Rectifying the long-term climate fluctuations in
the Milankovitch Bands
by
Wenwei Pan
S.B. in Atmospheric Physics
Beijing University,China
(1988)
Submitted to the Center for Meteorology and Oceanography
Department of Earth, Atmospheric and Planetary Sciences
in partial fulfillment of the requirements for the degree of
Master of Science in Meteorology
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 1992
@ Massachusetts Institute of Technology 1992. All rights reserved.
Author ................................................
Center for Meteorology and Oceanography
Department of Earth, Atmospheric and Planetary Sciences
,August 31, 1992
Certified by...
. .
.
,... ........... . .
Richard S. Lindzen
Sloan Professor in Meteorology
,..
..
Thesis Supervisor
Accepted by
'......................
................
Thomas H. Jordan
Chairman, Departmental Committee on Graduate Students
OF TF.NNIPGY
M
Rectifying the long-term climate fluctuations in the
Milankovitch Bands
by
Wenwei Pan
Submitted to the Center for Meteorology and Oceanography
Department of Earth, Atmospheric and Planetary Sciences
on August 31, 1992, in partial fulfillment of the
requirements for the degree of
Master of Science in Meteorology
Abstract
In this thesis we address the ice age problem in light of our recent knowledge of
both geological evidence on climate spanning the late Pleistocene and the intensity
of Hadley circulation in modulating the mid-latitude wave activity and high-latitude
temperature.
Since the radiative forcing at the top of the atmosphere is mathematically modulated by the eccentricity and the global ice volume measured by the 8018 is inversely
correlated with the eccentricity[30], we argue that the 100kyr period in the Late Pleistocene 8018 records is the result of rectification of the radiative forcing. Based on the
physics of the climate system, we have constructed a new rectifier which takes into
account the Hadley winter heat flux.
This rectifier has only two free adjustable parameters, and under certain range it
can well detect the low frequencies from the high frequency radiative forcing by comparing the rectified time series with those of the eccentricity. It is further compared
with a commonly used rectifier due to Imbrie and Imbrie[31]. By combining the rectified Hadley winter heat flux with the traditional Milankovitch high-latitude summer
insolation into a linear differential model, we simulated the ice volume fluctuations
during the last 700,000 years. The results are further compared with the SPECMAP
stack data[32].
The results demonstrated the importance of rectification mechanism in understanding the late Pleistocene ice ages while the role of the Hadley circulation is
manifested through the simple physically based rectifier.
Thesis Supervisor: Richard S. Lindzen
Title: Sloan Professor in Meteorology
Acknowledgments
I am deeply indebted to my advisor, Prof. Richard S. Lindzen, for his continuing
inspiration and insightful guidance, which help me build up the ability of doing independent research, throughout the course of this thesis.
Also I would like to thank Shuntai Zhou, Zhongxiang Wu and Nilton Renno for
their great friendship and assistance. Special thanks also go to Jane McNabb at the
CMPO headquater, who gave me a lot of help to face the entire new environment
after I came here; to Diana Spiegel, who gave me a lot of assistance to encounter the
problems in my computer work.
Finally I would like to thank my wife, Yang Zhong, and my parents for their
wonderful love, care and encouragement.
Contents
Abstract
2
Acknowledgments
3
List of Figures
6
1 Introduction
2
9
1.1
The specific problems . . .... . . . . . . . . . . . . . . . . . . . . . .
9
1.2
Review of current research involving the first problem . . . . . . . . .
10
1.3
Review of current research involving the second and third problems .
12
1.4
Outline of the thesis
17
. . . . . . . . . . . . . . . . . . . . . . . . . . .
The importance of the Hadley circulation in modulating the midlatitude baroclinic wave activity and the winter extratropical temperature
3
19
2.1
The importance of the Hadley circulation in the present climate
2.2
The determination of the Hadley intensity . . . . . . . . . . . . . . .
21
2.3
The insufficiencies in the present ice-climate link approaches . . . . .
24
. .
Parameterization of the intensity of the Hadley circulation
3.1
26
Seasonal energy balance model and the results . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .. . . .
19
26
3.1.1
M odel choice
. .
26
3.1.2
Model description . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.1.3
Model limitations . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2
Orbital parameters and the incident solar radiation at the top of the
atm osphere
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.3
M odel results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.4
Parameterization of the intensity of Hadley circulation
34
3.4.1
. . . . . . . .
The dependence of the Hadley intensity on the displacement of
the maximum temperature . . . . . . . . . . . . . . . . . . . .
3.4.2
34
Parameterization of the intensity of the maximum Hadley winter heat flux . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 A new nonlinear rectifier
37
38
4.1
Equilibrium models vs. differential models . . . . . . . . . . . . . . .
38
4.2
Constructing a new rectifier . . . . . . . . . . . . . . . . . . . . . . .
39
4.3
Rectifying the lower frequencies of orbital forcing
. . . . . . . . . . .
41
4.4
Modeling the global ice volume fluctuations
. . . . . . . . . . . . . .
47
5 Summary and implications
52
Appendix
53
Bibliography
55
List of Figures
1-1
Change in oceanic heat transport required to produce a given change
in globally averaged surface temperature, as computed from Rind and
Chandler's atmospheric GCM(plotted from Rind and Chandler's Table
6[58]). The best-fit cubic curve through the points is shown. .....
1-2
15
Oxygen isotope records(and inferred temperature) for the past 70Myr
from low-latitude surface waters of the Pacific(A) and from ocean deep
waters as observed in the South Atlantic(+).Because ocean deep waters are formed at high latitude, the lower curve can be taken as representing the isotope composition of high latitude surface water. The
temperature scale is valid only in the absence of an Antarctic ice sheetroughly up to mid-Miocene time. Including a correction for the effect of
the ice sheet would raise more recent temperatures by ~ 30 C. (Figure
courtesy of Molnar and England[43].) . . . . . . . . . . . . . . . . . .
2-1
16
Numerical model results for streamfunction for (a) qo = 0 deg, (b)
00 = 2 deg, and (c)
4o
= 6 deg. Units are in 1010kgs- 1 and the contour
interval is 0.1 x 1010 kgs-' for (a) and (b); twice this value for (c)[41].
22
2-2 Equal-area results for asymmetric redistributions within the winter cell,
0e/90 is the heavy lines, and 9 /0 the thin line[26]. . . . . . . . . . . .
2-3
23
Calculated meridional streamfunctions in units of 1010kgs~' for Oo =
60. (a) A = 0, (b)A = 20/300 with an asymmetric redistribution over
both cells[26].
3-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Time series of the three orbital parameters from 3myr B.P. to 1kyr A.P.. 30
3-2
Power spectral density for the time series shown in Figure 3-1. .....
3-3
Zonal averaged land and ocean cover from Smithsonian Meteorological
Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-4
31
33
Time series of (a) the displacement of Tma, with v = 0; (b) the displacement of Tma, with v = .472;(c) the curvature at Tma, for the
period of 1000kyr B.P. to 100kyr A.P. . . . . . . . . . . . . . . . . . .
3-5
Numerical calculation for the maximum streamfunction of the Hadley
winter cell vs.
the displacement of maximum temperature during
northern hemisphere winter[41]. . . . . . . . . . . . . . . . . . . . . .
3-6
The time series of normalized
36
m only due to the change of 0 using
Equation 3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-7
35
36
Time series of parameterized maximum Hadley winter heat flux according to Equation 3.10 for the period of 1000kyr B.P. to 100kyr A.P.. 37
4-1
Eccentricity and global ice volume over the past 730,000 years. (A)
Variations in orbital eccentricity calculated by Berger[3]. (B) Oxygen
isotope curve for deep-sea core V28-238 from the Pacific Ocean.
4-2
. . .
40
The time series of glacial potential(left column) and their corresponding power spectral density(right column) with Fo = 2.0 for four choices
of c. The units for the horizontal coordinates are kiloyears in time for
the left column diagrams and 1/kiloyears in frequency for the right
column diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4-3
Same as in Figure 4-2 except with Fo = 2.5.
. . . . . . . . . . . . . .
43
4-4
Same as in Figure 4-2 except with F = 3.0.
. . . . . . . . . . . . . .
44
4-5 Same as in Figure 4-2 except with F0 = 3.5.
. . . . . . . . . . . . . .
45
4-6
Same as in Figure 4-2 except with Fo = 4.0.
. . . . . . . . . . . . . .
46
4-7
The time series of R and their corresponding power spectral density
with F0 = 3.442 for four choices of c. The units for the horizontal
coordinates are the same as in Figure 4-2.
. . . . . . . . . . . . . . .
48
4-8
The time series of y and their corresponding power spectral density
with Fo = 3.442, a = 17.kyr for four choices of c. The units for the
horizontal coordinates are the same as in Figure 4-2.
4-9
. . . . . . . . .
49
The time series of normalized eccentricity and its power spectral density (a); and normalized y and its corresponding power spectral density
with F6 = 3.442, c = 0.5, a = 17.kyr (b). The units for the horizontal
coordinates are the same as in Figure 4-2.
. . . . . . . . . . . . . . .
50
4-10 Time series of the averaged summer insolation from 60*N to the North
Pole for the period of 1000kyr B.P. to 100kyr A.P. and its associated
power spectral density. The units for the horizontal coordinates are
the same as in Figure 4-2.
. . . . . . . . . . . . . . . . . . . . . . . .
4-11 Time series of the normalized modeling result (dash line) and 60
50
1
of SPECMAP (solid line) for the past 700,000 years with a = 17kyr,
c =1.0, c2 = 0.012, FO = 3.442. . . . . . . . . . . . . . . . . . . . . .
51
4-12 Power spectral density of the modeling result (dash line) and the
SPECMAP record (solid line) for the past 700,000 years. . . . . . . .
51
Chapter 1
Introduction
1.1
The specific problems
It is been a long time unsolved scientific mystery regarding the mechanism of climatic
fluctuations in the Milankovitch bands, namely:
1. How can we explain the glacial cycles with 100kyr dominant periodicity over the
late Pleistocene[23][30]?
2. What determines the transition in the period(from dominant 41kyr period in
the early Pleistocene and late Pliocene to 100kyr period in the late Pleistocene),
amplitude(- 50% increase in late Pleistocene), and the mean value of the ice
fluctuations occurring at about 700kyr B.P., i.e., near Matuyama and Brunhes
magnetic polarity epochs boundary[43][62]?
3. What triggers the initiation of northern hemisphere significant glaciation around
2.5ma (1 ma refers to one million years ago, hereafter)[2][75][88][60]?
Most of the theoretical research in palaeoclimate has been concentrated on the late
Pleistocene, i.e., regarding the first problem above, because we have more available
data during this period. For the second and third problems, emphases are still around
observational analyses and associated hypotheses. However, any successful theory
about the ice age problem has to answer the three questions together. In the following
we will briefly review the proposed mechanisms on the three questions mentioned
above.
1.2
Review of current research involving the first
problem
The most accepted theory regarding the first problem is the astronomical one, which
attributes the climatic fluctuations to the changing of the Earth's orbital parameters, i.e., eccentricity(-100kyr, 400kyr periods), obliquity(-41kyr period), and the
precession of the equinoxes (~22kyr period). Although by comparing the time series
of ice volume with that of the eccentricity there is an inverse correlation between
eccentricity of the orbit and the global ice volume[30], it is commonly agreed that the
direct insolation forcing in the eccentricity period(~ 100kyr) is too weak to explain
the dominant Brunhes 100kyr glacial-interglacial response(The change of eccentricity
will only account for ~0.2% change in the total received radiation; cf. Chapter 3.2).
The essential question here is why we still have the dominant 100kyr periodicity in
the climate records. Does the 100kyr period have anything to do with the external
radiative forcing? Or is it the only result of the internal oscillation due to the complexity of the climate system? Around these questions a lot of works have been done.
To summarize the enormous works on this issue, it is useful to collect them according
to the following three categories:
1. Free oscillation of an internally driven, nonlinear system:
The amplitude and period are determined by the inherently nonlinear interactions in the air-sea-ice system, with orbital variations serving only to phase-lock
the variations at the preferred time scales. The argument is that low-frequency
(red noise) variance can be produced when white noise forcing, from a time
series with a short response time, is applied to another system with a relatively
long response time.
The studies fitting the second category are Sergin[72][73], Saltzman et al.[66],
Saltzman and Sutera[67], Chalikov and Verbitsky[8][9]. It is worth noting that
Saltzman's approach is fundamentally different from some other schemes. Instead of being based on the standard deductive modelling method, i.e., starting
from the fundamental statements of conservation of mass, energy, and momentum and successive averaging and parameterization of flux processes, it is based
on an inductive method through formulating a closed set of equations and tuning the modelling output to match the observations.
2. Resonance of a nonlinear system driven both externally and internally:
The proposed models are from Kallen et al.[33], Ghil and LeTreut[19], LeTreut
and Ghil[38] and LeTreut et al.[39], where the free response of the system is
dominated by oscillations near 10kyr. A nonlinear resonance mechanism that is
based on feedbacks between the model's ice sheet and other parts of the system
transfers energy from the precession band(where radiative forcing is strong) into
the 100kyr band(where this forcing is very weak).
The thing we must keep in mind is that even though sophisticated statistical
modelling is useful, it never forms a real alternative to physics-based models.
3. Ice volume fluctuations forced by secular change of incident solar radiation:
* Suarez & Held[79][80]: including ice-albedo feedback but the resulting impact is too small.
e Weertman[86], Pollard[57], and Pollard et al.[59]: ice-sheet model without
containing explicit representations of the physical processes, i.e., ice sheet
flow and accumulation and glacial isostatic adjustment.
* Oerlemans[50][51][52],
Birchfield et al.[4][5],
Pollard[58] and Hyde &
Peltier [28] [29]: including the nonlinear interactions between accumulation
and ablation of the ice sheet flow, elastic lithosphere and viscoelastic mantle. Oerleman[50] was the first to demonstrate that the feedback between
ice physics and glacial isostatic adjustment could lead to long period os-
cillations in predicted ice volume only when applying a relaxation time
of 10-20kyr which is approximately an order of magnitude longer than is
known to be reasonable on geophysical grounds. Birchfield[4] [5] employed
a similar model with a more appropriate isostatic adjustment time scale of
3kyr but failed to produce significant power in 100kyr-period. The third
one incorporated a more realistic representation of the glacial isostatic adjustment process and managed to get a promising result. However, the
model result is very sensitive to the choice of tuning parameters.
* Denton & Hughes[12], Ledley[40], Denton et al.[13], Peltier[55]: global
ice-sheet model(sea-level linkage between terrestrial ice-sheet model and
marine model), wherein the linkage occurs by direct atmospheric cooling
and by the impact of high-elevation ice sheets in the North Atlantic on the
westerly jet-stream flow.
1.3
Review of current research involving the second and third problems
For the mechanisms regarding the onset of northern hemisphere glaciation there are
many proposed explanations. One group of the explanations looks to changes in atmospheric composition, such as CO2, ozone or trace gasses[56], or in solar strength[54],
where the first one partly verifiable from geological data while the second one is
untestable. While Kennett and Thunell[34] suggested the increased volcanism during
the latest Cenozoic as a possible cause of glaciation.
Most of the other explanations call on tectonic changes as the trigger of onset of
significant northern hemisphere glaciation. Among them there are polar wandering
theory proposed by Ewing and Donn[17]; possible land-sea distribution link by [49];
changes in susceptibility to glaciation due to epeirogenic uplift of highland regions in
northern Canada by[18], Emiliani and Geiss[16], and Birchfield et al.[5]; changes in
oceanic heat and moisture fluxes induced by shoaling or deepening of critical oceanic
gateways such as the Panama Isthmus or the Bering Straits by Emiliani[15].
Birchfield and Weertman[4] proposed that the large-scale uplift in two key regions,
i.e., Tibet and Western North America, would enhance albedo-temperature feedback
on a globally significant scale. The cold air over the large, newly created region of
high-standing topography at middle latitudes provides a temperature discontinuity
that accelerates the normal southward movement of the snowline in autumn-winter.
This enlarged high-albedo surface in turn may lead to global cooling during the transitional and winter seasons and trigger the onset of significant northern hemisphere
glaciation.
Ruddiman et al.[64] and Ruddiman & Raymo[65] proposed that the late
Neogene(Pliocene-Miocene) uplift in southeast Asia and the American southwest results in greater meridionality of northern hemisphere winter jet stream and subsequently cooling over large areas of Canada and northwestern Europe, which can set
the stage for the growth of large continental ice sheets.
Raymo et al.[60] have also suggested the connections between orogeny and climate/ocean chemistry can trigger the initiation of northern hemisphere glaciation.
Namely, significant increases in uplift rates observed in the late Neogene have resulted
in a global increase in chemical weathering rates over the last 5my. The associated
increase in river dissolved fluxes would have increased ocean alkalinity and , therefore,
lowered oceanic and atmospheric C02 levels. Decreased atmospheric C02 may have
resulted in a cooling sufficient for the growth of northern hemisphere ice sheets.
However, Molnar and England[46] argue that most of the evidence used to infer late Cenozoic uplift could, in large part, be a consequence of the very climate
changes that this supposed uplift is thought to have caused. They propose the possibility that climate change, weathering, erosion and isostatic rebound might interact
in a system of positive feedback. Suppose increased weathering and erosion could
enhance a negative "greenhouse effect" as aforementioned; then, if climate change
accelerated erosion and weathering, the associated withdrawal of carbon dioxide from
the atmosphere might cause further global cooling. Increased rates of erosion and
isostatic rebound would alter the distribution of elevations by making the crests of
ranges higher than before. The prolonged duration of winter and snow cover in such
higher areas might lead to further reductions in temperature. In addition, mountain
ranges with high crests and deep valleys will perturb atmospheric circulation more
strongly than those with the same mean elevation but lower crests. Thus, if the rise
of crests of mountain ranges were to perturb the climate in such a way as to enhance
erosion, isostatic rebound would cause a further rise, and further perturbations to
the climate. By such feedback mechanism, the gradual regional uplift of the Earth's
surface in Asia, due to Cenozoic tectonics, might have led to accelerating climate
change and the illusion of the accelerated uplift of mountain ranges throughout the
world.
In addition, Rind and Chandler[61] emphasize the important role of the oceans
in understanding the climate changes. After employing an atmospheric GCM to the
oceanic simulation using two special techniques, namely, fixing sea surface temperatures at whatever values seem to fit a past or potential future climate and then
letting the atmospheric GCM run to equilibrium in order to get the oceanic heat
transport; altering the form of atmospheric GCM's lower boundary conditions to
specify oceanic heat transport in order to assess the response of climate to a given
amount of oceanic heat transport, they suggest that increased topography during the
late Cenozoic might have altered the surface wind stress in such a manner that led
to reduced oceanic heat transports; this effect would then need to be considered in
understanding the beginning of glaciations(Figure 1-1) provided this effect is not fully
compensated by changes in the atmosphere.
Both paleobotanical data and oxygen isotopes indicate a monotonic, but not
steady, cooling of the Earth over the past 50myr, which might be the consequence of the tectonic processes that built Tibet, the Himalayas and other high
terrains in Asia. Abrupt decreases in temperature occurred near Eocene/Oligocene
boundary(~ 36ma), at ~ 15ma in the Miocene epoch, and at
-
2.5ma near the
end of the Pliocene epoch, when continental glaciation in the northern hemisphere
began(Figure 1-2). Moreover, the decrease in temperature has been much greater at
high latitudes than at low latitudes. For example, oxygen isotopes from planktonic
Ctefaceous
d
fIC
Aqe
IcI Age
.
Surfaca temoerature cnange
4K)
Figure 1-1: Change in oceanic heat transport required to produce a given change
in globally averaged surface temperature, as computed from Rind and Chandler's
atmospheric GCM(plotted from Rind and Chandler's Table 6(58]). The best-fit cubic
curve through the points is shown.
foraminifera deposited at sub-Antarctic latitudes indicate a decrease in surface water temperature of ~ 120C since late Eocene time[76][74], but those from equatorial
latitudes indicate hardly any change in such temperature
[70][69];
also see Figure 1-
2). The combined effect of global cooling trend from the late Cenozoic and the increased latitudinal temperature gradient resulting from complex interactions within
the climate system determines the onset of northern hemisphere significant glaciation
around 2.5ma, although we do not know how these interactions set the latitudinal
temperature gradient of the Earth's surface. For example the regional response to
the late Cenozoic global cooling due to the tectonic processes might be quite different
because of the differences in regional land-sea distribution, emissivity, opacity, etc.,
thus setting the secular change of meridional temperature gradient as well as affecting
the global cooling itself. Increased temperature gradient leads to more accumulation
of ice at high latitudes due to both atmospheric and oceanic moisture fluxes, while
the global cooling trend prevents it from ablation in the warm seasons.
Thus we
think the change in latitudinal temperature gradient is most important because it
promotes an altered Hadley circulation, a changed eddy transport of latent heat and
sensible heat from low to high latitudes, a reorganized ocean heat transport due to
wind-driven circulation as well as thermohaline circulation.
3
10
049
liocene Miocene
20
Oligocene
4 01
r
Eocen
60
Maastr:chtian
Age (Myr)
Figure 1-2: Oxygen isotope records(and inferred temperature) for the past 70Myr
from low-latitude surface waters of the Pacific(A) and from ocean deep waters as
observed in the South Atlantic(+).Because ocean deep waters are formed at high
latitude, the lower curve can be taken as representing the isotope composition of
high latitude surface water. The temperature scale is valid only in the absence of
an Antarctic ice sheet-roughly up to mid-Miocene time. Including a correction for
the effect of the ice sheet would raise more recent temperatures by ~ 30 C. (Figure
courtesy of Molnar and England[43].)
As for the transition in period, amplitude and the mean value of the ice fluctuation around 700kyr, there are only a few proposed explanations. Ruddiman and
McIntyre[63] suggest that the transition may be linked to the onset of larger ice
sheets, whose southern limits in North America and Europe must have extented farther into middle latitudes, and thus come under the influence of stronger precessional
forcing. Kutzbach[37] proposes that it might in some way reflect an increased sensitivity of ice sheets to precessional insolation heating of low- and mid-latitude land
masses at latitudes south of the ice sheets. While Saltzman[68] implied a model of a
nonlinear oscillator driven by internal variabilities and demonstrated the possibility
of mid-Quaternary climatic transition.
A similar approach but based on a stochastic resonance model has been recently proposed by Matteucci[45].
By slowly changing the only free parameter of
the model, the system can undergo a pitchfork bifurcation and the bifurcation point
separates a linear regime(identified with the early Pleistocene) from a strongly nonlinear regime(the late Pleistocene) where the stochastic resonance mechanism produces
rapid and symmetric transitions between the stable steady states of the system. Both
Saltzman[68] and Matteucci[45] postulate tectonic forcing has altered the climatic
state resulting in a change of the stability properties of the system.
1.4
Outline of the thesis
In this thesis I focused myself on the first problem because we have better climatic
proxy data both in quantity and quality for the late Pleistocene than for the earlier times.
In Chapter 2 recent work on the determination of Hadley circulation
and its role in modulating mid-latitude wave activity and high-latitude temperature
is reviewed. Also we pointed out the insufficiencies in the present ice-climate link
approaches. In Chapter 3 a simple parameterization formula of the intensity of the
Hadley circulation is given based on both numerical results and analytical arguments.
In Chapter 4 a new nonlinear rectifier is constructed based on the physics of highlatitude snow accumulation and ablation, and by incorporating it with the traditional
Milankovitch radiative forcing into a differential model we simulated the global ice
volume fluctuations for the entire 700,000 years. Implications and summary of this
thesis are given in the last chapter.
Chapter
The importance of the Hadley
circulation in modulating the
mid-latitude baroclinic wave
activity and the winter
extratropical temperature
2.1
The importance of the Hadley circulation in
the present climate
The highly effective role of dynamic transport in determining the equator-to-pole
temperature distribution can be demonstrated using the equilibrium Energy Balance
Model(EBM):
(2.1)
I + F[T(y)] = QS(y)A,
with y = sin(latitude), T surface temperature(OC),
Q
the solar constant divided
by four (340.0Wm- 2 ), S(x) the latitudinal distribution of solar radiation, A the
absorptivity of the surface to incoming solar radiation.
Taking the annual average distribution of solar radiation,
S(y) = 1 + .45P 2 (y)
and the ice free absorptivity, A = 0.68, together with the present infrared radiation
flux linearization,
I = A + BT
where A(= 203.3W/m 2 ), B(= 2.09W/m 2oC) are empirical coefficients taken from
satellite data[78], gives an annual average equator temperature of 42* and pole temperature of -35*,
in the absence of heat flux divergence, that is with F[T] = 0. This
temperature difference of 770 indicates the importance of dynamic heat flux both in
reducing the equator-to-pole temperature difference to the present value near 400 and
in reducing the equator temperature to the present value near 270.
The work of Charney[10] and Eady[14] identified the meridional temperature gradient and the associated available potential energy as the energy source for unstable
baroclinic wave growth. Recent studies indicate that the subtropical jet is determined by mean-flow accelerations by the Hadley circulation and decelerations by
midlatitude eddies. Theoretical models of the Hadley circulation showed that in the
absence of eddies, the poleward angular momentum advection by the Hadley circulation could lead to a strong subtropical jet while the effect of midlatitude eddies is
to reduce the jet intensity to the observed one[25]. GCM study(1] and a series of observational studies[6] [35] [36][87] supported the relationship between tropical heating
maxima, subtropical winter jet maxima, and extratropical baroclinic activities.
Using a simplified zonally symmetric model Hou and Lindzen[27] showed that, in
the absence of waves, a stronger Hadley winter cell resulting from tropical heating
concentration can dramatically increase the baroclinicity and potential vorticity gradients in the winter extratropics. Very recently Hou[26] employed a simplified GCM
and showed that "a shift of the prescribed tropical heating toward the summer pole,
on time scales longer that a few weeks, leads to a more itense cross-equatorial winter
Hadley circulation, enhanced upper-level tropical easterlies, and a slightly stronger
subtropical winter jet, accompanied by a warming at the winter middle and high
latitudes as a result of increased dynamical heating."
Although the above conclusions are based on simplified models with prescribed
latent heating and need to be further tested in a more complex model, the importance
of the Hadley circulation in modulating the mid-latitude baroclinic wave activity and
the winter extratropical temperature is manifested. Before considering this hypothesis
in our paleoclimate research, we have to say something about the determination of
the Hadley intensity.
2.2
The determination of the Hadley intensity
As mentioned in Hack[21], there appear to be two conceptual views of the Hadley
circulation and the ITCZ. The first, considers the Hadley circulation and the ITCZ as
monthly(or seasonally) and zonally averaged phenomena. The second view is that the
Hadley circulation and the ITCZ are phenomena that can be identified on individual
weather maps and whose fluctuations can be followed day to day. Regardless of
which conceptual view is adopted, the intensity of Hadley circulation is physically
determined by the tropical diabatic heating concentration and displacement with
respect to the equator.
The difference in the strengths of the Hadley winter and summer cells when the
maximum heating being off the equator was first discussed by Lindzen and Hou[41].
This can be illustrated by comparing the June-August and the December-February
meridional circulation fields. In June, July and August most of the latent heat release
associated with the maximum heating occurs north of the equator. The meridional
circulation associated with this latent heat release exhibits a distinct preference for
the winter hemisphere cell. In December, January and February most of the latent
heat release occurs south of the equator, and the low level air is drawn into the
ascending region primarily from the north. This effect is clearly illustrated in the
observational analyses of Newell et al. (Figure 3-19)[47] and Oort(Figure F-44)[53].
The sensitivity of the Hadley circulation to tropical heating displacement is care-
~_oi
~
-
.29
01
17
0
-90;0-3
LATrI/J
30
60
930
,deareas)
Figure 2-1: Numerical model results for streamfunction for (a) 4o = 0 deg, (b) 4o =
2 deg, and (c) 40 = 6 deg. Units are in 10' 0 kgs-1 and the contour interval is 0.1 x
10' 0 kgs- 1 for (a) and (b); twice this value for (c)[41].
fully studied in [41].
They find that moving peak heating even 2 degrees off the
equator leads to profound asymmetries in the Hadley circulation, with the winter cell
amplifying greatly and the summer cell becoming negligible. An example of this is
shown in Figure 2-1, where the numerical model results for the streamfunction are
plotted for three cases of displacements.
The influence of concentrated heating on the Hadley circulation is investigated in
[27].
They found that in the case of heating symmetrically centered on the equator,
concentration unambiguously increases the intensity of the Hadley circulation by up
to a factor of 5. For heating centered off the equator, concentration of heating primarily from the winter side of maximum heating, which is consistent with the explicit
0
-.-..
-..--
40 -
1.30
a
A-10/3 00
A=20/300
-0380
A80/3
A...
-160/300
A
3
-6/
1.20[.
.10-
.00I-
'
.
0.90
3.80
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
.0
y
Figure 2-2: Equal-area results for asymmetric redistributions within the winter cell,
0,/90 is the heavy lines, and 9/9o the thin line[26].
is.--...F
- - -- - - -- - - -
A*
--n
--n
_13_ __3L
TIn
M.
_)
M._
_
_
_
_
_
L__
1.
_
_
3
_
_
_
_
REESEEn=
Figure 2-3: Calculated meridional streamfunctions in units of 1010kgs- 1 for 4o = 60.
(a) A = 0, (b)A = 20/300 with an asymmetric redistribution over both cells[26].
calculation and from ECMWF analyses, also leads to pronounced intensification of
the Hadley circulation. Further they found that agreement between the calculated
and observed Hadley intensity is achieved with mild concentration, consistent with the
observed zonally averaged precipitation. The nature of the concentrations is shown in
Figure 2-2, and Figure 2-3 shows the numerically calculated streamline distributions
for various degrees of concentration. The strength of the circulation for A = 20/300
is nearly 5 times that for A = 0.
2.3
The insufficiencies in the present ice-climate
link approaches
The several flaws in the ice-climate link approaches (cf. Chapter 1.2.3) stem from the
fact that they can not explain the following facts:
* Southern hemisphere paleoclimatic evidence.
GCM results indicate that the
climatic influence of large North Atlantic and European ice sheets is restricted
to Northern hemisphere, and that the expansion of the Antarctic ice sheets was
not large enough to influence the Southern hemisphere[44] [7].
e Observation of last termination about 14000 B.P. indicates near-synchroneity
in both hemispheres, which does not allow much time lag.
* A number of pre-Pleistocene climate records exhibit significant fluctuations at
100kyr, when there is little evidence for the presence of extensive ice sheets[11].
The above flaws in the ice sheet models might be related to the neglecting of global
heat transport. In fact, most of the previous ice sheet models assume that the climatic
response is forced by local radiative forcing at the top of the atmosphere and has
nothing to do with the meridional heat transport and a variety of processes relating
the surface temperature with the radiative forcing. Although the radiation field sets
the fluxes of energy at the top of the atmosphere and provides overall constraints upon
the behavior of the climate system, the climate system is for the most part heated from
the surface. Both observational and modelling results indicate that the high latitude
climate is determined by the combination of local radiative forcing, meridional heat
transport from lower latitudes by the atmosphere and oceans, vertical heat transport,
as well as the ocean memory through storing and releasing both heat and chemical
species(e.g., CO 2 ).
However, it is a well-known observational fact that the winter hemisphere is dynamically more active than the summer hemisphere.
The GCM experiments[20]
showed that the thermal tendency in the summer hemisphere depends more strongly
on the diabatic heating and dynamics, while it depends on horizontal advection in
the winter hemisphere. Based on linear model results for axisymmetric basic states,
it is suggested that the winter hemisphere is close to the "advective limit", while the
summer hemisphere close to "diabatic limit" [83]. Very crudely speaking, for example,
the high latitude surface is in near radiative-convective equilibrium during summer,
while it is in a high eddy regime during winter([47][22][81][82]).
Based on the above discussion, we assume that the high latitude winter surface
temperature is dominantly determined by eddy heat transport, which may be forced
by the tropical Hadley circulation; while the summer surface temperature is dominantly set up by diabatic heating, which is related to the summer radiative forcing.
One simple but realistic approach to modelling the high latitude ice-sheet response
to the external forcing is to take into account both winter heat transport and summer insolation. Before doing this, we have to parameterize the intensity of Hadley
circulation first.
Chapter 3
Parameterization of the intensity
of the Hadley circulation
The aim of this chapter is to parameterize the intensity of the Hadley circulation in
terms of the displacement of the surface temperature maximum. In the following we
are going to employ a simple seasonal EBM to translate the radiative forcing at the
top of the atmosphere to the surface temperature, which can be used to parameterize
the intensity of the Hadley circulation.
3.1
Seasonal energy balance model and the results
3.1.1
Model choice
There are several reasons for choosing North's seasonal energy balance model(SEBM,
hereafter)[48) rather than annual mean energy balance model or other kinds of
SEBMs. Firstly, the answer may lay in the Milankovitch hypothesis itself, namely,
that the latitudinal distribution of seasonal component of insolation would be more
important than annual component to the accumulation or ablation of ice sheets.
Moreover, in order to calculate the heat flux based on the asymmetric Hadley
circulation model rather than the traditional parameterization of heat flux involving
a relaxation toward the global mean or diffusion, we need a model that can resolve
the seasonal variation of surface temperature. At minimum we need a model with an
ocean mixed layer in order to have the heat capacity approriate to seasonal heating.
North's model has limited adjustable parameters which are always explicit, and
standard efficiency typical of spectral models which start with the largest and slowest
scales and add on as corrections the finer scale information. More specifically, in the
model the latitudinal distribution of the zonal average surface temperature is represented by a series of Legendre polynomials, while its time- dependence is represented
by a Fourier sine-cosine series. However for the present purpose, only mixed layer
is at issue and time harmonics are adquate. The Lengendre polynomial representation of the latitudinal distribution of the surface temperature is certainly not a must
here. Although the relaxation time over the oceans is on the order of years, and to
investigate a perturbation one must wait through many e-folding times by numerical
integration, here we may extract analytical "steady-state" solutions directly by just
solving algebraic equations.
3.1.2
Model description
In order to translate the solar forcing at the top of the atmosphere into the seasonal
surface temperature, we apply North's seasonal energy balance climate model, taking
into account the 75m ocean mixing layer and an idealized geometry of land-ocean
distribution.
The model equation is given by
OT
C(, 4)5t(x,
4, t) -
F[T(x, 4, t)]+ I = QS(x, t)A,
(3.1)
with x denoting sine of latitude, C(x, 4) the latitude- and longitude-dependent heat
capacity per unit area, T(x,
4, t)
surface temperature (0C),
Q the
solar constant
divided by four (340.0Wm- 2 ), S(x, t) the fraction of the incident solar flux received
by latitude x at time t and normalized so that the mean annual fraction integrated
over the hemisphere is unity, and A an absorptivity of the surface to incoming solar
radiation.
Since we will be primarily interested in the displacement of the solstitial temperature maximum from the equator, there is no point in including the parameterized
meridional heat flux divergence.
The outgoing infrared radiation is simply parameterized linearly in terms of surface
temperature,
I = A + BT,
(3.2)
with A = 203.3Wm-2 , B = 2.09Wm-2*C- 1 .
For the heat capacity, we follow North and Coakley[48] and take the thermal
response over land to be the heat capacity of an atmospheric column divided by the
radiation constant, CL/B = 0.16 year. Over water we take the thermal response to
be the heat capacity of the ocean 75m mixing layer divided by the radiation constant,
Cw/B = 4.7 years.
The interaction between land and ocean can be represented as g-(TL - Tw) and
(Tw - TL), where TL and Tw are the surface temperatures over land and ocean
respectively, v is the interaction coefficient. Both North & Coakley's experiment and
ours show that by increasing v one effectively couples the land to a large thermal
reservoir, and the ocean temperature amplitude is hardly affected while the land
amplitude is reduced. By tuning v we can bring the TL to some observed phase and
amplitude. Since as shown in Figure 3-4(b) the coupling between land and ocean has
small effect on the magnitude of the displacement and does not affect the variation
and secular pattern of the zonally averaged displacement, for simplicity we will neglect
the land-sea interaction and consider TL and Tw as functions only of latitude and
time. The zonally averaged temperature is given by
T(x,t) = fLTL + fwTw,
(3.3)
where fL and fw are fractions of land and ocean areas.
For the absorptivity A we take A=0.68 and don't include the albedo feedback
mechanism since it is irrelevant to the displacement.
3.1.3
Model limitations
The limitations in the above model are obvious. The model is one- level zonally
averaged energy balance model without any vertical and longitudinal structure as
well as detailed snow budget, and it's not been developed enough to incorporate
the ocean dynamics. Also we don't invoke such mechanisms as albedo feedback and
atmosphere's composition changes because at this stage we are only interested in
exploring the role of the displacement in the poleward heat flux. In addition, the
linearization of infrared radiation tends to overestimate the sensitivity in low latitudes
and underestimate it in high latitudes. These latitudinal differences in sensitivity are
related to changes in the tropospheric static stability[24].
3.2
Orbital parameters and the incident solar radiation at the top of the atmosphere
The energy available at any given latitude at the top of the atmosphere is a singlevalued function of the solar constant S., the semi-major axis a of the ecliptic, its
eccentricity e, its obliquity e and the longitude of the perihelion Co measured from
the moving vernal equinox. To determine the evolution of the external forcing thus
requires computing the long-term variations of three orbital parameters e, E and C.
We use the formulae given by Berger[3] to make the calculation. The resulting time
series for e, e, and esin C and their power spectrums are plotted in Figure 3-1 and
Figure 3-2 respectively.
The daily-mean incident solar flux at the top of the atmosphere at time t and
latitude
# is
Sca 2
S(t, 4; e, e,Co) = 4r2
5S[w(t; e,Cv), 4; e)
(3.4)
where w is the longitude of the earth with respect to the moving vernal equinox, S5
is the incident flux for a circular orbit, i.e., r = a, w = t; and
2
S(w,4; e) cos Od4
= 1.
(3.5)
24.6 r
24.4 F
24.2
24.0
-
23.8
0 23.6
23.4
. 23.2
23.0
22.8
0 22.6
22.4
22.2
22.0
-3000
-2500
-2000
-2566
-200
-500
-1000
-1500
in thousanas of yearsit-O at presentl
0
500
1000
56
166M
.060 r
.250
-
.240
.030
.020
.010
0
-.
010
-. 020
-. 030
-. 040
-. 050
-. 06
in
-566
-1on
-156g
thousands of yearsit=0 at presentI
6
.060 e~
.055
&
.050
.045
.040
.035
.030
.025
.020
j
.015
.010
2 -3000
-2500
-2000
in
-500
-1000
-1500
thousands of yearstt=0 at presenti
0
500
1000
Figure 3-1: Time series of the three orbital parameters from 3myr B.P. to 1kyr A.P..
106
105
0104
0
$103
U.
10
.1
a
.010
20
.020
.3
.030
00
20
.040
.050
.6
00
.
.060
.070
.0B0
frequency(cyclo/1000 years)
104
r
102
0
l
10-2|
0
.010
.020
.030
.040
.050
frequency(cycle/1000 yearsi
.060
.070
.080
0
.010
.020
.030
.040
.050
frequency(cycle/1000 years)
.060
.070
.080
U
U
10-2
Figure 3-2: Power spectral density for the time series shown in Fig.3.1.
The formulas for calculating S(t, 4; e, e, ) are given in [3].
There are several
properties in Equation 3.4 which need to be pointed out in order to understand the
evolution of S.
1. The annual global mean flux is only a function of e at a given time. Because
e varies from 0 to
~0.2% of
-
0.06, the variation of annual global mean flux is about
S.
2. The annual mean flux at a given latitude is a function of e and e. The annual
mean flux received by the atmosphere is further dependent on planetary albedo
which is a function of latitude and w.
3. For an eccentric orbit there is a symmetry for S:
S(w,
4; e, e,
) = S(w + 180,
-;
ee, C
+ 1800).
(3.6)
This tells us that at two given latitudes symmetric about the equator, the
incident solar fluxes at the top of the atmosphere for opposite seasons are exactly
out of phase. That is reasonable because the precessional effect only acts to
redistribute the incident solar radiation among different latitudes so that it
does not affect the annual mean flux at any given latitude[80][77].
Following North and Coakley[48], we decompose S(x, t) by Legendre polynomials
in latitude and Fourier series in time with the form of
4
4
S(x, t) = E Z(alk cos 2rkt + blA sin 27rkt)P(x),
(3.7)
1=0 k=0
where x denoting sine of latitude
3.3
4 same as in Equation 3.1.
Model results
By decomposing T(x, t) in the same manner as for S(x, t) we can solve Equation 3.1
simultaneously for all components.
Data from Smithsonian Meteorological Tables
.
---
o--
zonal averagma
tana cover%)
zonal averag.O ocean covert%)
C.
-:.0
-*.8 -0.6
-0. - §2
0
0.0
0.
.4
0.6
0.
1.0
sin(latitude)
Figure 3-3: Zonal averaged land and ocean cover from Smithsonian Meteorological
Tables.
In Figure 3-4 we show the time series of qo(the displacement of Tm..) and C(the
curvature of the temperature profile at Tm.,)
for constant land ocean distribution,
i.e., fL : fw = 0.2 : 0.8 for the tropical region, which are approximated from Figure 3-
3. In Figure 3-4(b) we also show the time series of 4o with land and ocean interaction
coefficient v = 0.472 as in [48].
Here Tm,
is the averaged surface temperature
maximum for the northern hemisphere winter(December, January and February).
There are several aspects from the diagram needed to be emphasized:
9 The displacement of maximum temperature in the northern hemisphere winter
always occur south of the equator, varying from about -2.5*
to -7.0*
with a
variation of ~ 100%, indicating significant secular shift of the displacement of
maximum heating.
* The variation of pole to equator temperature gradient(approximated proportional to the curvature at maximum heating as defined in Appendix) for the
calculated time domain are much less than that of the displacement of maximum heating, i.e., ~ 7%, indicating no significant change of the external forcing
in terms of the radiative equilibrium pole-to-equator surface temperature gradient. It should be mentioned that in reality the curvature depends mostly on
easterly waves and monsoons and is not at issue here but might be if monsoons
were alterred.
e Finally the dominant frequencies for the time series of displacement of maximum
temperature are 23kyr and 19kyr in precessional bands(near equator, insensitive
to obliquity), while 41kyr for the curvature at maximum temperature(mainly
depends on obliquity).
Parameterization of the intensity of Hadley
3.4
circulation
The dependence of the Hadley intensity on the dis-
3.4.1
placement of the maximum temperature
In this section we will apply the symmetric model results[41] to formulate an empirical formula relating the intensity of Hadley winter cell in terms of the maximum
streamfunction
f... with the
displacement of maximum temperature
4o
when the
concentration of the temperature profile is fixed.
Figure 3-5 is a plot of maximum streamfunction in the Hadley winter cell,
the latitude of the temperature maximum,
between
4maz(unit
ma£-.
4o,
kmax,
based on[41]. The empirical relation
: 10kg/s) and O is #/ma. (unit:10kg/s) and
40
is
(3.8)
4.60 + 1.2340o + 1.67 x 10-242 + 7.80 x 10-243 - 5.73 x 10-34.
The resulting time series of normalized
4
'ma
only due to the change of
Equation 3.8 are plotted in Figure 3-6, where we see
(~ 100%) in amplitude.
vs.
'bmax(t)
4o
using
has large variation
- -2.5
M
N-3.0
U
'-3.0
0 -4.5
.0 -5.5
C2-.-00
-;00
-800
-700
-600
--so@
-408
-300
1
in thoumanda of yearuzts at preseni
-200
-100
0
I
-100
0
100
o -2.0
U -2.5
4 -3.0
c -3.5
. -4.0
3
-4.5
-5.0
-5.5
-6.0
6.51
-
0-
-1000
-
C
-900
-800
-700
-600
-500
-400
-300
-200
in thousands of yearsit=0 at presentl
130.5
-131.5
I
S-132.5
I.'
-133.5
U3-136.5
-1000
-900
-800
-700
-600
-500
-400
-300
in thousands of yearsit=0 at present)
-200
-100
0
Figure 3-4: Time series of (a) the displacement of Tm., with v = 0; (b) the displacement of Tm,, with v = .472;(c) the curvature at Tm., for the period of 1000kyr B.P.
to 100kyr A.P.
10
40 .
25
*ata
trm nemencat reum.
-
1981
30
25
20 t
15t
10
0
0 -
6
2
4
0
8
latituce of Temp_ max(degree)
-2
10
Figure 3-5: Numerical calculation for the maximum streamfunction of the Hadley
winter cell vs. the displacement of maximum temperature during northern hemisphere
winter[99].
5.6 r
5.2
4.8
.4
2b
V
Z: .2.4
2.3
-100 0
-q0a
-800
-700
-boo
In
-500
thousanas of
Figure 3-6: The time series of normalized
Equation 3.8.
A
-- 00
yeara
1m
tO0
at
-220
-200
-100
a
prusentl
only due to the change of
#o using
110e
6.0
5.5
-4.0
1~2.5E
-10 0
-;00
-300
-700
-500
-600
In thousands of years
-300
-400
at present
it=0
i
-200
-100
100
0
Figure 3-7: Time series of parameterized maximum Hadley winter heat flux according
to Equation 3.10 for the period of 1000kyr B.P. to 100kyr A.P..
3.4.2
Parameterization of the intensity of the maximum
Hadley winter heat flux
One simple way to consider the dependence of Fma(the maximum Hadley winter
heat flux) on the displacement of the maximum heating is to just take Equation (.8)
as in Appendix:
Fma.
cC(-C)5 1 2 f
(40).
(3.9)
In Figure 3-7 we show the time series of so parameterized maximum Hadley winter
heat flux from 1000kyr B.P. to 100kyr A.P..
It is true that in order to take into account more accurately the change of Hadley
intensity due to the displacement and concentration of the tropical heating, we need
some kind of model which can model the secular change of tropical deep convection
pattern. As pointed out by Lindzen and Nigam(42] the tropical rainfall distributions
are tightly related to mixed moist boundary layer and are essential to the calculation
of the circulation aloft. However the secular variations of the tropical rainfall pattern
is hard to resolve in our simple model; and we assume that for the most part it is
linearly related to the radiative equilibrium temperature distribution.
Chapter 4
A new nonlinear rectifier
4.1
Equilibrium models vs. differential models
In the literature of paleoclimate modeling, there exist two conceptually different
groups: equilibrium models and differential models. Equilibrium models take the
form of y = f(x), where a system function (f) relates the equilibrium climatic state
(y) with the orbital boundary condition (x). In the simplest forms, the system function is used such as the input being a linear combination of insolation curves. This
approach has been widely used by geologists searching for correlations between geological records and Milankovitch orbital forcing. In more complex models, the system
function is produced from a set of differential equations which is assumed to control
the fast physics of the response process. The equilibrium climatic state is obtained
by integrating the equations to equilibrium at fixed values of x[79][71]. However, as
would be expected with an equilibrium model, the geological record of climate lags
significantly behind the model's output.
Differential models take the form of dy/dt = f(x, y), where y is usually used to
measure the ice-sheet response. One example of this approach is presented by Imbrie
and Imbrie[31], where the input x is parameterized as
X = 6 + ae sin(w -
4)
where a and
#
are adjustable parameters and e, e, and w are the orbital elements
previously defined. The parameter
4 controls
the phase of the precession effect, and
a controls the ratio of precession and obliquity effects. More examples can be found
in Chapter 1.2.3. It should be pointed out that most of the previous work assumes
that the ice sheet fluctuation in the high latitudes is determined by the local summer
insolation alone, a consistent consideration with Milankovitch's original hypothesis.
However, as we discussed in Chapter 2.2.3, the high-latitude snow cover is determined
by both accumulation during the cold season and melting during the warm season.
While it is safe to say that the melting of snow fall depends strongly on the summer
insolation, the acccumulation in the winter season depends strongly on the dynamics
and moisture supply. It is expected that too much or too little heat flux both reduce
the accumulation of snow fall in the high latitude winter, because they tend to favor
a warmer winter or a dryer winter. An intermediate optimal heat flux maximizes the
accumulation of snow cover. Based on such consideration, we are going to offer one
simple rectifier in the next section.
4.2
Constructing a new rectifier
As shown in [31], it is necessary to introduce some form of nonlinearity in order to
produce the 100kyr period in the model response. However, the nature of why we
need such nonlinearities is not clear, so far as we know. There exist several forms
of nonlinearity in the literature. The most popular used one is that associated with
different time scales for ice sheet growth and decay. This nonlinearity is supported by
both the theoretical arguments, suggesting that growth times of land-based ice sheets
are considerably longer than shrinkage times[85], and the isotopic curves suggesting
that major glaciations terminate faster than they begin. However, in order to produce
enough 100kyr power in the output, the relaxation time scales for the model are set
to be 42.5kyr for the accumulation and 10.6kyr for the melting[31), which are much
longer than the theoretical values[84].
We argue that the nature of nonlinearity in producing the lower frequency is
A
204
0,02-j
Years ago (x 10)
Figure 4-1: Eccentricity and global ice volume over the past 730,000 years. (A)
Variations in orbital eccentricity calculated by Berger[3]. (B) Oxygen isotope curve
for deep-sea core V28-238 from the Pacific Ocean.
rectification, analogous to that used in electronics, because the radiation forcing is
mathematically modulated by the eccentricity and also because of the inverse correlation between eccentricity and the global ice volume[30]. Figure 4-1[31] shows the
time series of eccentricity and global ice volume over the past 730,000 years. It is
clear that the timing of major deglaciations coincides with the timing of peak eccentricity. Although as we mentioned in Chapter 1.2, the direct insolation forcing in
the eccentricity period is too weak to explain the dominant 100kyr period in the late
Pleistocene proxy data, rectification can channel the energy into the 100kyr period
band. The question is what is the proper rectifier in the nature. A proper rectifier
must not only be physically reasonable but also effective in extracting the lower frequency information from the input. As we already discussed in the previous section, a
good candidate is the nonlinear interaction between the dynamical transport and ice
sheet accumulation and decay. We argue that only optimal winter heat flux ensures
the accumulation of snow cover, while too much of that will cause well above normal
warmer winter and therefore deteriorate the accumualtion of snow fall, and too little
of that will cause drier winter and therefore cut the source of snow fall. The wide
variation in
im
associated with eccentricity suggests that tile extremes are relevant.
Therefore the rectifier based on the above consideration is the following:
FR[F] = exp -(
Fo
C
)2
(4.1)
where R is called "glacial potential", F the maximum Hadley winter heat flux as
determined by Equation 3.9, Fo the optimal F corresponding to the maximum of R,
and c the e-folding half-width. There are two control parameters in Equation 4.1: Fo
and c. Here the larger R is associated with maximum accumulation.
In Figure 4-2-Figure 4-6 we show the resulting time series of R[F(t)] and their
corresponding power spectral density for different choices of control parameters.
Although the input of F as shown in Figure 3-7 only has the powers in the obliquity
and precession bands, the glacial potential R[F(t)] displays different power spectral
density patterns, varying from lacking of lower frequencies at 100kyr/400kyr periods
to lacking of the frequencies in the entire precession bands based on different choice
of free parameters of FO and c. Moreover, Fo is more effective to change the power
spectrum of the input than c.
4.3
Rectifying the lower frequencies of orbital
forcing
To illustrate the effectiveness of this rectifier in detecting the lower frequencies of
orbital forcing, we take a simple differential model and use the glacial potential as
the model input. The model equation is:
dt
+ y/a = R[F(t)]
(4.2)
where a is the relaxation time scale for the integration and t the time in kiloyear.
In Figure 4-7 we show the time series for R and the corresponding power spectral
density with FO = 3.442 for four choices of c. In Figure 4-8 the time series of y and the
corresponding power spectral density are shown with a = 17.kyr. In Figure 4-9 we
POWER OF R(FI FO-2.00.C-2.00
R[F] F0-2.00.C-2.00
1.0
CD
CD
.2
.7
.6
CD
D
D
CD
C
C
D
D
D
D
C
C
C
10
LO
D
D
C
CD
C
.9'
C
C
.
N
~
C
C
C
C
CD
C
D
~
D
D
.0
r..
D
CD
D
0
CD
CD
CD
CD
C
N
Cl)
'
N
CD
I~
CD
~
CD
CD
CD
C
.1
0
CD 71
Cn
r,
x
RIF) FO-2.00.C-1.
POWER OF R[F] F0-2.00.C-1.50
1 .0
.9
.8
.7
.6
>- .5
.4
.3
.2
1.0
.9
.8
.7
.6
.5
.4
.3
.2
JW
0
CD
N
.D
x
RIF] F0-2.00.C=1.00
1.0
.9
.8
.7
.6
-
.4
.3
.2
.1
.D
MD
IT
q
.D
C
MD
CI
.D
MD
10
.
0D
.
CD
CD
.
MD
a,
.
CD
M
.
-
POWER OF RIF] FO-2.00.C=1.00
1 .0
.9
.8
.7
.6
.5
.4
.3
.2
.5
MD
CD
CD
-
x
R(F] FO=2.00,C=0.50
IS!
19
G!.
POWER OF R[F] F0=2.00.C=0.50
1.0
.9
.8
.7
.6
1.0
.9
.8
.7
.6
>-.5
.4
.3
.2
.4
.3
.2
0.
CD
CD
19
CD
CD CD
CD
CD
CD
-
A
A
--.
0
Lfl
M
N
CD
a*
N
Figure 4-2: The time series of glacial potential(left column) and their corresponding
rower spectral densitv(right column) with F = 2.0 for four choices of c. The units
for the horizontal coordinates are kiloyears in time for the left column diagrams and
1/kiloyears in frequency for the right column diagrams.
(~
N)
U e ue
w ,w
. . . . . . . . . .. . . . . . . . .
-.
m
-
6
Le
u.
4 w
o
yI
.080
.090
.100
.080
.090
.100
.120
.130
.060
x .070
.080
.090
.100
.110
.120
.130
.140
x .070
.080
.090
.100
.110
.120
.130
.140
.140
.110
x .070
.060
.140
.130
.120
.110
x .070
.060
.050
.060
.050
.050
.040
.030
.020
.010
0
.050
. . . . . . . .. .. ... . . . . .
100-
0
C."-
.040
.030
.020
.010
0
100 1 . . . . .
0
-100
-200
-200
-100
-300
-400
-500
-300
-400
T
!x
.040
.040
N C.
. . . . . . . . . . . . . . . . . ..
C.)
-6
N)
-500
I'd
.030
.030
.010
0
.020
*
.020
.010
0
**D
CD
En
0
-100
-200
-200
-100
-300
-3001
-500
-400
nx
N
-400
-500
Y
u'
ox
'1
-600
'1
-600
-1000
-600
wD*,
-600
-4
-700
61
-700
i2
CD
H
04Q
. in
-700
-
k ) Cii
.
-700
.
-800
.0
*a
-800
*4
-J
-800
Ul
6.
-800
.-
-900
9
-900
ta
-900
D *0
-900
-
-1000
P.- 6U 0
Y
-1000
Nuia
Y
-1000
ea -
Y
N)
. . .....
m
*M
w.i
ut oL
-
J
o
6U
%.
0 ts
..... .. ....-
*
~-
N)'
i 61
Y
at
9
w
.030
.040
.050
.060
x .070
.080
.090
.100
.110
.120
.130
.140
.030
.040
.050
.060
.070
.080
.090
.100
.110
.120
.130
.140
.030
.040
.050
.060
x .070
.080
.090
.100
.110
.120
.130
.140
.939
.049
.050
.060
x .070
.089
.090
.100
.110
.120
.130
.140
0 6
.020
.J
.020
x
*) Ul 0
6
...................
.020
0
020
. .. .. .. ... ... .. ...
.010
o
.010
wa m
.010
Uo U' -.
. .... ........
..
NJ U
.010
*.--
y
0
0
By
-100
-100
-100
-100
00
-200
-200
-200
-200
100
-300
-300
-300
-300
100--
-400
-400
-400
-400.
1009
-500
-500
-500
-599
o
-600
'Oe
-600
-o
-600
l 6i
-600
*P.
-700
k
-700
'1--
-
-700
m
6
-700
o~
-800
U1
-800
a.
-800
N) C)
-899
-.
-900
*
-900
oW e
-900
-e
-900
UaU
-1000
N) Um
-1000
-~
-1000
*
-1000
iU
............
-1909
-
:0
-----------
N
c-i
In
C,,
IL
IL.
IA.
0
IAJ
U
0
ci-
*
('a
(-3
U
U'
C,
U
IL
IL.
U
U
w
U
a-
A
0 ~ a- .c m
0'
~-
U
-~
C,~ N
~o u~
-
OVL*
C ......
N -
otr L
Oct'
w r- .0 aLnIt
CM
m
Oc
V
a
060'
00 L
090
060
OOL'
OOL
060 *
L
060'
I
ta
080'
010'
OL0
C,
I
Ote.
SES
oit.
090.
SOe'x
i
090'
090*
100C
loe'
cc
SE:0
Oee.
OWl
oire.
Irs.
OLO*
I
. '
N .-.
let-
leg-
156-
OIL-
0~I
0 to
0
UIL
0
I
a.
SOL-
.
00 L-
.
eez-
.
Oez-
-U
U:
.'
OeeEs
.
USE:-
C
090.
090.
L,
ezt*
.. 0Il
. .
U 0-
o
.
.
OZL*
.
A
I. -.
.,
OWV
*
0
OWL
aa
u'*
C-
C
a
IL
Ca
-.
OLL1
OOL*
080'
090.
L
OE:9
0
006-
009-
U 0
0e9leg -U IL
O0L-
06-
009-
006-
BoI-
008-
0001c')
PLO
bU
14
R[FI F0-4.00.C-2.00
POWER OF R[F] FO-4.00.C-2.00
1 .0
.9
.8
.7
.6
.95
.85
.75
- .65
.45
.4
.3
.2
.35
0
.55
ao
M-
-
A'
a
X
- - .l
.
.
a
.).
ato
NC
')
MW
aao
a-
ao
a
~oa
00r
a
-
A
a
a
a
a
awa
a
.0
C.~
a
'
M
a
eN
-
v.
C'
POWER OF R[F] F0=4.00.C=1.00
RIF] F|-4.00.C-1.00
1.0
.9
.8
.7
.6
a
POWER OF RIF] F0=4.00.C=1.50
ta
.
.
In
q
a
R[F] F0=4.00.C=0.50
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
-
Nl
a
.5
.4
.3
.2
.1
0
i
a
a,
a
CD
a
r-
a
10
to
a
a
a
m'(2
a
aa
Nm
V)
Z'
In
r-
0
Co
m~a
N
x
RIF1 FO=4.00.C=0.50
1.0
.9
.8
.7
.6
POWER OF R[F1 F0=4.00.C=0.50
a
.9
.8.7 .6 .5 -
.4
.3
.2
.1
~'
r-
IA
'
.4
.3
C'-
. aaa
'
.1
a
i
am
a
gaA
a
a
a
.
a
IA
a
.
a
~ a~
a
11
Figure 4-6: Same as in Figure 4.2 except with Fo = 4.0.
46
a'
a.
a
N
C)
'
NC'
plot the time series of both eccentricity and y and their corresponding power spectral
density. For the convenience of comparison, we normalized the time series of both
eccentricity and y. It is clear that the rectifier effectively detects the lower frequencies
in the eccentricity bands.
4.4
Modeling the global ice volume fluctuations
In this section we assume that the global ice volume in the late Pleistocene is determined by both the rectified Hadley winter heat flux and northern hemisphere
high-latitude summer insolation. The governing equation for the global ice volume G
is:
dG
-=
(c1 * R[F(t), Fo, c] -
C2 *
1(t)
-
G)/a
(4.3)
where 1(t) is the averaged summer insolation (JUNE-AUGUST) from 60*N to the
North Pole at time t(in kiloyear B.P.). ci and c2 are two parameters controlling the
relative importance of R and I.
In Figure 4-10 we show the time series for the averaged summer insolation from
60*N to the North Pole and its scaled power spectral density. Figure 4-11 shows the
time series of modeling result G and 60" of SPECMAP for the last 700kyr. Figure
4-12 shows their corresponding power spectral density. The data are normalized such
that their standard deviations are unity for the convenience of comparison.
Despite the simplicity of our model, the result is promising in terms of both the
secular pattern and the power spectral density. We didn't make fine tuning of the
parameters because at this stage our goal is to illustrate the conceptual feasibility
of our approach rather than to exactly match the result with the data. Indeed our
simple model lacks a lot of potentially important processes which may determine the
detail response of the ice sheet. To name only a few of them, the process which
translates both the Hadley winter heat flux and the high-latitude summer insolation
into the budget of snow accumulation and ablation and the feedbacks within the
climate system are potentially important in our further modeling effort.
F0-3.44.C-1.50.C1-
1.0
.9
.8
.7
.6
.5
.4
.3
.2
1.0.C2-
0.0
1 .0
.9
.8
.7
.6
.5
.4
.3
.2
0
-
C14
m
in
.
10
N
. . . . . . . . .. . . . . . . . . . .
x
FO=3.44,C=1.20,C1= 1.0,C2-
1.0
.9
.8
.7
.6
>-
~N
OC.
03
.
.
.
.
.
.
l
.
0.0
1 .0
.9
.8
.7
.6
.5
.5
>-
.4
.3
.2
.4
.3
.2
.1
.1
0
0-
CO
r-
N
.0
a
10
M
m
CA
&'
F-
M
N
4.
C-0.9
.C10
1..C2
0.
0
wX
x
FO-3.44.C-0.90.Cl-
1.0
.9
.8
.7
.6
1.0.C2-
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
.5
.4
.3
.2
0
aC'
0
I
0
in
I
Cl
M
I
1a
I
IM
7
.
. .. .
.
.
.
....
tod
.
.
Ml
.
0'
0.0
mn
.... .
-
ts
.
. .
.
C
C
.
...
1 .0
.9
.9
.7
.6
.7
.6
.5
.4
.3
.2
1
0g
.4
.3 .2
1
0f
- '3~I
Figure
FO
as
4-7:
=
in
3.442
Figure
The
time
four
for
4-2.
series
choices
of
of
R
c.
and
The
their
units
corresponding
for
the
power
horizontal
inCl
0
spectral
coordinates
N
~
0
~
density
are
-
with
the
same
N
Ml
-W
FO-3.44.C-1.50.C1-
1.0.C2-
0.0.a-17.0
FO-3.44.C-1.50.C1-
1.0
1.0
.4
.3
.4
.3
.2
1.0.C2-
0.0.a-17.0
.2
.11
0
.1
-
0
~~~~
(N
.0 m
U~
F0.=3.44,C1 .20.Cl=
x
1 .0,C2=. 0.0.&=17.0
F0=3.44.C=1.20.C1=
1 .0
1 .0
.11
Al
,NAr
.4
.
~
N
M
M
M
M
M
N
'
10.C2= 0.0.a=17.0
.6
.4
.3
.2
.3
.2
.1
0
a,
g
F
1.0
3
4-
.0
.
X
c1.
,
20
-
W0
01 0
U
x
1.0
.9
.8
.4
.3
.2
.5
.4
U
.2
U
.1
0
U U
U1
O2
U
U
P0-3.44,C=0.60,C1=
.1
LO
.0
x
.1
.1
.2
VN
FO-3.44.C-0.90,CI- 1.0,C2- 0.0.a..7.0
.2 .3
U
-M
N. 7
k
U
x
1 .0.C2=
/\
t
I A/~
U
7
U
0.0.a=17.0
F0-3.44.C=0.60.C1=
. .
1 0 C2= 0 0 &=17 0
.
\w16d~~
.1
U
0
U.0
N
U
0
0
U
~
N
~
,
W
c'
.-
x
(
(2
.
Z-
.
r
.
I
0
.
.
.
.
.
x
Figure 4-8: The time series of y and their corresponding power spectral density with
1o = 3.442, a = 17.kyr for jour choices of c. The units for the norizontal coordinates
are the same as in Figure 4-2.
49
N.
.
3
3M
a
M
3
3
V1~.
3
3
c
inrCi
o
M
3
Ca
3
M
3
0'
31
M
-
Mi
NN
J-
Ln
r,
00.
x
x
Figure 4-9: The time series of normalized eccentricity and its power spectral density (a); and normalized y and its corresponding power spectral density with F =
3.442, c = 0.5, a = 17.kyr (b). The units for the horizontal coordinates are the same
as in Figure 4-2.
60N-90N AVE. SUM. INSOL.
430
SCALED POWER SPEC. DENSITY
._..1.0
_t
-9
420
.8
-
410
400
370
360
370
.2
.3
MN
r-In
CiM0 M
M'
ta
M
CI
M
C
M
M
qr
x
x
o
a,
M
3
M
10
M
M
GN
C1'L
-
M
0M
)
NI)
x
x
Figure 4-10: Time series of the averaged summer insolation from 60*N to the North
Pole for the period of 1000kvr B.P. to 100kvr A.P. and its associated oower sDectral
uensiy. Lie units ior
ine norzonai cooroinates aZe
Lie same as in z igure q-2.
+2.5
+1.3
-2. 5
-700
I
\1
0.00
-800
-400
-500
-
-200
-300
-100
0
time in kiloyear(t=O at present)
Figure 4-11: Time series of the normalized modeling result (dash line) and 6018
of SPECMAP (solid line) for the past 700,000 years with a = 17kyr, ci = 1.0,
c2 = 0.012, Fo = 3.442.
1.00
0.75
w
"4V 0.25-
0.00
I l
-
L -1
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
frequency(1/ky)
Figure 4-12: Power spectral density of the modeling result (dash line) and the
SPECMAP record (solid line) for the past 700,000 years.
Chapter 5
Summary and implications
A study of the ice age problem has been presented in this thesis in light of the recent
knowledge of both geological evidence on climate spanning the late Pleistocene and
the intensity of Hadley circulation in modulating the wave transport and dynamic
heating at the mid- and high-latitude.
Based on the seasonal energy balance model and the Hadley circulation theory,
we find that there is considerablly large variation in the tropical forcing in terms of
the intensity of the Hadley winter heat flux, which may determine the mid-latitude
wave activity and high-latitude temperature. A new rectifier based on the dependence
of high-latitude snow accumulation and ablation on the Hadley winter heat flux is
tested in detecting the 100kyr period from the orbital forcing and in simulating the
global ice volume fluctuation for the entire late Pleistocene. The modeling result is
compared with the SPECMAP 80
18
data in both time and frequency domains.
The implication of this study is that the tropical forcing may be very important in
modeling the late Pleistocene ice volume fluctuation. And the final success might lie
in the appropriate treatment of translating both high latitude summer insolation and
the winter heat flux induced by the tropical Hadley circulation into high-latitude snow
accumulation and ablation. In that case a more sophiscated model which includes
the budget of high-latitude snow cover is a must.
Appendix
Here we consider
do = 0, and the radiative equilibrium temperature
T(y) = To(1 + A/3
distribution is
(5.1)
Ay 2 )
-
where To is the reference temperature, y = sin(latitude), A = --
(d)
|y=o=
(C is the curvature at Tmaz).
The mean temperature within the Hadley domain is
T ~ T(y)dy/(2yH).
(5.2)
-9H
The heat flux F satisfies
T(y) - T
F(y = ±YH)
=
0.
(5.3)
(5.4)
So we have the latitudinal distribution of F(y)
F(y) =
ATO (y3 - y
(5.5)
where A > 0 and thus F(y) > 0(0 < y < YH).
Also ATO = -C/2 so Equation 5.5 becomes
F(y) =
67
2
HY
)
(5(ya
(5.6)
y.
The Fma, occurs when "F(y) = 0 i.e.y-=
Fma, oC (-C)5
where noting in the symmetric case YH oC A/
2
So the resulting Fma, is
/2
(5.7)
oc (-C) 1 / 2 [25].
Combining the above formula with f(#o)(normalized empirical relation between
?pma,
and
do,
cf. Equation 3.8 in Chapter 3.4.1, we have the highly empirical formula
Fmax(#o, C)
oc
(-C)5 / 2 f(o0 ).
(5.8)
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