SPECTRAL CORRELATION TESTS OF AN
EDDY HEAT FLUX PARAMETERIZATION
by
JOHN NEWELL MCHENRY
B.A., DePauw University
(1977)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May, 1980
@
Massachusetts Institute of Technology 1979
Signature of Author
J~
V..T
Department of' MeteorolcJy
May 9, 1980
Certified by
Thesj
Supervisor
Accepted by
Chairman, Departmental Committee
FI-
U LAItT
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SPECTRAL CORRELATION TESTS OF AN
EDDY HEAT FLUX PARAMETERIZATION
by
JOHN NEWELL MCHENRY
Submitted to the Department of Meteorology
on May 9, 1980 in Partial Fulfillment of the
Requirements for the Degree of Master of Science
ABSTRACT
A two-layer -model processes eight years of
twice daily geopotentjal height data to compute
the total eddy flux of- sensible heat and its comebination of predictors in a parameterizattop representative of those -used in climate models, The
parameterization assumes that the eddy flux is
proportional to a-s7tability coefficient, X, times
the square of the meridional temperature gradient,
Zonal Fourier decomposition and temporal analysis
of covariance are emplovea to determinie. the correlation spectra of the parameterization, thereby
testing its validity for various zonal- spatial and
temporal scales. This procedure- is performed for four
10 degree wide latitude bands centered at 35, 45,55,
and 65 degrees north,
The results show- that the parameteri'zation
models the seasonally forced zonally avera2ed eddy
flux well, in agreement with previous studies, and
reveals additional seasonal forcing on a.few other
large zonal scales as well, We cannot determine
whether the-stability coefficient improves the
parameterization in comparison with another which
does not include- it.
Baroclinic adjustment, whereby the eddies
adjust the gradient as- a result of baroclinic- instabilities. is revealed in the negative correlations
found on short time-scales. It--is shown that adjust-
ment may be occurring on zonal scales as small as
the relevant zonal scale of the eddies themselves.
By identifying the most favored time scale as that
scale which exhibits the most negative correlation
for a given zonal scale, we can show that that time
scale retrogresses as the eddy size decreases. Eventually the eddies become too small, and hence too
short-lived, to be observed. We identify four days as
an overall favored time scale in which baroclinic eddies
adjust the gradient.
THESIS SUPERVISOR: Edward N. Lorenz
TITLE: Professor of Meteorology
ACKNOWLEDGEMENTS
I would like to thank Professor E.N. Lorenz who advised
the research and writing of this thesis.
Professor Peter
H. Stone also provided considerable feedback during the
course of the project.
Thanks also are extended to Maurice Blackman for providing spherical harmonic data tapes and a program for converting grid point geopotential heights into spherical
harmonics.
In addition, the consultants at the NCAR com-
puting facility were invaluable in lending advice on the
use of NCAR's CDE 7600 and peripheral devices.
Diana Spiegel
should also be mentioned for her help in debugging programs
and using the department's Harris terminal system.
Finally, thanks go out to Isabel Kole who drafted the
figures.
TABLE OF CONTENTS
Page
TITLE
1
ABSTRACT
2
ACKNOWLEDGEMENTS
4
TABLE OF CONTENTS
LIST OF TABLES
7
LIST OF FIGURES
8
CHAPTER ONE.
11
1.1
1.2
1.3
The Eddy Flux Parameterization Problem
Forced Versus Free Eddy Behavior and
Baroclinic Adjustment
Objectives of this Research
CHAPTER TWO.
2.1
2.2
MODELLING THE EDDY FLUX AND ITS PREDICTORS
Choice of Specific Parameterization for
Study and Analysis Method
A Two-Layer Format for Processing Geopotential
Height Data
2.2.1 Critical Shear
2.2.2 Mass-Weighted Meridional Temperature
Gradient
2.2.3 Actual Shear
2.2.4 Eddy Flux
13
14
17
17
22
23
24
24
26
29
Data Sources and Description
Initial Computational Work
Computation by the Two-Layer Model
Spectral Decomposition and Analysis of
Variance
3.4.1 Fourier Analysis in Latitude Bands
3.4.2 Poor Mans Time Spectral Analysis
29
30
32
CHAPTER FOUR. DESCRIPTION OF CORRELATION SPECTRA
4.1
11
DATA ACQUISITION AND ANALYSIS
CHAPTER THREE.
3.1
3.2
3.3
3.4
INTRODUCTION
Determination of Degrees of Freedom and
Description of Spectral Tables
44
47
48
51
51
Page
4.2
4.3
4.4
4.5
4.6
Correlation Spectra for Individual
Wavenumbers and Time Scales by Latitude
Band
Correlation Spectra for the Sum over
all Wavenumbers
Correlation Spectra for the Sum over all
Time Scales
Total Correlations by Latitude Band
Variance and Covariance Spectra
CHAPTER FIVE. DISCUSSION OF RESULTS AND CONCLUSIONS
5.1
5.2
5.3
65
91
96
101
101
104
Forced and Free Regimes
104
Baroclinic Adjustment: A Free Regime Process 105
Validity of the Eddy Flux Parameterization:
Eddy Flux Predictors
109
CHAPTER SIX.
6.1
6.2
6.3
SUGGESTIONS FOR CONTINUING RESEARCH
114
Regional Analysis
Multiple Correlation Studies
General Uses of the Program SPECTRA
114
117
118
Computer Program SPECTRA: A Listing
Additional Spectral Output
119
129
APPENDICES
A.
B.
REFERENCES
136
LIST OF TABLES
Page
Table
3.1
3.2
4.1
Two-Layer Model Programming,
Programs CRTSHR, DAGRI3
33
Two-Layer Model Programming, Program
FINAL
43
Degrees of Freedom and 95%, 99%
Confidence Limits for Correlation
Coefficients
66
5.1
Physical Size of Wavenumber Domain
107
5.2
Correlation Coefficient Comparison:
Lorenz (1979) vs. McHenry
112
LIST OF FIGURES
Figure
3.1
Page
Sample Section of an Input Height
Grid, 500 mb, Dec. 1, 1962, OZ
34
Critical Shear, Sample Output,
5/9/64 00Z
35
3.3
Sample Output 200 AT, 5/9/64 OOZ
36
3.4
Sample Output Tl*, 5/9/64 OZ
37
3.5
Sample Output T 3 *, 5/9/64 OOZ
38
3.6
Sample Output, V1 *, 7/9/66 OZ
40
3.7
Sample Output, V3*, 7/9/66 OOZ
41
3.8
Sample Output, U 1 - U 3 , 7/9/66 OOZ
42
3.9
Sample Output, X * (AT)2,
45
3.10
Sample Output, Eddy Flux, 4/l/63
4.1
Components ((Am,Am) + (Bm,Bm))k in
3.2
Analysis of Variance of X
Latitude Band 1.
4/1/63 OOZ
OOZ
46
(AT)2 for
53
4.2
Continuation of 4.1
54
4.3
Continuation of 4.1
55
4.4
Components ((Cm'Cm) + (DmDm))k in
Analysis of Variance of Eddy Flux for
Latitude Band 1
56
4.5
Continuation of 4.4
57
4.6
Continuation of 4.4
58
4.7
Components ((AmCm) + (BmDm )k in Analysis
of Covariance of Eddy Flux with X
for Latitude Band 1
4.8
Continuation of 4.7
(AT)2
59
60
List of Figures
(continued)
Figure
Page
4.9
Continuation of 4.7
4.10
Correlation Coefficient Spectrum, Latitude Band 1
62
4.11
Continuation of 4.10
63
4.12
Correlation Spectra by Individual
Wavenumber and Time Scale K
67
4.12.1
4.12.2
4.12.3
4.12.4
4.12.5
4.12.6
4.12.7
4.12.8
4.12.9
4.12.10
4.12.11
4.12.12
4.12.13
4.12.14
4.12.15
4.12.16
4.12.17
4.12.18
4.12.1
4.12.20
4.12.21
4.12.22
4.12.23
4.12.24
Wavenumnber
4.13
Correlation Spectra for the Sum Over all
Wavenumbers
4.13.1
4.13.2
4.13.3
4.13.4
Latitude
4.14
Correlation Spectra for the Sum Over all
Time Scales
4.14.1
4.14.2
4.14.3
4.14.4
Latitude Band 1
Latitude
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"I
"
"
"
"
"
"I
"
"
"n
"
"
"
"'
"
"
"
"
"
"
"
"I
"
"
"
"
"
"
"
"
"
2
"I
"
"I
"
3
4
"
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
1
Band
"
"t
"
Band
"
2
"
3
"
4
97
98
99
100
10
List of Figures (continued)
Figure
Page
4.15
Total Correlation Coefficient
102
5.1
Parametric Plot of Preferred Eddy Time
Scale in Baroclinic Adjustment
110
CHAPTER ONE
INTRODUCTION
1.1
The Eddy Flux Parameterization Problem
In seeking a closed system of equations to describe
climate, the equations of motion along with the thermodynamic equation must be solved in time averaged form.
Often
the climate is defined as a zonal mean process, and the
Each independ-
equations may be spatially averaged as well.
ent variable is usually thought of as the sum of average
and eddy parts which are substituted into the equations,
then the equations themselves are averaged.
In the case of
the mean Navier-Stokes equations, the average product of
stochastic variables appears.
These terms are known as
Reynold's stresses, which describe the transport of turbulent momentum.
Similar terms appear in the thermodynamic
equation; these are known as eddy heat fluxes.
In principle
the eddy heat fluxes can be solved for from the original
equations.
It is, however, a difficult process because
the non-linear terms in the momentum equations generate
higher order deviations.
To get around this closure prob-
lem, current research focuses on parameterization of the
eddy heat flux in terms of the mean atmospheric structure.
Usually the parameterizations include the mean meridional
temperature gradient and other quantities derived from the
12
physical reasons for the existence of the eddies.
Climate models which include the total heat transport
must contain parameterizations which accurately measure the
contribution of the eddies to that total.
Current para-
meterizations only roughly model the gross eddy transport.
The key problem is thus finding improved parameterizations.
In order to accomplish this, the physical behavior of the
eddies in relation to their predictors must be examined.
Because the eddy behavior is a function of spatial and
temporal scales, one parameterization may work well when
the averaging time is long, say, several months, but wouldn't
work at all for short time scales, say, less than a month.
Furthermore, zonal averaging smooths out the contributions
of eddies of smaller zonal scale which exist because of locally unstable baroclinic zones.
Because differing regions
of the earth exhibit differing climates, parameterizations
without zonal dependence yield little information for specific geographic regions.
Hence no parameterization can cap-
ture all of the physics of the growth and decay of eddies.
Solutions to the problem hinge on several approaches.
First, what does theory (particularly baroclinic instability
theory and scale analysis) suggest?
the data reveal?
Secondly, what does
Finally, how well do particular theories
describe the actual behavior of the eddies, and what can
we learn about the eddies from them?
on answers to this last question.
Our study will focus
1.2
Forced Versus Free Eddy Behavior and
Baroclinic Adjustment
Much is already known about the behavior of the eddies.
Lorenz (1979) showed that for the zonally averaged case,
modelling eddy transport as heat diffusion works well only
for the meridional planetary scale when the time scale is
inclusive of seasonal forcing.
For shorter or longer time
scales and smaller meridional scales, the variations in eddy
flux were free--unrelated to external seasonal forcing--and
thus unable to be accounted for by diffusive theory alone.
When we speak of free variations, we will be talking about
the short period (less than a month) variety.
Stone (1978) has termed the zonally averaged free eddy
behavior "baroclinic adjustment."
In this process, the
eddies grow in response to baroclinic instabilities unrelated to the absolute localvalue of the temperature gradient--but ultimately affect it by efficiently transporting
heat.
In contrast for the seasonal case, changes in the
sun's inclination force changes in the gradient which force
the eddies--a diffusive type transport.
Lorenz was the
first to identify this difference between forced and free
regimes.
Current theories of parameterization are derived for
forced variations only and in general apply to the zonally
averaged case.
Hence, heat transported by eddies as a re-
sult of free variationis is not well modelled.
In most
instances the parameterization seeks to account for the
gross effect of the transport due to free variations by defining an eddy diffusivity (in analogy to a thermal diffusivity) which weights the forcing by the gradient.
The form
of this coefficient may be simple or elaborate depending on
which theory was used to derive it.
One by product of these theories, and baroclinic instability theory in general, is the prediction of the relevant zonal scales of the eddies.
instance,
Held (1978a) suggests, for
that the zonal extent of an eddy may depend upon
the vertical extent of its associated baroclinic wave.
Another by-product is the well-known result that the 6-effect
serves to stabilize the growth of baroclinic waves.
two-layer model (.Phillips, 1954),
In the
6 is important in deter-
mining the shear at which the model becomes baroclinically
unstable.
"Two-layer" eddies can grow and transport heat
when the actual shear exceeds the critical shear.
Both of
these considerations have been taken into account in theories
which adjust the form of the eddy diffusivity mentioned
above.
1.3
Objectives of this Research
By using actual data to test a particular parameteri-
zation, much can be learned about (1) the parameterization
itself, and (2) the behavior of the eddies in relation to
its predictors.
By studying one particular parameterization we are
limited in our ability to compare with other parameterizations.
However, we can assess its ability to model the eddy
flux in comparison to that for which it was designed.
To
determine how a particular parameterization functions is
to determine the spatial and temporal scales over which
actual data assess its validity.
That is, for what space
and time scales is the eddy flux well-positively correlated
with its combination of predictors in the parameterization
of question?
The eddy behavior in relation to its predictors can
be understood, on the other hand, by examining the entire
range of correlations through the various space and time
scales available in the data base.
Previous studies have concentrated on zonally averaged
data.
Stone (1978) identified the negative correlations on
short time scales between the eddy flux and meridional temperature gradients as baroclinic adjustment, but used only
a small data base--three Januaries. Lorenz (1979),
although
making use of 10 years of twice daily data, again studied
the zonally averaged case in looking at meridional scales.
Eddies adjust on short time scales (free variations)
and are forced on long time scales.
The zonal scale of an
eddy can be significantly smaller (determined by the relevant radius of deformation) than the planetary.
By studying
the zonally-dependent space-time correlation spectrum be-
16
tween the eddy flux and its predictors, we should be able
to observe adjustment on short time scales within several
zonal wavenumbers in the form of significant negative correlations.
Adjustment has never been observed on scales
smaller than wavenumber 0.
Also, we should be able to ob-
serve forced eddy behavior on seasonal time scales for at
least wavenumber 0.
No study has sought to tie together ad-
justment and flux-gradient forcing so succinctly.
Further-
more, we can look at the correlation spectrum as a function
of latitude and gain knowledge about the relationship between
adjustment regimes, forced regimes and the scales of the
eddies themselves.
The primary advantage of a spectral correlation test
is -that it will uncover which zonal spatial scales yield the
most significant temporal correlations.
Available to us
are 10 years of twice daily geopotential height observations
at 850 mb,
500 mb and 300 mb (see Section 3.11.
By arranging
the data in a two-layer format, modelling a parameterization based on the two-layer model and constructing its
correlation spectrum as a function of latitude, the freeforced space-time behavior of the "two-layer" eddies should
be adequately revealed.
CHAPTER TWO
MODELLING THE EDDY FLUX AND ITS PREDICTORS
2.1
Choice of Specific Parameterization for Study
and Analysis Method
In choosing the particular parameterization for this
study, we must select one that can be adapted to the twolayer model, since our data are available in that format.
The parameter dependences should include meridional temperature gradient and an eddy diffusivity which accounts for
the s-effect (because we are interested in latitudinal variations), and which takes into account the zonal scale of the
free regime eddies.
A priori, then, we choose a parameter-
ization which is likely to reveal the free-forced regime
eddy behavior as a function of zonal scale and latitude.
It is convenient to follow Held (1978b).
We write the
eddy heat flux as
v''
= -D
(2.1)
Dy
where D is the eddy diffusivity with units of velocity time
length.
D = V * L
(2.2)
Stone (1972) assumes the velocity scale is determined by the
vertical shear times the scale height, V
length scale is approximated by L = NH/f.
=
H
and the
Here N is the
1/2
Brunt-Vaisali frequency, N a
parameter.
2-
, and f is the Coriolis
Stone's choice of length scale is the Rossby
radius corresponding to the scale height.
By the thermal
wind relation, 3zu a ay so that when D is substituted into
(2.1),
o2
VIa
V'O' a
( -
1/2(23
(2.3)
af
- a affects the growth of baroclinic waves, one
ay
would expect it to affect the form of the diffusivity (2.2).
Because
Held (1978b) has argued using scale analysis that (2.3) should
be modified such that
- 2
V'6' a
[ J 1/2
F (x)
(2.4)
where X = h, H being the scale height, and h is the vertical
scale of the most unstable baroclinic mode.
scale is defined by h = (f2 -)/2.
X >> 1; conversely X << 1 for large 6.
This vertical
When 6 is small
Held's arguments
proceed to deduce the following form for F (X) in the limits
6 large and 6
-+
small:
F (X) =
1
X3
small
6large
For small 6 (6 decreases northward) (2.4) reduces to (2.3);
for large 6 (2.4) is changed significantly.
The problem
arises because scale analysis doesn't predict the form of
F 3 X) for realistic atmospheric values of 0.
atmosphere, h ~ H so that X is near 1.
In the real
A plot of F 3 vs. X
looks as follows:
z
-I
Assume for the real atmosphere that F 3(x)aX.
case,
If this is the
(2.4) becomes
____
E
V'
2
,-1/2
a
-l
(2.5)
This is a complex parameterization to attempt to study with
a large data set.
If we ignore the contribution of the
static stability to the 1/2 power in (2.5),
it both simpli-
fies the form of the diffusivity and leads us to Saltzman's
(1967) parameterization, which says
D a
- B
(2.6)
20
where B is
a stability coefficient.
B is directly analogous to X.
X
V''
In
the two layer model,
Thus we choose to test
-l
2
(2.7)
both because it is simpler than (2.5) and representative of
In (2.7),
(2.6).
NH/f.
the zonal scale is no longer explicitly
This is a concession, but we expect it to make little
difference in our analysis.
Equation (2.7) is easily adapted to suit a two-layer
Let subscripts 1 and 3 denote the upper and lower
model.
layers respectively.
Then X a
U., -U3
R(6 -63
2
, where Uc
U
Stone,
1978.)
jI
13zJ
is given by
is the critical shear.
(See
When (U -U3 )/Uc > 1 the model is baroclini-
cally unstable and waves may grow.
Hence X is just a
Saltzman stability coefficient in the two-layer sense.
In order to test (2.7) spectrally, we must be able to
compute the instantaneous values of each side as functions
of X, $,
t.
For the left hand side, we want the mass weighted
vertically integrated eddy flux of sensible heat--call it
V*T*.
Here V = [V] + V*, T = T* + [T] where [
a zonal average.
Thus V*T* (A,$,t) is the instantaneous
transport desired.
Let AT be the mass weighted tropospheric
temperature gradient.
V*T*(o,$,t)
)] indicates
Then
(AT)
2
U
- U3
(X,4,t)
1U
c
(2.8)
Equation (2.8) is
appropriate for testing with our
data set and accurately represents the physics in (2.7) provided the correlations are not performed point by point.
To avoid this, the quantities in (2.8) will each be smoothed
latitudinally over the width of 4 independent 100 wide latitude bands.
The term V*T* will represent the average flux
through the band.
Similarly (AT)2 will be defined as 200
AT centered at the band center, and (U1 -U 3 )/Uc will also be
smoothed across the band width.
The smoothing process,
along with a 200 wide AT, is necessary because small scale
gradients do not force eddy fluxes to any great degree.
It
also eliminates noise.
Spectral decomposition of (2.8) will thus take place
in 4 latitude bands chosen to be centered at 350, 450, 550,
and 650 N.
Fast Fourier analysis will be used to compute
the zonal Fourier coefficients of each side of (2.8) as
functions of time.
Then, the poor man's time spectral ana-
lysis described by Lorenz (1979) will be used to construct
the correlation spectrum of Fourier coefficients as functions
of time.
Thus, the spatial scales will be resolved as well--
the zonally averaged case so often studied just falls out
as the wavenumber 0 correlation spectrum.
Identically cal-
culated spectra are produced independently for the four latitude bands, allowing interpretation of results to span the
midlatitudes.
Sections 3.4.1 and 3.4.2 will discuss this
analysis in detail.
A priori we can say something about what we expect to
learn from studying (2.8).
First of all, we would expect
it to model forced variations for the zonally averaged case
very well.
We suspect that the X parameter by itself is
not a very good eddy diffusivity, because diffusive assumptions like (2.6) are simpler than those derived from complex
baroclinic theory.
Because X is on average near 1 (the
long-term mean shear approximates the long term critical
shear),
we can project that for seasonally forced variations
our correlations will resemble these between the flux and
gradient squared alone.
We hope that they would be slightly
better, but we have limited means with which to verify this.
We expect that (2.8) will reveal nicely baroclinic
adjustment.
Because X automatically includes a in its cri-
tical shear, we believe the negative correlations indicative
of adjustment may be enhanced by the inclusion of X.
We
expect to be able to detect adjustment for a range of wavenumbers, but perhaps not as well as if the zonal scale were
explicitly included as in (2.5)..
On this last point, how-
ever, we can only conjecture.
2.2
A Two-Layer Format for Processing
Geopotential Height Data
We seek to compute all of the quantities in
(81 using
a two-layer model given height data at 300, 500 and 850 mb.
We model the atmosphere as follows:
23
level
300 mb
400
0
--------------------------------
T1 ,0
U1
500
675
T3 ,0 31 U3
-- interface
3- ~layer 3
4
850
In
1
2
---------------------------------
layer 1
some
sense
is a crude model because there is some
it
eddy transport above 300 mb. (particularly when the tropopause
is higher than that) and below 850 mb.
We are actually
looking at a slice which includes about 80% of the mass of
the troposphere
2.2.1
Critical Shear
To compute this we must compute the temperature hydrostatically:
(i
T
T
3
=
-J2,4
R
A .2z
(2.9)
in
5
In( 8.57
5
(2.10)
stands for the height difference between the i,j
where A.
1,J
layers. The potential temperatures 6 and 03 are just
R/C
6=T
l
1p)
s
--
p
(2.11)
24
R/C
6
= T3 (-
(2.12)
P
where p 1 = 400 mb, ps = 1000 mb, and p 3 = 675 mb.
tical shear, given by Uc = 2 (0 -63)
c f
1 3
The cri-
is then determined as
a function of latitude.
2.2.2
Let AT
Mass Weighted Meridienal Temperature Gradient
stand for the temperature difference at layer
i along a 200 latitude stretch of a given longitude X
.
The
mass weighted gradient is given by
AT
m
(2.13)
2AT 1 + 3.5 AT 3
=
5.5
Eq. (2.13) is computed using (2.9) and (2.10).
2.2.3
Actual Shear
We want to compute the velocities U 1 and U 3 in the two
layers from the geopotential height data.
It is convenient
to follow Lorenz (1979) in first computing a geostrophic
streamfunction and then getting the velocity fields.
From the divergence equation,
u - V(f+c)
=t -x
2
V [(gz) +
1
- o
u - U]
+
(f+c)c
-
VW
-*
+ V - F
-
(2.14)
include only the terms
^
+
0 = -k x u
Vf + f-
2
gz
(215)
(2.15)
where C =
-
= k -
(V x u).
Assuming that we can repre-
sent the velocity by a stream function $ such that
u,
(
,
ition.
, we have V
Then C = V $ by defin-
Hence,
gV 2 z= (-k x u)
Now
6 = 0.
=
= aaf
and
f=
ax
gV 2z =
gV 2z
=
0.
-
Vf + fV2
(2.16)
Then
u
ay
+ fV2
(2.17)
+ fV 2$
(2.18)
- Vf + fV - Vp = V -(fv$)
or
gV 2z = V
Finally,
g?2z = 20V -
(2.19)
(2.20)
(sin $V$)
A procedure for solving (2.20) for $ when z is expressed in spherical harmonics is described in Lorenz (1978).
Equation (2.20) is more complex than the simple geostrophic
relationship gz = f$ because it includes a effects--this is
exactly what we desire.
To get the stream.function at levels 1 and 3 we compute
$0
$1
+
2
~3l
, U=
2
$2
*3 =
+ $2
y
*4
,
U3
ay
(2.21)
We can then use a centered finite difference scheme to get
the velocities u1 and u 3 from $l and 4$
3.
(u1 - u 3 ) is the actual shear desired.
Hence, finally,
Combining the results
u1
of (2.21) with the critical shear calculation yields
1
-u
u~
C
3
one part of the right hand side of (2.8).
2.2.4
Eddy Flux
We want to compute the total eddy flux representing
the integrated value through the depth of the model:
300 C
V*T* =
--
V*T*dp
(2.22)
850
27r
( )dX so that
As before [( )] =
V* =V-
[V]
T*= T
[T].
(2.23)
T* is simply computed from (2.9) and (2.10).
The choice of
a stream function to represent the velocity presupposes a
non-divergent field; hence there is no mean meridional signal resolved by this method.
This is convenient again be-
cause we are only interested in the eddy part of the V field.
Hence
(2.24)
V * =
1
V
V3
ax
a 3
ax
27 A~~~
hold for the eddy velocities at levels 1 and 3.
Combining
(2.23) and (2.24), we compute the two-layer finite sum representing (2.22):
3
V*T* =
C
l
V.*T.*A.p,
1
A
1
= 200
A 3 = 350
(2.25)
This completes all of the model equations needed to
process the height data into the quantities found in (2.8).
The next chapter will discuss data acquisition and handling,
the programming of these equations, and the equations and
programming of the statistical analysis by spectral decomposition.
CHAPTER THREE
DATA ACQUISITIONMODELAND STATISTICAL ANALYSIS
3.1
Data Sources and Description
All the data were derived from synoptic analyses of
the 850, 500 and 300 mb geopotential height field made at
the National Meteorological Center, although it originated
in the different formshaving been stored on tapes at the
Initially available to us were
NCAR computing facility.
previously analyzed height fields at 500, 850 mb in the form
of triangularly truncated spherical harmonics (Leight, 1974).
The series are defined by:
L
Z(X,$,t)
n
-
Cmn(t)cos mX + Smn(t)sin m]
=
n=0 m=0
x P
(sin $)
(3.1)
Lorenz (1978) documents the spherical harmonic data set at
500 mb completely, the 850 mb set is identical in structure.
Here Z is the geopotential height, X is longitude, $ latitude, t time, and P n the associated Legendre function of
degree n and order (or wavenumber) m.
cated triangularly at L = 18.
The series were trun-
Maurice Blackmon provided
NCAR tapes B6902 and B10618 containing the coefficients Cmn
and Smn, 100 cosines and 90 sines represent each field.
30
These two datasets spanned 10 years and 1 month beginning
Dec 1, 1962.
In order to employ our model, data was acquired through
the NCAR tape library for the 300 mb geopotential heights
in 400 word packed binary.
Tape C4650 was used; the dataset
on this tape began before Dec 1, 1962 and extended through
12z Nov 21,
3.2
1971.
Initial Computational Work
The task of organizing the massive quantity of data
(over 13 million data pieces) needed for this study was formidable.
Over half the computing time was consumed by this
stage of the project.
In order to calculate, via the model,
the quantities in (2.8) six tapes--3 of the heights themselves and 3 of their spherical harmonic representations-were needed.
The grid size was chosen to be 50 x 5.6250 (latitude
x longitude) extending from 250N to 750N.
latitudes and 64 longitudes.
This provided 11
The longitudinal spacing was
chosen because fast Fourier analysis is convenient for powers
of 2, and in foresight of the upcoming longitudinal Fourier
decomposition.
The latitudinal range was chosen so that
200 wide AT's were available for the four latitude bands,
centered at 350,
450,
550, 65g.
For convenience the length of the study was chosen to
begin Dec 1, 1962 and end Nov 21, 1971 so that the datasets
31
were exactly compatable.
This meant that observations were
lopped off the beginning of tape C4650, and lopped of the
end of the two spherical harmonics tapes.
A program was written to reconstruct the height grids
(50 x 5.6250) from their spherical harmonic coefficients.
These grids then became inputs to the parts of the model
which used direct heights for calculation.
Thus, tapes of
the 850 and 500 mb height grids and their spherical harmonic
coefficients (4 tapes total) were acquired rather easily.
Producing the raw 300 mb data was much more tedious.
Package routines at NCAR were used to unpack the binary data
on tape C4650 and convert it to grids like those at the
other two levels.
It was then discovered that the 300 mb
data contained 200 (out of 6556) missing observations.
Pro-
grams were then written to hydrostatically interpolate the
300 mb heights for the scattered missing observations from
the 850 and 500 mb grids,
and then insert the interpolated
data into their missing slots.
Once this was accomplished,
we had a complete tape of the 300 mb height grids spanning
6556 observations.
The reverse procedure then had to be undertaken to produce the spherical harmonic coefficients from the height
grids'at 300 mb.
Maurice Blackman provided a program suited
for our needs, and after adapting it to our data structure,
it was used to accomplish our purpose.
The problems presented by massive quantities of data
were significant.
For example,
the 300 mb spherical harmonics
had to be produced in 5 separate pieces because of the limitations on tape storage capacity.
Program efficiency was a
high priority because somewhat complex calculations were repeated thousands of times.
Hence, small test data sets were
constructed and used to test almost every program of significant length.
Ensuring that each data set was identically
formatted was a must, so that in merging the levels we could
be certain that all 3 levels always contained observations
made at the same date and hour.
3.3
Computation by the Two-Layer Model
Three programs, named CRTSHR, DAGRI3, and FINAL, were
used to compute and store the quantities in (2.8).
Table
3.1 describes the functions of the programs CRTSHR and DAGRI3.
Program CRTSHR produced the quantities from the input
height grids as shown in Table 3.1
A sample section of an
input height grid is shown in Fig 3.1.
The programming al-
gorithms used in CRTSHR are quite simple and available from
the author.
Sample outputs of critical shear, 200 AT, T* and T* are
shown for 5/9/64 at OOZ in figures 3.2 - 3.5 respectively.
One item to be noted is that the critical shear is a much
stronger function of latitude, due to the a-effect, than
longitude.
Variations in Uc longitudinally are solely due
to static stability differences.
Sample outputs were thor-
oughly checked for consistencies with realistic meteorological
Table 3.1
Two-Layer Model Programming,
Programs CRTSHR, DAGRI3
Quantity
Equation in Text
Input
Produced by PrQgram
Height Grids
CRTSHR
Height Grids
CRTSHR
(.2.9)
Height Grids
CRTSHR
(2.10)
Height Grids
CRTSHR
(.2.24)-
Spherical Harmonics
DAGRI3
V* Level 3
(2.24)-
Spherical Harmonics
DAGRI 3
Actual Shear
(2.21)
Spherical Harmonics
DAGRI3
Critical Shear
(2. 11)_,
200 Temperate
Gradients
(2.13).
T* Level 1
(2.23),
T* Level 3
(2.23),
V* Level 1
(.2. 12)-
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values.
The program accurately produced temperature depar-
tures such that [T*]
= 0.
The logic involved in program DAGRI3 was quite a bit
more complex than CRTSHR.
ical harmonic inputs.
DAGRI3 handled all of the spher-
Programming for the solution of (2.20)
followed the algorithm for its solution described in Lorenz
(1978).
Once the stream functions were produced, DAGRI3
used a centered-finite difference scheme to produce zonal
velocities and the meridional velocity departures.
Figures
3.6 - 3.8 show sample outputs for V*, V* and the actual shear
between the layers, Ul-U 3 , for July 9, 1966 at OOZ.
A good check on the accuracy of the algorithm for
solving (2.20)
is
the sum of V* or V* around a latitude
1
3
circle, because
0.
In all cases checked, that sum
was zero or within .1 or .2 meters/second of zero.
Program FINAL merged the outputs of CRTSHR and DAGRI3
into the quantities of eq.
(2.8),
those outputs having been
stored separately on the NCAR computing facility mass store
device (TMS-4).
Table 3.2 shows how this was accomplished.
The X parameter, hereafter referred to as X, defined by
U -U3
for the two layer model, was calculated after
1U
X
c
the shear and critical shears were smoothed latitudinally.
Once this was done, the 200 AT was squared and multiplied
by X to produce the right hand side of (2.8) for each of the
four latitude bands.
Statistics were kept on Uc because
numerous small U C's would have abnormally dominated the
-1.
V-t
N
CD
-4.2
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U.
15 7.5.,.
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Table 3.2
Two-Layer Model Programming, Program FINAL
Quantity Produced
Equation in Text
Eddy Flux
(2.25)
X Parameter Times
Temperature Gradient
Squared
(2.8)
Inputs
20 0 AT, U 1
-
U 3 , UC
44
Fourier spectrum to be produced.
It was found that only
.0015% of the smoothed Uc's were less than .1 in absolute
value.
(A small percentage were negative.)
FINAL calculated the eddy flux from (2.25) for each
grid point, then did the 100 smoothing for each latitude
band.
In each case, the smoothing was performed by averaging
the three values at, and 50 either side of, the latitude at
the center of the band.
Figures 3.9 and 3.10 display sample output from FINAL,
eddy flux in joules/m/sec and X times (AT)2 in
April 1,
3.4
K 2,
for
1963 at OOZ.
Spectral Decomposition and Analysis of Variance
Program SPECTRA was written to produce the variance,
covariance, and correlation coefficient spectra for eq. (2.8).
Two major algorithms are found within.
SPECTRA first pro-
duces the zonal Fourier representations of the left and right
hand sides of (2.8).
Then, using the coefficients as inputs,
it produces the space-time spectra desired by making use of
Lorenz' (1979) poor man's spectral analysis of variance.
In
order to make use of the poor man's analysis technique, the
time series were required to be in powers of two.
Following
Lorenz, we truncated the output data of the program FINAL
for input into the program SPECTRA.
Each year thus became
256 days in length (Dec 1 - Aug 13, except in leap years),
,D
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47
and we made use of the first four and last four years only,
discarding Dec 1, 1966 - Aug 13, 1967.
More discussion about
this point follows below.
3.4.1
Fourier Analysis in Latitude Bands
The left and right hand sides of (2.8) were analyzed
into zonal Fourier series as follows, where m = zonal wavenumber, X = longitude, * = latitude, t = time.
X -(AT)
2
N
(X,$,t) = A(4,t) +
(Am ($,t) cos MX
m=1
(3.2)
+ B (4,t) sin mX)
N
Eddy Flux (X,$,t) = C (4,t) + m (Cm($,t)cos mX
m=1
+ Dm($,t) sin mX)
(3.3)
Figures 3.9 and 3.10 show the exact input structure of the
data in (3.2) and (3.3), which is also just the output of
program FINAL.
Fast Fourier analysis yields the Fourier
transforms for 32 wavenumbers (1/2 the number of input points)
from which the coefficients can be directly calculated.
For
each observation and latitude band, 32 sine coefficients and
32 cosine coefficients are produced.
The A
and C0 terms
are the zonally averaged cases, the sine coefficients being
zero for m = 0 and m = 32.
Hence, (3.2) and (3.3) produce
time series of Fourier coefficients for introduction into
48
the temporal analysis scheme.
3.4.2
Poor Man's Time Spectral Analysis
Lorenz (1979) describes this nifty analysis of variance
which is performed separately for each spatial wavenumber
and latitude band.
The reader should refer to his paper for
more complete details; we shall just summarize it
here.
Assume that the data to be processed consists of a 2K
successive values, and we wish to find the covariance specWe define
trum of the scalar variables X and Y.
[XY]k
-l
.I
it
j=0
-
~1
L=O
X(j+i)]1
-,-l
Y,~
L=0
where k = Q,....K; L = 2K; and the data are
ginning at 0 running through L -
J =L/Z.
1.
(Yj+i ,
(34
subscripted be-
We note that k = 2k and
Then
(XY)k
[XIYlk
by definition, for k < K.
-
(3.5)
[X,YJk+1
Observing that [X,Y]
is the mean
product of X and Y, while [X,Y]K is the product of the means,
we find the covariance as a sum:
K-1
I
(XY)k =
[XOY] 0
-
(3.6)
[XY]K.
k=0
when Y = X, we calculate the variance spectrum.
Our obser-
vations are twice daily so that (X,Y)k is the covariance of
2k-1 day averages within 2k day periods, k=O,...,K-l.
The
49
analysis scans the spectrum with overlapping filters whose
corresponding periods center about 1 day, 2 days, etc. up to
1024 days for our data set.
down to (X,Y)
The components (X,Y)8 , (X,Y)7
select contributions to the covariance within
years, higher order components average between years.
In
this way the seasonal cycle (forced eddy behaviorl is pronounced, and there is no interruption in the spectrum as there
would be in a conventional time series analysis.
this reason the data was organized into 2
It is for
day (29 observa-
The seasonal cycle is
tions) blocks stacked year to year.
thus represented as the contrast between winter (Dec 1 April 7) and summer (April 8 - Aug 131.
Shorter period cycles
are represented as contrasts between, for example, the
first and last halves of summer and winter, etc.
period cycles--2 9,
210 and 2
Longer
days--are contrasts between
years.
For our case, the X's are the Am (,t)
the Y's are the Cm (,t)
and Dm(_$,t)..
and
and BM (,tj,
The components of var-
iance of the eddy flux can thus be written as ((Cm ,Cm) +
DmI,D m)k-'
in analogy with (3.6).
energy spectrum.)
are ((A ,A ) + (B
(This is not the eddy flux
The components of variance of X -(AT)
mB ))k
ance are just ((AmCm)
2
and the components of the covari-
+ (Bm,Dm))k. Finally, the relevant
correlation coefficient is given by
A ,CM) + (Bm ,DM)
mk
((AmAm
+ (Bm,BM)
(Cm Cm) + (Dm,Dm
k
It should be understood that no statistics are produced across
scale:
only coefficients with the same zonal wavenumber are
correlated.
Thus, in producing analyses of variance in time
for each zonal scale, the spatial scales are resolved as well.
We can, however, produce correlation coefficients representing the sum over all zonal wavenumbers for a given time
scale, or, the sum over all k for a given m.
We can also
produce a total correlation-coefficient by computing the total
variances and covariances, which are their respective sums
over all space and time scales analyzed.
These sum and total
statistics will be discussed in the next chapter.
Lorenz
(1979) describes the algorithm by which the
analysis is programmed.
SPECTRA follows his logic by intro-
ducing the proper Fourier coeffiicents as they are produced
directly into the analysis of variance.
read, the entire analysis is complete.
Once the dataset is
This eliminates the
need to store the coefficients before processing by the
analysis of variance.
ing of the code.
Appendix A provides a FORTRAN IV list-
CHAPTER FOUR
DESCRIPTION OF CORRELATION SPECTRA
In this and the next chapter we will describe and discuss the results of our study.
This chapter will be devoted
to determination of the degrees of freedom in the various
space and time scales in the problem, a description of the
raw output of the program SPECTRA, and an organized graphical
presentation of the correlation spectra themselves.
Chapter
Five will discuss the results and conclusions to be drawn
from the output and grahpical analyses.
4.1
Determination of Degrees of Freedom and
Description of Spectral Tables
In order to have some measure of the statistical signi-
ficance of the correlation spectra, it is necessary to analyze the number of degrees of freedom involved in calculating
the coefficients.
The most simple approach is to, assume
statistical independence in the observations.
Recalling
that for our case the observations are time series of Fourier
coefficients, we see that to accurately measure the interdependence of successive observations is difficult.
Meteorolo-
gical data sets are known to exhibit autocorrelation in time
because, for instance, today's 500 mb map is not independent
of yesterday's.
est time scales
In our case, however,
except for the small-
(k = 0,1,2 where the period = 2k ),
we can
assume independence, because processes whose characteristic
scales are about 4 days or greater are essentially independent.
With thousands of observations at short time scales
in our data set, an adjustment in degrees of freedom via
calculation of an autocorrelation function would increase
Hence
the 95% and 99% confidence values only a small amount.
we choose to assume statistical independence and use caution
in making our conclusions.
The analysis ((3.4) , (3.5) and (3.6)) compptes the contribution
of the covariance of pairs of data to the total covariance.
Each pair has one degree of freedom for zonal wavenumber
m = 0 or m = 32, and 2 degrees of freedom for 0 < m < 32.
This is because only cosine coefficients contribute when
m = 0 or 32, but both sine and cosine coefficients contribute
(independently) to the covariance when 0 < m < 32.
Hence,
for k = 0, there are 2048 pairs of observations, and when
m = 0 or 32 there are 2048 degrees of freedom.
When, for
k = 0, 0 < m < 32, we have 4096 degrees of freedom.
Figures 4.1 - 4.11 display the spectral tables output
by the program SPECTRA for latitude band 1 ($ = 350).
Sel-
ected tables for latitude bands 2, 3, and 4 (4 = 450, 550
and 650, respectively) may be found in Appendix B.
The first
nine tables consist of the analyses of variance and covariance, and the last two of the correlation spectrum.
In each
case wavenumber m proceeds down the left-hand column, time
period k proceeds across the top.
In the analyses of variance
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9
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COEFFS ?0I% SUM OVERO K
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E
________
_______
64
and covariance, k = 12 is actually the squares or products
of the means, and to its right TOTAL shows the mean products
for the wavenumbers.
At the bottom of the wavenumber column,
TOTAL gives the sum of the variances or covariances over all
wavenumbers for a given k.
Looking down column k = 12, and
over from row TOTAL, we find the grand square or product of
the means, and the grand total is just the total mean product.
The total variance or covariance is just the grand total
minus the grand square or product.
We will briefly discuss
the variance and covariance spectra in section 4.6.
Figures 4.10 and 4.11 display the correlation spectrum
for the band centered at $ = 350.
The rightmost column en-
titled COEFFS FOR SUM OVER k FOR EACH WAVENUMBER contains the
coefficients derived from the two rightmost columns of the
analyses of variance and covariance (see discussion above.)
The bottom most row contains the coefficients for the sum
over m at each k, these being derived from the bottom TOTAL
row mentioned above.
The TOTAL COEFFICIENT results from the
grand totals in the other analyses.
The correlation coeffi-
cients are all calculated according to (3.71 or modifications
thereof.
Before describing all of the correlation spectra in
the next few sections, we should conclude our discussion
about degrees of freedom, and indicate the 95% and 99% confidence limits on the correlation coefficients.
To determine
the degrees of freedom assuming independence for the sums
65
of the k's at each m, just add them up for the individual
k's.
Similarly add to get the total for the sum over the
m's at each k (again assuming independence).
The total number
of degrees of freedom is computed by adding once again across
the other scale.
Table 4.1 lists the degrees of freedom in
all possible combinations and the total, which is 253890.
It also gives the value of the correlation coefficient significant at the 95% and 99% confidence limits for an equaltails test of hypothesis rmk = 0.
These limits are either
taken directly or interpolated using the inverse square root
formula from G.W. Snedecor, Statistical Methods, 4th ed.,
Ames, Iowa, Iowa State College Press, 1946, p. 351.
Thus, a
correlation coefficient exceeding one or both limits is significant at the given limit assuming statistical independence.
4.2
Correlation Spectra for Individual Wavenumbers
and Time Scales by Latitude Band
In this and the next three sections, the important
parts of the correlation spectra will be described by means
of graphical analysis.
Because there is a massive amount of
information to be studied, this section will fully describe
only spectra for wavenumbers 0 through 5.
Wavenumber 5 in-
cludes zonal scales ranging from about 6600 KM at 350 to
3400 KM at 650.
This analysis should adequately justify the
conclusions to be discussed in the next chapter.
Figures 4.12.1 - 4.12.24 present the correlation spectra
by individual wavenumber and time scale for latitude bands 1
0
or
32
O<M<32
Sum
over
M
Table 4.1
Degrees of Freedom and 95%,
99% Confidence
Limits for Correlation Coefficients
2048
.043
.057
2048
1024
.061
.080
1
.061
1024
512
.087
.114
2
.114
.087
512
256
.122
.160
3
.022
.029
7936
.160
.122
256
128
.172
.225
4
.041
.032
3968
.225
.172
128
.315
.242
64
5
.058
.044
1984
.315
.242
64
.435
.338
32
6
.081
.063
992
.435
.338
32
.590
.468
16
7
.115
.088
496
.590
.468
16
.765
.632
8
.162
.123
248
.765
.732
8
.917
.811
4
.229
.175
124
.917
.811
4
.990
.950
2
.320
.246
62
.990
.950
2
1.000
.997
1
.0050
.0039
Total
253890
.028
.022
8190
.040
.031
4095
99%
95%
D.O.F.
99%
95%
D.O.F.
99%
95%
D.O.F.
Sum
Over
K
D.O.F.
95%
99%
4096
.043
.080
15872
.016
.020
11
D.O.F.
.031
.057
31744
.011
.014
10
95%
.040
63488
.0078
.0102
9
99%
126976
.0056
.0072
8
D.O.F.
95%
99%
1.0
C
0
6-
0
K-
0
1
2
3
4
5
6
7
8
9
10
11
Figs. 4.12: Correlation Spectra by Individual Wavenumber
and Time Scale K.
68
-
-.
0
-2-
3-
.7-
-4-
-.
8-
-9-
K-+
0
1
2
3
4
5
6
Fig.
7
4.12.2
8
9
10
69
Within
years
Between
years
99% Limit
95%
.9 .8-
WAVE NUMBER 0
LATITUDE BAND
3
.6 -
4 .3-/
-,e,
-
/
0
.Ift
o
2i
C
-%5%
95%
99
-. 9-
K,
O
1
2
3
4
5
6
L imi
%
7
Fig. 4.12.3
8
9
10
11
70
I
I
I
I
I
I
I
I
I
Within
yeors
i
Between
years
99% Limit
.-
1,D0
.9 -/
951
WAVE NUMBER
LATITUDE BAND
0
4
.6 -
.5-
.3-
0
-.
-.
-. 8
-
95%
S99%
-o
K-
1
Limit
0
1
2
3
4
5
Fig.
6
7
4.12.4
8
9
10
If
C
0)
'4~4~
C
0
0
4-
a)
L.
o-.2
0
K-.
0
1
2
3
4
5
Fig.
6
7
4.12.5
8
9
10
I.0
.9
.8
.7
.6
.5
.4
.3
C
o,.2
.5.I
0
C
0-,
-1.01
K-+.
0
2
3
4
5
Fig.
7
6
4.12.6
8
9
10
It
73
C
C
O-.2
K-4
0
1
2
3
4
5
Fig.
6
7
4.12.7
8
9
10
11
74
0
4-.I
0
-0
C
O-.
-1.01
...
K-?
0
I
2
3
4
5
6
Fig.4.12.8
7
8
9
iO
75
99%
-9
Limit
.9WAVE NUMBER
2
LATITUDE BAND
I
.8
.6-
-4-
'00I,
-.6 -I\'Or0
000
N
Nb
-.
7-
\
990/
K+
0
1
2
3
4
5
Fig.
6
7
4.12.9
8
\
Limit
9
0
11
76
99% Lim'ir
-
.9.8
LATITUDE
/95*
2
WAVE NUMBER
BAND
2
.7-
1
.6
/
.5
.4-
o -o38C).!D
300
Lmt
-9
-.
40
-6-K
-
-
4-.-
-).1-
\
.-
K -p
0
I
2
3
4
5
Fig.
6
990%
7
4.12.10
Limit\\
8
9
10O1
-
I.
.-
k-4
I
Z
3
4
5
Fig.
6
7
4.12.11
8
9
10
Ii
78
ppI
p
Within
years
t
Between
years
1.O-
99% Limit
.9.8
WAVE NUMBER
LATITUDE BAND
2
4
.6-1
.54-
o
-
-
cO
o-.2
C) -. 6 -
-
N4.
U-
-4-
-. 9-1.
K-+
99% Limit
P
0
I
1
I
I
P
I
I
I
4
2
3
4
5
6
7
8
Fig.
4.12.12
P
9
10
11
79
1.0 99%
.9-
WAVE NUMBER
3
95%
I
BAND
LATITUDE
L imit
.5.4-
0/1
c
-.3
-.
4p
00,
,7-
\
-8-\
Limit
-. 9 -99%
10K
K-
9~
0
2
1
3
4
5
6
7
a
9
0
3
4
5
6
7
83
9
10
Fig.
4.12.13
11
A)
0C
OA
o
-.
0-
-1.0
K-1
1.L
0
I
2
3
4
8
6
7
5
Fig. 4.12.14
9
10
81
Within
years
Between
years
1.0 -
99% Limit
.9-
.8 ~
WAVE NUMBER
LATITUDE
3
BAND
3
.7-
1
.6
.5A
.3-
L;-4 o
6-6-
-.-
-
,
100
C-0)
0
\-
"
op
--
0
\
-
.0
C
-4-
-8
99% Limit
9 -
-10
K-
0
1
2
3
4
5
Fig.
6
7
4. 12.15
8
9
10
1
82
Within
years
1.0 -
Between
years
-
Limit
99 %
.9 .8 ~WAVE NUMBER
LATITUDE
3
BAND
.7
4
9/
~
.6
/
-.3 -/
.4-.5-
99% Limit
-10
K-
0
1
2
3
5
4
Fig.
6
4.12.16
7
8
9
10
11
C
'I.,
0
K+
0
1
2
3
4
Fig.
5
6
4.12.17
7
8
9
10
84
Within
years
Between
years
1.099% Limit
.9.
.8-
WAVENUMBER
4
BAND
LATITUDE
2
/1
.7
.6.5.4.3>.2
OOle
Ntt
tf
1
0
o-.
K-
O
5-\
1
2
3
4
5
Fig.
6
7
4.12.18
8
9
10
11
85
-1.01
K-+
'
0
1
2
3
4
8
7
6
5
Fig. 4.12.19
9
10
86
1.0 -.
99% Limit
.98
WAVENUMBER
/
4
LATITUDE BAND
/
4
7-
.5-
--
0
.62
-.
7
000
9~
8-\
-.
99%
K-+
I
2
3
4
Fig.
5
6
7
4.12.20
Limit
8
9
10
11
87
Within
years
Between
years
1.0 /
99 % Limit
.9-
954
.8 -
WAVENUMBER
LATITUDE
5
BAND
/
I
.7 -
7/
/
.5
.4
4-.-
/
/
- 'Ole
0
-5-6-
-.8-\
99% LOmit
-. 9
-101
'K+-,
0'
0
1
2
2
3
3
4
4
5
5
Fig.
6
7
4.12.21
8
9
10
11
88
Within
years
Between
years
99% Limit
.9-95% Li mit
.8
WAVENUMBER
5
LATITUDE BAND
.7-
Z,
.6-
.5 -
1
.4
6-
.-
-.
0
o
,
--
7p
-
95
o
-. 9
-1.6
Ko
99% Limit
0
A -- -
I
-101
K4
Liit
1
2
3
4
5
Fig.
6
7
4.12.22
8
9
I
10
11
V
89
Within
years
Between
years
1.0-
99/% Limit
.9
95% Limit
WAVENUMBER
5
LATITUDE BAND
3
.6 .5 -
-. I-
00I,
.44/
o
-.3/
--
/
-8
-
-s
95% Limit
\
.9
-99%
Limit
0
W.
Oi
2
3
4
5
Fig. 4.12.23
8
9
10
K
I.0
.9
.8
.7
.6
.5
.4
.3
0
C
0
0
-.
K+
0
I
2
3
4
5
Fig.
6
7
4.12.24
8
9
10
11
through 4.
The confidence limits appear dashed while the
coefficients have been connected by solid lines.
When k < 8
the coefficients describe within year processes, when k > 8
they are interannual.
Hence, we have separated the solid
lines to denote these two temporal areas.
Fig 4.12.1 shows
three significant negative coefficients and two more negative ones for k < 4, 3 significant positive coefficients and
another small positive one for 8 < k < 5.
significant occurs.
For k > 8 nothing
The' coefficient for k = 11, m = 0 must
be 1.0 because there is only 1 degree of freedom there.
For
large k, say k > 9, there are fairly few degrees of freedom
so that decisive conclusions about statistical trends are
difficult to make.
Rather than discuss each graph we will point out a few
recognizable traits.
In all cases except fig 4.12.24 (wave-
number 5, latitude band 4),
on short time scales.
we observe negative correlations
Many are significant at the 99% level.
Secondly, we observe significant positive correlations (exception:
fig 4.12.24) only on long time scales (k = 6,7,8),
and only for selected zonal scales.
Finally,
evidence for significant correlation in
there is
little
the k > 8 range of
the spectrum.
4.3
Correlation Spectra for the Sum Over All Wavenumbers
Figures 4.13.1 - 4.13.4 display these spectra summed
over all wavenumbers.
The graphs are identical in structure
92
C
'O .05
C
0
O-
K+
I
2
3
4
5
6
7
8
9
10
if
.2
.1
05
4>-
O-0
c)
0
0
C
O
-,05
C
0)
C,.O
0
T0
O
K-+
O
I
2
3
4
5
6
7
8
9
10
1I
95
C
05
C
0
O
0
-. 05
96
to those of figure 4.12.
Strikingly, all four latitudes
show negative correlations on short time scales and positive
correlations on long time scales (k = 6,7,8),
although the
In con-
trend is less distinct in the two higher latitudes.
trast to figures 4.12, we do observe significant correlations
for the between years scales; however, caution is the word
here because of the small number of degrees of freedom.
4.4
Correlation Spectra for the Sum Over All Time Scales
These four spectra, depicted in figs. 4.14.1 -
4.14.4,
display correlation coefficient as a function of wavenumber
for the sum over all time scales.
The reader should observe
the expanded ordinate scale between 0 and .1.
They describe,
in essencer whether positive or negative correlations dominate for the entire range of time scales.
Each spectrum is
somewhat different; however, in all cases positive correlations are the rule for the zonally averaged flow.
Negative
correlations are observed to dominate in the 2-5 wavenumber
range with a few exceptions.
Caution should be taken in
broadly interpreting these spectra because they include
between year processes (k > 8) which are not of significance
for this particular study.
Latitude band 4 (650) departs
from the patterns established at lower latitudes with high
positive correlations between wavenumbers 9 and 17.
Referr-
ing to Appendix B, output for latitude band 4, we can see
that this is caused by a concentrated group of high positive
97
.9-
Figs, 4.14: Correlation Spectra
the sum over all time scales
.8 -for
'7 ~
LATITUDE
BAND
.6
.5
.4.3.2 -.9
99%
Limit
C
Q)
95595%
99% Limit
Limit
0
0,C -
-3-WAVE
NUMBER
-. 4 '0
2
4
6
8
10
12
14
16
Fig. 4.14.1
18
20
22
24
2,6
28
30
32
98
1.0
.9
.8
LATITUDE
BAND
2
.6
.5.4.3
.2
.1
99% Limit
\
95% Limit
95%
Limiti
L
0
C)
Limit
99%
-. 2 --
-4
.
-~WAVE
z
NUMBE R-
I
2 - 4
I
I
6
I
I
8
uj---
-.
i i
95./
10
I
12
Fig.
14
16
4.14.2
18
20
22
24
9I
26-
I
1i
28
i
30
1
52
99
Fig.
4.14.3
100
Fig.
4.14.4
101
correlations within these wavenumbers on short time scales.
At 650, the Zonal Scale is at most 1800 KM (m = 9) which
approaches the reliable resolution limit of the data.
(Hypo-
thetically that limit is 2fra cos 650 = 512 KM, but realisti32
cally it is probably twice or even three times that large.)
Hence, not too much stock can be placed in significant correlations near this resolution limit.
(See Table 5.1, which
gives the physical size of the wavenumber domains as a function of latitude.)
4.5
Total Correlations by Latitude Band
Figure 4.15 depicts the total correlation coefficient
between the eddy flux and X* (AT) 2 as a function of latitude.
Little information is to be drawn from this graph because
the absolute values of the coefficients are so small, even
though they exceed the 99% confidence limit twice.
One would
not expect a high positive correlation between datasets
which are negatively correlated on certain scales and positively correlated on others.
These are probably the least
meaningful results.
4.6
Variance and Covariance Spectra
Here we will briefly touch on the variance and covari-
ance spectra found in Figures 4.1 - 4.9. In general, the
.2
variance of both X- (AT) and the eddy flux drops off with
increasing wavenumber.
Relative maxima may be found, for
102
.04-
.03-
02-
99% Limit
-
-
----
-
95 % Limit
0
95% Limit
..
0
99% Limit
01 -
-02-
-03-
-04-
-. 5
350
450
55*
Fig. 4.15
650
LATITUDE
Total Correlations by
Latitude Band
103
instance,
for 1 < m < 9 and k = 0 in
the X
-
(AT)
2
variance.
Another relative maximum may be found at m = 0, k = 8.
In
the eddy flux case, the variance maximum occurs at k = 2,
m = 9.
The relative maxima described here refer to latitude
350N only.
Lorenz (1979) reports similar findings.
In his
case the variance in zonally averaged temperature decreased
as the meridional scale decreased.
Figures 4.7 - 4.9 reveal
that the components of the covariance also decrease in magnitude as wavenumber increases.
Positive maxima occur near
m = 0 and k = 8, while negative maxima occur near m = 0,
k = 2.
104
CHAPTER FIVE
DISCUSSION OF RESULTS AND CONCLUSIONS
5.1
Forced and Free Regimes
We can make use of the figures in the last chapter to
draw some conclusions about the forced and free regimes revealed in the parameterization (2.8).
In our case, because
we correlate the eddy flux with the square of the temperature
gradient, positive correlations imply that the variations
are forced.
For free variations the correlation must be
negative, although it is possible to have a negative correlation even though some forcing is present (Lorenz, 1979).
For the zonally averaged case, free regimes appear on short
time scales and forced regimes appear on seasonal scales.
This result solidly verifies both baroclinic adjustment and
seasonal forcing within the spectre of the same data set.
For planetary and smaller scale wavenumbers (through
m = 5),
forcing at the seasonal time scale is confined to:
wavenumber 1 at 350 and 450, wavenumber 2 at 350 and 450,
wavenumber 3 at 550 and 650 and wavenumber 4 at 450
al).
(margin-
Because the total eddy flux is modeled, both station-
ary and transient eddies are included.
The wavenumber 3
forcing at high latitudes may very well be due to the topographical (land mass/ocean) forcing of the stationary eddies.
Flux variations which can be explained by this para-
105
meterization (i.e.,
which have significant correlations,
either positive or negative) are, however, primarily free.
At the seasonal time scale, many of the figures reveal no
Wavenumber 5 at 650 and wavenum-
significant correlations.
ber 3 at 450 are the only seasonal time scales which are
marked by negative correlations.
As mentioned briefly in
the last chapter, it is striking that significant negative
correlations occur on all short period time scales except
one.
We conclude that we have observed what we set out to
observe:
the eddies also adjust the gradient on scales
smaller than the zonally averaged scale!
We feel that we have been successful in effectively
displaying the behavior of the eddies by spectrally testing
(2.8) on real data.
Additionally, we can assess more com-
pletely baroclinic adjustment, and make some general conclusions about the validity of (2.8).
This we will do in the
next two sections.
5.2
Baroclinic Adjustment:
A Free Regime Process
By studying the latitudinal and temporal dependence
of the eddies for short time scales we can learn a great
deal about baroclinic adjustment.
the favored time scale.
First, we will discuss
This is the period at which the
eddies are most negatively correlated with X* (AT)2.
Sec-
ondly, we can examine the zonal scale of the eddies as a
function of latitude.
Finally we will describe how the
106
characteristic adjustment period (= 2k days) is affected by
the zonal scale on which the adjustment is occurring.
With the exception of the zonally averaged case, the
peak negative correlation begins at k = 3 (8 days) and retrogresses with increasing wavenumber until it reaches 1 day
for m = 5, and finally dissappears altogether at 650 for m =
5.
For the zonally averaged case, the favored time scale is
4 days.
This agrees with Lorenz' (1979) results.
An average
of the zonal and smaller scales could very well suggest 4 days
as an overall favored time scale as well.
We can offer a
simple explanation for the observed retrogression, but most
do so in light of the zonal scale of the eddies'as a function of latitude.
Table 5.1 illustrates the physical size of the various
wavenumber domains as a function of latitude.
Referring to
Fig 4.10, we see that adjustment is observed up to m = 11
for k = 0.
Here, we define observation of adjustment to
include the highest wavenumber exhibiting significant negative correlations for time scales such that k < 4, and within
the reliable resolution limit of the data.
to a physical scale of about 3000 KM.
This corresponds
Comparing this with
the highest wavenumber for which adjustment is observed at
650 (see Appendix B) on the same time scale, we get m = 3.
This corresponds to the physical scale of about 5600 KM.
Doing the same for the middle two latitudes yields physical
scale "lower limits" of about 3800 KM ($ = 550) and 4000 KM
107
Table 5.1
Wavenumber Scales (KM.)*
450
550
650
32780
28300
22980
16930
16390
14150
11490
8465
10927
9433
7660
5643
8195
7075
5745
4232
6556
5660
4596
3386
5463
4717
3830
2822
4683
4043
3283
2419
4098
3537
2872
2116
3642
3144
2553
1881
10
3278
2830
2298
1693
11
2980
2573
2089
1539
350
$=
(b = 35 0
* Computed using
a = 6.371 x 106m
2ff = 6.28318
108
($ = 45 0).
A reasonable conclusion is that the zonal scale
of the eddies has a lower size limit at around 4000 KM, for
time scale k = 0 (1 day fluctuations).
This result is con-
sistent with baroclinic theory.
As the time scale increases, so does the "lower limit"
on the eddy size observed to baroclinically adjust.
This is
exemplified very nicely at 550 where for k = 0, m = 6 represented the smallest adjusting eddy, whereas for k = 3, m = 1
is the smallest adjusting eddy.
This transition is very
smooth (refer to Appendix B) for $ = 550 and less smooth but
still quite noticeable at all other latitudes.
In light of this we can offer an explanation for the
retrogression observed in the preferred time scale.
As we
move north, the time scale of an eddy at any given wavenumber
decreases because the eddy size at that wavenumber decreases
towards its lower physical size limit.
At any one latitude,
the time scale of an eddy decreases as zonal wavenumber increases because the physical size decreases with increasing
wavenumber.
Two limitations on this process exist.
First,
the eddies cannot be smaller than their characteristic length
scale.
In (2.51 that scale is HN/f, where H is the scale
height, N the Brdint-Vaisdld frequency and f the coriolis parameter.
Hence, in fig. 4.12.24,
the physical size may be
3300 KM) too small and no baroclinic
(m = 5, $ = 650 or
-
eddy is observed.
Secondly, their time scale cannot greatly
exceed k = 3 (k = 4 is about maximum) because other physical
processes (e.g. stationary eddies) take over the burden of
109
the heat transport.
We can think of this retrogression in a different way.
An eddy of small size is rapidly mixed into eddies of larger
sizes --
hence to observe their transport we must search on
short time scales.
However, a large eddy retains its singu-
lar features longer.
The largest baroclinic eddies thus
exhibit the longest time scales--up to the limit of about
k = 4 (16 days).
We are satisfied that we have learned a great deal about
the zonal scales on which adjustment takes place, and realize
that this is due in part to the unique spherical geometry of
the earth.
In conclusion of this section, Figure 5.1 dis-
plays a parametric plot of the preferred eddy time scale in
baroclinic adjustment, revealing its retrogressive behavior.
5.3
Validity of the Eddy Flux Parameterization:
Eddy Flux Predictors
We can make some general conjectures about the validity
of (2.81 in modelling forced variations, which is what it
is designed to do.
Lorenz (1979) correlated (essentially)
the eddy transport with the temperature gradient only, whereas our correlations involved an eddy diffusivity which is
supposed to correct for the effect of the free variations
by weighting the forced variations.
By comparing his corre-
lations with ours we will get a gross measure of the effect
of the X parameter on the eddy flux parameterization.
Before doing this we should make clear that the two
110
Parametric plot of preferred
Eddy time scale , function of
latitude and wavenumber
8.
M
Z
76 -5-
IOX-
0
4-I
I
0
retrogression
z
0
3
I
I
2
2-
3
2
2
2
3
3
3
3
I
I
I
350
I
450
550
650
LATITUDE
Fig. 5.1 Parametric Plot of
preferred eddy time scale in
Baroclinic Adjustment*
*k is plotted, time scale is 2
days
111
studies are not perfectly compatible.
First, Lorenz corre-
lated the two quantities by meridional spectral analysis
making use of the Legendre representations of the heat transport and temperature gradient.
meridional planetary scale (n
This effectively isolated the
2, where n- is the degree of the Legendre
polynomial) which proved to be the scale at which forced variations dominated.
Our study correlated the quantities
directly within latitude bands, without spectral meridional
analysis.
Stone and Miller (1979) point out that the charac-
teri.stic meridional scale of the fluxes is basically the
planetary scale when the fluxes are zonally and monthly averaged.
Hence, we can be certain that the basic scale involved
in our data for m = 0 and k = 7 or 8 (greater than Monthlong averages) is the planetary--but our data do not isolate
this scale from the others.
Hence, small contributions from
other scales would decrease our correlations relative to
Lorenz's.
Secondly, our data admit latitudinal dependences which
Noting that we can compare only the m = 0
Lorenz's does not.
results, for each time scale we have four correlation coefficients to Lorenz's one.
Table 5.2 presents the comparison,
where Lorenz's coefficients have been calculated from the
tables in his paper.
In his case, negative correlations
indicate forcing of the flux by the gradient.
Lorenz's coefficients are markedly higher than ours for
both time scales.
We cannot assess whether this is because
112
Table 5.2
Correlation Coefficient Comparison:
Lorenz (1979) vs. McHenry
m = 0
Lorenz
McHenry
k = 8
350
.9834
450
.9653
550
.8935
.9131
650
-. 996
.1-i
k = 7
350
.7813
450
.6983
550
.5688
650
.0596
-. 893
113
his parameterization is better than ours, though, because of
the above mentioned reason.
What we had hoped to show--that
ours is better than his--we cannot do either.
We should conclude by mentioning that we set out to
test a parameterization which both took account of the a-effect
and contained some measure of the zonal scale in its eddy
diffusivity.
In fact, (2.8) neglects this horizontal scale.
We have shown, though, that (2.8) models forced variations
well, but cannot determine the impact of X on its ability to
model those variations.
In addition, we have presented a graphical analysis
which displays both adjustment and seasonal forcing succintly.
We have documented the retrogressive behavior of the characteristic adjustment time scale with the physical eddy size.
Finally, we believe the inclusion of X in (2.8) enabled us
to better detect the overall behavior of the free regime
eddies.
.a
114
CHAPTER SIX
SUGGESTIONS FOR CONTINUING RESEARCH
In this chapter we will outline several ideas for research related to our study.
In the first two sections we
will discuss proposals directly related to this research,
and in the last section we'll talk about general uses of the
program SPECTRA.
6.1
Regional Analysis
Instead of performing correlations in wavenumber space,
they could be performed by defining a regional eddy flux and
examining specific geographic regions of varying size.
The
primary advantage of this would lie in better understanding
of free variations, because forced variations occur on only
the two or three largest longitudinal scales (m = 0,1,2).
In this analysis the eddy fluxes are defined regionally
and correlations are performed only over the regions of definition.
This procedure is undertaken because it is logical
to assume that eddy transport (primarily due to transient
eddies, in this case) in a region on one side of the globe
is not a result of instabilities on the other side, and
therefore does not adjust the gradient on the other side.
So
long as the regions are greater in extent than the appropriate length scale of the transport, illuminating results
should be expected.
115
In order to begin, we define
<( )>
=
X 2 ~ Xl
co1
a cose
( ) dA
(6.1)
as an average over the part of a latitude circle spanned by
2
longitudes A,
The regions are thus of extent a.cos
(A2 ~A1 ) in the zonal direction.
= T -
T
X
V
4
We now define:
<T>
(6.2)
=
V
-
<V>
where the ( )X denotes the departure of the field from its
average < >.
Assuming a non-divergent geostrophic solution,
as before, we find that <V> / 0, and the eddy flux for the
region, call it EFR, now applies only to the region between
A1 and A2'
To perform the correlations, we might average EFR over
the region, getting <EFR> as a function of time.
We could
compute and average X over the region, and define AT to span
its
meridional extent.
series to be correlated.
Each region would yield two time
If we chose A2
~
1 =
, we could
mweol
1
2
perform correlations over regions of identical size, ob-
taining one correlation table for each set of regions of the
same size studied.
By choosing A2 - A -m 2
1
coulddaalso
we
o
V-weo
directly compare results with the present study for the
free variations of short period.
116
In defining a regional flux, we must account for the
fact that the total eddy flux,
{
V*T*dX, is not conserved
0
when the definitions (6.2) are used.
A simple expression is
now derived which relates the total eddy fluxes as defined
each way.
As before, let
<()>
1
A2 ~
a cos $
fly
( )X = departure from regional average
(6.3)
[ ] = zonal average
( )* = departure from zonal average
The total fluxes are
<VT> = <V><T> + <VXTX>
(region)
(6.4)
[VT] =
[VI [T] +
[V*T*]
(zonal)
(6.5)
[VT] =
[<VT>]
[<V><T>]
and
Now,
=
=
[<V>][<T>]
+ [<VXTX >]
+ [<V>*<V>*] + [<VX T >1
(6.6)
Combining the expressions (6.5) and (6.6) we get
[VT]
=
[VI [T] + [<V>*<T>*]
+ [<VX T X>]
(6.7)
Again combining, this time (6.7) and (6.5) we arrive at the
result desired:
117
(6.5)
[V*T*] = [<V>*<T>*] + [V XT]
We have made use of the fact that [< >1
=
[ ].
As it stands
(6.8) relates the eddy fluxes as defined both ways, integrated around a latitude circle.
The difference term
[<V>*<T>*], is the eddy flux resulting from the departure
of the regional averages from the zonal averages.
We sus-
pect this term involves zonal planetary scale waves, and
hence would reveal forced variations if studied over seasonal
time scales.
6.2
Multiple Correlation Studies
Because our research did not reveal the affect of the
inclusion of an eddy diffusivity on the parameterization
(2.8), we feel that a multiple correlation study would be
of value.
It would be a simple modification to produce
three time series:
eddy flux, X, and (AT)2 , analyze their
zonal Fourier components, and correlate them jointly and by
pairs.
We suspect the results would indicate that X en-
hances the negative correlations on short time scales, but
has only a small effect on seasonal scales since on average
X is close to unity.
This would confirm the suspected in-
ability of X by itself to model short period free variations,
and would point to more complex parameterizations for improved results.
118
6.3
General Uses of the Program SPECTRA
SPECTRA is a very versatile computer program which
can be used to temporally correlate the Fourier coefficients
of any two datasets.
A.
It is listed in FORTRAN IV in Appendix
The restrictions are such that the number of datapoints
in a set be in powers of two, and that the number of sets
in a time series be in powers of two.
It is generally easy
to interpolate, or truncate, meteorological data sets to fit
these requirements.
Instructions are provided in the comment cards on the
meaning of the control parameters--there are only three--and
on how to adjust the FORMAT statements to fit these parameters.
We have found the program to be highly efficient
and therefore inexpensive to use.
SPECTRA produces the variance, covariance and correlation coefficient spectra of the input data sets.
We can
think of many opportunities for its use which would involve
only adjustment of the input data sets themselves.
For in-
stance, little is known about the cross-correlation spectrum
of the eddy heat flux itself.
By inputing V*(tl and T*tt),
this spectrum would be produced in the output table entitled
COMPONENTS IN THE ANALYSIS OF COVARIANCE.
119
APPENDIX A
COMPUTER PROGRAM SPECTRA:
LISTING
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APPENDIX B
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