The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Did A Jormungand State Exist?
An investigation using the Budyko-Widiasih model

By Christopher Rackauckas
Oberlin College

March 13, 2013

By Christopher Rackauckas

Oberlin College

1/44The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Outline

Our investigation will be conducted as follows:

1

Begin by looking at the scientic data.

2

Introduce the Budyko-Widiasih model and its conclusions.

3

Examine the results of numerical simulations to the
Budyko-Widiasih model.

4

Find an approximation to solutions to the Budyko-Widiasih
model.

By Christopher Rackauckas

Oberlin College

2/44The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

The Possibilities
Geological and paleomagnetic evidence indicates that glaciers
grew near the equator during the last two Neoproterozoic
glacial periods.
There are dierent hypotheses as to the exact nature of these
glaciations:

1

The Snowball Earth Hypothesis: Glaciers covered the entiretyof the Earth's surface.

2

The Slushball Earth Hypothesis: Continents completely
covered in ice, belt of free ocean.

3

The Thin-Ice Hypothesis: Glaciers completely covered theEarth, but the ice is thin at the tropics.

4

The Jormungand Hypothesis: Glaciers mostly covered theEarth, ice is not snow covered near the tropics, and a belt of
free ocean existed.

By Christopher Rackauckas

Oberlin College

3/44The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Evidence for Extreme Glaciations
The magnetic orientations of rocks tell us the continents werenear the equator 750 million and 580 million years ago.
But there was glacial debris on these continents during thisperiod:Glaciers today only survive 5,000 meters above sea level (4,000
in the last ice age).
These contained iron-rich rocks which implies little to nooxygen in the oceans and atmosphere.
Rocks known to form in warm water accumulated just after
the glaciers receded (evidence for strong hysteresis).
Just after the proposed glaciation is the Cambrian Explosion.
Reference:

Snowball Earth, Scientic American, Homan andSchrag.
By Christopher Rackauckas

Oberlin College

4/44The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

The Snowball Earth Hypothesis

Glaciers covered the entirety of the Earth's surface.
Life survived in small communities near hot springs.
The isolation explains the Cambrian Explosion.

CO built up because of the lack of silicate weathering caused2

the abrupt change.

By Christopher Rackauckas

Oberlin College

5/44The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Evidence Against the Snowball Earth Hypothesis

There is evidence that many sponges survived the
Neoproterozoic glaciations.
There is evidence that photosynthetic eukaryotes thrived both
before and immediate after the Snowball episodes.
New evidence that life can survive under miles of glaciers does
not apply to complex life.
Reference: The Jormungand Glocal Climate State and Implications
for Neoproterozoic Glaciations, Abbot et al.

By Christopher Rackauckas

Oberlin College

6/44The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

The Slushball Earth Hypothesis

Continents completely covered in ice, belt of free ocean.
This would allow complex life and photosynthetic eukaryotes
survive.
The Slushball models do not seem to have a strong enoughhysteresis to account for the

CO measurements.
2

The Slushball Model of Liu and Peltier (2010) occur with CO2
O(100 − 1000) ppmv
Measurements by Bao et al. indicate values one or two ordersof magnitude more!

By Christopher Rackauckas

Oberlin College

7/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

The Thin-Ice Hypothesis

Glaciers completely covered the Earth, but the ice is thin at
the tropics.
The thin ice would allow photosynthetically active radiation topenetrate to the ocean below.
Such a solution is possible if bare sea ice has a hightransmissivity and an albedo lower than that of snow coveredice.
Has not been found in a global climate model so far, and there
is debate as to whether the parameter regime is realistic

By Christopher Rackauckas

Oberlin College

8/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

The Jormungand Hypothesis

Mixture of Slushball Earth and Thin-Ice.
Ice sheets almost cover the entire Earth, though those near thetropics are not covered in snow.
This is a solution that global climate models have found.
There is strong hysteresis associated with the Jormungand
state, which is to say that the Jormungand state and one orboth of the other states are stable for a wide range of pCO2.
(Abbot 2011)

By Christopher Rackauckas

Oberlin College

9/44The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Introduction to the Budyko-Widiasih Model

The Budyko-Widiasih Model is an Energy Balance Model(EBM).
It is designed to examine the movement of the ice-line.
As a lower order model, dynamical systems theory can be used
to verify the existence of a Jormungand state.

By Christopher Rackauckas

Oberlin College

10/44The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

The Budyko-Widiasih Modely is the sine of the compliment of the polar angle (the latitude
as written from 0 to 1).

η is the ice-line, the latitude of furthest extent of the Earth's
polar glaciers.

T (y , η) is the annual average surface temperature as afunction of latitude and the ice-line.

M is the meridional heat transport, the transport of heat fromone latitude to another.
These quantities are related in the following manner:

R

∂T
∂t
∂η
∂t

=

Ein − Eout − M ,= (T (η, η) − Tc ).

T (η, η) = (limy →η− T (η, η) + limy →η+ T (η, η)).
Tc is the critical temperature to melt the glaciers, −10◦ C .
1
2

By Christopher Rackauckas

Oberlin College

11/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

The Energy Terms

Ein = Qs (y )(1 − α(y , η)), Eout = A + BT , M = C (T − T ).
s describes the distribution of the insolation as a function oflatitude. It can be well-approximated by a quadratic.

α is the albedo of a latitude as a function of the ice-line.
´
T = 01 T (y )dy , the average temperature of the Earth's
surface.

Eout is the Budyko-Sellers model veried by satellite data.
M is a relaxation to the mean.
By Christopher Rackauckas

Oberlin College

12/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

The Albedo Function

The Budyko model used the albedo function as follows:



αs ,
α(y , η) = αw ,1
(αs + αw ),
2

y >η
y <η
y = η,

where αs > αw .

By Christopher Rackauckas

Oberlin College

13/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Solution to the Budyko-Widiasih Model
We are looking for a solution to understand the dynamics of the
ice-line, that is a function

h that satises∂η= h(η).
∂t

Solutions to the Budyko-Widiasih Model satisfyh(η) =

QB +C



s (η)(1 − α(η, η)) +
ˆ 1

where

α(η) =

C(1 − α(η))
B


−

A
− Tc ,Bα(y , η)s (y )dy .

0

By Christopher Rackauckas

Oberlin College

14/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

McGehee-Widiasih
McGehee and Widiasih utilized the Budyko albedo function andsolved for an approximation to∂η= h(η).
∂t

By Christopher Rackauckas

Oberlin College

15/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

An Extension to the Albedo Function

We wish to introduce the idea of bare sea ice into the albedo
function.
As noted before, the albedo of bare sea ice is less than that ofsnow covered ice.
When the ice-line grows past a certain latitude ρ, it enters the
Hadley cell circulation zone.
This would lead to more evaporation than precipitation leading
to bare sea ice.

By Christopher Rackauckas

Oberlin College

16/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Abbot's Albedo Function
Abbot et al. introduced the idea using the following albedo
function:α2 (y ),α(y , η) = 12 (αw + α2 (η)),


αw ,

y >η
y =η
y < η,

where
1

1

2

2

α2 (η) = (αs + αi ) + (αs − αi ) tanh(y − ρ).

By Christopher Rackauckas

Oberlin College

17/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Numerical Solution Using Abbot's Albedo Function

By Christopher Rackauckas

Oberlin College

18/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Hadley Cells

By Christopher Rackauckas

Oberlin College

19/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Hadley Cell Eect

By Christopher Rackauckas

Oberlin College

20/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Hadley Cell Eect (Continued)

By Christopher Rackauckas

Oberlin College

21/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Hadley Cell Intensication
Poulsen and Jacob examined the Hadley cells' circulation at
the onset of Snowball Earth

They concluded that the Hadley cell circulation abruptly
intensies and then abruptly weakens.
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Oberlin College

22/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Resulting Albedo Function
The resulting albedo function would then be complex:

As the ice-line heads towards the equator, the ice in the
Hadley cell area would be mostly bare due to the evaporation.
How much the Hadley cell eect increases eects how close to
α the albedo becomes.
i

We can then understand the system by bounding its
possibilities between two models:An albedo function which becomes instantly α in the Hadleycell zone due to increased Hadley cell eect.
The albedo in the Hadley cell zone changes linearly from α to
α.
i

s

i

The albedo of the Earth's system will be underestimated in therst function, resulting in a maximum equilibrium ice-line.

By Christopher Rackauckas

Oberlin College

23/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

The Instant Jormungand Albedo Function

The Instant Jormungand Albedo Function is dened as:

η<ρ


αs ,





αi ,
α(y , η) = αw ,


1


2 (αs + αb (η)),
12 (αi + αw ),

η>ρ
y >ρ

η<y <ρ
y <η
y =ρ
y =η



αs ,
α(y , η) = αw ,12 (αs + αw ),

y >η
y <η
y =η

where αw < αi < αs .

By Christopher Rackauckas

Oberlin College

24/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Instant Jormungand Albedo Function Solution

By Christopher Rackauckas

Oberlin College

25/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

The Linear Jormungand Albedo Function
The Linear Jormungand albedo function is dened as:

η<ρ


αs ,



αb (η),α(y , η) = αw ,
 1 (αs + αb (η)),

2


1
2 (αb (η) + αw ),

η>ρ
y >ρ

η<y <ρ
y <η
y =ρ
y =η



αs ,
α(y , η) = αw ,12 (αs + αw ),

y >η
y <η
y =η

where

αb (η) =αs − αiη + αi ,ρand αw < αi < αs .
By Christopher Rackauckas

Oberlin College

26/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Linear Jormungand Albedo Function Graphs

By Christopher Rackauckas

Oberlin College

27/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Linear Jormungand Albedo Function Solution

By Christopher Rackauckas

Oberlin College

28/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Stable Equilibrium Result

The Budyko-Widiasih suggests that when taking into accountthe change due to the weakening (and eventual halt) of the
Hadley cells, the stable large ice-line solution is a Jormungand
state.
This runs counter to the thin-ice and Snowball Earthhypotheses.
The bifurcation diagram shows that the Jormungand model
can produce the necessary hysteresis.

By Christopher Rackauckas

Oberlin College

29/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Hysteresis of the Jormungand Model

We model the eect of increasing
term

A.

CO levels as decreasing the2

A required for an equilibrium η:BCA(η) =Qs (η)1 − α(η, η)) + Q (1 − α(η)) − BTc .
B +C
B
We can solve for the value ofWe can then use a bifurcation diagram to analyze the eect on
the stable ice-line solutions.

By Christopher Rackauckas

Oberlin College

30/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Bifurcation Diagrams

By Christopher Rackauckas

Oberlin College

31/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Problem Statement

We wish to nd an analytical approximation toh(η) for the

Jormungand Linear model which does not require a numericalintegration.
We will begin following McGehee-Widiasih and end using a
fast-slow approximation.
Note: McGehee-Widiasih have already solve
our goal is to solve

By Christopher Rackauckash(η) piecewise.

Oberlin Collegeh(η) for η > ρ, so

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Conclusion

Step 1: Split the Temperature Function

Let



U (y , t ),
y <η



V (y , t ),
η<y <ρ
T (y , t ) = W (y , t ),
y ≥ρ



 1 (U (η, t ) + V (η, t ), y = η.
2

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Oberlin College

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Conclusion

Step 2: Second-Order Legendre Approximation

AssumeU (y , t ) = u (t )p (y ) + u (t )p (y )V (y , t ) = v (t )p (y ) + v (t )p (y )W (y , t ) = w (t )p (y ) + w (t )p (y ),
where

0

0

2

2

0

0

2

2

0

0

2

2

p (y ) = 1 and p (y ) = (3y − 1).
0

By Christopher Rackauckas

2

1

2

2

Oberlin College

34/44

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Conclusion

Step 3: Write the Model in Terms of the u's, v's, and w's(T (η, η) − Tc )

η̇

=

u˙0

=

R

.
.
.

.
.
.

.
.
.

w˙2

=

R

1

1

(Q (1 − αw ) − A − (B + C )u0 + CT (η))(Qs2 (1 − αs ) − (B + C )w2 ,where

T (η, η)

=

T (η)

=

k

=

1
2

(u0 + v0 ) +1
2(u2 + v2 )p2 (η),

η u0 − (η − ρ)v0 +1
2

By Christopher Rackauckas

12(η 3 − η)u2 − ( (η 3 − η) − k )v2 + (1 − ρ)w0 − kw2 ,12(ρ3 − ρ).

Oberlin College

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Conclusion

Substitutions
Repeatedly substitute in functions of the u's, v's, and w's toeliminate variables and receive variables with solutions.
Many variables could then be written like:

ė =

1

R

((2Q (αs − αw ) − (B + C )e )which collapse over time to a single value. Thus our system
becomesη̇ = (T (η, η) − Tc ),

ȧ
ż

=
=

1

R
1

R

Q (1 −12

1(αs + (αw + αi (η)))) − A − (B + C )a + CT ,2

(Q (αi (η) − αw ) − (B + C )z ),By Christopher Rackauckas

Oberlin College

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The Scientic Debate

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Conclusion

Fast-Slow Approximation

McGehee-Widiasih estimates  ≈ 3.9 × 10

−13 .

Thus η is a slow variable while the others are fast variables.
We will use the fast and slow subsystems to understand thesolution.
Reference: Christopher Jones, Geometric Singular Perturbation
Theory.

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Oberlin College

37/44

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Conclusion

Fast-Slow Hypotheses
We wish to use the theory of Fast-Slow Systems to solve for theinvariant manifold.
Notice we can write our system asẋẏf (x , y , )
= g (x , y , )=x are the fast variables and y are the slow variables.
∞ since they are polynomials of the
Notice that f , g ∈ C
wherevariables.
Let

M be any compact subset of {(x , y ) ; f (x , y , ) = 0}.
0

Thus M0 is a subset of {(x , y ) : x = h0 (y )} where h0 (y ) isdened for y ∈ K , a compact domain of R.

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Oberlin College

38/44

The Scientic Debate

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Analytical Approximation

Conclusion

Fast-Slow Overview

Given that the previous hypotheses are satised, Fenichel's
Theorems assert that:M that lies within O() from M and
M . Moreover it is locally invariant under

There exists a manifold
is dieomorphic to00the ow dened by our system.
We can writeM = {(x , y ) : x = h (y )} and thus we canwriteẏ = g (h (y ), y , ) = g (h (y ), y , 0) + O()0

By Christopher Rackauckas

Oberlin College

39/44

The Scientic Debate

The Budyko-Widiasih Model

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Analytical Approximation

Conclusion

Fast Subsystem
The Fast Subsystem is described by the system:

η̇ =

0

ȧ

1

ż

=
=

R
1

R

Q (1 −12

1(αs + (αw + αi (η)))) − A − (B + C )a + CT ,2

(Q (αi (η) − αw ) − (B + C )z ).

From this we can solve for the manifold
Since this implies η =M.
0constant , we can easily solve for a and z

on the manifold.

By Christopher Rackauckas

Oberlin College

40/44The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Slow Subsystem

Using the values from the manifold forg (h (η), η, 0) = a +0

1
4

a and z , we can solve

1(e − z ) + (u2 + d − s2 z )p2 (η) − Tc .
2

We can approximate the ow on the manifold

M by notingη̇ = g (h (η), η, ),g (h (η), η, 0) + O(),11= a + (e − z ) + (u + d − s z )p (η) − Tc + O().
=04

By Christopher Rackauckas

2

2

2

Oberlin College

2

41/44

The Scientic Debate

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Numerical Solutions

Analytical Approximation

Conclusionh(η)

By Christopher Rackauckas

Oberlin College

42/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Conclusion

The scientic record is ambiguous between a Snowball Earthstate and a Jormungand state.
Taking into account the eect of the Hadley cells on equatorial
ice-sheets we see the Budyko-Widiasih model gives a
Jormungand state solution.
We can receive an approximation to the solution using
fast-slow theorems.

By Christopher Rackauckas

Oberlin College

43/44

The Scientic Debate

The Budyko-Widiasih Model

Numerical Solutions

Analytical Approximation

Conclusion

Further Analysis

Avenues of further analysis include:
An investigation of the models against the climate record.

McGehee and Lehman analyzed the Budyko-Widiasih model
against the climate record
Can do a comparative time series analysis
Specically look at the regime switching or bifurcation.
More investigations with GCMs.

By Christopher Rackauckas

Oberlin College

44/44