A Paleoclimate Model of Ice-Albedo Feedback Forced by …€¦ · A Paleoclimate Model of Ice-Albedo Feedback Forced by Variations in Earth’s Orbit∗ Richard McGehee† and Clarence

SIAM J. APPLIED DYNAMICAL SYSTEMS c© 2012 Society for Industrial and Applied MathematicsVol. 11, No. 2, pp. 684–707

A Paleoclimate Model of Ice-Albedo Feedback Forced by Variations in Earth’sOrbit∗

Richard McGehee† and Clarence Lehman‡

Abstract. Earth undergoes long-term temperature cycles alternating between glacial and interglacial episodes.It is widely accepted that changes in Earth’s orbit and rotation axis cause variations in solar inputwhich drive the glacial cycles. However, classic papers have clearly established that the response ofEarth’s climate system to orbital forcing is not a simple linear phenomenon and must include non-linear feedback mechanisms. One of these mechanisms is ice-albedo feedback, which can be modeledas a dynamical system. When combined with the cycles in the orbital elements and compared withthe climate data, the model confirms that ice-albedo feedback is an important component of Earth’sclimate.

Key words. dynamical systems, paleoclimate, Milankovitch cycles, ice-albedo feedback

AMS subject classifications. 37N99, 86A40

DOI. 10.1137/10079879X

1. Introduction. For millions of years, glaciers have been a permanent feature on Earth’ssurface. The climate has cycled through relatively warm periods, such as we are experiencingnow with glaciers confined mostly to Antarctica and Greenland, to relatively cold periods withvast ice sheets extending over the continents.

The sun provides the energy powering Earth’s weather and climate. The incoming solarradiation, or insolation, varies over several time scales. Changes in Earth’s orbit and rotationaxis, called Milankovitch cycles, cause variations in the insolation at time scales comparableto the glacial cycles. These cycles contain three components: (1) the eccentricity of Earth’selliptical orbit around the sun; (2) the tilt, or obliquity, of Earth’s axis of rotation; and(3) the precession of Earth’s rotation axis. These components will be discussed in detail below.Many studies, dating back to the mid–nineteenth century, have shown that the Milankovitchcycles affect Earth’s climate [7]. In a seminal paper, Hays, Imbrie, and Shackleton [2] firmlyestablished that the Milankovitch cycles contribute substantially to the glacial cycles.

Since the incoming radiation is a fairly complicated function of both time and space, it isoften convenient to reduce it to a single quantity. A quantity traditionally used is the dailyaverage insolation at a particular latitude on a particular day, typically at 65◦ north on thesummer solstice [7, 11], a quantity we will denote by Q65. This quantity changes slightly from

∗Received by the editors June 15, 2010; accepted for publication (in revised form) by M. Zeeman January 31,2012; published electronically May 31, 2012.

http://www.siam.org/journals/siads/11-2/79879.html†School of Mathematics, University of Minnesota, Minneapolis, MN 55455 ([email protected]). This author

was supported in part by NSF grant DMS-0940366.‡Department of Ecology, Evolution and Behavior, University of Minnesota, Minneapolis, MN 55455

([email protected]). This author was supported in part by a Resident Fellowship Grant from the University ofMinnesota’s Institute on the Environment.

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ICE-ALBEDO FEEDBACK 685

year to year, depending on the components of the Milankovitch cycles, and can be computedand analyzed as a time series.

As measured by the power spectrum of Q65, the major contribution to variations in inso-lation is caused by the precession, with changes in the obliquity contributing less, and changesin the eccentricity contributing the least [2]. Using ocean sediment data from the last 468,000years, Hays, Imbrie, and Shackleton noted that the major components of climate variationare in the frequencies associated with the eccentricity, followed by frequencies associated withobliquity, with frequencies associated with precession forming the smallest component, theexact opposite of the order found in Q65 [2]. (See Figure 1.) They ascribed this reversal ofthe contributions to possible nonlinear effects in climate dynamics [2].

precession

obliquity

eccentricity

eccentricity

obliquity

precession

Q65 Hays climate data

obliquity

eccentricity

precession

Zachos climate data

obliquity

eccentricity

precession

Model Prediction

Figure 1. Relative contributions of the Milankovitch components to insolation and climate.

In a more recent study, using ocean sediment data from the last 4,500,000 years, Zachoset al. [16] reached a somewhat different conclusion. They found the strongest contributionto climate response in frequencies associated with obliquity, followed by eccentricity, withnegligible contribution from precession [16]; see Figure 1. One should note that the discrepancybetween the Hays climate data and the Zachos climate data is not simply a question of betterdata collected between 1976 and 2001. The data show that the frequencies associated witheccentricity are more pronounced during the last million years. Indeed, an analysis of themodern data over the last million years still yields the result described by [2] and illustratedin Figure 1.

Actual climate dynamics are very complex, involving much more than insolation and cer-tainly much more than insolation distilled down to a single quantity, Q65. Recent advancesin modeling the relationship between Milankovitch cycles and glacial cycles have involvedthe introduction of triggering mechanisms, whereby changes in insolation caused by the Mi-lankovitch cycles trigger glacial retreats, which then continue due to feedback effects [3, 5, 10].Other studies have replaced Q65 with the total insolation during the summer [4, 6].

One feedback mechanism is the ice-albedo effect. The reflectivity of Earth’s surface, calledthe albedo, is different for ice than for water or land. If more of Earth is covered with ice, thenmore of the solar energy is reflected back into space, leaving less to heat Earth. On the otherhand, if less ice covers Earth, more solar energy is absorbed. This effect constitutes positivefeedback. As the ice advances, more solar energy is reflected, cooling Earth and causing the iceto advance further. Conversely, if the ice is receding, more energy is absorbed, heating Earthand melting more ice. The concept that ice-albedo feedback is a mechanism that amplifies theeffect of Milankovitch cycles on climate originated in the mid–nineteenth century as well [7].

The ice-albedo feedback can be modeled using ideas dating back to Budyko [1] and Sell-ers [13]. The model is a dynamical system with a state space consisting of the annual average

686 R. McGEHEE AND C. LEHMAN

temperature as a function of latitude. The energy input is the annual insolation as a functionof latitude. Ice is assumed to form if the annual average temperature falls below a certaincritical value and to melt if it rises above the critical value. The amount of insolation absorbedat a given latitude depends on whether there is ice at that latitude. Reradiation is a functionof Earth surface temperature and hence depends on latitude. Heat transfer between latitudesis modeled simply as a linear transfer of energy from the warmer latitudes to the cooler ones.The details of this model are described in section 4.

The annual average temperature as a function of latitude and time can be computedusing the Milankovitch cycles to determine the input to the ice-albedo feedback model. Thelocation of the ice boundary and the global mean surface temperature (GMT) as functions oftime can also be computed. The model predicts that the ice boundary and the GMT dependmostly on the obliquity, followed by eccentricity, with no dependence on precession. Thus themodel is consistent with the last 4,500,000 years of data, as analyzed by Zachos et al. (SeeFigure 1.) It is also consistent with recent studies showing that the glacial cycles are triggeredby obliquity [5] and that they are driven by the summer average insolation [6].

Although the model correctly predicts the relative contribution of the components of theMilankovitch cycles to Earth’s climate, there are large discrepancies between the predictedclimate and the data, suggesting that other mechanisms play a larger role than that of thealbedo or perhaps a synergistic role, amplifying the effect of ice-albedo feedback.

2. The climate record. On the glacial time scale, the most complete climate record isprovided by ocean sediment data. Small organisms called foraminifera form calcium carbonateshells which are fossilized in the sediments. The concentration of oxygen isotope 18O inthe shells was fixed at the time the organism was living and depends on both the isotopeconcentration and the temperature of the sea water. Since the 18O isotope is heavier thanthe more common 16O isotope, water molecules evaporating from the ocean are less likelyto contain 18O than those left behind in the ocean. Therefore, the concentration of 18O insea water is higher during glacial periods, when a larger portion of 16O is locked up in theglaciers. Also, the uptake of 18O by the foraminifera is higher when the water temperatureis colder. As a result, the 18O concentration in the fossils is a proxy for climate: higher 18Oconcentrations correspond to colder climates. Since the effect due to temperature is smallerthan the effect due to concentration during the glacial cycles [12], we assume that the 18Ofossil record primarily indicates ice volume.

The 18O record for the last 5 million years is shown in Figure 2. The vertical axis labeledδ18O indicates a change from a fixed standard. Note that the vertical scale is reversed; δ18Odecreases vertically, so that values higher on the vertical axis indicate higher temperatures.Data are taken from Lisecki and Raymo [9].

It will be convenient to use the geologic time scale for various time periods represented inFigure 2. The Pliocene Epoch runs from about 5.3 million years ago to about 2.6 million yearsago. The Pleistocene Epoch runs from the end of the Pliocene to about 12,000 years ago. Wewill use the term “early Pliocene” for the beginning of the Pliocene up until about 3.6 millionyears ago, and the term “late Pleistocene” for the last million years of the Pleistocene. Thesetime periods are indicated in Figure 2.


2.5

3.0

3.5

4.0

4.5

5.0

5.5

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

18O

Kyr

Early PlioceneLate

Pleistocene

Figure 2. Climate record for last 5.32 million years (data from [9]).

3. Milankovitch cycles. The incoming solar radiation at a point on Earth’s surface varieswith time, most notably, daily cycles of night and day and yearly cycles of changing seasons.There are also longer-term cycles caused by variations in Earth’s orbit and rotation axis.These are the so-called Milankovitch cycles, whose three components are (1) eccentricity,(2) obliquity, and (3) precession.

3.1. Eccentricity. Earth moves around the sun on an approximately elliptical orbit. Theorbit changes over geologic time due primarily to the influence of the other planets. Aswill be discussed below, the main component of Earth’s orbit affecting the global annualinsolation, averaged over the entire surface of Earth over an entire year, is the eccentricity ofthe ellipse. Over the last 5.5 million years, the eccentricity has ranged from almost zero (anearly circular orbit) to about 0.06. The current value is about 0.017. Using the equations ofcelestial mechanics, Earth’s orbit can be computed both backward and forward in time. Theeccentricity for the past 5.5 million years is shown in Figure 3. Here we are using computationsdue to Laskar et al. [8].

0

0.01

0.02

0.03

0.04

0.05

0.06

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

eccentricity

Kyr

Figure 3. Eccentricity over the last 5.5 million years.

The graph shows a period of about 100 kyr superimposed on a period of about 400 kyr.The power spectrum of this time series is shown in Figure 4, which is computed using thediscrete Fourier transform algorithm in MATLAB. Note that the apparent 100 kyr cycle breaksinto a 95 kyr cycle and a smaller 125 kyr cycle.


Figure 4. The power spectrum of the eccentricity over the last 5.5 million years.

22.0

22.5

23.0

23.5

24.0

24.5

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

obliquity(degrees)

Kyr

Figure 5. Obliquity over the last 5.5 million years.

3.2. Obliquity. The obliquity is the angle between Earth’s rotational axis and the vectorperpendicular to the plane of the ecliptic (the plane containing Earth’s orbit). The obliquityvaries over geologic time between about 22◦ and about 24.5◦. The current value is about 23.5◦.Changes in the obliquity have an important effect on Earth’s climate because the intensityof the solar input near Earth’s poles depends on the obliquity. Higher values of the obliquityproduce more solar energy hitting the North Pole during the northern summer and the SouthPole during the northern winter.

The obliquity can also be computed using equations of classical mechanics. Figure 5 showsthe obliquity of Earth’s rotation axis for the last 5.5 million years. Again we are using thecomputations of Laskar et al. [8].

The graph shows a dominant frequency with a modulating amplitude. The power spec-trum is shown in Figure 6, where we see that the dominant frequency has a period of about41 kyr. The long-period modulation seen in Figure 5 is probably a beat between the two closefrequencies seen in Figure 6.

3.3. Precession. The axis of Earth’s rotation changes due to small differences in thegravitational force on Earth’s equatorial bulge. Like a spinning top, Earth’s rotation axisprecesses about the vector perpendicular to the plane of the ecliptic.

The actual precession in a fixed reference plane is not the important variable for Earth’sweather and climate. Instead, the appropriate precession variable is the longitudinal angle be-tween the rotation vector and the major axis of Earth’s elliptical orbit. This angle determines


Figure 6. The power spectrum of the obliquity over the last 5.5 million years.

0.06

0.04

0.02

0

0.02

0.04

0.06

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

precessio

ninde

x

Kyr

Figure 7. Precession index over the last 5.5 million years.

Figure 8. The power spectrum of the precession index over the last 5.5 million years.

where the seasons occur along the ellipse and affects the relative insolation during winter andsummer months.

In the computation of the insolation quantity Q65, the precession shows up as the productof the eccentricity and the sine of the precession angle, called the precession index. This graphis shown in Figure 7.

The amplitude modulation is the effect of multiplying by the eccentricity. The underlyingprecession cycle has a period of about 23 kyr, although it is actually composed of threedominant periods, as shown in Figure 8. As is the case for Figures 5 and 6, the amplitude


420

460

500

540

580

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

W/m

2

Kyr

Figure 9. Daily average insolation at 65◦ N on summer solstice.

Figure 10. The power spectrum of the daily insolation at 65◦ N on summer solstice for the last 5.5 millionyears.

modulation shows up as a beat between the frequencies seen in Figure 8.

3.4. Daily insolation at 65◦N. All of these variables together affect Earth’s weather andclimate. It is sometimes useful to measure their effect on insolation by considering a singlevariable, the daily average insolation at 65◦north latitude on the summer solstice, denotedQ65. This quantity is a function of the three Milankovitch variables and hence varies withtime. Its values for the last 5.5 million years are shown in Figure 9.

A superficial inspection of the graph leads to the speculation that this insolation closelytracks the precession index. This observation is confirmed by the power spectrum, as canbe seen in Figure 10. Note the prevalence of the frequencies clustered around a period of23 kyr, in a pattern very similar to that of the precession index seen in Figure 8. Note alsothe frequencies around a period of 41 kyr, in the same pattern as those of the obliquity seenin Figure 6. Finally, note the lack of any discernible frequencies associated with eccentricity.

3.5. The climate record. In a highly cited work [2], Hays, Imbrie, and Shackleton relatedthe Milankovitch cycles to the climate of the last 468,000 years, using data from ocean sedimentcore samples. Analyzing the power spectrum of the data, they concluded that

“. . . climatic variance of these records is concentrated in three discrete spectral peaksat periods of 23,000, 42,000, and approximately 100,000 years. These peaks correspondto the dominant periods of the Earth’s solar orbit, and contain respectively about 10,25, and 50 percent of the climatic variance.” [2]


On the other hand, analyzing the power spectrum of the solar forcing, they found the oppositeorder for the variance of the forcing, as illustrated in Figure 1. It should be noted that theyanalyzed as solar forcing both the insolation at 60◦ N at summer solstice and the insolationat 50◦ S at winter solstice, instead of the usual 65◦ N at summer solstice. However, aswe saw in the previous section and illustrated in Figure 10, the conclusion holds as well for65◦ N: the dominant variance occurs at periods associated with precession, followed by periodsassociated with obliquity. Periods associated with eccentricity contain a negligible portion ofthe variance.

In a more recent work, Zachos et al. [16] repeated the previous approach using much moreextensive data. In the record of the last 4.5 million years, they found the dominant periodsin the power spectrum of the data to be those associated with obliquity, followed by thoseassociated with eccentricity. This conclusion is illustrated in Figure 1.

The climate data used by Zachos et al. is essentially the same as that shown above fromLisiecki and Raymo. The power spectrum is shown in Figure 11. Note the dominance of thefrequency corresponding to a period of 41 kyr, and note the resemblance to the spectrum of theobliquity shown in Figure 6. Note also the suite of frequencies associated with periods around100 kyr, corresponding to the spectrum associated with eccentricity and shown in Figure 4.We also see a small suite of frequencies around 23 kyr, presumably a small contribution fromprecession. A linear trend was subtracted from the data before the power spectrum wascomputed using MATLAB’s discrete Fourier transform function.

0.00 0.01 0.02 0.03 0.04 0.05 0.06

power

frequency (1/Kyr)

100 Kyr 41 Kyr 23 Kyr

Figure 11. The power spectrum of the Lisiecki–Raymo stack.

As mentioned in the introduction, several theories have been proposed to explain thedominance of the obliquity cycles in the climate data. The model analyzed below providesa possible explanation based on the effect of ice-albedo feedback. The parameters in themodel are all annual averages. As will be seen below, the contribution of precession cancelswhen the average annual insolation as a function of latitude is computed. A similar effectoccurs when computing the seasonal average insolation [4, 6]. Although theories based onseasonal averages probably produce more complete explanations of the process governingglacial retreats, it seems to us that a simpler model may also have some merit.

4. Ice-albedo feedback. Models exploring the effect of ice-albedo feedback on Earth’s icecover date back to Budyko [1] and Sellers [13]. Many authors have explored similar models.Here we use a version due to Tung [14].


The basic variable is the annual average surface temperature T as a function of latitude.The dynamical equation can be written

(1) R∂T

∂t= Qs (y) (1− α (y, η))− (A+BT ) + C

(T − T

),

where y is the sine of latitude and T is the annual average surface temperature as a functionof y and time t. This equation is an example of an energy balance model, since it is of theform

R∂T

∂t= heat imbalance.

The units on both sides of the equation are Watts per square meter (Wm−2). The quantityR is the specific heat of Earth’s surface, measured in units of Watts per square meter perdegree centigrade. The actual value of R is irrelevant in this paper, because we consider onlyequilibrium solutions.

The right-hand side of (1) breaks into

heat imbalance = insolation − reflection − reradiation + transport.

4.1. Insolation. The annual average incoming solar radiation is given by the term

insolation = Qs (y).

The quantity Q is the global annual average insolation, while s (y) is the relative insolationnormalized to satisfy

(2)

∫ 1

0s (y) dy = 1.

We can think of s (y) as the distribution of the incoming solar radiation as a function oflatitude, averaged over an entire year. The variable y is chosen instead of the actual latitudeso that, by Archimedes, the annual GMT can be written simply as

T (t) =

∫ 1

0T (y, t) dy.

Note that we are assuming symmetry with respect to the equator, so the variable y takesvalues between 0 and 1.

4.2. Reflection. The incoming solar energy reflected back into space is given by the term

reflection = Qs (y)α (y, η) .

It is assumed that there is a single ice line at y = η, that ice covers the surface for y > η,and that the surface is ice-free for y < η. The albedo, α (y, η), has one value, α1, where thesurface is ice-free, and another value, α2, where the surface is ice covered. Thus,

(3) α (y, η) =

{α1, y < η,α2, y > η.

Since the albedo is the proportion of energy reflected back into space, the annual rate at whichsolar energy is absorbed by Earth’s surface is Qs (y) (1− α (y, η)).


4.3. Reradiation. The energy reradiated into space at longer wavelengths is approximatedlinearly by the term

reradiation = A+BT.

This term includes many phenomena, all wrapped into a single linear term. The basic effectis the emission of heat from Earth’s surface, which depends on the surface temperature. Butbefore the heat escapes into space, some of it is absorbed by greenhouse gasses and returned tothe surface. The reradiation term A+BT is the net loss of energy from the surface to space.The parameters A and B used below were determined empirically from satellite data [14].

4.4. Transport. The energy transported from warmer latitudes to cooler latitudes is ap-proximated by the term

transport = C(T − T

).

Once again, many phenomena are included in this single linear term, mostly atmospheric andoceanic circulation. Since all these phenomena are averaged over an entire year, it is perhapsnot a terrible simplification to assume that all parts of the surface are attempting to reach theGMT and that the annual heat transfer is proportional to the difference between the annualGMT and the annual mean temperature at a particular latitude.

4.5. Equilibrium solution. Following Tung [14], we use these values for the constantsintroduced above:

Q = 343 Wm−2, A = 202 Wm−2,

B = 1.9Wm−2(◦C)−1, C = 3.04 Wm−2(◦C)−1,

α1 = 0.32, α2 = 0.62.

We now look for an equilibrium solution T ∗η (y) having a single ice line at y = η. This

equilibrium will satisfy

(4) Qs (y) (1− α (y, η))− (A+BT ∗

η (y))+ C

(T ∗η − T ∗

η (y))= 0.

Integrating (4) yields

∫ 1

0

(Qs (y) (1− α (y, η))− (

A+BT ∗η (y)

)+ C

(T ∗η − T ∗

η (y)))

dy = 0,

which simplifies to

(5) Q (1− α (η))−A−BT ∗η = 0,

where

α (η) =

∫ 1

0α (y, η) s (y) dy =

∫ η

0α1s (y) dy +

∫ 1

ηα2s (y) dy.

Using the normalization of s (y) given by (2), we can write

α (η) = α2 − (α2 − α1)S (η),


where

S(η) =

∫ η

0s(y)dy.

Thus, if we know the ice line η, we can solve (5) for the GMT,

(6) T ∗η =

1

B(Q (1− α (η))−A),

and, using this equation and (4), we can solve for the equilibrium temperature profile,

(7) T ∗η (y) =

1

B + C

(Qs (y) (1− α (y, η))−A+ CT ∗

η

),

where α (y, η) is given by (3) and T ∗η is given by (6). Note that the discontinuity of the albedo

produces an equilibrium temperature profile that is discontinuous across the ice boundary,despite the assumption of heat transport between latitudes. The heat transport is not modeledas a diffusion process, which would produce a continuous temperature profile, but is insteadmodeled as a relaxation toward the GMT.

Continuing to follow Tung [14], we assume that, at equilibrium, the average temperatureacross the ice line is

Tc = −10◦C.

Since s is continuous, we have

T ∗η (η−) =

1

B + C

(Qs (η) (1− α1)−A+ CT ∗

η

),

T ∗η (η+) =

1

B + C

(Qs (η) (1− α2)−A+ CT ∗

η

),

so the ice line condition becomes

(8) Tc =T ∗η (η−) + T ∗

η (η+)

2=

1

B +C

(Qs (η) (1− α0)−A+ CT ∗

η

),

where

α0 =α1 + α2

2.

Combining (8) and (6), we have

1

B +C

(Qs (η) (1− α0)−A+

C

B(Q (1− α (η))−A)

)= Tc,

which reduces to

(9)Q

B + C

(s (η) (1− α0) +

C

B(1− α (η))

)− A

B− Tc = 0.

This is the equation we will solve numerically for the location of the ice line η. Once we havecomputed the location of the ice line, the GMT follows from (6), and the temperature profilefollows from (7).


5. Insolation. We turn now to the insolation Qs (y) as it appears in the ice-albedo feed-back model. We will compute the global insolation Q and the insolation distribution functions (y) in terms of Earth’s eccentricity, obliquity, and precession. Let K be the solar output inWatts (Joules per second). At a distance r from the sun, the solar intensity is

K

4πr2Wm−2.

The average annual insolation is different for different points on Earth’s surface. To computethese quantities, we use spherical coordinates to describe a point on the surface of Earth:

u =

⎡⎣u1u2u3

⎤⎦ =

⎡⎣cosϕ cos γcosϕ sin γ

sinϕ

⎤⎦ ,

where γ is the longitude and ϕ is the latitude. Note that u is a unit vector perpendicular tothe surface of Earth.

We introduce coordinates (x, y, z) with the sun at the origin. The (x, y)-plane is theplane of Earth’s orbit (the ecliptic), the x-axis is the major axis of Earth’s elliptical orbit,and the positive x-axis points to the aphelion (the point on the ellipse furthest from thesun). We will refer to these coordinates as the “ecliptic coordinates.” We will also use polarcoordinates in the (x, y)-plane, assuming that the position of Earth in the ecliptic coordinatesis (r (t), θ (t), 0).

The following orthogonal matrix rotates the earth to an obliquity angle of β:

U1 (β) =

⎡⎣ cos β 0 sin β

0 1 0− sin β 0 cos β

⎤⎦ .

It will be useful below to introduce the variables u, ϕ, and γ by

u =

⎡⎣cos ϕ cos γcos ϕ sin γ

sin ϕ

⎤⎦ = U1 (β) u .

One can think of ϕ and γ as latitude and longitude on Earth’s surface measured not withrespect to the axis of rotation but with respect to a vector perpendicular to the ecliptic plane.

We also introduce the following matrix, which rotates the earth to a precession angle ofρ :

U2 (ρ) =

⎡⎣cos ρ − sin ρ 0sin ρ cos ρ 00 0 1

⎤⎦ .

In the ecliptic coordinates, the point on the surface of Earth becomes

⎡⎣r cos θr sin θ

0

⎤⎦+ U2 (ρ) u =

⎡⎣r cos θr sin θ

0

⎤⎦+ U2 (ρ)U1 (β)u .


The insolation at this point is proportional to the cosine of the angle between the vectorpointing from the sun to Earth and the vector perpendicular to Earth’s surface. Becauseof the way we have chosen the coordinates, the scalar product of these two vectors will bepositive at night, when the insolation is zero, and negative during the day, when the insolationis positive. Therefore the insolation (in Wm−2) at the point on Earth’s surface is the positivepart of

(10) − K

4πr2[cos θ sin θ 0

]U2 (ρ)U1 (β)u.

If we multiply out the matrices, this quantity becomes

− K

4πr2(cosϕ (cos β cos (θ − ρ) cos γ + sin (θ − ρ) sin γ) + sinϕ sin β cos (θ − ρ)) .

Therefore, the insolation at any given point on Earth’s surface (ϕ, γ) (latitude, longitude) atany given point (r, θ) along the orbit is

I(β, ρ, r, θ, ϕ, γ) =K

4πr2[− cosϕ

(cosβ cos(θ − ρ) cos γ + sin(θ − ρ) sin γ

)− sinϕ sin β cos(θ − ρ)

]+,

where β is the obliquity and ρ is the precession angle.We are interested in the annual average. Backing up to expression (10), we can write

I =

[− K

4πr2[cos θ sin θ 0

]U2 (ρ) u

]+.

The insolation then becomes the positive part of

− K

4πr2

⎡⎣ cos θ cos ρ+ sin θ sin ρ− cos θ sin ρ+ sin θ cos ρ

0

⎤⎦T ⎡⎣cos ϕ cos γcos ϕ sin γ

sin ϕ

⎤⎦

= − K

4πr2cos ϕ cos (θ − ρ− γ) .

Thus we can write the insolation at any given point on Earth’s surface as

I (ρ, r, θ, ϕ, γ) =K

4πr2cos ϕ[− cos (θ − ρ− γ)]+,

where ϕ is the latitude and γ is the longitude of the point on Earth’s surface with respect tothe ecliptic coordinates and where ρ, r, and θ are as before. Note that the explicit dependenceon the obliquity has disappeared in these coordinates.

We now think of eccentricity, obliquity, and precession as constant and integrate theinsolation over a year. The only variables depending on time are r and θ, which move alongan ellipse. The yearly average is

I (ρ, ϕ, γ) =1

P

∫ P

0

K

4πr(t)2cos ϕ[− cos (θ (t)− ρ− γ)]+dt

=K cos ϕ

4πP

∫ P

0

[− cos (θ (t)− ρ− γ)]+

r(t)2dt,


where P is the period of Earth’s orbit. Recalling Kepler’s second law of planetary motion, weintroduce the orbital specific angular momentum,

Ω = r(t)2dθ

dt,

and write the integral in terms of θ:

I (ρ, ϕ, γ) =K cos ϕ

4πPΩ

∫ 2π

0[− cos (θ − ρ− γ)]+dθ

=K cos ϕ

4πPΩ

∫ π/2

−π/2cos θdθ =

K cos ϕ

2πPΩ.

Note that the precession angle and the longitude both disappear completely from this equa-tion. The annual average insolation depends only on the period and the angular momentumof Earth’s orbit and the latitude of the point on Earth’s surface, measured in the eclipticcoordinates.

Returning to Earth’s coordinates, recall that

u = U1 (β) u ,

which implies that

sin ϕ = − sinβ cosϕ cos γ + cos β sinϕ

and gives us an annual average insolation of

K cos ϕ

2πPΩ=

K

√1− (− sinβ cosϕ cos γ + cos β sinϕ)2

2πPΩ.

We have so far completely ignored Earth’s rotation, focusing instead on a particular latitudeand longitude on a nonrotating Earth. If we now average over the longitude γ, we find theannual average insolation at a given latitude on a rotating Earth:

I =K

2πPΩ

1

2π

∫ 2π

0

√1− (sin β cosϕ cos γ − cosβ sinϕ)2dγ

=K

4π2PΩ

∫ 2π

0

√1− (sin β cosϕ cos γ − cosβ sinϕ)2dγ .

Note that this quantity is independent of the precession angle ρ. Note also that it is an evenfunction of ϕ.

In the ice-albedo model, this quantity I that we just computed is the same as Qs (y),where y = sinϕ and the function s is normalized so that

∫ 10 s (y) dy = 1. Therefore,

Qs (y) =K

4π2PΩ

∫ 2π

0

√1−

(√1− y2 sin β cos γ − y cos β

)2dγ.


00.20.40.60.81

1.21.4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 12. Insolation distribution as a function of y.

Integrating over y, we have

2Q = Q

∫ 1

−1s (y) dy

=K

4π2PΩ

∫ 1

−1

∫ 2π

0

√1−

(√1− y2 sinβ cos γ − y cos β

)2dγdy

=K

4π2PΩ

∫ π/2

−π/2

∫ 2π

0

√1− (cosϕ sin β cos γ − sinϕ cos β)2dγ cosϕdϕ.

Returning to the ecliptic coordinates (ϕ, γ) and noting that cosϕdϕdγ = cos ϕ dϕdγ, we have

2Q =K

4π2PΩ

∫ π/2

−π/2

∫ 2π

0cos ϕ cos ϕ dγdϕ

=K

2πPΩ

∫ π/2

−π/2cos2ϕ dϕ =

K

2πPΩ

π

2=

K

4PΩ,

and so

Q =K

8PΩ

and

s (y) =2

π2

∫ 2π

0

√1−


)2dγ.

The graph of this function is shown in Figure 12 for the current value of obliquity β = 23.5◦.Kepler’s third law tells us that

P ∼ a−3/2 ,

where a is the semimajor axis of Earth’s orbit. We also have that

Ω ∼ a−1/2√

1− e2,

where e is the eccentricity of Earth’s orbit. According to Laskar et al. [8], the semimajor axisis essentially constant. Therefore, we can write

Q = Q (e) =K

8PΩ=

Q0√1− e2

.


Using a current eccentricity of 0.0167 and the current value of Q = 343 given by Tung [14],we see that

Q0 = 343√

1− 0.01672 = 342.95,

which is the value we use in the computations below.

6. Model simulation.

6.1. Equilibrium ice line and GMT. In section 4 we derived the ice-albedo model (9)which can be solved for the ice line y = η. For convenience, we repeat it here:

(11) h (η, e, β) =Q (e)

B + C

(s (η, β) (1− α0) +

C

B(1− α (η, β))

)− A

B− Tc = 0,

where

α (η, β) = α2 − (α2 − α1)

∫ η

0s (y, β) dy.

Note that we are explicitly displaying the dependence on Earth’s eccentricity e and obliquityβ.

We also derived (6), which expresses the GMT in terms of the location of the ice line:

(12) T ∗η (e, β) =

1

B(Q (e) (1− α (η, β))−A) .

In the previous section, we computed the global annual mean insolation Q and the insolationdistribution function s (y) in terms of e and β:

Q (e) =Q0√1− e2

,

s (y, β) =2

π2

∫ 2π

0

√1−


)2dγ.

These equations are inserted into (11) to get an implicit equation for the ice line η in termsof the eccentricity and obliquity.

The current value for Earth’s eccentricity is e = 0.0167, while the current value of theobliquity is β = 23.5◦ = 0.410 radians. For these values, the graph of h (η, e, β) is shownin Figure 13 as the solid line. Note the two zeros, one at about η = 0.92 and one at aboutη = 0.26. The higher value corresponds to a stable ice line, while the lower value is unstable.For a discussion of the stability, see Tung [14] or Widiasih [15].

Figure 13 also shows the graph of h for two other values of the parameters. The dottedline corresponds to eccentricity e = 0 and obliquity β = 22◦ = 0.384 radians, while the dashedline corresponds to eccentricity e = 0.06 and obliquity β = 24.5◦ = 0.428 radians. Thesevalues were chosen to illustrate the extremes over the glacial cycles. It is perhaps surprisingthat the variation of the stable equilibrium over glacial extremes is so small, given that, at theglacial maxima, the ice covered about 30% of Earth’s surface. This small variation indicatesthe existence of feedback mechanisms other than ice-albedo and is discussed below.

Laskar et al. [8] provide tables for the eccentricity and obliquity for the past 5.5 millionyears. For each value of eccentricity and obliquity, we use Newton’s method to solve (11) for


10

5

0

5

10

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 13. The graph of h as a function of η. The solid line is for current values of eccentricity andobliquity, the dotted line is for e = 0 and β = 22◦, and the dashed line is for e = 0.06 and β = 24.5◦. The zeroscorrespond to equilibria of the ice line.

0.91

0.92

0.93

0.94

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

IceLin

e

time (Kyr)

Figure 14. Ice line location from the ice-albedo feedback model.

the ice line η, which we can plug into (12) to derive the GMT. We use a continuation methodto follow the stable value of η through changes in e and β.

We are assuming that the differential equation (1) comes to equilibrium quickly comparedwith the geologic time scale, so that it is necessary only to compute the equilibrium solutionfor each value of eccentricity and obliquity.

6.2. The model results. The results of the computation of the ice line for the past 5.32million years are shown in Figure 14.

We can compare the model output to the actual data shown in Figure 2. Under theassumption that the ice line is closely related to ice volume and therefore to the δ18O record,Figure 14 should bear some resemblance to Figure 2. At first glance, these graphs seemunrelated. The differences will be discussed below, but first we consider the similarities byinspecting the power spectrum, shown in Figure 15.

Note that the dominant period is at 41 kyr, exactly the dominant period for the obliquity.Indeed, one sees small spikes at frequencies of about 0.019 about 0.035, which also correspondto spikes seen in the obliquity spectrum (Figure 6). Although barely visible in Figure 15,there are also small spikes at the periods around 400 kyr and 100 kyr, reflecting the effect ofthe eccentricity. A comparison of this figure with Figure 8 shows an absence of frequenciesassociated with precession—not a surprise, since the effect due to precession cancels out of themodel, which considers only annual averages. Thus we see the dominance of obliquity, with avery small contribution from eccentricity, and an absence of precession. Although the actualpowers are substantially different, these relative contributions by the Milankovitch cycles are


0 0.01 0.02 0.03 0.04 0.05 0.06

power

frequency (1/Kyr)

400 Kyr 100 Kyr 41 Kyr 23 Kyr

Figure 15. Power spectrum of the ice line computation.

13.8

14

14.2

14.4

14.6

14.8

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

GMT(°C)

time (Kyr)

Figure 16. GMT as computed in the ice-albedo feedback model.

0 0.01 0.02 0.03 0.04 0.05 0.06

power

frequency (1/Kyr)

400 Kyr 100 Kyr 41 Kyr 23 Kyr

Figure 17. Power spectrum of the GMT computation.

in exactly the same order as they are in the climate data, as shown in the power spectrumillustrated in Figure 11.

Recall from our discussion in section 2 that, although the biggest contribution to the δ18Orecord comes from the ice volume, the record is also affected by the deep ocean temperature.The ice-albedo feedback model readily allows for the computation of the GMT. Although thecomputed GMT is related to surface temperature rather than to deep ocean temperature, itstill may be reflected in the δ18O record.

Figure 16 shows the GMT computed from the model, and Figure 17 shows its powerspectrum. Note that, although the GMT is still dominated by the obliquity cycles, thereis a substantial contribution from eccentricity. The model seems to indicate that, although


the eccentricity has a smaller effect on climate than the obliquity, its effect on the GMT isstronger than its effect on ice volume.

Note that both the computed ice boundary and the computed GMT show the same relativecontributions from the orbital parameters. Thus we have established the main point of thepaper: a simple model of ice-albedo feedback is sufficient to explain the relative contributionsof the Milankovitch cycles in the climate data. However, as noted above, the model is farfrom providing a complete explanation of the data.

7. Comparison of the model with the data.

7.1. Amplitude comparison. Although the ice-albedo feedback model provides an expla-nation for the frequencies found in the climate record, it does not do very well at explainingthe amplitude. To compare the amplitudes, we need to transform the model output of the iceboundary to the climate data of δ18O. We will compare each to sea level.

First recall that η corresponds to area, measured in units where 1 corresponds to the sur-face area of Earth. Assuming that the ice sheets have a constant thickness, then η correspondsto ice volume. Since δ18O also corresponds to ice volume, we assume that there is a linearrelation between η and δ18O. Also, approximating with a constant surface area for the ocean,the sea level corresponds to volume of water and hence, inversely, to the volume of ice.

We use the estimate that, if all the ice sheets melted, the sea level would rise about 65meters, corresponding to an ice boundary of η = 1. According to the model, the current valueof η is about 0.92. Therefore, the conversion factor is 65/0.08 = 810 meters per unit area(whole globe) of ice loss.

The current value of δ18O is about 3.2, while the value at the last glacial maximum wasabout 5.0 [9]. If we use the estimate that the sea level at the last glacial maximum was about125 meters below what it is today, we have a conversion factor of 125/1.8 = 70 meters of sealevel drop per unit of δ18O.

Thus the conversion factor from η to δ18O is 810/70 = 11.7 units of δ18O per unit area.Since the current value of δ18O is 3.2, while the current value of η is 0.92, we have theconversion formula

δ18O = 3.2 − 11.7(η − 0.92),

which we used to produce Figure 18.

2.5

3

3.5

4

4.5

5

5.5

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

18O

Kyr

Figure 18. The model result (red) superimposed on the climate record (blue).


2.5

3

3.5

4

4.5

5

5.5

1000 900 800 700 600 500 400 300 200 100 0

18O

Kyr

Figure 19. Detail of Figure 18 for the late Pleistocene.

2.5

3

3.5

5500 5000 4500 4000 3500

18O

Kyr

Figure 20. Detail of Figure 18 for the early Pliocence.

Figure 18 illustrates how far the model is from actually replicating the climate record,especially for the late Pleistocene (shown in detail in Figure 19), indicating that there are otherphenomena at work much more powerful than ice-albedo feedback. However, as illustrated inFigure 20, the model more closely replicates the amplitude for the early Pliocene.

7.2. Delay. If one examines Figures 19 and 20 carefully, one can see a slight delay betweenthe model output and the δ18O data. We can quantify this delay by maximizing the correlationbetween the delayed model output and the data. For the early Pliocene, this delay is about2.5 kyr, as shown in Figure 21, while for the late Pleistocene, this delay is about 7 kyr, shownin Figure 22.

Lisiecki and Raymo noted a similar delay between the obliquity cycles and the δ18O data,increasing from the early Pliocene to the late Pleistocene [9]. They attributed the increaseddelay to the increased ice volume in the late Pleistocene, noting that it would take longer forthe massive amounts of ice to respond to changes in the insolation. Indeed, they hypothesizedsuch delays as part of the “tuning” of their age model, so it is not surprising that these delaysshow up again in the cross-correlation of their data with the ice-albedo feedback model, which,as noted above, responds primarily to the obliquity cycles.

Our analysis tracks only the equilibrium solution of the Budyko model. Widiasih intro-duced a dynamic ice line to this model [15], which would cause a delay between the obliquityforcing and the model response.


0.2

0

0.2

0.4

0.6

5 0 5 10 15 20

correlation

delay (Kyr)

Figure 21. Cross-correlation between the model output and the climate data as a function of the delay forthe early Pliocene. For this and the following figure, positive delay means the data lags the model.

0.2

0

0.2

0.4

0.6

5 0 5 10 15 20

correlation

delay (Kyr)

Figure 22. Cross-correlation between the model output and the climate data as a function of the delay forthe late Pleistocene.

7.3. Skewness. Even with just a cursory look at the data during the Pleistocene, onenotices that the ice retreats faster than it advances. Huybers has quantified this observationby computing the statistical quantity of “skewness” for the derivative of the data [5]. Skewnessis a measure of the departure of the distribution from symmetry about the mean. Huybersfound that the skewness of the derivative of the δ18O data increases during the Pleistocene,indicating an increasing asymmetry between the rate of glaciation and the rate of deglaciation.

Figure 23 shows the cumulative distribution function (cdf) for the derivative of the δ18Odata for the late Pleistocene. The data are normalized so that the mean is zero and thestandard deviation is one, so the horizontal scale is standard deviation. The vertical axis isthe probability that the derivative of the δ18O data is below the corresponding value on thehorizontal scale. Using the standard definition of coefficient of skewness as the third momentscaled by the standard deviation, we find that the skewness of the derivative of the δ18O datain the late Pleistocene is −0.46, meaning that the slopes are skewed toward negative values.Recall that the δ18O is a measure of ice volume, so a negative slope corresponds to decreasingice volume, or deglaciation. Thus a negative skewness indicates that the deglaciations proceedfaster than the glaciations.

A simpler measure of the asymmetry of the data is “Pearson’s second moment of skewness,”


0

0.25

0.5

0.75

1

3 2 1 0 1 2 3

cumulativeprob

ability

slope (standarddeviations frommean)

Figure 23. The cdf for the derivative of the δ18O data during the late Pleistocene. The data are scaled sothat the mean is zero and the standard deviation is one. The vertical line indicates the median.

0

0.25

0.5

0.75

1

3 2 1 0 1 2 3

cumulativeprob

ability


Figure 24. The cdf for the derivative of the model output of the ice line during the late Pleistocene.

which is just 3 (x− x)/σ, where x is the mean, x is the median, and σ is the standard deviation.Since the data in the figure are scaled so that the mean is zero and the standard deviationis one, this skewness measure is simply −3x. The median is shown as the vertical line inFigure 23, occurring at 0.106. Thus Pearson’s second moment of skewness is −0.32, alsoindicating that the deglaciations are faster than the glaciations.

Figure 24 is the corresponding graph for the ice line model. We expect no skewness fromthe model, since the Milankovitch cycles are symmetric and since the relatively small forcingamplitude means that the system can be expected to respond linearly. Indeed, we see none:the coefficient of skewness is −0.001, while Pearson’s second moment of skewness is +0.03,both effectively zero, meaning no difference between the rates of glaciation and deglaciation.Thus the model does not reproduce the shape of the δ18O data for the late Pleistocene.

However, the early Pliocene is a different story. Figure 25 is the corresponding graph forthe δ18O data for this period. Here the coefficient of skewness is +0.06, while Pearson’s secondmoment of skewness is +0.007, both effectively zero. Figure 26 is the corresponding graph forthe model ice line. Here the coefficient of skewness is +0.001, and Pearson’s second momentof skewness is −0.03, again both effectively zero. Thus, for the early Pliocene, neither thedata nor the model indicates an asymmetry between glaciation and deglaciation.


0

0.25

0.5

0.75

1

3 2 1 0 1 2 3

cumulativeprob

ability


Figure 25. The cdf for the derivative of the δ18O data during the early Pliocene.

0

0.25

0.5

0.75

1

3 2 1 0 1 2 3

cumulativeprob

ability


Figure 26. The cdf for the derivative of the model output of the ice line during the early Pliocene.

8. Summary. The Budyko model of ice-albedo feedback accurately reproduces one ofthe main features of the climate data: the dominant frequency in the climate time series isthe frequency associated with the obliquity cycles of the Milankovitch forcing. This featureextends for at least the last five million years, from the early Pliocene (5.3 Myr to 3.6 Myrago) to the late Pleistocene (1 Myr to 12 kyr ago). This reproduction, of course, does notimply that ice-albedo feedback is the main cause of the dominance of obliquity in the climatedata; it implies only that our analysis does not rule out ice-albedo feedback as a significantfactor in Earth’s climate.

The model fails to reproduce many of the important properties of the climate data, par-ticularly for the late Pleistocene. The amplitude of the δ18O cycles during the late Pleistoceneis much larger than that produced by the model. There is a delay of 7 kyr between the cy-cles produced by the model and the corresponding cycles present in the data. The data arestrongly skewed toward rapid deglaciations and slow glaciations, while the model output isunskewed.

On the other hand, the model more accurately reproduces the data from the early Pliocene.The amplitude is approximately correct, the delay is only 2.5 kyr, and neither the data northe model output is significantly skewed.

The late Pleistocene is characterized by massive ice sheets on the continents of the north-ern hemisphere, while the ice of the early Pliocene was mostly in Antarctica [16]. Clearly


something changed between the Pliocene and the Pleistocene that had nothing to do withMilankovitch cycles. Our analysis indicates that ice-albedo feedback may be a factor for thePleistocene glacial cycles, but it is not the dominant one. However, it might have played amuch more significant role during the early Pliocene.

Acknowledgments. The authors thank Mary Lou Zeeman for reading various drafts ofthe manuscript and suggesting many important changes. Many other improvements weresuggested by the referees. McGehee also thanks Jacques Laskar for enlightening conversa-tions and for his hospitality during a visit to the Observatoire de Paris, and he thanks RayPierrehumbert for help with the geological interpretations.

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[3] A. M. Hogg, Glacial cycles and carbon dioxide: A conceptual model, Geophys. Res. Lett., 35 (2008),L01701.

[4] P. Huybers, Early pleistocene glacial cycles and the integrated summer insolation forcing, Science, 313(2006), pp. 508–511.

[5] P. Huybers, Glacial variability over the last two million years: An extended depth-derived agemodel,continuous obliquity pacing, and the pleistocene progression, Quaternary Sci. Rev., 26 (2007), pp. 37–55.

[6] P. Huybers and E. Tziperman, Integrated summer insolation forcing and 40,000-year glacial cycles:The perspective from an ice-sheet/energy-balance model, Paleoceanography, 23 (2008), PA1208.

[7] J. Imbrie and K. P. Imbrie, Ice Ages: Solving the Mystery, Harvard University Press, Cambridge, MA,1979.

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[10] D. Paillard, Glacial cycles: Toward a new paradigm, Rev. Geophys., 39 (2001), pp. 325–346.[11] J. R. Petit, J. Jouzel, D. Raynaud, N. I. Barkov, J.-M. Barnola, I. Basile, J. C. M. Bender,

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[13] W. D. Sellers, A global climatic model based on the energy balance of the earth-atmosphere system, J.Appl. Meteorol., 8 (1969), pp. 392–400.

[14] K. K. Tung, Topics in Mathematical Modeling, Princeton University Press, Princeton, NJ, 2007.[15] E. Widiasih, Dynamics of Budyko’s energy balance model, SIAM Appl. Dyn. Syst., submitted.[16] J. Zachos, M. Pagani, L. Sloan, E. Thomas, and K. Billups, Trends, rhythms, and aberrations in

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A Paleoclimate Model of Ice-Albedo Feedback Forced by …€¦ · A Paleoclimate Model of Ice-Albedo Feedback Forced by Variations in Earth’s Orbit∗ Richard McGehee† and Clarence