S( f ) (8.37). The Earth maintains a mean global tem-

S( f ) = C

perature of approximately 15 —¦ C through a bal-

10

ance between incoming solar radiation and long-

S( f ) ∝ f ’2 wavelength outgoing radiation. Long-wavelength

10

outgoing radiation is characterized by the Stefan--

10 Boltzmann Law, in which outgoing radiated en-

ergy depends on the fourth power of absolute

10

temperature. For small temperature ¬‚uctuations

10 10

f

10

around an equilibrium temperature, the T 4 Boltz-

mann dependence may be linearized to yield the

Fig 8.23 Example of coherence resonance in Eq. (8.37). (a)

¬nite-difference equation (North et al., 1981)

Time series of Eq. (8.37) with „ = 600, = ’0.1, and

D = 0.1. The time series of model output exhibits a cycle

Tn

= ’B T n

with an average period of 300 time steps. (b) Average power C (8.38)

t

spectrum of 1000 samples of the model illustrated in (a). The

power spectrum has a dominant peak with a period of „/2

where n is an index of the time step, C is

and smaller-amplitude periodicities at odd harmonics of the

the heat capacity per unit surface area of the

dominant periodicity, superposed on a “background”

atmosphere-ocean-cryosphere system, B is the co-

spectrum which is constant at small frequencies and

ef¬cient of temperature dependence of outgo-

proportional to f ’2 at larger frequencies. From Pelletier

ing radiation, and T n is the temperature differ-

(2003).

ence from equilibrium. This equation character-

izes a negative feedback: global warming (cool-

time series in Figure 8.23a has two stable states ing) resulting from short-term climatic ¬‚uctua-

at T = +1 and T = ’1. The system switches be- tions results in more (less) outgoing radiation. C

tween these states with a period equal to „/2, may be estimated from the speci¬c heat of wa-

ter, 4.2 — 103 J/kg/—¦ C, and the mass of the oceans,

despite the absence of any periodic terms in the model

1.4 — 1021 kg, and the surface area of the Earth,

equation. In Figure 8.23 we have illustrated the

5.1 — 1014 m2 , to obtain C = 1.2 — 1010 J/—¦ C, as-

negative-feedback case by choosing < 0. Posi-

tive feedback produces a similar output, but with suming that the heat capacity of the climate sys-

a dominant period equal to „ rather than „/2 tem is dominated by the oceans. The value of B

(Tsimring and Pikovsky, 2001). Less-dominant pe- is constrained from satellite measurements to be

2.1 J/s/m2 /—¦ C (North, 1991). The ratio of B to C

riodicities also occur at odd harmonics of the

dominant periodicity (i.e. periods of „ /6, „ /10, is de¬ned to be c 1 in this model and de¬nes

„ /14, etc.) in Figure 8.23b. The ˜˜background™™ a radiative-damping time scale for the climate

8.9 COHERENCE RESONANCE AND THE TIMING OF ICE AGES 213

outgoing

(a) (b) incoming

”T ”T

”t ”T

”T ”t

”t

”t

Th

interglacial

glacial glacial interglacial

=

+

T T

broad expansion

in northern Canada

narrow expansion

in southern Canada

(c) net change (d) W/bedrock deflection

”T ”T

”T = c |v|

”t ”t

”t 3

interglacial

glacial

T |v| T

wavelength outgoing radiation. This variability

Fig 8.24 Schematic diagram of processes included in our

may be modeled by introducing a random noise

radiation-balance model. Each graph is a sketch of the

in Eq. (8.38) to obtain (Hasselman, 1971; North

functional relationship between temperature change, T / t,

et al., 1981)

and global temperature, T , for each process. (a) A high global

temperature results in more outgoing long-wavelength

Tn

= ’c 1 T n + D ·n (8.39)

radiation (and relative cooling). This effect is proportional to

t

the temperature difference from equilibrium where the

where · is a white noise with standard deviation

constant of proportionality, c 1 , is equal to the radiation

damping time, constrained to be c 1 = 180 yr as described in equal to D . Short-term internal temperature vari-

the text. (b) Lower global temperatures result in increased ations generate signi¬cant low-frequency temper-

albedo (and relative cooling) with a function that depends on ature variations because the climate system inte-

the ice sheet distribution in Figure 8.25. (c) Long-wavelength

grates the short-term random variations through

outgoing radiation and the ice-albedo feedback taken

their effect on the outgoing radiation ¬‚ux. The

together yield a system with two ¬xed points, corresponding

result of this process is Brownian-walk behavior

to glacial and interglacial states. (d) Bedrock subsidence and

for global temperature. Equation (8.39) has the

rebound modify the effects of the ice-albedo feedback by

behavior of a damped Brownian walk, however, be-

shifting the curve towards greater relative warming by an

amount proportional to |v |. From Pelletier (2003). cause the equation also includes a negative feed-

back term.

The growth of ice sheets provides a positive

system equal to 180 yr (the inverse of c 1 ). The feedback that dominates the negative feedback

simple linear relationship between temperature of outgoing radiation if ice sheets grow nonlin-

change and mean global temperature in Eq. (8.38) early with global temperature. The effects of the

is plotted in Figure 8.24a. ice--albedo feedback on global climate may be

The negative feedback of Eq. (8.38) is balanced modeled by including a term in Eq. (8.38), a(T ),

by natural variability in Earth™s radiation balance which re¬‚ects the dependence of Earth™s albedo

driven by short-term (decadal and centennial) on global temperature, to obtain

¬‚uctuations in atmospheric temperature. These Tn

= ’c 1 T n ’ c 2 a(T n ) (8.40)

¬‚uctuations create random variability in long- t

214 STOCHASTIC PROCESSES

We assume that the modern interglacial state is Vostok data provide the independent data to de-

a steady state of the climate system. This provides termine a(T ). The albedo data were obtained by

multiplying each 1—¦ — 1—¦ grid square in Peltier™s

the constraint a(0) = 0 and ¬xes the temperature

of any other stable states of the system to be with data by its area and the amount of incident solar

respect to an origin at the modern interglacial radiation it receives to obtain the relative change

temperature. in albedo as a function of global temperature:

Stable states of Eq. (8.40) are determined by 180 90

setting the right hand side equal to zero. T = a(T ) = ai, j (T ) cos2 ( j) (8.41)

i=’180 j=’90

0, the modern interglacial, is a ¬xed point be-

cause we de¬ned a(0) = 0. Whether or not an- where ai, j = 0.25 if the grid square is ice-free and

ai, j = 0.85 if the grid square is ice-covered.

other ¬xed point exists depends on how a(T ) in-

creases with decreasing T . If the Earth™s albedo Equation (8.41) provides an estimate of the rel-

increases greater than linearly with T , the pos- ative change in Earth™s albedo due to the pres-

itive ice--albedo feedback dominates the system. ence of Late-Pleistocene continental ice sheets.

A dominant ice--albedo feedback leads to an ice- The albedo difference between modern and LGM

covered Earth if a(T ) increases greater than lin- conditions calculated in Eq. (8.41) equals 10% of

early for all T . However, if the ice--albedo feed- the modern albedo assuming that ice-free areas

back increases less than linearly for any T , the are also continuously snow free as well. 10% is

combination of outgoing radiation and the ice-- an overestimate, however, because many areas as-

albedo feedback will be negative or self-limiting sumed to be ice-free in Eq. (8.41) have seasonal

for those values of T . snow and ice cover. Accounting for modern sea-

Determining the form of a(T ) is complicated sonal snow and sea-ice cover, Oerlemans (1980a)

by uncertainties in the geographic distribution of estimated that the albedo difference between the

ice as a function of global temperature. Most nu- LGM and the present was approximately 5%. We

merical ice-sheet-climate models determine a(T ) have used this value to scale the albedo curve to

by ice-sheet dynamics in their full complexity, a maximum change of 5% between the present

including internal ¬‚ow, basal sliding, and mass and the LGM. In other words, we have used Oer-

wasting. The complexity of these models results lemans™ albedo estimate for the total change be-

in dozens of free parameters for the motion of tween the LGM and the present, and Peltier™s

continental ice sheets. These processes, especially reconstructions to estimate the relative change

basal sliding, are poorly constrained. The distri- through time. The result is plotted in Figure 8.25.

bution of sea ice as a function of global tempera- The data of Figure 8.25 are well approximated

ture is also highly uncertain. As an alternative to by a square-root function:

modeling ice-sheet dynamics, the extent of land 1

(T h ’ T ) 2 T < Th

if

ice through time during the last deglaciation a(T ) = (8.42)

T ≥ Th

0 if

may be inferred from the pattern of postglacial

where T h = ’1—¦ C (relative to the modern tem-

rebound (Peltier, 1994). Peltier™s reconstruction

provides ice extents at 1-kyr intervals from the perature) is the temperature at which subpolar

Last Glacial Maximum (LGM) to the present. His ice-sheet growth is initiated in Hudson Bay. The

reconstruction provides data on ice-sheet topog- value of the coef¬cient of a(T ), c 2 , is given by

raphy as well as extent, but only the data on ice-

Q a(T g ) ’ a(T h )

c2 =

sheet extent are necessary to constrain variations (8.43)

C (T h ’ T g ) 1

2

in albedo. We can estimate the relationship be-

where T g is the temperature of the full glacial

tween global albedo and temperature from the

climate state. Q , the incoming solar energy, is

LGM to the present using Peltier™s ice extents and

equal to 340 J/s/m2 (North, 1991). Equation (8.43)

the Vostok time series. First, a global-temperature

yields c 2 = 5.4 — 10’10 —¦ C1/2 /s. Scaled to the ra-

time series can be constructed by averaging the

diative time scale of 180 yr this is c 2 = 3.0 —¦ C1/2 .

Vostok time series in 1-kyr intervals centered

The relationship between incoming energy and

on each integer value of kyr. The time-averaged

8.9 COHERENCE RESONANCE AND THE TIMING OF ICE AGES 215

continues only as long as ice sheets expand non-

6

linearly with global temperature. The fact that

5 ice sheets grow preferentially on the continents

eventually weakens the positive feedback as the

4 North American ice sheets expand into a taper-

”a

ing continent. The result is a second stable state

(%) 3

characterized by an equilibrium between the ice-

-albedo feedback and the negative feedback of

2

outgoing radiation. The square-root form may be

approximately understood by considering the

1

constraints on the North American ice sheets (in-

0 cluding the Laurentide and former ice sheets) as

’10 8 6 4 2 0

they expand from their initial location in Hudson

”T (°C)

Bay. When the North American ice sheet is ex-

panding southward and westward across Canada

Fig 8.25 Observed relationship between albedo and global

during its initial growth, the Earth™s albedo is

temperature based on Peltier™s (1994) gravitationally

self-consistent sea-level inversion and Vostok temperatures sensitive to small changes in global temperature.

for the same time periods. The data closely approximate a The result is a strong positive feedback between

square-root function between albedo and global temperature cooler temperatures and larger ice sheets. The ice

1

(a(T ) = c 2 (Th ’ T ) 2 with Th = ’1). The shape of this curve

sheet eventually reaches its maximum westward

largely re¬‚ects the sensitivity of the Laurentide ice sheet

extent in northern Canada as the global temper-

coverage to global temperatures as the ice sheet retreated

ature further drops, however. If we assume that

westward and southward from Hudson Bay during the later

ice sheets expand preferentially on land, the ice-

stages of retreat. Earlier stages of retreat were not as

-albedo feedback weakens at this stage because a

sensitive because ablation resulted in limited retreat due to

further decrease in global temperature results in

the tapered ice sheet geometry in southern Canada. From

Pelletier (2003). smaller changes in ice-sheet extent because ex-

pansion is limited to a southerly direction and

a tapering North America. The combined effect

global temperature is plotted in Figure 8.24b. The of long-wavelength outgoing radiation and the

combination of the ice--albedo feedback and long- ice--albedo feedback is positive and of large mag-

wavelength outgoing radiation are illustrated in nitude for small ice sheets, but becomes progres-

Figure 8.24c. The combined effect of these pro- sively weaker as ice sheets grow in extent and are

cesses results in two ¬xed points in the model eventually geographically limited in where they

system corresponding to the full glacial and can expand. As global temperature continues to

interglacial states. drop, a balance is eventually achieved between

Bistability in the climate system is based on the long-wavelength outgoing radiation and the

a balance between the negative feedback of long- increase in Earth™s albedo. This is the full-glacial

wavelength outgoing radiation and the positive climate.

ice--albedo feedback. The negative feedback of Figure 8.24c illustrates the radiation balance

outgoing radiation is dominant in an Earth with- of the Earth as a function of global temperature

out extensive subpolar ice sheets. When large for the combination of long-wavelength outgoing

continental ice sheets expand in North Amer- radiation and the ice--albedo feedback. The ¬xed

ica and Northern Europe, however, temperature points are obtained by setting Eq. (8.43) equal to

drops result in an albedo increase large enough zero to obtain

to dominate the negative feedback of outgoing 1

T = c 2 (’1 ’ T ) 2 (8.44)

radiation. The net positive feedback driven by ice-

with c 2 = 3.0 —¦ C1/2 . This quadratic equation has

sheet expansion sends the climate system into a

two roots at T = ’1.1 and T = ’8.0 —¦ C. The root

self-enhancing feedback of colder temperatures

at T = ’8.0 —¦ C is a stable ¬xed point; cooling

and larger ice sheets. This feedback, however,

216 STOCHASTIC PROCESSES

below this point leads to a net warming and a re- 2.0

turn to the ¬xed point. The root at T = ’1.1 —¦ C is

an unstable ¬xed point. Equation (8.44) predicts

h

two stable states of the Late-Pleistocene climate

(km)

system: an interglacial climate with T = 0 —¦ C and

a full-glacial climate with T = ’8 —¦ C. This analy- 0.0

sis is independent of the system dynamics and

depends only on equilibria of Eq. (8.40).

The ¬‚exural isostatic de¬‚ection of the litho- slope = 0.001

sphere beneath an ice sheet may have at least

slope = 0.002

two effects on the geometry of the ice sheet.

First, lithospheric subsidence may lower accumu- 0 2000

1000

lation rates by decreasing the ice-sheet elevation. x (km)

This ˜˜load-accumulation™™ feedback is perhaps the

Fig 8.26 Steady-state ice-surface pro¬les of a threshold-

most well-studied effect of lithospheric de¬‚ection

sliding ice sheet with different scenarios of bedrock

on climate (North, 1991). The strength of this

topography, calculated from Eqs. (8.45) and (8.46). The

feedback, however, is limited by another effect

elevation of the ice surface and bedrock topography are

of lithospheric subsidence: changes in lateral ice-

shown for ¬‚at bedrock (solid line), and uniformly sloping

¬‚ow velocities. An ice sheet may be considered bedrock (dipping away from the margin) with a slope of

to be a conveyor belt in which ice ¬‚ows from ± = 0.002 (small-dashed line), and ± = 0.004 (long-dashed

zones of accumulation to zones of ablation. In line). As the bedrock depression deepens, the ice surface

steady-state, this ¬‚ow occurs without a change elevation decreases, but not as much as the bedrock. The

in surface topography or ice thickness. In fact, result is a counter¬‚ow within the ice sheet that causes

thickening or thinning of the ice sheet depending on whether

an increase in accumulation may be completely

the lithosphere is subsiding or rebounding. From Pelletier

accommodated by an increase in lateral ice-¬‚ow

(2003).

velocities such that no change in ice volume oc-

curs. This possibility highlights the dif¬culty of

modeling ice-sheet response to the small changes ing on its bed. On a ¬‚at bed, the pro¬le of a

in insolation experienced during the Pleistocene. threshold-sliding ice sheet is given by

In particular, a small change in basal-sliding pa-

h= 2h o (L ’ x) (8.45)

rameters may greatly change the sensitivity of ice

sheets to external forcing. Basal-sliding parame- and the thickness for an ice sheet on a sloping

ters are poorly known, introducing great uncer- base of gradient ± is

tainty into forward models of climate based on

h ho

ice-sheet dynamics. x’L = + 2 ln |±h ’ h o | (8.46)

± ±

Although the absolute values of basal-sliding

where L is the length of the ice sheet and h o is

parameters are poorly known, the relative effects

the maximum thickness of the ice sheet on a ¬‚at

of lithospheric de¬‚ection on ice-sheet sliding may

surface. h o is a function of the sliding shear stress

be determined more precisely. Lithospheric sub-

if we assume a sliding rather than a plastic ice

sidence perturbs a steady-state ice sheet by in-

sheet. To illustrate the effects of lithospheric de-

creasing the basal slope of the ice sheet, re-

¬‚ection on the ice-sheet thickness, we have plot-

ducing the gravitational force available to drive

ted ice-sheet pro¬les from Eqs. (8.45) and (8.46)

lateral ice ¬‚ow. This ¬‚ow reduction thickens the

for a ¬‚at surface and for surfaces with ± = 0.002

ice even under conditions of uniform mass bal-

and 0.004 in Figure 8.26. These pro¬les illustrate

ance. Conversely, lithospheric rebound enhances

that as the lithosphere subsides, an ice sheet

lateral ice-¬‚ow velocities and results in ice-sheet

will thicken because lateral ice ¬‚ow is inhibited

thinning. The effects of lithospheric subsidence

by the increase in basal slope. Conversely, litho-

and rebound may be illustrated by considering

spheric rebound leads to ice-sheet thinning.

the steady-state geometry of an ice sheet slid-

8.9 COHERENCE RESONANCE AND THE TIMING OF ICE AGES 217

glaciation deglaciation Fig 8.27 Schematic illustration of

the coupling between ice sheet

(d) flow-driven thinning only

(a) flow-driven thickening only

adjustment by lithospheric

de¬‚ection and ice-sheet advance or

retreat. The initial bedrock and

ice-sheet geometry is shown

(dashed line) along with the ¬nal

geometry (solid line) in each case.

v During glaciation: (a) Bedrock

v

subsidence considered alone leads

to ice-sheet thickening. (b)

viscoelastic rebound

viscoelastic subsidence Accumulation growth considered

alone leads to uniform thickening

(b) accumulation growth only (e) ablation decay only (assuming uniform accumulation

rate along the ice sheet) with a

corresponding advance. (c) Both

”agrowth ” adecay processes taken together act

destructively because some

accumulation growth is canceled out

by subsidence, reducing the effective

albedo change. During deglaciation:

(d) Bedrock rebound alone leads to

(c) thickening plus growth (f) thinning plus decay ice-sheet thinning. (e) Ablation alone

leads to uniform thinning of the ice

sheet with a corresponding retreat.

” anet ” anet (f) Together these processes act

more retreat constructively to modify albedo

less advance

than decay only

than growth only because ablation results in a greater

retreat for a thinner ice sheet.

v

Modi¬ed from Pelletier (2003).

v

Ice-sheet thickening and thinning, in turn, effects of subsidence-induced thickening and ac-

modify the effects of changes in global tempera- cumulation growth taken together are illustrated

ture to retard glaciations and accelerate deglacia- in Figure 8.27c. The combination results in a

tions. This coupling may be called the ˜˜load- smaller ice-sheet advance relative to the case

advance™™ feedback and is illustrated in Figure with accumulation alone. In cases in which ac-

8.27. Figures 8.27a--8.27c illustrate how a de- cumulation is not uniform (i.e. accumulation in-