AN OCEAN-ATMOSPHERE ENERGY CLIMATE MODEL
by
LONG SANG CHIU
B.S.,University of Miami
(1974)
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF SCIENCE
at the
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
August, 1980
Massachuetts Institute of Technology
Signature of Author....
..............
Department of meteorology and Phy sical
Certified by........
...........
Accepted by.........
Oceanography
August, 1980
........
Reginald E. Newell
Thesis Supervisor
....................
Chairman,Departmental
OF 4NO
MO,
LIB
Y
AR:)
Peter H. Stone
raduate Committee
AN OCEAN-ATMOSPHERE ENERGY CLIMATE MODEL
by
LONG SANG CHIU
Submitted to the Department of
Meteorology and Physical Oceanography
on August,1980 in partial fulfillment of the
requirements for the degree of Doctor of Science in
Meteorology and Physical Oceanography
Abstract
oceanic
and techniques for estimating
The mechanisms of
heat transport are reviewed. The global pattern of heat transport
derived from direct estimates showed a northward transport in the
a southward transport in the Pacific. This pattern
and
Atlantic
abyssal
the
of
Arons' model
and
is consistent with Stommel
the mean
that
showed
estimates
direct
The
circulation.
the
global
contributor in
meridional circulation is the main
heat
transport.
oceanic
oceanic mean
Oceanic heat transport associated with the
a simple six-box
in
is
parameterized
circulation
meridional
and Budyko
Sellers
annual average energy model similar to the
the
in
layers
three
of
consists
model
The
type.
the
vertical,representing an atmospheric layer and two layers in
surface layer and the deep ocean,and two
ocean,representing the
high latitudes.
low and
blocks in the horizontal,representing
calculated using empirical relations from
are
fluxes
Radiative
include
heat
fluxes
climatology. Atmospheric
day
present
flux
and latent heat flux which depends on the
eddy
baroclinic
mean temperature of the atmosphere. The formation of bottom water
circulation,is
a mean oceanic
at high latitudes,which forces
to the surface cooling of the high latitude oceans.
proportional
by
Temperature-albedo feedback(TAF) is incorporated in the model
the atmospheric
of
terms
in
ice-line
the
parameterizing
temperature at high latitudes.
in
different
variations
The sensitivity of the model to
parameters is examined. To better understand i) the TAF,ii)latent
iii)
effect of diffusive oceanic heat
heat
flux feedback and
with the oceanic mean
associated
transport and iv) heat transport
on
models based
energy models,six
meridional circulation on
of the full model is examined. It is found
simplified versions
the system.
that TAF is the most important positive feedback in
heat flux feedback enhances the TAF although the effect
Latent
-
2 -
is small. The inclusion of linear diffusion in the ocean in the
two layer models does not change significantly the sensitivity of
the model to variations in the solar constant. If the heat
transport associated with the mean meridional circulation is
included (three layer models),such sensitivity is reduced. Hence
it is suggested that the transport by the mean meridional
circulation is an important dynamical feedback in the climate
system.
different
models
to
small
The adjustment of the
perturbations about the equilibrium state is examined. Adjustment
time scales in the atmospheric model are 100 and 15 days. They
correspond to the radiative relaxation and energy redistribution
time scales. For the models which include an oceanic layer with
diffusive heat transport,the adjustment times are increased to
~20 and 3 years,due primarily to the increase in the heat
capacity of the system. Two new
modes,which
characterize
atmospheric temperature changes are found. They have relatively
short time scales,about 3 and 5 days,and are dependent mainly on
the efficiency of air-sea heat exchange.
The response of the full model to i) variations in solar
constant with TAF and ii) to variations of the ice-line with no
change in the solar constant is examined. It is found that the
response of the model to both variations are qualitatively
similar. This result is relevant to the climatic theories of the
ice-ages.
The assumption about the rate of formation of bottom water
in the model limits its ability to forecast temperature changes
in the deep ocean during an ice age. The physics which controls
the change is put into better perspective.
Thesis Supervisor: Dr. Reginald E. Newell
Title: Professor of Meteorology
Acknowledgements
I am pleased to acknowledge those who have contributed through the
course of education.
Special thanks are due to Professor Reginald Newell, my thesis
supervisor, who first suggested the topic and has provided guidance and
stimulating ideas as the work progressed.
Thanks are also due to Professor
Peter Stone for his continued interest and helpful suggestions.
Careful
reviews and critical comments on the first draft of this thesis by him and
Professor Ray Pierrehumbert are gratefully acknowledged.
I have also
benefitted from discussions with Professors Henry Stommel, Erik MolloChristensen and Eugene Rivas.
I enjoyed the friendship and many discussions with many members of
the Meteorology department, in particular Charles Lin, Lin Ho, Ron Errico,
Jim Fullmer and Alfredo Navato.
Susan Ary, Kathy Huber and Lynn Egan
have provided continual moral and technical support.
I thank my undergraduate advisors Professors G. Alexandrakis and
H. Gordon who provided guidance and care and have helped
lay a good
foundation for my graduate studies.
I thank members of my family, grand parents, uncles and aunts,
who from thousands of miles away, have given me continued moral
support.
I am especially thankful to my father, who after the death of my mother,
has unyieldingly taken up the dual role of both parents.
To him this thesis
is humbly dedicated.
Financial support from the National Science Foundation during my
stay at M.I.T. is gratefully acknowledged.
- 4 -
TABLE OF CONTENTS
page
ABSTRACT
2
ACKNOWLEDGEMENT
4
TABLE OF CONTENTS
5
LIST OF TABLES
7
LIST OF FIGURES
8
1. INTRODUCTION
9
2. ESTIMATES OF OCEAN HEAT FLUX
12
2.1 Introduction
12
2.2 Residual Method
13
2.3 Surface balance 1Method
15
2.4 Mechanisms of Ocean Heat Transport
17
2.5 Direct Estimates
19
2.6 Heat transport associated with the net meridional flow
21
2.7 Transport of fresh water
24
2.8 Discussion
26
3 THE MODEL
29
3.1 General Description
29
3.2 Governing Equations
31
3.3 Flux Parameterization
36
3.3.1 Short Wave
36
3.3.2 Long Wave
39
3.3.3 Dynamical Fluxes
41
3.3.4 Bottom Water Formation
45
3.4 The Choice of Ky, Kz
50
3.5 Comparison with Observation
58
3.6 Energy Balance at Different Levels
60
4. LINEAR ADJUSTMENT AND SENSITIVITY
62
4.1 Methodology
62
4.2 Temperature-Albedo and Latent Heat Flux Feedback
65
4.3 One Layer Models
67
4.4 Two and Three Layer Models
75
4.5 Parametric Dependence of Adjustment Time Scales
80
4.6 Comparison of Models
84
5. DISCUSSION
89
5.1 Changes in Climate and Solar Constant
89
5.2 ' Simulating' the Ice-Ages
92
5.3 The Deep Ocean Response
95
6. SUMMARY AND CONCLUSION
99
APPENDICES
103
TABLES
117
FIGURES
143
REFERENCES
152
-6-
List of Tables
Table 2.1: Summary of results of direct calculation of ocean heat transport
Table 2.2: Precipitation,Evaporation,Runoff and Transport of Fresh water
Table 3.1: Latitudinal distribution of short wave components.
Table 3.2: Short wave parameters used in the model.
Table 3.3: Solutions for selected values of K and K z
y
Table 3.4: Comparison of model result with observation.
Table 4.1: List of models.
Table 4.2: Eigenmodes of one layer model
Table 4.3a,b:Eigenvalues,eigenmodes and sensitivity of model III.
Table 4.4a,b:Eigenvalues,eigenmodes and sensitivity of model IV.
Table 4.5a,b:Eigenvalues,eigenmodes and sensitivity of model V.
Table 4.6a,b:Eigenvalues,eigenmodes and sensitivity of model VI.
Table 4.7a,b:Eigenvalues,eigenmodes and sensitivity of model VII.
Table 4.8a,b:Eigenvalues,eigenmodes and sensitivity of model VIII.
Table 4.9: Sensitivity of eigenvalues for model VI.
Table 4.10:Sensitivity (
)of models I-VIII to parameter variations.
Table 4.11: Ice line and Global sensitivity of models I-VIII.
Table 5.1:
Sensitivity of model VIII to solar constant variation.
Table 5.2:
Sensitivity of model VIII to ice-line variation.
-
7 -
List of Figures:
Figure 2.1: Schmatic showing the components of fresh water balance
for a volume of sea water.
Figure 2.2: Oceanic transport of fresh water and atmospheric moisture
transport.
Figure 2.3: Schematic showing the direction of ocean heat transport.
Figure 3.1:
Schematic of the Model showing the fluxes of energy.
Figure 3.2:
Graph of outgoing long wave radiation verses surface temperature.
Figure 3.3:
Seasonal variations of the Ratio of latent to sensible
heat flux.
Figure 3.4: A north-south section of temperature and density in the Atlantic.
Figure 3.5: Plot of (Ky,K z ) pairs which satisfy Ro=0.3 and T4-T 6=0
Figure 4.1:
Sea level temperature at edge of sea ice.
-
8 -
NOTATIONS
2. el5W/m**2= 2.0x1015W/m 2
50*N= 50 0N
5*C= 5 0 C
B
* T= BX T
1.
Introduction
There now exists a number of simple climate
have
been
used
to
system to
variations
Dickinson
(1974)
which
which
examine the response of the Earth's climate
of
have
external
reviewed
conditions.
the
Pioneering works in this area have
models
models
Schneider
subject
been
and
comprehensively.
Sellers'
and
Budyko's
predict a 1.6-2% decrease in the solar constant is
sufficient to lead to an ice covered earth.
In their models, the
amount of solar radiation entering the earth-atmosphere system is
determined by the albedo at the surface,
i.
e.
ice-covered areas
reflect more solar radiation than ice-free areas. The
terrestrial
radiation
linear function
effective
of
emitted
the
emitting
layer
to
surface
must
space
air
be
amount
is parameterized as a
temperature.
situated
Since
higher
the
up in the
atmosphere, the underlying assumption is that the lapse
constant.
of
rate
is
Differential heating between low and high latitudes is
offset by dynamical fluxes. The ice-line is determined internally
as the latitude where
certain value.
the
surface
temperature
drops
below
With this set up , ice-albedo temperature
a
feedback
is incorporated.
The results of these simple
generated
a
lot
of
interest.
energy
climate
Faegre(1972)
using
formulation, with an empirically derived relation
temperature,
present
models
of
a
have
similar
albedo
on
found numerically five steady state solutions to the
condition of insolation.
Spectral solution(expansion in
terms of the Legendre polynomials) showed that there exists three
solutions to the model for the present condition
- 9 -
of
insolation.
The
first
two
solutions, corresponding to
and an ice covered earth,
are
the third solution,
while
stable
having its
to
the present climate
small
perturbations,
ice-line situated somewhere
in between the present and a total ice cover, is unstable to such
perturbations.
The sensitivity of the amount of ice cover
in
the
insolation
is
to
dependent on the parameterization of the
dynamical fluxes. In Sellers' and Budyko's models, the
flux
is
assumed
to
be
a
linear
function
dynamical
of the meridional
temperature gradient. From baroclinic theory, it was
the
variations
shown
that
dependence is quadratic rather than linear(Green 1970, Stone
1972). The quadratic dependence tends to decrease the sensitivity
(Stone
1973).
sensitivity
Lindzen
is
and
Farrell(1977)
showed
that
this
determined by the latitude range over which heat
transport is acting to smooth out temperature difference and
effectiveness
with
which
this
smoothing occurs. When the heat
flux associated with the Hadley circulation
effect
is
to
decrease
Hadley adjustment
the
the
sensitivity
is
introduced,
its
. The validity of the
parameterization has been examined
by
Warren
and Schneider(1979).
A large portion of the meridional heat flux is carried
the
by
ocean currents. Early estimates showed that the oceans carry
about 20% of the total flux(Sverdrup 1957). More recent estimates
show that the oceanic transport is about 50% of the total at
latitudes(Oort and Vonder Haar 1976).
of
low
While the parameterization
the atmospheric heat flux has grown to such sophistication as
to include baroclinic
eddy,
-
latent
10 -
heat
flux
and
heat
flux
associated
with
the Hadley circulation, the parameterization of
oceanic flux has remained one of Fickian diffusion in most energy
models(e. g.
examine
the
Sellers 1973).
various
The theme
of
this
thesis
is
to
heat transport mechanisms in the ocean and
try to parameterize, in a simple way,
the important oceanic heat
transport processes so that the effect of oceanic heat
transport
on the sensitivity of simple energy models can be assessed.
- 11 -
2.Estimates of ocean heat flux
2.1 Introduction
Because of the sphericity of
incoming
is
the
in
the
atmosphere
and
Much of this radiation
at
the
In
energy
atmosphere
1971,
Oort
oceanic variables suffer
coverage;
with
therefore
particular time.
resort
to
transport
and
from
terms
by
have
been
Rasmusson 1971). Measurements of
lack
of
continuity
and
global
direct calculation of oceanic heat flux can
only be made for a particular oceanographic section
to
a
and the oceans being the dominant modes (Sellers
1965). Statistics of the atmospheric transport
computed(Oort
maintain
must be transported through the envelope of the
earth -atmosphere system to higher latitudes
the
wave
the annual mean, there is a net radiative surplus
at low latitudes and a deficit at high latitudes. To
balance,
is
Earth's surface. The
absorbed radiation is returned to space in the form of long
radiation.
of
amount
solar radiation incident at low latitudes per unit area
larger than that at high latitudes.
absorbed
the
earth,
taken
at
a
Consideration of the global ocean heat flux has
other
means.
In
estimates of oceanic heat flux from
the following, we shall review
1)residue
calculations,
surface energy balance and 3) direct calculations.
-
12 -
2)
2.2 Residual method
From the observed or calculated distribution
of
the
net
radiative flux at the top of the atmosphere, the total meridional
flux
required
for balance is obtained.
By subtracting the heat
flux of the atmosphere, the oceanic heat flux is
residue.
Using
the
observed
computed
in
the
2.1e15 W at ~30*N,
the
Northern
compared
atmospheric flux.
Newell et al.
location
Using the net calculated radiative flux
heat
flux
using
the
observed
oceanic
flux
for
the
of
flux
total
show
the
net
flux,
such
found
calculated
required
calculations
for
a
the
can
be
large
balance is obtained as an
cal
cm**-2
min**-1(~7W
radiation budget, the probable error in the
flux required is ~1.2e15W at 20*N. The
interannual
also
Hemisphere. His results are in
variability
(Oort
combined will lead to uncertainties in
large
They
e15W at 30*S. Trenberth
integral. For a probable error of 0. 01
in
The
in the region of overlap.
Errors associated with
total
~2.4
radiative
Southern
agreement with Newell et al.
m**-2)
for
magnitude of the maximum agrees with those of Oort
southward
the
all
to a maximum of 2.1 e15W at 50*N
(1979),
since
at
Hemisphere, with a maximum of about
and Vonder Haar in the Northern Hemisphere.
maximum
flux
(1974) computed the oceanic heat flux to 30*S.
and
a
distribution of the net radiative
flux, Vonder Haar and Oort(1973) found a northward
latitudes
as
1977).
atmospheric
The
two
fluxes
factors
the oceanic heat flux
as
as the flux itself. While this method gives an estimate of
the magnitude of
the
ocean
-
heat
13 -
flux,
it
fails
to
provide
information
on
balance method,
the
in different oceans. The surface
transport
described next,
-
provides such a means.
14 -
2.3 Surface balance method
ocean
Sverdrup(1957) was the first to calculate the
flux
heat
using this method. Based on the net heat input at the ocean
surface, the flux divergence can
there
if
column
no
is
long
divergence can be integrated
of integration is
that
assumption
prescribed.
there
an
for
computed
oceanic
term change in temperature. The
to yield the heat flux if
a constant
data
Using Budyko's(1956)
no
the
of
northern boundary
is
be
and
the
oceanic heat flux at the extreme
calculated
Bryan(1962)
oceans,
the
heat flux in the north Pacific and Atlantic. It is found
oceanic
that there is a northward transport in the north Atlantic, with a
maximum of about 1.0e15W at 30*N, but a
transport
southward
in
the Pacific, with a maximum of 2.0e15W at the equator.
Largely in connection with
of
maps
Year,
surface
the
heat
have
been
oceanic
heat
components
Emig(1967) recalculated
updated(Budyko 1963).
Geophysical
the International
the
flux. The Atlantic was found to be carrying heat poleward in both
hemispheres.
and
Pacific
The
southward heat flux in
the
Indian ocean combined to yield a
northern
and
subtropics
southward
transports in the Southern Hemisphere.
More recent estimates by Hastenrath(1980)
heat
north,
transport
at
showed that
all latitudes in the Atlantic is
with a maximum of about 1.6e15W at 20*N
and
towards the
towards
south in the Indian ocean with a maximum of 0.7e15W at 15*S.
flux
in
the
Pacific
the
the
Heat
is directed away from the equator, with a
maximum northward flux of 1.1e15W at 30*N and
-
15 -
a
southward
heat
flux of 2.1
e15W at ~20*S.
The seasonal cycle of the surface
heat to the atmosphere, but there
gain
by
Clark 1967).
exported
budget
has
been
During most of the year, the area north of ~20*N loses
studied.
heat
heat
to
is
an
energy
surplus(i.
the oceans) during the Summer season(Bunker 1976,
This energy surplus
other
latitudes.
goes
Using
into
the
storage
or
The
is
found
in
Weak
cold
warm
is
dominated
by
is
dominated
by
advection suggesting that the heat transport in this region
is equatorward in the Summer.
computation of the storage,
in
(heat
advection, particularly in the Kuroshio region from
fall to Winter. But in the Summer, this region
cold
advection
the eastern part of the North Pacific
throughout the year. The western north Pacific
strong
north
difference, between the net surface heating and the
storage is attributed to advection.
divergence)
is
observed thermocline
structure, Clark(l. c. ) computed the storage term for the
Pacific.
e.
Because of the uncertainty in
Clark did not place
this particular result.
-
16 -
the
much confidence
2.4 Mechanisms of Ocean Heat Transport
From our present knowledge of the oceans,
the
following
mechanisms of heat transport may be operative.
1. Transport associated
through
an
with
the
net
meridional
volume
flow
oceanic section. This term is usually not considered
by assuming that there is no net meridional flow.
As
there
are
imbalances of precipitation, river runoff and evaporation between
latitudes, the net meridional flow is nonzero.
2.
Transport associated with the mean
abyssal
circulation,
with
a
meridional
poleward
cell
of
the
drift(if it is a direct
cell) of warm surface water and a deep return flow of cold water.
3. Transport associated with the horizontal gyres
basins
which
basically
the
ocean
transport water in a boundary current and a return
flow at a different temperature in the interior.
are
in
isothermal
below
1Km,
As
the
oceans
this
transport
must be
drifts
generated
by
the
Transport associated with eddy motions within the ocean.
The
restricted to the upper ocean.
4.
Transport associated with
action of
5.
heat
wind in
Ekman
the surface layer.
transport associated with the detachment of a cyclonic ring
of cold
water
0.05e15W(Newton
from
the
1961).
Gulf
Since
the large scale circulation
energy,
baroclinic
Stream
estimated
to
be
~
the available potential energy in
far
instability
-
is
17 -
exceeds
that
may
important
be
of
the
kinetic
in the heat
transport processes. Applying the theorem of
to
a
flat
bottom
Charney
and
ocean, Gill et al(1974) showed that the most
favored condition for baroclinic instability corresponds
with
a
Stern
to
one
westward surface current, and isopycnal surfaces.sloping
upward towards the equator. As this instability tends to suppress
the tilting,
the net effect is
a heat transfer equatorward.
Since salinity also enters in
density
field, the density,
the
determination
of
phase. Stommel et al.
fields
are
(1977) reported a geostrophic eddy
with large heat transport. The density field is maintained
salt transport in the same direction.
-
the
temperature and salinity fields can
be quite different when the temperature and salinity
out
of
18 -
by
a
2.5 Direct Estimates
Direct estimates of the oceanic heat flux are hampered
by
the lack of concomitant measurements of temperature and velocity.
From
hydrographic
data,
only
the
geostrophic
calculated through the thermal wind relation.
integrated
The shear
mass
study
properties. As it is difficult to find
would satisfy all the water mass properties,
always
can
unique.
the
of
the
a level that
choice
is
not
Recently, Wunsch(1977) mathematically formulated
the problem of determining the general circulation of the
as
be
to yield the geostrophic velocity by choosing a level
of no motion. This level is usually chosen from a
water
shear can be
oceans
an inverse problem. The solution that is most compatible with
the distribution of temperature and density turns out to
with
a
reference
be
one
level that minimizes the barotropic energy of
the oceans.
Heat flux associated with the wind driven gyre circulation
can be estimated by assuming a rectangular
interior
flow in the ocean,
velocity
profile:the
calculated by the Sverdrup formula,
which carries the mean temperature
of
the
ocean
interior,
is
compensated by a return flow in the boundary at the mean boundary
temperature.
Very
often
climatological
winds
particular oceanographic section taken at a
are
particular
used for a
time
of
the year.
Because of
eddies
are
not
the
spacing
resolved.
between
hydrographic
stations,
The errors in direct estimates have
-
19 -
been examined in detail by
involved,
the
direct
Bennett(1978).
estimates
Despite
the
errors
have the distinct advantage of
yielding the relative role of the different transport mechanisms.
This physical insight is more valuable
than
knowledge
of
the
magnitude of the transport itself.
Table 2.1 summarizes the results of
heat
transport
from
Bryan(1962)
of
northwards.
studies
two
estimates
The heat
positive
directed
involve
when
twelve
hydrographic
sections taken within the period from March to October.
sections
at
16*S
in
the
Atlantic,
two
factor
of
two,
both
northwards. The section at 32*N in the Pacific showed a
southward transport of ~1.2e15W.
August,
The
taken thirty years apart,
showed heat transports that differed by a
directed
of
Bennett(1978).
1.0e15W,
fluxes are given in units
The
and
direct
the
direction
of
As
this
section
is
taken
in
transport is consistent with Clark's
result of a cold advection in the Summer in the Pacific.
Bennett's results are given in
represent
the
range
assumptions
about
assumption
affects
the
the
of
heat
width
flux
of
the
parentheses.
estimates
boundary
The
for
numbers
different
current.
This
heat transport associated with the wind
driven gyre circulation. The two sections in the Indian
ocean,
taken some thirty years apart, show a northward transport.
From the somewhat scattered data, the results of the direct
calculations showed a northward transport in the Indian ocean and
Atlantic to about 40*N, a southward transport in the Pacific.
-
20 -
2.6 Heat transport associated with the net meridional flow
Starr(1951) developed an expression for the
transport
across
envelope
from
a
altitude
basic
thermodynamics.
When
that the
transport
density,
Co
wall
of
to
internal
energy
earth's
fluid
hydrodynamics
and
within
principles
applied
total
the
of
the ocean, Jung(1952)
Co f
energy,
T,
argued
where
P
is
is specific heat and T the absolute temperature, is
the dominant term.
Let us consider
extends
from
the
a
volume
northern
of
ocean
f
water
which
boundary of an ocean where the mass
flux is negligible small, to a lower latitude where the
section
,
area is Al (see figure 2.1).
is definded so that marginal seas
vertical
The area at the surface A2
are
included.
The
flux
of
energy across Al is
Denoting <
> as average over Al,
and <
>
as
departure
from
the sectional average, FT , can be written as
FT O C < eV>< T>A, tSSAVCo<Pfv > TT
(2.1)
The first term on the RHS of (2.1)
meridional
flow
at
the
mean
is the transport
temperature
meridional
flow.
the
net
of the section. The
second term represents transports by mechanisms
net
by
A
other
than
the
This term has been computed in the direct
estimates.
Conservation of mass requires that
(2.2)
- 21 -
where
v
is the velocity vector. When integrated over the volume 4
and the divergence theorem used, we get in the steady state
At f
(2.3)
where w
kr -d A
I'
A2
A
is the upward velocity. The term on the RHS of (2.3) can
be evaluated as
JJ%wJS
re)dA
+-ffr
dr
where p, r, e, are
rates
of
precipitation,
river
runoff
are density of the appropiate quantities.
evaporationand ?, Jfr,
Evaporation, except possibly for the generation of
aerosols
in
the
process,
water from the ocean.
From
their
Junge and Werby (1958) have measured major
analysis,
1.001 g/cc. The density of
(Holland
1978)
marine
can be considered a removal of pure
chemical constituents of rain water over the
States.
and
compared
the
river
to
continental
United
density of rain water is
runoff
is
~
also
~
1.001g/cc
typical sea water density of 1.027
g/cc. If we assume that
where
£f is 1 g/cc, equation (2.3) can be written as
A
(2.4)>
-4
+
where P, R, E are the integrated
runoff
and
-E)-
rate
of
precipitation,
evaportation and F is the total fresh water input at
the surface A2.
The energy transport
associated
meridional flow is
(2.5)
river
Co
fV>) T >,
=
Ca
F
T>
with
the
net
Since(T-O, the direction of fresh water tranpsport determines the
direction of energy transport associated with the net
meridional
flow. Typical values of fresh water transport are 10el2m**3/year.
For
a typical mean temperature of 280K (7*C), the heat transport
is ~ 0.4e15 W which is comparable in magnitude to heat
associated
with
the
second
term in (2.1)
calculations.
9
transport
obtained from direct
2.7 The transport of fresh water
The studies of E, P, R entail a large body of data.
compilation
by
Baumgartner
and
Reichel(1975)
of
The
the
global
distribution of the terms are used here. Table 2.2 summarizes the
precipitation(P),
evaporation(E),
and
runoff(R)
volumes
in
Km**3/year for every 10* latitude belts for the three oceans from
Baumgartner
F,
is
and Reichel. Fresh water transport at any latitude,
calculated
evaporation
by
summing
precipitation,
water
the equation of
integrated
over a polar cap,
and
moisture flux in
between
boundaries.
by
water
substance
the transport of fresh water in
by
rivers
of
transports
over
the atmosphere,
land.
of
moisture
in
2.2 compares the fresh
which involves
the
water
transport
in
over
precipitation
divergence of moisturen in
water
the
the
the
Figure
oceans(present
It is
to note the agreement between the curves as they are
derived from completely different data sets.
fresh
is
correlation
study) and in the atmosphere(from Starr and Peixoto 1971).
evaporation
We
Direct estimates of the
the moisture and velocity field, have been made.
interesting
The
flows into the Atlantic.
conservation
oceans must be compensated
atmosphere
northern
output from the polar oceans is 3.0el2m**3/year.
assumed that the polar oceans
If
less
from the northern most boundary of the oceans, i. e.
a no flux condition is assumed at the
fresh
runoff
in
the
the ocean.
in
the
atmosphere
Precipitation
As there
subtropics,
and
is
excess
there
is
convergence
of
exceeds evaporation at
the high latitudes and at the Equator, hence there is convergence
-
24 -
of atmospheric moisture and divergence of oceanic fresh water.
It can be seen that there is an
excess
of
fresh
water
input in the Pacific and a deficit in the Atlantic and the Indian
ocean. Hence there is southward flux in the Pacific. This flux of
fresh
water from the Pacific is discharged into the Atlantic and
Indian ocean at the
southern
latitudes
connected.
-
25 -
where
the
oceans
are
2.8 Discussion
Figure 2.3 compares the direction of ocean heat
obtained
from
various estimates.
transport
The estimates of Bryan(1962),
Emig(1967) and Hastenrath(1980) from surface balance calculations
are indicated by
B(Bryan),
estimates
of
estimate).
Results
Bryan
E(Emig)
and
Bennett
from
the
and
are
present
H(Hastenrath).
Direct
indicated with D(direct
consideration
of
the
transport by the net meridional flow in indicated by P.
It can be seen that the heat transport associated with the
net meridional flow,
calculated
from
P,
is
in
the
same
direction
as
those
the direct method except for the North Atlantic
section.
There is agreement in
between
the
direct
the
estimates
direction
(with
the
of
net
correction,P) and the other estimates in the
low
latitudes
and
heat
transport
meridional flow
North
Atlantic
at
in the South Pacific. In the South Atlantic,
all estimates showed a northward heat flux except that of Emig.
Because there is no atmospheric path of salt, Stommel
Csanady (1980)
in
the
oceans
showed that the salinity- temperature distribution
is
determined by the ratio of the flux of fresh
water and heat in the oceans.
moisture
and
heat(derived
Using the total observed
from
residual
and
flux
Atlantic,
of
surface balance
method) they showed that while there is a northward heat flux
the
and
in
the heat transport in the Pacific is to the south
at ~40*N using Sellers' estimate of heat flux.
The more interesting results from the direct estimates are
-
26 -
the assessment
of
the
According
mechanisms.
relative
include the heat transport associated
flow,
heat
transport
of
the
with
with
associated
the
with warm surface water flowing in
circulation,
accompanied
the
transport
direct calculations which do not
the
to
importance
net
meridional
mean
meridional
one
direction,
flow, is the dominant contributor in
by deep return
the Atlantic, Indian and North Pacific sections. It is
only
to
the
transport
the gyre in the South
with
associated
secondary
Pacific. This picture of global heat transport, with a
heat
in
flux
the
Atlantic
and
southward heat flux in the
a
(1960)
Pacific, is consistent with Stommel and Arons'
the
abyssal
circulation.
Atlantic and
Therefore
the
They
Weddell
there
is
a
Sea
pointed
are
northward
northward
out
sources
flow
model
that
of
of
the North
bottom
water.
of surface water in the
Atlantic and a southward deep flow. In the absence of any
bottom
water
in the
source,
the
meridional
flow pattern is
reversed
Pacific.
The lack of a source of deep water in the North Pacific is
usually attributed to the low salinity at
salinity,
sea water.
cooling
more
The
precipitation
low
and
and
heat
surface
At
low
salinity
may
be
due
to
excess
over evaporation in the region. Hence
runoff
budget,
surface.
is required to raise the density of the
not only does evaporation and
water
the
it
precipitation
affect
the
global
is also important in determining the
deep ocean circulation.
Jung(1952) was first to point out the
importance
of
the
mean meridional circulation in the heat transport. The agreements
-
27 -
between
the
direct
estimates,
the
calculation of Stommel and
Csanady and Stommel and Arons' model suggests the
this
mechanism.
If
heat
transport
associated
importance
with the mean
meridional circulation of in the ocean is to be parameterized
an
energy
model
for
climatic
incorporated.
-
28 -
of
in
studies, the deep ocean must be
3.The Model
3.1 General description
layer in
the atmosphere and two layers in
one
representing
the ocean,
with depth h and the deep ocean; and two blocks
thermocline
the
vertical:
the
The model consists of three layers in
latitudes.The
in the horizontal representing low and high
boxes
are labelled 1,...,6 as shown in figure 3.1
The idea of box models in oceanography originates from the
fact that there exists identifiable water masses. Because of
short
identifiable on a
there
Nonetheless,
exist
domains where different mechanisms are dominant. At the
physical
low latitudes, there is
a surplus of
solar
incoming
radiation
long wave radiation whereas there is a deficit at
outgoing
over
basis.
permanent
air masses are less
processes,
atmospheric
of
duration
the
high latitudes.
meridional
in
layers
chapter
in
We stressed
circulation
the
parameterize
ocean
this
is
the
the
the
importance
heat
transport
minimum
the atmosphere,
the
model
a
mean
processes. Two
number
required
to
equatorward flow in the deep ocean, upwelling
ocean.
Box
surface
layer,exists
7, describing the location of bottom
water formation, provides a source region
Such
the
with sinking of cold water
at low latitudes, and a return flow in the
in
of
mechanism. A mean circulation, similar to the
Hadley circulation in
at high latitudes,
in
2
for
such
convection.
a circulation is asymmetrical, with rising motion occupying
much larger area than the sinking motion (Stommel 1962, Rossby
at
amplitude
This
latitude
of the recirculation
latitude
of
predominantly
chosen
y is the sine of the latitude. This corresponds
y=0.75,where
to a latitude of ~ 50*.
the
low and high latitudes is
A dividing line between
1965).
maximum
the
of
picked
so
can be fully captured.
energy
eddy
is
in
flux
type.
the
Furthermore,
that
It
is
the
also
atmosphere,
paleoclimatic
records have revealed that the mean ice line rarely comes further
equatorward than this latitude. An ice sheet, of fractional
at
high
line
area
latitudes, is present in box 2. Presently, the mean ice
is
at
situated
~72*N
in
the
Northern
Hemisphere,
corresponding to y=0.95 and Tj =0.2.
The choice of using
meridional
coordinate
sine
the
of
the
latitude
as
our
the latitude belts. The areas of
weights
the low and high latitudes are
A1=0.75(2 I
R**2) ;A2=0.25(2
TI R**2)
where R is the radius of the Earth. If a horizontal length
Lx
is
defined,
the
length
of
the low and high latitudes are
L1=A1/Lx,L2=A2/Lx.A meridional length scale
Ly=(L1+L2)/2.
and
partial
derivative
finite differences.
-
30 -
scale
can
Ly can
be
defined
be approximated by
3.2 Governing equation
If we consider latent heat of water vapor as a form of
internal energy, the energy equation for a Boussinesq, plane
stratified atmosphere is
10
((c99
(3.0)
(f
A
)
rF' +Fr)
is density, Cp is specific heat at constant pressure,
where
8
(
L
is potential temperature,
latent heat of vaporization,
is specific humidity, which equals
RH rs(p,T), RH is
the relative humidity, assumed constant and r s the mixing ratio
at saturation,
and
Tr Y,
I and 1 are meridional and vertical coordinates
fF
are zonal averages of meridional and vertical
fluxes of sensible and latent heat, F~dis the zonal average
radiative heat flux.
For a Boussinesq atmosphere,
where
T
Expanding
where
'd the adiabatic lapse rate.
is temperature, and
r
in terms of a Taylor series, we get
Te is the zonal mean equilibrium temperature and
departure from it.
The term
-
the Clausius-Clapeyron relation.
can be evaluated through
For a hydrostatic atmosphere
with a constant lapse rate
^(VTLc
1
- 31 -
T'
t) T
the lapse rate
r
where T is the surface temperature and
T'
can be written as
T(Y-,jt.)
T5,,t) - T0t)3
r
-
LHS of equation (3.0) when integrated over an atmospheric
column gives
where Ps is the surface pressure,
is the acceleration due to gravity,
g
] is the vertical averaging operator.
and [
If we consider annual conditions, the boundary conditions
The
are no flux at the equator and at the pole (y=0, y=l).
vertical velocity vanishes at the top of the atmosphere, so the
dynamical flux of energy also vanish there.
When equation (3.0) is integrated over the volume defined
by boxes 1 and 2, the energy equation for the appropiate boxes are
(3.1)
s
3 F Yz.7S
Y=.7
(.(32)
+
where (
FY+I
L
C
t f
L
Z
=-7
-B
+
ft
5Fr"
,,
Z. <
-
=
i denotes horizontal averaging over box i, and the
L
approximation
s
--
aT
a-"
-
32 -
is used
F
L
a Z;,
where (
) denote horizontal averages over box i,
fF jP.,is
horizontal
the integrated
the
y=0.75,i.e.
at
flux
boundary,
is
f
the vertical
flux,
Ti's are horizontal
and
averages
of surface air temperature for box i, i=1,2.
Jung (1952) pointed out that the energy
is
ocean
dominated
in
transport
the
the transport of internal energy. If we
by
assumed that the composition of sea water remains unchanged, then
the internal energy is
C T-
USea Wter
where Co
is
the
heat
specific
of
sea
water.
The
boundary
conditions for the oceanic boxes are
i)
no flux at the pole and equator,
level
ii) the thermocline level z=-h is defined to be the
radiative
fluxes are negligible. Short wave radiation is trapped
within the top few centimeters. Turbulence
greater
where
depths.
This
turbulent mixing
mixes
this
down
to
depth is typically 20m
(Kraus and Turner 1967).
iii) neglecting geothermal heating there is no flux at the bottom
of the ocean,
z=-H.
Assuming that the temperature structures remains
unchanged
within each oceanic box, the energy equation for boxes 3 to 6 are
---- 3
(3.3)
Colo 0
j>
Li
-
33 -
103
I Z=Lk
t
L 27-H.7
(H4i>TF-)
(3.5) Cccqo-
(3.6)
where
4-
-)
° -
c!
-
_
_
C-('(F
Vo is density of sea water,assumed constant,
is depth of the surface layer
H
is the total depth of the ocean and equals 4 Km, and
>
HF
-H
--7s
and Ti, i=3,4 are average sea level temperatures for boxes 3
and
4,
Ti, i=5,6 are average temperatures at z=-h for boxes 5 and 6.
The observed thermocline depth varies latitudinally.
a
minimum
of
From
about 60m at the equator, it reaches a maximum of
about 500m in mid- latitudes, h is assumed
constant
and
equals
200m in the model.
The radiative balance differs greatly between ice-free and
ice- covered regions. Vowinckel and
Orvig(1970)
indicated
that
while infrared cooling from an ice-covered area is less than that
from
an
ice-free
area,
the turbulent transfer of sensible and
- 34 -
latent heat is
The
ice
almost totally suppressed
sheet
in
in the presence of
the model serves two purposes: it provides a
surface of high albedo for short wave reflection and acts
insulating
lid
for
ice.
turbulent
fluxes
between
atmosphere. Ice volume is not predicted in the
the
model.
as
ocean
To
an
and
close
the system of equations,the fluxes at the different boundaries of
the
boxes
must
be
parameterized as a function of the internal
variable.
-
35 -
3.3 Flux Parameterization
The radiative flux consists of both short wave(solar)
and
long wave (terrestrial) radiation.
3.3.1 Short wave
The distribution of solar radiation incident on the
normalized
atmosphere,
by
So=Sc/4
Sc
where
top
is
of
the
the
solar
constant,can be approximated to 2% accuracy by
I ±S2 Pz )
S(r,)=
where
(North 1975).
reflectivity
.482-
-
5,
Considering
R,
the
atmosphere
slab,
having
transmissivity T, and absorptivity A, it
can be
shown that, in the presence of a reflecting
the
as
a
underlying
surface,
transmissivity and absorptivity are
effective reflectivity,
(appendix A3)
T7- .
T
+
T
5
I-
Rc
T
A(
I-R
)
where o is the surface albedo.
The model consists of only ocean and ice.
the
ocean
surface
is
~0.07
and
is
The
highly
dependent(Kondrateyv 1969). In order that the model
be
compared
albedo
zenith
results
of
angle
can
with other models(e.g. Budyko 1969) the latitudinal
-
36 -
dependence of
(M
assumed
O(
(
and
the
In table 3.1,
ice-covered regions.
albedo
albedoes
surface
with
function,
is
albedo
surface
the
to
step
a
be
for ice-free and
surface
average
zonal
for the Northern Hemisphere from Sellers (1965) is
)
shown in column 4.
The
disposition
solar
the
in
distribution
has been calculated by a number of investigators(e.g.
atmosphere
London 1957,Katayama 1967,Sasamori et al 1972).The components
radiation
solar
of
-reflection,transmission,absorption-
are
cloud
type,
of
dependent on the distribution
cloud
amount,
chemical constituents of the atmosphere, zenith angle and surface
From
albedo.
a model atmosphere, London (loc cit) calculated
the reflection,absorption by the atmosphere and by the underlying
surface for every 10* latitude belts for the Northern Hemisphere.
model
previous
than
smaller
that
showed
measurements
Satellite
the
in
shown
table
(
latitude
in the atmosphere(a),
absorption at
and absorptivity(A)
in
for all latitudes can be calculated using the
that
The
slab
the
poles.
absorptivity
slab
transmissivity
maximum(minimum) at ~10-20*N,
the
(T)
appendix A2.The results are summarized in table 3.1.
It can be seen
constant.
the
t ),are those derived from London,andO(s, the surface
albedo from Sellers,the slab reflectivity (R),transmissivity
formula
is
Assuming that the ratio between the solar
3.1.
reflection(r),absorption
surface
is
calculation(Ellis and Vonder Haar
The observed planetary albedo for every 10*
1976).
albedo
planetary
and
A
remains
(reflectivity)
decreases(increases)
fairly
has
a
towards
The area weighted average of the various parameters
to be used in the model for low and high latitudes are
-
37 -
shown
in
table 3.2.
The ice-free surface albedo (cr(
the low latitude surface
is
albedo
calculated
albedo, 0.1 .
from
the
assumed to be equal to
is
The
area
ice-covered
surface
weighted average surface
albedo of the high latitudes,
(-
t ) ,
- rgo; = o.22
This gives o(X.0.7. The ice-free planetary
albedo
are
0.43 for low and high latitudes and 0.58 for ice-covered
These
numbers
are
and
regions.
0.32 and 0.69 for ice-free and
to
compared
0.28
ice-covered regions in Budyko's model.
The amount of solar radiation absorbed
at
low
and
high
latitudes are
C0, =
where at.,
g0 5,
t
i
dware atmospheric absorptivity
for
ice-free
high latitude, for ice-covered high latitude, and
75
S1,
-
{1)
5,
.1/
-
38 -
low
and
Total absorption at the surface for low and high latitudes are
s~tL,
I
t
5S, ( -t
where
' )
5,2i-n)
Y
,andtare effective transmissivity
into
the
underlying
surface at low and high latitudes.
3.3.2 Long wave
The long wave flux at the top of the atmosphere is assumed
to be a linear function of surface temperature
I = A + B*
T
Since the effective emitting layer must be situated higher up
the
is
the underlying assumption is
atmosphere,
constant.
in
that the lapse rate
This simple linear relation proves to be better than
the fourth power law of blackbody radiation since temperature and
An
moisture content are positively correlated in the atmosphere.
increase
in
will
temperature
the
increase
atmospheric column and hence reduce the
to
dependence
the
of
opacity
a
power
less than four.
Using
the
moisture, Rodgers(1967)
of the atmosphere.
function
of
distribution
observed
calculated
flux
seems
to
as
a
temperature for four seasons from Rodgers.
Except for the scatters for the high temperature in the
there
and
temperature
calculated the long wave flux at the top
Figure 3.2 shows the
surface
of
tropics,
be a linear relation between the long wave flux
and temperature.
Warren and Schneider(1979) correlated
-
39 -
the
infrared
flux
from satellite measurements with surface temperature for
derived
different zonal belts.In the annual case, the optimal values of A
and B differ depending on whether or not the Antarctic values are
if
included. There is considerable scatter
monthly
values
are
2.1.
This
analysed.
Rodgers'
The values of B from
value
is
close
to
is
calculation
those obtained in one dimensional
radiative
convective models which assumes constant cloud altitude.
value
(-1.6)
temperature is
B=1.55
is
assumed ( e.g.
these
Cess 1976).
if
models
The
small
fixed cloud
values
A=211
and
North's model is adopted for the present model.
in
used
from
obtained
A
Fixed cloud temperature is
therefore implicitly assumed.
The effective long wave radiation from the surface, Fo, is
the difference between the black-body
investigators have related the
atmosphere
to
the
clear
Go,for
back(counter)radiation,
radiation
6*T
and
the
Previous
skies.
from
back(counter)radiation
the
pressure at the surface. Although the
vapor
functional form fitted quite well to data, the coefficients which
entered the formula differed from case to case. On
of
a
collection
of
reexamination
sets , Swinbank (1963) suggested the
data
following
Go = -a
where
+ b 6$
T**4
Go = back radiation from clear skies at screen
height
in
W/m**2,
T =screen height temperature and
6=
Stefan
Boltzmann
-
40 -
constant
which
equals
5.77e-8
1.195.
This fit
result
also
and
is
showed that relationship between the back radiation
vapor
surface
the
His
investigations.
previous
than
better
170.9 and
are
for a and b
values
Numerical
W/(m**2)(K**4).
was
pressure
a
through
established
correlation between temperature and humidity.
The effective long wave radiation for clear skies is
FO =
C. = .
T-
In the presence of cloud
and
a
, (- T4-
temperature
jump
the
between
screen height and the underlying surface, the effective long wave
flux at the surface is
Fo(,- Cq
C"Z= )
where
is
C
a constant,
+
6-TS(T,
- T)
usually taken as 0.76,
C the fractional cloud cover,
6the
emissivity of the surface,
the temperature of the surface.
average
and
fractional cloud cover
are it
the top of the atmosphere z= 00
The
area
=0.48 and 'C
high latitudes (from Sellers 1965).
and
taken as 0.95,
weighted
=0.62 for low
The radiative fluxes at
and at the surface
z=0 are
3.3.3 Dynamical Fluxes.
The exchange of sensible heat and latent heat between
ocean
and
atmosphere
the
is calculated using the conventional bulk
aerodynamic formulation
-
j
- 41 -
Vc, CT-(T
T)+ Lk.I
t
T
(T))
where Cp = drag coefficient =1.5 e-3
V = r.m.s wind speed = 5 m/s
r(T) = mixing ratio at temperature T
1 )=
mixing ratio at saturation
L = latent heat of vaporization.
There is however evidence that the
for
transfer
coefficients
heat is different from that for moisture, and that they both
depend on
wind
Businger
1973).
speed
and
atmospheric
heat
1976,
It should be pointed out that this formulation
a
is a diagnostic rather than
latent
stability(Bunker
exchange
prognostic
one.
The
amount
of
is limited by the energy available to the
ocean surface, and does not increase
as
indefinitely
the
wind
speed increases.
The Clausius-Clapeyron
vapor
pressure
to
relates
equation
the
saturation
temperature. By fitting the equation to
the
vapor pressure data, Cess(1973) obtained
_5365/T
(
=.2
( te)
e.
with T in K. Since the mixing ratio
e
mass of water vapor
mass of dry air
-
where 0.622 is the ratio of the molecular weight of
to
that
of
dry
air,
and
P
is
the
water
vapor
pressure of dry air (in
Atmospheres),an expression relating the temperature to the mixing
ratio
at the surface (1 atmosphere)
-- r) P
I.
-
is
A11Pe
42 -
- 5 3 9 5/
be
to
assumed
where RH is the relative humidity,
in
80%
the
model.
In Budyko's model, the net heat transport is parameterized
as a linear function of temperature. This
the
observations
well
quite
fits
parameterization
at the poles and near the
except
equator(Schneider and Warren 1979).Since the total heat transport
atmospheric
in this model is a combination of both
oceanic
of individual heat
parameterization
different
transport,
heat
and
transport processes are deemed necessary.
energy
The conversion of available potential
kinetic
to
energy throught the action of baroclinic eddies is most important
at
Observations (Oort 1971) showed that
to high latitudes.
mid
~50*N,the transport in the atmosphere is
at
predominantly
of
the eddy type. The choice of the boundary allows us to ignore the
circulation
mean
constant,
meridional
temperature
heat
the
on
flux
quadratic(Green 1970,Stone
is
gradient
stability is held
static
the
baroclinic
of
dependence
the
If
transport.
1972),i.e.
RT'
'
= E (
T)
The model's atmospheric sensible
proportional
heat
flux
is
assumed
to
be
to the square of the temperature difference between
low and high latitudes.The constant of proportionality E is found
to
be
about
tropospheric
6-8e17
cm**3/(s
K)
temperature gradients
Oort and Rasmusson (1971),
from
43 -
for
at ~50*N (Clapp 1970).
we calculated
-
observation
a
surface
midFrom
temperature
difference between low and high latitudes as 22K. For a v'T' of ~
10K
m/sec,
the eddy coefficient E' is 5.2e17 cm**3/s K. A value
6e17 cm**3/s K is
The flux of
atmospheric
heat
adopted.
contributes
moisture
flux
in
the
cit).The latent heat flux is
loc
(Oort
significantly
parameterized, by assuming that the departure of the mixing ratio
from
its
zonal
mean
is
proportional
to
the
temperature
departure (Leovy 1973, Mullen 1979)
If we multiply by
the
expression
relates
flux is
which
the
and
integrate
vertically,
an
latent heat flux to the sensible
obtained (see appendix A3).
f
As a(Ts)
velocity
is
L
Lv'r'
T.
a slowly varying function of temperature,
variation in Bo is due to changes in the
specific
most of the
humidity.
We
shall assume that a(Ts) is constant.Figure 3.3 shows the seasonal
variations
of
the surface mixing ratio [R] and the ratio of the
total flux of latent heat to sensible heat at 50*N from Oort
Rasmusson(l.c.).
and
The correlation between the curves is 0.97. The
temperature Ts in our parameterization is assumed to be the
surface temperature in our model,
T
L .
i.e.
L+*m
L, + L-
- 44 -
mean
3.3.4 Bottom Water formation
We have emphasized the importance of the
circulation
in
the
heat
budget
of
cold water sphere of the world oceans is
The
meridional
the global heat budget. This circulation is also
important in the maintenance of
ocean.
mean
the
continually
nourished with cold water that is formed at the surface.
As
the
are heated at low latitudes and cooled at high latitudes,
oceans
the coldest (warmest) and most (least) saline
near
deep
waters
are
found
the surface. From the observed distribution of salinity and
temperature at depth, Stommel and Arons(1960) estimated that most
of the bottom waters are formed in the
Sea.
Weddell
Gordon(1
9 7 1)discussed
North
Atlantic
in
important
role
accentuates
the
such
case
while
a
of
sea
ice
mechanical
process.
cooling
formation)
stirring
by
(or
must
wind
salt
play an
probably
Formation under ice shelves comes
about when freezing of sea water under
addition
the
the possible mechanisms for the
production of Antarctic bottom water. Surface
addition,
and
ice
shelves
of salt to the supercooled water below.
results
in
Because of the
nonlinearity of the equation of state of sea water, the mixing of
two water types of the same density but
temperature
can
result
in
a
different
salinity
and
mixture that is denser than both
mother types. This phenomena is commonly known
as
"Cabbelling".
Foster (1972) pointed out that such a process may be important in
the formation of bottom water in the Antarctic.
Since salinity is involved in the process described above,
we resort to other means of parameterizing the process of
water
formation
as
salinity
bottom
is not included in the model. The
- 45 -
formation of deep water has been observed in the Mediterranean in
detail (MODEC 1970).The Mediterranean Sea is characterized
shallowest in the center.
it is
a
all year round. The thermocline is tilted so that
gyre
cyclonic
by
It
is
therefore
gravitationally
more stable at the rim than at the center. Winter cooling reduces
Instability occurred at
the stability everywhere in the surface.
the
of
center
gyre while other parts of the gyre are only
the
and
slightly perturbed. Deep convection occurs in narrow regions
mixes
the water down to the bottom where it spreads over the sea
an isothermal column is left behind
floor. As a result
in
the
core of the convecting regions.
Regardless of the process, the net effect of bottom
formation
is
a transfer of surface cooling to the layers below.
As we pointed out earlier,the
brings
water
of
formation
bottom
water
that
cold water into the abyss is necessary in maintaining its
heat budget.
To parameterize the process of formation,
column
we
consider
water in box 4, occupying a fractional area o of high
of
latitudes. This region represents the core of convection
neutrally
a
stable.
instability
Static
sets
in
and
is
for any excess
surface cooling. An amount of water, at temperature
T6 typical of
conservation
the
region
sink
will
to
the
bottom.
Mass
requires that the displaced water be replenished by
the surrounding water. Assuming that
this
is
a
self-adjusting
process, the integrated energy equation for the column is
co,,'
at
5
F, +
o
t
-
46 -
ST
- 5, T
is the temperature characteristic of bottom water,
where T
the
amount
latitude
of
bottom
F--
area,and
formed,
water
is
normalized
net
the
by
S8
the high
and
(radiative
flux
is
across the ocean-atmosphere interface. This formation
dynamical)
area is assumed to be situated in the
ice-free
areas,
net
the
flux across the ocean-atmosphere interface is
FT=
where
(iSZ.
is the average solar transmission
high latitudes
in
the
ice-free
and equals
2-
and , and ,
earlier.
are radiative and surface turbulent fluxes
defined
The effect of small scale turbulence and diffusion have
been neglected.
The temperature of the bottom water is quite variable(e.g.
Pierson and Neumann 1966) Presently, it is
passage.
upper
reported
that
the
Drake
as low as -2.3*C have been observed at the
Temperature
boundary
at
0.4*C
of
the
Ross
the
bulk
of
Sea
shelf
Antarctic
water.
water
has
Gordon(1975)
an
average
temperature of about -1.0*C.Actually at a temperature below
sea
water
of
any
salinity
will
-1*C,
freeze before it reaches its
maximum density under normal atmospheric pressure (Pounder 1965).
This value for T
could
is adopted and
be
variable
The
choice
assumed
constant
although
it
during climatic changes as suggested by Weyl
(1968).
of
the
fractional
- 47 -
area
over
which
such
convection
occurs,
assumption
of
,
the
is
discussed
formation
process
later. The self-adjusting
gives
us
a
diagnostic
equation for So ,i.e.
F,
5o
This mass flux, which is proportional to the surface
cooling
at
high latitudes, is our mean meridional circulation. Its effect on
the
energy
budget
of
water at temperature at
box 6, say,
S,
is to transfer
amount of
into, and the same amount at T out of
T.
box 6,i.e.
cc a(H-0
This
convective
-. + ST -
process
constitutes
both
T
a
horizontal
and
vertical dynamical heat flux for each box in the ocean.
A note about the energy balance in box
only
a fraction (1-oC)
4.Box
4
occupies
of the high latitude area. The
turbulent
transfer at the ocean-atmosphere interface is suppressed
in
the
presence of ice, hence the effective turbulent transfer for box 4
is
S - (I- 71 -
)
.
The residue of the oceanic heat
other
bulk
transport
by
mechanisms
than the mean meridional circulation is parameterized as a
diffusion.
Vertical
diffusion
is
parameterized
in
a
similar fashion.
co
Cfo~
<F
< F>
,
-
(T- T
+ Cor KY(-h)(T
S, T3 +
-
Q,f<
0
=
=
F~ 48
-
S
, oy
+
Cc,
k4
(T 3
)
-T)
+ cof.k, I-s)CT- T6,
The
horizontal
parameters
in
and
the
vertical
model
diffusivity
and
section.
-
49 -
will
be
(Ky,Kz)
discussed
are
tuning
in the next
3.4 The choice of Ky,Kz
Substituting the heat flux parameterization
3.1-3.6,the energy equations
for the six boxes
into equations
units
are,in
of
W/m**2,
Co~_TI
(3.7)
=
S t
(3.8)
(3.9)
I-2.
: t,-
Cs t
+ f"
I - I,
+
A1
(3.10)
)
-
•
,)
T 5r
(3.11)
(3.12)
e-
D
o
$+
o-
Kd
S (1i)
+
- T)
T-(o
+ K, ( 5
4 - K,
rate
of
o
) (TA,
T3
formation
water is
(3.13)
L)
= K6ST5-T) -4
+ (--T-3
- T -T6
The diagnostic equation for the
- f - f,)
-
50 -
+ S6 (I 4-
T )
(T
+ k,)(T -T,)+
.if (,- t -
T
TL4 + (K*-SXT
S0
L
, + (5
f -
+Jf, -
-
T5 )
Sv
of
-%)
bottom
where CaL = Cp Ps/g (1 +
)
=columnar heat capacity of the atmosphere,
= columnar heat capacity of
Cs= CoI'i
the
ocean
surface
layer,
Cp= Cofo(H-h)= columnar heat capacity of the deep ocean,
= long wave flux across the atmosphere ocean interface,
+
=(170.9-.195(1)C1
Z=
E6.T
turbulent fluxes of sensible and
the ocean-atmosphere
-
C(T
latent
heat
across
interface
T )+ LfLT)-
4-~
-T)
(T+
CTJJ}
j
CCfA fCzV
= normalized eddy flux of sensible heat and moisture
= E (1+Bo)
Bo=
M)
L (T
where E= E C
(T1-T2)**2,
(T
)
, h
"YCX
K=
L,
;
Ky=horizontal
diffusion
Kz=vertical
diffusion
L,
coefficient,
k, Co fo
v
a*I
coefficient,
= ratio of low latitude to high latitude areas
= ratio of the depth of the deep ocean
ocean=
to
the
surface
//L
= solar transmission into the ice-free high latitude per
unit area -
°
o tw
52$
= solar transmission into box 4 per unit area
%(-
-
d-L) +
S~tA
"L
In the steady state, equation 3.7-3.13 constitutes
of
non-linear
algebraic
equations
- 51
a
set
for T1,...,T6 if the set of
parameters are prescribed. Newton's method, described in appendix
A4,
is used to calculate numerically the steady state result.
The possible values of Ky and Kz cover a wide range.
a
balance
between
the
advective
and
diffusive
be
obtained
through
a
knowledge
of
processes,
coefficients
estimates for the horizontal and vertical diffusion
can
From
the distribution of
oceanographic properties (e.g. salinity ,temperature).The results
of
such
investigation
Sverdrup(1942),
have
Neumann
been
summarized
to
about
in
and Pierson(1966) and Veronis(1975).The
vertical diffusivity varies from about 0.01 in the
water
e.g.
still
Danish
100 cm**2/s in the equatorial Atlantic.Robinson
and Stommel(1959) have argued that Kz must be about 1 cm**2/s
order
the
for
to have a thermocline.Munk(1966) has used
ocean
temperature and salinity profiles in
the Pacific to conclude that
the
a value of Kz=1.3 cm**2/s is not inconsistent with
of
distribution
in
in a one-dimensional
properties
oceanographic
observed
model.
The
California
horizontal
Current
(Stommel
1958).Data
suggests
that
diffusivity
to
about
from
mesoscale
2e8
the
varies
from
cm**2/s
in
mid-ocean
motions
2e6
the
in
the Gulf stream
dynamic
experiment
an effective horizontal
have
diffusivity of 1e7 cm**2/s in the upper ocean(Rhine 1977).
The anthropogenic production of tritium as a result
of
a
series of nuclear tests in the fifties and sixties gave rise to a
new
tool
for
the
study
tritiated water enters the
of
oceanic
oceans
dependent fashion that it becomes
- 52 -
in
exchange
a
time
processes. The
and
latitudinal
a unique clock and dye tracer.
observed distribution of tritium in the Sargasso Sea
Taking
the
and an
empirical
between
relation
and
concentration
tritium
temperature, Rooth and Ostlund(1972) tried to separate the effect
of
diffusion from that of advection. Assuming that the transient
time scale is
life
long compared to the half
they
tritium,
of
at an upper bound for Kz and Ky of 0.2 and 1.5e7 cm**2/s
arrived
the
respectively. Similar values have been obtained by analysing
distribution
tritium
the
in
out
pointed
paper)
Brocker(in a comment following Veronis 1975
tritium distribution is still in a transient state and
the
that
Suess 1975).
and
Pacific(Michel
that the values they obtained are likely to be lower bounds.
We approach the problem differently. At
total
heat
the
equilibrium,
transport across any latitude is governed by the net
radiative heat flux at the the top of the stmosphere. Stone(1978)
have been specified correctly, the magnitude of the total
albedo
observations,
Vonder
dynamics.
internal
transport is independent of
and
Haar
Suomi(1971)
portion
of
heat
total
From
satellite
estimated that the
e15
total flux across 50*N is 9e19 cal/day(4.5
the
the
and
parameters
external
the
as
pointed out that so long
W).Let
Ro
be
flux carried by the ocean. Residue
calculations(described in chapter 2) showed that Ro is about
0.3
at 50*N(Vonder Haar and Oort 1973).
section
Figure 3.5 depicts a north-south
and
density
in
the
mid
Atlantic
from
Wust
temperature distribution, it can be seen that
poleward
of
~50*
the
of
temperature
(1935).From the
water
column
is almost isothermal whereas at low latitudes
the isotherms are almost
horizontal.
- 53 -
At
around
50-60*,
there
exists
a
set
of closely packed isotherms and isopycnals in the
top Km, sloping downward and
isopycnals
are
less
equatorward.
than
that
The
slopes
of
the
of the isotherms, suggesting a
possible transfer of cold water downward and equatorward if water
motions are confined to isopycnal surfaces.
If
we
require
that
the
solution
be
consistent
with
observation, namely
1) the portion of oceanic heat flux is that derived from residual
calculation ,
Ro=0.3
(A)
and 2) that the high latitude oceans are isothermal,
T4 -T6 = 0
(B)
then for a given value of Kz,there exists a unique
pair(Ky, o
)
that satisfies these requirements.
The equilibrium solutions are calculated for
,
Kz=0.1
,10 ,cm**2/s
1.0
1e6 < Ky < 2e8 cm**2/s
0.05 <
Given
statically
Kz
and
Ky,
the
< 0.13
oL
high
unstable, (T4-T6)<O,ifoL
latitude
oceans
will
be
is too small. An increase in O
(hence the amplitude of convection) will increase
the
stability
for
different
as well as the overall ocean heat transport, Ro.
Table 3.3 shows the steady state solutions
Kz's
that
are
closest
to
fulfilling the requirements (A) and
(B).Exact solutions are not
available
-
54 -
since
the
steady
state
are
solutions
calculated
for
discrete
of Ky and 0(
values
.
Despite the wide range which Ky and Kz cover, the atmospheric and
The
invariant.
almost
the deep ocean at low latitudes,T5,increases for
of
temperature
are
temperatures
surface
oceanic
larger values of Kz.The rate of bottom water formation, Sg
seems
to increase asymptotically to a value of 18e6 m**3/s(o = 0.07) as
increased. Figure 3.7 shows the three values of Kz and Ky
is
Kz
satisfy
which
Estimates of
(A)and (B).
distribution of tritium for the Atlantic
a
large
from
Kz
and
the
)
seem
It
are included for comparison.
and Pacific
that
Ky
obvious
value of Kz is accompanied by a small value of Ky
Note also that as Ky is increased,the portion of
and vice versa.
heat flux associated with the mean
gets
circulation
meridional
smaller.
18e12m**3/s, is the maximum amount
The rate of production,-
allowable in the model without violating
discuss
the
later,
determined by
are
parameters
surface
air
specified.
If
the
certain amount,
is
model
temperature
if
the
radiative
we impose the constraint that a
air
temperatures are determined. This,
the
sea
surface
through
the
atmosphere,
the
in turn, determines
sea-air
exchange
since the remainder of the meridional flux, FO ,must
be transferred into the low latitude oceans and the
out of the high latitude oceans.This amount,
by
we
the
the
formulation
As
in
FA , of the flux be carried by
temperatures
(B).
and
flux
meridional
total
(A)
same
amount
FO , is transported
the mean meridional circulation (MMC) and diffusion(i.e. heat
flux mechanisms other than the mean meridional
-
55 -
circulation).
In
the
case
when
the
MMC
dominates
the
oceanic
heat
transport,Ky=0.The MMC carries warm surface water (at temperature
T3)poleward and cold deep water (at temperature
T6,which
equals
In Seller's(1969) model which included a one-layer
ocean,
T 4 ,by (B) ) equatorward.Hence
FO
=SSS A
where
a
diffusivity,
Ky,
of 5e8 cm**2/s was used. In his model,
vertical diffusivity,
Kz, is effectively
zero.
His
choice
the
of
parameters are therefore not inconsistent with ours.
Inspection of temperature maps shows that the
of
the
level
of maximum temperature gradient is ~5-10*C at low
latitudes.The pair(Kz,Ky)=(1,1.6 1e6) cm**/s is
sensitivity
temperature
chosen
for
the
The solution for this set of parameters is
studies.
taken to be our equilibrium state.
The actual mechanism of vertical diffusion in
is
not
fully
Robinson
and
structures are also reproduced in
1959).Jenkins(1979)
surfaces
in
atmospheric
ocean
the
found
that
Sargasso
Sea
variables(temperature
surface
oceans
Besides the thermohaline circulation
understood.
model proposed by
the
perturbations
are
Stommel(1959),thermocline
like
pure advective models(Welander
salinity anomalies on isopycnal
correlated
in
significantly
with
Valentia),suggesting that
advected
along
isopycnal
surfaces.The distribution of tritium in the same area can also be
fitted
vertical
better
to
diffusive
a
lateral advective model
If
model.
-
56 -
the
tracers
than a continuous
are
advected
quasi-horizontally along isopycnal surfaces, the Kz values can be
viewed
as
an
effective transfer coefficient as a result of the
tilting of the isopycnal
surfaces.
-
57 -
3.5 Comparison with observation
The steady state
solution
for
cm**2/s are summarized in table 3.4.
=
Ky
1.6e6
and
Kz=1.0
The steady state results are
compared with the appropriate observed variables.
those
The radiative fluxes (a,t,I,)compare favorably with
derived
Satellite
from
observation(Ellis
and
Vonder
1976).The solar radiation absorption at the surface are
Haar
compared
those values at 25 and 55*N in the Pacific (Clark 1967).The
with
long wave and turbulent fluxes (r,f)at the surface
values
Clark's
with
are
compared
at the same latitudes.The Bowen ratio(i.e.
the ratio of sensible to latent flux at the surface ) are 0.2 and
0.7 for low and high latitudes in the model,
0.3 in the Pacific at 25 and 55*N.
to
the
use
of
the
that
while
same drag coefficient for both fluxes. The
for
latent
warmer than the air by 1-5*C(Bunker
The total meridional
than
larger
that
flux
less than the value
mean
we
heat is larger when the sea is
1976).
calculated
in
the
model
is
observed at 45*N(Oort 1971).The ratio between
latent and sensible heat agrees well with observation
tuned slightly.
and
The discrepancy is attributed
coefficient for sensible heat transfer is
assumed
0.1
compare to
as
it
is
The oceanic heat flux is mostly attributed to the
meridional
nonetheless not
circulation
inconsistent
by
with
our choice of Kz.The choice is
oceanographic
measurements.
The temperatures T1,T2,T3,are identified as the sea level and sea
surface
temperatures.They
and Hsiung 1978).
compare
well with observation(Newell
The temperatures at the high
are consistent with figure 3.5 (Wust 1928).
-
58 -
latitude
oceans
For Kz= 1 cm**2/s,the model calculated a
of bottom water of ~18e6m**3/s.
an
amount
of
20e6m**3/s.
rate
Stommel and Arons(1960) estimated
Dynamic
oceanic sections showed a transport
from
production
calculations
of
Antarctic
at
different
water
bottom
one to a few million cubic meters per second (warren 1970).
Based on a salt
budget
calculation
of
the
Antarctic
waters,
Gordon(1971) estimated a rate between 20-50 million m**3/s.
- 59 -
3.6 Energy balance at different levels
it
results,
model
In order to better understand the
is
useful to look at the energy balance at different levels. Summing
equations 3.7-3.12,an equation governing the energy change of the
system is obtained.
T
T
(3.14)
=,
-
+ (
+
- I
)
Equation 3.14 states that the total energy of the system can only
atmosphere.
imbalance at the top of the
be changed by radiative
The imbalance are only functions of the atmospheric temperatures.
Term
(1)
is the net radiative flux into the low latitude areas,
W/m**2.In the steady
and has a value of ~81
must
it
state
be
balanced by term (2).
If
for
al,tl,a2,t2,are
some
unknown
decreased
reason,
total
the
the
radiative
radiative
decreased. In the steady state, changes in the
at
the system is
in
energy
fluxes
fluxes
the top of the atmosphere must be reflected in changes in the
surface temperature.
Summing equation
an
3.10-3.12
equation
for
the
ocean
temperature is
{Cs(TT 3 V(1-0()T
4
i
(4)
(3)
Term
(3),which
radiative
and
consists
turbulent
+ T)J
c(TsT
of
solar
fluxes,rl,
- 60 -
transmission
tl,into
and
fl,out of the low latitude
is
only
the
deep
ocean
the
Note that the energy of
ocean,is 24W/m**2.
changed by fluxes at the surface.
Summing equations 3.11 and 3.12 an equation for
is
ocean temperature
( C (Ts
S T,
T))=
T,
-
)
K, (T 3 - Ts)
4-
(7)
(f)
the
If we define a mean temperature of
deep
and
ocean
denote
deviations from the mean by primed quantities, then
T6
Both term (5)
positive.
and
Term(5)
<
<
o
negative
are
(6)
T5
remains
(7)
term
while
represents the downwelling of cold water into
the deep oceans and term(6) upwelling into the ocean surface.This
convective
cooling
heating(term 7)
offsetted
is
by
to maintain a balance.
of
The deep ocean contains the largest volume
hence
the
largest
effect of the deep
candidate
diffusive
downward
inertia
thermal
ocean
has
been
water
in the system.The flywheel
as
considered
for internal causes of climatic change.
about the ice ages,for instance, hinges on
the
a
61 -
possible
Newell's idea
beating
the convective(term 5 and 6) and diffusive terms(term 7).
-
and
between
4.Linear Adjustment and Sensitivity
In
parameter
this
chapter
we
will
examine
the
sensitivity
to
variations and adjustment to small perturbations about
versions.
a steady state of the model and some of its simplified
4.1 Methodology
The model can in general be represented as
=
(4.1)
F
i=1,
)
...
, n
where n is the number of variables(six for the model we described
) and x = ( x ,
in chapter 3.
equations(nonlinear)
...
, x.)
describing
the
time
variables.We consider small perturbations
state x .Linearization of (4.1)
x
set
the
of
evolution
the
about
of
the
steady
gives
X(
(4.2)
is the Jacobian evaluated
- (o)
xj
At
.Equation (4.2)
solutions of the form e
where
The problem
reduces
at
x o .We
look
for
becomes
J
-
(4.3)
matrix,
Fi(x)
and
to
finding
J .The solution of x
eigenvalues
can be represented as
L L(t) Ut
-
of
62 -
the
Jacobian
of
U- is the i'th eigenvector
where
eigenvalue )L
the
with
Jacobian
the
associated
.The Ci's are constants determined from
initial condition.
the
achieve
examine the
external
steady
present
sensitivity
to
solution
to
tuned
must therefore
solution.We
state
the
of
been
have
A number of parameters in the model
in
changes
the
It is customary to change a parameter by a
parameters.
We
fraction of its value and study the steady state so achieved.
approach
the problem by examining the slope of the tangent plane
of the solution in parameter space.
The set of n equations in the steady state are
o= F ( )
Each of the equations represents a surface in phase space
The
steady
state
is
determined
by
surface. To examine the behavior of
space,
we
look
for
variations
x.
of
the intersection of all n
the
solution
of
Fi's
in
parameter
along
their
surfaces.Treating an external parameter p as a variable, we get
K.O,
Rearranging terms and taking the limit 4p
the
change
an
expression
for
of the i'th component of temperature for a change in
the parameter p
is obtained.
The sensitivity of x 's
to pK is
determined
- 63 -
by the sensitivity of
parameterization
exists
only
if
and the
the
Jacobian
Jacobian
is
of
the
J
system.
invertible.This holds if the
linearized set of equations are linearly independent.
This approach has the advantage that once the Jacobian and
its inverse is computed, the sensitivity
external
parameter
can
be
multiplication.As discussed in
of
obtained
appendix A4,
the
simply
model
by
- 64 -
any
matrix
the convergence
steady state is not always guaranteed.Hence the customary
may involve a number of trials and errors.
to
to a
method
4.2 Temperature albedo and latent flux feedback
The contention that
latitudes
is
lower
surface
temperature
at
high
accompanied by high surface albedo constitutes the
temperature albedo feedback
(TAF).If
the
temperature
at
high
latitudes is reduced, the albedo increases due to the increase of
ice
cover.This
reduces
the
solar radiation absorbed and hence
reduces the temperature further.
To incorporate the temperature albedo
simple
models,
feedback
in
these
the ice line must be parameterized as a function
of the internal variables.
It
is
usually assumed that the ice is
situated at a latitude where the air temperature
falls
below
a
certain temperature.
Figure 4.1
shows
latitude
of
the
function
of
longitude
October.In
The
data
ice
January,
for
the
surface
air
limit
in
Southern
for
the
the
figure
the
months
temperature
of
at
the
Hemisphere as a
April,
July
and
Antarctic is almost free of sea ice.
4.1
is
taken
from
Schultz
and
Gates(1971-74).It can be seen that the temperature at the edge of
ice is quite variable, but generally in the range of -2 to -12*C.
Inherent in these box models is the
gradient
temperature
assumption
that
the
is constant within each box.Taking the ice
cover as a linear function of the local air temperature(T2),
where the subscript p denotes the steady
assumption
that
the parameter
the
high
state
value,
and
the ice edge temperature is -10*C(Budyko
can be evaluated.
latitude
box
the
1969),
When the air temperature
at
has reached a temperature of -10*C, the
-
65 -
high latitude box must be half covered with ice, i.e.
. =
-.. 0
o-
=-1.7*C has been used.
where T2p
assumed
ice line
on
ice
--2
0.5
0- -0 T
WL_
or
oT
or
the
c
It can be seen that
the
lower
edge temperature, the weaker dependence of the
local
temperature(i.e.
is
smaller)hence
the
The latent flux feedback is incorporated in the model
via
weaker the TAF.
its
parameterization
atmosphere(Bo(Xm))
in
terms
(see section
of
the mean temperature of the
3.3.3,
also
appendix
A3).
A
reduction in X2 is accompanied by an increase in ice cover, hence
increasing
the
amount
of
short
temperature of the system (as we
portion
wave
show
reflection and the mean
later).This
reduces
the
of latent flux and hence the total energy transported to
high latitudes, thus reducing X2 further.
-
66 -
4.3 A one-layer model
Table 4.1 shows the
list
of
models
we
have
examined,
arranged in order of their complexity.
To
consider
illustrate
one
table 4.1,
the
techniques
described
earlier,
we
layer models(atmospheric layer), model I and II in
where
analytical
solution
can
be
obtained.In
the
absence of an oceanic layer, the energy equations that govern the
atmospheric
temperatures
at
low
and
high
latitudes
equation 3.7 and 3.8 with rl+fl replaced by tl,
r2+f2(1-
(
from
)
by
t2) are (1 )
c, t :a,
(4.5)
C -,
(4.6)
where
4- &
2-+t4
is the atmospheric heat flux and equals
E (1
and x
-(A ---
--a + t£ -(A + X,)
-a
E
+,
+ B.(,x)) (X,-
XL)I
, X. are temperatures at low and high latitudes,
xm is the area weighted average temperature
X,
=
-X,+
X I)/(
of
61-1 )
the
atmosphere.
,
(1) For calculations with the one and two layer models, it has
been assumed that Cal
= Ca2 = Cp Ps/g, i.e. the effect of
moisture in determining the heat capacity has been neglected.
This
assumption
introduces
no
error in the sensitivity
calculations. We shall also show in section 4.5 (table 9.4a) that
while the atmospheric heat capacity is inversely proportional
to
the short time scales (/*-1,A2),
the distribution of weights in
the eigenmodes are unaffected. The longer time scales are also
relatively unchanged. This moisture effect is however included in
the calculation in model VIII where we wee that it increases the
time scales of the atmospheric modes by about 50%.
-
67 -
An equation governing the mean temperature xm is
(4.7)
4M
where
Ym - (A + BXm)
im is the mean solar heating and equals
Subtracting
(4.6)
(4.5),
from
an
for
equation
the
horizontal temperature gradient is obtained.
xX+
+
-
where x = x, -x, is the horizontal temperature gradient,
S= (a~+t,)- (4+
the atmosphere and
o
t~)
is
= E
With values of a ,
t
,
the differential
+
)L
a ,
t,
4-
A,
o
solar
heating
of
)
B obtained in
the
full
model substituted in, the mean temperature of the atmosphere is
-A
X
This is
= 12.5 C
the same as the mean atmospheric temperature of the three
layer model.
We note that this must be the case since the
energy
the
on
top
of
the
temperatures(see
atmosphere
section
depends
3.7).
-
68
The
only
balance
at
the atmospheric
equilibrium
temperature
gradient is
8-
-
The positive root is chosen since X
is positive by definition.
I
In the absence of atmospheric
dynamics,
=0
o(
and
the
radiative equilibrium temperature gradients is
X
XRE
In the absence of
9 OC
latent
heat
flux,
Bo(Xm)=O.The
temperature
gradient for this dry atmosphere is
25. 3 "C
if the same value of eddy diffusion used in the three layer model
is used.
The inclusion of latent flux
gradient
since the effective diffusion coefficient is larger.The
temperature gradient for a
moist
reduces
atmosphere,
the
temperature
Bo=0.4(from
with
table 3.4), is
<
Xm=21.9*C
Xp
This can be compared to the model
al.(1965)who
find
that
the
inclusion
results
of
Manabe
of a hydrology cycle in
their GCM reduces the meridional temperature gradient
total
flux
remains
determined by
atmosphere
the
the
relatively
net
total
constant.As
radiative
meridional
budget
et
at
the
the
while
the
total flux is
top
of
the
flux is rather insensitive to
internal dynamics(Stone 1978).
In order that the result is comparable to the full
the
coefficient
of
diffusion
-
is
69 -
tuned
to
achieve
model,
the same
atmospheric temperature structure
as
that
found
in
the
full
model, i.e. X =18.7*C with Bo unequal to zero.
Linearization of (4.5) and (4.6)
arranging
and
in
the
form of (4.3) we get
XL
(4.8)
where
C& is the columnar heat capacity =p
= 2 E (1+Bo)(x, -x
5385
)
i
a rL
and
g
Ps
is
the
change
latitudes(normalized
of
solar
by
local temperature X .As
radiation
So)
associated
derived in appendix
incident
with
A5,
on
high
a change in the
g
is
positive
since an increase in the high latitude temperature is accompanied
by
a
retreat
of
the
ice-line
and
hence
a reduction in the
reflected solar radiation back to space.
The indical equation is
(4.9)
+
It can be shown that all
s
A's
+(
o
of (4.8) are real(appendix A6);i.e.
there exists no oscillatory behavior of the perturbations.
In the absence of temperature albedo feedback(NTAF) Y{=0.2
-
70 -
and g=0O.The roots of (4.9)
-
_
-
are
-_
____
C
_____
2C,
A_
The inverse of the roots( [.,
relaxation
-
)
C
correspond
to
the
radiative
and energy redistribution time scales derived in Held
and Suarez(1974).The radiative relaxation time scale
interpreted
from
perturbations
(4.7).It
is
the
adjustment
of the mean temperature of
the albedo is fixed(i.e. Y{
the
is
time
scale for
atmosphere
when
is constant).
The associated eigenvectors U +,(_ for L and
A
are
: )-
L(
readily
I)
The eigenmode that corresponds to that of radiative relaxation
U
is
characterized
latitudes
while
characterized
by
that
by
identical changes in both low and high
for
opposite
redistribution LU
energy
changes
in
,
is
low and high latitudes,
presumably due to the effect of heat fluxes
that
tend
to
cool
down the low latitudes and warm up the high latitudes.Note that q
does
not
enter the solution.In the case of NTAF, the eigenmodes
are unaffected by the
parameterization of the latent flux.
G=5.976, q=0.373, g So=1.064 W/(m**2
/- and
K),
numerical
values
With
for
?L.are
X
=-3.69(yr)
corresponding to an e-folding time of 98 days,
AL
=-27.1(yr)
corresponding to an
e-folding
time
of
13.1
days.(see table 4.1)
In the case of TAF, the system can become unstable if this
-
71 -
feedback is strong enough (appendix A6).
We can examine how the flux of
adjustment
(i.e.
process
with
TAF
latent
heat
affects
by i)computing the
A's with q=0O
Bo constant, model I) and ii) with q unequal to
zero(model
II).Table 4.2a summarizes the results of calculations for i)
ii),
the
and
together with the case of NTAF.
It can be seen that the inclusion of the TAF
magnitudes
of
A's
q.In
feedback,
radiative
the
the eigenvalues, i.e. adjustment to perturbations
are less rapid.Note that the sum of the
the
reduces
presence
of
relaxation
latent
(energy
flux
is
redistribution)
independent
of
adjustment
to
processes
are
faster(slower).(see table 4.2)
The radiative relaxation times are 74,
NTAF,
TAF
and
TAF
with
latent
redistribution times are 12,
a
vertical
of
the
static
that adjustments
13.5 and 13.1days
energy
respectively.For
section 9.3.4).In a radiative
to perturbations
stability
present model
of
)
the
dynamical
stability of the Earth, Stone(1972) found
damping time of about 34 days.
static
feedback.The
for
scale height of ~8 Km, the radiative relaxation time
is ~120 days(Goody 1964,
model
flux
94 and 98 days
the
are damped
It
can
atmosphere
adjustment
are
be
is
oscillations,
shown
that
constant
simple
if
( as
damping
with a
the
in
the
with
no
oscillation in his model.
Table
temperature
4.2b
with
summarizes
respect
to
-
the
each
72 -
sensitivity(changes
parameter)
of the model
in
to
variations in the solar constant, So, the distribution
insolation,
S;the
long
wave
parameters,
A
and
of
solar
B;and
the
atmospheric diffusion coefficient E.
For an increase in So, the
m**-2)
with
NTAF,
.67*C/((W m**-2)
,
.53
changes
.41
*C/(W
.55*C/(W m**-2) with TAF and .52 and
with TAF and latent feedback.
The parameter S governs the
insolation.
.46,
are
Its
role
is
latitudinal
similar
to
that
distribution
of
of
the obliquity
angle.If S is increased, more(less) solar radiation is
deposited
at
low(high) latitudes.The consequence is a temperature increase
at
low
latitudes
and
enhances(reduces)the
a
decrease
the
mean
increase
at
low
an increase in
temperature
reduction at high latitudes
is
effective
heat
meridional
temperature gradient.
Increases in
temperatures.
an
atmospheric
latitudes.
TAF
more
is
the
the
the obliquity
atmosphere
than
angle
since the
compensated
by
the
latent flux included, the
increased,
hence
reducing
the
the long wave parameters A and B reduces
the
For
changes are smaller
So.Since
of
latitudes.With
transport
high
at high(low ) latitudes. As pointed
change
out by Cess and Wronka(1979),
increases
at
a
change
of 1 W/m**2 in A, the temperature
than those associated with
increase
in
temperature,
both
latent
parameters
flux
the same change in
reduces
feedback
the
increases
mean
the
sensitivity.
The eddy diffusivity E is a measure of the
efficiency
of
the atmosphere to transport heat.In the case of NTAF, an increase
in
E
lowers(raises)
the
temperature at low(high)latitudes.The
-
73 -
mean temperature is unchanged.With TAF, the temperature change at
high latitudes is accompanied by ice retreat and
short
wave
absorption.The
shared by both low and
extra
high
warming
latitudes
to
hence
increase
derived
from TAF is
a
temperature
give
increase globally.
Changes in the obliquity and efficiency of heat
do
not
change
the
mean
temperature
of
the
substantially whereas changes in So, A and B do.It can
that
the
effect
transport
atmosphere
be
noted
of the latent flux feedback is to increase the
sensitivity of the model to mechanisms that tend
mean temperature of the system.
-
74 -
to
change
the
4.4 Two and three layer models
The
separate
of
parameterization
the
oceanic
and
heat fluxes enables us to evaluate the impact of the
atmospheric
inclusion of an active ocean in simple energy climate models.
From the set of equations
full
(section
model
3.7-3.13
which
describes
the
it can be seen that if Kv and Sg are
3.5),
set to zero, the deep layers of the oceans are decoupled from the
upper layers.The set of
describes
(3.7-3.10)
equations
with
,
S
Kv=O
a two layer model which includes the atmospheric and a
shallow oceanic layer.
The governing equations for the two layer model are
t
(4.10)
-I± I +T
(4,11)
5
(4.12)
) t
(4.13)
,
L4
C
,
-
2-,
-(,-n)
± K(T
3
-T
4
)
The oceanic layer can act as an active medium for meridional heat
transport
depending
or
on
simply
the
a
source
choice
-
of
75 -
of
K,
moisture
the
for
evaporation,
meridional
diffusion
If K=O, the model ocean acts simply as
coefficient in the ocean.
a moisture source with no meridional transport (swamp).
From equation (4.12), we note that the total amount of long
is
latitudes
available
If
tl
rl+fl
the
air
temperature
given,
is
temperature
is determined via(4.14).
similar
Using
regions.
sea
maximum
the
surface
Although the equations used
here are developed for an areal average they must
local
low
tl.
at the surface,
(4.14)
at
surface
short wave radiation
of
amount
the
by
limited
the
at
wave(rl), sensible and latent flux(fl)
empirical
hold
also
for
relations
radiative and turbulent fluxes and specifying an air
in
the
temperature
of 27*C and a relative humidity of 70%, Newell (1979) showed that
the
present
day maximum sea surface temperature is ~ 30*C.
(4.14),
the
maximum
for
value
is
or
This is
the
temperature,
E
air
=0).
A
7
relevant to the problem of climatic changes as changes in
the sea surface temperature and surface air
region
From
(4.14).
when the radiative and turbulent fluxes
maximum
are in equilibrium(i.e.
from
rl+fl is tl.As horizontal heat
maximum
the
transport could only reduce
temperature
estimated
be
can
maximum air temperature
The
of
their
maximum
could
yield
in
the
about
the
temperature
information
variation of the partitioning of the short wave absorption in
atmosphere and at the surface.
-
76 -
the
Models III and IV corresponds to models where
act
as
4.1).Models V and VI corresponds to
table
swamp(see
a
oceans
the
with
models with diffusive transport and VII and VIII
diffusive
and MMC transport.The difference between III and IV (V and VI) is
the inclusion of the latent flux feedback.
In section 3.5 the steady state solution to the full model
Ky,
parameters
the
tuning
is calculated by
Kz
the
that
so
constraints (A) , the portion of oceanic heat flux Ro is constant
and
the
(B),
and
0.3,
equals
oceans remains
latitude
high
isothermal, are satisfied.In model VII, the
values
of
Ky,
Kz,
obtained from such tuning are used and are treated as constants .
From table 4.7a and b, it can
at
stability
statically
static
the
For a small increase in
the changes in T and T
constant,
that
the high latitude oceans is extremely sensitive to
in the solar constant.
changes
seen
be
unstable
physical unreality,
Because
ocean.
this
opposite,
are
is
model
of
not
this
the solar
imply
which
a
sensitivity and
considered
detail
in
further.
There is no reason to presume the constancy of the portion
of oceanic heat flux, Ro , in
Newell's
idea
ice
about the
climatic
a
ages
change.For
actually hinges on Ro being
variable.In calculations associated with model
(A)
is
relaxed
while
(B),
that
VIII,
constraint
the high latitude oceans are
The assumption (A)
isothermal , is maintained.
instance,
illiminates
the
possibility of a statically unstable ocean.
The prognostic equation for the deep ocean
-
77 -
temperature
at
high
latitudes
T (3.12) becomes a diagnostic equation.for the rate of
bottom water formation(S ).
This is the amount that is needed to
maintain the static stability of
fractional
area
of
the
formation, oL
,
high
latitude
(or
the
oceans.The
fraction surface
cooling at high latitudes that must be transferred to
below)
can
be
via
calculated
the
layer
set of 6 prognostic
(3.13).The
equations(3.7-3.12) in model VII for the variables T ,
...
,
now replaced by the set of 5 prognostic equations (3.7-3.11)
a diagnostic equation (3.13) for the variables T ,
...
,
T
is
and
T , "C )in
model VIII.The underlying assumption in this model is that enough
bottom water has to be formed to maintain the static stability at
the high latitude oceans.
In all the simplified versions(1 and 2 layer models),
of
coefficients
atmospheric
the
and oceanic diffusion are tuned so
that the temperatures are the same as those obtained in the
full
model.
The sensitivity to parameter variations of models III-VIII
and
the
linear
calculated.The
adjustment
elements
for
times
eigenvalues
calculated by the QR method(appendix A4).
the
sensitivity
and
linear
are
of
the
Corresponding
Jacobian
to each
adjustment time scales are
calculated for a control model with no TAF.
adjustment
III-VII
of the Jacobian matrix are approximated
by finite different quotients and
model,
model
Since
instantaneous
is assumed for the high latitude oceans, there exists
five modes for model VIII. The moisture effect in determining the
atmospheric heat capacity is included for this
calculation.
The
results of such calculations are summarized in tables 4.3-4.8.The
- 78 -
sensitivity
although
of
later
parameters,
the model to a number of parameters are included
discussions
will
So .
- 79 -
focus
only
the
external
4.5 Parametric dependence of adjustment time scales
In this section we
examine
the
of
dependence
time
scales of the one and two layer models on the various parameters.
From tables 4.3-4.6, it can
be
seen
that
there
exists
of adjustment time scales. In the absence of TAF, the
(2)
first two adjustment times are"3 and 5 days.The eigenmode of the
separation
former
characterized by opposite changes while the latter is
is
characterized by changes that are of the same sign in
almost no change in
two component(atmospheric temperatures) with
the
and fourth components(oceanic temperatures).The third
third
eigenmode, with an e-folding time of ~ 3 years
by
first
the
changes
opposite
characterized
is
the low and high latitude temperatures
in
while changes in the atmospheric and oceanic temperatures in
low
and
high
latitudes
within
the
same
the
latitude box are in
phase.The fourth eigenmode, with an e-folding time of - 16 years,
sign
is characterized by changes of the same
weight
and
equal
almost
all components.In the presence of TAF, the adjustment
in
times become less rapid.The e-folding time of the fourth mode(the
longest adjustment time) is increased if latent flux feedback
included.The
general
of
characteristics
the
eigenmodes
is
are
relatively unchanged with TAF.
The time evolution of small perturbations
in
the
system
can be described in terms of the eigenmodes.With perturbations in
the
atmospheric
temperatures
are
excited.These perturbations
perturbations
modes are
(2)
in
the
first
rapidly
two
modes
damped(3-5
can
be
days).With
the oceanic temperatures, the third and fourth
necessarily
excited.These
see footnote on page 67.
-
80 -
perturbations,
with
an
e-folding
time
in the range of 3-20 years, will persist in both
Hence perturbations in
the atmospheric and oceanic temperatures.
the
temperature
oceanic
have
will
lasting
on
impact
its
atmospheric counterpart.
The eigenmodes of the three layer model are shown in table
4.7.The eigenmode with an e-folding time of 3.5 and 5.5 days have
similar characteristic
eigenmodes
to
those
of
the
two
layer
model.But
with e-folding time in the intermediate range(1.4 and
time
9.7 years) have dissimilar feature to those (with e-folding
3
and
the
years)of
20
two
layer model.The eigenmode with an
e-folding time of ~300 years is characterized by major changes in
the six'th component(high latitude
with
eigenmode
the
longest
deep
adjustment
ocean
temperature).The
time
(1370 years ) is
the
characterized by major changes in the fifth component,
ocean
temperature
deep
at low latitudes.Box 5, the deep ocean at low
and
latitudes contains the largest volume of water in the system
the
hence
largest
heat
capacity.
A
long
time scale is also
present in model VIII. This mode is characterized by
changes
in
the deep ocean as well.
We noted in section 4.3 that the radiative relaxation
energy
redistribution
time scales are
100 and 13
days in the
rise
one layer model.The inclusion of an oceanic layer gives
two
longer
time
and 16 years ),
scales(3
low
and
a
number
changes
high latitudes and global concomitant changes)
and two shorter time scales (
models,
to
with characteristics
similar to those derived in the one layer model(opposite
between
and
3-5 days
).
In
the
two
layer
of parameters have been introdued.It would be
-
81 -
interesting to examine the dependence of the time scales
various
parameters.This
on
the
was done, thought not quite exactly, by
calculating the eigenvalues of the steady states for changes in a
particular parameter.
Table
model
VI
4.9 shows the eigenvalues of the
with
TAF
for
exchange coefficient in
flux
exchange
a
steady
the bulk aerodynamic air
formulation,
K,
sea
turbulent
the oceanic diffusivity,
oceanic
E, the
layer.
can be seen that the most rapid adjustment times scales ( 1 ,
-2 )iare
very sensitivity to changes in
transfer
coefficient,
which
heat exchanges between the
increased(the
C,
the
bulk
aerodynamic
is a measure of the efficiency of
atmosphere
and
the
sea.
If
C
is
exchange process is more efficient) the adjustment
i.e.
processes becomes more rapid,
sensitive
1 and
the magnitudes of both
2 are increased. The longer adjustment time scales (
most
of
50% change in the parameters C, the
atmospheric diffusivity, and h, the depth of the
It
state
3
,
4)
is
to h, the surface layer depth. If h is increased
(decreased) to 300 m(100m), the e-folding time of the fourth mode
is 30 years(10 years).It therefore
adjustment
time
seems
that
the
most
rapid
scales are strongly dependent on the efficiency
of air sea exchange while the slowest adjustment time ( A 3 ,
are dependent on
atmosphere
h. Since the
corresponds
meter column of water,
to
columnar
heat
capacity
A)-1
4
of
)
the
roughly the heat capacity of a three
the surface layer depth
determines
to
a
large extent the columnar heat capacity of the two layer model.
The third and fourth modes in the
- 82 -
two
layer
models
are
analogous
to
the energy redistribution and radiative relaxation
time scales in the one layer model.The
first
two
modes,
which
characterize atmospheric temperatures, are therefore new findings
in this model.
We have neglected the effect of
the
atmospheric
heat
capacity
moisture
the
determining
for the one and two layer model
calculations. Table 4.9a summarizes the
and
in
adjustment
time
scales
eigenmodes for a 50% change in the dry atmospheric heat
capacity. It can be seen that while the time scales are inversely
proportional to the atmospheric heat capacity, the eigenmodes are
almost unchanged. The longer time
scales
( A3, /4)
are
also
unaffected. It can therefore be said that the neglect of moisture
in
the atmospheric heat capacity does not change
the longer time scale modes significantly.
-
83 -
4.6 Comparison of models
In section 3.6, we showed that the energy
depends
only
of
the
system
on the atmospheric temperature.In order to compare
the sensitivity of the model to different parameters, we define
where Tm is the area weighted average
Pin
(Y
units
of
the
air.
of *C, can be interpreted as the change of the mean
air temperature in hundredths for a
parameter.
temperature
The
values
for
one-percent
the
change
different
in
the
parameters
and
different models are listed in table 4.10.It can be seen that
is the largest among
((So)
i.e.,
all the parameters,
the
mean
temperature is most sensitive to the Solar Constant.
Aroused by the
Budyko
to
surprising
results
of
the
Sellers
and
models that a small change in the solar constant can lead
an ice-covered earth, energy models developed along this line
have been examined in detail for their sensitivity to changes
solar
constant.This
is
done,
for example, by calculating the
equilibrium ice line as a function of the
the
sake
of
sensitivity,
model
comparison,
in
let
us
solar
define
constant.
the
For
ice-line
, , as the fractional change in the ice- free areas
at high latitude associated with a change in the solar constant
which is simply the slope of the equilibrium curve.If3 >0,
an
increase
in
i.e.
the solar constant is associated with less ice,
- 84 -
can
be
as follows:if small perturbations in the system cause
understood
a
cover,
ice
the
a decrease (increase) of
requires
stable.This
be
to
said
is
solution
the equilibrium
the
state
perturbed
larger (smaller) solar constant for equilibrium.This
energy deficit (surplus) drives the ice line back to its original
states,
stable
position.The converse holds ifOq O.For
we
say
that a model is more sensitive if Iis larger.
of the ice-line in
From the parameterization
can be expressed as function of
the
section 4.2,
temperature
air
at
high latitudes
p
a
)T
bS
So0
)T
)5.
'I-
T
In the absence of atmospheric fluxes, the ice line sensitivity of
the one
can
model
layer
calculated
be
by
substituting
the
Jacobian in (4.8), with q=G=O into (4.4) with p =So.With
tt,
a
_
.377
So
we get
(X,
as°
6- s°
-
t+
so
)
775
This solution is stable since the positive feedback due to TAF is
compensated by the negative feedback of long wave emission.
sensitivity
is
)qfor
decreased
if
dynamical
fluxes are introduced.
the different models that have been considered can
be calculated from tables 4.2-4.8.The results are
table
4.10.
This
Together
with table 4.1,
85 -
in
they provide some insight
about the physics that controls the sensitivity.
-
summarized
denote the ice lire
e (()
difference
between
sensitivity
for
model
i.The
model I and II(V and VI) is the inclusion of
latent flux feedback.We noted that
that is , the sensitivity of the model with a swamp is
that
with
an
close
active ocean.This is surprising since the oceanic
heat transport is not modelled in the model with a swamrr.The
value
for
model
VII
low
VII and VIII shows that the MMC in the oceanic
transport reduce the sensitivity more.
models
to
The low
value
of Oi[ for
and VIII suggest that the MMC transport is efficient
in smoothing out temperature gradients.
Model I II and
V
do
not
incorporate
the
latent
that
the
inclusion
feedback increases the sensitivity as P?
for
models
feedback.The
result
also
showed
feedback is less than those without.
- 86 -
flux
of this
with
this
As we showed in section (3.6), the
atmosphere
depends
system
temperature.Another measure
of
only
energy
on
model
of 'the
the
air
surface
sensitivity,
(
is the global sensitivity,
feedback comparison,
earth
useful
for
, (Schneider
and Mass 1975)
where Tm is the mean surface air temperature.In
TAF,
the
absence
of
a) )can be evaluated from equation (3.14).In the
(denoted
steady state equation (3.14) is equivalent to (4.7)
4 -(A +em)
0
An expression for Tm is
6
which gives
13
AS.
The
value
calculated
of
PO
from
is
in
so
excellent
empirically
agreement
determined
The
relation of Lian and Cess(1977).
global
with
results
temperature
sensitivity,
albedo
from
table 4.10 of the different models, is compared with other models
in table 4.11.
The parameter'y,
defined as
K-
- 87 -
is a measure of the effect of TAF.If
is
large
global
then
the
global
Lian and Cess(l.c.)
for
models
III-
is
VI,
is
no
TAF.If V
in hand.The inclusion
instances,
the
enhances
other models, as calculated in
for
for
included
value
comparison.The
ofY
which is in the general range of .23-.27,
the
place them in the category with
which
hand
all
in
feedback,
sensitivity
there
is strong.It can be seen that both
and ice line sensitivity go
of the latent flux
TAF.The
feedback
=0,
model
of
Lian
and
Cess,
employed empirical albedo-temperature relationship derived
from observation, and the GCM of
Wetherald
and
Manabe
(1975).
These set of models are less sensitive to variations in the solar
constant than Sellers' and Budyko's model.
the
oceanic
heat
such sensitivity is
With the inclusion of
transport by the mean meridional circulation,
reduced by
-
about 5%.
88 -
the
5 .Discussion
5.1 Changes in climate and solar constant
solar
the
in
variations
any
refutes
Radiometric
constant.
constant
solar
measurements over the past years showed that the
or
supports
that
evidence
observational
no
is
There
over
the
period
1969-76(Wilson and Hickey 1977), To better understand
the
effect
has
of
within
to
unchanged
remained
.75%
a variation in the global climate, we shall examine the
such
response of the model to such changes.
for
to
increases from 0.2
are
latitudes
temperature
atmospheric
-1.89
are
latituds
at
changes
and
are
changes
temperature
those
than
-1.02*C
at
latitudes
surface
-.54*C.Lindzen
and
-1.71
high
and
ocean
the
while
The
latitudes.
high
low
low
at
changes
temperature
The
0.237
larger
VIII
in the solar constant.The ice cover
change
percent
one
a
model
of
Table 5.1 shows the steady state solution
and
pointed out that this feature is common in energy
Farrell(1977)
reduction
the
constant,
models.For a decrease of the solar
in
absolute magnitude is larger at low than at high latitudes simply
of
because
the
sphericity
the
of
Earth.In
absence
the
of
a
meridional transport, radiative adjustment alone would produce
change
larger
at
low
latitudes.
effect
The
of
dynamical
transport is to reduce the temperature at low latitudes further.
Reconstruction of sea surface paleotemperature showed that
during an
ice
age,temperature
changes
at
low
latitudes
are
smaller than those at high latitudes(CLIMAP 1976).Introducing the
so
called Hadley adjustment Lindzen and Farrell (l.c.) were able
- 89 -
than
to reproduce tropical changes that are less
high
latitude
changes.The Hadley adjustment parameterization has been commented
Warren and Schneider(1979).In a model where the planetary
by
Schneider
temperaure,
the
of
function
a
as
aldedo is parameterized
local
surface
and Gal-Chen(1973) were able to reproduce
the same feature.
While the results of Sellers and Budyko suggested that the
have
others
pointed
constant,
as the cause of the ice
amount
the
that
argued
ages.Simpson(1934)
increase
an
to
solar
the
of
reduction
a
to
ice ages could be due
is
cover
ice
of
on the amount of moisture available.For an increase in
dependent
the solar constant,
transport,thus carrying more moisture for the
moisture
the
in
increase
an
by
accompanied
tropics,
build
the
over
evaporation
increased
is
there
up
the
of
glaciers.While Simpson's idea has neglect ed the thermodynamics of
ice
formation,the
increase in moisture transport is
in
attained
our model.For a 1% increase in the solar constant,the atmospheric
e 15W, of which
flux increases from 3.67 to 4.02
is
the
latent
solar
~1.24
W
e15
flux, compared to 1.04 if there is no change in
heat
constant.This
20%
increase
in
the
moisture
flux
provides the mechanism which is required by the Simpson theory.
For a decrease in the solar consta nt,the oceanic transport
increases
drastically
from
the
total
increase,although
1.5
to
2.0
meridional flux
,almost
is
slightly.The steady state response of model VI (two
diffusive
in
the
oceanic
solar
a
30%
increased only
layers
with
and latent heat flux) to a one percent change
constant
has
also
- 90 -
been
calculated(result
not
shown).An
increase
in
the oceanic flux was also found although
the change is only ~3%.From table 5.1,it can
be
seen
that
the
increase in oceanic flux in the three layer model (model VIII) is
due
entirely
to
the
enhancement
transport is actually decreased
constant.
for
of
a
the
MMC.The
decrease
in
diffusive
the
solar
The increase in the rate of bottom water formation is
attributed to the increase cooling (r2+f2) at high
fractionl area of formation,do,
-
is reduced.
91 -
latitudes.The
5.2 "Simulating" the ice ages
General circulation models have been used to simulated the
the
reconstruction of surface temperature,surface albedo
CLIMAP
to
0.1
of
cover
state solution of model VIII for a prescribed ice
and
model
our
the ice line.Table 5.2 shows the steady
in
variations
much
but
similar
a
performed
1974).We
1977,Williams
simplified experiment by examining the sensitivity of
VIII
and
boundary conditions(Gates 1976,Manabe
and ice topography as
Hahn
using
B.P.
global distribution of climatic variables for 18,000
the case of 1 =0.3 (ice age) the atmosphere and the
0.3.For
than
ocean are colder
Y1 =0.2
with
case
the
,table
(present
3.4).Because of the decrease in atmospheric temperature gradient,
the
atmospheric
decreases.Such
flux
mostly
is
decrease
a
reflected in the decrease in the moisture flux as Bo,the ratio of
latent to sensible flux,is reduced.
These changes are consistent
with results from the GCM simulations cited above.
Because
associated
of
the
tropics
with lower sea surface temperature,Kraus(1973) argued
that the mid-tropospheric meridional
have
in
evaporation
decreased
the
decrease
a
decreased.With
temperature
in
the
gradient
must
baroclinicity,the
atmospheric circulation was weaker and hence a weaker wind driven
ocean circulation.
transport
is
In
decreased.
calculation,the
our
This
diffusive
diffusive
transport
mechanisms other than MMC.The transport associated with
oceanic
is due to
the
MMC
is enhanced.This intensification is due entirely to the increased
cooling at high laitudes.
The total energy flux is increased.
In comparing tables 5.1 and 5.2,we noted that the
-
92 -
changes
in
temperature
the
all
flux
and
components for an increase(
cover
ice
prescribed
the
for
changes
of
solar
the
0.1(0.3).Hence
of
external
temperature and flux changes due to an
the
as
decrease) of the solar constant is in the same direction
cause(variation
due to
constant) is indistinguishable from those
variation in the internal variable,the ice cover.
In
section 4.2,the ice cover , '
as
parameterized
,is
a
function of the air temperature at high latitudes
T
0(+ +Z.
(5.1)
3 is evaluated by assuming a
The parameter
calculate
We
temperature
meridional
gradient within each box and an ice-edge temperature
temperature
of -10*C.
constant
the
and
parameter
ice
edge
temperature T2 for the
required if the steady state
caseN =0.3(0.1)
the
is a solution to (5.1).This can be done,e.g.
for
the case of rY =0.3,by solving the following pairs of simultaneous
equations.
0,___
T
0(%_
T, - T;ice e e
5
=-2.22.The solution is
where Tzp=-1.73;Tz 1
-
=_.2.0 5"CI
-ie e
-
3.2 c
The results for both calculations are included in
required
ice
edge
temperature
- 93 -
is
table
5.2.The
~-3.2*C for both cases.This
4.1).The
figure
observation(see
from
is five to six
parameter
the
of
magnitude
range
of
temperature is not entirely out
temperature
times larger than that evaluated for an ice edge
-10*C.The
strong
dependence
of
ice
the
of
on the local
cover
temperature enhances the TAF.
The
linear
dependence
of
the
TAF
on
the
ice
cover
parameterization is derived in appendix A5, and the condition for
linear
instability
of
one-layer model derived in appendix
the
A6.With a five fold increase in the TAF(gSo),the one layer
becomes
unstable
for such a parameterization.
two and three layer models is worthy
-
94 -
model
The stability of
of further investigations.
5.3 The deep ocean response
The deep ocean is not
theories about the cause
deep
climatic
deep ocean with a decrease in the
Worthington
intensity of the deep circulation while
that
water.We
shall
examine
and
examine
paleotemperature
of
is
the
cooling results in colder and denser
intense
the
energy
for the ice ages.The theories of Newell
a warmer
to
point
Weyl
opinion
previous
any
The changes in the deep ocean are relevant to
inertia.
and
in
volume,the deep ocean has a large thermal
huge
its
models.With
included
the
ocean
deep
of
record
the
physical implications of the
model response in this light.
The
means
various
oceanic
of
reconstruction
their
and
limitations
for
the
records
have
been
paleoclimatic
reviewed by Savin(1977).Sea surface paleotemperature records
be
derived from the distribution of planktonic(surface dwelling)
distribution
of
benthic(deep
generally agreed that the
during
deep
from
inferred
be
foraminifera while that of the deep ocean can
the
can
dwelling) foraminifera.It is
ocean
cooled
down
(by
~15*C)
the Tertiary 63 to ~1 million years before present (Savin
1977,Shackleton 1979).
Complete records of the Pleistocene
are
given
in
isotopic
temperature
Emiliani(1955) and Emiliani and Shackleton(1974).
Shackleton(1967) pointed out that most of
the
variance
in
the
isotopic record is due to isotopic changes in sea water and hence
the
ice
isotopic temperature curve is better interpreted as a global
volume
curve.From
foraminifera,maps
of
the
the
SST
- 95 -
distribution
at
18,000
of
planktonic
years B.P. have been
The sea surface was found to be
constructed.
18,000
at
colder
B.P. than at present.There is ,nonetheless,not a coherent picture
for the deep ocean at 18,000 B.P.
Atlantic
North
the
masses:
water
of
the
between
the
NADW
to correlate negatively(Lohmann 1978).
found
AABW,are
the
of
carbonate and oxygen-18 isotope, taken at the Rio
Grande Rise(~30*S) which marks the transition
and
and the
Water(NADW)
The downcore distribution
Antarctic Bottom Water(AABW).
percentage
Deep
major
two
The present day deep ocean is characterized by
Since AABW is more corrosive to carbonate ,the correlation can be
during
the
glacial
last
other benthic foraminfera,taken at the
of
abundance
period.The
AABW
of
increase
interpreted as an
same site,bears no resemblance to the oxygen-18 record.
distribution
The
foraminifera
during
faunal
of
assemblages
the glacial period indicated a cessation of
Arctic Bottom Water and a diminution of NADW
1974,1979;Streeter
formation(Schnitker
and Shackleton 1979), Because of the dominance
of the warm dwelling species Schnitker(1974) suggested
abyssal
water
in
the
Atlantic
North
oxygen-18
the
signals
in
the
that
the
was probably warmer.The
contrary was suggested on a comparison of the absolute
of
benthic
of
magnitude
benthic species between the
Atlantic and Pacific cores(Streeter and Shackleton 1979).
We showed in section 3.6 that the temperature of the
ocean
is
maintained by diffusion and convection.Emiliani (1954)
interpreted the deep water temperature as representative
situation
deep
at
the
high
latitude
surface
interpretation is in line with assumptions in our
- 96 -
oceans.
model
of
the
Such
VIII.In
this model, the high latitude oceans are isothermal.Therefore box
with
the
surface.From the results of the two numerical experiments in
the
5,
deep
the
ocean,
is
in
communication
direct
previous sections,it was found that the deep ocean is colder when
is
there
increase
attributed to an
age"
an"ice
extent(table 5.1 and 5.2).This cooling is
ice
larger
component.During
convective
the
in
the MMC is more intense.The increase in the rate of
surface
bottom water formation is due entirely to an increase in
cooling 'of the high latitude oceans.
Since this amount of bottom
water is that which is needed to maintain the static stability of
the
latitude
high
oceans,such an increase is necessary for any
increase in the surface cooling in our model.
actually
ocean
warmed
up
deep
the
if
model
Although we cannot conclude from this
or cooled down during an ice age,the
been
physics that controls the temperature of the deep ocean has
diffusion
illuminated.Downward
the deep
for
warming
provides
ocean.During the last ice age when the sea surface was colder,the
warming
due
controlling
to
diffusion
mechanism
(Kz)
must
have
remained
component(formation of bottom water)
deep
for
cooling
the
ocean.The strength of such cooling is dependent on the rate
of bottom water formation(SB ).If this rate
the
convective
unchanged.The
provides
the
if
reduced
been
amount
is
age.Newell(1974)
the
to
of heat loss from the surface at the high latitudes,
(as our model VIII),the rate must have increased
because
proportional
suggested
is
rate
formation( oC )
is
that
decreased
decreased
- 97 -
the
as
due
deep
the
to
during
ocean
larger
ice
warmer
was
effective
a
an
area
sea
of
ice
extent.Weyl(1968) suggested that the temperature of formation
)
was
higher
because
the
area
latitude closer to the equator.
deep
ocean
due
of
(TS
formation is located at a
Worthington's
idea
of
a
cold
to intense cooling is consistent with our model
assumptions.
- 98 -
6. Summary and Conclusions
fall
which
transport
into
the
oceanic
of
estimates
various
the
reviewing
On
categories
method,ii) surface balance method and iii) direct
find
residual
i)
of
heat
calculation,we
even in the sense of transport among them. In
disagreement
the
the direct calculations, heat tranpsort associated with
net
meridional flow is left out. The heat flux can be estimated via a
knowledge of the fresh water transport.
The
surface
the
from
transport,calculated
balance
of
the
pattern
from
the
distribution
of
agreement
evaporation,precipitation and river runoff,is in close
with
water
fresh
oceanic
of
distribution
global
atmospheric moisture transport calculated
of
moisture
atmospheric
content
and
circulation. This finding has strong implications on the accuracy
for the observations presently available.
It is found that there is a southward flux of fresh
in
the
Pacific
and
a northward flux in the Indian ocean and a
major part of the Atlantic.
the
reinforces
The associated heat flux
heat flux derived from direct estimates. Results from direct
estimates, which indicated a northward heat flux in the
and
a
southward
Atlantic
heat flux in the Pacific, showed that the heat
transport associated with the mean meridional circulation is
dominant mechanism of transport.
the
water
temperature
and
salinity
Heat flux calculation, based on
distributions in the oceans at ~
40*N showed a similar pattern (Stommel
global
the
and
Csanady
1980).
This
pattern of heat transport is also consistent with Stommel
and Arons' model of the abyssal circulation and the
- 99 -
distribution
of
and salinity in the oceans. It is thus suggested
temperature
that the mean meridional circulation is the most important in the
of
global consideration
oceanic
the
although
transport
heat
possibility of the dominance of other mechanisms in local regions
is not ruled out.
terms
The mean meridional circulation is parameterized in
of
the rate of bottom water formation. This rate is proportional
From
to the surface heat loss from the high latitude oceans.
of
analysis
an
equations in our model,the cooling of the deep
the
ocean due to the formation of bottom water at high latitudes must
be balanced by the downward diffusion of heat at
If
so
tuned
which control the deep ocean temperature are
parameters
the
that
agree
results
model
the
of
magnitude
observations,the
latitudes.
low
present
with
and
horizontal
the
vertical
derived
diffusivity used are in the general range of those
day
from
analyses of the distribution of tritium. The rate of bottom water
formation
agreement with
and
Stommel
is
model
the
in
calculated
Arons'
~18e6 m**3/s, in close
(1960)
and
Gordon's(1971)
estimate. This amount is close to the maximum amount allowable in
the
model
if
all
oceanic
the
flux
heat
is due to the mean
meridional circulation.
Because of the dependence of the long wave flux at the top
of the atmosphere on the surface temperature,the
system
is
temperature
atmosphere
energy
of
the
determined by the surface temperatures. The meridional
gradient
to
dependent
is
distribute
heat.
At
on
the
ability
radiative
the
equilibrium,the
temperature between low and high latitudes is ~90*C.
- 100 -
of
Atmospheric
the
reduces
dynamics
gradient to ~20*C.
If latent heat flux is
included,the effect is to reduce the temperature gradient more.
is
perturbations
The
examined.
small
adjustment,one
of
modes
to
models
The adjustment of the one and two layer
common
radiative relaxation and one of energy redistribution,are
to
sets
both
models. Temperature -albedo feedback tends to
of
reducing the rate of radiative relaxation.
time
scales
effect on
But its
the
are small. These modes are dependent on
in
exist
the heat capacity of the system. Other modes
layer
feedback
TAF,increasing the rate of energy distribution and
the
adjustment
flux
heat
decrease the rate of adjustment. The latent
modifies
of
the
two
models. These new modes,which characterize the atmospheric
sensitive
the
to
of
efficiency
time
shorter
temperature changes,have a much
They
scale.
are
transfer at the air-sea
heat
interface. From the distribution of the weights in each component
of the eigenmodes,it is concluded ,on
while
atmospheric
effect on
impact
on
the
consideration,that
energy
are rapidly damped with little
perturbations
will
perturbations
ocean,oceanic
lasting
have
When the deep ocean is
the atmospheric temperatures.
included, long adjustment time scales of ~ 1000 years exists. The
associated eigenmodes characterize
in
changes
the
ocean
deep
which has the largest thermal inertia in the system.
to
The sensitivity of the models
parameters
is
It
examined.
are
variations
found
temperatures(which characterize the energy
most
sensitive
to
variations
of
the
various
the
system)
air
are
in the solar constant. Among the
three sets of models the sensivity of the two
-
that
in
101 -
layer
models
are
to
comparable
use
which
models
the
empirically
temperature
albedo relations and the general
of
and
Manabe
Inclusion of the
Sellers' and Budyko's models.
circulation
the
reduces
models
circulation
less
are
They
Wetherald.
determined
sensitivity further.
sensitive than the
mean
meridional
Latent heat flux
feedback increases the sensitivity in all cases. This increase is
small.
We examine the
associated
variations
with
constant. The changes are in
GCM's
for
the
simulation
in
full
the
in
changes
the
VIII)
model(model
ice line and a fixed solar
with
agreement
the
from
results
of the atmospheric circulation using
surface conditions prevalent at 18,000 years B. P. . However,
we
are unable to distinguish qualitatively the response of the model
to variations in the ice-line from that due to a variation in the
solar
constant.
This
is
important
to
the causes of climatic
changes.
ocean
We cannot conclude from this model whether the deep
during
an ice age is warmer or colder than it is today.
But the
parameters which are relevant to the deep ocean temperatures
are
put forth. To better understand the role of the deep ocean during
climatic
changes,
the variability of these parameters is worthy
of further investigation.
- 102 -
Appendix Al:List of symbols
symbol
0L
section
3.3.2
constant in long wave formulation
3.3.1
short wave absorption at low ,high latitudes
(as
3.3.1
effective absorptivity
A
3.3.2-
constant in long wave formulation
A,
3.1
area of low latitudes
A-
3.1
area of high latitudes
A
3.3.1
slab absorptivity
b
3.3.2
long wave flux
6
3.3.2
13
3.3.3
ratio of latent to sensible heat
iC
3.2
specific heat of air at constant pressure
C,
3.4
heat capacity of atmosphere
C1
3.4
heat capacity of surface ocean
C,
3.4
heat capacity of deep ocean
Co
3.2
specific heat of water
Cp
3.3.3
drag coefficient
2.6
Evaporation
a,
..
E 3.3.3
F,
F/
I1
II
II
atmospheric diffusivity
T,Fr,2.6
flux of salt,temperature,density
F
2.6
flux of fresh water
,F
3.2
horizontal,vertical dynamical fluxes
yd3.2
<F
fIII
formulation constant
PFr, 93.2
5,
radiative fluxes
meridional fluxes in the surface ,deep ocean
3.3.3
turbulent fluxes accross air sea interface
4.3
change of eddy flux w.r.t.temperature
4.3
change of short wave absorption w.r.t.temperature
-103-
3.3.2
back radiation from clear skies
3.2
depth of surface ocean layer ,deep layer
3.3.2
long wave flux at the top of the atmosphere
4.1
jacobian
3.3.4
horizontal ,vertical diffusion coefficient in ocean
K, Kv
3.4
bulb diffusion diffusion coefficients
L
3.3.3
latent heat
3.1
length scale of low and high latitudes
3.1
meridional length scale
2.6
flux of density,salt
3.1
surface pressure
2.6
precipitation
4.3
change of latent heat flux w.r.t. temperature
2.6
runoff
3.3.3
mixing ratios
3.3.2
long wave flux at the surface
3.3.1
slab reflectivity
2.6
salinity
2.6
salinity of evaporate,precipitate,river runoff
3.4
rate of bottom water
3.3.1
solar constant
3.3.1
insolation at low,high latituditudes,over ice.
2.6
mean salinity of oceans
3.3.1
short wave absorption at the surface at low
and high latitudes
3.1
time
3.1,2.6
temperature
3.4
solar transmission into box 4
3.3.3
tropopause-surface temperature difference
3.3.4
temperature of bottom water
- in/I -
H
L
5 ris
Me,
S
5,, S, S
So
Br
formation
$,t
"F
3.3.1
slab transmissivity
Ct
2.6
exchange volume
if
2.6,3.1
meridional velocity
U
4.3
eigenvector
2.6,3.1
east west coordinate
4.2
mean air temperture
3.1
sine of the latitude
w
3.3.1
surface albedo,of ice-free,of ice-coverd regions
(
4.6
ice-line,global sensitivity
3.1
ratio of depth of deep ocean to surface ocean
3.3.2
emissivity
"
2.6
sigma
"
3.4
ratio of low to high latitude area
--
3.3.2
stefan Boltzmann constant
S
4.6
measure of TAF
4.3
differential solar heating
o;
/
- 105 -
Appendix A2: Derivation of effective reflection, transmission, absorption
coefficients of a medium on top of a reflecting surface.
Consider a unit beam of light incident on the medium.
is reflected, T transmitted and A absorbed.
04ST
absorbed,
Of the amount of T transmitted,
is reflected back to the medium. Of these
T~$sT
R of the beam
oS
,
AT5S
is
transmitted to space and T0IR again reflected back
The light traces are depicted below
to surface.
Tz
ds
The effective reflection is the sum of all reflected rays and it turn out to
be a geometric sum
2-
a result due to Rasool and Schneider (1974).
Similarly the effective absorption
is
-
A + AT
+ AT
- 106 -
-
-
iAs
and transmission is
'I
T-F+
,
+ T ,'
. + -" -
the amount which transmits into the underlying surface is
±7t
since
t'
(-.1,
, it is easy to show that
Ri-A +T
The inverse relations of the dependence of the slab reflectivity,
transmissivity and absorptivity are
CIS
I- d
-
107 -
Appendix A3: Formulation of the latent heat flux
Assuming that the departure of mixing ratio r' is proportional to
the departure of temperature T' from their zonal means through the
Clausius-Clapeyron relation, we get, if we assume constant relative humidity, RH
r,()
T
RH
where T is the mean zonal temperature. Multiplication by v', the meridional
velocity, an expression relating the flux of sensible and latent heat is
)T
The ratio of
the vertically averaged flux of latent heat to sensibel heat is
14
RiH
J
L v'r' Ji
J0o
where H is the
v'T' (z)
From Oort and Rasmusson (1971),
scale height.
has a maximum in the lower troposphere and a secondary maximum
at about 200 mb.
where [
I'T' J+C
If we assumed that v'T' is constant, then
] denoted vertically averaged quantities.
From the Clasuius-
-
Clapeyron relation,
'CT)
where p is pressure.
For p in atmospheres, a=1.36X106 and b=5385K.
Hence
(A3.1)
IT
F
- 108 -
For a hydrostatic atmosphere with a constant lapse rate, the pressure
and temperature can be written as
T,t) = TS()- rt
where P , T are the surface pressure
s
s
and temperature, F the lapse rate
and H the pressure scale height.
Substitution of the above relations into equation (A3.1) gives
brs
(A3.2)
P
P
b
T'-
where the variation of the pressure scale height with temperature is
small and neglected.
For
l = 6 K/Km, T = 285K,
rt
for 0 (z<
Linearization of (A3.2) gives
IbI-
(--
-)
e
(A3.3)
PS T,
)T
T2
where
h=
s
is the moisture scale height, and
bp
-4
K1
T2r
The vertical average of (A3.3) is
(A3.4)
,TJ-
_
67d
-
109 -
(I-+
I
H
/ (HoP
PT rc,"
H,
i(
h,)
H z
h ±)s~
Ps~
where
al
5 -
L)
H-k
k
For values of
20 Km and
given before, h,v2.5 Km, Hv8Km, and h
iL
r and
is
the dominant term in equation(A3.4)
-
SH
r(
- )
Table A3.1 shows the values of a(T s ) and r(T s ) for T s=265, 285, 295K.
It can be seen that a(T s ) is a slowly varying function of temperature
with respect to r(Ts).
We shall assume that a(Ts ) is constant in our
model and equals .022K-l
model and equals .022K.
Table A3.1:
a(Ts)
and r(T ) as a function of T
s
T
a(Ts)
oK
OK-1
g/Kg
265
.0216
2.1
285
.0225
8.6
295
.0229
16.4
r(Ts)
- 110 -
s
T
Appendix A 4 : Numerical techniques
i) Steady State Calculation
In the steady state,(
e
written as
=0, the system of equations can be
a
There exists a number of numerical techniques for solving sets of nonlinear
equations (e.g. Carnahan 1969).
particular problem.
The method used is often dependent on the
Newton's method has been used here.
an approximation to the zero of F(~ at step k.
about
7
3f
T
An expansion of
be
F
gives
FIT o F(T.4-o
J)
where
Let
--
cf
.
is the Jacobian evaluated at
from
The departure of
the zero of the equation can be approximated as
~J
-,
r
Successive approximations to the zero can be obtained by adjusting the k th
approximation
7-r t-/
I'
Convergence is guaranteed if there exists a region in the neighborhood of
the zero such that the function is smooth (i.e. the sum of the row elements of
the Jacobian is some positive number less than unity), and if the initial guess
falls within such a region (Carnahan et al, loc cit).
The disadvantages of
the Newton's method is that it requires computation of derivatives.
- 111 -
For a
system of n equations, n
2
partial derivatives must be evaluated.
Robinson
(1966) has developed an iterative algorithm where the Jacobian is replaced
by difference quotients.
If the function is twice differentiable in the
neighborhood of its zero, an initial guess sufficiently close to the zero
Subroutine ZEROIN
will necessarily converge to the zero of the function.
in the MIT MATH program library was available to carry out such an iteration.
A check for convergence is
where
L
is
a user supplied tolerance,
taken as 5*10 - 1
3
Lacking a priori knowledge of the steady state, a number of trials
were made that did not lead to a solution.
For guesses sufficiently close
to the zero, about 4-10 iterations are needed.
Some initial guesses led
to a steady state where SB =0 in the model, i.e. the oceans are statically
unstable.
These solutions have to be discarded on physical grounds.
- 112 -
ii) Jacobian and Eigenvalues
The elements of the Jacobians are approximated by finite difference
quotients
F.
)-
F(
-
2A
where
?C
is the steady state solution and
- is a vetor whose
( o .. ,
components are zero except the j th component, i.e.
, o,.-
To see if the finite difference quotients are good approximations to
the partial derivatives, two calculations with i)
were made.
=0.001 and ii) S=(0.001)2
The magnitude of the nonzero elements are in the range 100 to 1
in units of w/(m 2 oK).
The results from the two calculations showed that they
agree to within three decimal places.
EISPACK, which is a collection of FORTRAN subroutines for solving
eigensystem problems, is used to calculate the eigenvalues and eigenvectors.
The QR algorithm, described e.g. in Dahlquist and Bjorck (1974), is used
for calculating the eigenvalues of real general matrics
- 113 -
Appendix A5: Evaluation of g (reference to section 4b)
The incoming solar radiation at high latitudes is
J
where
is the co-albedo over ice-fere regions and equals 6a
'. is the co-albedo over ice-covered regions and equals
5
and
C
ti
a+
5"1 are normalized average solar incidence in ice-free and
ice-covered high latitude regions as defined in section 2c. With
denoting a small change in
5
(A5.1)
, we get
674
From the fact that the total solar radiation incident on high
latitudes remains constant
we get
IOv)
-
Substituting this relation into the RHS of
(A5.2)
5 jX)o
$lYh
4h(5
Al
, we get
5
is parameterized as a linear function of the
If the ice-cover
local temperature as described in section
oz-
(thenT,)
then,
6r
4.1.
I(
aI S
- 114 -
f
In the limit
-
, the partial derivative of solar insolation
o
is
at high latitude with respect to
The change of the solar absorption associated with a change of the
temperature
T.
is dependent on 1) the difference between the co-albedoes
of the ice-free and ice-covered regions, 2) the latitudinal distribution
of solar radiation and, 3) the parameterization of the ice-line.
From section 2c,
J
f-s
ir+
jf
1,(5jf )J
vV
2-
5
= -0.482, is a measure of the differential distribution of solar
insolation. The change of
,e
0.25,
Since
5
'
with respect to
>
0,
is
0, i.e., the average solar
insolation on the ice-covered regions increases as the ice-line is pushed
equatorward.
Because a
7
, and
a
0
, the increase of
temperature at high latitude is accompanied by a retreat of the ice-line,
For
therefore less radiation is reflected back to space.
= 0.42,
.
7
= 0.2,
1.064 W/m2K
t
=
0.554 and
*
- 115 -
=
.175,
w = 0.57,
we get
Appen dix A6 :
Necessary Condition for Linear Instability of the Steady
State in the presence of Temperature-albedo Feedback.
From equation (4.6), the indical equation is
) t
b
(Co.)+
C =
o
where
CSo
The discriminant
Z
'S are real.
Hence, all
, after some manipulation, is
=
In the absence of this feedback,
the solution is
-
.-
bi
0, or
( A6.
)
then
X+
If
6 7
2-
__
In the presence of this feedback, 8
If
,
t2 T )
L.
GC' a
6~LtB
0,
20
7,
c
is positive.
0,
i
t
then there exists at least one negative root,
C L
0 since the product of the roots A+/ _
that C <
= 0,
A
.
- C
In order that A4>
.
0,
The condition
0 is
'
( A6 . 2)
-4
)
Hence a necessary condition for linear instability of the steady state is
when either ( A6.1) or ( A6.2) holds.
This is possible since the absolute
value of g is determined by the parameterization of the ice-line ( see
Appendix
A5).
- 116 -
Table 2.1: Summary of results of direct calculation of Ocean heat transport.
Latitude
Heat transport in 1015 Watts,positive northward
Indian
Pacific
Atlantic
400 N
-0.0
36 N
0.7
320 N
-1.2
00
160S
1.3;0.6
240S
0.3;(0.3,0.6)
(-1.2,-0.2)
280S
320 S
430 S
(0.2,0.7)
(0.5,0.6)
(1.6,1.8)
(-0.2,0.4)
- 117 -
Table 2.2:Precipitation,Evaporation,Runoff and Transport of fresh water
ii n the Oceans.
Latitude
E
P
R
Atlantic
Pacific
Indian
E
P
F
R
F
P
E
R
-3.0
80°N
70°N
300 N
0.5
2.2
1.8
7.9
1.0
00
9.3
0.5
100S
16.8 13.9
0.8
0
20 S
12.7
8.7
0
50 S
2.0
-1.7
19.5
0.5
15.8
0.1
8.2 15.2
0.0
23.1
10.0
0.0
20.6
5.2
0.0
16.7
1.3
0.9
13.9
300 S
S0
40
1.9
12.5
600 S
9.1
700 S
3.2
4.7
-0.8
3.5
2.4
1.2
7.5
2.5
1.2
-7.1
5.6
4.3
1.4
11.9
6.9
0
6.8
7.2
1.8
-12.9
12.6 14.4
0.4 -11.5
6.7 14.6
0.6
12.1 23.1
2.1
5.2 13.9
0.7
27.0 28.2
1.0
6.7 12.0
1.0
47.1
26.9
1.6 -24.4
9.9
8.0
4.5
22.6
25.6
2.4
4.0
8.7
6.1
22.2
26.9
0.1 -19.3
1.5
9.8
0.3
3.1
9.9
0.2
-2.7
-2.5
-23.9
18.5 22.3
16.1
0.2
0.5
-0.1
100 N10N 4.9
0.9
0.2
50°N
400 N
1.1
0.5
60°N
20ON
F
15.3
16.6
0.3 -14.6
0.8
5.7
9.1
0.7
14.5
11.0
0.0 -19.0
7.3
5.9
0.0
11.0
5.8
-24.4
5.4
3.2
0.0
4.5
2.1
0.0 -26.7
2.1
1.0
0.0
0.9
0.3
0.5 -28.0
0.4
0.1
0.5
800S
P:Precipitation volume in 10 12m3/year
E:Evaporation volume in 10
R:Runoff volume in 10
12 3
m /year
12 3
m /year
F:Fresh water Transport in 10 12m 3/year,positive northward.
- 118 -
-3.3
-5.6
-8.3
-9.7
-2.5
5.4
9.7
3.4
2.6
10.5
17.1
19.8
18.4
16.1
14.9
14.1
Table 3.1:Latitudinal distribution of short wave components.
R
T
A
.08
.23
.58
.19
.25
.09
.21
.61
.18
.18
.27
.10
.24
.59
.17
.52
.17
.31
.10
.28
-. 56
40-50 0 N
.47
.18
.36
.12
.33
.51
.16
0
N
50-60
.42
.17
.41
.14
.38
.46
.16
60-70N
.36
.19
.45
.24
.41
.42
.17
70-800 N
.25
.21
.54
.46
.47
.36
.17
80-90N
.21
.20
.59
.61
.46
.39
.15
Latitude
t
a
r
0-100 N
.54
.20
.25
10-200 N
.56
.19
20-30°N
.55
30-400 N
5S
.16
t:fractional absorption at the surface(from London 1957)
a:fractional absorption in the atmosphere (London 1957),
r:planetary albedo (Ellis and Vondar Haar 1976),
d-:surface albedo(Sellers 1965),
R:deduced slab reflectivity
T:deduced slab transmissivity
A:deduced slab absorptivity.
-
119 -
Table 3.2:Short wave parameters used in the model
for low and high latitudes
low latitude
high latitude
R
.25
.41
T
.57
.42
A
.18
.17
f.
.285
.427
--
.58
TX
- 120 -
Table 3.3:Solutions for selected values of Ky and Kz.
Kz
cm2/s
Ky
cm /s
10.
6 10 5
1.0
0.10
1.6 106
3.6 107
Temperature OC
1.1.
h.l.
1.1.
h.l.
1.1.
h.1.
surface air
17.26
-1.72
17.27
-1.73
17.29
-1.77
sea surface
19.85
0.17
19.86
0.15
19.87
0.09
deep ocean
16.41
0.17
6.06
0.15
0.29
0.09
Fluxes
1015W
Atmosphere
3.66
3.67
3.69
Ocean
1.61
1.60
1.58
MMC
1.52
1.50
1.17
Diffusion
0.09
0.10
0.41
Ro
.306
.304
.300
>_
.067
.066
.050
18.3
18.1
14.0
Rate of bottom
water formatign
10 m 3 /s
h.l.:high latitudes
1.1: low latitudes.
- 121 -
Table 3.4:Comparison of model result with observation.
Parameter
model
unit
low lat.
W/m 2
W/m2
short wave
flux a. i t
observation
high lat. low lat.
265
126
194
84
Plantary
"f
albedo
high lat.
270
124
173(250 N)
.31
source
67(550 N)
.30
E&VH(1976)
Clark (1967)
E&VH(1976)
2
W/m
238
208
246
191
E&VH(1976)
W/m
185
107
194(25*N)
106(55 0N)
Clark (196,7)
long wave r
W/m 2
61
75
54(25*N)
46(550 N)
Clark(1967)
4
W/m2
125
44
140(25 0 N)
60(55 0 N)
Clark(1967)
.2
.7
.3
Clark (1967)
Long waveflux
surface flux
C +
turbulent
Bowen ratio
2
-
Meridional flux 1015 W
5.2
4.5(450 N)
Oort(1971)
3.7
3.0(45ON)
Oort (1971)
atmospheric
ratio of latent
to sensible heat
.4
-
1015 W
1.5
MMC
1015W
1.4
diffusion
1015 W
0.1
Oceanic
Temperature
.4 (5 0 N)
Oort(1971)
1.6(45ON)
VH&O (1973)
oC
air
17.3
-1.73
ocean surface
'19.9
0.15
deep ocean
Rate of bottom
10 m /s
water formed
6.1
.15
17.4(30*N) -3.15(600 N)
Sellers (1965)
20.8(320 N)
N&H(1978)
5-10
Wust(fig 3.5)
18.1
20
Stommel(1960)
20-50
Gordon(1972)
*the effective transfer is (l-T)or 80% of 44W/m 2
Sources: E&VH Ellis and Vondar Haar;N&H Newell and Hsiung
- 122 -
Table 4.1:List of models
Model
Structure
Latent flux
feedback
no
Role of Ocean
Kz=(=0
Ky=O
--
I
1-layer
II
1-layer
yes
III
2-layer
no
Swamp
yes
yes
IV
2-layer
yes
Swamp
yes
yes
V
2-layer
no
diffusive
yes
no
yes
no
VI
2-layer
yes
VII
3-layer
yes
VIII
3-layer
yes
OL constant
--
3
ytransport
diffusive
and MMC
transport
- 123 -
yes
no
Table 4.2: Eigenmodes of the one layer model
Model
I
yes
TAF
no
yes
yes
latent flux feedback
no
no
yes
X
(yr-1)
-4.9 -29.2
74
e-folding time (days)
12.5
1
-1
1)( 3)
U
-3.86
94
-3.70
-26.7
98
13.5
13.1
.57 -.35
-. 36
.93/
.65
.75)
-27.1
.8 )(.94)
Table 4.1b: Sensitivity of the one layer model
bYl
)XL
Parameter p
-
So
oC/(Wm
-2
2)
S
.46
1C
10.8
.41
-6.9
A
OC/(Wm- 2 )
B
°C/(Wm-K-l)
E
oC/(Wm-2K-2)-52.9 158.8
-.64
-.64
-8.57 -6.57
.53
.55
.52
.67
9.52 -9.37
9.4
-.77
-. 75 -1.04
-.87
-8.33
-9.84 -8.84
-9.62 -10.93
-22.29 213.27
-23.13 221.27
- 124 -
Table 4.3a:Eigenvalues,eigenmodes and sensitivity of model III without TAF
-111.5
3.3 days
e-folding time
u
parameter unit
16.1 years
.77
.26
-.28
-.64
.96
.41
-. 51
-.00
-.00
-.29
-.43
.00
-.Ol
.81
-.55
.50
magnitude
2
.498
15.833
.612
-. 214
.575
-.500
1.228
-. 525
-. 525
-.434
-. 510
-. 576
-. 476
-.476
-.463
.433
.00
.00
-. 967
-. 253
OC/(Wm- 2 K- 2 ) 169.74
-.288
.685
-.189
.665
S
oC
A
oC/(Wm-
E
3.3 years
-.062
.535
oC/(Wm
C
5.0 days
-.304
.845
S
B
-72.65
0
2
)
)
C/(Wm-2K- 1)
il""
14.65
- 125 -
.462
.502
Table 4.3b:same as 4.3a except with TAF
-111.4
e-folding time
(K
S
o
unit
-.049
5.1 d
3.4 y
20.4 y
.77
.26
-.30
(-.461
.40
-. 53
-.63
.96
-.00
-.00
-.
01
aX
magnitude
1.086
-.291
3.3 d
.00
parameter
-72.02
.501
-. 391
.80
-. 59J
3x
.517
.429
.545
S
16.077
.556
-. 282
.528
-.577
A
1.600
-.491
-.538
-. 406
-. 552
B
18.581
-.536
-. 502
-.443
-. 515
C
.433
.00
.00
-. 967
-. 254
E
223.035
-.060
.696
-. 050
.714
units :same as in table 4.2
- 126 -
Table 4.4a:Eigenvalues
,eigenmodes
A
and sensitivity of model IV,no TAF
-112.9
e-folding time
-71.51
3.2 d
5.1 d
.75
.24
-.66
.97
-. 00
-. 00
.00
parameter
unit
-.01
-.309
-. 062
3.2 y
16.1 y
-. 27
.41
-. 29
.82
-.44
-.55
-.38
-.59
X3
magnitude
S
.884
.491
.536
.425
.539
S
14.972
.628
-.171
.592
-. 474
A
1.289
-. 480
-.561
-.397
-. 545
B
15.301
-. 530
-.571
-. 438
-. 506
C
.433
.00
.00
-. 967
-. 254
.685
-. 188
.665
o
E
169.01
-.228
Units: same as in Table 4.2
- 127 -
Table 4.4b:Same as 4.4a except with TAF
-112.81
e-folding time
parameter
magnitude
-71.00
3.2 d
5.1 d
.75
.23
-.298
-.047
3.4 y
21.3 y
f-.30
.41
.97
.40
.56
-.00
-.00
-.32
.35
.00
-.01
.80
.62
x)
1.179
.454
.553
.389
.579
15.156
.582
-. 236
.554
-. 546
1.741
-. 444
-. 571
-. 367
-.585
20.227
-.486
-. 542
-.402
-.555
.0
.0
-. 967
-.256
.696
-. 050
.713
.433
228.325
-. 061
Units: same as in Table 4.2
- 128 -
Table 4.5a:Eigenvalues,eigenmodes and sensitivity of model V,without TAF.
-104.97
e-folding time
magnitude
S
0
S
OC(Wm-2 )-1
OC-1
- 2
A
oC(Wm
-1
-. 318
-. 063
3.5 d
5.3 d
3.1 y
15.87 y
.87
.17
-.27
-.51
.98
.49
-. 51
-.00
-.00
-.28
-. 45
.00
-.01
.77
-. 53
-. 48
parameter
-68.35
DX,
X3
.834
.547
.493
.476
.481
16.449
.616
-. 287
.568
-. 464
1.213
-. 534
-. 526
-. 445
-. 490
14.431
-. 591
-. 463
-. 493
-. 439
.013
-. 039
-.837
-. 545
B
oC2 (Wm-2 )-1
C
oC/(Wm-2K -l)
.469
E
oC/(Wm-2K- 2)
191.395
-. 241
.723
-. 191
.618
K
OC/(Wm-2K -
9.314
-. 177
.532
-.196
.805
I
)
- 129 -
Table 4.5b:same as 4.5a ,except with TAF.
A
-104.69
e-folding time
3.5 d
5.6 d
.89
.16
-. 307
-. 051
3.3 y
19.7 y
-. 29
-. 48
-. 45
S98
.49
-. 54
-. 01
-.00
-. 30 i
-.43
.00
-. 01
.76
-. 55
1.055
.515
.527
.446
.506
16.738
.544
-. 382
.507
-. 549
1.561
-. 503
-. 553
-. 421
-. 514
17.976
-. 555
-. 503
-. 463
-. 473
.476
.004
-. 052
-. 833
-. 550
249.349
-. 065
.751
-. 046
.655
11.359
-. 051
.590
-. 082
.801
Ui
parameter
-65.29
magnitude
Units: same as in table 4.5a
- 130 -
Table
4
.6 a:Eigenvalues,eigenmodes and sensitivity of model VI,without TAF.
-105.96
e-folding time
3.4 d
it
parameter
-67.53
5.4 d
-.322
-.063
3.1 y
15.9 y
/ .47
.85
/ .16
-.27
-. 51
.98
.49
.54
.41
-.01
-.00
-.28
.00
-. 01
.78
.56
.529
.446
.509
magnitude
S
.862
S
15.732
.629
-. 256
.582
-.447
A
1.258
-. 497
-. 558
-. 416
-. 517
B
14.887
-. 555
-. 504
-. 463
-.472
C
.469
.013
-. 038
- .838
-. 544
E
190. 980
-. 241
.723
-. 191
.618
K
9.246
-. 177
.531
-. 196
.805
.511
units: same as in Table 4.5a
- 131 -
Table 4.6b:same as
4 .6
a,except
SA-
with TAF.
-105.73
e-folding time
3.5 d
.87
5.7 d
'
.15
-. 311
-.049
3.2 y
20.4 y
-
29
r .44
-.48
.98
.49
.58
-.01
.00
-.30
.38
.00
parameter
-64.61
-.01
.76
.57
)(3
magnitude
3
S
1.125
.477
.562
.414
.534
S
16.007
.563
-. 352
.525
--. 532
A
1.667
-. 465
-. 584
-. 389
-. 539
B
19.189
-. 514
-. 543
-.429
-. 506
C
.476
.005
-. 053
-. 833
-. 550
E
254.734
-. 065
.751
-. 046
.655
K
11.47
-. 052
.593
-. 081
.799
o
units : same as in Table 4.5a.
- 132 -
Table 4.7a:Eigenvalues,eigenmodes and sensitivity of model VII,without TAF.
-104.14
Ae-folding time
3.5 d
.85
-66.47
5.5 d
-. 719
-.103
-.0033
1.4 y
9.7 y
303 y
-. 16 "
.19
-. 52
-.99
-.00
.00
.00
.00
-.80
.00
.00
.00
.54
.72
'
'
-.00073
1470 y
f-.07
.22
.52
.19
-. 01
.06
.21
.67
-.07
.20
-.03
.01
.01
-. 00
-.01
-. 37
.00
-.00
-.00
-. 92
.15
.49
-.03
.83
1.29
-9.64
3.51
-1.22
-.36
-13.45
-3.88
-
I-.92
.23 ,
parameter
so
S
8.07
1 .45
7.13
A
-.78
-. 25
-.69
-9.89
-2.62
-. 054
.162
-8.69
47.79
-2.96
-1.13
3.42
-1.26
.02
.35
-.466
-15.93
-.007
.04
.018
-19.5
-.008
10-5
Units: same as in Table 4.5a for So ,S,A,B,C,E,K.
for parameter Kv is oC(Wm-2K-)
for parameter h is °C/m
- 133 -
-. 546
194.8
.24
-. 150
62.83
10.17
213.54
.033
23.08
3.33
_lO - 4
10-3
-. 004
5.19
Table 4.7b:same as 4.7a,except with TAF
NL
e-folding time
parameter
-103.9
3.5 d
-63.4
-.70
5.8 d
.87
S.15
-.49
.99
-.00
1.43 y
-.10
I0 y
-.0034
298 y
-.0007
1408 y
.20
.72
-. 54
.20
-.07
-.01
-.00
.21
.66
-. 21
-.07
-.00
.00
-. 79
-.02
.00
.01
.00
-.00
.00
-.01
-.92
-.37
.00
-.00
.00
-.00
i -.23
.92
(-.23
(-.07
ay4
parameter
.578
.17
.53
8.41
1.67
7.44
-.835
-.28
-.74
-10.51
-. 016
-4.77
-.34
-.002
10-5
-3.01
0.187
-9.25
-.02
1.34
-1.33
.26
-14.60
.02
7.19
-17.88
3.92
-. 54
5.31
.024
-. 003
-10 -4
10 - 5
10~
units: same as in table 4.7a
- 134 -
8.97
.03
-.43
54.9
.90
.033
-10 -4
-.47
216.7
-8.60
.26
3.31
-.39
-4.25
-.13
69.64
214.0
23.1
3.33
10-3
10
-. 004
Table 4.8a: Sensitivity of model VIII, without TAF.
(yr)
x
-67.2
-1i
e-folding time
AL
-51.7
992yr
5.7yr
29.4yr
-.43
.38
.72
-.64
-.22
.93
.17
-.39
-.18
0
-0
.67
-.59
-.20
-0
-0
-.27
-.16
.05
-.92
-0
0
-.06
.00
)X4
parameter
.498
.285
.448
.181
-2.41
S
8.88
-.98
8.04
A
-.711
-.448
-. 617
C
-.001
7. id
U
L
B
-.034
5.4d
.90
So
-.174
-9.18
-4.76
.077
-.026
-7.89
-. 434
-. 266
-2.91
-.111
.575
-5.02
-.836
-9.39
-.385
.011
-.190
-. 016
-. 168
-.006
E
-29.85
89.54
-18.42
43.98
115.81
3.263
K
-36.35
109.04
-40.39
165.84
-210.03
8.255
25.76
19.91
.131
K
v
-.56
.0007
1.69
-.0022
-.627
.0008
units: same as in table 4.7a
- 135 -
-.0034
.0046
-.0002
Table 4.8b: same as 4.8a, except with TAF.
(yr)-l
e-folding time
-66.0
-50.1
5.5d
7.3d
A4
-. 174
-.031
-.001
5.7yr
31.4yr
1061. 8yr
-.49
.32
.72
-.65
-.24
.86
.94
.18
-.41
-.20
.67
-.59
-.22
-.17
0
-0
-.00
-.06
-. 27
-0
.00*
.00
.06
i I
-. 91
)L
parameter
So
.562
S
8.65
A
-.813
B
-10.26
C
-. 008
E
-9.455
K
K
v
-11.52
.346
-1.19
-. 545
-5.79
.093
108.86
132.57
-. 178
.0002
2.059
-. 0027
.506
7.84
-. 708
-8.87
-. 418
-. 237
-18.25
-.283
.0003
units: same as in table 4.7a
- 136 -
.216
-2.53
-. 321
-3.51
-.102
.681
.0118
-5.39
-.195
-1.01
-. 018
-11.18
-. 189
-. 356
-.006
55.12
149.47
3.669
179.41
-169.03
8.749
20.55
.138
2.787
.0037
.0037
-.00018
Table 4.9:Sensitivity of eigenvalues for model VI
Model parameter
ij
A3
-1
year
C
E
K
h
-1
A4
-1
year
year
-1
year
+50%
-141.25
-82.07
-.332
-.049
-50%
-71.57
-45.62
-.276
-.049
+50 %
-110.10
-73.36
-.363
-.051
-50%
-102.31
-48.35
-.238
-.046
+50%
-104.56
-65.93
-.339
-.051
-50%
-107.10
-63.45
-.282
-.047
+50 %
-105.37
-64.42
-.208
-.033
-50%
-106.79
-65.19
-.618
-.097
- 137 -
Table 4.9a: Sensitivity of eigenvalues and eigenmodes to atmospheric heat
capacity for model VI.
+50%
-
(yr)- 1
-43.3
-70.8
-. 310
-.049
.44
.87
.15
-.29
-.49
.99
.49
.57
.38
-0
-.0
-.29
0
-.0
.76
.57
-. 312
-. 049
.44
-50%
h
(yr) - 1
-128.6
-210.4
.87
.15
-.28
-.48
.99
.49
.57
0
-0
-.29
.38
-0
.76
.57
-0
- 138 -
((b) of models I-VIII to parameter variations.
Table 4.10:Sensitivity
model
I
II
III
IV
186
183
189
V
VI
VII
187
159
parameter
S
0
180
2.31
S
2.39
A
-167
-173
B
-14.9
-15.4
1.4
1.4
2.68
-168
2.76
-175
-15.2
1.6
-15.7
2.0
183
2.52
-170
-174
-15.1
3.24
-157
-147
-15.5
-13.4
.31
.16
1.3
1.3
.38
.99
.53
.003
.10
-
.0009
.076
K
-
-.000
unit:OC
- 139 -
-14.2
-.04
.52
h
3.00
-.04
K
v
2.58
170
-. 105
Table 4.11:Ice line and Golbal sensitivity
Model
One layer
Radiative
equilibrium
I1C)
.796
('d
c)
148
.0199
180.3
.22
II
.0241
187
.26
III
.0202
183
.23
IV
.0235
188
.27
V
.0200
183
.23
VI
.0228
187
.26
VII
.0061
159
.07
VIII
.0125
170
.14
Budyko(1969)
155
400
1.58
Sellers (1969)
150
325
1.17
Wetherald
and Manabe(1975)
146
185
.27
Lian and Cess(1977)
147
184
.25
- 140 -
Table 5.1 :Sensitivity of model VIII to solar constant variation.
(w(Win
2
2
t2
(Win2 )
r
(Win
2
fo
)
)
2)
(Win2)
71
59
T2
T3
T4
T5
T6
119
131
2)
(Wi(Wn )
241
210
235
207
(OC)
19.13
-.41
(0C)
21.53
1.09
18.15
-.39
8.55
1.09
4.06
-.39
(OC)
.152
ice cover
Atmospheric flux
(lo15w)
(1015W)
4.02
.45
Oceanic flux
(10 15W)
MMC flux
fractional formation
area
6 3
.237
3.24
.37
1.13
1.95
.115
(10 m /s)
-2.75
2.04
5.27
Total flux
15.37
1.25
.12
Diffusion
Rate of formation S
.312
.309
12
T1
193
196
planetary albedo
1
-1%
+1%
Solar constant change
12.5
- 141 -
.08
5.28
.036
25.7
Table 5.2:Sensitivity of model VIII to ice-line variation.
a1
t1
r
-2
)
a2
(Wm
t
(Wm-2)
-2
2
0.3
0.1
prescribed ice line
r1
r2
-2
(Wm )
fl
f2
(Wm-2)
70
42
70
194
88
194
61
74
126
40
62
123
.313
.308
planetary albedo
239
2)
I1
12
T1
T2
(OC)
17.76
T3
T4
(OC)
20.30
T5
T6
(CC)
6.92
(Wmi
209
207
237
16.73
-2.22
.45
19.38
-.11
.45
5.22
-.11
-1.24
ice cover
3.7
Atmospheric fluxes (1015W)
B
.41
o
oceanic fluxes
(1015 W)
MMC
fractional formation
area
Rate of formation SB
1.80
1.33
1.70
5.14
Total flux
.075
6Rate
of
formation
3S
(10 m /s)
- 142 -
.39
1.44
.11
Diffusion
3.6
15.6
.10
5.41
.055
21.0
Figure 2.1:Schematic showing the components of fresh water balance
for a volume of sea water.
P
E
A 3~A
Ai
- 143
-
Figure 2.2 :Oceanic transport of fresh water(solid line) and atmospheric moisture transport(dotted).
M(d
-)
0
0)
r-4
o
zH)
k~
-10
4-J
44
4)
14
0C
0
-20
-30
905S
800
700
600
500
400
300
200
100 S
00
Latitude
10ON
200
300
400
500
600
700
800
90°N
Figure 2.3:Schematic showing the direction of Ocean heat transport
from different estimates.
90 NI
PBEH
H
/3
60 NI
300 N
0ONl
P
D
H
1
I
I
I
D
4
30°S-
1~
D
600 S
P
E
INDIAN
PACIFIC
ATLANTIC
900S
Arrows indicate direction of transport from estimates of Bryan (B),Emig(E),
Hastenrath( H ),direct calculation(D) and the present study(P).
- 145 -
Figure 3.1:Schematic of the Model showing the fluxes of energy.
Z=
2
t
C
II t
22
T,
ICE-
,
,
SHEET
=0
3
-7
Z
4-41
=-k
S6
== 0.0
Y= .75
Y -1.0
low latitudes
High latitudes
are short wave
I9 I2 are long wave fluxes at the top of the atmsophere,a ,a2 and t ,t
at the surface;
flux
wave
long
are
rl,
surface;
the
at
and
atmosphere
absorption in the
F '
and
flux
heat
atmospheric
is te
flf
2 are turbulent fluxes at the surface,
and p~ horizontal and vertical heat fluxes in the ocean
E
-.1-
EI
220
210200-
o
°
190
180
260
270
,
280
TEMPERATURE
Figure 3.2:Graph of outgoing long wave(I)
290
IN'K
verses surface temperature.(data from Rodgers 1967)
300
Figure 3.3 :Seasonal Variations of the Ratio of Latent to Sensible heat Flux
(solid) and surface mixing ratio (dashed).
9
0.9
8
0.7
7
30.6
6
z
o
4J
a
a,
j0.5
0
.a)
o
0.4
4o
r
2
0.2
I
0.1
Month of the year
- 148 -
Figure 3.4: A north-south Section
0
I
2
3.
4
O
2
3
.1n
of Temperature(top) and density(bottom) in the Atlantic(after Wust 1928).
10
Figure 3.5:Plot of (Ky,Kz) pairs which satisfy Ro=0.3 and T -T6=0.
The (Ky,Kz) pairs deduced from Tritium distribution for
the Atlantic (A) and Pacific (P) are included for comparison.
10-
*
z~T
0.1I
_
5
T4.1010
4.
4.10
Ky
IN CM S
Y¥
- 150 -
7
4.10
1Pigure
4.1
: Sea level temperature
"
oAPR
APR
2 .-
at edge of sea ice(data from Schultz & Gates)
i
oJUL
-4 SOCT
x
'
-6'C
)
-8-I10
-12
-14
4
I
180W
I
140
I100
60
longitude
20 *W
20 0 E
60
100
140
'5
180 0 E
REFERENCES
Baumgartner,A. and E. Reichel, 1975: The world water balance: mean annual
global continental and maritime precipitation, evaporation and runoff,
Elsevier, 179 pp.
Bennett,A.F. 1978: Poleward heat fluxes in the Southern Hemisphere Oceans.
J. Phys. Oceans., 8, 785-789.
Bolin,B and H. Stommel, 1961: On the abyssal circulation of the world
ocean IV: Origin and rate of circulation of the deep water as
determined with the aid of tracers. Deep Sea Res., 8, 95-110.
Bryan,K. 1962: Measurements of meridional heat transport by ocean
currents. J. Geoph. Res., 67, 3403-3414.
Budyko,M.I. 1956: The heat budget of the earth's surface ( translated
by Nina A. Stepanova, U.S. Dept. of Commence, Washington D.C. 259pp)
Budyko,M.I. 1969: The effect of solar radiation variation on the climate
of the earth. Tellus, 21, 612-619.
Budyko,M.I.,N.A. Epinora, L.I. Zubenok and L.A. Strokina 1963: The heat
balance of the earth's surface, Izv.Akad Nauk, SSSR, SER.Geograf.,
no 1, 6-16.
Bunker,A.F. 1976: Computation of surface energy flux and annual airsea cycles of the North Atlantic Ocean. Mon. Wea. Rev., no 104,
1122-40.
Businger,J.A. 1973: Turbulent transfer in the atmosphere surface layer.
Workshop on Micrometeorology, Haugen D.A., ed, AMS, 392 pp.
Carnahan.B. and H.A. Luther and J.O. Wilkes, 1969: Applied Numerical
Methods . John Wiley and Sons, New York, 604 pp.
Cess,R.D. 1974: Radiative transfer due to atmospheric water vapor:
Global consideration of the earth's energy balance. J.Quart.
Spectrosc. Radiat. Transfer, 14, 861-871.
Cess,R.D. 1976: Climate changes: An appraisal of atmospheric feedback
mechanism employing zonal climatology. J. Atmo. Sci., 33, 1831-1843.
Cess,R.D. and J.C.Wronka, 1979: Ice ages and the Milankovitch theory:,
A study of interactive climate feedback mechanisms.Tellus, 31, 185-192.
Clapp,P.F. 1970: Parameterization of a macro-scale transient heat transport
for use in the mean model of the general circulation. J. Atmo. Sci.,
9, 554-563.
Clarke,N.E. 1967: An investigation of large scale heat transfer processes
and fluctuation on sea surface temperature in the Northern Pacific.
Ph.D. thesis, Dept. Meteorology, M.I.T.
- 152-
CLIMAP project members, 1976:
Science, 191, 405-412.
The surface of the ice age Earth,
Dahlqist,G. & Bjorck,A., 1969: Numerical Methods, Series in Automatic
computation, Prentice Hall, 573 pp.
Ellis,J.S. & Vonder Haar,T.H., 1976:
Zonal Average Earth Radiation
Budget Measurements from Satellites from Climate Studies.
Atmosphere Science paper #240, Dept. of Atmo. Science,
Colorado State University, US ISSN 0067-0304.
Emig,M., 1967: Heat transport by ocean currents,
J. Geoph. Res., 72,2519-29.
Emiliani,C., 1954: Temperature of Pacific Bottom Waters and Polar
Superficial Waters during the Tertiary. Science, 119, 853-855.
Emiliani,C.,
1955: Pleistocene temperature.
J. Geol, 63, 538-579.
Emiliani,C. & Shackleton N.J., 1974: The Brunhes Epoch: Isotopic
Paliotemperatures and Geochronology. Science, 183, 511-514.
Faegre, A., 1972: An intransitive model of the earth atmosphere ocean
system. J. Appl. Meteor, 11, 4-6.
Foster, T.D., 1972: Haline convection in Ploynyas and leads.
J. Phys. Oceanogr., 2, 462-469.
Friedrich,H. & Levitus S., 1972: An Approximation to the Equation of
State of Sea Water Suitable for Numerical ocean Models.
J. Phys. Oceano., 2, 514-517.
Gal-Chen,T. & Schneider,S.H., 1976: Energy Balance Climate Modeling:
Comparison of radiative and dynamic feedback mechanisms.
Tellus, 18, 108-120.
Gates,W.L., 1976: The Numerical Simulation of Ice-age Climate with a
Global General Circulation Model. J. Atmos. Sci., 33, 1844-1873.
GillA.E., Green J.S.A., & Simmon A.J., 1974: Energy partitioning in
the large scale ocean circulation and production of mid-ocean eddies.
Deep Sea Res., 21, 499-528.
Goody,R.M., 1964: Atmospheric Radiation. Oxford, Clarendon Press 436pp.
- 153 -
Gordon,A.L. and Taylor,H.W., 1975:
Science, 147, 346-347.
Gordon,A.L. 1975:
Circulation.
Seasonal change of Antartic Sea Ice.
General Ocean Circulation.
Numerical Models of Ocean
Nat'l Academy of Sciences, 364pp.
Gordon,A.L. 1971: Recent Physical Oceanographic Studies of Antartic Waters.
Research in the Antartics, Quam L.O., ed., AAAS, publication no. 93.
Gordon,A.L. 1971: Oceanography of Antartic Waters. Antartic Oceanology I.
Antartic Research Ser., 15, 169-203, Reid, ed., AGU, Wash., D.C.
Green,J.S.A. 1970: Transfer properties of the large scale eddies and the
general circulation of the atmosphere. Quart. J. Roy. Met. Soc.,
96, 157-185.
Hastenrath,S. 1980:
Heat Budget of Tropic Ocean and Atmosphere.
Oceano., 10, no.2, 159-170.
J. Phys.
Hays,J.D., Imbrie,and Shackleton,N.J., 1976: Variation in Earth's Orbit:
Pace-maker of the ice ages.
Science, 194, 1121-32.
Held,I.M. and Suarez,M.J., 1974: Simple albedo feedback models of the
ice-caps.
Tellus, XXVI, 6, 613-628.
Holland,H.D. 1978: The Chemistry of the Atmosphere and Oceans.
Wiley-Interscience, 351pp.
Jenkins,W.J. 1979: Tritium and Helium in the Sargasso Sea, Woods Hole
Contribution no. 4321, Woods Hole Oceanographic Institution, Woods
Hole, Mass.
Jenkins,W.J. 1979:
On the Climatology of an Oceanic Subtropical Gyre.
Woods Hole Contribution no. 4438, Woods Hole Oceanographic Inst.
Jung,G.H. 1952: Note on the Meridional Transport of Energy by the Oceans.
J. Marine Res., 11, 139-146.
Jung,C.E. and Werby R.T., 1958: The concentration of chloride, sodium,
potassium and sulphate in rain-water over the United States.
J. Meteor., 15, 417-425.
- 154 -
Katayama,A., 1967:
On the radiative Budget of the troposphere over the
Northern Hemisphere (II). J. Meteor. Soc. Japan, 45, 1-25.
Kondratyev,K.Y., 1969: Radiation in the atmosphere.
Geophysical Series, Vol. 12, Academic Press.
International
Kondratyev,K.Y. & L.N. Dyachenko, 1971: Comparison of Satellite and
Calculation charts for the Earth's radiation Budget. Space Research,
XI, 651-660.
Kraus,E.B.,1973: Comparison between Ice age & Present General Circulation.
Nature, 245, 129-133.
Kraus,E.B. & Turner,J.S., 1967: A one-dimensional model of the seasonal
thermocline: II The general theory and its consequence. Tellus,
19, 98-105.
Leovy,C.B., 1973: Exchange of water vapor between the atmosphere and
surface of Mars. Icarus, 18, 120-125.
Lian,M.S. & Cess,R.D., 1977:
of ice-albeds feedback.
Energy Balance Climate Models:
J. Atmos. Sci., 34, 1058-62.
A reappraisal
Lindzen,R.S. & Farrell,B., 1977: Some Realistic Modifications of Simple
Climate Models. J. Atmos. Sci., 34, 1487-1501.
Lohmann,G.P., 1978: Response of the Deep Sea to Ice Ages.
21, no. 4, 58-64.
Oceanus,
London,J. & Sasamori,T., 1971: Radiative Energy Budget of the Atmosphere,
Space Research, XI, 639-49.
London,J., 1957: A study of the atmospheric heat balance. College of
Engineering, Res. Dir., Dept. of Meteor and Oceanog., N.Y.U.
Final Report AFC-TR-57-287, OTS PB 129551, 99 pp.
Simulation of the tropical climate of an
Manabe,S. & Hahn,D., 1977:
ice age. J.
Geophy. Res., 82, no. 27, 3889-3911.
MEDOC,Group, 1970: Observation of formation of deep water in the Mediterranean Sea, 1969. Nature, 227, 1037-1040.
Michel,R.L. & Suess,G.E., 1975: Bomb Tritium in the Pacific Ocean.
J. Geophy. Res., 80, 4139-52.
latitude
Mullen,A.B. 1979: A mechanistic model for midmean temperature structure. Sc. D. thesis, Dept Meteorology, M.I.T.
-
I
K --
Neumann,G. and Pierson W.J.Jr.,
Prentice Hall, 545 pp.
1966: Principle of Physical Oceanography,
Newell,R.E. 1974: Changes in the poleward energy flux by the atmosphere
and ocean as a possible cause for the ice ages. quart. Res., 4, 117-127.
Newell,R.E. 1979: Climate and the Ocean. American Scientist, 67, 405-416.
Newell ,R.E. and J. Hsiung, 1979: Fluctuation in zonal mean sea surface
temperature. Rivisita Italiana di Geofisica e Scienze Affini,
V, 151-155.
Newell,R.E.,J.W. Kidson, D.G. Vincent and G.J. Boer, 1974: The general
circulation of the Tropical Atmosphere, Vol 2, M.I.T. Press,
Cambridge, London, England.
Newton,C., 1961: Estimates of vertical motion and meridional heat exchange
in Gulf Stream eddies, and a comparison with atmospheric disturbances.
J.Geophy. Res., 66, no. 3, 853-869.
North, G.R., 1975: Theory of energy-balance climate models.
J. Atmos. Sci., 32, 2033-2043.
Oort, A.H. 1977: The interannual variability of atmospheric circulation
statistics. NOAA Professional paper 8, 76 pp.
Oort, A.H. 1971: The observed annual cycle in the meridional transport
of atmospheric energy. J.Atmos. Sci., 28, 325-339.
Oort, A.H. and E.M. Rasmusson, 1971: Atmospheric circulation statistics.
N.O.A.A. Professional paper 5, 323 pp.
Pounder, El R. 1965: The physics of Ice, Pergamon Press.
Rasool, S.I. and S.H. Schneider, 1971: Atmospheric CO 2 and aerosols,
effect of large increase on global climate. Science, 173, 138-141.
Rhine.P.B. 1977: The dynamics of unsteady currents. The sea: ideas and
observations on progress in the study of the sea, E.D. Goldberg
Ed., Wiley and Son. 189-318.
Robinson, S.M. 1966: Interpolative solution of systems of nonlinear
equations. SIAM J. Numer. Anal., 3, no.4, 650-658.
Robinson,A. and Stommel,H. 1959: The oceanic thermocline and the associated
thermohaline circulation. Tellus, 11, 295-308.
Rodgers, C.D., 1967: The radiative heat budget of the troposphere and
lower stratosphere. Planetary circulation project report no. 22,
Dept. Meteorology, M.I.T. 99pp.
Rooth,C.G. and H.G. Oslund, 1972: Penetration of tritium into Atlantic
thermocline. Deep Sea Res., 19, 481-492.
Rossby,H.T. 1965: On thermal convection driven by non-uniform heating
from below. Deep Sea Res., 12,1-16.
- 156 -
Saltzmann, B., 1977: Global mass and energy requirement for global oscillation
and their implication for mean ocean temperature oscillation.
Tellus, 29, 205-212.
Savin,S.M., 1977: The History of the Earth's Surface Temperature during the
Past 100 Million Years. Ann. Rev. Earth Planet Sci., 5, 319-55.
Schneider,S.H. & Dickinson,R.E., 1974: Climate modelling, Review of
geophysics and space physics. 12, 447-493.
Schneider,S. & Gal-Chen,T., 1973: Numerical Experiments in Climate Stability.
J. Geophy. Res., 78, 6182-94.
Schnitker,D., 1974:
120,000 Years.
West Atlantic Abyssal Circulation during the Pass
Nature, 248, 385-387.
Schnitker,D., 1979:
The Deep Waters of the Western North Atlantic during
the Past 24,000 Years, and the re-initiation of the Western Boundary
C rrent. Marine Micropaleontology, 4, 265-280.
Schliltz,C. & Gates,W.L., 1971-1974: Global Climatic Data for Surface,
800 mb, 400 mb. Jan. April, July, Oct., ARPA no. 189-1, RAND.
Sellers,W.D., 1973:
241-254.
A new global climate model. J. Appl. Meteor., 12,
Sellers,W.D., 1969: A global climate model based on energy balance of
the earth atmosphere system. J. Appl. Meteor., 8, 392-400.
Sellers,W.D., 1965: Physical Climatology.
Chicago London.
Univ. Of Chicago Press.
Shackleton,N.J., 1967: Oxygen Isotope Analyses and Pleistocene Temperature
Re-assessed. Nature., 215, 15-17.
Shackleton,N.J., 1978: Evolution of the Earth's Climate during the
Tertiary Era. Evolution of Planetary Atmosphere and Climatology of
the Earth, CNDS, Nice, 573 pp.
Simpson,G.C., 1934: World Climate during Quaternary Periods.
Quart.J.Roy.Met.Soc.,60,425-471.
Starr,V.P. & Peixoto,J.P., 1971: Ploe to Pole Eddy Transport of Water Vapor
in the Atmosphere during the IGY. Arch. Met. Geoph. Biokl. Ser.,
A, 20, 85-114.
Starr,V.P., 1951: Application of energy principles to the general circulation,
Compendium of Meteorology, American Meteorology Society.
Stommel,H.M. and G.T.Csanady, 1980: A relation between the T-S curve and global
heat and atmospheric water transports. J.Geophy.Res.,85,no cl,495-501.
- 157 -
Stommel,H., 1962: On the smallness of sinking regions in the oceans.
Proc. Nat. Acad. Sci., Wash., 48, 766-772.
Stommel,H. and Aron,A.B., 1960: On the abyssal circulation of the world
ocean II, An idealized model of the circulation pattern and amplitude
in oceanic basin.
Deep Sea Res., 6, 217-283.
Stommel,H., Meincke,J. and Zenk,W., 1977:
Polymode News, 22.
New Animals for the Eddy Zoo
Stone,P.H., 1978: Constraints on dynamical transport of energy on a spherical
planet.
Dyn. Atmos. Oceans., 2, 123-139.
Stone,P. 1973: The effect of large scale eddy on climate change.
J. Atmos. Sci., 30, 521-529.
Stone,P. 1972: A simplified radiative dynamical model for the static stability
of rotating atmosphere.
J. Atmos. Sci., 29, 405-418.
Streeter, S.S. and Shackleton,N.J., 1979: Paleo-circulation of the deep
North Atlantic: 150,000 yrs record of benthic foraminifera and oxygen-18.
Science, 203, 168-171.
Suarez,M. and Held,I., 1976: Modelling climatic response to orbital parameter
variations.
Nature, 263, 46-47.
Sverdrup,H.U. 1957: Oceanography,
Akademie-Verlag, 608-670.
Handbuck der Physik, vol. 48, Berlin,
Sverdrup,H.V., Johnson, M.W. and Fleming,R.H., 1942:
The Oceans, Prentice Hall
Swinbank,W.C., 1963: Long wave radiation from clear skies.
Roy. Met. Soc., 89, 339-348.
Quart. J.
Trenberth,K.E. 1979:
Mean annual poleward energy transport by the oceans
in the southern atmosphere.
Dyn. Atmos. Oceans., 4, 57-64.
USSR Navy, 1976: Atlas of the oceans, (in Russian), Ministry of Defense,
USSR Navy, Pergamon Press.
Vondar Haar, T.V. and Qort,A.H., 1973: New estimates of annual poleward
energy transport by northern hemisphere oceans.
J. Phys. Ocean.,
3, 169-172.
Vondar Haar,T.H. and Suomi,V.E., 1971: Measurements of the earth's radiation
budget from satellites during a five-year period Part I: Extended
time and space mean.
J. Atmos. Sci., 28, 305-314.
Vowinckel,E. and Orvig,S., 1970: The climate of the north polar basin,
World Survey of Climatology, vol. 14, Elsevier Publishing Co.
- 158 -
Warren,W.R., 1970: Northward transport of Antartic bottom water in the
Western Atlantic Ocean.
Deep Sea Res., 17, 367-71.
Warren,S. and Schneider,S., 1979: Seasonal simulation as a test for
uncertainties in the parameterizations of a Budyko-Sellers zonal
climate model.
J. Atmos. Sci., 36, 1377-91.
Welander,P., 1959: An advective model of the ocean thermocline.
Tellus, 11, 309-318.
Wetherald,R.T. and Manabe,S., 1975: The effect of changing the solar constant
on the climate of a general circulation model.
J. Atmos. Sci., 32,
2044-2059.
Williams,J., Barry,R.G. and Washington, W.M., 1974:
Simulation of the
atmospheric circulation using the NCAR global circulation model with
ice age boundary conditions.
J. Appl. Meteor., 13, 305-317.
Wilson, R.C. and Hickey,J.R., 1977: Implication for variation in the solar
output in cycle 20. The Solar Output and its Variations,
O.R. White, ed., Colorado Assn. Univ. Press, Boulder, 5 2 6 pp.
Washington, L.V., 1968: Genesis and evolution of water masses.
Climate Change, Meteorological Monographs, 8, 63-67.
Causes of
Wunsch,C., 1977:
Determining the general circulation of the oceans:
a preliminary discussion.
Science, 196, 871-875.
Wust,G., 1935: The stratosphere of the Atlantic Ocean vol. I, Berlin 1935.
translated by Emery E.J., Amerind Pvt. Ltd., New Delhi,1978.
- 159 -