dynamics of the budyko's energy balance model: instability ... … · instability of ice free...

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


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


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!)


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


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


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


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.


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.


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]}


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.


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.


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.



The starting Temperature Profile is T (y , 0) = 34− 54y2, the bluecurve, and the starting ice line is the X at 0.1.



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).


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.


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])]


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.


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.


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?


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


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.


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.


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

http://www.ipcc.ch/publications_and_data/ar4/wg1/en/faq-1-1.html


Milankovitch Cycle

dynamics of the budyko's energy balance model: instability ... … · instability of ice free...

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