An Approximation of the Budyko-Widiasih Model with the Jormungand Albedo Function
Chris Rackauckas, Oberlin College. Advisor: James Walsh

The Budyko-Widiasih Model

Simulation

Geometric Singular Perturbation Theory Approximation

Energy Balance Models (EBM) are conceptual models that are developed to analyze thelong-term behavior of specific system variables. The Budyko-Widiasih model describes
the Earth’s ice-albedo feedback system and incorporates the dynamics of the ice line, theoutermost edge of the polar glaciers. It is defined by the Budyko Model and Widiasih’s iceline condition.
The Budyko Model is defined as follows:

We wish to understand the equilibrium solution of the ice line. We can solve an attracting
one-dimensional curve

Since 0 <   1, the functions can be approximated away from the equilibrium solutions
by the fast subsystem:R∂T∗h(η) = T (η, η) − Tc =Q
B+C(s(η)(1 − α(η, η)) +CB(1 − α(η))) −ABdη− Tc= Qs(y)(1 − α(y, η)) − (A + BT ) + C(T − T )).

Z 1α(y, η)s(y)dy.

α(η) =dηdtT (η, η) =1∗2 y→η−y→η +=Rz(η)

2(αs +2(αw + αi (η)))) − A − (B + C)a + C T̄ (η)),(Q(αi (η) − αw ) − (B + C)z).

=Q(αi (η) − αw )

,

B+C
1a(η)

=

(Q(1 −B

121(αs +2(αw + αi (η))) − A + Cµ(η)).

We can thus solve for the attracting curve
01h (η) = a +4(e − z) +12(u2 + d − s2 z)p2 (η) − Tc ,Results from geometric singular perturbation theory show that h0 is an O() approximation
to h. Equilibrium ice lines are found when h(η) = 0. Below is a graph of the h function
we receive from this approximation. The same analysis on this graph shows the lowest
equilibrium, the Jormungand state, is a stable equilibrium.

This curve can be shown to satisfy the equation

dt

∗

R
1

1Thus, since η is a constant, we can solve for the equilibrium values of z and a:= (T (η, η) − Tc ).

( lim T (y, η) + lim T (y, η))dtdz(Q(1 −1Below we show the resulting h(η) found computationally using MATLAB.

dηT (η, η) is defined as the average of the temperatures just above and just below the iceline, or more rigorously

1

dt

00

Therefore the equation C(T − T ) describes the meridional heat transport. The constant
C is determined from data.
We will denote the ice line as η ∈ [0, 1]. Widiasih’s ice line condition models the change
in the ice line as a function of the temperature:dtda0,=where∂tT is the annual average surface temperature as a function latitude. The latitude is describedby the variable y which is defined as the sine of the compliment of the polar angle. TheBudyko model assumes that the temperature is distributed symmetrically across the equatorand thus it suffices to consider y ∈ [0, 1] where y = 0 is the equator and y = 1 is theNorth Pole. Q is the global average incoming solar radiation (insolation) while s describes
the distribution of the insolation as a function of the latitude. The albedo α describes thepercent of the Sun’s energy reflected by the Earth. Thus the energy that enters the Earth’ssystem at a given latitude is Qs(y)(1 − α(y, η)).
The process through which the Earth radiates energy is complicated by the greenhouseeffect. The Budyko model approximates the outgoing radiation using the equation A + BT
where A and B are determined by satellite data. The last part of the Budyko model
describes the transport of energy between latitudes, also known as the meridonal heattransport. This process is complex and involves winds and ocean currents but its effects
can be approximated by a simple linear relaxation to the mean. Thus we define T as themean Earth temperature:
Z 1
T =T (y)dy.

== h(η).

Thus the equilibrium ice line latitudes are found by setting h(η) = 0. These results show
the existence of three equilibrium solutions. h(η) > 0 leads to the melting of ice which
decreases the ice line. h(η) < 0 leads to the freezing of water which increases the ice line.
From these facts we see that the solution η ≈ .06 is a stable equilibrium, η ≈ .65 is an
unstable equilibrium, and η ≈ .95 is a stable equilibrium.

The constant Tc denotes the critical temperature for melting the glaciers.

Our goal is to understand the equilibrium behavior of the ice line.

Solving the Model
Previous research has solved for the case where η > ρ. Thus we solve for the equilibrium
where η < ρ. Define the temperature piecewise asThe Jormungand Albedo Function
The albedo α is the percent of solar radiation reflected off the Earth’s surface. In ourmodel, α is described by the Jormungand Albedo Function. The Jormungand Albedo
Function assumes the Earth’s surface is covered with water. Sea ice is only covered withsnow if there is sufficient precipitation. An argument can be made to show that the parts ofthe glaciers that extend below the latitude ρ = .35 would not be covered in snow. Denotethe albedo of snow covered ice as αs , non-snow covered ice as αi , and the albedo of water
as αw . We define the albedo function as:

U (y)V (y)
T (y) =

W (y)

1
(U (η) + V (η))
2

Approximate the functions U , V , and W as quadratic functions andsubstitute to write the model asdη=η<ρ αsα(y, η) = αi (η)
αwy>η
ρ<y<ηy<ρη>ρ
(αsα(y, η) =αw

dtday>η
y<η

(Tb − Tc ),1=dtdzR
1=dti
where αi (η) = αs −α
η + αi and αw < αi < αs .
ρy<η
η<y<ρy≥ρ
y=ηR(Q(1 −121(αs +2(αw + αi (η)))) − A − (B + C)a + C T̄ (η)),(Q(αi (η) − αw ) − (B + C)z),

where a and z are functions of η, U , V , and W .

References•James Walsh, Richard McGehee. Climate Modeling, Dynamically Speaking.
http://www.oberlin.edu/math/faculty/walsh/walshpubl.html•Richard McGehee, Esther R. Widiasih A simplification of Budyko’s ice-albedo feedback model
http://www.math.umn.edu/˜mcgehee/publications/McGeheeWidiasih2012/index.html•Christopher K.R.T. Jones. Geometric Singular Perturbation Theory
Dynamical Systems, Lecture Notes in Mathematics Volume 1609, 1995, pp 44-118.