An Estimate of the Upwelling Rate in the Tropical
Pacific Ocean
by
Sarah Louise Samuel
B.A., Somerville College, Oxford University
(1996)
MRes., The University of Edinburgh
(1997)
Submitted to the Department of Earth, Atmospheric and Planetary
Sciences
in partial fulfillment of the requirements for the degree of
Master of Science in Climate Physics and Chemistry
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
October 1999
Massachusetts Institute of Technology 1999. All right rsrved.
@©
MASSACHUSETTS INSTITUTE
I
JA
A uthor ..............
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&.-.,.
- .. . . .-
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iRIE
Department of Earth, Atmospheric and Plan tary Sciences
/ ) / OIober 21, 1999
77
Certified by ............
.......
darl I. Wunsch
Professor
Thesis Supervisor
................................................
Ronald G. Prinn
Head, Department of Earth, Atmospheric and Planetary Sciences
A ccepted by .........
An Estimate of the Upwelling Rate in the
T ropical
Pacific
Ocean
by
Sarah Louise Samuel
B.A., Somerville College, Oxford University
(1996)
MRes., The University of Edinburgh
(1997)
Submitted to the Department of Earth, Atmospheric and Planetary Sciences
on October 21, 1999, in partial fulfillment of the
requirements for the degree of
Master of Science in Climate Physics and Chemistry
Abstract
An inverse box model of the tropical Pacific Ocean from 321S - 10N is constructed
from two zonal and six meridional hydrographic sections. This data is supplemented
with LADCP data close to the equator where geostrophy fails. A consistent solution
is found despite the presence of a number of mid-ocean crossing points and the data
being spread over many years and seasons. The total upwelling across the Oo = 23.5
isopycnal surface in a 60 latitude band centered on the equator is estimated to be
55 ± 27Sv. The zonal mean cross-isopycnal velocity for the same surface in the same
latitude band is estimated to be 6.88 t 3.23 x 10 4 cms- 1 .
The addition of radiocarbon data places a strong constraint on the vertical transfers in the model and significantly reduces the error on the estimated vertical transport and velocity. When radiocarbon constraints are included, the upwelling across
the oo = 23.5 isopycnal surface in the equatorial zone is estimated to be 52 ± 16Sv
and the zonal mean cross-isopycnal velocity across the same surface is estimated as
7.15 t 1.90 x 10- 4 cms-1 .
That a consistent solution can be found is encouraging but it remains unclear
whether one-time data is representative of mean conditions in a region which is known
to be highly variable.
Thesis Supervisor: Carl I. Wunsch
Title: Professor
Acknowledgments
Although the cover page of this thesis bears only one name, I would not have been
able to complete this task without the support and guidance, scientific or otherwise,
of many people over many years.
First, I must thank those who gave me help specifically associated with this work.
Carl Wunsch, my thesis advisor, has provided support throughout my time as a graduate student at MIT and has shown great patience whilst I grappled with problems.
Robert Key kindly provided the radiocarbon data and a wealth of advice on how to
deal with it; Eric Firing supplied the LADCP data and Susan Wijffels provided the
mean hydrographic data. I am enormously indebted to Alexandre Ganachaud who
wrote the wondrous Hydrosys Matlab routines and saved me from many months of
programming and to Alison MacDonald who gave me thoughtful advice and positive
guidance. This work was supported by Caltech J.P.L. contract 958125; NSF/WOCE
contract OCE-9730071 and NASA/WOCE contract NAG5-7857.
Many others have been essential to the completion of this project in many ways:
Scott Sewell gave a sympathetic ear and advice (not to mention chocolate) when I
was at my most despairing and uttered the phrase that has been my guiding light
for the last 9 months: "...you'll have your masters, you'll be in Fiji and you'll be a
happy woman". Patricia Kassis (nee Bradshaw) did most of my problem sets for me
and taught me everything I know about "Them Broncos!". One (or all) of Veronique
Bugnion, Will Heres and Jessica Neu were always willing to be distracted from their
work on the 17th floor and Mary Elliff frequently helped out with particularly tricky
problems, such as sending a fax. Gerard Roe was always willing to give sage advice,
especially over a pint and Christopher Barrett rarely refused an invitation to The Misery. Thanks are also due to Rebecca Morss, Mort Webster, Nili Harnik, Constantine
Giannitsis, Geri Campos and Jenn Jamieson.
The staff and volunteers of Youth Enrichment Services enormously enriched my
life and provided a retreat from the rigours of academic study. The "Wednesday
night fitters" including Steve Murray, Trish Gallagher, Dee Cassidy, Jeff Mills, Verena
Jones, Coleen Sebastien, Sarah Olson et al. made hectic evenings in the shop great
fun, and Richard Williams, Mary Crowther and Eric Pickney were always kind and
appreciative. The kids were always challenging and even more rewarding-especially
Carlos and Trisha, my two top pupils, who inspired me with their enthusiasm and
joy.
Many friends in the UK and other exotic locales (notably Fiji) have supplied
encouragement, Ribena and Marmite as need. My sister, Claire Samuel, frequently
sent chocolate and was always ready to commiserate and my parents, Ronald and
Ann Samuel, continue to tolerate their daughters' apparent disinclination to become
gainfully employed. And to fulfill a two-year old promise, thanks to Caroline Ross
for the tea, Vix Snell for the right sleeve and Claire Marshall for the left sleeve.
Finally, my thanks to three people who I have shared so much with: Antonio
Irranca has been a constant companion over the last three years - thanks for being
the froth on the cappucino of my life; Molly Frey was a wonderful roommate and
continues to supply many happy evenings of conversation and friendship and Albert
Fischer, what can I say? Friend, technical support, travel consultant and so much
more - Thank you.
Contents
1 Introduction
11
1.1
Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.2
Outline of thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2 Steady State Model
2.1
2.2
2.3
Initial Flow Estimates
14
. . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.1
Hydrographic data . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.2
LADCP data . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.1.3
Ekman fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.1.4
Indonesian Throughflow
. . . . . . . . . . . . . . . . . . . . .
19
2.1.5
Validity of assumptions . . . . . . . . . . . . . . . . . . . . . .
19
Model Formulation....... . . . . . . . . .
. . . . . . . . . . . .
19
. . . . . . . . . . . . . . . . . . . . .
19
. . . . . . . . . . . . . . . . . . .
21
2.2.1
Model Areas. . . . .
2.2.2
Model Equations.. . . .
2.2.3
Estimated variances
. . . . . . . . . . . . . . . . . . . . . . .
23
2.2.4
Initial imbalances . . . . . . . . . . . . . . . . . . . . . . . . .
23
Model solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3.1
Simplified model
. . . . . . . . . . . . . . . . . . . . . . . . .
27
2.3.2
Full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.4
3
2.3.3
Horizontal Circulation . . . . . . . . . . . . . . . . . . . . . .
40
2.3.4
Vertical transports . . . . . . . . . . . . . . . . . . . . . . . .
58
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Summary
61
Time Dependent Model
3.1
3.2
3.3
Model Equations for A 4 C . . . . . . . . . . . . . . . . . . . . . . . .
14
Catm . . . . . . . . . . . . . . . . . . . . .
62
. . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.1.1
Time history of A
3.1.2
Invasion rate
3.1.3
Time history of A
. . . .
63
3.1.4
Total Carbon . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.1.5
Time history of subsurface AMC
. . . . . . . . . . . . . . . .
63
3.1.6
Errors in AM4 C terms . . . . . . . . . . . . . . . . . . . . . . .
64
Model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
14
Csu- .
.. .. .
..
.. .
- - ..
3.2.1
Horizontal circulation.
. . . . . . . . . . . . . . . . . . . . . .
64
3.2.2
Vertical Transports . . . . . . . . . . . . . . . . . . . . . . . .
74
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
Summary
80
4 Discussion
4.1
Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
Model improvements . . . . . . . . . . . . . . . . . . . . . . .
80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.1.1
4.2
61
Further work
List of Figures
2-1 Hydrographic sections
. . . . . .
2-2 Model areas ...............
. . . . . . . . . .
20
2-3a Initial imbalances in areas 1-12
. . . . . . . . . .
24
2-3b Initial imbalances in areas 13-24. . . . . . . . . . . . . . . . . . . . .
25
2-4 Model results for simplified model using typical const raints . . . . . .
28
2-5 Model results for simplified model using modified cor straints . . . . .
29
2-6a Reference level velocities on zonal sections
. . . . . . . . . .
31
2-6b Reference level velocities on meridional sections . . . . . . . . . . . .
32
2-6c Model residuals in areas 1-9 . . . . . . . . . . . . . . . . . . . . . . .
33
2-6d Model residuals in areas 10-18 . . . . . . . . . . . . . . . . . . . . . .
34
2-6e Model residuals in areas 19-24 . . . . . . . . . . . . . . . . . . . . . .
35
2-6f Cross-isopycnal velocities in areas 1-9 . . . . . . . . . . . . . . . . . .
36
2-6g Cross-isopycnal velocities in areas 10-18
37
. . . . . . . . . . . . . . . .
2-6h Cross-isopycnal transports in areas 1-9 . . . ...
. . . . . . . . . .
38
2-6i Cross-isopycnal transports in areas 10-18 . . . . . . . . . . . . . . . .
39
2-7a Reference level velocities along zonal sections . . . . . . . . . . . . . .
41
2-7b Reference level velocities along meridional sections . . . . . . . . . . .
42
2-7c Model residuals in areas 1-9 . . . . . . . . . . . . . . . . . . . . . . .
43
2-7d Model residuals in areas 10-18 .
2-7e Model residuals in areas 19-24 .
2-7f Vertical velocities in areas 1-9 . . . . . . . . . . . . . . . . . . . . . .
46
2-7g Vertical velocities in areas 10-18 . . . . . . . . . . . . . . . . . . . . .
47
2-7h Cross-isopycnal transports in areas 1-9 . . . . . . . . . . . . . . . . .
48
2-7i Cross-isopycnal transports in areas 10-18 . . . . . . . . . . . . . . . .
49
2-8
Transport in the thermocline layer . . . . . . . . . . . . . . . . . . . .
51
2-9
Equatorial currents along P16 and P17 . . . . . . . . . . . . . . . . .
52
2-10 Ekman transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
2-11 Transport in the intermediate layer . . . . . . . . . . . . . . . . . . .
55
. . . . . . . . . . . . . . . . . . . . . . .
56
2-13 Cumulative transports in the deep layer . . . . . . . . . . . . . . . . .
57
3-la Reference level velocities along zonal sections . . . . . . . . . . . . . .
65
3-1b Reference level velocities along meridional sections . . . . . . . . . . .
66
3-1c Model residuals in areas 1-9 . . . . . . . . . . . . . . . . . . . . . . .
67
3-1d Model residuals in areas 10-18 . . . . . . . . . . . . . . . . . . . . . .
68
3-le Model residuals in areas 19-24 . . . . . . . . . . . . . . . . . . . . . .
69
3-1f Vertical velocities in areas 1-9 . . . . . . . . . . . . . . . . . . . . . .
70
3-1g Vertical velocities in areas 10-18 . . . . . . . . . . . . . . . . . . . . .
71
3-1h Cross-isopycnal transports in areas 1-9 . . . . . . . . . . . . . . . . .
72
. . . . . . . . . .
73
3-2a Transport in the thermocline layer . . . . . . . . . . . . . . . . . . . .
75
3-2b Transport in the intermediate layer . . . . . . . . . . . . . . . . . . .
76
. . . . . . . . . . . . . . . . . . . . . . .
77
3-2d Cumulative transport in the deep layer . . . . . . . . . . . . . . . . .
78
2-12 Transport in the deep layer
3-1i Cross-isopycnal transports in areas 10-18. .
3-2c Transport in the deep layer
.
List of Tables
2.1
M odel layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.2
Summary of constraints typically applied to inverse problems . . . . .
26
2.3
Cross-isopycnal transport in steady state model . . . . . . . . . . . .
58
2.4
Cross-isopycnal transport in steady state model . . . . . . . . . . . .
59
2.5
Summary of constraints applied . . . . . . . . . . . . . . . . . . . . .
60
3.1
Cross-isopycnal transport in time dependent model . . . . . . . . . .
74
3.2
Cross-isopycnal transport in time dependent model . . . . . . . . . .
79
Chapter 1
Introduction
Understanding the role of the oceans in the global carbon cycle is key to understanding
the climate system as a whole. The equatorial Pacific Ocean is particularly significant
since this region is the major ocean source of carbon dioxide to the atmosphere [Tans
et al.,
1990].
CO 2 is released from the ocean when deep waters upwell and are
warmed at the ocean surface and so a good estimate of vertical transport is necessary
as this is the major control on the surface exchange.
1.1
Previous work
Vertical velocities are small and difficult to measure directly so estimates of upwelling
are either inferred from other observations or calculated in general circulation models.
The highest upwelling rates are confined to a narrow band centered on the equator
and away from the equator there is downwelling and so estimates are strongly effected
by the area over which they are made.
Quay et al. [1983] made an estimate using measurements of "C made during the
NORPAX Shuttle experiment in a two-dimensional (north-south and depth) multi-
layer mixing model. They estimated a upwelling transport rate of 47±13Sv associated
with a vertical velocity of 110 ± 30myr-1 (approximately 3.5 ± 1 x 10- 4 cms- 1) across
the base of the mixed layer in the region 5S - 4'N and 170'E - 100 0W. Bryden and
Brady [1985] estimated a vertical transport of 22 Sv and velocity of 2.9 x 10- 3 cms
1
from a three-dimensional diagnostic model applied to data in the region 50 N - 5S
and 150'W - 110'W and Wijffels [1993] estimated 60 Sv and velocities of order 2 x
10- 4 cms- 1 in an area from 15'S - 8"N and 165 0 E to the eastern boundary from an
inverse model.
Studies which have focused on the region closer to the equator typically give higher
estimates of the maximum vertical transfer. Philander et al. [1987] diagnosed vertical
velocities of over 400 cmday
1
(around 5 x 10- 3 cms 1 ) and vertical transport in a
5' latitude band centered on the equator of some 114 Sv. Poulain [1993] focused
on a 20km latitude band either side of the equator and derived an estimate of 1.5 2 x 10- 3cms- 1 by requiring the vertical velocity to balance the horizontal divergence
computed from the trajectories of satellite-tracked surface drifters and a more recent
study by Harrison[1996] estimated 200cmday-
1
between 2'S - 20N at 1400 W using
a state-of-the-art ocean circulation model.
1.2
Outline of thesis
In this work, a new estimate of vertical transport is made based on the recent measurements of radiocarbon from the WOCE Pacific Ocean radiocarbon program [Key,
1996; Key et al.,
1996; Stuiver et al.,
1996]. The atmospheric testing of nuclear
weapons in the 1960s introduced a large perturbation to the natural
14
C cycle. Bomb-
"C is absorbed by the oceans at the surface and its vertical distribution in the ocean
may be considered as a balance between the rate of invasion and the rate of upwelling
and advection of bomb-1 4 C free waters.
In Chapter 2, the setup of the basic mass, heat and salt conserving model is
described and some results presented.
The full model including the radiocarbon
constraints is discussed in chapter 3. Limitations of the current model and possible
improvements and future work are discussed in chapter 4.
Chapter 2
Steady State Model
An inverse box model is used to obtain an estimate of the circulation in the tropical Pacific Ocean. Hydrographic and lowered acoustic Doppler current profilemeter
(LADCP) data is used as to make the initial estimate of the flow field and the system
solved using Gauss-Markov estimation. An estimate of the zonal mean equatorial
upwelling and cross-isopycnal velocity is made.
2.1
2.1.1
Initial Flow Estimates
Hydrographic data
The hydrographic data used in this model consisted of two basin-wide zonal sections
and six meridional sections. All of these were collected as part of the World Ocean
Circulation Experiment (WOCE) and in this thesis I shall refer to them using their
WOCE labels. The positions of the sections along with the dates of the cruises and
the section labels are shown in Figure 2-1. The stations at the western end of P4
were replaced with mean stations since the Mindanao current was anomalously strong
when the measurements for P4 were made [Wijffels, 1993; Wijffels et al., 1995].
150N
P4: Feb/06/89 - May/19/89
..................
CON
0
0
-O-
C:4)
Z!
CL
10.
204S
'
-.
-.
30 S 37OP6:
120E
1200E
160 0E
Apr/05/92/2JJ1/30/92
1600W
800W 700W
120 0W
ON
LL
the
Figure 2-1: Map showing the positions of the hydrographic data sections used along withQO<
a
0
for each section anda. the dates of the cruises
WOCE labels
i)I
LL
The temperature and salinity data were interpolated onto a standard grid of 50 db
intervals from 0 db to 8000 db. Geostrophic velocities, perpendicular to each section,
were calculated using the thermal wind equations:
dz
(2. 1a)
Opdz
*f p z,, OX
(2.1b)
ugj=
Vg =
where: g
acceleration due to gravity
f(= 2Q sin 0) Coriolis
parameter,
(Q is the Earth's rate of rotation and 0 is latitude)
p
density
x~y
zonal and meridional directions
z
depth
ug, oV9
zonal and meridional geostrophic velocities at depth z
relative to those at depth zo
The reference level, z0 , was taken as 3000 db *, the approximate depth of the
boundary between the slow moving deep and bottom waters of the Pacific. Temperature and salinity for a station pair is taken to be the mean of the two stations.
2.1.2
LADCP data
Close to the equator, geostrophy cannot be applied in this form since
f becomes
very
small. However, in the long term mean, geostrophy is an adequate approximation
for the zonal component of the flow [Pedlosky, 1996]. The equatorial geostrophic
*The small difference between depth in m and pressure in db is neglected
1
IMM
IMMIW
WIN,
balance, equation 2.2, is obtained by differentiating equation 2.1a with respect to y
[Arthur, 1960].
u
where
#
=
gis
9
-
(2.2)
P
p#80 y2
the meridional gradient of the Coriolis parameter and p is pressure.
No reasonable estimate of the equatorial flow field could be found even when the
data were smoothed using either a quadratic fit [Johnson and Toole,
1993] or a
quartic fit [Wijffels, 1993]. This is unsurprising since, in previous work, this balance
has been applied to composite hydrographic sections where mean observations have
been derived from many different surveys in different years and seasons. The data
used here are a one-time survey and the equatorial geostrophic balance is particularly
sensitive to variability and error in the observations: a change in dynamic height of
only 0.OlJkg- 1 I at the equator results in a change in geostrophic velocity of 7.1cms-1
[Bryden and Brady, 1985].
As a result of these difficulties, the flow field within 30 of the equator was estimated
using the LADCP data collected on each cruise [Firing et al.,
1998]. These are
measurements of absolute velocity and this was accounted for in the model by setting
the variance of the reference level velocity to zero in the columns where this data was
used.
Defined 'sections'
Two zonal 'sections' were defined at
±30
using the stations from each meridional
section that was closest to the given latitude and the geostrophic velocities calculated
using equation 2.1b. It was assumed that the the station pair properties for the most
easterly pair represented the ocean state right up to the eastern boundary. These
tlJkg-1 = 1 'dynamic decimeter' = 10 dyn cm
sections are likely to be the least satisfactory in the model since each consists of only
six data points spread over several years in time. The section at 3"N will be referred
to as P41 and that at 3S as P42.
2.1.3
Ekman fluxes
The Ekman transport across each section, or section segment, was estimated using the
Southampton Oceanography Centre (SOC) monthly wind stress climatology [Josey
et al., 1996]. The fields were derived from the Comprehensive Atmosphere-Ocean
Dataset (COADS) Release la and supplemented with information about observing
practices on different ships from the World Meteorological Organisation (WMO)
[ibid.].
The data consisted of the monthly mean northward and eastward compo-
nents of wind stress on a 1 latitude by 1 longitude grid from 31.5"S to 9.5 0 N and
from 120 0 E to 70 0W. From this, the annual mean wind stress was computed and the
Ekman flux calculated using the equations
N
TEkxT~kf=
_1
where TEk
f/
N
TEky
i1
-TPAX~
f
23
2-3)
Ekman transport normal to the stress direction
N
number of stations in section segment
T
wind stress
Azi, Ay,
width of gridbox i
The annual mean wind stress is the most appropriate to use since I am assuming the
hydrography represents a mean state [Jayne, 1999].
2.1.4
Indonesian Throughflow
The published range of estimates for the flow through the Indonesian Passages is
large-Fieux et al. [1996] gave a range of 1-22 Sv. In this model, I use the most
recent available estimate of 14 ± 4Sv [Ganachaud, 1999].
2.1.5
Validity of assumptions
All of the above discussion assumes that the initial data are representative of the
long term mean and that the circulation is in geostrophic balance. Both of these
assumptions are suspect. The tropical Pacific Ocean is known to be a region of very
high variability and the dataset may not even be self consistent, since the observations
were taken in different seasons and in different years. Takahashi et al. [1997] list as
El-Nifno periods: Oct 91 - May 92; Oct 92 - Oct 93 and Apr 94 - Feb 95 so sections
P10 and P19 were both taken during El-Nifios. During such episodes, upwelling in
the eastern Pacific is much weaker than usual and temperatures in the upper water
column are increased. The equatorial regions are also the area where geostrophy is
most difficult to apply. The geostrophic balance is not valid at the equator and is very
noisy close to the equator due to the small size of
f
and the signal of ageostrophic
motions in the temperature and salinity fields. Nevertheless, in order to proceed,
these assumptions will be kept.
2.2
2.2.1
Model Formulation
Model Areas
The region covered by the model was divided into 24 overlapping areas. These areas
are shown in Figure 2-2. Area 24 is the whole region (i.e. the area bounded by
150 N
....
:2
1O N
)
*
. . .....
100
mwg~
........ .................
20
-19
21
22
200S 23
304S -1___.....I..
3 7
0
11 0 B200 E
160 0 E
160OW
1204W
804W70*W
Figure 2-2: Maps showing the areas defined in the model. Area 24 is the whole region
bounded by P4, P6 and the two continents
sections P4 and P6 and the continents).
Each area was then divided into 14 layers in the vertical. The bounding isopycnal
surfaces were based on those of Wijffels [1993] and Tsimplis et al. [1998] and are
shown in Table 2.1
Layer number IUprBudy
I Lower Boundary
Boundary
Boundary [Lower
Layer number Uppersurface
uO = 23.5
= 23.5
o = 24.5
00o = 25.5
o = 26.2
o = 26.7
a0 = 27.1
ao = 24.5
00
ao = 25.5
o=
26.2
ao = 26.7
oo = 27.1
36.5
36.75
36.85
or=
36.5
36.75
02=
o2 = 36.85
o-2 = 36.95
o2 =
2=
2=
36.95
45.81
a4 = 45.85
o4 = 45.93
a2
=
a4 =
o4 = 45.81
u4 = 45.85
bottom
bottom
o-4 = 45.93
surface
Table 2.1: The labels and bounding pressure surfaces for the model layers
Model Equations
2.2.2
Balance equations for mass, salt and heat may be written for each of the boxes defined
and are of the form:
pj(q)C 3 (q)(v, + bj )jAa3 (q) + WBiACB
j
-
TA
T
+
q
where: j
number of station pairs bounding box
q
number of standard depths in layer
C
temperature or salinity
b
reference level velocity
6
±1 to define positive direction as into the box
Aa
vertical interface area
Z(TEk) k =
k
0
(2.4)
tuB, WT
vertical velocity across bottom and top of box
A
horizontal area of box
CB, CT mean temperature and salinity over bottom and top of box
k
number of Ekman grid points bounding box
and other variables have been previously defined.
For each box, this may be written in vector form as
ab + n =
where a
=
b =
(
p1(q)C1(q)61 Aa1(q)
b1 b2 --- b
(2.5)
-..Ep(q)Cj(q)6jAaj(q)
ACB
ACT
OT)
WB
E E p7(q)C (q)v, 63Aaj(q)+ Ek(TEk)k
j q
and
n represents noise due to uncertainty in the observations
Writing an equation like this for each box and for each conserved quantity gives
a set of simultaneous equations which can be represented as a matrix equation:
Ab+n=F
(2.6)
There are a variety of methods that can be used to solve such a system. Here,
it was decided to use Gauss-Markov estimation which gives an estimate, b, which
minimizes the mean square difference between the estimated solution and the true
solution (i.e. minimizes
(
b))
-
) The expressions for the solutions are as follows
[Wunsch, 1996]:
b
n={I -
=
RbbA T (ARbAT + Rnn)
ARbA T (ARbAT + RFn
F
(2.7)
F
(2.8)
P
=
Rb,
-
RbbAT (ARbAT + R
I
-
ARA
Pnn =
R,
T
(ARbAT +
{I - ARMAT
(ARbAT +
ARb
) 1
(2.9)
x
)R 1
(2.10)
b and ii are the estimated solution (the reference level velocities and vertical velocities) and the model residuals (remaining box imbalances). P(=< (b - b)(b - b)T >)
is the solution uncertainty (i.e. the dispersion of b about the true solution) and
Pnn(=< (f - n)(fi - n)T >) is the uncertainty of the residuals. Rbb and Rnn are
described in the next section.
2.2.3
Estimated variances
Gauss-Markov estimation requires the a priorispecification of the expected dispersion
of the solution (Rb, =< bbT >) and the residuals (Rn, =< nn T >). Physically, the
diagonals of these matrices are the expected order of magnitude of the reference
level and vertical velocities and the error to within which a box is considered to
be in balance.
Rb,
and Rn, were both initially defined as diagonal matrices and
so implicitly include the assumption that there is no correlation in noise between
station pairs or between boxes. A typical set of variances, based on those which
have been used in prior work was used as a starting point for finding an accurate
set of constraints for this particular problem. These are summarised in Table 2.2.3.
In preliminary models, the only change made to these constraints was that variance
of the reference level velocities within 30 of the equator were set to zero since the
LADCP data is absolute velocity.
2.2.4
Initial imbalances
Figure 2-3b shows the initial layer and top to bottom imbalance in each of the areas.
-50
0
50
Area 1
100
-50
0
-20
0
20
Area 4
40
-40
-20
0
Area 7
10
-10
50
Area 2
100
-50
0
20
-50
0
Area 3
0
Area 5
-150
-100
-50
Area 8
50
50
100
Area 6
0
-40
-20
0
20
0
20
Area 9
M
-50
0
50
Area 10
100
-200
-100
0
Area 11
100
-40
-20
Area 12
Figure 2-3a: Initial imbalances (in Sv) in each layer and top to bottom (layer 15) for
areas 1-12.
24
-20
0
Area 13
20
-50
0
50
Area 14
100
-20
ILI
0
Area 15
10
20
0
Area 18
40
-10
U
-40
-20
0
Area 16
20
-10
0
10
Area 17
20
-20
II
I
U
U
I
-20
20
0
Area 19
40
-50
-50
0
50
Area 22
100
-50
0
50
Area 20
100
-200
0
Area 23
50
-50
-100
0
Area 21
0
50
Area 24
Figure 2-3b: As for Figure 2-3a but for areas 13-24
100
100
Reference level velocities
Vertical velocities
box imbalances
except
±1cms-1
±1 x 10 4 cms-1
boxes in contact with surface
±1Sv
±2Sv
Table 2.2: Summary of constraints typically applied to inverse problems
It can be seen that there are only a few boxes in which the constraints are met
by the relative velocity field alone. Part of this imbalance arises from errors at the
points where the hydrographic sections cross in mid-ocean. Seasonal variability and
the presence of eddies can cause large discrepancies. In most areas, the imbalance is
distributed through the water column, although a small imbalance in each layer can
cumulatively give a large top to bottom imbalance. However, in area 9 the imbalance
is dominantly in the surface layer. The imbalances in areas 20 and 21 have a similar
structure but opposite sign suggesting that it is the boundary between these two areas
(P17) that is dominating the balance in these regions. It would appear that this is
mostly a result of transports in the equatorial region since the imbalances in areas 10
and 11 also show a similar structure with opposite signs but areas 4 and 5 and areas
15 and 16 do not.
2.3
Model solutions
Several sets of constraints were tested to attempt to find a solution that may be
considered to be consistent with the specified statistics. One definition of consistent
may be that all the unknowns must come within the specified constraints. However,
if the initial variances are considered to be one standard deviation then a consistent
solution may be one for which more than 67% of the unknowns are found to be within
the specified limits. The acceptability of a solution may also depend on which boxes
do not meet the constraints, since it would be anticipated that larger areas should
be closer to balance. For example, a solution for which 80% of the boxes are within
the specified constraints may be accepted if all boxes in areas 19-24 are in balance
whilst those in smaller areas are less well balanced, but may be rejected if the larger
boxes are outside the constraints but more of the smaller areas meet the constraint.
Similarly, poorly balanced equatorial boxes maybe deemed more acceptable than poor
balance in other boxes due to the highly variable and energetic flow in the equatorial
zone.
2.3.1
Simplified model
As a first check, a simplified model consisting only of sections P4 and P6 was constructed to see if a reasonable solution could be found for the region as a whole. Figure
2-4 shows the solution using the 'typical' constraints shown in Table 2.2.3.
It can be
seen that all boxes balance to within the specified constraints, except level 6 which
has a residual of -1.22 Sv, but this is still an acceptable residual. The cross-isopycnal
velocity for the lowest isopycnal (o = 45.93) is the only velocity to fall outside the
constraints but the cross-isopycnal transports all appear to be a reasonable order of
magnitude given the size of the region and that strong upwelling in the upper layers
would be anticipated in the equatorial zone.
A solution was also found in which the cross-isopycnal velocities were constrained
to be 1 x 10- 5 cms
1
(other constraints were the same as in the previous case) and
this is shown in Figure 2-5. In this case, both the surface layer and layer 6 are not
quite in balance with residuals of 3.63 Sv and -1.26 Sv (compared with the specified
2 Sv and 1 Sv). Only the transfer across the lowest isopycnal is strongly affected and
this is now close to zero compared with about 8Sv in the previous solution. The only
obvious difference in the reference level velocities for the two solutions is the increased
I
I
140
160
I
I
0.5-
0-0.5
I
I
-1
120
I
I
I
220
200
180
P4 (cms-)
along
velocities
level
Reference
260
240
280
0.5|-
-0.51-
11111
10
,,,,II
11,11P
1
1I'll11,1111,
.....11,
......
...
...................
"
It-
-15
- 0
I
I,,, , , I111
I
J
300
250
'
200
Reference level velocities along P6 (cms-)
15'0
23.50
1-
23.50
2-
24.50
3-
25.50
4
26.20
26.20
26.70
26.70
7-
27.10
27.10
8
36.50
36.50
9-
36.75
36.75
36.85
36.85
56-
1011
12
36.95
13-
45.81
14-
45.85
15.
-10
10
10
Model residuals (Sv)
24.50
25.50
36.95
I
-
L]
45.81
45.85
45.93
45.93
2
Cross isopycnal velocities (104cms-l)
-11
20
0
-2 0
Cross-isopycnal transports (Sv)
Figure 2-4: Model solution for the simplified model using typical constraints. Dotted
lines are the a priori specified variances and the shaded region on the lower left and
middle panels is the solution uncertainty
28
0.5 0-0.5 -
-1'
120
I
I
140
160
I
I
I
180
200
220
Reference level velocities along P4 (crns-)
I
I
240
260
280
0.5 F
0.5
-1L
200
250
Reference level velocities along P6 (cms1 )
150
23.50
1-
300
.
23.50 --
3-
24.50 25.50-
25.50 -
4
26.20-
26.20
26.70-
26.70U
7-
27.10
27.10
8-
36.50-
36.50-
9-
36.75-
36.75
36.85
36.85
12
36.95
36.95-
13-
45.81 -
14
45.85
2-
56-
1011
15-10
24.50 -
-
45.81
-
]
45.8.-1
45.85
45.93
0
10
Model residuals (Sv)
45.93
-0.5
0
0.5
Cross isopycnal velocities (10-4cms-)
-20
0
20
Cross-isopycnal transports (Sv)
Figure 2-5: As for Figure 2-4 but using modified constraints.
velocities on P4 around 180'W, but the magnitude of these is still well within the
constraints.
2.3.2
Full model
Having achieved reasonable solutions for the single area model, solutions of the full
model with 24 areas were investigated.
Figure 2-6a - 2-6i show the solution for the model using the set of typical constraints summarised in Table 2.2.3. In this case, 44% of model residuals, 80% of the
reference level velocities and 45% of the cross-isopycnal velocities were within the
specified constraints. This solution is clearly not consistent with the specified statistics since not even 2/3 of the estimates are within the expected bounds. However,
examination of this solution may give an indication of which constraints should be
modified.
The largest reference level velocities are found at the western end of P4 and this
may be related to the large uncertainty associated with Box 1 due to the Indonesian
Throughflow. Also, on P4 around 110'W, which is close to the intersection of P4 with
P18, and on P6 between around 130"W (close to P17) and 110"W (close to P18) there
are clusters of large velocities. Over the whole region (i.e. area 24), balance is fairly
closely achieved but in the smaller areas, fewer layers are in balance. This is to be
expected, if no systematic errors are present in the data, it is more likely that balance
would be achieved over a larger region. Many of the cross-isopycnal transports appear
to be rather large (greater than 10 Sv) particularly in area 11. This may be related
to the strong box balance constraints rather than the specified magnitude of w.
The model constraints were relaxed and another solution found. The box balance
constraints were such that in areas 19 - 24, boxes were required to balance to within
2 Sv and to within 3 Sv for boxes in contact with the surface. The constraints
5-
0-
120
140
180
160
240
220
200
260
28 0
1
0-
-2-
140
160
180
200
220
240
260
28 0
160
180
200
220
240
260
280
0
-1
-2 I-3'
140
321S0-
--1-2-3 -
150
200
250
300
Figure 2-6a: Reference level velocities in cms-I along sections (from the top) P4,
P41, P42 and P6 for full model using typical constraints. The dotted lines are the
specified variances.
0
-4
-
L
-2
4
2
0
-2
-2'-35
0
2
4
6
8
11
-
-
2-
-30
-25
I
I
-20
-15
-10
I
I
-5
I
0
5
I
I
10
0-
-2
-35
-4L-
-35
-2-2L
-35
-3
-30
-25
-20
-15
-10
-5
0
5
10
-30
-25
-20
-15
-10
-5
0
5
10
-30
-25
-20
I
-
I
-15
I
I
-10
-5
I
I
0
5
10
Figure 2-6b: As for figure 2-6a but for sections (from the top) P1O, P13, P16, P17,
P18 and P19
13
14
15 L
-10
0
10
Area 1
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15 _
-20
10
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-10
0
Area 5
5
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-5
10
Area 8
20
-10
0
-10
8
9
10
11
12
13
14
15
-10
10
0
20
-5
5
-10
Area 4
-5
0
Area 7
0
10
20
Area 2
0
0
0
Area 6
5
10
Area 9
20
Figure 2-6c: Model residual in Sv for areas 1-9 from full model using typical constraints. The solid lines are the model residuals; the dotted lines are the a priori
specified variances and the shaded areas are the residual uncertainties (the square
root of the diagonal elements of P,,).
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-40
-20
|
-20
-10
:
0
Area 10
0
Area 13
20
10
1
2
3
4
5
6
7
8
9
10
1
12
13
14
15
-10
-20
0
-10
10
Area 11
0
Area 14
20
-10
0
10
Area 12
10
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-5
0
5
Area 15
1:
2
3
4
5
6
7
8
9
10
11
-5
0
Area 16
5
-5
0
Area 17
5
I'
12
13
14
15
-5
Area 18
Figure 2-6d: As for Figure 2-6c but for areas 10-18
34
89.
1011 -
1213
14
15
-5
-15
-10
Area 19
-5
Area 20
0
5
5
10
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15[
-10
0
10
-5
0
Area 22
Area 21
-15
-10
-5
Area 23
0
5
12.
34
5
6
7
8
9
10
11
12
13
14
15
-10
-5
Area 24
Figure 2-6e: As for Figure 2-6c but for areas 19-24
Area 2
Area 1
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-5
23.50
24.50
25.5026.20.
26.70
27.10
36.50
36.75
36.85
36.95
45.81 -45.81
45.85
45.93
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
5
-5
0
Area 4
Area 3
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
5
-1
0
Area 5
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95-
2
0
Area 7
45.85
45.932
-5
0
5
23.50
24.50.
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
10
-2
Area 8
Figure 2-6f: Cross-isopycnal, velocities in
10-4 cms
0
Area 6
0
2
4
Area 9
1
for areas 1-9 from full model
using typical constraints. The solid lines are the vertical velocities; the dotted lines are
the a priori specified variances and the shaded regions are the solution uncertainties
(the square root of the diagonal elements of P...)
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-20
0
-10
23.50
24.50:
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-20
10
Area 10
0
Area 11
23.50
24.50:
25.50:
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
20
-5
23.50
24.50
23.50
24.50
23.50
24.50
25.50
25.50
25.50
26.20
26.20
26.20
26.70
27.10
26.70
27.10
26.70
27.10
36.50
36.75
36.85
36.50
36.75
36.85
36.50
36.75
36.85
36.95
36.95
36.95
45.81
45.85
45.93
-2
10
-10
23.50
.
23.50
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
:
24.50
25.50
26.20
26.70
27.10
36.50
36.75
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.85
:
36.85
36.95
45.81
45.85
45.93
-2
0
Area 16
36.95
45.81
45.85.
45.93
-1
2
0
Area14
0
Area 17
0
Area 12
:
45.81
45.85
45.93
2
-2
45.81
45.85
45.93
-20
0
Area 13
.
Area 15
36.95
45.81
45.85
45.93
1
-4
-2
0
Area 18
Figure 2-6g: As for figure 2-6f but for areas 10-18.
2
23.5024.50-
23.50
24.50
25.50
26.20
26.70
25.50
26.20-
26.70
36.50
27.10
36.50
36.75
36.75
27.10
36.85
36.95
45.81
45.85
45.93
-10
24.50
25.50
20
[
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
5
-10
LIZZm
|
36.50
36.75
36.85
0
45.81
45.85
45.93[
-5
0
Area 4
23.50
LI
LII]
24.50
25.50
26.20
26.70
L2II
27.10
36.50
36.75
36.85
36.95
45.81
LI]1
I]
[1
El
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-4
LI
-2
0
Area 7
-10
0
10
L
45.93
-10
0
Area 3
Area 2
l
36.95
45.93
-20
10
Area 1
[I]
26.20
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.85
0
L
27.10
36.95
45.81 -
23.50
26.70
25.50L
26.20
26.70
E
36.85
EI
23.50
24.50D
L1
El]
LIII
LI
45.85
45.93,
-10
2
23.50
10
L
24.50
25.50L
26.20
26.70L
27.10
36.50
36.75
36.85
36.95
45.81
0
Area 5
[
45.85
45.93
-2
0
Area 6
23.50
26.70
27.10
0
25.50
[L
LII]
LIIZ
LI
0
(}
L]
0
26.20
L
L
E
24.50
L
[0
10
Area 8
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-10
20
2
[
El
I
0
10
20
Area 9
Figure 2-6h: Cross-isopycnal transports in Sv for areas 1-9 from full model using
typical constraints.
23.5024.50
25.50 -I
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
El
Fl
El
[]
E|ZI]
45.85
45.93
-40
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-10
.
0
-20
Area 10
23.50|
24.5025.50-
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93 _
-50
20
26.20L
26.70
27.10
36.50
36.85
36.95
45.81
45.85
45.93|
-10
0
Area 11
LII
23.50
LIII]
24.50
F25.50
El
26.20
F-1 26.70
27.10 ~LIIIZZJ
F-1
Lii
36.50
36.75
LI
36.85
36.95
0
EZZI
45.81
45.85
45.93
0
-20
5
0
-5
Area 14
Area 1.
23.50-
23.5024.50
25.50
23.50
24.50
25.50l
rn
~rn
El
36.75
36.85
36.95
45.81
45.85
LIII]
45.93
-5
0
5
Area 15
10
-5
0
Area 18
5
26.70
27.10
36.50
36.75
36.85
36.85
36.95
0
Area 16
27.10
36.50
26.20I
LI
36.75
L
24.50L
25.50E
El
El
El
36.50
10
26.70
El
26.20
26.70
27.10
0
Area 12
26.20
m
45.81
45.85
45.93[
-10
El
36.75
36.95
-l
45.81
45.85
-5
0
Area 17
5
45.93
-10
Figure 2-6i: As for Figure 2-6h but for areas 10-18
were relaxed still further for areas 1-18, with balance required to within 5 Sv for
each box and 10 Sv for those boxes in contact with the surface. The variance of
the cross-isopycnal velocities was increased to 5 x 10- 4 cms- 1 for most layers and to
5 x 10 3 cms
1
for the upper five layers in the equatorial zone. This was done since
upwelling is typically much stronger in the equatorial region as a result of the Ekman
divergence. In this solution, 84% of model residuals, 99% of reference level velocities
and 87% of cross-isopycnal velocities were within the specified range. The solution is
shown in Figures 2-7a - 2-7i-these are the same as for Figures 2-6a-2-6i but with
the modified constraints.
Whilst some of the reference level velocities are outside
the specified range, none are of an unacceptable size: the maximum magnitude is
1.48 cms
1.
The balance achieved in areas 1-18 is now much improved but many
layers in areas 20, 21 and 23 have residuals larger than those specified. It may be
that given that this is one-time data from a highly variable region that this is the
best that can be achieved. The cross-isopycnal velocities are all of an appropriate
order of magnitude.
2.3.3
Horizontal Circulation
Simple circulation diagrams showing the transport across each of the section segments
were constructed for three layers. These layers were defined as:
Thermocline: au < 26.7 (model layers 1-5)
Intermediate: ao > 26.7;
Deep:
U2
>
92
<
36.95 (model layers 6-10)
36.95 (model layers 11-14)
The transports shown should be considered as an indication of the relative size
of the transports as errors on some sections may be large. This is discussed further
below.
10.5-
-0.5-1 -1.5
-
120
140
160
200
180
240
220
260
280
1 0.5 -
u-
-0.5 -
1 .5
'-
140
180
200
220
240
260
280
180
200
220
240
260
280
-0.51
-1
-
140
160
0.5 -
-0.5
150
I
I
200
250
Figure 2-7a: Reference level velocities along zonal sections
0-
-4
1
I
I
I
I
I
I
I
I
-0
1
CD
-1'-
IL
-11
-35
-30
-25
-20
-15
-10
-5
0
5
10
-30
-25
-20
-15
-10
-5
0
5
10
-30
-25
-20
-15
-10
-5
0
5
10
-30
-25
-20
-15
-10
-5
0
5
10
0-
-35
0-
-1'-
-35
0
-1
-3 5
-
Figure 2-7b: Reference level velocities along meridional sections
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-20
1
2345
6
7
8
9
10
11
12
13
14
151
-20
0
Area 1
20
-20
0
Area 2
13
20
0
10
11
12
13
14
15I
-20
Area 6
Area 5
1
2
3
4
5
6
7
8
9
10
11
2
3
4
5
6
7
8
9
10
12
13
12
13
14
15
-20
20
7
8
9
11
12
0
Area 4
-20
1
2
3
4
5
6
1
2
3
4
5
6
7
8
9
10
14
15
-20
20
14
15
0
Area 7
20
-20
0
-20
Area 8
Figure 2-7c: Model residuals in areas 1-9
072
Area 9
20
-40
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15,
-40
-20
0
Area 10
-20
0
Area 13
20
-50
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15,
-50
0
Area 11
0
Area 14
50
-50
50
1:
2
3
4
5
6
7
8
9
10
11
12
13
14
15,
-20
1
2
3
1
2
3
1
2
3
4
4
4
5
6
7
8
9
10
11
12
13
14
15
-50
5
6
7
8
9
10
11
12
13
14
15
-20
5
6
7
8
9
10
11
12
13
14
15
-20
[
0
Area 16
50
0
Area 17
20
Figure 2-7d: Model residuals in areas 10-18
0
Area 12
50
|
0
Area 15
20
0
Area 18
20
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-20
-10
0
Area 20
10
20
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-2 0
-10
0
Area 22
10
2
10
2
3
4
5
6
7
8
9
10
11
12
13
14
15.
-20
0
Area 24
10
20
0
Area 19
1
2345
6
7
8
9
10
11
12
13
14
15
-20
-10
0
Area 21
10
1
15 L
-10
-5
0
Area 23
5
,l
-10
Figure 2-7e: Model residuals in areas 19-24
23.5024.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-10
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-5
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-5
0
'
10
Area 1
0
Area 4
0
Area 7
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
20
-10
0
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
5
-5
5
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
10
-5
:
:
:
-5
'
Area 2
23.50
24.50:
25.50:
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
5
-5
0
Area 5
23.50
24.50
25.50
26.20
26.70-..
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
5
-50
-
0
Area 8
23.50
24.50
25.50
26.20
26.70..
27.1036.5036.75
36.85
36.95
45.81
45.85
45.93
50
-50
Figure 2-7f: Vertical velocities in areas 1-9
0
Area 3
5
0
Area 6
5
0
Area 9
50
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.931
-50
23.50
Area 10
23.5024.5025.5026.2026.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93-50
23.5024.5025.5026.2026.70
27.10
36.50
36.75
36.8536.95
45.81
45.85
45.93
-5
0
Area 13
0
Area 16
50
-5
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-5
5
0
Area 11
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-50
50
23.5024.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-10
0
Area 14
0
Area 17
0
Area 12
-5
5
Figure 2-7g: Vertical velocities in areas 10-18
50
0
Area 15
0
Area 18
5
23.50-
23.50]
24.50
24.50
25.5026.2026.70
27.10
36.50
36.75
25.50,
26.20-
23.50
24.50
25.505
rn
26.20
26.70
26.70
36.95
45.81
45.85
45.93
-20
20
-20
23.50-
24.50
25.50
26.2026.70
27.10
36.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.75
36.85
36.95
36.85
36.95
45.81
45.81
45.85
45.93
45.85
45.93
0
Area 4
23.50
24.50
5
-10
45.85
1
45.93
0
10
-10
E
L
L
0
10
20
Area 3
23.50-[
24.50-[
25.50.[
EJ
m
26.2026.70
[]
-
27.10
36.50
36.75
36.85
36.95
0
10
45.81
45.85
45.93
20
-10
FE
NE
a
E]
.
-5
|
0
5
10
20
Area 6
23.50.
24.50-
26.20
25.50
26.20
26.70
26.70
27.10
36.50
27.10
36.50
36.75
36.85
MEl
36.75
36.85
36.95
-10
l
A
Area 5
i
25.50
IE
0
Area 2
23.50 -
-5
45.81
E
45.85
45.93[
0
Area 1
]
-i
36.50
36.75
36.85
36.95
45.81
9
0
27.10
27.10
36.50
36.75
36.85
36.95-
36.85
[
36.95
45.81
45.85
45.93
45.81a
45.85
45.93[
-2
0
Area 7
2
-10
0
Area 8
10
-10
0
Area 9
Figure 2-7h: Cross-isopycnal transports in areas 1-9
23.50-
23.50-
24.50
25.50
26.20
26.70
27.10
23.5024.50
25.50-
25.50
26.20
26.70
26.70
27.10
36.50
36.75
36.85
L7
36.50
36.75
36.75
36.85
36.85
36.95
36.95
45.81
45.81
0
Area 10
20
0
Area 11
50
23.50
24.50-
25.50
25.50L
26.20
26.70
27.10
36.50
26.70
27.10
36.50 -
36.75
24.50
-10
0
Area 14
El
10
45.81
10
45.93
-20
-10
El
25.50J
L
26.20
I
26.70
27.10
36.50
36.75
36.50
36.75
45.81
45.85
45.93
-10
0
Area 15
36.95
23.50
24.50
27.10
36.85
36.95
20
45.85
25.50,
26.20
26.70
10
Area 12
36.85
[LI
45.81
45.85
0
36.75
LI]
36.85
36.95
23.50
45.93
-10
23.5024.5026.20 .
0
Area 13
0
45.81 45.85
45.93[
-50
45.93[
-20
0
36.95
45.85
-20
F-7
26.20,
27.10
36.50
45.85
45.93[
-40
24.50
36.85
36.95
[III
LIII
45.81
45.85
0
10
Area 16
20
0
Area 17
45.93 -20
0
-10
0
Area 18
Figure 2-7i: Cross-isopycnal transports in areas 10-18
10
Thermocline
The transport across each of the section segments in the thermocline are shown in
Figure 2-8.
In general, the large-scale circulation shows the expected features. In the equatorial boxes, there is eastward transport which is consistent with the presence of a
strong equatorial undercurrent (EUC). The exception to this is across the equatorial
segment of P16 where the transport is westward. This reversal may occur because
the EUC is anomalously weak on this section or because other westward currents are
unusually strong. Figure 2-9 shows the currents in the upper 500m in the equatorial zone across P16 with those on P17 shown for comparison. On the P16 section,
the EUC is of about the expected order of magnitude but more significantly, the
South Equatorial Current (SEC), is strong beneath the EUC between the equator
and 3"N. This would account for the anomalous westward thermocline transport on
this section.
In the northern zone (areas 1-7), the flow is also mostly eastward which is consistent with the expected transport of the North Equatorial Counter Current (NECC).
The general pattern of northward flow across the eastern parts of P4 and southward
across the western part fits with the idea of water being recirculated between the
NECC and the westward North Equatorial Current (NEC) north of 10'N. In the
southern region (boxes 14-18), the westward flow across the zonal sections, together
with the northward transport across the eastern part of P6 and southward transport
across the western part is consistent with the sense of the subtropical gyre.
The Ekman component of the transports is shown separately in Figure 2-10. The
meridional Ekman transports close to the equator (i.e. on sections P4, P41 and P42)
are 1-2 orders of magnitude larger than those across P6 showing how much more
important Ekman transport is in the equatorial regions compared with subtropical
0
20S-
23
-
25
35
-110
--5
30*S
5
3 0"%E 120*E
160*E
3
160*W
6
120*W
4
5
80*W 70*W
Figure 2-8: Transport in Sv in the thermocline layer. The bold numbers close to the arrows are
the approximate magnitude of the transport in the direction of the arrow. The smaller, italic
numbers are the approximate upwelling through the base of the layer. Positive numbers are
upward into the layer.
F0.2
-250 -300
0.2
-350 -
-0.
0.4
-400
-.0.2
4500.2
-500
-3
-2
-1
0
1
0.
.6
0-
3
2
-50
0.
-100
0M0.2
0.
-200-250 --0.2
-300-
-350-400
-0
-45000
-500
-3
-2
-1
0
1
2
3
Figure 2-9: Comparison of the currents in the upper equatorial region of sections P16
(upper panel) and P17 (lower panel). The velocities shown are in ms- 1 .
150 N
10ON 00
104S8
-C,
2048 300S370E
160 0E
1604W
120*W
Figure 2-10: Ekman transport in Sv in the surface layer.
804W 70*W
regions.
There is a strong meridional divergence in the equatorial zone which is
reduced or reversed when the geostrophic flow is also included. The zonal Ekman
transport is also 1-2 orders of magnitude smaller than the meridional transport close
to the equator as would be expected since the winds are mostly zonal.
Intermediate
The transports of intermediate waters are shown in Figure 2-11.
Some of the
transports, particularly across P41 and P42, appear to be rather large. However,
it should be remembered that along these two sections the specified variance in the
reference level velocities of 1 cms- 1 is equivalent to a variance in the transport of
intermediate water across a station pair of up to 100 Sv since the stations are very
widely spaced. It is difficult to discern any large-scale pattern of the flow in this
layer. Wijffels [1993] found that the flow in intermediate layers is dominated by large
geostrophic eddies making it difficult to identify features of the mean flow despite
using time-mean hydrographic data.
Deep
The deep water circulation is shown in Figure 2-12. Again, it is difficult to discern any
large-scale features of the flow and in particular, the expected deep western boundary
current is not apparent in this representation.
Figure 2-13 shows the cumulative
transports from east to west across sections P4 and P6. These plots show a generally
northward transport across the western end and a generally southward transport
across the eastern end of the two sections which is consistent with conventional ideas
of deep water circulation in this basin.
1 0N
17
10
5
30
141 64
64
211
21
31
27
0* -
20
40
17
+-26
14-
6
:53
7
4t
+2413'
70
47
45
19
50
10*S
2
0b
OS-
200S -
30S
30
37OoE 120 0E
8
26
-44
2
1600E
160OW
6
40+-
29
294
120OW
4
____
804W 70*W
Figure 2-11: Transport in Sv in the intermediate layer. The numbers are the approximate
magnitude of the transport in the direction of the arrows. Upwelling is not shown but that from
the top of the layer would be the same as shown in Figure 2-8 and that through the base of the
layer the same as that shown in Figure 2-12
10ON
2
9
5
10
75N27 131
1-I
T'
-12
2321
1
00
8-
14 2
3
4
1
1
23
-
-0 10
7
12
21
+
2
6
4-2
8
+3
20OS-
8-80
38
43
1
10
5
- 1
-8
304S-
12
370
1 0 E 120 0 E
1600E
7
160OW
120OW
4
80OW 70OW
Figure 2-12: Transport in Sv in the deep layer. Bold numbers are the magnitude of the transport
in the direction of the arrow. Smaller, italic numbers are the approximate magnitude of the
upwelling through the top of the layer. Positive numbers are upwards, out of the layer.
3025201510 -
5 -0 +.-.--....-...-...
-5
-10L
140
160
200
180
220
240
260
105 .
~ . ........
~~ ~ ~ ~ ........
........
..........................
..............
-5 -10-15160
180
200
220
240
260
280
Figure 2-13: Transports across sections P4 (upper panel) and P6 (lower panel) in the deep layer in
Sv. The solid line is the cumulative transport summed from east to west and the dotted line is
the transport between each station pair
Isopycnal
-o = 23.5
Cross-isopycnal transport in Sv
55
Transport uncertainty in Sv
27
o-o = 24.5
oo = 25.5
26
16
24
23
-o= 26.2
8
21
26.7
-6
20
o-o = 27.1
0
18
=36.5
o-2 = 36.75
U2 = 36.85
36.95
o2
o-4 = 45.81
o-4 = 45.85
6
8
14
18
17
17
19
16
9
-4
15
13
-=
02
Table 2.3: Cross-isopycnal transport in steady state model
2.3.4
Vertical transports
In general, there is strong upwelling in the upper layers of the equatorial boxes and
downwelling in the upper layers of the northern and southern boxes. The total crossisopycnal transports for the equatorial zone is shown in Table 2.3. The upwelling is
strongest across the o = 23.5 isopycnal. This is not surprising since the upwelling is
dominantly required to balance the Ekman divergence from the equatorial boxes and
in this model the Ekman transport is assumed to be entirely in the surface layer. It
is difficult to make a comparison of these estimates with those mentioned in section
1.1 since a different area is covered but this estimate is of an order of magnitude
consistent with previous estimates.
Table 2.4 shows the mean cross-isopycnal velocity across each isopycnal in the
equatorial zone. The mean vertical velocities are at the low end of the range of estimates given in section 1.1 but are a reasonable magnitude. It should be remembered
that these are not strictly vertical velocity but the velocity perpendicular to the isopy-
Isopycnal
Cross-isopycnal velocity (10- 4 cms-')
Uncertainty (10- 4 cms- 1 )
uo
ao
o0
o
co
O
=
=
=
=
=
=
23.5
24.5
25.5
26.2
26.7
27.1
6.88
4.54
3.93
3.10
-1.11
0.02
3.23
3.04
2.91
2.70
2.45
2.10
a2
= 36.5
a2
=36.75
-0.86
0.48
0.46
1.97
-0.28
-2.06
2.03
2.00
1.98
2.39
2.27
2.03
36.85
36.95
a4 =45.81
a4 = 45.85
U2 =
U2
=
Table 2.4: Cross-isopycnal transport in steady state model
cnal surface so it is perhaps not surprising that these estimates are at the smaller end
of the expected range.
Although the zonal mean vertical transport and velocity is upwards, there is significant downwelling in areas 8 and 11 (Figure 2-7i). This is unexpected, Philander
et al. [1987] in a modelling study found upwelling at all meridions, and it is not clear
whether the downwelling at the base of the thermocline seen here is a real feature
of the the steady circulation or related to some problem with the input data. The
downwelling in area 8 may be a real feature - water may be upwelled from the EUC in
the east Pacific, recirculated through surface currents and downwelled back into the
EUC in the western Pacific. The sense of the transports shown in Figure 2-8 suggest
that such a pathway is feasible to the south of the equator, through the SEC, since
there is generally southward transport across the eastern end of P42 and northward
transport into area 8 across P42. It would seem more likely that the downwelling
in area 11 is the result of inadequate data - there are large residuals of the same
sense in the top five model layers so that over the whole thermocline there is a strong
Reference level velocities
Vertical velocities
except
box imbalances (boxes 1-18)
except
box imbalances (boxes 19-24)
except
±lcms 1
upper 5 layers in equatorial zone
±5 x 10 4 CMS 1
±5 x 10 3 CMS 1
boxes in contact with surface
boxes in contact with surface
±5Sv
±lOSv
±2Sv
±3Sv
Table 2.5: Summary of the constraints applied to the model
divergence in this area.
2.4
Summary
An inverse box model has been setup, incorporating mass, salt and heat conservation
statements and solved using the constraints shown in Table 2.4.
The circulation pattern found is in good agreement with previous descriptions of
the circulation of the region and the estimates of the zonal mean vertical transport
of 55 ± 27 Sv and vertical velocity of 6.88 ± 3.23 x 10-4 cms- 1 are also of the same
order of magnitude of prior estimates.
Chapter 3
Time Dependent Model
3.1
Model Equations for A 14 C
Bomb radiocarbon is a transient tracer and so the time dependent version of the
general advective tracer equation must be used (equation 3.1).
aC
t)
af + v.VC = Q(x,
(3.1)
Following the method of Wunsch [1984], this equation may be integrated over volume
and time to give equation 3.2.
2
A"4C?
t
dt +wB A
1tit
d
t2
A14CB~t
12
=-
(a4C(t )
2
-
pjvj Aaj
A14C(t 1 )] V
SCT (t) dt
dt ~WT A
t2
A14Cj (t)
1.1E~
ECO2
2
[A
[t
dt -
Fek
t~
Ctmt)
-
A14Cj(t) dt
Au 4 Csurf(t)] dt
(3.2)
The LHS and the first two terms on the RHS are the same as those appearing in the
steady state equation (Equation 2.4). The additional two terms are the change in
the radiocarbon inventory of the box and an atmospheric source term where E is the
invasion rate and ECO2 is the total carbon.
3.1.1
Time history of Z 14Catm
The atmospheric A
14
C is based on Figure 17 of Broecker et al. [1995]. The curve for
the global mean tropospheric AC1 4
is used here since the model domain spans the
northern and southern hemispheres. The rise in concentration from 0%c in 1955 to
a peak value of 725%o in 1965 was represented by a quadratic function which was
judged to give a better representation than either linear or exponential rises. The
decay from the peak value to about 125%o in 1992 was modelled as an exponential
function. This was written as
7.25(t - 1955)2
1955 < t < 1965
(3.3)
Catm(t) = 725e(1965-t)/T
1965 < t < 1992
(3.4)
A 1 4Catm(t) =
A
14
T
3.1.2
= 14.2yrs
Invasion rate
Wunsch [1984] used a value of E = 16 ± 4molm-2yr- 1 based on a survey of prior
estimates. Quay et al. [1983] estimated E = 15 ± 3molm-2yr- 1 as giving the best
model fit to observations and Broecker et al. [1985] estimated E
=
19.4molm-2yr-1
for the Pacific Ocean by requiring that the change in ocean bomb-1 4 C inventory is
equal to the bomb-1 4 C input to the ocean but gave no estimate of the possible error.
The estimate of Wunsch [1984] is adopted here.
3.1.3
Time history of
14Cuf
There is no significant change in surface bomb AMC between the GEOSECS survey
and WOCE [Peng et al., 1998] and so again the method of Wunsch [1984] is adopted
here to describe A 14 Csuf. It is assumed that A 4 Csurf rises linearly from its prebomb value to its post-bomb value between 1959 and 1968 (as given in equation 3.5)
and remains constant from 1968 to 1992.
A 1 4 Csurf (t)
= A1 4 Csurf,pre +
A"C.,~t)=
3.1.4
MCary,,
(1968
-
t)
(A14
srpotA4Cufre
(1968- - 195)
+(1968
1959) (MCsur,p,,st
-
A 1MCsuf,,,e)
(35
(3.5)
Total Carbon
Only a small error is introduced by assuming that ECO 2 is constant over the whole
region. Wunsch [1984] used a value of 2.1 molm- 3 and Broecker et al. [1985] used
ECO 2 = 2.15 molm-3. A more recent estimate study of Jain et al. [1995] used a
model estimated value of 2.1 molm-3. These estimates are all in good agreement and
here ECO 2 = 2.1molm- 3 is used.
3.1.5
Time history of subsurface A14C
The subsurface A 14 C was assumed to rise linearly from its pre-bomb value to its
present value over a nine year period. The time at which the values start to rise
is taken to be a linear function of depth, ranging from 1959 at the surface to 1992
at 600m, the calculated mean penetration depth. In the upper 600m, the pre-bomb
A14C values were calculated using the correlation with silicate described by Broecker
et al. [1995]. The mean penetration depth was calculated by finding the depth at
which the reconstructed pre-bomb A 1 4 C profile intersected the observed A 1 4 C profile
in each station pair and then taking a mean value over all station pairs.
3.1.6
Errors in A 14 C terms
Errors in the radiocarbon budget arise from uncertainty in the mass budget, the
atmospheric source term and the volume storage term. The mass imbalance is kept
the same as in the previous model. The source terms and volume storage terms are
both assumed to have an error of 25%.
3.2
Model results
A solution was found with the same constraints as described in the previous chapter.
In this case, 83% of model residuals, 96% of reference level velocities and 85% of
cross-isopycnal velocities were within the specified constraints. This is not quite as
high as that achieved in the previous model, as would be expected since another set of
constraints (the
1
C data) has been added. However, the solution is still considered
to be acceptable and the details of the solution are shown in Figures 3-1a - 3-li.
These are the same as Figures 2-6a-2-6i except the model includes the radiocarbon
data and the modified constraints are used.
The solution is very similar to
that described in the previous chapter. It appears that the uncertainty in the model
residuals, particularly in the upper layers, is increased. It would be expected that
the inclusion of the radiocarbon constraints would have the most impact in the upper
layers since it is assumed that there has been no change in the radiocarbon distribution
in the deeper layers and so no new information is included for these layers. The most
significant difference can be seen in the vertical velocity profiles.
3.2.1
Horizontal circulation
The circulation in the thermocline, intermediate and deep layers (as defined in the
previous chapter) are shown in Figure 3-2a, 3-2b and 3-2c and the cumulative trans-
2
1
.
0
-1
-2
20
140
160
180
200
220
240
260
280
1
0.5
0
-0.5
-1
160
40
180
200
220
240
260
28 0
1
0.5
-
.
.
-
.
0
.IT
0-
-0.5
-
-II
-1
-1.5
1 40
160
180
200
220
240
260
280
2
1.5
1
0.5 --0
0
-0.5
-1
1
200
250
Figure 3-la: Reference level velocities along zonal sections
300
-L
0-
-4
-2
0
2
-4
-2
0
2
I
I
I
8
6
4
8
6
4
I
I
I
I
0
5
0
5
1-
0-
-135
-30
-25
-20
-10
-15
-5
10
1-
0
I
-135
-30
I
I
I
I
I
-25
-20
-15
-10
-5
1
I
10
I
0-
-2
-35
-30
I
-25
I
-20
I
-15
-10
I
I
-5
I
0
I
51
I
C-0 0
I
I
-3 5
-25
-20
I
I
-15
-10
I
-5
I
I
0
5
Figure 3-1b: Reference level velocities along meridional sections
66
10
1:
2J
3J
4
5
6
7
8
9
10
11
12
13
14
15
-20
0
20
Area 1
40
7
8
9
10
11
12
13
14
15
-40
0
-20
40
0
Area 5
-20
Area 4
1
23:
4,
5
6
7
8
9
10
11
12
13
14
15
-20
.. :
:
0
Area 7
20
-20
20
-20
0
20
40
Area 2
20
-20
0
20
Area 8
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-20
Figure 3-ic: Model residuals in areas 1-9
67
0
Area 6
0
Area 9
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15.
-40
-20
0
20
1
23
4
5
6
7
8
9
10
11
12
13
14
15
-20
Area 10
0
Area 11
20
1
23
4
5
6
7
8
9
10
11
12
13
14
15,
-20
1
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-40
20
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-40
20
2
3
4
56
7
8
9
10
1112
13
14
15
-20
20
1
23
4
5
6
7
8
9
10
11
12
13
14
15
-20
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-20
1
2
3
45
6
7
8
9
10
11
12
13
14
15
-20
-20
0
Area 13
0
Area 16
-20
0
Area 14
0
Area 17
Figure 3-1d: Model residuals in areas 10-18
68
0
Area 12
20
0
Area15
20
0
Area 18
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-1 0
~T1
-5
1
23456
7
8
9
10
11
12
13
14
15r
-1 0
0
Area 19
5
0
Area 20
5
10
-10
-5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15_
-10
-5
0
Area 22
5
10
-10
-5
0
Area 24
5
10
...............................
-5
0
Area 21
5
10
Area 23
Figure 3-le: Model residuals in areas 19-24
69
23.5024.5025.5026.2026.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93-10
Area 1
Area2
Area 3
Area 4
Area 5
Area 6
-5
0
Area 7
23.50
24.5025.5026.2026.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.935
-50
0
Area 8
23.50
24.50
25.50
26.20
26.70
27.1036.50
36.75
36.85
36.95
45.81
45.85
45.93
-50
Figure 3-1f: Vertical velocities in areas 1-9
0
Area 9
50
45.93--50
0
Area 10
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-50
50
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-5
50
Area 13
0
Area 16
23.50
24.50
25.5026.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-5
5
0
Area 11
0
Area 14
0
Area 17
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-50
50
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-5
5
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-5
5
Figure 3-1g: Vertical velocities in areas 10-18
0
Area 12
50
0
Area 15
5
0
Area 18
5
23.50
23.50
24.50
25.50
24.50
Li1ul
L! ll]
D
-10
0
36.85
36.95
45.81
36.95
45.85
45.93
10
-20
E
-10
E]
45.81
Gl
45.85
0
10
23.50-
-l 24.50
25.50
26.20
L~hl-i
Lilh~-i
Li~-i
E-l
L-]
-i 26.70
27.10
36.50
36.75
36.85
36.95
45.93
-10
0
Area 3
El
El
25.50L
El
26.20
26.70
27.10
36.50 36.75
0
L
|
L
El
36.85
36.95
0
45.81
0
10
45.85
45.93
20
-10
-5
0
5
10
Area 9
20
Area 5
Lilih
Luhihu-l
LiLi-i
LI0
23.50
24.50
25.50
-i 26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
2
-10
Area 6
0
0
0
0
23.50
24.50
- 25.5026.20
26.70
27.10
10
Area 8
Fii
l
LI
36.50
36.75
36.85
0
*L
0
36.95
45.81
45.85
20
45.93
-10
0
L
0
Figure 3-1h: Cross-isopycnal transports in areas 1-9
72
10
23.50
24.50.
- 45.81
45.85
45.93[
-10
Area 7
El
26.70
27.10
Area 2
ElmEl
-2
26.20[
36.85
Area 4
23.5024.5025.5026.2026.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-4
25.50-
36.50
36.75
Area 1
23.50
24.50
25.50
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93[
-5
24.50L
LI
FI
27.10
36.50
36.75
0
23.50|
E]
25.5026.20
26.70
26.20
26.70
27.10
36.50
36.75
36.85
36.95
45.81
45.85
45.93[
-20
D
23.5024.50
25.50
23.5024.5025.50
26.20
26.20
26.70
27.10
26.70
27.10
36.50
I
36.50
36.75
36.85
36.95
45.81
45.85
45.93
-40
|
36.95
45.81
.
-20
20
m
pl
-
27.10
36.50
36.75
36.75
36.85
36.95
45.81
45.85
45.93 r
10
-10
5
36.85
36.95
0
45.81
45.85
45.93
-10
|
0
Area 13
23.50.
1
24.50
25.50
26.20
26.70
27.10
36.50
36.75
El
E
36.85
36.95
45.81
45.85
-10
0
Area 16
23.50.
24.5025.50.
26.20
26.70
27.10
36.50
36.75 36.85
36.95
45.81
45.85
45.93f
10
-5
36.75
36.85
45.81
i
20
0
Area 11
-20
23.50
24.50
25.50
26.20
26.70
27.10
36.50
El]
36.95
45.85
45.93
0
Area 10
0
36.50
2 EE
RI~i
36.75
36.85
23.50
24.50
25.50
26.20
26.70
45.93f
-20
23.50I
24.50
25.50
26.20 -E
26.70
27.10
0
E
f
45.85
45.93
40
-10
|
0
Area 12
10
m
0
ie5
amE Mid
M
RaM
M eW#
0
10
Area 14
36.85 LW
36.95
45.81
45.85
45.93
-10
20
-5
0
Area 15
5
23.50
24.50R~is%
25.50-
l@
26.20
26.70
27.10
36.50
36.75
SaE
EE
M
3
5
0
Area 17
36.85
36.95
45.81
45.85
45.93
-20
10
0
1
-10
0
Area 18
Figure 3-li: Cross-isopycnal transports in areas 10-18
10
Isopycnal
Cross-isopycnal transport in Sv
Transport uncertainty in Sv
ao = 23.5
52
16
u= 24.5
uo = 25.5
-o= 26.2
23
4
0
8
12
11
-= 26.7
-= 27.1
-2 =36.5
U2 = 36.75
U2 = 36.85
-10
-1
5
8
13
11
12
13
13
13
u2=
36.95
17
14
U4 =
o4 =
45.81
45.85
8
-2
11
9
Table 3.1: Cross-isopycnal transport in time dependent model
ports across P4 and P6 in the deep layer in Figure 3-2d.
These are very similar
to the circulation pattern found in the steady-state version of the model. These have
been described in the previous chapter and are not discussed again here.
3.2.2
Vertical Transports
The total cross-isopycnal transport for the equatorial zone is shown in Table 3.1.
The magnitude of the upwelling is very similar to that found in the steady state
model but the uncertainties are significantly reduced. This is to be expected because
radiocarbon is a strong constraint on vertical transfer since the observed distribution
is mostly the result of a balance between the downward diffusion of bomb-radiocarbon
and the upwelling of bomb-"C free waters.
Table 3.2 shows the zonal mean cross-isopycnal velocity in the equatorial region.
Again, compared with the results from the previous model, the uncertainties are
strongly reduced, particularly in the upper layers. The upwelling across the isopycnals
20*S -526--
-5
0
30"S-
8
310*E 120*E
160*E
25+-
37
4
160*W
8
3
4
80"W 70*W
120*W
Figure 3-2a: Transport in the thermocline layer, as for Figure 2-8 but for model including
14
C.
3740
1
1
1O0E 1200E
1
1
160 0E
/I
160 0W
120OW
800W 70OW
Figure 3-2b: Transport in the intermediate layer as for Figure 2-11 except that transports through
the top of the layer are the same as those shown in Figure 3-2a and those through the bottom of
the layer as in Figure 3-2c.
+17
2043043-
33*
17
O1'i0E 120 0E
160 0E
15
-t
5
10
160OW
120OW
I
+
6
4
80*W 70OW
Figure 3-2c: Transport in the deep layer as for Figure 2-12 but for model with 'IC constraints.
20
V
4
10-
-10-20160
140
200
180
220
240
260
1000
50
..
-..
--.
--- ---- --...............---
-..
..
--.
...
-.....-
-5
-10
-15-20-25160
180
200
220
240
260
280
Figure 3-2d: Cumulative transport in the deep layer as for Figure 2-13 except for model including
constraints.
14
C
velocities
Isopycnal
ao = 23.5
7.15
1.90
ao = 24.5
3.97
0.80
uO =
cO =
co =
o =
1.25
0.53
-0.74
-0.89
1.51
1.42
1.50
1.53
36.5
o2 = 36.75
-1.34
0.22
1.56
1.61
= 36.85
0.30
1.66
36.95
o4 = 45.81
o4 = 45.85
1.59
-0.64
-1.91
2.11
2.16
1.86
25.5
26.2
26.7
27.1
U2 =
U2
U2 =
Table 3.2: Cross-isopycnal transport in time dependent model
00
= 25.5 and uo = 26.2 is much less than in the previous model suggesting that
most of the water that upwells into the mixed layer comes from the middle of the
thermocline rather than the base of the thermocline.
3.3
Summary
The model described in the previous chapter has been modified to include radiocarbon
constraints.
The horizontal circulation pattern is very similar to that previously
found, but the radiocarbon has provided a strong constraint on the vertical transfers
in the surface layers and has significantly reduced the estimated error. The maximum
total upwelling in the equatorial zone is now estimated to be 52 ± 16Sv and the
4
maximum zonal mean cross-isopycnal velocity is estimated as 7.15 ± 1.90 x 10- cms--
Chapter 4
Discussion
4.1
Summary of results
An inverse box model has been set up using the new WOCE hydrographic, LADCP
and radiocarbon data. The maximum upwelling in the equatorial Pacific is found
to be 52 ± 16Sv. The inclusion of the radiocarbon data significantly reduced the
error on the estimates of upwelling rate in the upper layers. However, the question
still remains of how well the input one-time data represents the mean state of the
ocean in a region of such high variability. Firing et al. [1998] note that individual
sections from the Pacific Equatorial Ocean Dynamics (PEQUOD) program differed
substantially from the mean and it would seem likely that the same is also true of
the WOCE sections.
4.1.1
Model improvements
There are several improvements which could be made to this model which may help
to improve the estimate of the equatorial upwelling. Flux constraints were not used
and the addition of these may have a significant impact on the transports found. The
north Pacific Ocean is an almost closed basin, the flow through the Bering Straits is
of order 1 Sv, therefore any mass flux out of the Pacific, that is the Pacific-Indian
throughflow, must be balanced by a similar flux into the South Pacific. One would
anticipate that the net mass flux across sections P6 and P42 was of the order of
15 Sv and that across P41 and P4 of only around 1 Sv. Greater independence of
the box-balance equations could be achieved by using salt and temperature anomaly
equations. The variations of salinity and temperature are only a small percentage of
the total salt or heat content and so the conservation statements for these are almost
the same as those for mass [ Wijffels, 1993]. A more rigorous estimate of the expected
variances could also be made. In this model, the specified statistics were based on
the constraints used in previous work and were then modified somewhat arbitrarily
to account for the large variability known to exist in this region. An examination
of the sources of error, such as that of Ganachaud [1999] who proposes a variability
of ±20 Sv at a latitude of 50, may significantly change some of the constraints used
here. The pre-processing of the radiocarbon data could be improved - the method of
interpolation used here was the simplest possible. An improved interpolation method,
for example using objective mapping, may give a more reliable estimate of the
14 C
distribution, particularly along the sections where the data are sparse (P4, P41 and
P42). Finally, in this version of the model, only one term for cross-isopycnal transfers
was included and this was assumed to represent all the transport across a surface due
to advection, diffusion and mixing. However, given that the radiocarbon distribution
is a balance between two opposite processes and the upwelling of
4
C poor waters -
the downward diffusion of bomb- "C
it is probably more appropriate to include
at least two terms to represent the cross-isopycnal transports.
4.2
Further work
As well as the possible model improvements that have been described in the previous
section, there are several ways that this work could be further developed.
A global radiocarbon dataset will eventually be available from the WOCE program
and this would allow the model to be extended to cover all of the oceans. If the model
could be well enough constrained, it would be possible to infer the invasion rate rather
than specifying it and so give a direct estimate of the air-sea flux of C0 2 , although an
estimate of the sea-air flux would also be required to estimate the net rate of uptake
of CO 2 by the oceans.
One issue that has not been addressed here is that of how to deal with the midocean crossing points. In this case, at the crossing points there were stations available
on both sections at almost the same point in space and so, when estimating the
geostrophic velocity, stations from different sections were not matched up. However,
there are inconsistencies between the data taken from different sections, particularly
in the surface and upper water column measurements which vary most strongly on
a seasonal basis. For example, at the eastern end of P6 (taken during the southern
hemisphere winter) the surface stations are in model layer 3 whilst at the southern
end of sections P18 and P19 (taken during the autumn) the surface stations are in
model layer 2. This may just be related to the seasonal shift or there could be some
more systematic inconsistency between the sections.
Given the large variability that exists in this region, it seems that the compilation
of time mean hydrographic data is the only way in which a good estimate of the mean
circulation of this section of the ocean could be achieved. Further, separate datasets
could be complied for measurements taken during El-Niio or La-Nifia periods and
solutions from models for the separate datasets compared to give an indication of
the difference in the circulation pattern during these episodes. By separating out the
El-Nifio data from other data it may be possible to place tighter constraints on the
models since El-Nifno is the major source of interannual variability.
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