__
_~_I__~_.I^IYi-_I-LI , *.--.~^-i-~-i~3
P-b(
THE EFFECT OF WIND MEASUREMENT ERRORS ON
LINEAR SIMULATICNS OF EQUATORIAL CIRCULATICNS
by
Robert Kuklinski
B.S., Worcester Polytechnic Institute
(1981)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS OF THE
DEGREE OF
MASTER OF SCIENCE
IN METEOROLOGY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1984
(
Massachusetts Institute of Technology
1984
Signature of Author _
Department of Earth, Atmospheric, and Planetary Sciences
May 11, 1984
Certified by
I7
)1 /
Mark A. Cane
Thesis Supervisor
Accepted by
Theodore R. Madden
Chairman, Departmental Committee on Graduate Students
MASS
A
UGB
AUG
"
ft p.;
TABLE OF CONTENTS
ABSTRACT
3
ACKNOWLEDGEMENTS
5
CHAPTER 1:
INTRODUCTION
6
CHAPTER 2:
POINT FORCING
9
CHAPTER 3:
RAY THEORY
20
CHAPTER 4:
PATCH FORCING
29
CHAPTER 5:
AN ANALYSIS OF THE EFFECTS OF MEASUREMENT ERRORS
ON EXPECTED CIRCULATIONS IN THE EQUATORIAL PACIFIC
36
CHAPTER 6:
AN ANALYSIS OF THE EFFECTS OF MEASUREMENT ERRORS
ON EXPECTED CIRCULATIONS IN THE INDIAN OCEAN
50
CHAPTER 7:
DISCUSSION
55
REFERENCES
62
APPENDIX
1
63
APPENDIX 2
64
APPENDIX 3
71
FIGURES
79
3
THE EFFECT OF WIND MEASUREMENT ERRORS ON
LINEAR SIMULATIONS OF EQUATORIAL CIRCULATIONS
by
ROBERT KUKLINSKI
Submitted to the
Department of Earth, Atmospheric, and Planetary Sciences
on May 11, 1984 in partial fulfillment of the requirements
for the Degree of Master of Science in Meteoroloqy
ABSTRACT
This paper examines the response of the linear shallow water
equations on a zonally bounded equatorial beta-plane subject to low
frequency periodic zonal wind stress. The general solutions, following
Cane and Sarachik [1981], consist of sums of the eastward propagating
equatorial Kelvin mode and a number of westward propagating Rossby
modes.
To explore the response of a basin to small scale forcing the zonal
wind stress is idealized as a delta function. The response is
dominated by the sum of directly forced Rossby modes. It is
demonstrated that ray theory can be used to duplicate the point source
results. In both cases regions of intense response foci are found at a
distance CT/4 westward of and at the equatorial image point of the
forcing , where T is the period of the forcing and C is the Kelvin
wave speed. Regions that experience a very weak response called shadow
zones are also found.
The point source solution is used as a Green's function to obtain
the response of a basin to unit forcings over an area. We find that
features predicted by point forcing and ray theory appear in our
"patch" forced problem. As the area we force over grows the higher
Rossby modes are damped and the important response becomes confined
closer to the equator.
We use the patch solution and superposition to examine the companion
problem of the effect of forcing across a basin on a single
observation point. The important influence on the observation point
comes from four sources.
1. Strong influence from local effects.
2. Moderate influence from Rossby waves forced within 10* of the
equator to the east of the observation point.
3. Intense influence from inverse focus points CT/4 east of the
observation point
4. Moderate influence from Kelvin waves forced along the equator to
the west of the forcing.
J~CYI~~ ~_L^--C-I.L~_I
._.li.l.ii~U
The patch influence functions for the first three baroclinic modes
are used to examine the importance of wind measurement error in the
modeling of sea surface height in the equatorial Pacific and Indian
Oceans. The wind errors over the basins are represented by a
difference field of two surface wind analyses. FNOC (Fleet Naval
Oceangraphic Center) and NMC (National Meteorlogical Center) winds on
a 2.5"x2.5" grid are used in the Pacific. ECM (European Meteorlogical
°
Center) data and a wind analysis from Wiley and Hinton on a 2°x3 grid
are used in the Indian Ocean. In the Pacific during the January
1982-June 1983 El Nino event, small sea surface errors are found in
the eastern Pacific and within one degree of the equator. Serious
errors are found north of the equator in the center of the basin near
Fanning (9.1cm mean and 48.7cm variance deviation from zero sea
surface error). North and south of the equator in the east the errors
are also large; (near Rabaul 12.4cm mean + 101.4cm variance). In the
Indian Ocean from December 1978 November 1979 large errors from the
equator northward into the Arabian Sea are associated with the
seasonal monsoon circulation.
Thesis Supervisor:
Title:
Dr. Mark Cane
Associate Professor of Meteorology
ACKNOWLEDGEMENTS
I would like to thank my advisor, Mark Cane for the guidance and
freedom he provided me with during my stay at MIT. I would also like
to thank Ed Harrison for serving as a substitute advisor when Mark was
away. Members of the MIT staff and fellow graduate students Mike, Ron,
Mark, Haim, Steve G., Steve Z., Wes, and Sean provided assistance and
insightful disscusions. Most of all I would like to thank my wife Anne
H. Kuklinski for the help, support, and love she has given me over
the past two years.
INTRODUCTION
The theoretical study of forced planetary scale waves in the tropics
has
attention
received considerable
periodic
horizontal wind
stress
are
recently. These waves
studied
using the
forced by
equatorially
scaled shallow water equations:
Ut -
Here
U,V,
height
and H
YV + Hx = e i wtJx(X,Y)
(1.1)
Vt + YU + Hy = eiwt .y(X,Y)
(1.2)
Ht + Ux + Vy = 0
(1.3)
are
the
respectively. The
zonal
velocity,
meridional
velocity,
and
(1.1-1.3) can be
dimensional equivalent to
recovered with the scaling:
T'
0
= TmT
Tm =
[1/)C]
' = LmX
Lm =
[C/)]0. 5
.
5
(1.4)
(1.5)
Here T' and X' are the dimensional time and distance; Lm and Tm the
equatorial distance and time scales and C the Kelvin wave speed.
If we confine our analysis to low frequency, small zonal wavenumber
waves,
the system (1.1-1.3) may be rewritten as:
.. . ~_-PI
L-~l TY-I^^-L^-~_
-~-P*I(
WP-LIIIII
~I~*--L~___~-
= ei
YV + Hx
Ut -
w
t JTx(X,Y)
(1.6)
(1.7)
YU + Hy = 0
=
Ht + U x + Vy
We
are
We
- <<1.
find
(1.7)
equation
the
that
and the
(1.8)
w =
with
motions
examining
only
0
forcing
meridional
1/)x=
O(v)
must
enter
at
in
at 0(E)
zonal forcing
and
O(E)
equation
,here
0(1)
So
(1.6).
in
if
stress components are assumed to act on a scale of 1 to 5 degrees of
latitude only the zonal stress need be retained.
Studies
Cane
[1983];
system
sum meridional modes.
that
low frequency
and others have explored
Gent,O'Neill,and
& 1981];
[1976,1977
by Cane and Sarachik
constructed from
are
The solutions
to this
solutions
the sum of the equatorial Kelvin mode and a sum of Rossby modes that
have their
turning
points equatorward of
frequency.
bounded
by
play an
important part
X=0
and
and
Cane
forcing
X=Xe
in
solved
Sarachik
unbounded
but
the
the turning latitude at
solution
(1.6-1.9)
meridionally. The
in
that
they
in
a
the
basin
boundaries
require
return
flow into the interior.
This paper will use these earlier results to examine the response of
the equatorial
be
idealized
provide
ocean to a periodic
as a delta
a Green's
function
function
to
zonal wind stress.
to
study
examine
more
The stress will
small scale
general
forcing
forcings.
and
Forcing
over an area is a more realistic situation but the response is more
complex. Features from the simpler
delta response will be useful
in
analyzing the more complicated patch forcing case.
An analysis of the character of the response is an important first
step in an attempt to understand how errors in the measurement of wind
stress are transformed into errors in the equatorial height field. The
is of
height
field
model
because
in
particular
most
cases
interest
the
using a shallow water
when
simulates
model
the
observed
properties of the height field better than the observed properties of
the currents.
The
paper
is
organized
as
follows:
In chapter
2 the
zonal
wind
streus is idealized as a delta function in both x and y. An analysis
along with plots of the
of the response of the U, V, and H fields
fields in a 6000km wide basin is provided. Chapter 3 uses ray theory
on Rossby waves and compares the solutions
to the delta results. In
chapter 4 the point source solution is used as a Green's function to
solve for forcing over an area rather than at just a single point. The
results of the previous chapters are used to analyze the effects
of
wind measurement error on sea surface height in the Pacific and Indian
Oceans
in chapters 5 and 6 respectively. The
summarized
in
the
seventh
chapter
and
are
important results
followed
containing notation and mathematical derivations.
by
are
appendices
POINT FORCING
The problem of the response of a linear baroclinic equatorial ocean
to
a
Cane
by
examined
was
constructed
is
solution
Their
(1981).
stress
wind
zonal
periodic
the
from
and
Sarachik
forced
periodic
solution in the absence of boundaries plus a free wave solution added
on
to
Kelvin
and sums
waves
conditions.
boundary
the
satisfy
of
The
Rossby waves.
long
solutions
These
consist
of
boundary conditions
are:
at
U=O
fJ ay=
means
(2.1)
Physically
Kelvin
boundary are reflected as a sum of
boundary
U=O
boundary
would
Rossby
waves.
approximation
(1977)
have
that
would
imply
be
reflected
Since
short
shown
our
Rossby
that
the
(2.2)
X=O
at
that
(2.1)
X=Xe
incident
waves
on
the
eastern
long Rossby waves. At the western
long
waves
Rossby
incident
on
the
the
Kelvin
wave
and
a sum of
short
solution
makes
use
of
the
long
wave
as
waves
are not
correct
long
present. Cane
wave
and
Sarachik
boundary condition
for
__I ___I~_~L
this
case
is
given by
The short Rossby waves
(2.2).
form a boundary
layer that when integrated meridionally can carry no net mass zonally,
leaving the Kelvin wave as the only means of returning meridionally
integrated zonal mass flux incident on the western boundary.
We wish to examine the effect of a small scale periodic disturbance
on a model basin. To accomplish this the model is forced by a zonal
wind stress idealized as a delta function of the form:
(2.3)
Jx(X'Y) = 6(X-X,)6(Y-Y,)
The general form of the solution may be expressed as the sum of forced
and free terms:
=
H
V
+
p
(2.4)
V
H
H
-LoJcL
F/F
The free and forced solutions independently satisfy equation (2.1).
p is a constant chosen to satisfy boundary condition
at X=Xe,
X=Xe;
(2.2). Since U=O
equation (1.7) implies that the height is independent of y at
hence
p determines
the amplitude
of
the height
field at the
eastern boundary.
The solution is
derived
write the complete result:
in detail
in appendix 2;
here we simply
U
v =
e
4
*-4x1
)( tR
4m(m+1)Rmu(Y )R e- m ''
[
-
M(Y)M
]
(2.5)
H
+
=iwXe
(M +
[ e'
me
2
t
t=i
M
Rmu
[2_nRmu(Y*)e
=iwe"
=ix
+ M(Y)m
rh(I)
where
M
Rmh
(2.5) is valid in the region west of the forcing
the east of the forcing
M
iY
(X < X,).
To
(X > X.) only the free term is nonzero and we
are left with (2.6).
U
v
=
[ e -+
t
(M +
2
m.Rme-2(m+1)(
-@
]
]
(2.6)
H
the
notatation for
(2.5-2.6) is given
in appendix
1 with R m
and M
denoting the Rossby and Kelvin modes. The Rossby and Kelvin terms are
defined in terms of the normalized scalar function
representing the mth Hermite polynomial.
fm~(Y),
with Hm(Y)
~II____
j_~/_
TM(Y)
The
complexity
of
the
=
[-F
2mm!
solutions
]'Hf(Y)e
makes
them
(2.7)
-
difficult
to
analyze
analytically. An asymptotic form of the eigenvalue will simplify the
analysis. Morse and Feshbach
(1953) showed that for
large N and near
the equator TY,(Y) may be approximated by:
Y
= (42/7) (2m-Y
2
)cos[0.5Yi-2m-YL
-MT/2
+ msin (Y/fim)]
(2.8)
We may use (2.8) to see which terms in the solution are important if
we consider several modes. The mth Rossby mode written in asymptotic
form for U and H respectively are:
.1
R,
=4~(2m-Y 2 )1 [(-(2m+1)
R,
= 2 (2m-Y
2)
] + (2m+1) [D]sin[
[(2m+1-0.5(2m-Y2)-)Ycos[
2)
[D] = 0.5(2m-Y
a
-1)Ycos[
may also be expressed in
-
0.5Y (2m-Y
2)
] + [D]sin[
]]
]]
(2.9)
(2.10)
)
I + m(2m-YX
asymptotic form using Stirling's formula.
13
alm
a
=
[(2m+1)!][2m!I
=
[427r(2m+1)
aq.
in
The formulae
interested
in
]
e
m
(2.11)
e
(2.8-2.11) are valid for Y2 < 2m. Since we are only
the
response
Order
satisfactory.
= 27r
azr = 0
of
in
the
magnitude
equatorial
statements
regions
using
these
are
these asymptotic
forms will highlight the structure of (2.5-2.6).
The behavior of free waves was examined in Cane and Moore 1981.
higher m terms in the sums were shown to be small
least as fast as m "
).
The
(they decrease at
This allowed the expressions to be accurately
approximated as infinite sums which can written in closed form as:
-itan[-2i(-c)]
U
e '(-iw
V
cos[-2i(-
]e-
L-
iwYsect [-2i( -Q)] (2.12)
0
H
free
p has a form which allows us to use the Cane and Moore sum to rewrite
it in the simplier form:
p TAN2 ]
p = -we- cos2 [-2wX,] tan[-2wX ]e
O(z
74n l
(
(2.13)
14
(2.12-2.13) we observe that the free wave times p is an order
From
one
except
term
where
+-k=1,3,5...
is
problem
near
the
zonal
where
wX= Rk/4
near
forced
is unbounded or
velocity
is
p
large
Trk/4
w(X-Xe)=
at
points
resonance
(i)
the
and
(ii)
entire
if
the
free
wave term is important. All three fields oscillate rapidly away from
the equator.
Along the equator we find:
M =
R
0
=
These allow the order of
The terms
terms
FoflOE
= ZO(mX)
+
=0
HFO
= ZO(mM7)
m" ,
[0.5;
2
1i2.
(2.14)
(2.15)
V
like
M =
0(1)
+ 0(1)
in the zonal velocity sums
decay
)
forced terms to be written as:
1.
U
O(m
RM=
O(m'1r)
and
meridional
the
while the height
grow like m
velocity
is identically
zero. The higher modes dominate the forced zonal velocity term. The U
solution then should be very sensitive to the value we chose for N,
the number of Rossby modes.
Away from the equator we find:
T=
O(mi )
RM R O(m
)
R
O(m6'
)
(2.16)
1_~ I~X___ r_~
____;~1~__~~111_
The
equator,
while
confined
near
Physically
equator.
the
us
tells
this
the
response off
is
velocity
zonal
the
of
response
principal
the
largest
its
experience
will
then
field
height
would
we
that
expect the group velocity of a packet of Rossby waves to speed up near
the
equator
slow
and
turning
the
near
down
forced
The
latitudes.
meridional velocity has terms which grow like m but are multipied by w
which is assumed to be small.
figures
All
[2.1-2.20].
figures
grid. For a representative case will take C=1.0
speeds
for
1.7,
2.8,
the
first
1.1,
0.8,
baroclinic
five
0.6
and
(m/s)
a
from
drawn
are
may be seen in
solutions
the delta forced
The general character of
modes
51
by
X
41
by
(typical Kelvin wave
in the
respectively);
Indian Ocean are
as
N=50
large
a
(our
number of Rossby modes, and we will examine a semi-annual forcing
period T=186.2).
north
and
south
Our model basin size
of
equator
the
roughly
or
Ocean. In nondimensional units the
6000km east-west
is
basin
the
size
28.4 by
is
Y
of
and 2200km
Indian
the
10.4. We examine
the response of the basin to delta forcings at various positions. The
nondimensional
which
to
relative amplitudes of the responses reveal
regions
the
in
basin
are
by
influenced
the degree
forcings.
the
Dimensional values depend on scaling assumptions, such as the depth of
the fluid over which the wind stress acts.
the
dimensional
values
here
because
the
It is not necessary to use
nondimensional
values
are
adequate in our analysis.
Figures
[2.1-2.3]
illustrate the amplitude of the
free U, V,
and H
fields. The general response of the three fields is contained within a
sinusoidal
envelope
that
is
T/4
periodic.
The
outer
edge
of
this
envelope represents the paths of the highest mode Rossby waves.
The principal response of the height field is within the envelope.
important and the response at the edge of
The higher modes are less
the envelope is smaller than in the interior of the envelope. We see
equatorward
that
the
latitudes
turning
the
of
is
field
height
a
constant at the eastern boundary.
The
shown by
as
velocity
zonal
[X-Xe]) times the height field. At
and
-itan(2w
is just
(2.12)
equation
(X-Xe)= - Tr/4w and -
rr3/4w (=4000
50km) on the equator we see the singularity in the U field. At
these points all Rossby modes are present and in phase. 've call this
region of intense response the focus. We also observe that in regions
where
the
of
response
the
free
height
field
important,
is
the
amplitude of the free zonal velocity is small.
The
meridional
free
modes.
equator
along the
is zero
It
the
and
is dominated
velocity
Kelvin
wave
because T
equator
is
term
by
Rossby
the
higher
has
a zero on the
by
definition
zero
for
the
meridional velocity.
Figure
problem
[2.4]
were
denominator of
is a plot of the amplitude that p would have if the
forced
various
at
(2.13) appoaches
plot looks like a constant
field. The
points
1 like m
in
the
basin.
Since
the
; the structure of this
(w2) times a displaced free zonal velocity
frequency parameter wX,
here
replaces w(X-Xe ).
All
our
solutions are a superposition of free and forced terms. If the problem
is forced near XA = 7r/4w
p will
be
large
and
the
or Tr3/4w
free wave
( 2000 or 5950km) on the equator,
will
dominate
the
solution.
To
examine the structure of the forced solution alone we need to force in
regions where p is small.
We will examine
show the amplitude of the U,
[2.5-2.7]
Figures
comparison
in
p is
Here
(X,,Y,)=(5000,0).
for
small
free wave terms are
small and the
terms.
the forced solution
with the forced
and H fields
V,
that result from forcing at this point. The general response of these
that originate at the forcing points.
fields are sinusodial envelopes
These
envelopes
for
waveguide
a
are
sum
The
waves.
Rossby
the
of
forced Rossby modes dominates the solutions. While the Kelvin wave is
small
is
it
present
the
to
compared
zonal
The
Rossby modes.
sum of
velocity response is large near the equator. We observe a focus on the
approximately
equator
4000km
east
of
forcing
the
point.
The
height
field is characterized by large responses away from the equator at the
peaks
the curving U
of
and V
V
waveguides.
exhibits
a
significant
response along the entire waveguide with the exception of the equator
along which the forced meridional velocity is zero.
The position of the forced response pattern depends on the location
of
forcing
the
pattern
entire
point. We
moves
observe
intact
in
with
figures
[2.5,2.8,2.9]
forcing
the
point,
that
with
the
the
westernmost portion of the pattern dissappearing as the forcing point
is moved westward. We also observe that no important response appears
east of the forcing point. This indicates
response,
of
number
the
free
modes
wave
or
is
for
that relative to the forced
inconsequential when considering a large
forcings
away
from
the
resonance
points.
The
temporal
variations
in
the
solutions
are
easily
analyzed.
For
the
real V
field
figures
[2.10-2.12]
show for
the various
times
a
qualitatively similar picture. Waves are forced at (5000,0) and travel
north
equator and
toward the
At
(1000,0).
the
the waveguide. They are turned back
following
and south west
are intense
specified
times
near
focus
the
the v
at approximately
while
static
is
waveguide
individual waves moving in it cause a rise and fall in the value of V
at a specific point. Since our interest lies in the gross response of
a
a
to
basin
forcing, we
on
attention
concentrate
will
amplitude
plots.
of
Results
[2.13-2.15].
forcing
the
off
equator
are
shown
in
figures
The zonal structure of the response is the same as for
the equatorial forcing but the meridional structure is altered. The
forced solutions are image symetric about the equator.
at a point
response
(X%,Y ) and observe a response W(X,Y)
W(X,-Y)
to
forcing
at
a point
(X, ,-Y,).
So if
we force
we would observe a
The
fields
still
follow waveguides which demonstrate the dominance of the forced Rossby
sums. The focus appears at the equatorial image point roughly 4000km
east of the forcing point. The waveguides become contorted sinusodial
patterns around a line that runs from the forcing point to the focus.
The inviscid results are sensitive to the number of Rossby modes.
Figures
number
[2.16-2.20]
depict
the
delta
forced
solutions
for a modest
(N=7) of modes. Since the turning latitudes are lower for the
lower modes, we observe that the responses are trapped closer to the
equator. If we force poleward of the turning latitudes, the response
is very weak. The contours used to plot figure
stronger than those in figure
[2.20].
(2.15] are ninety times
In general equatorward of the
19
turning
latitudes
the
response
is
also
weaker
but,
the
familiar
pattern of curving waveguides still appears. The waveguides are not as
compact
as
in
similar.
The
response
near
the
case
height
the
of
field
the
for
turning
higher
modes
example
still
latitudes
of
the
but
are
qualitatively
experiences
highest modes.
a
strong
In the U
field we see the region 4000km east of the forcing point where a large
sum of Rossby modes will create a focus.
To
this
point
singularities
in
the
have
only
in the free wave
forced
Although
we
our
solutions
are
understanding
a
of
considered
inviscid
results.
The
and the dominance of the higher modes
result
how
of
our
friction
neglect
works
on
of
the
friction.
model
is
incomplete, we know it is present and should be accounted for. Adding
a Rayleigh
friction
will
keep our
free
U bounded
and will
damp
the
higher modes. We may accomplish this by replacing w with:
w = w - iR
As
we
stated
the
inviscid
allowing higher modes
focus. Figures
is
taken as
unrealistic
in
to dominate and retaining a singularity at
the
[2.21,2.22]
w/10 and
results
(2.17)
are
somewhat
have friction in them. The spindown time R
w/100
( 5 and
50 years for this
case).
We see
that friction acts to localize the response at the forcing point. When
it
is
moderate,
waves. When we
R=
w/10,
the
focus
use a small amount
of
isn't
reached
damping;
by
w/100=R;
most
the
of
focus
present, but is appreciably smaller than in the inviscid case.
the
is
20
RAY THEORY
Ray theory
problem.
packet
of
provides us with
The
shows
theory
waves
will
an
alternative way of
asymptotically
follow. Whitham
trajectories
the
(1961)
has
the
examining
shown
that
that
a
these
trajectories called 'rays' satisfy the equations:
(3.1)
DY
Dt
31
DX
Dt
ak
(3.2)
(3.3)
Dk
Dt
Here k is
the
ax
Dl
= -80a
Dt
3Y
Dw
Dt
at
(3.5)
the zonal wavenumber, 1
frequency.
The
rays
are
(3.4)
is the meridional wavenumber
and
because they
are
important
energy paths for a packet of waves. The
physically
theory only tells use where
packets of wave will travel and doesn't distingush between free and
forced waves. We will use the theory then to describe the 'action' of
the packet of Rossby waves emminating from a forcing point.
We
begin
with
the
system
of
equations
which
(1.1-1.3)
can
be
combined into a single equation for V.
-Vttt +Vxx t +Vyy t
Equation
(3.6)
_y 2 Vt
has a degenerate
(3.6)
=0
+Vx
solution V = 0 which
is
the Kelvin
wave. Ray theory won't provide us with any new information on it since
it
is
non-dispersive and we know exactly how
enerqy from east to west along the equator.
it
behaves:
it
carries
Our asymptotic analysis
has shown that the Kelvin terms are small compared to a sum of Rossby
modes
for
large N so here we consider the behavior
Rossby terms. By only considering
variations
in
X are smaller
than
low
of the dominant
frequencies and
those in
Y,
assuming the
(3.6) can be simplified
to:
Vyyt
-y 2 Vt
+Vx
=0
(3.7)
we now use WKB theory and assume V has the form
V=Aei (x y It
(3.8)
where the
total
phase c is
the amplitude A is
defined by
= 1, cx =k,
,
and
t
= -w. If
assumed to vary slowly compared to the phase we are
left to highest order with the dispersion relation:
W
We
=
o(X,Y,l,k)
substitute the dispersion
=
2
-k/(1
relation
+
2
)
(3.9)
(3.9) into the ray equations
(3.1-3.5) to obtain the set of equations;
DY
=
21w 2
Dt
(3.11)
DX
Dt
=
Dk
=
0
Dl
=
-21w
k
Dt
Dw
(3.10)
k
(3.12)
2
(3.13)
k
=
0
(3.14)
Dt
The solutions are easily found to be:
W=
k
WO
= k0
(3.15)
(3.16)
23
X
=
wt/k + X 0
(3.17)
Y
=
[-k/w] sin[2w 2 t/k + cO]
(3.18)
1
=
[-k/w] cos[2w2t/k + ao]
(3.19)
Each ray path then has a characteristic
(3.17)
that
shows
a
disturbance
may
(3.15) and (3.16).
w and k by
only
initial position since only negative k is
move
westward
from
its
allowed by the dispersion
relation.
We
want to examine
rays from a point disturbance. That is to say
given an
initial position
paths
(X,Y)
in
space.
(X 0 ,Y0 )
k,
given a
At t=0,
to plot the
ray
) we may determine
the
and w we want
(X,Y
0
constant ao.
0a =
sin-l[Yo(-k/w) ]
(3.20)
wt/k + X 0
(3.21)
then X and Y may be determined by:
X
Y
=
=
(-k/w) sin[2tw 2 /k + sin-l(yO(-k/w)
t appears only as a parameter
)]
(3.22)
and can be eliminated. Equatorial wave
theory suggests the notation:
-k/w
= 2m + 1
(3.23)
24
so:
Y =
+ sin-
+ [2m+1]ksin[2w(X-Xo)
(3.24)
(Y/(2m+1))]
(3.24) may expressed in the simple form:
(3.25)
Y = + [2m+1-YO]zsin[2w(X-XO)] + Yocos[2w(X-XO)]
The area over which a ray acts is called a 'ray tube'. Ray tubes are
in effect energy channels. Hence regions in which few rays penetrate
called 'shadow zones' will be regions of a weak relative response to a
forcing. Regions in which rays are packed together called
then
are
expected
to
experience
a
strong
response
and
'caustics'
if
several
adjacent rays come together and cross at a focus we could expect the
wave amplitude to become immense.
The foci may be found from (3.25) by finding points on the ray paths
that are multivalued in M. Those are points on which:
sin(2w(X-XO))
2w(X-XO)
= Nr
= 0
(3.26)
25
The focus then is located at:
(X -
XO) = AX = NT/4
(3.27)
or in dimensional units at:
AX
=
CTN/4
(3.28)
=
YO(-1)
(3.29)
in terms of Y the focus is at:
Y
Figures
initial
[3.1,3.4]
points
are
(5000,0)
plots
and
of
ray
paths
(5000,1000).
that
The
emanate
plots
use
from
the
scaling as the previous figures but the number of modes used is
the
same
30.
We
see that the rays are packed along the outer edge of the sine packet
and congregate
as
they near
the
foci.
We
expect
then
to have
our
largest response at the focus and in the areas that are shaded in the
figures.
Ray
theory
tells
us
that
forcing occur in two regions, one
which energy streams
north-west
important
adjacent to
responses
to
the forcing
and south-west of
a point
point
the source
in
and a
second region adjacent to the equatorial image point CT/4 km west of
~~~_~_li
26
the
that
implies
theory also
is
energy
in which
source
concentrated
will
energy
at
be absent
a
focus
in
point.
the shadow
Ray
zone
along the line connecting the focus and the source. These ray paths do
not
for
account
the behavior
of the Kelvin wave term which will be
important if the forcing is near the eauator.
Ray theory may also be used to find an energy density in (X,Y) space
for a packet of Rossby waves. Along a given ray the magnitude of the
group velocity times the energy density
is a constant. So the energy
density may be expressed as:
(kO)/
<E>
Where
IC)
the group velocity and
CD is
(3.30)
E(kg) is a constant for that ray.
may be found from the dispersion relation (3.9).
C,
=
i +
j
=
-(1
1221
+Y2)
222
1 + 2kl(1 2 +Y22-
j
(3.31)
then:
<E> = e(k
0
) [(12+y 2 ) -
2
+ 4k 2 12 (1 2 +y 2 )-
-
(3.32)
which may be simplified with (3.18-3.19) and (3.23) to become:
<E>=
(2m+1) [1+4w
2
(2m+1-Y 2 )]-
(3.33)
27
Instead of using the mode number m , we could find expression for the
energy density in terms of the initial position, space coordinates,
frequency, and the initial energy. From (3.25) without quantization we
find:
Y
+ ((-k/w)-YO) sin2wAX
=
+
Yocos2wAX
(3.34)
and see immediately that.
-k/w = [Y-YOcos2wAX]2[sin2AX]-
2
(3.35)
+
The energy density defined only along a ray path then is given as:
<E>
If
draw
(Y
S,.Jti
we had an explicit
a
continuous
expression
energy
using
the discrete
modes. The
for E(ko)
density
approximation from modal theory for
e(k
+ Y)
2
plot.
C(ko).
figures
(3.36)
0)
we could use
We
Figures
have
a
(3.36) to
discrete
[3.3,3.6] are drawn
show the energy density
that
would result from equally spaced rays. At the turning latitudes we see
a lot of energy. This is
time
caused by the fact that the waves take a long
to turn. Near the foci the actual amount of energy per
ray is
small because the waves move through these regions quickly. The total
amount of enerqy here is expected to be large though because the ray
tubes come together and the area over which
the energy acts becomes
_ 1_1_ --I~LIIIDLCII~IC_~--~II_~---
vanishingly
small.
In
short
the
two
factors
that
govern
energy
density are the area over which the ray tube acts and the intensity of
the
energy
along
the
tube.
Both
factors
are
significant
for
our
problem.
Cane and Sarachik (1981)
showed that ray theory is inadequate if
the Kelvin wave or reflected Rossby waves are important.
total
[(U2
energy
V2
+
H2
+
)/2]
from
modal
The actual
theory
should
resemble the ray theory results in our case since both the Kelvin and
free
terms
Figures
the
are small
[3.2,
results
compared
to
the
sum of
3.5] are the total energy plots.
is clear.
We
see
in
both
forced
Rossby
modes.
The agreement between
plots
that
the
largest
concentration of energy is near
the foci and turning latitudes.
We
also
the
of
observe
figures.
regions
in
same
location
These plots illustrate how effective
determining
method.
shadow
the
essence
of
the
solutions
with
in
both
sets
ray theory can be in
a
simple,
analytic
^_U~__
1______ ~*__~__~~___LI~I~_Fm____
29
PATCH FORCING
We have
analyzed
distrubance.
the response of a model basin to a point source
With
the
exception
earthquakes or nuclear explosions,
of
cataclysmic
events
such
as
the forcing of long period waves in
the real ocean occurs over a region rather than at a single point. The
point forced solutions
(2.5-2.6) may be used as a Green's function and
integrated over a forcing region to obtain the response of the basin
to a regional forcing. We will examined the response of the basin to a
square patch of a unit zonal periodic wind stress.
That is
:
U,V,H [X,Y]
=
ff
TX(X,Y,)G,.,
(X,Y,X,,Y,)dX dY
(4.1)
< AY
(4.2)
FOCg¢N(r
We take :
T
=
1
0
IX-X)
< AX ;
elsewhere
IY-Yj
The
complete
region X <
U
(X'
derivation
- AX)
[ 2w- e
V
H
[
[7,
T'
.)e
Rsin((2m+1)
(erf(Yn)-erf(Ys))
R e
( M + E 2a
[-iw-e
3. The
results
- 2(
-i W
M sin.
-
for
the
40
]
Yn = Y'+AY
X'=X
S= wax
Ys = Y'-AY
y' =y
=
-((Yn)-(Ys)
[erf(Yn)-erf(Ys)]sin4,,+
+
4~(
(4.3)
-
I' = iwX'
I
p =2ie
[e E I
S2w-7
e
p
Appendix
are:
4t
[ 2mlw
=
is given
(2m+1)-F'f
Isin(.,(2m+1))e
for the region X > (X' + AX)
U
V
H
p"1
'4
)]
(4.4)
and for the region
IV=
H
iw[-i
I
-
(X' - AX) < X < (X' + AX)
+
Re
( M + E 2a
T
e
J
=
i
J
)]
Re
i w- 1eor . -IP RA
e4r Z I JRAdY
i-le
-I
]]i1e
-7'[erf(Yn)-erf(Ys)]M[1-e
J2
=
-erf(Y)
-7
[erf (Yn)-erf (Ys) ]e
[erf(Yn) -erf (Ys) ] M[ 1-e
-- (
] iW--le4T
The solution is defined in three distinct regions; for X <
entire forced portion is
(4.5)
J2 +
+
Eelt; ~,r
X-AX the
X-AX <X <X+AX only a portion of the
forced term is experienced ; and west of the forcing region only the
free term remains.
An
asymptotic
analysis
will
prove
useful
in
illuminating
the
structure of the 'box' solution. The only new terms that appear in the
solutions
are the
integral of the RR, and the R,
terms.
For
large N
using equation (2.8) we find that:
(YT(Yn) -
(Ys)+(2m+1)-fy)
1
%s
O(mf)
(4.6)
The order of
the forced terms that are
the
forced and observed on
equator may be estimated using (4.6).
We
U
=
V
=
H
=
that
note
0(1/wim)
not
(4.7)
O(1/w m2 )
velocity field and like
is
as sensitive
O(1/w)
0
modes
higher
the
+
to
(m-
2)
+
0(1/w)
are
damped
the
zonal
the height field. The response then
in
higher modes
the
in
(m-)
like
as the delta case
is.
By
forcing over a patch rather than at a single point phase interference
has destroyed the high frequency response within the forcing region.
We are still left with an important forced reponse though because of
the (w-1 ) factor which by assumption must be large.
To obtain the
box
we performed
solutions
X,
an
introduced a [w(2m+1)]-1 factor to the Rossby terms.
condition
wavenumber
(3.23)
(2m+1)
as
a
function
which
Our quantization
the
of
meridional
1 ; and the distance from the equator Y. At the turning
latitudes we know
delta
expresses
integral,
forced
that
Rossby
Our
1=0.
terms
by
box
solution
Y2 . This
importance of the higher modes is
reduced.
then
is equivalent
It
divided
the
to saying
the
has
should be noted that the
strongest response occurs very near the equator in the patch case.
The sum of forced Rossby modes no longer completely dominates the
solution.-The Kelvin wave term is the same order as the Rossby term
as in
if,
and #g
we assume that we force over a large box(
(4.7),
is
an
one
order
area is
small( O(25x25km))
O(w-1).
A small box
case and
solutions
to
see
the
Kelvin
in both the U and H fields. If the forcing
response on the equator
delta
expect
term. We would
0(4°x4°))
order one and not
the Kelvin wave term is
in this case then should strongly resemble the
by
be dominated
by
normalized
AXAY
the
Rossby
approach
the
terms.
The
box
delta
forced
forcing
solutions
(2.3-2.4) in the limit as AX and AY approach zero.
a
is multiplied by p instead of p .
in the delta case.
the same as
the box case is
The free wave in
It
p is simply the Kelvin wave and its
Rossby reflection at the eastern boundary times a periodic function of
the
position
forcing
zonal
and
basin
zonal
the
length.
When
we
a
integrate p
and Y * to find p we again see that the higher
over X,
Rossby modes are attenuated. We expect then that p will be large only
near
the
equator
that
and
the
focusing effect
is
tempered by the
attenuation of the higher modes.
The relative
importence of the free wave
much greater than in
much smaller.
in
the box case should be
the delta case because the forced box solution is
p will determine how much the free wave contributes to
the total response.
Figures
[4.1-4.12]
illustrate
the
inviscid
the box solutions.
The
amplitude of all fields are displayed. The box solution's sensitivity
to AX and AY is
examined along with the number of Rossby modes and the
position of the forcing.
The solutions are normalized by AXAY.
~ul~--rr.^--~-~~Y~LCaaaar*lh
ii'~~
~~LII--*-~-~~ *
Figures
first
a
[4.1-4.3] are the p terms for three different box sizes.
is
a
very
small
non-dimensional units).
box
20km
by
20km
(0.09
equator
figure
in
[4.2]
0.09
in
It is small enough to resemble p . We observe
a weak focus and note the only real difference is
the
by
The
regions
where
p is
shows a 2* by 2* box.
small.
As
that p is
an
large on
intermediate case
Here the focus is
weak as
are the
higher Rossby modes that allowed p to be large away from the equator.
The last figure is
for a 4"
by 4
box (1.05 by 1.05 in
nondimensional
p is large only within 5°
units. No evidence of a focus is observed.
of the equator. Outside of the focus regions p is larger than near
the equator.
As the box
size grows
, parts of
the box extend into
regions where only high 2ossby modes are present.
less energetic,
p
The high modes are
so only the parts of the box in which the lower modes
u6
are present contribute significantly to p . We see then that p will be
large only near the equator no matter how large of an area we force
over
because
forcing
away
from
the
equator
can
only produce
weak
influences.
and H fields for a 4
The U, V,
shown
in
figures
[4.4-4.6].
The
box centered at
by 4'
effect
of
the
(5000,0)
are
of
the
attenuation
higher modes is seen. The zonal velocity response is large west of the
forcing across the equator. The maxima of the zonal response are still
near the forcing point and at the focus.
[4.7]
has
a
equator. The
forcing.
The
more
main
of
the
response
equator. The response of
extends
focus
and
height
field
pronounced
response
The 2 ° by 2* box
of
the
height
farther
is also
field
is
large
still
in
figure
from
the
near
the
off
the
the meridional velocity is similar to the
delta case but it is weaker by a factor of 10.
rCL-rP
~V^-U4n+~-ura~~~~
response with
The
N=7 modes
shown
is
in figures
The
[4.8-4.10].
general response is similar to the N=50 case. Unlike the delta case
our
the number
to
sensitive
less
is
solution
the
of modes because
higher modes are attenuated.
Results of forcings in the western half of the basin are shown in
figures
[4.11-4.12].
see
We
appears to the east of the
to
contributes
wave
the
part
eastern
superposition
of
of
the
free
response
velocity
indicates that the Kelvin
near
response
total
equator.
the
The
is difficult to see for forcings in
basin
and
zonal
free
forcing. This
contribution of the Kelvin wave
the
a
that
because
forced
it
responses
appears
in
as
part
the
which
of
a
forced
response is very complicated near the equator.
may add
We
given
friction
box
solution with
amount of friction will not alter results
did
in
the
damped.
It
will
it
to the
delta
still
case
because
prove
useful
the
to
highest
examine
equation
(2.17).
A
as impressively as
modes
what
are
already
effect
some
arbitary spindown time will have on the model solutions.
Figures [4.13-4.14] show the responses with friction, corresponding
to a spindown time of 25 years. The zonal velocity response is only
slightly affected indicating, that the higher modes don't contribute
much to the solution.
36
AN ANALYSIS OF THE EFFECT OF WIND MEASUREMENT ERRORS
ON EXPECTED CIRCULATIONS IN THE EQUATORIAL PACIFIC
To this point we have examined the effects of an isolated regions of
periodic zonal wind stress on a model basin. We will use the results
of the previous chapters to determine the response at a single point
to lonq period waves
allow
us to
gauge
forced
the
throughout a basin. This analysis
importance
of
errors
in
will
the measurement
of
surface winds in the modeling of sea surface height in the equatorial
Pacific.
The forcing over the real ocean is basin wide. The response of the
by
a
field
of
wind
then
is
governed
stresses.
The model we developed
is
linear.
Linearity allows
us to
represent
the height or current at a single location as that
which
ocean
at
a
single
station
would result from a sum of individual forcings. The delta case assumes
an impulse forcing is representitive of the forcing that occurs within
a region. The patch forcing assumes that the stress is uniform over a
region. We will divide a basin into a discrete number of regions and
use either the delta or patch assumption to examine the total effect
of basin wide forcing at a station.
The patch and delta solutions may be used to calculate a response
(X,Y). The response function tells
function for an observation point
in the
forced
influence waves
us how much
will have
basin
the
on
observation point. We find the response function by first dividing the
basin into a grid of forcing points. For the delta case each forcing
point represents the forcing over the area between adjacent points. In
the
uniform
of
region
rectangular
that
stress
halfway
extends
grid
each
at
to
find
,Y,).
W(w,X,Y,M,X,
or
,X,Y,M,AX,AY,X',Y')
point
The
the
to
function
response
the
a
(4.3-4.5) and
adjacent grid points. We fix X and Y and solve equations
(2.5-2.6)
of
center
the
is
point
forcing
each
case
forcing
patch
W(w
function
response
is
and a single baroclinic mode
defined for a single forcing frequency
M.
To solve the equations we could have taken our transform in space
instead of time, but for the problem under consideration a frequency
of
analysis
is
errors
wind
a more
insightful
and
straightforward
approach. The time dependence of the problem is naturally periodic in
the
sense
that
We would
seasonally cyclic.
periodic
winds
the
that
hence
spatially;
not
our
are
errors
contain
the
measurement
expect
the
wind
errors
analysis
is
done
in
the
be
to
as
frequency
domain.
In determining the response function we assumed a unit forcing over
a region. It is necessary then to weight the response function by the
observed
forcing
each
at
frequency
to
determine
sea
the
surface
height. By superposition the total sea surface error at a station will
be
a product
stress
summed
of
the
over
response function
all
forcing
and
frequencies,
the
error
in the
baroclinic
zonal
modes,
and
_i~~I-aU~ieriu~DI*O
---
I- L~rxbnu
forcing points. The total error, n ,at a station is written as:
nj(X,Y,t)
E7Z
n W(w,Y,YX,,YM) T (W,X),,Y)e
=
A
(5.1)
nis the sea level scale that translates a height into sea level.
We will model the Pacific Ocean as a rectangular basin with straight
north-
south
unbounded
aligned
at the north
for the Pacific
this
size
coastlines
to
that
120E
and
70W.
from 30N to 30S.
south
available
is
be
and
at
over a 2.50
by 2.5"
The
basin
wind data
Surface
grid.
is
We will
adopt
case
this
based on observations of Eriksen et al
1983
of
our
forcing
grid.
In
the
patch
yields 68 by 24 2.50 boxes of uniform stress.
The
scaling we
use
is
using CTD casts to determine reasonable stratification profiles in the
Pacific at
179W near the Equator. We scale distance by L
T
. The wind stress is
of
the
mixed
layer.
these assumptions
and time by
assummed to act as a body force over the depth
Below
the
the sea level
mixed
scale
layer
n
is
felt.
With
defined, following
Cane
no
stress
is
(1983) by:
n
A,(O)
is
the surface
=
[T/(p,
amplitude
gD)]
of
A2(0)
L.t
the horizontal
(5.2)
structure
function
which is constant in the mixed layer, D is the depth of the ocean and
p, is the density of the water.
39
The stress at the surface is parameterized with
T V=
(5.3)
p CO U J
Co is the drag coeffeicent taken to be 1.8X10-3, ),is
U
is
the
zonal
component of
the
surface
wind,
the air density,
and
UIJ
is
the
magnitude of the surface wind. With these assumptions and a zonal wind
stress of one dyne we have the scaling parameters that appear in Table
for
5-1
from
any
the
first
focus
shows that
away
baroclinic modes
will
four baroclinic modes. Table
points
the
first
and
second
5-1
dominate the sea surface height over most of the basin.
TABLE
5-1
T (days)
A"(0)
C (m/s)
L (km)
1
2.91
361
1.43
4.22
1.436
2
1.78
282
1.83
4.02
1.026
3
1.13
225
2.30
2.05
0.212
4
0.83
190
2.66
1.61
0.110
MODE
We will examine the sea surface errors at 10 islands located near
the equator in the Pacific. The names and location of these stations
s~U_
~_l_/i__lL _i~__l__
__II~_~_
I~IX~_
~_
~L___Z_~~1~
40
are listed in Table 5-2 and illustrated in figure
[5.1]. These islands
have tidal gauges which recorded the sea surface height during the El
Nino event of January 1982 - June 1983.
TABLE
5-2
LATITUDE
LONGITUDE
GALAPAGOS
iS
90W
CHRISTMAS
2N
157W
FANNING
4N
159W
JARVIS
.5S
161W
CANTON
3S
172W
8.5N
168E
NAURU
15
167E
TRUK
7N
151E
RABAUL
4S
152E
MALAKAL
7N
134E
STATION
KWAJALEIN
For each island we must develop a set of response functions
(one for
each forcing frequency). We will drive the model with a time series of
18
monthly mean
functions
stress
are valid
for
errors
at each
a single
forcing point. The
frequency
so
the
time
response
series
of
stress errors must be transformed into the frequency domain to find
the magnitude of the forcing
at
that
frequency. The
time series
is
____~~5~ah~
resolved in a dc(w=0),
with
9 positive, and 9 negative forcing frequencies
the Fourier transform. Since we
negative
frequency
stresses
are transforming real
data, the
are simply the complex conjugates of the
positive frequencies.
A
Tx (X
,Y*,t) -----
(X AY',A W)
Tx
w=27/T
T= ,+-18/n
n=1,..9
4
X
T = Forcing Periods = -,18,9,6,18/4,18/5,3,18/7,18/8,2 Months
is W(w) = W*
The response function also possesses this property, that
(-w) Since the product of complex conjugates is the complex conjugate
the
of
product, we
need
frequencies
frequencies.
by
the
consider
positive
and
dc
forcing
for the negative frequencies when we sum over
frequencies. We account
all
only
introducing
By symmetry the
a
factor
of
2
response sums
imaginary
the
in
to
positive
zero and we
are left with a real time series of sea surface errors.
Figures
Fanning
for
the
forcing cases.
typical
show the amplitude of
[5.2-5.16]
first
We
baroclinic
use this
mode
for
station to
response functions. We
assume a
the response functions
both
the
delta
and
at
patch
illustrate the properties of
spindown time of
5 years
in
20.
The
number of modes is not a critical parameter except very close to
the
all
the
response
observation
assumption
point.
functions.
points
we
used
for
long
does
20 modes appear
not
The
number
forcing
damp
of
periods.
the
modes
Rossby
The
solutions
is
Rayleigh
at
the
friction
observation
to be a value for N that keeps the importance
?
.-
of the response function near the obervation point reasonable. In any
event the number of modes does not significantly effect our analysis.
delta
The
patch
and
response
similar
are
functions
with
the
exception of the dc forcing. The dc response is a local phenomenom in
larger
over a much
It is felt
the patch case.
region
in the delta
case. In the patch case we see that the only important forcing occurs
a line
about
the
Unless
that
wind
runs
north
are
errors
south of
and
dramatically
the observation point.
incoherent
meridionally,
significant north south phase cancellation keeps the importance of the
response about this line small. The delta case contains this feature
along with a significant response to the east of Fanning. At the other
frequencies the delta influence is felt further north and south of the
observation point because as we discussed in the previous chapter the
patch
case
attenuates
the
higher modes
that
are
off
important
the
equator.
We
observe
at
all
forcing
frequencies
Fanning
is
significantly
affected by forcing immediately adjacent to the island and from Rossby
of the equator to the east of the island. For
waves forced within 10
forcing
periods
shorter
than
18/4 months
we see
a maximum
in the
response function at the equatorial image point to and CT/4 east of
Fanning. This maximum, called the inverse focus, corresponds to the
place where Rossby waves that focus at Fanning originated. For forcing
periods shorter than 18/8 months we see a second inverse focus at the
same latitude CT/2 eastward of Fanning. This maximum corresponds to
the
region
in
which
Rossby
waves
that have
their
second
focus
at
Fanning originated. The second inverse focus is weaker than the first
because of the inclusion of friction.
43
To
influence
frequencies
the
for
functions
the
height
the
at
that
several
The
response
features
similar
influence
in the basin that strongly
the regions
to
To sum up the response
those of those of the first baroclinic mode.
function shows
island.
the
reveal
baroclinic modes
other
at
boundary
western
field
and Rossby
Kelvin waves
equator,
the
off
reflected
are
that
waves
near
of Fanning
west
the
the station are:
1. a strong local response at the observation point
2. a moderate broad response to the east of the
island within 10* of the equator
3. a moderate response west of the island along
the equator
4. intense responses from inverse focus points
east of the island
importance
The
the
location
depends
on
Figures
[5.17-5.22]
For stations
felt.
For
Galapagos,
in
show
of
of
in
Kelvin
these
the
response
the east away
stations
the
each
of
the
wave
the
and
station
functions
for
from the equator
east
near
influence
at
contributions
the
is
forcing
some
station
each
frequency.
stations.
other
only local effects
are
at
the
off
the
equator,
large.
such
as
Stations
equator in the central and western Pacific are influenced primarily by
Rossby waves. Kwajalein, Truk, and Rabaul
response functions show the
strong influence of the Rossby waves. In the center of the basin near
the equator
at Jarvis we see that
to the response.
all
four of
the effects contribute
----I
-~ir^L-rMa.C~
I---------------~
-~^~----- ----
44
The
errors
surface
sea
be
now
may
determined
if
we
have
a
representation for the stress errors over the basin. We will take a
field of two frequently utilized surface wind analyses to
difference
into
the basin. These are translated
over
the wind error
represent
stress errors with the parameterization
This parameterization
(5.3).
of the stress as the zonal velocity squared will likely lead to large
stress errors in regions of strong zonal wind. The wind fields we used
were the
FNOC
x2.5* grid
Center)
analysis on a 2.5"
(Fleet Naval Oceanqraphic Center)
(National Meteorlogical
(120E-70W and 30N-30S) and the NMC
surface wind
over
region
same
the
on
a
5"x5*
grid.
These
analyses are monthly mean winds for 18 months from January 1982 - June
1983. The NMC data was linearly interpolated down to a 2.5°x2.5" grid
to conform to the FNOC analysis.
(in dynes) are shown in
The monthly NMC surface zonal stress fields
figures
[5.23-5.31].
The negative sign
indicates
stress
imparted on
the ocean by an easterly wind. During the first six months of 1982 the
Pacific was forced by a typical wind stress pattern. Strong easterlies
between 5N-25N across the basin exert a strong negative zonal stress
during the period. Near
15N 170W we see stress over 2 dynes in March.
Near the equator the stresses are generally weak. South of the equator
the easternly stress pattern is broken from the western boundary to
approximately 170E. At the western
oscillates
from
approximately
10S
boundary moderate westerly stress
to
10N
during
a typical
yearly
cycle. From July 1982 to April 1983 anomalous westerly winds are seen
to extend far into the basin. These anomalous westerlies drove Kelvin
waves
across
the
basin
that
caused
dramatic
30-40cm
rises
in
sea
surface height in the eastern Pacific during the El Nino event. The
extent of the westerly stress was enormous. In September 1982 westerly
mean monthly wind stress ranged from the western boundary to 120W. By
April 1983 the forcing over the basin reverted back to a more normal
condition.
difference
The
representation
the
field
of
the
wind
of
NMC
and
measurement
difference field is not a true error
FNOC
zonal
error
over
is
stresses
the
basin.
a
The
field. Small differences could
occur in data poor regions where both fields are erroneous. The entire
measurement
error
could
be
contained
one
in
data
set,
but
at
present it is impossible to say which data set is better. The response
then
of
field.
wind
the
is
basin
to
uncertain
order
the
are likely to be
These differences
of
this
large in regions of high
gradients, large wind variability where the wind
difficult to measure, or where
there is
difference
is
inherently
little data. The degree to
which these are well correlated spatailly or temporally will determine
the magnitude of the sea surface errors.
The amplitude of the Fourier transform of the stress error
6].
are shown in figures[5.32-5.3
fields
The largest errors appear from 10N to
20N across the basin in the dc frequency (the dc component is just the
mean of the error time series).
Some of the errors are over 1/2 dyne
in this region. The errors for
the higher frequencies are generally
smaller
than the
errors
frequencies we observe
easterly wind
belts
at
for
the
lower
that the largest
15N
and
15S
forcing
frequencies.
errors occur
across
At
all
in the strong
the basin. The
largest
errors on the equator are near 130W for the six month period forcing.
_r____lU~IXXIII__~~-_~_
These transform plots are multipied point by point with the response
function
and
and
modes,
baroclinic
frequencies,
all
over
summed
forcing point to find the total error.
Figures [5.37-5.46] show time series of predicted sea surface errors
islands.
each of the
at
In the sea surface error plots,
the dotted
line with the longer space is the error in the first baroclinic mode
in the delta case. The dotted line with the shorter space is the error
in the first three baroclinic modes in the delta case. The thin black
line represents the patch error due to the first baroclinic mode. The
thick black
line
to
due
the error
is
the
sum of
first
the
three
baroclinic modes in the patch case. The first baroclinic mode in the
patch case contains much of the information that is in the sum of the
first three modes.
We have seen that amplitude of the response functions for the delta
individual stations however small
forcing are similar. At
and patch
differences in the response functions cause significant variations in
the
sea
total
actually
area
Energetics
obviously
on
surface
averages
smaller
influence
error.
so
scales
Wind
the
stress data
case
patch
than
is
the comparison of
is
resolved
at
grid
more
by the
delta and patch
points
are
appropriate.
patch
would
results at
a
point station. At Nauru the error plots bear faint resemblence to each
other.
The
delta
response functions
are
inadequate
for
determining
actual sea surface error and we will rely only on the patch case.
The means and variances of the sea surface errors from zero at each
island are listed in Table 5-3. The error in the first three modes is
greater than in the first mode alone at each station. Rather than the
~
i~L
il)*^--~--.--_1111*1~
Y -..IXUI.~IYil
errors of the baroclinic modes summing out of phase to produce smaller
errors, we see the opposite is true.
TABLE
5-3
SAMPLE VARIANCE
MEAN ERROR
STATION
1st mode
1st 3 modes
1st mode
1st 3 modes
GALAPAGOS
2.26
3.27
1.87
4.43
CHRISTMAS
3.99
6.14
5.00
10.76
FANNING
6.45
9.08
24.89
48.72
JARVIS
3.02
4.00
3.75
6.01
CANTON
4.44
7.13
10.34
27.63
KWAJALEIN
6.32
8.58
20.71
38.16
NAURU
2.92
4.99
6.63
16.71
TRUK
7.37
9.08
18.08
30.10
RABAUL
8.30
12.35
40.97
101.39
MALAKAL
9.47
11.48
31.98
62.31
The smallest sea surface errors occur over the eastern Pacific. Here
the sea surface height is effected primarily by Kelvin waves. At the
Galapagos
the influence function for a 9 month period forcing
shows
that errors in the zonal stress near the equator from 130E to 70W will
be
important.
The
stress
errors
over
the
equator
were
generally
small. The sea surface error at the Galapagos reaches a maximum of 8
jl__lU_~~
I~CI~I
48
cm in May of 1983. The maximum El Nino signiture in this region was
over 35cm. Wind measurement errors are not important in the modeling
of sea surface height in the eastern Pacific during this time.
In
equator.
Fanning
At
are
troublesome
error.
Pacific
central
the
Nearer
a mean
error
9.08
of
away
important
are
errors
is
cm
the
cm variance
and
equator
at Christmas
the errors
still significant. Within half a degree of
found.
are
the equator
the
from
Even more
this
variablity of
48.72
the
the
the
but
smaller
in the same
longitude belt the cyclic six months error pattern seen at Christmas
at Jarvis but not nearly
and Fanning is present
Busalacchi
important
1984
in a hindcast of
discrepancies
between
the
1982-1983 El Nino event
observed
and
and
as large. Cane
modeled
sea
found
level
at
Fanning and Chirstmas. The results suggest that it is likely that the
problems they had modeling sea surface at Christmas and Fanning were
caused by an inaccurate wind analysis to the east of the stations. It
is
obvious
from
the
fact
that
the
errors
in
this
longitude
belt
decrease as we near the equator that the important source of error at
Fanning and Christmas comes from Rossby waves
the islands.
forced to the east of
One wind analysis we used resolved the winds poorly to
the east of the islands and this error manifested itself as large sea
surface errors off the equator near Fanning and Christmas. If the wind
errors came from west of the islands they be would carried by Kelvin
waves and the maximum sea surface error would appear on the equator.
I~~
"--I~4RII"*L~"LLL-I~X~Y"~I^-~
49
Over the Western Pacific the errors are generally worse to the south
of the equator and as we move away from the equator to the north. At
large mean errors of 12.35 cm are seen. The
Rabaul south of equator
observed sea surface signal here ranged from 20 to
we drove
If
event.
with
a model
either
of
these
15 cm during the
wind fields
and
compared the results to the observations wind measurement errors are
large enough that
it would be impossible to in any sense verify the
model here.
In general the sea surface errors are not significant in the eastern
Pacific or near
central
and
the equator. North and south of
western
Pacific
regions that are primarily
east
of
the
stations.
The
the
errors
are
the equator
important.
influenced by Rossby waves
winds
in
the
regions
in the
These
are
forced to the
influencing
these
stations are generally strong easterlies. Large sea surface errors may
result in regions influenced by areas of strong zonal winds containing
relatively modest errors.
50
AN ANALYSIS OF THE EFFECT OF WIND MEASUREMENT ERRORS
ON EXPECTED CIRCULATIONS IN THE INDIAN OCEAN
We will examine the sea surface errors in the Indian Ocean. Unlike
the Pacific, the
Indian Ocean
is
generally
devoid of
islands
that
serve as an observation network of sea surface heights. We have chosen
10 points near the equator scattered across the Indian basin to serve
as
our
observation
network.
These
stations
are
identified
with
a
letter and their position is shown in figure [6.1].
The Indian Ocean is modeled from 20N to 20S and from 96E to 45E. Our
forcing grid is a 20 latitude by 30 longitude grid. We have chosen our
forcing grid to conform to available wind analyses. We use the same
model and assumptions we used in the Pacific with the exception of a
typical stratification value. This correspondingly changes several of
the scaling parameters that appear in Table 6-1. The higher baroclinic
modes
are
insignificant.
The
contribution
mode appears to be the most important.
of
the
second
baroclinic
TABLE
MODE
C (m/s)
L (km)
6-1
T (days)
A (0)
n
1
2.80
356
1.47
3.9
1.104
2
1.73
280
1.87
5.5
1.726
3
1.11
224
2.34
2.5
0.286
4
0.80
187
2.75
2.1
0.168
represented by
a difference
errors will
The wind stress
field of
two frequently
The
(5.3).
wind
fields
again be
used wind
we used
fields
by equation
parameterized
used were
a European Meteorlogical
Center (ECM) analysis and a wind analysis from Wiley and Hinton. These
analyses provided winds over the basin for the period December 1978 to
November 1979 for a 20 x 3"
dealing
with
a
12
latitude x longitude grid. Because we are
month
time
series
we
will
only
examine
a
dc,12,6,4,3,12/5,and 2 month period forcing. The wind stress over land
areas
The
(India, Africa, and Madagascar) was taken to be zero.
ECM zonal wind stresses
[6.2-6.13].
over
the
The dotted line shows stress
easterly wind
and
the
solid
lines
basin
are
shown
in figures
imparted on the surface by
by westerly wind.
South
of
the
52
equator
the winds
equator
the
are
easterly
stress
wind
pattern
throughtout the
shows
the
year. North
strong
seasonal
of
the
monsoon
circulation. In June we see zonal stress of almost 3 dynes at 10N near
the entrance of the Arabian Sea. In the eastern Indian Ocean north of
the equator strong cyclonic activity is present in May.
The
Fourier
transform
of
the
stress
errors
are
shown
in
figures[6.14-6.17]. The errors are over 0.5 dynes in the Arabian Sea
at all forcing frequencies. The ECM stress
is calculated from 5 day
averages while the Wiley and Hinton data comes from monthly means that
supress short time cyclonic activity. We see errors
in the mouth of
the Bay of Bengal because of this. South of the equator in the strong
easterly wind belt the errors
are also
large. Near
the equator
the
errors are generally small.
Table 6-2 lists the location and the mean and variance of the errors
in
the
Kelvin
Indian
wave
Ocean.
terms
By
should
locating
be
large
the
stations on
because
the
the
Kelvin
equator
wave
the
decays
exponentially away from the equator. The first baroclinic mode in the
Indian Ocean is not as important as it was in the Pacific as seen in
the Table.
53
TABLE
STATION
LATITUDE LONGITUDE
6-2
SAMPLE VARIANCE
MEAN ERROR
1st mode 1st 3 modes
Figures
1st mode 1st 3 modes
0
91E
2.05
3.63
2.13
6.88
7N
91E
3.50
6.78
4.53
13.60
5S
91E
3.29
9.00
5.10
21.63
0
72E
1.39
2.98
1.51
13.51
5S
72E
4.14
11.01
5.83
19.78
6N
70E
4.38
11.39
8.78
42.19
0
54E
3.38
6.79
6.66
54.47
6N
52E
9.78
13.26
26.65
142.52
10S
49E
2.03
3.98
1.71
11.10
0
49E
3.06
6.29
5.09
44.86
response
functions
[6.18-6.39]
are
and
sea surface
error
plots for stations in the Indian Ocean. The response function of the
second baroclinic mode at station A shows the strong influence of the
Kelvin waves. The response functions
in the Indian Ocean posses
the
same properties that those in the Pacific did.
The errors over
the Indian Ocean are huge from the equator
north
into the Arabian Sea. The fluctuations in sea surface height are large
54
in the Arabian Sea. In some places as much as a meter. At stations Z,
B, and W, Rossby waves forced north and east of the stations in the
errors are very important.
of large
regions
The
sea surface
errors
reflect the monsoon cycle from April to July at each of the stations.
Our use of only zonal stress in this region also poses a problem since
the
seasonal
errors
are
Ocean.
Here
locally
monsoon
seen
also
circulation
north
of
is
the
strickly
not
equator
in
the
large
Important
central
errors
Indian
are
felt
easternmost
region
of
at the north central
Indian
station. At the equator in the
the
the
zonal.
central Indian Ocean at station X the cyclonic errors are felt. Rossby
waves
carry
nonexistent
large
sea
errors
from
surface error
the
plot
east
that
in June
distrubs
a
and July. The
virtually
smallest
errors appear in the east where the winds are lighter.
The errors are generally weaker near the equator and worse off the
equator especially
in the Arabian Sea. The
than those seen in the Pacific. The
errors
Indian errors are larger
are
larger
because
the
forcing over the basin is so much larger. In modeling the sea surface
fluctuations in the Indian Ocean we have seen that wind errors can be
serious. The wind analyses we used could be responsible for 13 cm mean
errors in the Arabian Sea.
r----I-~-Y-IV"-~~
IW
- YY *-(-E-Ys~~~~n
DISCUSSION
We have explored the response of the linear shallow water equations
in
a
unbounded
meridionally
basin
to
an
oscillatory
zonal
wind
stress. Solutions are found making use of the long wave assumption for
a single
low frequency w and include
directly
forced
term.
The
free
and
both a free basin wave and a
forced
terms
consist
the
of
equatorial Kelvin wave and a sum of Rossby modes. The highest number
of modes is a parameter we choose.
The zonal wind stress is first idealized
as a delta function in X
and Y, to measure the response to very small scale forcings and to
obtain a Green's function for more general types of forcing. Some ray
theory results demonstrate that the essense of the delta solutions may
be
captured using simple asymptotic methods. The
examining
these solutions
is useful
insight
gained
in understanding the case
in
of a
spatially uniform stress over a rectangular region.
The point source solution is dominated by the sum of forced Rossby
modes
independent
of
forcing location. The
free wave term is small
enough that no important response east of the forcing is found. The
important
response
is
contained within a
sinusoidal
envelope whose
)i- LI
__U__XII~ _II_______Z-~-Ytiili~L
56
and outer edges represent
inner
the lowest
and highest mode
Rossby
waves
travel
waves respectively.
at
originates
envelope
The
the
and
point
forcing
westward in two branches toward the turning latitudes. At the turning
latitudes the Rossby waves are forced to turn back toward the equator
to conserve vorticity. All the modes in both branches come together at
the equatorial image point of the forcing point. The envelope there is
focus. The
a single point called the
of
is a region
focus
intense
response. Past the focus the envelope again diverges in two branches.
the
Between
lowest
in
modes
two
the
branches
no
Rossby
waves
penetrate. This region experiences a weak response and is called the
shadow zone.
The response of the U,V,and H fields within the envelope differ. The
meridional velocity is dominated by the higher modes. The
important
response is found along the outer edges of the waveguide throughout
three
fields.
The
zonal
modes. The response near
is
velocity response
meridional
the basin. The
velocity
is
also
the weakest of
dominated
the turning latitudes
by
the
the higher
is small. The height
field is less dependent on the higher modes so the response is larger
near the
inner part of the
envelope then
in either of
the current
fields. The response of the height field is smallest near the equator
and largest near the turning latitudes.
Ray theory allows us to find the location of the focus relative to
the location of the forcing. If the forcing is at a point (X,Y) then
the
first
boundary,
focus
appears
at
C
(X-CT/4,-Y): here
is
(X=O) is
the
western
the
Kelvin wave speed, and T is the period of the forcing. The shadow zone
is
shown
to
be
centered
about
the
line
connecting
the
focus
and
forcing points. Ray theory was also useful in explaining the spatial
energy
density. We
find
(because many modes
that
influence
energy
is concentrated
a small
area)
and
at
near
the
the
focus
turning
latitudes (because the waves move so slowly here).
While point
forcing
and ray theory provide simple
solutions, they
also posses some unattractive properties. The high mode
response
is
very sensitive to the presence of a frictional mechanism. We added a
Rayleigh friction term saw that it acted to localize the response near
the source. The inviscid solution is sensitve to the number of Rossby
modes and the viscid solution to
the value of
the Rayleigh damping
term.
The delta
over
an
solution is
arbitary
function and
used as a Green's
rectangle
to
integration introduces a factor of
derive
the
(2m+1)-1
patch
integratred
solution.
The
in the Rossby terms which
damps the higher modes. This is equivalent, throught the quantization
condition, to dividing the Rossby terms by Y 2 . The major response in
the box solution is
closer
to the
equator
than
it
is
in the delta
case. Phase interference within the box has destroyed the higher mode
response,
and
away
from
the
forcing
region
the
response
is
built
principally from the lower modes. The box solution sensitivity to the
number of modes included in the sums is less.
The box size is a critical parameter that characterizes the type of
response we expect to find.
ILli _^llj~_~~XI~
^E
Il_
58
Case 1: small scale forcing ; 25km > AX, 25km > AY:
For forcings over areas this size or smaller the response is
virtually identical to the predictions of ray theory or delta
forcing.
Case 2:
large scale forcing
: 40
< AX, 4* < AY:
The foci no longer exist. The Kelvin and free wave terms are
important. The important response independent of the forcing
location is trapped near the equator. Beyond 100 north and
south of the equator the response is insignificant. The reason
is that parts of a patch that lie away from the equator respond
only in terms of higher modes which are small. The only important
response then occurs if parts of the box extends into the low
latitudes. The response then is only felt in these low latitudes.
Case 3: imtermediate forcing : 40 > AX > 25km, 4° > AY > 25km:
The foci are observed but not as large as in the small scale
forcing case. We also see traces of shadow zones. The response
away from the equator decreases as the box grows.
The size of the box used will be determined by the resolution of the
available data. The resolution required by case 1 is difficult to find
and
since we are exploring a simple linear model it is doubtful the
response of the real ocean is as simple as the point source solution.
- ~II--LI-_III
L_
rUI_ __
~__1____1_1__111111_La~~-*
59
Case
3
with
unrealistic.
uniform
Since
forcing
wind
over
stress
a
huge
data
at
area
2.5*
is
also
intervals
somewhat
is
often
available we conclude that the intermediate case is the most useful as
well as the most practical to compare will observations.
Linear dynamics allow us to examine the effect of basin wide forcing
on a single point. The basin is assumed to be divided into a grid of
boxes with uniform stress. A superposition of patch solutions yields
the
influence of
response
at
baroclinic
basin
a single
modes,
wide forcing on
is a sum over
point
and
a
stress
We
points.
single
point.
all forcing
find
that
The
total
frequencies,
independent
of
station location four distinct regions in the basin strongly influence
the height field at the observation point. Near the observation point
any wind forcing will have a strong influence. A broad region to the
east of the observation point within 10
influence on
the station. This
is the
of the equator has a moderate
region in which Rossby waves
originated that pass through the observation point. CT/4 east of the
observation point
at the equatorial
intense influence caused by the
region
where
Rossby
waves
that
image point to
inverse
focus
is
focus
at
the
the forcing
seen. This
is
observation
the
the
point
originated. Moderate influence is exerted on stations near the equator
in
the eastern part of
the basin by Kelvin waves
forced
along the
equator to the west. The Kelvin waves were more important than when we
examined
isolated
forcings
because
each
isolated
patch
near
the
equator excites Kelvin waves in the same location. In effect at the
observation point we are seeing the evidence of a sum of Kelvin waves
along the equator.
____l_~L_~_~~_~___I ~__lrr~
60
We used the patch model to examine the problem of the effects of
wind measurement error on predicted sea surface height. We took the
Raleigh damping to have a spindown time of 5 years and used 20 Rossby
modes in the sums. These parameters do not have an important effect on
the results. As an estimate of
the wind error we took a difference
field of two common wind analyses.
In the Pacific we used FNOC
(Fleet Naval Oceanagraphic Center) and
NMC (National Meteorological Center) monthly mean winds on a 2.5'x2.50
grid during the Jan82-Jun83 El Nino event. During the entire period
insignificant errors are found in the eastern Pacific and within one
degree equator. Moderate sea surface errors are present north of the
equator in the western Pacific. In the central Pacific near Fanning
and in the western Pacific near Rabaul large errors are present. The
largest sea surface errors
the
stations.
dynamics
have
The
areas
smaller
are caused by poor wind analyses east of
that
errors
are
only
because
influenced
the
wind
by
Kelvin
analyses
over
wave
the
equator are generally good.
In the Indian Ocean we used ECM (European meteorological center) and
WH
(Wiley and Hinton) winds on a 3 x20
grid from Dec78-Nov79. Large
errors are found from the equator northward into the Arabian Sea and
along the equator in the central Indian Ocean. The monsoon circulation
causes the large errors in the Arabian Sea. Tropical cyclones in the
eastern
Indian
ocean
are
not
resolved well
in the ECM data.
causes errors in the sea surface height on the equator
This
in the center
of the basin. The errors in the winds south of the equator are large
near 5S but decrease further south of the equator.
61
The important point to retain from the error
differences
that
are present
in
frequently
analyses is that the
used
wind
analyses
are
significant enough to cause large spatially correlated differences in
equatorial
sea
surface
heights.
These
errors
are
large
enough
places to make the verification of numerical models impossible.
in
62
REFERENCES
Blandford, R. 1966. Mixed gravity-Rossby waves in the ocean. Deep Sea
Res., 26a, 1033-1050.
Bretherton, F. P. The general linearized theory of wave propagation. In:
W. H. Reid (editor) Mathematical Problems in the Geophysical Sciences.
Am. Math. Soc., Providence, R.I., 13: 61-102.
Cane, M.A., 1983. Modeling sea level during El Nino. J. Phys. Oc.
(to appear in)
Cane, M.A. and A.J. Busalacchi. Hindcast of the 1982-1983 Pacific
sea level. J. Phys. Oc. (submitted)
Cane, M.A. and D.W. Moore. 1981 A note on low frequency equatorial
basin modes. J. Phys. Ocean., 11, 1578-1584.
Cane, M.A. and E.S. Sarachik. 1976. Forced baroclinic ocean motions.
I. The linear equatorial unbounded case. J. Mar. Res., 34, 629-665.
-------1977. Forced baroclinic ocean motions. II. The linear
equatorial bounded case. J. Mar. Res., 35, 395-432.
1981. The response of a linear baroclinic equatorial ocean
-------to periodic forcing. J. Mar. Res., 39, 651-693.
Gent, P.R., K. O'Neill and M.A. Cane. 1983. A model of the semiannual
oscillation in the equatorial ocean. J. Phys. Oc. (in press)
LeBlond, P.H. and L.A. Mysak. 1978. Waves in the Ocean. Elsevier
Scientific New York, 602 pp.
Messiah A. 1961. Quantum Mechanics. North-Holland, 504 pp.
Morse P.M. and H. Feshbach. 1953. Methods of Theoretical Physics
1st Edn: McGraw-Hill, New York, 1978 pp.
Patton, R. J. A numerical model of equatorial waves with application
to the seasonal upwelling in the Gulf of Guinea. MS Thesis, MIT.
Pedlosky, J. 1979. Geophysical Fluid Dynamics. Springer-Verlag,
New York, 624 pp.
Scholf, P., D.T.L. Anderson and R. Smith. 1981. Beta-dipersion of
low frequency Rossby waves. Dyn. Atmos. and Oceans, 5, 187-214.
Whitham, G.B., 1960. Anote on group velocity. J. Fluid Mech., 9,
347-352.
~--ilYY~
----XUICU- .I.~1^
.I._I__-11I~
~I~CJ
L~
i.i------
--^-~-~I~Y(I-VL
.. iT ~.- 1~4
63
APPENDIX 1
Normalized Hermite functions ym are given by:
-A
Ln4 f'rrt
3e
J4 H!
where Hm
(a1.1)
are Hermite polynomials of order (m i .
The oC m term is
defined as:
(A1.2)
for m = 0,1,2,...
Note that only odd ao's are defined since the even eigenfunctions ' are symmetric.
The Rossby and Kelvin modes are functions of the ' 's
are denoted by Rm(}
Rossby terms.
and Mk.
Here ()
They are given by:
-I
IM ____
-
and
denotes the U or H
V
_I~L_~_~
__~_I
X___~..
I
64
APPENDIX 2
The solution to the shallow water model
(1.6-1.8)
satisfying
boundary conditions (2.1-2.2) is given in (Cane and Sarachik
81)
as:
L
~-- e
CA(x\M/
uJ
+M1
4
(-,,
(A2.1)
*~hl
PL "
~o
Li
AO
$ ~ 'Zo(m
rC\=
Here:
(A ,C'
e-
[,.,] 11
"-"I
o4
<I
eC
m
N-
1_ ,,,jE
(jXY
--
eAe~cS'-f~
o
(A2.2)
4('-sA
~(~rcs+l~
rcrl~u
S,
F
7 -0.R
0o
(Am* 1) A,
1%d*
[F(X', - ) I F(x; TIc
(A2.3)
(A2.4)
(A2.5)
We take the forcing to have the form:
R
F
(,-1 S(b-X,
7\ =
(A2.6)
We now may evaluate (A2.1) explicitly beginning with (A2.4)
and (A2.5).
S0S(i-mA
( -X
-'000
-, (~l- ×,y
SH -1, A,
,
b,(;i)
So:
Similarly:
=
5
(X
t
io
I
-
X } M
(
X)
(A2.7)
r-
'YXd
4, J
st..
Xto)
rr,'-~
(A2.7)
(A2.8)
Y2vk4~
and (A2.8) may now be used in (A2.2)
find Ak(x) and Am(x)
and (A2.3) to
ab"~=----~I-"I~-UL"~-I
~-sllr~
T~YI)C
II~-_
66
r' (j
-
I"
l(V/- X X) h,
,
H1 X) C
Lt c
'3
or equivalently:
-5~E
%OV
eA'01/
x lu2,(yy\
ck ,()\
&
e
j X -Y)A/
(A2.9)
Similarly:
SO
~1V\
0(
4r,%
(r,14
L+M(M411
y(~.
eAt-)
l~tlw
*-*
jv
Gtj)
LY
(A2.10)
Expressions (A2.9) and (A2.10) are substituted into (A2.1)
to obtain the solution:
GJL1
))
,I
Ck2
-.A j-2m-fI)%
UXXt
-Aw
M~4
R~ul
------.
~-~---i
then becomes:
N
L ts
=A
e
r(2*I
Sw
i,
h/
I- I
XA
3
-w
e
Z
is%}tc
e^
2
*
wk 4r
-j (M+iVA 4
~zo F\lrc\~(\
N
.4
4-4L.JX4
-x
S2
I-
4
Then for
L
}w
e
n
o
V
(
'
AIK(I04.
- 7 (A2.11)
i
e
-(rV,(A?
*
e
e;w(r~x\
2'
-42O
~-plc~z~
r
i
S*
O
' -
This is simplified with the definitions:
and redefine j as
-ZI
,~,R,,lr,\-l '""~dw
k(1
r"L-
XA-4
l
e - I(1Pn+0
_I
Wca~
--iu~i-~~
--m---ix~--L---u-~-n~-
68
obtained by
-+(t ; this
The final form using the above symbols is
4 - (PX
front the common factor Q
taking in
leaves:
For x<x*
e t- x4)
M
I-,V%
IEA2
(A2.12)
v
For the region x, <x Am=Ak=O so we are left with only the
solution.
'free'
For X : >It
I"]
41
e
~2
t
"B*
-106.1l
v%: I~
M~t
(A2.13)
_
C\
To calculate our V we use equation (3):
HT4
UX
lAJ
&
V = - SL
\V/=
o0:
VI
2--()\ (m4'
S
V
6%
~
4J9
~
? L~~t~Zlrr~C"(IM4
1
-
I
OX)l
3F~,(y\M
*-OKIE~4
\
(A2.1 4)
Ci~~XIII~-OYTL-YI--_I_~
^_ Yllir
-L '.?"~'--i~b"-L1-iri--
69
(A2.14) may be simplified easily:
n~JeQT-"''C ~ RMV~
~=-S
N
-E4
("
(04- k
-I(M-tII ( 0 - W)
-4.(2r -+JjQMV I
So
V=*JPAJ
e
(A2.15)
4R J"r
SLP
t(2I\PI d
.I
Using the properties of Hermite polynomials we may solve
the integrals exactly.
S-j
i~v*l'\,td,%J
(A2. 16)
-- ir:%--u,-jre, c-~e~l
$O ThEt Fro
X< ,
-- 2
VJ
- ,
4
(A2.17)
~~_ --~~ll_-_I~L~_
~L._-Y-LUII
..-
70
And for x> x
Am=Ak=0.
e Jh(|
e
So
w)mI
(A2.18)
~rr-l-r
- ^Lnmrr~arrrrr~r;slara~~.r~--,x~l.
~
r*PLI~-~I
IU;I~X~-~N)-II~Ll~a_~--*1C~Y~
APPENDIX 3
We want to force in
a box centered at a point (xy').
Take
U=X
First we will examine the response away from the forcing;
that is in the regions x'+ - xx
>x.
and x'-x
Then
First solve for the U and H fields for x'- x zx.
(A3.2)
rY~;~C~C
t~
1-*
Integrating over x x first
5~e
O.J
' c-0\
_
___ I111~.
~-LI--L-----L
-LI
_.-.~..I
1I~..-.li~.illX~i.--~~X
I~
LI
i------YI*L~
72
S'~ F(S"~r,~,\e'~*~d'\3~3,
F(T XJ*~X
r'-s
x -ttr)
i
F('rY,%1f'\
kL&
-
x'-WC
-e
-
t~
-J
L:Z
7
Let: wSx=
( and since x'
we will call:
v
"<)I,
e~ 11)A
is just the center of our 'no
x'=.
So
F (1,x,
e
2
"
$1S', y
(A3.3)
Similarly
2
I4 IC
(A3.4)
The results of (A3.3) and (A3.4)
Li/
are substituted into (A3.2).
(su-(* ' i(,<e .J
%
(^-.11) e"'-,11
IE
I-r
(A3.5)
xl-~X
I~l~-L
~s~rrm1_LllliYYLi-(-ILII--\lillll-LL~
73
Now integrate over y,
"I
S -?2 ~
HYo
-~$
I
c'
IJ
1-1 1-1 = W1PI
The integral is just an error function so
-r4SI
i l
C1 *
-
Th\
ER
ERF(Yu
S1i'tr 1
'TE
(A3.6)
L
W'N1
(~
Our final solutions are found by substituting (A3.7) and
(A3.6) into (A3.5)
and noting that Am=ak=O for
(x'+
x)
>
to obtain:
For x
< (x'-
U)Ar~J
5x)
.
e 0 -09 -r
1
-5
v
Ct~uu-\t~r\
~
4
PI
r1
~4~115
t-~
:Pel~\-CF11
P
sSvt.
I ~Q
(A3.8)
i
~
~ --- --------
Il^a~- rauiL1~
I~-
74
For x > (x'+ $ x):
eOT
llil=3
~
-
Cz- -2.c
(A3.9)
'eAj
R.
Here
-ECF(ti
__________
e-z^t% p
oV\ 11
I -
(A3.1o)
-i1
(
m
The solution for V may be found using
v-=
CLSeL
H
This calculation was done with different constants in
(A2.14)-(A2.16) and the results are analogous.
The V solu-
tions then are found by replacing
to obtain for x 4(x'-
5
-
2 -A
x)
-2 1141
Cr
-A:'
(A3.11)
4
U
- k,
C< 'm,
ur;r;i~c~ .
I~..-^~-- --I---~--
75
And for x)(x'+. x)
V
^
* 2
C
(A3.12)
The solutions have been of the form:
xZxt
field = forced +
x >x,
field =
free
free
So if we want a solution within the box we will have
Field =
For x'- 5 x- x< x'+
Forced
N 't
5
ix~~
Free d
04
~
x.
Using the results from the previous calculation for (U,H)
we see
S)
I~~
x4~cC
ZKr
k
a
a
I-r!h &%--%
THI
Yt~n~
~
7,Jy\
hFF)fF(
T
7
(A3.13)
-Yizpe
W.
-It''
".~\~~q*
S
c''
~" 4
--X~'Lut
~r~-+-
IUULI^-~I~-~
-;-;ululm-^xr-rc~~-XdrrY'X
Il-~flll^--a~
76
5
All we have to evaluate directly are the
S Cb,4,
integralsa.
Qdxl
2-AC)
A
(A3.14)
-,e
Similarly
t-2WV1'
So substituting (A3.14) and (A3.1'
10)
1.i A
V-
(A3.15)
into (A3.13) we have:
1
4EeF(wl-
/%IY\
~i~'"~'-li
Zj
f
L
(A3.16)
Ff (%15MKI
-
011-
--
(4<
The solution for V is complicated by the 1 that appears
in the brackets but much of it has been calculated before
in
(A2.14-A2.16)
r
,cJce
~-C-ii
i
.-
L~-~gl
I-r*LI---I..^~ --~~LC~X~1
77
For the Rossby terms in the forced part:
(-I-4HtA
-4
AJ
k 1 c 24l\
-olW kWA
H ~(2~n~~ R~ul -~W Y\ I ensy
e/'g
-
Vr,
'42.1-'. %~H
gL-
e
F~
(A3.17)
We also have a non-zero contribution from the forced Kelvin
term.
j
-iv~r~
%Ad
lv~A
L n~
t~~L
( VK
) I
- 3e Y(
U"'
2..
which was shown in (A3.6) to be just an error function.
So
A
VK
~
A
(A3.18)
I ;; __XX___n*_____lqiijjlL__~I_
I I~-_i_? DI(_.iC....
i^..-~
78
So combining our results
For x- 5x<x< x+ 5x:
i
\1~1~\I
CY11T\~1
2v.Z4%\N
SIv91,JK7 x
2~~~
4'
('-')
4-1C
4p - OX)
-
OV
j
i
CJ
I-]
K FI F -^l --FPFr
N
---( 1
(A3.1 9)
FIGURE
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2--8
2-9
2-10
2-11
2-12
2-13
2-14
2-15
2-16
2-17
2-18
2-19
2-20
2-21
2-22
3-1
3-2
3-3
3--4
3-5
3-6
4-1
4-2
4-3
4-4
4-5
4-6
4-7
4-8
4-9
4-10
4-11
4-12
4-13
4-14
F IELD
FREE U
FREE V
FREE
H
U
V
H
U
U
REAL V
REAL V
REAL V
U
U
U
U
V
H
U
U
U
U
RAY PATHS
TOTAL ENERGY
;E>
RAY PATHS
TOTAL ENERGY
U
V
H
U
U
V
H
U
U
U
U
# MODES
50
50
50
50
50
50
50
50o
50
50
50
50
50
50
50
7
7
7
7
7
50
50
30
30
30
30
30
30
50
50
50
50
50
50
50
7
7
7
50
50
50
50
X
5000
5000
5000
3500
1000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
5000
1000
1000
5000
5000
TIME
0
0
0
0
0
0
rr/
0
0
500
-500
1000
0
0
0
500
1000
0
0
0
0
0
1000
1000
1000
0
0
0
0
0
0
0
0
500
0
1000
CONTOUR INTERVAL
5.0
5.0
5.0
0.02
9.0
9.0
9.0
9.0
9.0
9.0
9.0
9.0
9.0
9.0
9.0
2.0
1.0
2.0
2.0
0. 1
4. 5
9.0
60. 0
60. 0
10
111. 2
222. 4
222. 4
222 4
222. 4
111.2
222 4
222. 4
222. 4
222. 4
222 4
222. 4
222. 4
25
25
60. 0
60. 0
0.04
0.04
0.04
1. 5
1. 5
1. 5
1. 5
1.0
0. 5
1.0
1. 5
1. 5
1.5
1.0
FIGURE
5-1
5-2
5-3
5-4
5-5
5-6
5-7
5-8
5-9
5-10
5-11
5-12
5-13
5-14
5-15
5-16
5-17
5-18
5-19
5-20
5-21
5-22
5-23
5-24
5-25
5-26
5-27
5-28
5-29
5-30
5-31
5-32
5-33
5-34
5-35
5-36
5-37
5-38
5-39
5-40
5-41
5-42
5-43
5--44
5-45
5-46
PLOT
PACIFIC BASIN
FANN I NG RESPONSE FUNCTION
FANNING RESPONSE FUNCTION
FA tN I I RESPONSE FUNCTION
FANN INC RESPONSE FUNCTION
FANNINGIO
RFSPONSE FUNCTION
FANIN ING RESPONSE FUNCTION
I NO RESPONSE FUNCTION
FAIN
FANNING
FANN I NG RESPONSE FUNCTION
FANN I NG RESPONSE FUNCTION
FANN I NG RESPONSE FUNCTION
FANN I NG RESPONSE FUNCTION
FANN I N RESPONSE FUNCTION
FANNING RESPONSE FUNCTION
F ANN I NG RESPONSE FUNCTION
FA-N ING RESPONSE FUNCTION
GALAPAGOS RESPONSE FUNCTION
GALAPAGOS RESPONSE FUNCTION
JARVIS RESPONSE FUNCTION
KWAJALEIN RESPONSE FUNCTION
TRUK RESPONSE FUNCTION
RABAUL RESPONSE FUNCTION
NMC ZONAL STRESS
NMC ZONAL STRESS
NMC ZONAL STRESS
NMC ZONAL STRESS
NMC ZONAL STRESS
NMC ZONAL STRESS
NMC ZONAL STRESS
NiC ZONAL STRESS
NMC ZONAL STRESS
STRESS ERROR TRANSFORM
STRESS ERROR TRANSFORM
STRESS ERROR TRANSFORM
SIRESS ERROR TRANSFORM
STRESS ERROR TRANSFORM
GALAPAGOS SEA SURFACE ERROR
CHRISTMAS SEA SURFACE ERROR
FANNING SEA SURFACE ERROR
JARVIS SEA SURFACE ERROR
CANTON SEA SURFACE ERROR
KIAJALEIN SEA SURFACE ERROR
NAURU SEA SURFACE ERROR
TRUK SEA SURFACE ERROR
RARAUL SEA SURFACE ERROR
MALAKAL SEA SURFACE ERROR
TIME
BAROCLINIC MODE
PERIOD
18
9
.6
18/4
18/5
3
18/7
18/8
2
DELTA
DELTA
DELTA
DELTA
DELTA
CASE
CASE
CASE
CASE
CASE
18(months)
6
3
2
JAN
MAR
MAY
JUL
SEP
NOV
JAN
MAR
MAY
CONTOUR INTERVAL
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0. 1
0. 1
0. 1
0.25
0. 25(dynes)
0. 25
0. 25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.1
0. 1
0. 1
FIGURE
6-1
6-2
6-3
6-4
6-5
6-6
6-7
6-8
6-9
6-10
6-11
6-12
6-13
6-14
6-15
6-16
6-17
6-18
6-19
6-20
6-21
6-22
6-23
6-24
6-25
6-26
6-27
6--28
6-29
6-30
6-31
6--32
6--33
6-34
6-35
6-36
6-37
6-38
6-39
PLOT
INDIAN BASIN
ECM ZONAL STRESS
ECM ZONAL STRESS
ECM ZONAL STRESS
ECM ZONAL STRESS
ECHI ZONAL STRESS
ECtI ZONAL STRESS
ECtl ZONAL STRESS
ECtl ZONAL STRESS
ECHt ZONAL STRESS
ECtI ZONAL STRESS
ECMI ZONAL STRESS
ECtM ZONAL STRESS
STRESS ERROR TRANSFORM
STRESS ERROR TRANSFORM
STRESS ERROR IRANSFORM
STRESS ERROR TRANSFORM
A RESPONSE FUNCTION
A RESPONSE FUNCTION
A RESPONSE FUNCTION
A RESPONSE FUNCTION
H RESPONSE FUNCTION
X RESPONSE FUNCTION
F RESPONSE FUNCTION
F RESPONSE FUNCTION
B RESPONSE FUNCTION
B RESPONSE FUNCTION
RESPONSE FUNCTION
Q RESPONSE FUNCTION
A SEA SURFACE ERROR
T SEA SURFACE ERROR
H SEA SURFACE ERROR
X SEA SURFACE ERROR
F SEA SURFACE HEIGHT
SEA SURFACE ERROR
B SEA SURFACE ERROR
SEA SURFACE ERROR
Z SEA SURFACE ERROR
Z SEA SURFACE ERROR
TIME
BAROCLINIC MODE
DEC
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
PERIOD
CONTOUR INTERVAL
0.25(dynes)
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0. 1
0. 1
0. 1
0. 1
0. 1
0. 1
0. 1
0.1
0.4
0. 1
0.2
0.2
0.1
0. 1
0.4
0.4
2200
1100
0
-1100
-2200
1000
2000
3000
X - KM
4000
5000
6000
0009
000S
000Y
W)0 - X
000S
000z
0001
001-
ON
wrtt
UoC
0009
000S
000*
W)I - X
0001.
000a
0001
00
-
00I1
O Uot
oo~1
9
e
2200
1100
I'd
o
H
H
c+
IR)
mD
-1100
-2200
1000
2000
3000
X - KM
4000
5000
6000
:*zJ
*-
0009
000S
000*
W)I - X
0009-
000a
0001
0011-
a)
,I
cJ
o
Hr'O
00\
0
0011
00a
0009
000S
000t
W)l - X
000S
0003
000!
ooza-
a,
00QI-
r-p
O
O
4-' L
0
H
II
0
Od
o*
0011-
OOze
2200
1100
CD
-.
CD
0
II
H
OFj
oOH
Oct
p.,
0
-1100
-2200
1000
2000
3000
X - KM
4000
5000
6000
4O
009
OOOS
000*
W)I - X
000S
O00
0
0001
0
00-
,
0/0
O
°
•I
I
00(
00H0
)lllll
\
*1
iJ
0009
000S
ooot
101
-
X
000a
009
0001
0
-4
4
0
H
0O
H
H*
ooa;
2200
1100
-.
'0
H
0
OCD
OH
00
O
%-H*,'
CD
-1100
II hi
CO
c+
-2200
1000
2000
3000
X - KM
4000
5000
6000
2200
1100
II px
'J OH0
OD
0
00
OH
O H-
c+
II -d
-1100
4\4
-2200
1000
2000
3000
X --KM
4000
5000
6000
2200
1100
0 H
OI-J
0
o 0
O P.
11-Fd
C(
IItd
-1100
ci"
-2200
1000
2000
3000
X - KM
4000
5000
6000
0009
000S
000*
W)I
-
X
o0or
0003
0001
OOZZ4'O
RO
r--I0
oaat
0011-
C\l
P
c,'
OOZ
cd
0
•H II
O
o
3r
ool
o01aa
0009
00S
000*
Wl-x
0009
0002
0001
oaa-
4-H0
-0
C~j
0
o
oaa
0009
000S
oo
000
9-00
000a
0001
0
0011
Id
+30G
-H 0
(7
IO
C\J
0
0
~D00
0~
Oil
0
/a
FIGURE 2.16
Zonal Velocity Amplitude
.
-
A
(X.,Y,) = (5000,0) Modes=7
H)I
0
0
0
O
0
Lf
0
0
01
0
0
o
0
€3
m-q
98
FIGURE 2.17
0
c0
Meridional Velocity Amplitude
(X ,Y ) = (5000,0) Modes=7
0
c2
HN - A
O
O
O
0
O
O
0
O
O
O
O
O
0
0
0
O
0O
O
O
O
S
0009
000S
-
000t
I
Wl-x
OOOS:
I
0 of) I
*
000 T
I
rd
.H
2
cz
0
rdO0
c'J
.HL
oat
4,.O.
0-V
010
I
*
*
oa
I
O
OJ
10
-
0
0
0
0
A
I
I
I
o
O
O
WN o
(X.,Y,) = (5000,500) Modes=7
Zonal Velocity Amplitude
FIGURE 2.19
100
3
Cu
Cu
0
IHI)I-
.j
O
LI
.L
I0
A
I
0
I
Zonal Velocity Amplitude
(X.,Y.) = (5000,1000) Modes=7
FIGURE 2.20
101
0
!
Cu
Cu
0
0
0
00
o
0
I
2200
*
I
*
*-
I
*
IIII
*
I
*
m
I
D.0
1100
a
%so
CD
o-
00
O c+
O
3ct).
i
C)I-.
co
k-CFJ::L
-1100 F-
R/l
-2200
'
1000
--
lm.
2000
m
m
3000
X - KM
m
N
I|
I
4000
5000
L
6000
2200
1100 C
E0
0
0 oi
-2200I.
0
. 18.000
, ..
2000
000 ..
X
-1100 -
0
1000
2000
5000
X -
KM
600
KM
-
1 835.
.
-2200
. 5000
4000
D
4000
5000
6000
0009
000S
000t
H)I - X
0ot
0003
0001
00
-
o
co
II
*d
0011
ooa;
9
*
/7"
105
FIGURE 3.2
Total Energy
-
,-
(X.,Y.) = (5000,0)
C-c
%-6
WN - A
7
c*
0/
Ito
*
0
0
0O
o
C%.o
0009
000S
000*
14)1 - x
0001
000a
0001
OOZ?-
0011-P 0
02 0
(Do
qL'\
0
0
HX
0011
aON
0009
000S
o00t
H)I - X
000£
000a
000!
OOZa-
0
o
o
0
o
o
It
**
31
I
0011
ONI2t
0009
0009
000*
)0- X
0009-
000a
0001
001[-
o0
o
bOO
p
r-
O
II
H
0
out
000
0 1?E
w
0
2200
1100
0
0 tj
II (D
j
q
\j
0
-1100
-2200
os
1000
2000
3000
X - KM
4000
5000
6000
0009
000S
000*
W-
x
000i:
0003
001
ooaa-
oatr-
I
-P0
0
c00
3C
0011
oaa
0009
000S
000)
N)I -X
0009
000a
0001
ooaa-
oorr-H 0
o C\J
o
3r
-P0
0011i
-ooaa
9p
9
0009
ONGs
000*
W)g
-
x
00o9-
000z
000?
ooaa-
(1)
.-P0
H
[xl
rZ
o'-
0
Fz~
3C
100>
C) 0
oaa
113
FIGURE 4.4
DV
•
"OI
O
04
Patch Zonal Velocity
(X',Y')=(5000,O) Box=(4 0 x40)
0
00
I!
O
O
0
"oO
OO
3O
0
_
o0
I
I
'
o
'
114
FIGURE 4.5
0~
o
I
I
i
C>
0
(1,
D
I.
o
o
I
0
1
Patch Meridional Velocity
0
(X',Y')=(5000,0) Box=(4 0 x4 )
'
o0
- IM
HI - A
a
0
-i
i
-1
€:
0)
O
0c)
O
O
0
0
OI
0
" 41
8
i
1
0009
000G
00t
W) - X
0002
0003
000!
1-%
OOU00[1-
(DO
0 0o
II
*H
rzo
bOO
0
-~1
-s
0
00t8
O-
116
FIGURE 4.7
Patch Zonal Velocity
CO
S00
0d
0
O
b
0
,l
E
0
/o
b/
-OO
(X',Y')=(5000,O) Box=(2c x2o)
0
00
= i 0Ck
WN - A
-0
00
0
O
0
0
0
c-
oo
0009
e
000S
W
000*
- X
0002
0003
000T
0
0011-
c-II
C)0
N
DOlt%,.JQ
Q)-J
p~l
pq3
F9w
0
oo9
,t
go
C>
-0
S r1
0" a
oot'-
Oj2
G
00~
4'
C~0LC2
Ith
~
r
Ld
118
FIGURE 4.9
Co
fl
Patch Meridional Velocity
(X',Y')=(5000,O) Box=(4 0 x40) Modes=7
1 /I
i
O
O
O
O
0
0
0
Ocu
o
x
0009
00S
000)
W)I
-
X
000S
000a
0001
oaatId
0
0
ol-C
1~
r~1
+bO
d0
P-4Lr
oaa
9
w
0
0
1-4
.14
120
FIGURE 4.11
A
1-4
1-4
a
0
Patch Zonal Velocity
Box= (4 x40 )
(X',Y')=(1000,0)
0
HN-
.
Cu)
121
FIGURE 4.12
Patch Zonal Velocity
I
-
(1
A
°l
SO0
OCQ
(X',Y')=(1000,500) Box=(4'x40 )
0
_C0
W
.O
O
O
40
O
O
O
U)
0
oE
O
0O
0
O X
SO
O
0w
O
O
C)
o
C)
0
122
FIGURE 4.13
0"
,
.0
O
0
,0
Patch Zonal Velocity
(X',Y')=(5000,0) Box=(4 x4 ) R=25 years
S
I
0
I
0
o
0
0
O
0
0
0
C-
123
FIGURE 4.14
Patch Zonal Velocity
(X',Y ' )=(5000,1000) Box=(4 x40 ) R=25 years
0
CCo
0~0
ICd
O
O
O
0
O
0
O
0
O
0O
O
Cu
0O
0
30
t
5
H- ,
Hm
0
r
d
-5
a
a
-10
-15
0
c+
-20
-25
-30
-
120
137
154
171
222
188
205
EAST LONGITUDE
239
256
273
290
0
1
i
03
ei
.
-
_
125
FIGURE 5.2
Z~S~--------C----
_
aI
--
_n~
iV7
_-
!
CC
I
a
Fanning (4N,159W) Response Function
Forcing Period = DC
1
cl
C)
.
30ni
I
in
w
N
Cm
N
0-
CD
C,
0
126
FIGURE 5.3
"
C
iI I
Fanning (4N,159W) Response Function
Forcing Period = 18 Months
I
.)
0
C'
C/'
i
0
CC
U I
/1
/
127
FIGURE 5.4
f~~iC
000
o
C
/Q
1
0/)
I
060
'd
Fanning (4N, 159W) Response Function
Forcing Period = 9 Months
C3
C
0
CU
C-i
CUL
Z/
(Id
SI-q
cn
c1
UI
'T
128
FIGURE 5.5
II
/7
'
I
C:j
I
U
II
00
OI11V
VO
Forcing Period = 6 Months
Fanning (4N,159W) Response Function
~C3
ionlivi
0
or.
CU
LO
I-
0
CU
r Ln
0
Ci
¢1
I
0C
)1/
129
FIGURE 5.6
r
Q4c
j
7
r
I
I
3
C
0o
I
C
OCD
Fanning (4N,159W) Response Function
Forcing Period = 18/4 Months
'
p
0r3
3on11Y7
-I-
r)
0
0
r
0
I
0
a
a
130
FIGURE 5.7
0
a
CV
,
/
0
1c.1 I
0~
a
I
Fanning (4N,159W) Response Function
Forcing Period = 18/5 Months
Prl
0
Cu
U
I
I-.
Lf
4n*
LJ
O0
c'J
131
FIGURE 5.8
0
cs
U-,
z
-- I~~~V~4~Yn
-- -~rrx.--;n--^~----1-----
Fanning (4N,159W) Response Function
Forcing Period = 3 Months
O ganIUO
30.lh1*1v7
30
20
o
10
0
a
H
0
1
C O
-10
o-
-20
0
-30
-
120
154
222
188
EAST LONGITUDE
256
290
C
PC)
a
CY
133
FIGURE 5.10
C3
a
Ii
C
C
Fanning (4N,159W) Response Function
Forcing Period = 18/8 Months
C
r'J
3lI
JOrllI1LY7
'C)
30
20
10
o0
0
oOd O
-10
c+
CO
::s
-30 '120
154
222
1
188
LONGITUDE
EAST
256
290
135
FIGURE 5.12
_____I__UP__II__II~__..-~XI^-ii
Fanning (4N,159W) Response Function
,
I
0'
C;l
,
I
I
I
I
Delta Case - Forcing Period = DC
II
3on1l1¥7
I
C3Q
CU
LO
IC)
0
C'
CU1
C3C3
136
II
_i~l/~_~_~____~ _1~1
Response Function
FIGURE 5.13
Fanning (4N,159W)
(.
5_
3
------
/Z
o
I
I
3aniavi
tofl1EJto
C3
1I
I
Delta Case - Forcing Period = 18 Months
C3
C
_^__~_
LLn
Q
C".
UD
CU
CU
.
OD
137
FIGURE 5.14
Fanning (4N,159W) Response Function
i
II--
1/ '
0nIII I
C3,
1
.CU
Delta Case - Forcing Period = 6 Months
CU
0
C*
0
cu
138
FIGURE 5.15
Fanning (4N,159W) Response Function
II
I
Delta Case - Forcing Period = 3 Months
0
!
t
K)
SI-
C
30
20
C+
C1)
10
(D
hj
O --
0
(D
(D
-10
r-o
-20
Oc+
c+O
:v:
-30 120
154
222
188
EAST LONGITUDE
256
290
I"
C
140
FIGURE 5.17
---
'
Q
00
I
I
I
C,,
0I
Forcing Period = 9 Months
Galapagos (1S,90W) Response Function
I
03
~Oflhi1Y
I
I
a
*-
CU
0
3
3
141
FIGURE 5.18
C3
0
cow
cD
CUJ
CU
C
In
O
I_1114L__Y_____I~~C__~~
II1
CO(
Galapagos (1S,90W) Response Function
Forcing Period = 6 Months
3
3aI
It
/3
j,
142
FIGURE 5.19
OOC
(
I' IIN
0o
'
-"
IJ
U
Jarvis (O.5S,161W) Response Function
Forcing Period = 6 Months
/I
t
C
0
ow
C
0
0
0
143
FIGURE 5.20
Q40
0
P
C
0
0
il
Kwajalein (8.5N,168E) Response Function
Forcing Period = 9 Months
- SCU
3Dnli1vi
0
/t)
1
CU
I-C3
CM
In-
O
I
O
)
,
144
FIGURE 5.21
I
4.
I
0
I
0
0
CUo
ICU
I
Truk (7N,151E) Response Function
Forcing Period = 9 Months
I
O
0
C
0
I-
3-1
Uo
UD
)P
to
CU
145
FIGURE 5.22
DCd
I
I
Co
Rabaul (4S,152E) Response Function
Forcing Period = 18 Months
3Orit1¥7
I
in
0
CD
CU
o
z
-j
30
20
10
-10
-20
-30 -
120
154
188
222
EAST LONGITUDE
256
290
40
4
30
20
10
0
CD
F--
P,
CO
(D
-10 -(
-20
-30
120
154
222
188
EAST LONGITUDE
255
290
30
20
10
o
0-\
tI
CD
coc
-10
CO
p-
-20
-30
120
154
188
222
EAST LONGITUDE
256
290
I
In-,
I
,_..__._--'-
C3n
II
im~
I
oht
149
FIGURE 5.26
It
/1
/.
,
In'/
~
-
Il
I,/ ~li
(ca
I
3oninyii
C3
-i
.-
"s
/
jC3
L
I
I
OF
-x
NMC Zonal Stress (dynes)
July 1982
d/
N
/
\ )-~~~C~t
"
Ln
7
I
to "m
is
I
*
CS
. -*-.,
-
il
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/
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0
0
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cdI
cDJ~
30
20
10
N
cnr 0
C,~
CD
SCI)
CC
CD
-10
Fj
OD
(D
-20
-30 120
154
222
188
EAST LONGITUDE
256
290
30
o
-.
20
I--
0
Co
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-0-0
s
a-
I5
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EAST LNGITUDE
C,
0
(Dg
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20
(
154
... ...,d
-2.0
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AST
222
LnGITUDE~
~05Oa[7
25
290
Y( t
30
20
10
0
o
0
p-
D
LH
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LA
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120
154
222
188
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256
290
L
20
10
0
0
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H
CD
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120
154
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154
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222
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154
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256
290
0
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120
154
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168
EAST LONGITUDE
255
290
=xi
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154
222
188
EAST LONGITUDE
LtZ
256
290
40
I
25
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*
*
20
15
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5
(
H
0-10
co
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1
2
4
5
9 10 11
TIME IN MONTHS
12
13
14
15
1
17
18
d
25
20
15
10
a
5
p c+H"
FjS
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0
"
o -21
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0
1
2
3
4
5
5
7
8
9 10 11
TIME IN MONTHS
12
15
14
15
16
17
18
FO?
91
LI
91
ST
Vt £1
Z
SHLNOH NI 3HIL
It 01 6
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i
2
3
.
i
4
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5
.i
6
7
.
.
i
i
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8
TIME IN MONTHS
•
.
12
i
13
.
...
14
15
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15
17
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18
25
20
15
10
CDO
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1
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0
1
2
3
4
5
I
6
7
6
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•
9
10
11
TIME IN MONTHS
k
i
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I
.
13
14
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I
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0
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k
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0
1
2
5
4
5
6
7
MNH
9 10 11
TIME IN MONTHS
8
12
15
14
15
16
17
16
0H
U'
0
25
20
15
10
CDZ
-5
Ii
0-
Fj
-10
-15
-20
0
1
2
3
4
5
6
7
0
9 10 11
TIME IN MONTHS
12
13
14
15
16
17
18
m
91
Ll
91
ST
tI
t 1t
t
SHINOH NI 3HIL
01
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9
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8
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o
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kI
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3
kk
5
4
k-
6
7
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8
9 10 11
TIME IN MONTHS
12
13
1415
16
17
18
40
35
30
25
20
15
10
5
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o
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- 4
CDZ
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o'
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0
1
2
3
4
5
5
7
6
9
10
11
TIME IN MONTHS
12
13
14
15
15
17
18
--
20
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-
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.
I
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*
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45
I
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48
51
54
57
60
63
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,
56
69
72
.
75
EAST LONGITUDE
78
81
84
87
90
93
96
a
___ _ ___ ____ ____
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55 69 72 75
EAST LONGITUDE
...
78
L.
L.
81
U
84
-
U
87
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I
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.,
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93
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96
c.,
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---
46
51
54
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50
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65 59 72 75
EAST LONGITUDE
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78
81
64
87
90
93
96
20
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66 69 72 75
EAST LONGITUDE
78
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=
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20
45
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46
S
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c
--
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.
566 69 72 75
EAST LONGITUDE
-0 .8"
78
81
.I - .
.
-
.
84
87
90
('D
93
96
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20
15
10
5
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48
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54
57
50
553
66 69 72 75
EAST LONGITUDE
78
81
84
87
90
93
96
C3")
cu
V"
Q
W"
177
FIGURE 6.8
a
i)
I
-'-'
I II_
0tn
EMC Zonal Stress(dynes)
June 1979
fn
3 nI
30flhtIY7
-,-.
f3
i
O
Z
uC cD
L
VJ;
20
15
10
5
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45
48
51
" I I I"I '' ' 'i
69 72 75
54 57 60 563 6
EAST LONGITUDE
78
81
84
87
90
93
96
20
15
10
5
0
ct c+
-5
c
(8
O
-10
-15
45
48
51
54
57
60
53
6655 569 72 75
EAST LONGITUDE
78
81
84
87
90
93
96
0
-O
c
I
C0
CII
S
I
n
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C
180
FIGURE 6.11
0
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September 1979
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45
51
54
57
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L
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%
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63 66 69 72 75 78 81 84
- EAST LONGITUDE
i
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m
48
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93
96
20
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55 59 72 75
EAST LONGITUDE
S
78
i
81
•
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84
,
I
87
,,
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90
93
02
96
20
15
5
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54
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63
66 69 72 75
EAST LONGITUDE
78
.
'
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1
50
81
84
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87
90
93
f
96
40
15
2
10
)/3
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0
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, 02
.
00
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(
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45
46
51
-
54
57
60
63
55 69 72 75
EAST LONGITUDE
78
81
84
87
90
93
96
4
4
15
aN
S
1
0
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N
1
04
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0
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r
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t
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45
0.22
48
51
54
57
60
53
66 69 72 75
. EAST LONGITUDE
78
81
84
87
90
93
96
-j
\I
20
15
10
5
0OC
o
c+
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45
48
1 .
51
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54
57
60
63
e
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56 69 72 75
I EAST LONGITUDE
A
78
,I
81
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84
87
90
93
96
4
4
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4
20
15
10
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48
1.
51
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.
54
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I
57
.
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I
50
53
a
*.
.
*
66 59 72 75
EAST LONG ITUDE
78
81
04
07 90
. ,
93
96
20
15
10
0
ct-
.l e
0.2
,j
0
S-
0.2
-15
45 -4
54
IC.(D
Cl)
co
57
60
55 05
69
72
75
EAST LONGITUDE
78
~
o
-s--------a21
51
.
81
84
.
07
90
95
96
c
o
4
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15
20
0.4
ct
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I-
w
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45
4
51
54
57
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5
5
9
72
EAST
LONTDE
75
78
1
64
7
90
9
96
,,D
(
Id)
-15
45
48
51
54
57
50
53
5
69
72
75
EAST LONGITUDE
78
81
''
84
87
90
'
93
95
03'
190
FIGURE 6.21
'AIC
Cu
nolfV
0
I
I
Station A (0,91E) Response Function
Forcing Period = 2 Months
CU
3aninvi
I
CP
O
CP
D
UD
I-
rC1 cD
n
n
In
CU
I
0
In
CU
I"
I
in
-
191
FIGURE 6.22
0
I
in
I
iI
A
I
O
I
in
I
Station H (5S,91E) Response Function
Forcing Period = 4 Months
0
y1
~~Ir
OJqq
LO C
W
I--
m
CUc
l-F*0
z
mo
UD C
I Lln
I
20
15
10
Iz c+
H
00
5
0
(D --
H
o
-5
IICD
\j4Fd
-10
o0".
:
-15
-2 0
45
I
.
48
51
54
57
.
I
50
.
II
553
.
I
I
.
.
9 72 75
55
EAST LONG ITUDE
78
.
..
0 A.
81
84
87
90
93
96
m
r)
0
inI
iI
,
-
193
FIGURE 6.24
i
.
II
IC3
I
in=
0
In
L)
. I
Forcing Period = 12 Months
Station F (5S,72E) Response Function
,
Mode = 1
I)
30nlIIyV1
Q
c1
tn
"
u,
-
194
FIGURE 6.25
Response Function
in
Qi
in
Forcing Period = 12 Months
Station F (5S,72E)
Mode = 2
0
W"
3aonvi
r- C
L" LU
0
z
C"r-L
re**
UDLL
WD -.
UD
~jLn
I
C3
o W/
pc-
10
ctd
00
15
C30..II
=3
011
012
0
-1
45
48
51
54
57
50
53
55
59
72
75
EAST LONGITUDE
EAST LONGIUDE
78
81
84
07
90
93
96
20
15
0ow
10
CD P
H
II c+
0
5
I
OY
(d
-5
o 0
Pc
-10
o
-15
-2 0
45
.
48
I
I .
51 54
S e....
I . I
57 60
•
.
53
I
69 72 75
65
EAST LONGITUDE
78
81
84
87
90
"
93 96
Q
Cu
Ifl
U'W
SI
197
FIGURE 6.28
I
I
I
I
I
Station W (6N,52E) Response Function
Forcing Period * 6 Months
0
I
Ln
C)
to I-
Lf
Lt
0
Cu
n
-
U,
198
FIGURE 6.29
0
I
re
O0
K)
a'
0
W) W
et
IhcD
z
CU
0
CI-WP
CO
r-
In
V)
0*I
Ln
It)
In
in
I
reCU
I
Station Q (1OS,49E) Response Function
Forcing Period = 3 Months
0
-
3anhr1nV
30
25
20
iS
10
(D c+
c+
5
o
0
CD
-5
-10
0L'
'-1'j
-15
-20
-25
-30
0
I
I
1
2
.
I
I
I
3
4
I.
I
I
5
6
7
TIME IN MONTHS
I
8
9
10
11
12
4
30
25
20
15
Cl
10
Sc+
5
P-
-b-3
0
''
CD-)
-5
o
Iij~ .j
01
-10
-15
-20
-25
-30
FzJ
I
I
S1
2
3
4
5
6
7
TIME IN MONTHS
8
9
10
11
12,
Qr
v
II
1
O
6
SHINOH NI 3HII
S
9
L
a
O0
SI-
iZ-
o
S0
E 0
0
H
U
*H
0
C\j
O
al
o
0!
6
a
SHiNUN NI 3HIi
9
L
g
v
1
0
03-St-
*0
0o
St
I-I *U)
st
0?
S
NO
Ln
CU
o
Cd
.-
203
FIGURE 6.34
n
a
I
1n
C
Station F (5S,72E)
Sea Surface Error
a
W3
n
CdI
SLIn
I
a
M1
UDZ
I
Z
0
zW
25
CC
10
-5
-20 0
,
]I
•
I
.
I
.
|
,
i
, Ii
TIME
1
2
5
4
.
I
.
I
.
I
,
, .I
,
IN MONTHS
5
5
7
TIME IN MONTHS
8
9
10
11
12
(DN
4
6
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
I
-30
ST
2
4
5I
6
7
TIME IN MONTHS
8
9
10
11
12
O
L
m
10
m
L
206
FIGURE 6.37
uO 0
-II
iMm
) a0
!
I
Ll 0
Station W (6N,52E)
Sea Surface Error
C) m
M
wr
)qr w
mf
m
l
w
I
m
I
m
I
3
I
cn
LOZ
U.J
w
2
0
K)
Ll
CU
C
CU
W
-
207
FIQURE 6.38
KLI
0
I3
n
I
C3
C
I
Station Q (10S,49E)
Sea Surface Error
C3
r
NW3
)
I
I
C3
CU
Ln
CU
I
0
tn
I
ci
I.-
z
UO) 0
I
LDZ
w,
zc
__~_
_ _Il~q
25
20
15
iCf C)
10
c+
l0
-S
O
-0
-5
Ob
-10
-
-20
-25
-30 0
1
2
3
4
5
6
7
TIME IN MONTHS
8
9
10
11
12