F s t (F l t + BT(E, t)) incoming outgoing Major components: E has no spatial extent Seasonally-varying incoming solar radiation Ice-albedo feedback State-dependent definition of energy: Ocean mixed layer temperature (E > ) E ቊ Ice thickness E Sea ice thermodynamics (Eisenman and Wettlaufer 29)

11 Albedo (α) Arctic energy balance model de dt = 1 α E F s t (F l t + BT(E, t)) albedo 1 α i α o Energy (E) (Eisenman and Wettlaufer 29)

12 Albedo (α) Arctic energy balance model de dt = 1 α E F s t (F l t + BT(E, t)) albedo incoming solar radiation 1 4 α i F s 2 α o Energy (E) t 1 (Eisenman and Wettlaufer 29)

13 Albedo (α) Arctic energy balance model de dt = 1 α E F s t (F l t + BT(E, t)) albedo incoming outgoing solar longwave radiation radiation α i F s 2 F l 2 α o Energy (E) t 1 1 t (Eisenman and Wettlaufer 29)

14 Albedo (α) Arctic energy balance model de dt = 1 α E F s t (F l t + BT(E, t)) albedo incoming outgoing temperature solar longwave radiation radiation α i α o Energy (E) F s 2 t 1 F l 2 1 t T ቐ E (ocean) (melt) f(t, E) (ice) (Eisenman and Wettlaufer 29)

15 Energy (W m 2 ) Bifurcation analysis: motivation Perennially ice-free Seasonally ice-free Perennially ice-covered 25 (Greenhouse gases ) (Eisenman and Wettlaufer 29)

16 Energy (W m 2 ) Bifurcation analysis: motivation Smooth transition from current state to seasonally ice-free state 25 Perennially ice-free Seasonally ice-free Perennially ice-covered (Greenhouse gases ) (Eisenman and Wettlaufer 29)

17 Albedo (α) Arctic energy balance model de dt = 1 α E F s t (F l t + BT(E, t)) albedo incoming outgoing temperature solar longwave radiation radiation α i α o Energy (E) F s 2 t 1 F l 2 1 t T ቐ E (ocean) (melt) f(t, E) (ice) (Eisenman 212)

18 Albedo (α) (α) Albedo (α) (α) Arctic energy balance model de dt = 1 α E F s t (F l t + BT(E, t)) albedo incoming outgoing temperature solar longwave radiation radiation 1 1 α i ΔE α i α o Energy (E) α o Energy (E)

19 Solutions in the piecewise-constant albedo limit We can rewrite our model as: de dt = ቐ 1 + Δα F s t F l t + BT E, t, E > 1 Δα F s t F l t + BT E, t, E < = ቊ F +(t) BT E, t, E > F (t) BT E, t, E <

20 Solutions in the piecewise-constant albedo limit We can rewrite our model as: de dt = ቐ 1 + Δα F s t F l t + BT E, t, E > 1 Δα F s t F l t + BT E, t, E < = ቊ F +(t) BT E, t, E > F (t) BT E, t, E < = F + BE, E [no ice] F, E <, F > [ice melting] ζf ζ E, E <, F < [ice growing]

21 E Solutions in the piecewise-constant albedo limit We can rewrite our model as: de dt = ቐ 1 + Δα F s t F l t + BT E, t 1 Δα F s t F l t + BT E, t = ቊ F +(t) BT E, t, E > F (t) BT E, t, E < =.5 F + BE, E [no ice] F, F > E <, [ice melting] ζf ζ E, E <, F < [ice growing] t 1 Perennially ice-free Seasonally ice-free Perennially ice-covered

22 Piecewise-smooth dynamical systems No sliding Attracting Sliding Repelling sliding

23 E Piecewise-smooth dynamical systems No sliding Attracting Sliding Repelling sliding t

24 Repelling sliding intervals

25 Repelling sliding intervals 1. Width determined by F ± : 2. Affect stability of solutions 3. Introduce non-uniqueness de dt = ቊF +(t) BT E, t, E > F (t) BT E, t, E <

26 E F ± (t) 1. Sliding interval width de dt = ቊF +(t) BT E, t, E > F (t) BT E, t, E < F +.5 F.5 t t

27 2. Stability: Floquet multiplier P(E + ξ) E + ξ E P(E ) t t + 1 Floquet Multiplier = P E + ξ P(E ) ξ t +1 f = exp න t E τ, E dτ

28 Energy (min(e)) 2. Stability: Floquet multiplier 2ΔE L m (Greenhouse gases ) Perennially ice-free Seasonally ice-free Perennially ice-covered

29 Energy (min(e)) Albedo (α) 2. Stability: Floquet multiplier ΔE Energy (E) 2ΔE L m (Greenhouse gases ) Perennially ice-free Seasonally ice-free Perennially ice-covered

30 Floquet Multiplier Energy (min(e)) Albedo (α) 2. Stability: Floquet multiplier ΔE Energy (E) 2ΔE min(e) 2ΔE L m (Greenhouse gases ) Perennially ice-free Seasonally ice-free Perennially ice-covered

31 Floquet Multiplier 2. Stability: Floquet multiplier ΔE =.8 ΔE =.2 ΔE min(e) min(e) min(e)

32 Floquet Multiplier Log offm maximum 2. Stability: Floquet multiplier ΔE =.2 e1/ ΔE min(e) ΔE Maxima seem to increase like ek/ ΔE Using Filippov solutions (ΔE = ) to approximate the Floquet multiplier: t +1 f exp න t E τ, E dτ

33 Energy (min(e)) 3. Non-unique solutions Omit non-unique periodic solutions from bifurcation diagram Leads to bifurcation diagram with gaps Grazing-sliding bifurcations L m (Greenhouse gases )

34 Energy (W m 2 ) Bifurcation analysis: motivation Perennially ice-free Seasonally ice-free Perennially ice-covered 25 (Greenhouse gases ) (Eisenman and Wettlaufer 29)

35 Energy (W m 2 ) Bifurcation analysis: motivation Smooth transition from current state to seasonally ice-free state 25 Perennially ice-free Seasonally ice-free Perennially ice-covered (Greenhouse gases ) (Eisenman and Wettlaufer 29)

36 Energy (min(e)) Energy (min(e)) Energy (min(e)) Comparing piecewise-smooth to Filippov ΔE =.8 ΔE =.2 Filippov (ΔE ) 2ΔE L m L m L m

37 Energy (min(e)) Energy (min(e)) Energy (min(e)) Comparing piecewise-smooth to Filippov ΔE =.8 ΔE =.2 Filippov (ΔE ) 2ΔE L m L m L m Motivating Question: Can we move the location of the tiny hysteresis loop?

38 F ± (t) E Relationship between sliding intervals and gap size Sliding width comes from F ± : de dt = ቊF +(t) BT E, t, E > F (t) BT E, t, E < 1.5 F + 1 L m = 1.1 L m =.925 F.5 t.5 t

39 F ± (t) Energy (min(e)) Relationship between sliding intervals and gap size Sliding width comes from F ± : de dt = ቊF +(t) BT E, t, E > F (t) BT E, t, E < F + L m = 1.1 L m =.925 F t L m

40 F ± (t) Relationship between sliding intervals and gap size Map F ± to a standard form: F ± = 1 ± Δ α 1 S a cos2πt L m + L a cos2π t φ F ± = F ± + F ± cos 2πt ψ ± Δψ = ψ + ψ Δψ t

41 Energy (min(e)) F ± (t) Varying Δψ don t change with Δψ Δψ t (Default value) L m Δψ

42 Energy (min(e)) F ± (t) Varying Δψ Δψ t L m Δψ

43 Energy (min(e)) F ± (t) Varying Δψ Δψ t L m Δψ

44 Energy (min(e)) F ± (t) Varying Δψ Δψ t L m Δψ

45 Energy (min(e)) F ± (t) Varying Δψ Δψ t L m Δψ

46 Energy (min(e)) Energy (min(e)) Energy (min(e)) Energy (min(e)) F ± (t) F ± (t) Δψ =.21 (Original parameters) Δψ =.51 small sliding t t small gap L m L m smooth transition to seasonal state L m L m

47 Size of the jump: Δ(min(E))

48 Thanks to: Mary Silber (Statistics, U Chicago) Dorian Abbot (Geophysical Sciences, U Chicago)