<< . .

. 41
( : 51)



. . >>

10
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-

<< . .

. 41
( : 51)



. . >>