MODELLING THE SYNOPTIC SCALE RELATIONSHIP
BETWEEN EDDY HEAT FLUX AND
THE MERIDIONAL TEMPERATURE GRADIENT
by
STEVEN JOHN GHAN
B.S., University of Washington
(1979)
SUBMITTED TO THE DEPARTMENT OF
METEOROLOGY AND PHYSICAL OCEANOGRAPHY
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE
DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1981
Q
Massachusetts Institute of Technology 1981
Signature of Author.......... ............................
Department of Meteorology and Physical Oceanography
7
May 8,1981
Certified by............
.....
.. ..........
........ .....
Peter H. Stone
Thesis Supervisor
Accepted by.......................
..........
..............
Ravond T. Pierrehumbert
on Graduate Students
Chairman, Depar
T
NARITUTE
mpRARIES
2
MODELLING THE SYNOPTIC SCALE RELATIONSHIP
BETWEEN EDDY HEAT FLUX AND
THE MERIDIONAL TEMPERATURE GRADIENT
by
STEVEN JOHN GHAN
Submitted to the Department of Meteorology and Physical
Oceanography on May 8, 1981 in partial fulfillment of the
requirements for the degree of Master of Science.
ABSTRACT
A simple statistical-dynamical model relating synoptic
scale changes in the meridional
temperature gradient to
changes in the meridional eddy sensible heat flux is
developed. The model solution is compared with observations
relating the flux to the stability parameter for the two
layer model which, assuming the variance of the critical
shear in the two layer model is negligible, is equivalent to
the meridional temperature gradient. The comparison suggests
that a diabatic time scale of about one day is appropriate
for perturbations due to changes in the flux, and that
roughly one half of the variance of the temperature gradient
can be ascribed to processes with time scales much less than
the synoptic time scale. The possibility that variations in
the critical shear are important is discussed.
Feedback of the temperature gradient on the flux is
added to the model. Three model parameters emerge which, if
properly tuned, could yield significantly better results
than the model without feedback. Although this provides
supporting evidence for the presence of feedback, other more
justifiable mechanisms yield similar results.
Thesis Supervisor:
Peter Hunter Stone
Title:
Professor of Meteorology
TABLE OF CONTENTS
page
I
INTRODUCTION
4
II
OBSERVATIONS
6
III
MODEL EQUATIONS
12
IV
BASIC MODEL SOLUTION
27
V
BASIC MODEL RESULTS
30
VI
FEEDBACK MODEL
44
VII
CONCLUSIONS
53
APPENDIX
55
ACKNOWLEDGEMENTS
62
REFERENCES
63
I
INTRODUCTION
studies
Modelling
meridional
the
and
flux
heat
sensible
eddy
meridional
their
temperature gradient have in the past concentrated on
time
relationship.
mean
long as
scale
time
the
the
between
relationship
the
of
Ensemble averaging of at least as
eddies
transporting
the
of
is
implicit in mixing length parameterizations of the heat flux
in
terms
the temperature gradient (Green, 1970; Stone,
of
1972). An examination of the time dependent (synoptic scale)
relationship between
gradient
also
is
the
Such
warranted.
temperature
the
and
flux
heat
study should reveal
a
something about the processes which maintain the
relationship.
Recently
the
denoted by S) have shown how
governing
of
order
equations
the
the behavior of the heat flux and the temperature
gradient can be
autocorrelation
estimated
by
functions,
examining
their
their
i.e.,
respective
time
behavior. This information is then used to test the
of
finite
mean
hereafter
(1981,
al.,
et
Stone
time
dependent
results
amplitude calculations of baroclinic instability
(Pedlosky, 1979).
Modelling the observed correlation functions
be
can
also
fruitful. In particular, modelling the cross correlation
function for the heat flux and the temperature gradient
can
reveal specific details about the processes relating the two
variables.
Model
parameters
can
be
tuned
to
match the
observations, thus allowing one to measure both the relative
importance of different processes and their respective
time
5
scales.
Although
the physics involved in such modelling is
general
included in
circulation
models,
attempt
no
been made to explicitly model the synoptic scale
previously
relationship between the meridional eddy sensible heat
and
the
has
meridional
flux
gradient as seen in their
temperature
cross correlation function.
In this paper simple linearized models of the
relationship
between
the
heat
flux
the temperature
and
gradient are developed. The emphasis is on
temporal
reproducing
the
observed auto and cross correlation functions, although some
attention
is
devoted
to modelling variance. In section II
model
the observed correlation functions are described. The
equations
are developed in section III, while in section IV
the basic model solution is presented. These
discussed
in
solutions
are
section V. The possibility of feedback of the
temperature gradient on the flux is
considered
VI. Conclusions are presented in section VII.
in
section
II OBSERVATIONS
The data
used
in
this
study
is
that
used
in
S,
consisting of twice daily observations for three consecutive
Januaries
(1973, 1974, 1975) of the total tropospheric mean
eddy sensible heat flux across selected midlatitude circles
Zal'01
C0.5
J (o
SPd
Po
IV
dp
T
and the stability parameter for the two layer model
(,t)
11.2
=
[
L
2
where a is the earth's radius, cp is the
specific
heat
constant pressure, g is the gravitational acceleration, 0
the
latitude,
po
and
p. are
pressures,
respectively,
meridional
velocities,
u
the
and
surface
v
respectively,
are
at
is
and tropopause
the
zonal
and
T is the temperature
and
-
(e,
U
11.3
)
is the critical shear for the two layer model,
the coriolis parameter, 1
where
f
is
is the meridional gradient of f, R
is the ideal gas constant for dry air and 8 is the potential
temperature.
Square
brackets
denote
the
zonal
mean,
asterisks deviations from the zonal mean. The subscripts one
and
two
denote
tropospheric
related
to
the
mean,
the
mass
weighted
respectively.
meridional
upper
The
temperature
and
shear
gradient
thermal wind relation for the two layer model
'lower
u,by
u2
is
the
the
z,-
thickness
Because per cent variations of
pressure.
mid- tropospheric
the
of
those
than
less
are
z.
the
pM is
and
height
geopotential
the
is
z
where
temperature gradient we will often refer to the shear as the
the critical shear as the static
and
gradient
temperature
stability, with the appropriate scaling factors implied.
for
stability parameter, averaging the
the
and
flux
the
functions
correlation
S calculated the auto and cross
latitudes
respective functions over the three Januaries and
46N, 50N and 54N. For the cross correlation function (figure
1) lag is defined such that the stability parameter lags the
lags.
positive
for
flux
most notable feature is the
The
significant correlation for lags of zero, one half
and
A cubic spline fit to the cross correlations indicates
day.
the strongest negative correlation occurs at a lag of
one
one
Note
day.
half
also
lag is the cross
no
for
that
about
correlation significantly positive (the 95% confidence level
correlation is 0.22).
The auto
correlation
functions
for
the
flux and the stability parameter are shown in figures 2
and
3,
respectively.
shown
Also
are
auto
correlation
functions corresponding to first order Markov processes (red
noise)
with
respectively.
correlation
time
scales
As
discussed
function
for
one and one and a half days,
of
by
S,
although
the
auto
the flux is best fit by a second
order Markov process, a first order process is still a
good
fit.
We may consider variations in the stability parameter
0
I,00
.50
-2
3
LAG (DAYS)
Figure 1.
and
Cross correlation averaged over
latitudes 46N, 50N and 54N between the flux
the stability parameter j
lag
data
three
(in
days).
points.
The
(two-sided) is 0.22.
(t +
-)
as a
Januaries
c-(t)
function
of
and
the
Dashed line is a cubic spline fit to the
95%
confidence
level
correlation
z
Iw
0
4
LAG
Figure 2.
and
(DAYS)
Auto correlation averaged
over
three
Januaries
latitudes 46N, 50N and 54N of the flux as a function of
the lag
i . Dashed line is the auto correlation for
noise with a time scale of one day.
a
red
10
1
o
-
,
0
0
4
LAG
Figure 3.
(DAYS)
Auto correlation averaged
over
three
Januaries
and latitudes 46N, 50N and 54N of the stability parameter as
a
function
of
the
lag
t.
Dashed
line
is
the
correlation for a red noise with a time scale of one
half days.
auto
and
a
provided
gradient
the critical shear is
of
variance
the
temperature
meridional
the
in
variations
to
due
to be
According
negligible compared to the variance of the shear.
about
to S, the root variance of the stability parameter is
-I
m s ,
1.7
comparable
to the mean value. Since-'1the January
mean shear in midlatitudes (50N) is about 10 m s , if all of
in
changes
to
the variance of the stability parameter was due
the shear the rms deviation of the shear, or temperature
gradient, would be no more than
20%
of
mean
January
the
value. Since the January mean critical shear in midlatitudes
is
stability
m s
10
about
also
rms
an
deviation
of
January
mean
of only 20% of the
&,-E,
the static
value
is
enough for variations in the critical shear to be important.
Such synoptic variations are certainly conceivable, but have
never
been
studied.
Although
temporal relationship between the
study
modelling
a
flux
and
the
of the
stability
parameter would be illuminating in its own right and was the
original
assume
motivation
that
negligible,
the
so
for
variance
that
this
of
study, we shall tentatively
the
critical
shear
is
variations in the stability parameter
are due to variations in the temperature gradient. The above
observations are then relevant to the problem at
we proceed with the modelling.
hand,
and
III MODEL EQUATIONS
the
on
relationship
This study examines the temporal
synoptic time scale of two variables: the zonal and vertical
mean
gradient.
dependent
time
two
least
at
Consequently,
flux and temperature
heat
sensible
eddy
meridional
equations are required to model such a relationship.
in the temperature gradient to changes in
changes
relating
is
2)
limitations of the model,
complete
to 1) point out some of the
here
presented
a
1979),
1972; Lorenz,
the flux is not new (Ston e,
derivation
(111.8)
equation
model
the
of
form
the
Although
first
provide
estimates
of
parameters and 3) suggest which approximations should
model
be modified to yield more realistic results.
the
Assuming only that
ideal
in
gas
is
atmosphere
a
hydrostatic
thin shell, its kinetic energy negligible
a
compared with its static energy, the zonal mean equation
energy
be
may
conservation
of
in the form (Hantel,
written
1976)
III.1
j[cpT 4
specific
energy
humidity,
and
Decomposing
F,
±
heat
the
is
the
=
h
fields
[v
0
latent
the
where L is
Lc]
of
C4
LJ
+I
vaporization,
q
is
the
cpT + gz + Lq is the moist static
net
downward
into
their
components yields
111.2
cosd
C4
radiative
zonal
mean
flux.
and eddy
energy
To be useful a model equation derived from this
equation must include both the meridional eddy sensible heat
neglected.
be
may
all
while
larger than these must be retained,
smaller
All terms
time change of the sensible heat.
the
and
flux
of Hantel (1976)
results
The
much
terms
indicate the dominant energy balance in the atmosphere to be
vertical
between radiative flux divergence and
moist
eddy
energy flux convergence. The remaining difference in
static
meridional
midlatitudes is balanced largely by
convergence
flux
moist
eddy
according to Oort
which,
static
energy
(1971),
is dominated in winter by meridional
sensible
eddy
divergence
and
vertical eddy moist static energy flux convergence must
the
heat
be
retained,
while meridional eddy latent heat (in winter)
and geopotential
(all
In
neglected.
flux
Radiative
convergence.
flux
seasons)
by
assuming
the
effects
its
latent
may
heat
be
flux
latent
eddy
meridional
summer
convergence may not be neglected, but
incorporated
convergence
flux
can
be
to
be
flux
proportional to the meridional eddy sensible heat flux.
Although
meridional
advection
of
static
moist
energy
when
(Newell
considering
consequence
the
circulation is generally smaller in midlatitudes
than meridional eddy heat flux convergence, it
negligible
by
of
et al.,
temporal
thermal
1974).
not
be
This is especially so
because,
variations
wind
may
balance,
the
as
a
meridional
circulation in midlatitudes is forced by friction, diabatics
and eddy fluxes of heat and
momentun,
all
of
which
have
temporal
significant
variance.
meridibnal circulation forced by
forcing
by
friction
the
Therefore,
and
diabatics
eddy
heat
incorporated within the model, the effects of
be
can
flux
Although the effect of the
eddy
and
of
effects
momentum
flux
not.
can
the meridional circulation are
tentatively neglected, but are discussed further in
section
V.
Neglecting spherical effects, the energy
equation
may
now be written
111.3
fq.[c kK+
+-
c+
= 0.
E 11F,
Integrating from the top of the surface layer to the top
of
the atmosphere yields
o
4-LCF,Co - ( cojI
>
(
where
c~.
1
Temporal changes in the vertical mean
neglected
latent
heat
may
be
the water precipitates immediately upon
provided
convection from the surface. This is never strictly true
course; a certain time lag is involved.
of
If this lag is much
shorter than the synoptic time scale, the lag is negligible.
Since
convective
clouds typically develop over time scales
of a few hours, such
precipitation
within the
is
an
approximation
is
valid
provided
convective in origin and moist convection
troposphere
begins
immediately
after
surface
convection. According to the GFDL climate model (Miyakoda et
approximately
1969),
al.,
of
half
precipitation is
the
subgrid scale. Unfortunately, since precipitation is such an
grid
important part of the moist static energy balance, the
and its associated synoptic time scale
precipitation
scale
can not
arbitrary
included
the
in
later
can
the dry static
release
not.
energy
modelling),
scale
synoptic
the
Although this suggests we consider
budget
to
with
deal
heat
latent
explicitly, we shall continue with the moist static
energy budget because it affords us an a priori estimate
the
diabatic
time
scale.
Therefore,
precipitation is convective, and that a
of
an
can be treated as a white noise (and will be
lag
precipitation
of
convection
moist
Whereas
neglected.
be
we
assume
of
all
amount
significant
it follows immediately after convection from the surface
(we can relax this constraint if we integrate only from
the
top of the mixed layer rather than from the surface layer).
The remaining assumptions are all directed at
the
diabatics,
i.e.,
relating
the surface convective fluxes and the
radiative flux divergence, to the vertical mean temperature.
Newtonian cooling after Spiegel (1957) models the
radiative
flux divergence:
III.5
where ?-,is
radiative
EL
the
F, P.) - F, (o)]----7)
radiative
equilibrium
cooling
temperature
time
and
consistent
Tr is
with
observed surface temperature. Appropriate values for ?
the
the
will
be discussed later in this section. Simple drag
laws
model
the convective fluxes of sensible and latent heat at the top
of the surface layer:
I
I
- I
-,
111.6
where 2
scale
lis the convective time scale, H =
=
height,
cD is
the
subscripts
g
and
of
the
surface
longer
than
is
implicit
in
the
qs(T ).
surface
specific
humidity
that
the
The
ensemble
synoptic
the
at
ground
the
is
ground
just
time
the
temperature
Assuming the relative humidity r at the top
of
the
layer is constant and neglecting spatial variations
in the radiative cooling and
energy
layer.
mixing length expressions. The
appropriate specific humidity at
saturation
is a
the time scale of the eddies within
the surface layer but much shorter than
scale
the
o denote values at the ground and at the
top of the surface layer, respectively. Note
averaging
1u.1
drag coefficient and
typical wind speed at the top
is
equation
convective
differentiated
with
time
scales,
respect
to
the
y = a
becomes
<to)
111.7
o]
#<TrT>
The only remaining problem is to relate the temperature
at the top
temperature.
of
the
As
surface
a
first
layer
to
the
approximation,
vertical
mean
we shall assume
independent
the
about
perturbations
height,
of
time
so
mean
that
the required relation is
be
can
straightforward. In fact, temperature perturbations
Since the model results are quite sensitive
shallow.
quite
be
to
temperature
perturbations
to this condition, the possibility of shallow
is discussed in more detail in section V.
With these approximations the energy
reduces
equation
to
a a ,:JN
111.8
-:7<V 'V-r f
T
is
S= _
where
ZTe> is
time
diabatic
the
scale.
equilibrium
radiative-convective
of
sort
a
Te - 7 >
temperature consistent with the observed ground temoperature
and assumed
unforced
i.e.,
scales,
time
Te)
seasonal cycle in
can
(note
constant
here
that
short
we
enough
consider
that
so
is not resolved). Such an
the
assumption
be justified for the synoptic time scale over the ocean
as follows. The heat capacity of an infinite column
air
only
is
Te over
equivalent
the
temperature
ocean
of
dry
to that of a two meter column of water.
can
be
considered
constant
if
the
of this layer is constant for the time scale of
interest, i.e., one day. This is true if the time
scale
in
which this layer mixes with a much deeper layer is much less
than one day. According to Ekman layer theory, the dynamical
time
scale
of
the
entire
mixed
layer (N~100 meters) in
midlatitudes is about one day, so that the mixing time of
ten
meter
a
layer (much deeper than two meters) is one tenth
T e may
negligible.
of a day, clearly
then
be
considered
constant over the ocean for the synoptic time scale. Because
generally has a much smaller heat capacity than oceans
land
the
for
scale,
time
synoptic
be
T. cannot
considered
constant over land. However, the oceans comprise the greater
part of the zonal mean surface, so that Te
is
in the zonal mean
approximately constant. The actual value of
important in the time dependent calculations; it
TE) is not
only
need
be consistent with the time mean flux and temperature.
negative
additional
one
only
With
assumption
observed
the
at zero lag between the eddy sensible
correlation
from
heat flux and the temperature gradient may be' derived
the
in the same manner as Lorenz (1979).
equation,
energy
time
This assumption is that the flux is well correlated in
with
the
laplacian
of
flux.
the
This is not altogether
obvious because variations of the flux occur in general on a
If the temporal variance of the
spectrum of spatial scales.
flux as a function of meridional scale decreased only slowly
with
decreasing
laplacian
of
meridional
the
flux,
scale,
would
flux and
different
the
which accentuates smaller scales,
would peak at a scale well removed from that
One
of
variance
the
of
flux.
the
then expect significant correlations between the
the
flux
scales
laplacian
only
if
fluxes
of
widely
were well correlated in time, an unlikely
proposition. In fact, when we correlate the
flux
with
the
finite difference equivalent of the flux laplacian using the
data
of S, we find significant negative correlations at all
the
the
in
were
data
latitude,
that
latitude is heavily weighted by the flux at
if
at a given
laplacian
flux
the
of
equivalent
difference
finite
the
since
One can argue that
4 ).
latitudes (figure
level one should expect
noise
significant, non-physical correlations. However, calculation
the
of
deviation
standard
variance
the
that
5) indicates
unless
tha t,
are
scales
is
yield
to
sufficient
correlations. One must then come to
significant
the conclusion
meridional
difference
finite
the
of the flux laplacian from the same data (figure
equivalent
physically
of
fluxes
of
different
widely
(a truly
correlated
significantly
remarkable result), the variance of the flux is confined
a
narrow
band
meridional
of
meridional
,
scale
so
We will henceforth
scales.
assume that the var iance of the flux
to
dominated
is
by
one
that the flux correlates perfectly
with the laplacian of the flux.
The perturbati on flux laplacian then equals a
constant
the
times
perturbation
The
flux.
negative
resulting
perturbation energy equation becomes
where primes denote deviations from the time mean. Note that
the constant of
flux
proportionality
between
perturbation
and the perturbation flux laplacian is not necessarily
the same as that corresponding to the
time
proportionality
for
the
the
constant
of
mean.
the
Whereas
time
mean
corresponds to the planetary scale (Stone, 1978), the proper
0
-
I
I
I
I
i
a
z
O
.J
0
-1
I
3'"
8
a
- -
* -- 50.r
50
LATITUDE
62
Figure 4.
Correlation averaged over three Januaries between
the flux
S
of
g
&
the
(
= 80
(
,t) and the finite
(d
flux laplacian
+a
-t
,t) as a function of
latitude.
(two-sided) is 0.22.
The
95%
difference
,t) - 2
the
confidence
equivalent
9(
latitude
d ,t)
c
,
+
for
level correlation
21
12
O
O
w
I-
0
0
I
I
I
I
I
62
50
LATITUDE
38
Figure 5. Standard deviation averaged over
three
Januaries
laplacian as
of the finite difference equivalent of the -, flux
0
a
function
of
latitude.
Units
are
m s
C,
assuming
tropospheric depth of (generously) 1000 mb. The noise
is approximately four m s
C.
a
level
22
known.
value for the constant for perturbations is not well
Although
theoretical studies have considered the meridional
Simmons,
1974;
scale of these perturbations (Stone,
1974;
the value for D used in our modelling will
Pedlosky,1975a),
be empirically determined.
equation
Multiplying
perturbation
the
by
111.9
temperature gradient and averaging in time yields
D
III.10
(V
7 V>
>
49T>'
o
of
magnitude
the
the
of
behavior
gradient must be negatively
temperature
whatever
correlated, independent of
the
and
where overbars denote the time mean. The heat flux
the
governs
equation
Note that the negative correlation
flux.
depends on the presence of diabatics.
According to S, the flux can be
Markov
order
process.
Although
the
flux
a
as
modelled
first
is best fit in
winter by a second order process, a first order process,
red
or
noise, is also a good fit. Figure 2 shows that the auto
resembles
correlation function for the flux in winter
that
of a red noise with a time scale of about one day.
Theoretical justification for the red noise
derives
from
Pedlosky (1979),
hypothesis
in which he considers finite
amplitude dynamics of a weakly unstable baroclinic wave in a
atmosphere
continuous
friction
at
the
on
surface
a 3 -plane,
and
internal
with
both
damping. Pedlosky
derives an equation for the wave amplitude of the form
III.11
d
Ekman
V is the growth rate from linear theory and A.-is
where
the
equilibrium amplitude. Since the flux is second order in the
wave amplitude, the flux is governed by
Fe
2.
111.12
< vMT
F
F = F,+
F'. yields
V-
III.13
/
According to S, the rms deviation of the flux in
of
35%
about
mean
the
value.
winter
heat
flux
scale
is
Linearization of the flux about the mean
small.
is then justified in winter but perhaps not in summer.
time
is
One expects this value to
increase in the summer, when stationary eddy
relatively
mean
the
about
Linearization
>.
where
The
of the flux can be identified with one half the
inverse growth rate which, according to Eady's model, yields
a value in midlatitudes in winter of about a day and a half.
This is in approximate
agreement
with
the
observed
time
scale of about one day. One expects the time scale in summer
to be larger because of the weaker temperature gradient.
Adding white noise forcing to both the equation for the
for
flux and the equation
the
temperature
gradient,
the
flux,
and
model equations may be written
dF'
11.14
d
where G=jCT- , 2V
E;
GT
G 4
weF
i5
is
-a
the time scale of the
and Eg are white noises.
The source of the white noise
forcing
of
the temperature gradient was discussed earlier;
the white noise forcing of the flux might be due to resonant
triad interactions of baroclinic waves (Loesch, 1974).
Introducing the non-dimensional quantities
-'-
F'/Fe
=
111.15
yields the non-dimensional model equations
III.16
where
111.17
the
Although
of
S
is
not.
In
parameterization of
1972), 9 is
of
interpretation
case
the
the
equivalent
flux
to
of
from
L
r
obvious,
that
the
mixing
length
Eady's
model
(Stone,
an order one constant times the
ratio squared of the deformation radius
scale of the flux.
" is
to
the
meridional
The scale of the flux may be found from
L F!
-
25
ratio
computing the
rms
the
of
-IT
to
found
is
D
deviation
flux
the
of
to the rms deviation of the flux. For the data of
laplacian
S,
weighted
variance
flux. The value of D is found empirically by
the
of
scale
the
of
where L4 is the half-wavelength
-Z
corresponding
The
m .
1.8X10
be
half-wavelength is about 2300 km.
must
To calculate the diabatic time scale we
the
and
convective
scales.
time
radiative
estimate
Hicks (1972)
-3
c,
for
provides an average value
about
of
but
1.4x10
-
, depending on the
to 4Y10
finds values ranging from 4x10
stability of the surface layer. Typical 10 meter wind speeds
are
5 m s , so that a first estimate of the convective time
scale is about ten days. The
time
radiative
scale
varies
depending upon the height and vertical scale of the
widely,
temperature perturbations, and the
nature
of
the
surface
Prinn (1977) calculated values ranging from one half
below.
day for shallow perturbations immediately above a conducting
surface
from
perturbations
largest value,
value of
January
have
assumed
temperature
to be independent of height we shall take the
one
time
radiative
we
Since
surface.
the
for deep perturbations well removed
month
one
to
For
scale.
is
r-
as
month,
0.7
a
first
estimate
of
the
a relative humidity of 90% the
at
0 C,
appropriate
for
the
mean. The corresponding diabatic time scale is then
5 days.
Although we could use the results from Eady's model
find
9
,
there
really
to
is no point in doing so since the
value of the meridional
scale
is
empirically
determined.
Therefore we use the observed values for January of
-,
0
F = 20 m s C
G = 4Y10
C m
t-,= 1 day
to yield
= 0.2
S= 0.8.
With these first estimates of the model parameters we
proceed to solve for the correlation functions.
shall
27
IV
BASIC MODEL SOLUTION
Although one could just
easily
as
finite
the
solve
difference equivalent of the model equations, we present the
of
solution
system.
continuous
the
This
reasonable
is
small
provided the time scale of the white noise forcing is
but finite. The model equations are again (dropping primes)
d-f -
IV.1
+
6
e
-+-
IV.2
The auto covariance function for the flux is
multiplying equation IV.1 evaluated at time t +
'"
found
by
(~)>0)by
the flux at time t and averaging in time, yielding
or
IV.3
where
To
find
evaluated
the
cross
at time t
correlation,
+
Z
multiply
equation
IV.2
(anyZ ) by the flux at time t and
average, yielding
Substituting
(f,f,-
) from equation IV.3 into the
solution
~~C
3
'a~-j £cS
J
general
28
yields
et
(-,o
ce Matching solutions at
=
tfF
0 yields
Matching solutions at"Z:= 0 yields
*C4
IV.4
O)
T
The auto covariance function for the temperature gradient is
found by multiplying equation IV.2 evaluated at time
t + 1-
by the temperature gradient at time t and averaging,
(1 ,O)
yielding
Substituting
-(g,f,"Z) from equation IV.4 into the
general
solution
,
c
-
e
(,-&>i-eJI
yields
IV.5
To close the problem we
need
an
expression
relating
the
variance of the temperature gradient to that of the flux.
In the
temperature
special
gradient
case
6
=
0
the
variance
of
the
may be related to the variance of the
29
flux
IV.2
equation
multiplying
by
the
by
temperature
gradient and averaging, yielding
IV. 6
n0
) -or
fnctons)ten becom
a
The auto and cross correlation functions then become
- eIV. 7
~ic
IV.8
(If
(
Zf= /
\e
irI-
YfsI
IV. 9
-r,51 a)
'0
where
P
T2
KY
) --
(
T/x
,
/
2)
e2t
I
1(,
0),o
,,
Note that these solutions depend only on
.
jZ2Q
V
BASIC MODEL RESULTS
Solutions of the
temperature
auto
correlation
function
for
the
gradient and the cross correlation function for
the flux and the temperature gradient are shown in figures 6
and 7, respectively, for different values of
non-dimensional
the
flux,
time
which
non-dimensional
is
lag
has
one
be
day,
thought
values
of
i
given
differences,
of
the
functions
the
first
reveals
estimate
0.2. The modelled temperature gradient is much
than
persistent
maximum
is
negative
temperature
observed,
correlation
gradient
is
the
of in terms of days.
Comparison with the observed correlation
significant
Since
been scaled by the time scale of
about
may
.
much
while
between
the
the
more
modelled lag of
flux
and
the
later than observed. Either
some other process must be included in the model
or
larger
values of X must be justified.
We therefore include the white
temperature
variance
gradient
in
the
noise
model.
forcing
of
the
This adds additional
(g,g,O) to the temperature gradient which depends
.
on both the magnitude and time scale of
the
forcing.
The
total variance of the temperature gradient is then
V.1
where
_
a ~3
-
is the ratio of the variance of the
temperature gradient due to direct white
the
noise
forcing
variance due to forcing by the flux. Although
to
the flux
auto covariance function and the cross covariance function
z
O
_j
-J
w
00
CFigure
6.
0
Figure
6.
3
LAG
Model
auto
correlation
for
the
temperature
gradient as a function of the non-dimensional lag for
and different values of
( .
'
=-0
z
O
I-
w
0.4
0.6
1.0
2.0
-1
0
-2
3
LAG
Figure 7.
temperature
lag for
Model cross correlation
gradient
as
for
the
flux
and
the
a function of the non-dimensional
= 0 and different values of
.
are independent of .4(g,g,O) and hence 9 , the structure
of
auto covariance function and the magnitude of the cross
the
of
the temperature gradient. As
of
the
function
corrlation
the
for
I increases, the magnitude
decreases,
function
correlation
cross
forcing
noise
covariance function will be altered by white
auto
and
temperature gradient becomes
more like that of a red noise. In the limit
'-.
the cross
correlation function becomes zero while the auto correlation
function for the temperature gradient is that of a red noise
2t.
with time scale
function
Since
the
for the temperature gradient is similar to that of
red noise with a time scale of one
estimate
correlation
auto
observed
half
our
days,
diabatic time scale is clearly too large.
the
of
a
and
The diabatic time scale must be at least as small as one and
a half days and is probably smaller to allow for
reasonable
values of S .
Although other processes
shall
for
structure
of
determined by
correlation
moment
the
the
X,
can
cross
may
assume
still
are
they
correlation
be
important
not.
function
Since
is
we
the
fully
the observed lag of the strongest negative
be used to estimate
. From equation IV.9
we find that this lag is given by
V.2
This function is shown in figure 8. Since the
is one half day, we estimate that Y
observed
lag
is about one. Because
2
0
0
Figure 8.
between
of
9 .
Non-dimensional
2
lag
of
strongest
correlation
the flux and the temperature gradient as a function
s_~---~ri--X-l.-
~-I-IX--I----L--
.._il-rrrrrrnar~--+-
22 days
0
I
I
I
I
5
0
eb
Figure 9.
Dimensional lag of
(DAYS)
strongest
correlation
as
a
function of the time scale of the flux, for different values
of the diabatic time scale. Units are in days.
there is some flexibility concerning the proper
the
time
scale
different
of
the
~z, .
o,
dimensional
Figure
9
lag
shows
of
2o
time
scales
scale (
greater
strongest
as a function
?'bfor different values of the diabatic time
flux
for
of the flux we should check the effects of
b on
correlation
choice
scale.
For
than or order the diabatic time
0(1)) the lag of strongest correlation is nearly
independent
of
the
flux
time
scale
and
is
given
approximately by one half the diabatic time scale.
We can then be fairly confident that the diabatic
time
scale for perturbations in the atmosphere in midlatitudes in
winter
is
about one day. This is considerably shorter than
our first estimate of the diabatic time scale
We
(five
days).
consider two possible explanations. First, variations in
the
critical
shear
in
variations
the
may
not
shear.
be
By
compared
negligible
we
definition
expect
to
the
diabatic time scale for the critical shear to be at least as
small
as
one
half
that
appropriate
for
the
entire
atmospheric
depth. Furthermore, moist convection within the
troposphere
can
diabatics.
Although
be
an
extremely
source
efficient
we have no way of a priori estimating
the diabatic time scale associated with moist convection,
value
one
of
day
that
explanation
is
temperature
gradient
shallow.
of
is
a
certainly reasonable. The alternate
perturbations
associated
with
in
the
the
meridional
flux are quite
For perturbations which decay exponentially with a
scale height h the convective time scale is reduced
by
the
the
from
H/(h+H)
factor
scale
time
Oort & Rasmussen (1971)
from
flux
eddy
transient
homogeneity. Observations of the zonal mean
heat
vertical
assuming
show that the heat
flux in midlatitudes in winter decays exponentially from 850
Therefore,
mb with a scale height somewhat less than H.
convective
time
reasonable.
In
scale
time
perturbations is also shorter. Prinn (1977)
shallow
radiative
scale
time
day
a
about
of
seems
days
five
radiative
the
addition,
than
less
of
scale
a
and
a
for
found a
half
for
well removed from the surface with a vertical
perturbations
wavelength of 3 km (admittedly shallow).
The
corresponding
diabatic time scale becomes about one day.
With the value of
to
return
we
justified,
of about one well established
5
discussion of 3 .
our
and
Figure 10
shows the model auto correlation function of the temperature
Comparison
with
the
indicates that
S
independently
estimate
the observed vs.
the
Since
auto
observed
of order unity
is
1
the modelled cross
be
We
appropriate.
can
correlation
functions.
observed
K = 1.0
function,
we
about three, in reasonable agreement with
the previous estimate of
'
.
Such a value seems large
temperature
function
correlation
modelled cross correlation function for
to
1 .
by comparing the magnitudes of
"
is about twice the magnitude of the
expect
of
values
different
and
equal to one
"
gradient for
gradient,
for
suggesting
random
that
forcing
of
the
we should consider
variations in the critical shear as well as the shear.
Z
0
I
-J
0
0
Figure
10.
LAG
Model
auto
correlation
for
the
temperature
gradient as a function of the non-dimensional lag for Y = 1
and different values of "
.
39
the
in
organized
well
often
is
Random moist convection
but poorly organized over the meridional scales of
vertical
interest. However, it need not be well organized. Consider a
A e,
change
random
at
troposphere
in the vertical
upper
the
a given latitude, corresponding to a change
the
in
change
shear
the
maximum
The
of 8G1/2.
temperature
mean
which
over
L
scale
of
temperature
the
in
be
will
comparable to the change in the critical shear is given by
observed
the
to
comparable
about
is
which in midlatitudes
2400
Since
km.
is
this
scale of the flux we conclude
as
that random moist convection affects the shear
as
well
the critical shear for the meridional scales of interest.
auto
The model has reproduced the observed
and
cross
correlation functions for winter fairly well with reasonable
values.
parameter
We
can
some confidence make some
with
time
predictions for summer as well. We expect the diabatic
to be smaller in summer than in winter because latent
scale
heat convection is more efficient at the higher temperatures
The lag
associated with summer.
of
strongest
should then be smaller than it is in winter.
mean
weaker
temperature
Because of the
gradient in summer we also expect
the time scale of the flux to be larger than in
should
that
X
This
alone
stronger
in
correlation
winter,
so
be much larger than in winter (perhaps
2).
should
be
suggests
summer.
the
correlations
cross
However,
we
also
expect
'S
to be
_I~
.
____~~Il-i~~~LI~_
larger, so that the correlations may in fact be weaker.
and temperature gradient
flux
the
of
variances
relative
due to
forcing
S
(
flux
the
by
empirically established values for
gradient
temperature
flux.
non-dimensional
=
be
should
The
observed
variance of the flux was found to be
the
variance
variations
the
of
the
in
non-dimensio nal
of
the
non-dimensional
root
that
0.6
while,
0.35
assuming
is due only to
par ameter
stability
observed
the
grad ient,
temperature
of the temperature gradient
variance
root
non-dimensional
f of 0.8 and 1 .0,
the
of
the
using
then,
0)
& and
variance
root
the
respectively,
The
was found to be less than 0.2, or 0.6 that of the flux.
S
S = 0; for
agreement is only valid for
the
= 1
model
a non-dimensional root variance of the temperature
predicts
gradient of 0.8 that of the flux.
For
I
=
the
3
model
a value of 1.2. The mod el apparently overestimates
predicts
the effectiveness of the flux
gradient.
obvious
The
in
forcing
the
temperature
solution to the problem is that our
The
empirically determined value for D is too large.
level,
we
the variance of the temperature gradient is
of
all
If
V.1.
equation
agree with the relation expressed in
assume
observed
the
well
how
is
Another test of the model
smaller
although
laplacian,
is
not
than
negligible.
the
If
variance
the
noise
of
noise
the flux
level
negligible compared with the variance of the flux the proper
value
for
D
could be as small as one half our empirically
determined value.
we
Another possible explanation is that
must
include
the effects of the meridional circulation forced by the heat
flux. This could significantly decrease the effectiveness of
the flux in forcing the temperature gradient. We can roughly
this effect as follows. The zonally averaged zonal
estimate
momentum and thermodynamic equations may be approximated by
T]
V.4
:
VC
C
+
a
is
-
Substituting equations V.3 and V.4
the
into
C
(p)
stability.
static
the
of
measure
and
diabatics
is
Q
friction,
F, is
where
)(
L-V
--
V.3
thermal
wind
equation
V.5
<4
-~P L,' '~C~<D 'L3FT
dr ~-R
yields
is
identically
J
4
s-Y -r-Y]
o~-~~J -a[
Continuity
L
E ]
v.6
satisfied
-ALr~
if
meridional
the
streamfunction (I is defined by
J-= --Z7
(''3
Ev-]
Equation V.6 becomes
F L~
-t~~~ 43'h
- -,~R
~
5EJ2
TrI
]
-i
~tp
C~J
~
LQJ~3 .
j~~
42
The effects of forcing by individual terms may be considered
separately. For an idealized heat flux
the equation for the meridional streamfunction is
Assuming CF is constant and p = /p. where it
appears
as
a
coefficient yields
'TR
I-PP_
V
0'
0VP
. Z
R.L~*
L
The particular solution
IC=
4L~0-1 /7 P~ i:; n
2)
satisfying the boundary conditions
)17/-L
co
= Q,=ITZ/?
I
(4= O
1-z4O
S=
",
upon substitution yields
L~e F
,Lt
Cif
where
/
t~P~
8Ra;Po
oa
igl
~tZ
C
is the Brunt-Vaisala frequency,
,
N
deformation
radius
distance.
The
cIJ
cooling
>.O
0-AJ a
FO-
and
ratio
Lp
of
= --
the
is
the
L
pole
=
is
to equator
resulting
adiabatic
to the eddy heat flux convergence is
-
jc
-LZfP
Z
The effect of the meridional circulation forced by the
flux
can
the
heat
be incorporated in the model by simply decreasing
the value of
g:
corresponds
The observed half-wavelength of the flux
between four and five. Since we have found the
1
of
value
flux to be shallow, a
more
probably
for
value
m
accounted
the
predicts
than
this
circulation,
for
model
effect
is
three
25%.
Although
and
three,
was determined
because the final value of
With
results.
this
been
already
has
adjustment
the
model
a non-dimensional root variance of the temperature
gradient of 0.45 that of the flux for
1,
or
effectiveness of the diabatics is also decreased by the
meridional
by
two
of
than one. The effectiveness of the
accurate
flux should then be reduced by no more
the
a
to
0.9
both
for
I =
= 0,
0.6
for
3. Since the best estimate of
explanations
are
difference. The proper value for
required
to
resolve
S should then be 0.4.
.
g
=
is
the
VI
FEEDBACK MODEL
How might we further improve the model? The fact
according
to
S,
the flux is more accurately modelled as a
second order
Markov
feedback
the
of
that,
process
suggests
we
should
include
temperature gradient on the flux. In this
section we include such feedback, so that the
equation
for
the flux may be written
VI.1
r
where
=
is a
positive.
-
;
non-dimensional
One
possible
let the equilibrium
rather
-4
parameter
presumed
interpretation of
flux
F. depend
on
7
the
to
be
arises if we
instantaneous
than time mean value of the temperature gradient. In
particular, if the
equilibrium
flux
as
parameterized
by
mixing length arguments is
we have, upon linearization of equation 111.12,
which
upon
Clearly
such
an
length
assume
ensemble
scale
of
interpretation
equation
yields
non-dimensionalization
mixing
time
=k
FGE
2V F
$F=
VI.1.
instantaneous dependence is not valid for
parameterizations
which,
as
noted
above,
averaging over intervals comparable to the
the
flux.
of /1
as
this
Although
the
power
of
leaves
the
dependence of the
equilibrium flux on the temperature gradient in question, it
does not prohibit the possibility of some form of dependence
Therefore, although the appropriate
gradient.
temperature
value for nIY
not
is
a
known,
priori
To simplify the problem we shall
forcing
examine
shall
we
is non-zero.
/
solutions when
the
of
value
of the equilibrium flux on the instantaneous
white
neglect
noise
of the temperature gradient. The model equations in
non-dimensional form are then
VI.2
=
O
VI.3
-
+
-
-
fe
4
Equations relating the covariance functions are
VI.5
-
f )4a
)
VI.6
=
VI.7
-
Equations VI.4 - VI.7 form two sets of coupled equations for
: e
These
characteristic
yield
roots may be pure real
identical,
or
complex
separately in the
and
roots
p
The
form
=: -
distinct,
pure
conjugates. Each case
appendix.
the
of
Solutions
functions.
covariance
the
solutions
real
and
is considered
are
found
to
_
depend
on
only
Kr1
product
the
and
'
,
so
that a
K
equals
parameter study is feasible.
We have already considered the case in which
X
shows solutions of the cross correlation
11
Figure
function for
& = 1.0 and different values of
dramatically
alters
This
function.
X, .
positive
alter
the
non-dimensional
suggesting
whatever
Comparison
atmosphere.
lag
that
feedback
K
= 0 and
complex
does not significantly
Feedback
for
appropriate for
correlation
is not surprising since the characteristic
roots
correlation,
K( . Feedback
cross
the
of
form
the
= 1.0 are double roots for
e
roots for
order
is
K
and
( equals one
consider the case in which
unity.
that
established
having
unity,
one is appropriate for the atmosphere. We now
to
equal
order
is
Y
and
zero
negative
strongest
of
our
might
Y
of
choice
operate
is
the
in
with the observed cross correlation
function suggests a choice of
K.
near one
yield
would
a
more realistic modelled cross correlation function.
Figure 12 shows the modelled auto correlation
for the flux for
value
of
C
Y
near
= 1.0 and different values of
function
K . For a
unity the model solution is similar in
form to the observed auto correlation function. However, the
apparent time
scale
of
the
flux
is
much
shorter
when
feedback is included (by apparent time scale we mean the lag
at which the auto correlation reaches a value of 1/e).
suggests
that
our
estimate
This
of the time scale of the flux
from the observed auto correlation function
is too short.
i.-~~C1
Xlill-I.-^_I-~.Y.L^Li.. IUW~
LI-Y-~I
.~/i --
L47
5
0
-J
z
0
K=O
O
O
I_
m1
-2
O
Figure 11.
temperature
lag for
S
LAG
Model cross correlation
gradient
= 0,
as
for
2
the
and
flux
the
a function of the non-dimensional
Y = 1 and different values of K
.
48
z
0
_-J
0
O
C5
0
--
LAG
3
Figure 12. Model auto correlation for the flux as a function
of the non-dimensional lag for
values of K.
Y
= 0,
= 1 and
different
lag
non-dimensional
significantly
not
does
feedback
Since
of
strongest
the
affect
correlation
negative
between the flux and the temperature gradient we expect
valid.
The appropriate value for
This
complex characteristic roots.
of
range
'
should then be
larger
K
to get
This then requires a larger value for
one.
than
time scale of one day to remain
diabatic
the
of
estimate
for
values
K
considered for comparison
our
suggests
that
wide
a
and the flux time scale should be
with
observations.
than
Rather
carry out such a lengthy procedure we can go directly to the
observations
the time scale for the flux. For the flux
for
auto correlation function expressed in the form
2~*is the dimensional lag (in days), S
where
found
values
for
b = 1.316 day
= 1.189 day
= 0.865 radians
by fitting to the observed flux auto correlation at lags
'/z and
1 day.
of
Matching the form of the model solution with
the above form yields
or
2b - 1
d
50
For the above value for b and
whereas
Thus,
= 1 day we have
= 0.61
1,
the apparent time scale of the modelled flux
scale
underestimates the true time scale, the apparent time
of
day.
one
of
estimate
first
actually less than our
days,
2,
observed flux overestimates the true time scale. To
the
resolve this problem we consider the phase
model
the
According to
V
the
solution
.
may
phase
be
expressed as
yields
]
I)-
where p :-
For
= 0.61
X
K
and
= 1.0
the
model
9 = 2.045 radians. This is a significantly different
phase from the observed phase of 0.865 radians, and explains
why
the
and observed apparent time scales of the
modelled
flux relative to the true time scale are so different
that,
unless
(note
is much greater than one and X is small,
X
the modelled phase must be in the second or fourth quadrant,
whereas the observed phase is in the middle of the first
third
unless
model by adding feedback
correctly.
We
can
not
can
We
quadrant).
or
claim to have improved the
we
can
model
the
phase
do this by including white noise forcing
of the temperature gradient.
$
Consider the phase
for
the
temperature
gradient.
correlation function for
S.
The phase
of the auto correlation
'
Figure
= 1.0 and
is given by
13
function
shows this auto
different
values
of
rru
~II~__~CI
----~-i----ur.
Llil_
_jl~l~
I
z
0
w
0K=
O
O
5
LAG
30
Figure
as
gradient
"
= O,
Model
13.
?
a
auto
function
correlation
of
the
for
the
temperature
non-dimensional
= 1 and different values of
K
.
lag for
= 1.0 and
which for K
Y
= 0.61 yields
$
= 0.884
radians.
If the white noise forcing of the flux is very weak compared
with the white noise forcing of the temperature gradient, we
for
function
correlation
auto
the
expect
13.
including white noise forcing of the temperature
Thus,
gradient
to
figure
in
shown
resemble that of the temperature gradient
flux
the
only
not
will
function
correlation
yield
the
for
a
auto
realistic
more
temperature gradient, but a
for
more realistic auto correlation function
the
flux
as
well, complete with phase.
Although by tuning the three model parameters
and
S
find
may
one
do
observations, we
not
model
solutions
conclude
consistent
feedback
that
X ,
of
with
the
temperature gradient on the flux operates in the atmosphere.
The finite amplitude calculations of baroclinic stability in
the
absence of diabatics (Pedlosky, 1979) also yield second
order equations for the behavior of the flux.
al.
Pfeffer et
functions
for
the
(1980)
flux
and
cross
calculated
the
correlation
temperature gradient in
thermally driven rotating annulus experiments. The
cross
correlation
function (figure 11)
( K - 5) closely resembles the
function
in
the
geostrophic
modelled
for strong feedback
observed
turbulence
cross
correlation
regime. Although
feedback may be stronger in the annulus experiments than
the
in
atmosphere, it is more likely that the internal damping
is too weak in the annulus experiments.
_~ ~~_~_al~~~_l~
~_II__^_X~
_I_~_*__~__
~~~_
I _L~1___~___
VII
CONCLUSIONS
One can derive many of the
The fact that the flux behaves approximately as
principles.
be identified with one half the inverse
can
flux
the
scale
time
The
a red noise derives from Pedlosky (1979).
for
first
from
results
model
growth rate from baroclinic stability theory. Corrections to
the first estimate for the diabatic time scale
perturbations follows from the vertical
of
scale
vertical
wave
unstable
scale of the most
the
to
due
in
stability
baroclinic
theory. One could roughly estimate the amount of white noise
of the temperature gradient from the typical scales
forcing
of
moist
has
perturbations
been
considered
theoretically by Stone
Simmons (1974) and Pedlosky(1975a). The only missing
(1974),
considered
(1975b)
Pedlosky
the amplitudes of interacting
triads in a baroclinic current but found
the
on
flux.
the
element is a model of the white noise forcing of
depend
flux
of
scale
meridional
The
convection.
that
the
results
initial conditions. Further work is clearly
needed on this difficult problem.
The diabatic time scale which the modelling suggests is
perturbations
for
valid
surprisingly
due
the
eddy
flux
heat
is
short. Since the diabatic time scale chosen in
dynamical models is typically ten days or more, our
results
scale
is
appropriate. Since the diabatic time scale is comparable
to
suggest
the
that
advective
modelling
a
much
time
diabatics
shorter
scale,
in
the
numerical
diabatic
time
importance
models
of
of
properly
atmospheric
motions is readily apparent. We stress this because there is
room for improvement in modelling diabatics at NMC.
Finally, although for the meridional scales of interest
we
have
random
that
shown
meridional
the
as important as random forcing of
is
gradient
temperature
of
forcing
the static stability, we cannot neglect the variance of
critical
shear.
modelling
has
equation
then
One
has reason to question why the
successful.
so
been
the
We
suspect
that
an
very similar to that governing the behavior of the
temperature gradient also governs the behavior of the static
According
stability.
baroclinic
to
stability
theory
vertical eddy flux of sensible heat associated with synoptic
must
disturbances
coincide
with
meridional eddy sensible
In addition, diabatics should act to restore the
heat flux.
static stability, as well as the
temperature
gradient,
to
its respective equilibrium value. However, the diabatic time
scale
in
the
vertical
may not correspond to the diabatic
time scale in the horizontal.
variance
of
the
critical
Therefore,
shear
for
meridional
long
may
not
the
strictly
perturbations. Correlation functions
involving the temperature gradient and the static
individually
as
cannot be neglected, our
determination of the diabatic time scale
apply
as
stability
should be examined to separate meridional from
vertical perturbations.
_11~~
1_
(4_^_IJ^*__~~1I__I__YLLIIY~1--_Il-L~~I~X~
.L.I-._..~
.-li^.~
-.._L_.
- - .~-----------III~_~U~-(~
APPENDIX
The eight unknowns associated
the
with
general
four
solutions require eight constraints. Four constraints result
from the requirement that equations VI.4 - VI.7 hold for all
lags
or equal to zero, which is satisfied if
than
greater
results
the equations hold for zero lag. Another
requirement that the cross covariance functions
and
1
(g,f, t )
the
. (f,g,
)
zero lag. Two more constraints
at
match
from
follow from the requirement that
(which is independent of equations VI.4 - VI.7 and is
only
when
noise forcing of the temperature gradient
white
does not exist)
valid
hold
for
positive
all
lags.
The
final
constraint is that the flux auto covariance function at zero
lag
flux
the
match
variance.
All
constants
are
then
expressed in terms of the flux variance.
A
Real, Distinct Roots
For pure real and distinct characteristic roots p =
the general solutions are
)
if
A,
/-C,
a
31)
~I
d
~SrStl~eef
P-tt
4+-
+
3
e33
13 Ye .~
p-
_________I*~~~LYII__Y_
LI
56
The eight constraints are
+ E (A34 4,)
t4 +) 4,
( ptI±)
,
*f
-A4(A,-'-i,)
133
(1-f/')
=
+ S (oA+ 6)
+ (F
/4 ,-t
Az eg
S4(Ay+8,)=
o
=
6,
(s -
) 8, -
=o
d(A,
=
v4
=
.z
(F,
F
1-(f, f,o0)
The constants in terms of
are
P' I Pt- r~~
(pt-~~(pfpS~2)
p
(
--
Y)
I
(r
I
',oj
i (f(f)
131B-
3
-(e1o)
-p
1U
+r, )
J(+ Jo)
C '- )( " +)
£ (fCF, ., )
correlation functions
P-The become
<+
- (pt+ -)L.i
i+'
The correlation functions become
I
't
pl~s
'ii#~(pi~~- r) I+
")' -g +
C P'- p~)C ptp~tb~'l~l/~
fE?((f~t
P1 2
1' 4
Mje
P vt3
'
"
- KF
IP ) *&\l] e
to ( I
--
ptf>
E
E
-+
B
3
f (t,-+~(i)
+
e.
-P
"
4"
P'- c~
/O(IS1hZ)
/ 2
4
Qf-
Double Roots
For
real
double
characteristic
roots
solutions are
A,
A7-
A:3
- (1415'&)
p
pt
-t 81
-ep-
- S3 t e
P't
the
general
The eight constraints are
P a)
14A =
A, Lt8 ,
CP' r) 3 / g3-
/
4
4,
-O 0
- A,
A,
=
o).
/
(f,
The constants are
j
Zr
A,
p
p
IF
CCP+[)b/
1
Io
F(r, o
T
A,
BS
CB
S-1'
P4
t
s
-~
-
_
S2,,'
4
Ljy1
S(
(
C,o)
~4~_
__
I__^_L____IILVI___L~mi~__lill~~lll~
59
The correlation functions are
____-1
e)'
;t
(P'
L
cY4
rL
o
2
pI
e
474
tpc
( f-~/)
e- Pt
PI
k L
eP./4
pr4,9,'t)
C
Complex Roots
ip the general
For complex characteristic roots p = Pr
solutions are
- 7)
F
Prz L 4
.,
cos p;t 4,
£ (eig,h)r
C, S P: - +
The eight constraints are
(p,
(
4)
i)
,
p,i) AL
A3 =
Sp; g,
A_3 -/
4
p-)
k PL
(A
(A,
a-%
P;t 4
( p,-r
P: t )
G'Sip/"
-N
pi3
,
, -o
-sA,
= o
S('t
&A-1;
P,--a)
p~)
60
(pr+
')6K
-tp i[tJ
41\/%
P, a, * g1
( X- p,) 38,
PL4,+
= I (f,-, 0).
3
The constants become
Ad
X-'
3 1=
3
3r -t*2
, ]
i(#,
v /f-r) t9y\ I
X(
YCI4Y)+M
0,)
H(Io
L(i--) z -
.
N
z P;
I
,>o
A"L
- E( , , )
L ( / -'Y
g4\dZ
zp',b
I( ((o0)
64
"
E""~~i
0,
The correlation functions are then
/ ( ._, it )
-p,,t
.
"1 3"Y 4 ZE
,
"
e
/0 (f
,) -r)/
EY(14)g~
e-rt Es
-,0,
) -- e pIf? EC-0 S p-I
o ( 51 I"L
Y(1-
/4r
5
f)4SIK\
S&t1
IL±*
Rt -]
S
1_II__ _____I___I___1_~_L___CLY
62
ACKNOWLEDGEMENTS
The author would
Professor
like
to
express
gratitude
his
to
Peter Stone for suggesting this thesis topic, and
for his guidance and encouragement during the course of this
study. He also appreciates both the support
of
his
fellow
students, and the many stimulating discussions with them. He
thanks
his daughter Laila for being such a sweetie, and the
Carlins for being friends indeed. Most of all, he thanks his
wife April for bearing it all gracefully.
63
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_I~-
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. .i~~..I.~IX-il-*~.~-PI-^Y-n^LIII
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