dynamics of the budyko's energy balance model: instability ... … · instability of ice free...
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Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Dynamics of the Budyko’s Energy Balance Model:Instability of Ice Free Earth
Esther Widiasih
University of ArizonaMathematics of Climate Research Network-University of Arizona Node
also thanks to the Howard Hughes Medical Institute GrantSIAM
SIAM DS 2011Snowbird, UTMay 23, 2011
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
What is an Energy Balance Model?At the equilibrium:
energy received by the earth from the sun’s radiation=
energy reradiated back to space at the planet’s temperature
An Illustration from the IPCC AR-4
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
The Basic Principle of an Energy Balance Model for theEarth’s Climate
Energy Imbalance = Insolation-Reflection︸ ︷︷ ︸Short Wave
-Re radiation︸ ︷︷ ︸Long Wave
+Transport
Energy = Heat Capacity times TemperatureThe rate of change of the planet’s temperature is affected by
incoming solar radiation (insolation) and reflection,the planet’s outward radiation or re-radiation,
and the transport process.
Pioneers in climate: Mihail Budyko (A Russian Climatologist) andWilliam Sellers (from University of Arizona!)
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
The Ice Albedo FeedbackBudyko proposed an EBM that includes the ice albedo feedback tomodel the annual average temperature distribution.
Warmer Climate
Less Ice and Snow
More SunlightAbsorbed byLand and Sea
Colder Climate
More Ice and Snow
Less SunlightAbsorbed byLand and Sea
The Ice Albedo Feedback
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
The Variable in the Budyko’s Model:The Temperature Profile, T = T (y , t)
(Heat Capacity) x (Temperature) = Insolation - Reflection - Re radiation +Transport
Nθ
0 1
y = sin θ
y
θ = latitudey = sin(θ)
T (y , t) = latitudinal andannual averagetemperature aty , at time t
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Discrete Time Budyko’s EquationSince T = T (y , t) is an annual average, we will analyze the discrete timeequation.
RT (y , t + h)− T (y , t)
h=
Q · s(y) · (1− α(y , η))︸ ︷︷ ︸non-reflected insolation
− (A + B · T (y , t))︸ ︷︷ ︸reradiation
+ C ·(∫ 1
0T (y , t)dy − T (y , t)
)︸ ︷︷ ︸
energy gained from transport
(taken mostly from KK Tung, Topics in Mathematical Modelling, 2007)
Notes:R is the planet’s heat capacity.Q, A, B, C are all positive constants.s(y) = is the distribution of an annual incoming solar radiation.η is the location of the ice boundary; ice is formed at y when T (y , t) < −10oC
α(y , η) is an albedo function
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Smooth Albedo FunctionThe albedo function given an iceline η is
α(y , η) = 0.47 + 0.15 · [tanh(M[y − η])]
.The graph of the albedo function with η = 0.6 and M = 25, with icepoleward of η.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Static Large Ice Cap
RT (y , t + h)− T (y , t)
h= Q · s(y) · (1− α(y , η))− (A + B ·T ) + C ·
(T − T
)
The starting Temperature Profile is the same, ie the blue curveT (y)(0) = 34− 54y2 the starting ice line is the red X at 0.2, and the greenline is the ice forming critical temperature Tc = −10.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Local EquilibriaThe local equilibrium temperature with the ice line at a fixed η is
T ∗(y , η) =Q · s(y) · (1− a(y , η)) + C
∫ 10 T ∗(y , η)dy − A
B + C
=Q · s(y) · (1− a(y , η)) + C
B (g(η)− A)− A
B + C
where g(η) =∫ 1
0 Qs(y)(1− a(y , η))dy .
We wil identify the map T ∗(η) with the local equilibria set{T ∗(y , η) : y , η ∈ [0, 1]}
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Summary of the 1 timescale model
1. Too many equilibria.Given any ice line η, T ∗(y , η) is an equilibrium temperature profile.
2. Static ice line.The model lacks a mechanism for the ice line dynamics.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Equation for the Ice Line: The Debut
η(t + h)− η(t)
h= ε[T (η)− Tc ]
Basic Principles:
1. Ice line moves much slower than temperature profile.
2. When the ice line temperature is above some criticaltemperature, Tc , ice is too warm, therefore, the ice line retreats, orη → 1.
When the ice line temperature is below some critical temperature,Tc , ice is too cold, therefore, the ice line advances, or η → 0.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Simulations of the Two Time Scale Model
The starting temperature profile is T (y , 0) = 34− 54y2, the bluecurve, and the starting ice line is the X at 0.4.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Simulations of the Two Time Scale Model
The starting Temperature Profile is T (y , 0) = 34− 54y2, the bluecurve, and the starting ice line is the X at 0.1.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Simulations of the Two Time Scale Model
The starting Temperature Profile is T (y , 0) = 34− 54y2, the bluecurve, and the starting ice line is the X at 1, ie. at the pole (icefreeearth).
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
A Simulation with a Warm North Pole
The starting Temperature Profile is T (y , 0) = 34− 14y2, the bluecurve, and the starting ice line is the X at 1, the North Pole.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
The Equations of Two Time Scale Budyko’s EBM
∆hh T := F ([T , η]) = Qs(y)[1− α(y , η)]− (A + B · T ) + C
(T − T
)∆hh η := G ([T , η]) = ε (T (η)− Tc )
where ∆hh Z = Z (t + h)− Z (t). We will analyze the shift operator
map m associated with the difference operator ∆hh of this model.
m[T (y , t), η(t)] = [T (y , t + h), η(t + h)]
= [T (y , t), η(t)] + h · [F ([T (y , t), η]),G ([T (y , t), η])]
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
The Temperature Profile Function Space
Define:
B := {T : R→ R|T is bounded, Lipschitz continuous
with Lipschitz constant less than M =max slope of the albedo functionand ‖T‖∞ less than Q}
We use B with the sup norm as our temperature profile functionspace.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Inertial Manifold
TheoremFor a small ε, there exists an attracting invariant manifold forthe map m on B× R, that is,
1. There exists a Lipschitz continuous map
Φ∗ : R→ B
2. For any (T0, η0) ∈ B× R, the distancedist[mk (T0, η0), (Φ∗(η), η)] decreasesexponentially as k increases.
The proof is using the Hadamard’s graph transform method.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Instability of Ice Free Earth
Corollary
The invariant manifold Φ∗ is within a constant multiple of ε ofthe local equilibria T ∗, ||T ∗(η)− Φ∗(η)|| ≤ εM(Q+Tc )
B . Inparticular,
|T ∗(η)(η)− Φ∗(η)(η)| < εM(Q + Tc )
B.
Why do we care about this estimate?
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Instability of Ice Free Earth
Recall that the ice line dynamic is governed by the equation
η(t + h)− η(t)
h= ε[T (η, t)− Tc ]
Example: T ∗(η)(η) and Φ∗(η)(η) within 1oC
η(t)
η1 η2
0Eq.
1Pole
BIG ICE CAP UNSTABLE
SMALL ICE CAP STABLE
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Summary
1. The Budyko’s model coupled with an iceline equation has aone dimensional attracting invariant manifold
2. The 1-D invariant manifold suggests that ice free earth isunstable.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Some current to near future projects
1. Explore some physically ”reasonable” ε, with Prof. McGehee (UMN)and Jayna Resman (Smith College)
2. Add an equation for the evolution of the atmospheric CO2concentration (ie. greenhouse gas) to study the mid pleistoceneglacial cycles. Ref: Andy Hogg, 2007, with the MCRN Paleocarbonteam: Anna Barry (BU), Samantha Oestreicher (UMN), etc.
3. Study the existence of some terrestrial carbon source/ sink in themid-pleistocene glacial cycles, Samantha Oestricher.
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
References
• Budyko, MI. The effect of solar radiation on the climate of the earth, Tellus, 21,611-619, (1969).
• Tung, KK. Topics on Mathematical Modeling , Princeton University Press,(2007).
• North, GRTheory of Energy Balance Climate Models, in Journal of the AtmosphericSciences, Volume 36, Issue 11, 1975, online.
• Hogg, A. McC. Glacial cycles and carbon dioxide: A conceptual model,Geophys. Res. Lett., 35, L01701”, 2008.
• http://www.ipcc.ch/publications_and_data/ar4/wg1/en/faq-1-1.html
Budyko’s Energy Balance Model Dynamics, or, Lack Thereof Results Summary Future Directions References
Milankovitch Cycle