A Higher Order Closure Turbulence Model
of the Planetary Boundary Layer
by
JOSEPH STEPHEN SCIRE
B.S., Massachusetts Institute of Technology
(May, 1977)
Submitted in
Partial Fulfillment
of the Requirements of the
Degree of
Master of Science
at the
Massachusetts Institute of Technology
(October, 1978)
Signature of Author...8..y...........................
Department of Meteorology, October, 1978
Certified by................................................
Thesis Supervisor
Accepted by................................................
Chairman , Department Committee
on Graduate Students
2
A Higher Order Closure
Turbulence Model of the
Planetary Boundary Layer
by
JOSEPH STEPHEN SCIRE
Submitted to the Department of Meteorology
on October 10,
1978 in
partial fulfillment of
the requirements for the Degree of Master of Science
ABSTRACT
A higher order closure turbulence model, using
the full level 3 equations (Mellor and Yamada, 1974;
Yamada and Mellor, 1975) is described in detail.
A new
formulation for the length scale, Z, which appears in each
of the modeled terms, is employed.
Equilibrium boundary
conditions for the second moments are applied at the
lower boundary.
Day 33-34 of the Wangara experiment is simulated.
Surface temperature and mixing ratio are predicted with
a ground thermodynamics model.
The effect of the inclusion
of the Coriolis terms of the second moment equations on
the results is evaluated and is found to be small.
The similarity functions A, B, C, and D are
evaluated.
Vertically averaged variables are used in
the deficit relations (Arya, 1977, 1978).
With this
formulation, the similarity functions C and D are found to
be equal in the unstable boundary layer.
In the stable
boundary layer D appears to be smaller than C.
Name and Title of Thesis Supervisor:
Professor David Randall
Assistant Professor of Meteorology
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude
to Professor David Randall for his valuable guidance and
active interest throughout the course of this project.
I wish to thank Isabelle Kole for skillfully
drafting the figures and Joanne Klotz for expertly typing
the difficult material.
I would like to extend special thanks to my
fiancee,
Debby,
for her constant encouragement and moral
support, and to my family and friends for their encouragement throughout the preparation of this thesis.
TABLE OF CONTENTS
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ACKNOWLEDGEMENTS .
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TABLE OF CONTENTS
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5
7
ABSTRACT .
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LIST OF FIGURES
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LIST OF TABLES .
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1.
INTRODUCTION
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12
2.
DEVELOPMENT OF THE BASIC EQUATIONS .
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15
3.
4.
5.
2.1
Equations for the Mean Variables
15
2.2
Equations for the Second Moments
17
.. .
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3.1
Modeling Assumptions
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3.2
Level 3 Model Equations .
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3.3
Length Scale Formulation
.26
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..26
.39
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.46
BOUNDARY CONDITIONS
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52
4.1
Mean Variables
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52
4.2
Second Moments
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53
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57
SOLUTION OF THE EQUATIONS
5.1
Reduction of the Prognost ic Equations
into a Single Form . . . . . . . .
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.57
Finite-Difference Approximation
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59
SIMULATION OF DAY 33 OF THE WANGARA
EXPERIMENT . . . . . . . . . . . .
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63
5.2
6.
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MODELING OF THE EQUATIONS
6.1
Initial Conditions
6.2
Results .
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63
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63
Page
7.
8.
EFFECTS OF THE CORIOLIS TERMS ON TURBULENT
FLUXES IN THE PBL
. . . . . . . . . . .
.
EVALUATION OF THE SIMILARITY FUNCTIONS A, B,
C, and D .....
. . .. .
. . . . . ..
107
Definition of the Scales and Derivation
of A, B, C, and D . . . . . . . . . .
107
Results .
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114
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129
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131
APPENDIX B . . . . . . . . . . . . . . . . . . .
145
8.1
8.2
9.
SUMMARY
APPENDIX A .
APPENDIX C .
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149
REFERENCES .
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151
LIST OF FIGURES
Figure No
1
Page
Vertical profiles of the length scale,
2, for two values of a, based on the
formulation of Yamada and Mellor (1975)
2
3
4
5
6
7
8
9
10
11
12
13
.
.
49
Vertical profile on the length scale,
Z, based on the new formulation defined
by equations (65)
....
........
50
Initial profile for virtual temperature,
T... . . . . . . . . . . . . . . . . . . ..
64
Initialprofile
for water vapor mixing
ratio, R .........
. . . .
.
. .. . . ...
65
Initial profiles for the eastward velocity
component, u, and the northward velocity
component, v
... .........-.-.-.-.
66
Variation of the calculated mean velocity
component, u, as a function of time and
height . . . . . . . . . . . . . . . . . . .
67
Variation of the calculated mean velocity
component, v, as a function of time and
height . . . . . . . . . . . . . . . . . .
68
.
Variation of the calculated mean virtual
potential temperature, Ov, as a function
of time and height . . . . . . . . . . .
70
Calculated surface ® as a function of
v. . . . . . . . . .
time
71
Variation of the calculated mean water
vapor mixing ratio, R, as a function
of time and height . . . . . . . . . . .
73
Calculated surface R as a function of
time . . . . . . . . . . . . . . . . . .
74
Variation of the calculated values
of twice the turbulence kinetic energy,
q2, as a function of time and height . .
75
Terms of the turbulence kinetic energy
equation at 1300 hours, Day 33......
76
Page
Figure No.
14
15
Variation of the length scale, 2, as
. . . . . .
a function of time and height
.
77
.
79
Vertical profiles of q2 (twice the turbulence energy) for 1100hours and 1200 hours,
Day 33 . . . . . . . . . . . . . . . . . . .
80
The calculated boundary layer height, h,
. . . . . . . . . .
as a function of time
.
81
Variation of the calculated virtual
potential temperature variance, W5.,
as a function of time and height . . .
.
83
. .
84
Variation of the calculated water vapor
mixing ratio variance, ~rT, as a function
of time and height . . . . . . . . . . . . .
85
Virtual potential temperature profiles
for 1100 hours and 1200 hours,
16
17
18
19
20
21
Day 33
. .
. .
Terms of the virtual potential temperature variance equation at 1300 hours,
Day 33 . . . . . . . . . . . . . . . . .
Terms of the water vapor mixing ratio
.
86
perature covariance, r ' , as a function
of time and height . . . . . . . . . . . . .
87
Terms of the r'94 equation at 1300 hours,
Day 33 . . . . . . . . . . . . . . . . . . .
88
Variation of the calculated Reynolds
stress component, u'w', as a function
of time and height . . . . . . . . . .
90
variance equation at 1300 hours, Day 33
22
.
Variation of the calculated water vapor
mixing ratio - virtual potential tem-
23
24
25
26
.
Variation of the calculated Reynolds
stress component, v'w' , as a function
... . ........
of time and height
91
Variation of the heat flux component,
.
92
Variation of the moisture flux component,
w'r', as a function of time and height . . .
93
w'6},
27
. .
as a
function of time and height .
.
Figure No.
28
29
30
31
32
33
34
35
36
37
38
Page
Calculated surface heat flux, for two
values of the surface albedo, as a
function of time . . . . . . . . . . . .
Calculated surface moisture flux as
a function of time . . . . . . . . . .
. .
94
. . .
95
Calculated friction velocity, u,, as a
function of time . . . . . . . . . . . .
. .
96
The correlation coefficient for w' - 61
as a function of z/h . . . . . . . . . .
. .
97
Profiles of the Reynolds
u'w', at 1200 hours, Day
cases (Coriolis terms on
terms off) as a function
stress component,
33, for two
and Coriolis
of z/h
......
102
Profiles of the Reynolds stress component,
v'w', at 1200 hours, Day 33, for two cases
(Coriolis terms on and Coriolis terms
off) as a function of z/h . . . . . . . . .
103
Profiles of the heat flux component, w'Q4,
at 1200 hours, Day 33, for two cases
(Coriolis terms on and Coriolis terms off)
as a function of z/h . . . . . . . . . . . .
104
flux component,
33, for two
and Coriolis
of z/h
. . . . . .
105
Profiles of the moisture
w'r', at 1200 hours, Day
.cases (Coriolis terms on
terms off) as a function
Similarity function A as a function of
....................
h/L ........
... 122
Similarity function B as a function of
h/L ..
. o.
.
. ... .. .o..o. .
-
Similarity function C as a function of
h/L.....................................
124
39
Similarity function D as a function of
.125
- - . . . . . . o. .... ..
h/L .o. .
40a
Similarity function A as a function of
h/L and h/z for values of Iflh/u* > 0.1
.
123
.
126
10
Figure No.
40b
Page
Similarity function B asa function of
IfIh/u* > 0.1
126
Similarity function C as a function of
h/L and h/z for values of If h/u, > 0.1
127
h/L and h/z
41a
41b
for values of
Similarity function D as a function of
h/L and h/z
for values of Iflh/u* > 0.1
127
B-1
Staggered grid system use in the model . .
C-1
Flowchart of the level 3 model .
.
. .
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.
148
149
11
LIST OF TABLES
Page
Table No.
Unmodeled equations for the mean
variables and the second moments .
. .
II
Modeling assumptions .
III
Level 3 model equations
IV
Final level 3 model equations
V
Representation of the prognostic
equations in a standard form . . .
. .
VI
Similarity functions - Case A
.
VII
Similarity functions - Case B
VIII
Similarity functions - Case C
. .
IX
Similarity functions - Case D
.
X
Similarity Functions - Case E
. .
B-1
z-C values .
I
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27
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30
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40
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44
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58
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118
.......
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117
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119
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120
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121
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147
1.
INTRODUCTION
The random nature of turbulence makes the study
of turbulent flows difficult.
For this reason, it is
convenient to use a statistical approach to turbulence
problems, based on the concept of ensemble averaging.
An
ensemble average refers to an average taken over a collection of an infinite number of observations for which the*
mean conditions are identical.
Due to the randomness and
irregularity of turbulence, the details of each realization
are different even though the mean conditions are the same.
It is impossible to obtain an ensemble average from real
atmospheric data because mean conditions are never identical.
Certainly it is not possible to obtain an infinite number
of instantaneous measurements over which to average.
One must adopt an ergodic hypothesis, that is, an assumption
regarding the equivalence of different types of averages,
to establish the equality of an ensemble average over an
infinite number of observations with a time average over
an infinitely long averaging period under conditions
of stationary flow (Lumley and Panofsky, 1964; Busch, 1973).
A finite averaging time will yield an estimate with an
accuracy which increases as the order of the moment being
averaged decreases
(Wyngaard, 1973).
For the time scales
involved in atmospheric turbulence, it is possible to relate
ensemble averages to measurable time averages.
13
The statistical approach, however, results in a
situation in which the number of unknowns exceeds the
number of equations.
It is necessary, therefore, to make
simplifying modeling assumptions in order to obtain
closure.
Mellor and Yamada (1974) describe a hierarchy of
turbulence closure models.
Based on
a systematic simplifica-
tion of the appropriate equations, four model levels are
produced.
The most complex (level 4) model requires the
solution of prognostic simultaneous partial differential
equations for all of the components of the Reynolds stress
tensor, the heat flux vector, and the temperature variance,
as well as for the components of the mean flow and the
mean potential temperature.
If water vapor is to be
included in the model, additional equations for moisture
variables need to be solved.
The most simplified (level 1)
model is a set of diagnostic algebraic equations corresponding to a mixing length model.
The level 3 model, to be
used in this study, represents an intermediate degree of
complexity.
As shownby Mellor and Yamada,
the choice of
the level 3 model represents a compromise between the small
increase in relative accuracy obtained with a level 4 model
and the resulting large increase in computation time.
The level 3 model is a subset of a group of
models called Mean Turbulent Field (MTF) closure models
(Mellor and Herring, 1973).
MTF closure models consist
14
of two subsets, Mean Turbulent Energy (MTE) and Mean
Reynolds Stress (MRS) closure models.
Because the
turbulence energy in level 3 is calculated prognostically
while the individual components of the Reynolds stress
tensor are calculated diagnostically, the level 3 model
falls into the category of MTE closure.
Yamada and Mellor (1975) use the level 3 model
to simulate the Wangard boundary layer data.
The present
model differs from Yamada's and Mellor's model in several
important respects:
i)
ii)
iii)
iv)
v)
vi)
A ground thermodynamics model predicts
the surface temperature and mixing ratio.
The full level 3 moisture equations are
employed.
Equilibrium boundary conditions for the
prognostic turbulence variables are applied
at the lower boundary.
A new formulation for the length scale, Z,
is used.
The Coriolis terms are included.
A 5 second time step and a staggered grid
system are used.
The model is used to simulate Day 33-34 of the
Wangara experiment, and the results compared to those
of Yamada and Mellor (1975).
The effect of the Coriolis
terms on turbulent fluxes in the PBL is evaluated by
turning these terms on and off in the model and examining
the results.
Finally, the functions A, B, C, and D of
similarity theory are evaluated.
15
DEVELOPMENT OF THE BASIC EQUATIONS
2.
Equations for the Mean Variables
2.1
The variables of interest are the velocity components u. (i = 1,2,3),
potential temperature
0, pressure P,
The basic equations govern-
and water vapor mixing ratio R.
ing these variables are:
Continuity equation
auk
0
(1)
3xk
Momentum equations
au.
au 1
u
atxk
p+
3
0o
au.
0
3
3kt
k E
xk
axk
(2)
Thermodynamic energy equation
at
u
0+ = kT
Uk 3
Taxk
axk
) +
(3)
Water vapor equation
3R
u3R
+t uk
9
''x
R
x )
(4)
16
where S is the coefficient of thermal expansion,
fk =
(o,fy,f)
is the Coriolis parameter, v is the kinematic
viscosity, kt is the thermal diffusivity, p is the density,
n is the kinematic diffusivity for water vapor, and a is
the longwave flux divergence.
The Einstein summation con-
vention is employed so that whenever an index is repeated
in a term, summation is implied.
e jkt is the alternating
unit tensor and 6.. is the Kronecker delta.
l if
j,k,2Z = (1,2,3),
(2,3,1),
or (3,1,2)
0 if any index is repeated
Ejk
-1 if
j,k,Z = (3,2,1),
(2,1,3), or (1,3,2)
0 i f i /j
3
1 if i =j
The effects of evaporation and condensation of water are
not
included.
The Boussinesq approximation, in which
density is treated as constant except when it is multiplied
by g (in which case it is allowed to be temperature
dependent),has been used (Busch, 1973; Mellor, 1973).
Each variable can be represented as a sum of a
mean part and a fluctuating part.
u. = u.
1
1
U +
=
u!
1
e'(4b)
(4a)
17
P'
(4c)
R = R + r'
(4d)
P = P +
The overbar signifies an ensemble average.
To obtain equations for the mean variables, average equations
(1-4).
auk
=
0
(6)
k
4;t+
[u~k +F~lk k"
+O IUU.
1ap
x- +
I.
=
gO63
3
E
-jk9
k
au.
+3x
DO+at ax
3
+R a-
2.2
axk
(7)
aXk
kT T a
xk
=
[~uko + ue']
[R uk + r'u ]
=
(8)
xk)
a
(9)
Equations for the Second Moments
Equations (1-4) contain second order correlations
of perturbation quantities of the form uju.
In order
to evaluate these second moments, first obtain equations
for the fluctuating components by subtracting equations
(6-9) from equations (1-4).
Using equations (4a-4d) yields:
au'
-= 0
axk
-
(10)
x
+
[u.u' + u!uk + u!u' - uu]
pO
ao
at
+
x
3
jkk k
a [luke' + ulii + ue'
ax k
arl
at
a
xk
u'
r']
(11)
k
3xk
- u 6'] = kT
[ukr' + ukif + ukr' -
au'
a
f u' + v
- E
+
1
=
=
kak
(
k
-
(-)
k)
ax k
(12)
(13)
To obtain an equation for u!u!, multiply equation (11) by
I
J
u! and use the continuity equations
au+
ui
iat
+
a
u!
iak
u
-u!
+
ua
1
au!
aL (x1)
axk
axk
jk
+
and (11):
u
'
k
a
()
i ax.
p0
J
=
+
[u~u
(6)
Sgu.'6
i
3
1u~k
-E jkZ fku'u'
i Z
3.
J
(14)
19
A second equation is then obtained by switching the indexes i
and j in equation (14).
Adding this equation to equation (14),
and using continuity and averaging, yields the prognostic
second moment Reynolds stress equations:
+u.
- (u~u'.)
at
a
+
axk
__
I k
a3xk
1
i
+
--- l + uu
+ u!u
au.
a
(u!u!u ) = Sg[u!6'6
3
.J
1
+ u!6
u!a
u!
+ P~
PO
ax
Cik
+
au-J)
+
ku u
~-(~~~
a
xk
3]
u!p'
a
-E jkZfuuua
k i
__
uu
u
a
xk +k
- a3x.
3
a
x.
px
PO
p
00o
au!
uu!au!
2v
axk
xx
up'
1
auj
axk
axk
(15)
Equation (7)
requires that the second moment
u!u! be known in order to find u.
I
J
J
Equation (15),
an
equation for u!u!, contains the third moment uju!u .
equation for the
(n+l)th moment.
An
nth moment will contain a term with the
In other words, the number of equations is
less than the number of unknowns and the problem is
not closed.
In order to obtain closure, the unknown moments
are parameterized in terms of known quantities.
This
parameterization is, however, an approximation.
The
approximations are not necessarily more physically valid
than parameterizations of second moments in
quantities,
as in
less complex models.
terms of mean
However,
the fact
that the approximations are made at a higher order allows
one to hope that the results will be less sensitive to the
The results of models using this higher
parameterization.
order closure technique support this notion.
In the case i
# j,
p0 1uur
J represents a flux of j
momentum by the i turbulent component.
The interpretation
of the terms of equation (15) is as follows:
at
3
represents the local time rate of change of the
ensemble averaged turbulent momentum flux i3!Vu
j
(normalized by density)
au.
au.
u.'u
1
-a
~
+ u ug
3xku
1
k
represents the mechanical production of Reynolds
stress due to an interaction of the mean velocity
gradient with the Reynolds stress
uk axk- - ufu!
1
represents advection of Reynolds stress by the
mean wind
a
x
u ui)
k (u 13
is the triple correlation term which represents
the turbulent flux of ulu! by the fluctuating
component uk (i.e.,
turbulent diffusion)
Sg[u!6'6
.3
i
+ T'6
J
j
3.
i3.
represents the bouyant production (or destruction)
of Reynolds stress
Ejkk
fu'u'
ki Z +
f uu Z
ikYkj
represents the effect of Coriolis forces on the
Reynolds stress
1
[ a3
a(up') + a
(u p')]
represents the effect of the pressure perturbationvelocity perturbation correlation on Reynolds
stress destruction
au!
p
(
3
au'
+
3x
)ax
is the "energy redistribution" or "return to isotrcpy"
term representing the way the pressure-velocity
gradient correlation distributes energy among the
three energy components
22
au.'u!
ffi(..L.)
1
axk
-
au.
Du!
3a
axk
axk
1i
2v
aXk
represents the viscous diffusion and viscous dissipation of Reynolds stress
j,
In the case i =
equation (15) is an equation for
the i component of the turbulence energy. Summing the
2,2.
individual components, with q = u.' , gives the turbulence
kinetic energy equation.
a
2
u
(/2)
-
auu.
1
au!1
uk
k
Sgu6'63.
a
-
PO
a
I
U!
2u2
2
(xk
L
U
a
2
+
a
[
ak
2
au!1
(16)
The Coriolis terms do not appear in equation
(16)
because the Coriolis force cannot contribute to the total
turbulent kinetic energy.
Likewise, the energy redistribu-
tion term of the Reynolds stress equation is not present in
the total kinetic energy budget because its role is to
redistribute energy without contributing to the total.
The interpretation of the terms of equation (16) is
analogous to that of the Reynolds stress equation.
23
To obtain an equation for the heat flux vector
u!', first multiply equation (12) by u! and equation (11)
1
J
by e' and average to obtain
')u' --
'
a~(3'
+
at
[uku
uiau
kJ0' + UIN
a
au!
-
-
kT
au!
aut
-l
axk
uk
6
-
'
0
+ uu']
,
=k( T
axk
Txk
e
--xk
3
(17)
-
--
axk
axk
and,
au!
0' at
a__
=
a
axk
__
+I
a~l U!
+uk U
_
[u u 6' + u! '
30'
xk
xk
+ v -
-E jk f u'0'
k Z
a
xk
__
_
u
3O]
Ik
k
1 elap
o
+uk
U
+ ae2
a3
a!au!
(0' -- 1) - v -aak
xk x
e
(18)
Adding equations (17) and (18),
and using continuity,
yields the desired heat flux equation:
a
M ')
at J
+
a' [~u
ax k U
u
II
+ u u6k' - k u
kTu3 Dxk
axk
]
24
1+(p'e')
P0
ax
3
+ sjk
jk~kkuii'' r~ =
~
gO
2g'73 + p'p
3.
p0
- uaue
a k~
IU
+ v)
(k
Tax
ax.O
a,.J
au!
-
36'
x
k
u
a k
.
_
u
axk
(19)
An equation for the potential temperature variance 0'
is obtained by multiplying equation (12) by e',
using
continuity, and averaging.
3, 2 +
-~~
+kT
3
2
q
] = -2u O6
k [uu'
at
ae
a ( axk )
axk
T axk
ae
(20)
D
The terms of equations (19) and (20) are of the same form
Their interpretation is analogous.
as those of equation (15).
Equation (9) requires the turbulent moisture flux
u'r'.
An equation for u!r' is obtained by multiplying
equation (11) by r' and multiplying equation (13) by u3.
Averaging these equations and adding them yields:
a
t
(u'r')
+
-
[
k 3
r
+
u'
r
3
vr'
u
3 axk
a
axk
$
+
(o
0
(p'r') + ek
aXx
u r
- u u
= agr'6' 6
uxr'
au!
au.
__
k
+ p ar'
p 0 aX
3
k
k
(21)
An equation for the potential temperature-mixing
ratio covariance
O'r'
in equation (21).
is also necessary because it appears
Multiplying equation (12) by r' and
equation (13) by 6',
averaging, and adding yields:
a (6'r')
(3'3r' +6a
+
[uke'r' + uG'r'
at
axk
k
= -
(n+k)
30'
axk
3r'
*r-
ua'
-xk
-
j6'
-
DR
-
xk -
ae
r
k
-
ax
T
axk
3
r
-
An equation for the mixing ratio variance,
obtained by multiplying equation (13) by r'.
(22)
r'
,
is
After using
continuity and averaging:
a
(r
2
+
a
3
-
[ukr'
+ u r' i
a3r'2
2
=
- 2 ukr'
+
xk
r2n ar'
axk
3xk
2
axk
(23)
3.
3.1
MODELING OF THE EQUATIONS
The Modeling Assumptions
It is necessary to parameterize the unknown variables
of the equations in order to obtain closure.
It is also
desirable to neglect small terms that do not affect the
results so that unnecessary complexity is avoided and
computation time is minimized.
The system of equations con-
sists of equations (7-9),
and (19-23).. These
(15),
equations are summarized in Table I.
The terms to be
modeled have been doubly underlined.
The numbers
associated with each line correspond to the numbered modeling
assumptions which are collected into Table II.
The modeling assumptions in
types.
are of two
Table II
The first type concerns terms containing unknown
variables, i.e., variables for which there are no equations
in Table I expressing the variable in
known variables.
terms of only other
The parameterization of these terms is
required by closure considerations.
terms are of this type.
The triple correlation
The other type of assumption con-
cerns terms which are known (in terms of other variables)
but are small in
are difficult
the planetary boundary layer
(PBL),
or
(although possible given the set of equations
in Table I) to evaluate and are assumed small in the PBL.
The Coriolis terms are of the second type.
They have been
neglected in the models of Mellor and Yamada, 1974, 1975,
Table I
Unmodeled Equations for the Mean Variables and
Equathe Second Moments
4 -LI Equations for the mean flow
4 AA.
(7)
----
+A
)
30
i
ax
4AsA4,#
AoU+LAI
13
a. A~
xAr
P~
£t
+ CI
(8)
(9)
I
+
-
4
+C0
yk
I2 Ax + r AA#
-
O;k
a X-V
~Xic
(
Second moment equations
(15)
( "IA(-'
)+
AA -
I44
-
+
-x
4
E..i.A
~'~
I
1
-I
t-
Q~
4,C.dA4
4-
-IuX
1
o
I
,
'~
I
-
zC
%.>
?C,
(S)
(s)
-
(4,)
V
A-4AAj
Cr.)
(C)
Of
<E
I
t
~A4~Mj
-
4
(
(3)
Mkr
+
AA4k#
41 x Oc
~
I
_
4-
Equation
No.
(continued)
Table I
(19)
)
(A
y
4
+
-
1
G
e')
=
-
* o )X)
(A)
k.1
[
Z
E.C
Mi
OO
(me')
-
a G
A
I
A
+I
4.
x
Ad'e'
j
(,0)
- -
4
*t
'h"I
(20)
xAA
+0)
k Go
-t
Got
-
(itl
___
____
(k,
-
+
0 11
Ma
-.-
---
+
AA
A'
+
'
(,3)
(142
(on
<14f
(21)
'a
(AA
y
4
j' r
-A
+ #C
AA
? r
~~~~ '
a.
-At
AA! A4,t' P' )
4
2
-
--
-
( er'
~,
-
-A
Cu,
+ P,
(to 1
_____
r'
( vtk +
-
(Mc-
A4;'
?v-'
06
-
..
-.
+
-
ts
e'
8E
29
Equation
No.
Table I
(22)
4-A4-I
-
10'
yirII
(continued)
i''
AA(
Dxtr
+X
.A,
r04
+
(Z.3
(22.)
(+
KT)ae
C' M
61
r
-
)XAC
r.
6'?.~1
C
2 04 )
(23)
rb~r2
AMk'
TZ
(
\
V")
Ia
e
I.. -k
4
'~.'
ka
4.
-1
(MA
r"z
MS
~r'
Zr
(~z~)
Underlined terms have been modeled.
The numbers correspond
to the order in which tne modeling assumptions are listed
in Table II.
Table II
Modeling Assumptions
(1)
41 xx (
~AA~ M3
AA
)~I,
c~~
[
*
_____
(
0
(2)
rA.A
4
Ai
If
_C
(3)
)x;QdP
(4)
P
0
A4Z
A-M
-
;:
iJ Y
\
(5)
I
1
0
A4
I Z)
t
4 C.
2
AA.)
0
3
(6)
)C
A.
)44,0
(7)
I- zAl
(8)
k-4r*
(9)
~,c.
{ plot )
409
4
r0
A,(;
I
a YA
0
(3;;4
IAAi
xi2
A 4
(continued)
Table II
(1.0)
E~k
C
(11)
P'
-
.3)
.
-
X
.eJ
0k
I
A4~Q
(12)( +
O
go zL
I
(13)
'"k
(2.4)
14 ..-
(15)
2
x4c
-2G
P(
k
9 it
)-A
A2
(16)
-
(
IA /
4C)(q
;A4It 1'
- t A2. (
I
(17)
'D
ar'
(28
(?~)
-
-
0
+
M~
(continued)
Table II
O
(19)
(20)
~
r
la
k.A
.
AA; r
~y.)
(21)
3/
v\ + 0)
(22)
---
=
---
e' I
0
'(N
aee
(23)
(24)
'
(\
+
K-r
)
t
- --
- -
3%
(25)
--
V'
+,a
r
-
A1.
ar
-2k
(26)
xf
-%I
(27)
z v
.A%
Yxc )AC
Y'
r'G'
33
and Donaldson, 1973.
Wyngaard et al., 1974, however, discuss
the importance of the Coriolis terms in the Reynolds stress
In this study an option has been included in the
budget.
model to turn the Coriolis terms on or off to allow their
importance to be evaluated.
Both modeling options appear
in Table II.
One of the basic modeling assumptions contained
in the model is the energy redistribution assumption of
The total turbulence kinetic energy (per
Rotta (1951).
unit mass)
1
q
2
1
is
~~
= g(uj
The term
the sum of three components:
+u
2
2 +
+ u.)
p' (3u!/ax
+ au!/ax.) appears in the equations
for the individual components of the turbulence energy,
but not in the equation for the total turbulence kinetic
energy (equation 16).
As has already been noted, the role
of tnis term is to redistribute the energy among the three
components of energy without contributing to the total.
redistribution term is therefore modeled as:
au.
p' (--.
3x, + ~--)
ax.
I
6..
But
i
=
-3Z
1
(u!u!
1 j
-
32
au.
q2 ) + Cq 2(J
au.
+ ax.
i
where C is a constant and Z1 is a length scale which will
The
34
be prescribed.
Characteristic of this formulation is that a
departure from isotropy of the Reynolds stress tensor will
result in a tendency back toward isotropy.
"Return to isotropy" terms also appear in equations
(19) and (21) as p'
p0
36'
ax.3
and
p'
p0
ar'
axJ
respectively.
An analogous formulation is adopted for these terms:
PO
x
-
392
o
3
ax
m
2
j
A second length scale k2 is introduced.
It is necessary to model the triple correlation
term
representing a diffusion of Reynolds
u),
1
k
stress, in terms of known second moments.
j, and
struction in i,
k, using second moments, is:
-a
3xk
au! I
au
(u uau-
-- (U.!
axk
1 3
A symmetric con-
[-qX {(
1
3
) +
ax j
axk
(
)
ai
This formulation represents a down gradient diffusion of
Reynolds stress.
The
-
(u!u r')
triple correlation terms
(u! 6') and
are modeled as down gradient turbulent diffusion
of potential temperature-velocity covariance and mixing
ratio-velocity covariance, respectively.
e'
au
aue
-Xk(
-
axk
2
(u'u r')
a
ax
[-qX 2 (
-
2
aur'
i
ax j
+
a k
au! r'
)
+
axk
Analogously, the third moments involving potential
temperature variance-velocity correlation, potential
temperature-mixing ratio-velocity correlation and mixing
ratio variance are also modeled as down gradient turbulent
diffusion.
3xk axk
--
a
- xk
22
a
~
axkk
(UO ' ) = --
(u r ') =
2
q
3
-qX
a.
--- [-qX
xaxk
alr-
3r'
3
ax k
The variables A , A2 and X3 are diffusion length scales
which are prescribed.
Kolmogoroff (1941) hypothesized the isotropy of
small-scale turbulence.
In accordance with this widely
accepted hypothesis, the viscous dissipation term
2v(3u!/3x)
ju!Dk
3.xk (au!/ax)
is modeled as proportional to q3
for i=j, and is neglected for the nonisotropic components
j.
i /
au!
au'
3
-
2v-
The dissipation term kT(3O'/axk)(36'/axk)
is
similarly taken to be proportional to the potential temperature variance.
2
2ko36
kT
ael -2 q- 61
ax
A2
Analogously,
the dissipation terms for r'e'
and r'
V
are
modeled as:
(+k
)
''T' axk
211-
3r'
ar'
3xk
axk
ar
axk
-
2 a- r'6'
A2
2 q r"
A2
A1 and A2 are dissipation length scales which are
prescribed.
The remaining modeling assumptions in Table II
Zut
concern diffusion terms of the form - - [k U! 36 +
axk
3 axk
Ok
In the PBL these terms are relatively small, and we neglect
them.
37
a
[k u! 3'
T3 j
axk
au'
-J]
+ vo'
= 0
ak
au!
Sa
4
+r
[riu!
ax k
vr'
3ak
ar'
hout
a [rn6'ar'
36'
--- ]
+ k r'
T
axk
a
aXk
--- 1] =0
axk
[2T2
aXk
=0
axk
=0
Inserting the modeling assumptions (1-27) from
Table II into the appropriate equations in Table I will
yield the modeled level 4 equations.
The complete level
4 model consists of equations (7-9), and (24-29).
M
-1
~
V%~'
A(41
~MAA4
+
ZaJr
-,a;A4
+
V
AtZ
YAIi
*
71(
+
+.
+
A' A-
-
(
C~t
)x
D,
2-
1
kJ: 4 kM
II
J'
0m
4~x'
SC.A
4. '
(24)
)A
+
G '
;I.kj'
Zi AA)Q
-
Dci
D
'C I
3.4
~I'A
)
D
-o
IA.
.44.i
-A
0
-- ;e
G-
_1A
J
~*Ic
(25)
4
f '4 4 k G '
- x3
-7 x
3
*1
x?
xaI&
2t a
~
(26)
I
( .4
' b.)
ts~k
4.A
y',
D*r'
- 6 X,(
1)
0
f
r
ZPA4
V9
q
''
-
-----
4
~ E IA
.
i
-
A
, r'
A.A
D
'
I
r 'I
.
(27)
4-3
.41C '
'I
CAC
-
x--
a/1
!K
w 1k
.
,
r
-
z ";I- (v-Q
r lo
,
(28)
Ir
- 2
-A't
44k r
an7
pYKr
(29)
39
3.2
The Level 3 Model Equations
Mellor and Yamada (1974) make use of the fact that
the departure from isotropy of atmospheric turbulence is
small, to simplify equations (24-29) even further.
non-dimensional departures from isotropy, a..
u~u! E (6j + a. .)q2
13
a..
3
u!6'
I
b i q(
11
Defining
and b.:
=0
~71/2
)/
The level 3 and level 2 models neglect terms of order a 2
and b 2 .
The two levels are distinguished by the fact that
the tendency, advection, and diffusion terms are assumed
to be of order a or b in level 3 and order a2 or b2 in
level 2.
The result for the level 3 model of interest here
is that equations (24-29), representing 15 prognostic
equations for second moments, are reduced to 4 prognostic
equations with 11 diagnostic equations.
equations are for q
,
,
e'r'
and r'2.
The prognostic
The mean variables
require the solution of an additional five prognostic
equations.
The level 3 model, therefore, consists of a
total of 9 prognostic differential equations and 11 diagnostic
equations.
The level 3 equations are collected into
Table III.
Equations (26),
(28), and (29) are unchanged.
40
Table III
Level 3 Model Equations
a)
t
I
.44.
3A 1
3
'1
A4 I.A4.
-+
i.
a-
{
y'1
f(A4.#Aj it
+
_I9
<A,
-
~X'c
.AI'
)
r
+M ---
'
2.
(
1? 1d3A!0
-~~A
(31)
-Za
=
(26)
6
.A
-AA
--
'Ar9'
1 '
-
1(32)
1-
r 1
(28)
*~'~-
-wk
A -- (r'7)
) + M
4-Ic-00 A4
-t.-uj19" 9's i
A46
I
.'
* (-U.'Ac i ' - C 1bI& k')S
cvZt&
4~~r A~1Adj
~
3
=
(r''
----
Ai Ir
--
~
?XA
)7r~ e1z4
r --
-
(~% M~'Q'~'
+.Moe
I
.) Y,
(30)
U
S
4ko4
e
I
-A
=I -/
Xft
+ 2 (S z
.AA~
(Z
-
z
=..-
[bN3
a,
r
.J
~Z
..
.. A.6~
(33)
(pg'r
-
.At
(29)
41
In the PBL, the horizontal length scale is much
greater than the vertical scale height.
The final equations,
therefore, contain only z derivatives of perturbation
quantities.
Neglecting advection of mean velocity and
mean mixing ratio by the mean wind while retaining temperature advection (because it may be estimated by the thermal
wind relationship) will yield equations (34-37) for the
In this model, the vertical velocity,
mean variables.
w, has been set equal to zero at all levels.
Also,
virtual potential temperature, EvI is used to take into
account the presence of water vapor and its effect on density.
The term of equation (9) containing the kinematic diffusivity of water vapor is neglected.
au-
fv
at
az
a
av +-f
--
at
+
=-
~
-
a(WWI)
ax
v +
=)
-
ay
v
-
-a
p
ax
ap
(34)
1
ap
35
az
(w'O') + a
(36)
(37)
(w'r')
An estimation of the horizontal virtual potential
temperature gradient can be obtained as follows
(Hess,
1959).
=
u
Geostrophically,
g
V
=-
g
1
aP
--
fpo0
y
--
-
(38)
ax
1
fp
--
(39)
3x
fv
therefore,
g
- a inP
= RTax
ax
Taking a 2 derivative of this expression and using the
hydrostatic approximation,
a ln
ax
P
T
(-
R
axT ((
)
-
Therefore,
with
v
-X-
G = T
vv
aT
ax
)
-
v
afT
ax
-
fi
I
az
+ (1 + 0.61R)v z
d
fT
V
g
(40)
30
similarly,
ay
fT
g
g
au
(41)
-3z
(38-41) in equations (34-37)
The approximations
will yield the final equations (42-45) for the mean variables.
All the final equations appear in Table IV.
The final equations for the second moments are
obtained from the equations of Table III by neglecting
advection by the mean wind and retaining only vertical
derivatives of perturbation variables in an analogous
fashion with the procedure for the mean variable equations.
The results are equations
(46-61) in Table IV.
Equations
(46, 50, 51, and 52) represent only three independent
equations because q 2
2.
The Coriolis terms appear
in all the appropriate equations in Table IV.
Table IV
The Final Level 3 Model Equations
Equations for the Mean Variables
rZ A
-;7
=
-%
jT,
Z4
(42)
)
,1
(43)
IV'I AO
i
- -
(44)
;;6
- --
(
AA
(45)
')
Equations for the Second Moments
2L(f
A
3I )l~
w
(46)
1
2'
+ 2p MGI-
D IVr~..
) ',C -A
41
.
21C ( e,,,'1.)
(48)
w
41
"^-
&
.....
a~
Op
(47)
J-
~~~~
t
) ::
-222->-.~
r
-
--....... 1
Z
' .
-Z 2
'
- --
'r'
-
-
-
.- ---- -
2
--
.--
'G
(49)
eL-
-
as
/AA
-o
a I
AA
3m
+za1 av -
AA
195
3A
AA iAr
A
oe-
+V- *AAY
.AA
Lc
..--
A'Z -2
VAr
A -
AA v
)
AI
(51)
-AAf
(52)
-+fe Z
(53)
- -24
(54)
(55)
-
- C
3f5
..- A
64M4
1+1
--
AA IAO
z
ur'g,'I
(50)
ii-
-- Cf
Pa
+ 2
I
-
72
..--
Z^riA'
-
4-L*M4
"Ae,
(56)
AA
2.
p
~
~
2 --(57)
,~e
~;
6
--rs
T-
IzI
3A
--
l-,om
IV
(58)
2
-U
-
*
'AAr
2
VA
pa
'
i
. 4 ' r' I
-A
.4
tI
Ir1
tv'r
1
(59)
(60)
(61)
The Length Scale Formulation
3.3
The level 3 equations contain three types of length
The A's are dissipation length scales.
scales.
The
X's and Z's are diffusion and return-to-isotropy length
scales, respectively.
Every modeled term (see Table II)
contains one of these length scales.
Mellor and Yamada
(1974) assume all the length scales are proportional and
are given by:
(zl1
2 ,A1
,A2 1 '1 1 X2 'X3 )
=
(0.78, 0.79, 15.0, 8.0, 0.23, 0.23, 0.23)k
(62)
They evaluate k using Blackadar's interpolation formula
(Blackadar, 1962).
z =
(63)
kz
1 +
z
0
Therefore,
k
+kz
0
as
z+
as
z
0
+
Mellor and Yamada proposed equation (64)
formulation
for %
as a
based on the turbulence energy profile.
47
zqdz
a
t,=
d
f'qdz
,
(64)
a constant
In the analysis of Yamada and Mellor (1975), a
was assumed to be 0.10.
A sensitivity test, however, showed
that the mean variables were fairly insensitive to a 50%
reduction in
a.
The turbulence quantities, unfortunately,
are not insensitive to the value of a.
Using a typical early afternoon distribution of
2
q , 9(z)
is
evaluated for values for a of 0.05 and 0.10
(Figure 1).
The PBL top (h) is indicated.
is within 10% of Z
at about z/h
at z/h = 0.7 for a = 0.10.
The length scale
= 0.5 for a = 0.05 and
Above this, Z changes very
little.
A new determination of k
(equations 65) is proposed
which yields a Z(z) profile similar in shape in the PBL
to Deardorff's (1973, 1974) profile of the turbulence
energy dissipation length scale.
Figure 2 shows Z(z) for
the same q2 distribution as Figure 1.
h{zqdz
£
=
a(z)
h
0qdz
(65a)
48
where,
h
<. h
aiz
cX(z)
a1 -
=
h/2
/
-
z
) (-t
2)
=
(0.10,
< z
(65b)
< h
h < z
a2
a
h
(65c)
0.05)
As one moves from h/2 down to the ground, the length
scale decreases toward zero.
As the ground is approached,
the characteristic size of the turbulent eddies is limited
by z, the distance to the solid boundary.
approaches kz.
The length scale
For large z, the influence of the ground in
limiting the characteristic eddy size diminishes quickly.
As z increases beyond h/2, however, there is another factor
influencing the turbulence structure, and that is the
temperature inversion base at h.
eddies are probably found in
The larger turbulent
the middle of the PBL,
where
the distance away from any damping influence on their size
is
maximized.
Above h, Deardorff (1974)
points out that k
increases with height because the perturbation energy is
contained in gravity waves exhibting little diffusion or
dissipation of energy.
The perturbation energy contained
1
- -
2000
0
-
18Q
0.10
.05
16001400-
-1200-
E
h
1000 -
1800 600400200= KZ
s
0
10
20
LENGTH
Fig. 1.
40
30
SCALE
,
J
(m)
Vertical profiles of the length scale, Z,
for two values of a, based on the
formulation of Yamada and Mellor (1975).
2000
18001600-
14000-1200E
h
-
1000-
I800600400-
200-
0
10
20
LENGTH
Fig. 2.
30
SCALE,
40
f
(m)
Vertical profile of the length scale, Z,
based on the new formulation defined by
equations (65).
50
in this region is negligible compared to q2 in
the PBL,
and the length scale formulation above h has little effect
on the turbulence structure in the PBL.
The length scale formulation represented by
equation
(64)
(Figure 1) does not allow the stably stratified
layer above the PBL to have any influence on reducing the
characteristic eddy size (and therefore the length scale).
Equations (65), however, force Z to approach, in the
region z > h/2, a smaller, constant value.
For these
reasons, we have adopted this formulation for Z.
52
4.
4.1
BOUNDARY CONDITIONS
Mean Variables
It is necessary to provide an upper and a lower
boundary condition for each prognostic variable in order to
find solutions of the level 3 equations.
A staggered grid
is used in which the mean variables are defined at the integer
grid point levels.
Turbulence variables and z derivatives
of mean quantities are defined at the half integer grid
points.
The lowest integer grid point is at the roughness
height, z,(0.01 m).
The top of the grid (grid point
43.5) is at 2022 m.
Appendix B contains a more complete
description of the grid system.
The definition of the roughness height provides the
lower boundary conditions for u and v:
u (z ) = 0.0
(66)
v (z ) = 0.0
(67)
A simplified, single layer ground thermodynacmis
model (Ha forcing, Deardorff, 1978) was used to predict
the values of Gv and R at z
.
The ground temperature,
Tg, is predicted by the model based on net radiative heating
(or cooling), phase changes of the ground water (or frost),
and the heat flux at grid point 1/2
(0.07 m).
The value
of Gv (z ) is assumed to be equal to the virtual potential
ground temperature 0v (0).
)v(z0 ), therefore, is probably
somewhat too high during unstable conditions and slightly
underestimated during stable conditions.
The mixing ratio at z0 is determined utilizing soil
wetness parameters and the turbulent moisture flux at 0.07 m.
The upper boundary conditions are applied at the
The vertical virtual potential temperature
2022 m level.
gradient is assumed to approach a constant (stable) value.
The vertical derivatives of u, V, and R are assumed to be
zero (Yamada and Mellor, 1975; Deardorff, 1973).
Therefore,
at 2022 m,
v = 0.001
=v 0
4.2
(68)
k/m
R=
0
(69a,b,c)
Second Moments
The second moments requiring boundary conditions
ar: q 2 ,e'2 , r' 2 , and r'E'. The lower boundary conditions
are:
v
v
are obtained by assuming each of the preceding variables
is in equilibrium at the lowest half integer grid point
(0.07 m).
The time derivatives of equations (46-49),
therefore, vanish, yielding:
54
q2(0.07 m) =
-
2v'w'
-
a
[q
{-
(70)
3zV + 2Sgw'O' }
A2
,
(0.07 m)
2
r' (0.07 m)
= 2q{3
A
Tq -
=
[qX3
.
[
I[qX 3
3
3
(0.07 m)
[ql2
(71)
(72)
vaz
]
ar'
{
=
}V
- 2w''
]
7
A
r'e'
2u'w'
vaz
v
- w'
w'r'
}
(73)
The calculated value of the ratio
at z = 0.07 m, was sometimes in
- V)l/2,
v
excess of 1. Whenever this occurred, equation (73) was
r' v l/(r2
replaced by:
r''
=(r
- 61)1/2
v
z = 0.07 m.
This restriction of r'e'v was applied only at the lowest
grid point. Wyngaard et al. (1978) reported observed
r'-6O v correlation coefficients, above a warm evaporating
surface,
very close to unity.
Yamada and Mellor (1975) utilize boundary condi-
tions for q2 and 0'2
v of the form:
(74)
55
(75)
u2
q 2(0)
2
(0)
(C ,C2 ) = (0.61, 2.4),
where
2
u=
(76)
C2 H
=
2
2 1/2
+ (-v'w'
[(-u'w'(0))
(0))
]
,
and
H = -w''
(0)
The model with equations (70-71) as boundary conditions
2 2 2
2 2
yields ratios q /u, and e' ut/H within a few percent of
C
and C2 , respectively.
The equilibrium boundary condi-
tions, therefore, are consistent with observed surface
turbulence structure, yet do not require the use of
empirical constants.
The upper boundary conditions for the second
moments consist of the requirements that q ,
'
v
,
r
,
and r'Q'v vanish at the upper boundary (2022 m).
q2
0
r'
=0
,
2
,
0
rev =0
(76a,b)
(76c,d)
56
The 2022 m level is
sufficiently high to ensure the
PBL is contained within the grid, and the turbulence moments
can be expected to quickly approach zero outside of the
PBL.
57
SOLUTION OF THE EQUATIONS
5.
Reducing the Prognostic Equations
into a Single Form
5.1
When the mean velocity components are expressed in
complex form, u + iv, equations (42-49) can be reduced to
a single form:
S
at
=-__
az
(P
a
1 3z
)
-P
2
(77)
3
where $ is the variable to be incremented in time.
Table V summarizes the discussion to follow and contains
the P's ,for each $.
The Reynolds stress and heat flux equations
(50-58) are 9 simultaneous equations.
Yamada and
Mellor (1974) provide a solution of the equations with
the Coriolis terms omitted.
The Coriolis terms, however,
complicate these equations considerably.
Appendix A con-
tains the details of a solution of the 9 equations with
the Coriolis terms,
utilizing a back-substitution method.
The solution for u'w' and Vw'
is shown to reduce to
Yamada's and Mellor's solution in the special case of no
Coriolis terms.
Also contained in Appendix A is the
solution of the 3 moisture flux equations.
Due to the complexity of the expressions for u'w',
v'w', and w'0', the flux divergence terms of equations (42-44)
are identified with P 3 in Table V.
The moisture flux
Table V
Representation of the Prognostic Equations in a Standard Form
Equation
P2
123
No.
42, 43
vc
0
44
0v
0
P
ifvcg
if
fT
0
g
R
46
q2
1
v
qX
v
c
=u +iv,
__
A_
_
--w
cg
u
g
+ iv
-3
+ B
-2w'r'
A
v
vz
g
,
v =u
c
+ iv'
v
+ 2fgw'6'
__
__
v az
+ A
A2
(w'e')
-R
az
lo
-2wrv
A
3
az
2v'w'
__
__
9
r93
49
-
2q
q2
48
-
z
2q
qq-2uw'
617
u
0
K
w
____3
47
-
z
g
45
av
au
-
v
__
__
_
59
divergence term of equation (45), however, is written as:
(-w'r') =
(78)
R)
(K
where Kw and yR are evaluated in Appendix A.
The identification of the terms of the second
moment equations (46-49) with the P's is straightforward.
It was necessary, however, to write the w' r'
equation (48)
wir'
=
term of
as:
-
[Ar'O'
B]
where A and B are evaluated in Appendix A.
(79)
Equation (79)
allows the r'6'v dependence of w'r' to be evaluated implicitly,
thereby eliminating some numerical stability problems which
were encountered with the r'O'v equation.
Finite-Difference Approximation
5.2
A transformed coordinate system is used in the
model.
Vertical derivatives of any quantity, $,are evaluated
by:
azi.
=a.
j
j+1/2
-
1$
(80)
Equation (80) is the finite-difference form of equation (B-3)
(see Appendix B).
Equation (77) is approximated with the
following finite-difference equation:
k+l
k
2
{[(Pla) k+
+ (P a)
(P2 ) k+
-
(Yamada and Mellor,
+
-
[P a)
+
k +kl
+l
k
k
k
k+1
+ [(P1a) + (P1a)
]$
k~ + 1
k
k
+ 2(P 1a) J+ (P1a)
2P
+t
] k+
]
k+1
(81)
(P3 )
1975).
This finite difference scheme is an implicit scheme
2
with truncation errors of order 6t and (6x) (Richtmyer
and Morton,
1967).
The P1 for equation (45),
at the integer grid points,
levels.
i.e., Kw,
is not defined
but only at half integer grid
The terms of equation (81) involving [(P a)j+
+ (P a) -.]/2 are therefore replaced by (P a)j/
The Gaussian elimination method is used to solve
the implicit finite difference equation (from Richtmyer
and Morton, 1967).
Expressing equation (81) in the follow-
ing form:
-A
k k+l
k k+l
k k+l + B.k
-C $k 1
-3
3 J
1 j+1
=
D
k
where the coefficients A., B., C., and D
(82)
are all known
at time step k.
k+1
=
E
J
J
k+1 e
j+1
(83)
e
equation (83)
$k+1 + F
F.
=E.E
(
I
with j = j-1,
k+l
$.
Solutions are assumed to be of the form:
becomes:
(84)
1
Inserting equation (84) into equation (82)8,
A.
k+l
j
B.
J
-
k+1
C.E.
J J-1
j+1
+
D
+ C F j-
B.J
J J-l
- C.E.
(85)
and comparing equation (83) with (85) , gives expressions
for E. and F..
J
E. =
E
J
A.
B.
C IE
B - CE
(86)
D. + C.F.
j
B
-
C
(87)
E1
Applying the appropriate bottom boundary conditions
will yield E
and F1 .
Knowing E, and F1 , as well as the
coefficients A, B, C, and D, E2 and F 2 can be determined
62
with equations (86-87).
Applying this procedure for each
level, all the E's and F's can be determined.
The upper boundary conditions and equation (83) can
be used to determine $ at the top level.
Because all the
E's and F's are now known, equation (83) can be used to
determine all the $'s,
down to the bottom.
at time step k+l, from the top level
6.
6.1
SIMULATION OF DAY 33 OF THE WANGARA EXPERIMENT
Initial Conditions
Initial values for the mean variables, u, v,
v
and R are the observed values at 0900 hours of Day 33 of
the Wangara Experiment (Figures 3-5).
Clarke et al.
(1971)
tabulate values for these variables at 50 m intervals below
1000 m and at 100 m intervals between 1000 m and 2000 m.
Values at the grid levels are interpolated from the observed
values.
The Tv profile has been smoothed in the region
400-800 m to remove a slightly unstable lapse rate in that
area.
Initial values for the turbulence variables are
generated by running the model for 1 hour starting with
guessed values for the second moments (Yamada and Mellor,
1975).
The initial and subsequent values of the geostrophic
wind components, u
and v , were also calculated in the
same way as Yamada and Mellor, using their data.
6.2
Results
The time-height variation of the velocity components
u and v are presented in Figures 6 and 7.
Agreement with
the results of Yamada and Mellor and with observations is
good.
The nearly constant velocity profiles in the mixed
layer during the day and the development of a low-level
nocturnal jet are features of the velocity profiles noted
by Yamada and Mellor which are also apparent in Figures 6
and 7.
64
2000---'
18001600-
140012001000-
800-
600400-
200-
-
00
I
2
3
VIRTUAL
Fig. 3.
4
5
6
TEMPERATURE,
7
8
fv
(*C)
Initial profile for virtual temperature,
V
9
65
2000
1800 -
1600 -
1400"-
1200 -
E
1000 -
00-
400 -
200 -
0
2
1
WATER VAPOR
Fig.
4.
4
3
MIXING
RATIO,
R
5
(g/kg)
Initial profile for water vapor mixing
ratio,
R.
66
2000
1800160014001200E1000800
600-
400
200-
-3
-2
VELOCITY
Fig. 5.
-1
COMPONENTS
0
0 AND
1
V
m/ s)
Initial profiles for the eastward velocity
component, u!, and the northward velocity
component, v.
U
(m/s)
2000
1500
1-1000
-
500-
-2
-2
-4
-6
-8
-10
10-02
100 10
12
14
16
18
TIME
Fig. 6.
20
(HOUR)
22
24
DAY
2
4
6
8
33134
Variation of the calculated mean velocity component, u, as
a function of time and height.
-
10
12
14
V
(m/ s)
16
18
TIME
Fig. 7.
(HOUR)
20
22
24
DAY
33 34
2
4
6
Variation of the calculated mean velocity component
as a function of time and height.
8
69
Figure 8 contains the mean virtual potential temperature variation.
The rapid growth of the mixed layer with
nearly constant jv can be seen.
The surface layer is
super-adiabatic during the late -morning and afternoon
hours as solar radiation heats the ground.
surface inversion develops after sunset.
A strong
The time varia-
tion of the ground 0v, predicted by the ground thermodynamics
model is shown in Figure 9.
(1974) values is good.
Agreement with Deardorff's
The fall of the ground temperature
after sunset is halted by the freezing of ground water,
which releases latent heat.
The ice is slow to melt
during the morning of Day 34.
This is a deficiency of the
single layer ground model which keeps the ground temperature
at 0*C until the ground ice in the entire layer has melted.
Examination of the observed jv variation (Yamada
and Mellor, 1975) indicates that the daytime temperatures
in Figure 8 are slightly too high.
to the assumption that 0
This can be attributed
(ground) = 0v (z0 ).
temperature and heat flux at z
The air
are somewnat overestimated
during the day, yielding a warmer mixed layer.
an additional simulation was run with 0v (z)
As a test,
reduced by
about 8% (by increasing the ground albedo from 0.2 to 0.3).
The resulting Gv variation matched the observations very
closely
(also see Figure 28).
Yamada and Mellor reported that the longwave flux
divergence term of the 0v equation influenced the predicted
nu
(*C)
20
1500
1000
500
10
12
16
14
TIME
Fig. 8.
18
(HOURS)
20
22
24
DAY
33 34
2
4
6
Variation of the calculated mean virtual potential
temperature, jv, as a function of time and height.
8
71
40-
Deardorff
30-
(*C)
20-
10-
0I
10
I
I
12
I
I
I
14
I
16
I
I
18
I
S
20
I
I
22
I
I
24
|
|
t
2
4
6
DAY
TIME
Fig.
9.
(HOURS)
33,34
Calculated surface 0v as a function of time.
i
i
8
72
nighttime 0v values measurably.
Our neglect of the radia-
tion term explains the warmer nighttime 0
values in-
dicated in Figure 8.
The mean water vapor mixing
in Figure 10.
ratio,
R (z ) is shown in Figure 11.
R, is shown
Moisture in
the surface layer increases in the morning as the ground
water evaporates.
As the mixed layer develops from
1000-1200 hours, the surface moisture is carried upward
(resulting in
this time).
the bulge of the 3.5 and 4.0 contour lines at
The afternoon boundary layer dries out because
the moisture flux at the boundary layer top exceeds the
surface moisture flux as all the soil moisture evaporates.
The time-height variation of twice the turbulence
kinetic energy is shown in Figure 12.
The development of
the strong daytime turbulence is due to bouyant generation
(see Figure 13).
Stress production and diffusion are
negligible except close to the ground.
At the end of the
day bouyant generation becomes small or negative.
The
dissipation of turbulence energy, proportional to q3 is now
unopposed and quickly eliminates most of the turbulence.
A
second effect is the variation of the length scale (Figure 14).
As the level of turbulence in the boundary layer decreases,
the length scale also decreases.
Dissipation, being pro-
portional to l/Z, increases for a given value of q3
Figure 14 compares closely with the variation of Z calculated
by Yamada and Mellor (1975) in the region z < 500 m.
SQ/ Kg)
2000[
1
1
9
-11
1
1500
1.0
1000 -2.0
E-
-- 3.0
500
3.5
100-
4.0
10
-
10
12
14
16
TIME
Fig. 10.
18
(HOURS)
20
22
24
DAY
33|34
2
4
6
8
Variation of the calculated mean water vapor mixing ratio,
R, as a function of time and height.
74
I
I
I
I
I
I
I
-I
I
I
I
*
I
I
I
I
'
10
987o
Deardorff
10
12
(1974)
(9/kg)
5a
4
3-
14
16
TIME
Fig. 11.
18
(-HOURS)
20
22
24
2
4
6
DAY
33 34
Calculated surface R as a function of time.
8
2
10
12
14
TIME
Fig. 12.
(m 2-2
16
(HOURS)
18
20
22
24
DAY
33134
2
4
6
8
Variation of the calculated values of twice the
turbulence kinetic energy, q2 , as a function of time
and height.
DAY
33
1300 HRS
1.2
I.'
0.9
0.8
0.7
0.6
0.5
04
0.3
o.2
0.1
0.01-2.0
0
-1.0
X 10~
TERMS
-50
-40
-30
OF
THE
-20
TURBULENCE
-10
0
X 10-2 m2
Fig. 13.
0,4
2
0.8
1.6
2.0
-3
ENERGY
10
,-s
1.2
20
EQUATION
30
40
Terms of the turbulence kinetic energy equation
at 1300 hours, Day 33.
50
J(m)
2000
1500
1000
12
10
14
16
TIME
Fig.
14.
18
20
(HOUR S)
22
2
24
DAY
4
6
8
33134
Variation of the length scale,
and height.
k, as a function of time
Their formulation, however, continues to increase Z with
height, approaching a constant limiting value.
Equations (65)
force Z to decrease in the upper portion of the boundary
layer to a smaller, constant value in the stable region
above the boundary layer.
The boundary layer height, h,
tion of Z, is of interest
itself.
used in
the calcula-
It is defined here
as the layer of the atmosphere containing essentially all
the dissipation of turbulence kinetic energy.
The model
calculates h by integrating the dissipation of q 2 up from
the ground until adding the dissipation in the next grid
layer to the integration adds less than 1% of total inteAt this point, the inte-
grated amount below this level.
gration stops, and h is determined.
This definition of h
yields boundary layer heights at, or within, 1 grid point
of the base of the temperature inversion during the day,
yet also gives a reasonable estimate for the boundary layer
height at night, when identification of h from 0v or R
profiles is difficult.
Figure 15 shows h and the U
profiles for 1100 and 1200 hours.
The turbulence kinetic
energy profiles for these times is shown in Figure 16.
The evolution of the boundary layer height (Figure 17) is
interesting.
h grows rapidly in the morning hours, then
more slowly in the afternoon.
About an hour after sunset
the boundary layer height crashes to a small, more or less
constant value (108 m) through most of the night.
-~
E
I'dUU-
10000800h
60040020010
1200
1100
11
12
13
15
14
0"
Fig. 15.
16
17
18
19
20
21
(*C)
Virtual potential temperature profiles for 11O hours and 1200 hours,
Day 33. The calculated PBL top for
each time is indicated by h.
0.2 0.4 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
q 2 (m2 s.)
Fig.
16.
Vertical profiles of q2 (twice the
turbulence energy) for 1100 hours and
1200 hours, Day 33 . The calculated
PBL top for each time is indicated
by h.
h (m)
2000
r
1800
1600014001200
1000w800-
600-
400200-
F
10
12
14
16
1-i i1 i1 i i1 f i l l
18
20
TIME
Fig. 17.
22
24
2
g 4g
6
8
(HOURS)
The calculated boundary layer height, h, as a function
of time.
The time-height variation or the virtual potential
2
temperature variance, 0' , is shown in Figure 18. The
v
maximum values of el2 are found at the z grid level during
v
0
the day, and approximately 20-50 m above the ground at
night.
The e'
v
budget,
Figure 19,
shows a balance between
gradient production and dissipation of 6
v
entire boundary layer.
The r'
throughout the
time-height variation, Figure 20, shows
two areas of high mixing ratio variance, at the ground and
at the boundary layer top.
r'
,
The gradient production of
Figure 21, is strongest near the PBL top where the
R profile decreases rapidly with height.
important only around
above
h,
h and increase r'
Diffusion is.
tending to decrease r'
at
h.
just
The production of r'
is negligible throughout the mixed layer, except in a shallow
layer near the ground.
Figure 22 contains the mixing ratio-virtual potential
temperature correlation, r'e', as a function of time and
v
height.
The model yields positive values for r'0'v
throughout the boundary layer during the day.
In the stable
region above the boundary layer, where 6v rapidly increases
v
and R rapidly decreases with height, negative values of
r'6'v occur.
Throughout the shallow nighttime boundary layer
(about 100 m),
r'0' is negative.
Figure 23 reveals that
the dissipation term and the production term are in balance
12i
12
I
i
2000
I
y
2
(*K)
I I
S
S
I
I
S
I
1500-
E
1000-
-
.0.01
5000.05
10
5,
10
12
t.,
14
16
18
20
22
24
2
4
6
8
D AY
T IME
Fig. 18.
(HOUR S)334
Variation of the calculated virtual potential temperature variance, v'2 as a function of time and height.
0.05
00
1300 HRS
33
DAY
.Z 0,
h
-10
TERMS
002
j
h
5
I:
OF
I
0
2
4
X'10' 3
(*K
S
-2
-4
-6
-8
THE
I
i
I
I
F
I
II
PRODUCTION
200
-200
(0K)2
(*K
Fig. 19.
I
DIFFUSION
-400
8
EQUATION
8'
I
6
400
S'I
S
Terms of the virtual potential temperature
variance equation at 1300 hours, Day 33.
10
Kg-12
(1Kg
2000
1500
0-s
-10-s
--
1000
I
-0-.
10710
5001
100
10
--
1l0-
10
12
14
16
TIME
Fig.
20.
I8
20
(HOURS)
22
24
DAY
2
4
6
8
33134
Variation of the calculated water vapor mixing ratio
variance, r7, as a function of time and height.
DAY
33
1300 HRS
1.2
1.1
0.9
0.8
0.7
0.6
0.4
0.3
0.1
-10
-12
-14
{/
TERMS
----
L
h
-6
-8
OF
-4
THE
-2
2
0
X 1010
I
-18
-12
-9
-6
10
I
I
12
15
12
14
18
21
Af
-3
0
X 10~8
Fig. 21.
8
S-I
PRODUCTION
DISSIPATION
-15
I
I
-
-21
2
EQUATION
r
-
----
6
4
(Kg Kg~)
3
6
9
(Kg Kg -'I
s'
Terms of the water vapor mixing ratio variance
equation at 1300 hours, Day 33.
r'
e1
(0 K Kg Kgo'.)
201
1500
1000
DAY
TIME
Fig. 22.
(HOURS)
33 34
Variation of the calculated water vapor mixing ratiovirtual potential temperature covariance, r'ev, as a
T'
Areas of negative
function of time and height.
hatching.
are signified by
88
DAY
-10
-6
-8
TERMS
-4
OF
- -2
THE
1300 HRS
33
0
X 10-r
6
42
(* K S~ Kg Kg- )
8
10
EQUATION
r'v
-. 02h
.h
0
-2
-1
XIO'
Fig.
2 3.
2
0
(*K
s~ Kg Kg )
Terms of the r'6'V equation at 1300 hours, Day 3 3.
everywhere.
h
Dissipation and production change signs above
where r'O'V is negative.
The variation of the diagnostically determined
Reynolds stresses (u'w', v'w'), vertical heat flux
(w'e') and vertical moisture flux (w'r') as a function of
height and time are presented in
Figures 24-27.
Small
negative values of the heat flux (2-8% of the surface heat
flux) were calculated just above the afternoon boundary
layer top.
Yamada and Mellor (1975) reported downward heat
fluxes above the PBL top of a maximum of 2% of the surface
values.
Deardorff's (1974) model calculates an average
negative heat flux of 13% of the surface value.
The surface
(z
level) values of w'6', w'r'
,
and the friction velocity, u,, are shown in Figures 28-30.
The results of Deardorff (1974) and Yamada and Mellor
(1975) are also shown, when available, on these figures
for comparison.
with Deardorff's.
The surface moisture flux compares well
The calculated high values of w'O'v
have already been attributed to the assumption
v (z ) =
v
(ground).
Values of the individual turbulence energy components
in the unstable boundary layer show an anisotropic distribution.
equal.
The components u'
and v'
are approximately
The vertical energy component, w' , however, is
usually more than twice the other two components.
This
is due to the direct transfer of bouyant energy to the w'
M2 s'2 )
UI wi
E
10
12
14
16
18
20
22
24
2
4
6
8
DAY
TIME
Fig. 24.
(HOURS)
33134
Variation of the calculated Reynolds stress component,
u'w',
as a function of time and height.
v'iw'
( M2
''
)
2000
w
I'500
12
10
14
16
TIME
Fig.
25.
18
(HOURS)
20
22
24
DAY
2
4
6
8
33134
Variation of the calculated Reynolds stress component,
v'w', as a function of time and height.
w'
fv
I m S~'*K )
2000
1500
1000
E
500
100
10
TIME
Fig.
26.
(HOURS)
DAY
33134
Variation of the heat flux component, w'T',
function of time and height.
V
as a
w r'
X 10
(m s' g k
)
200
1500
1000
500
12
10
14
16
22
20
18
TIME
(HOURS)
24
2
4
6
8
DAY
Fig.
27.
33134
Variation of the moisture flux component, w'rT, as a
function of time and height.
10
12
TIME
Fig. 28.
14
16
18
(HOU RS)
20
22
24
TAY
33134
2
4
6
8
Calculated surface heat flux for two
values of the surface albedo, as a function
of time. The values of Deardorff (1974),
and Yamada and Mellor (1975) are also shown.
95
t
40
0
E 30
0
20
11.
I-
(n
0
-
10
12
TIME
Fig. 29.
14
16
(HOURS)
18
20
22
24
0 AY
33
2
4
6
34
Calculated surface moisture flux as
a function of time.
The results of
Deardorff (1974) are also shown.
8
l
1
0.50 -
i
l
t
$
I
0,45-
0.400.350,300,25
I-A
VAMAOAa MELLOR
0.20 -
(1975){
0.)
z0.15
S0.10
La.
0.0510
12
TIME
Fig. 30.
14
16
(HOURS)
18
20
22
24
2
4
6
DAY
33:34
Calculated friction velocity, u,, as
The results of
a function of time.
Yamada and Mellor (1975) are also
shown.
8
97
2,0
-Il
-0.8
-0.6
-0.4
-0,2
0.0
0.2
0.4
0.6
0.8
.
0.765
CORRELATION
Fig. 31.
COEFFICIENT
wI
&e
w'
6,'
The correlation coefficient for w'-6'
as a function of z/h, for 1300 hoursV
and 2000 hours, Day 33.
L.O
98
component (Deardorff, 1973).
The return-to-isotropy
(pressure correlation) term, proportional to 1/2,
to eliminate this anisotropy.
tries
The bouyant energy - w'
energy transfer is exemplified in Figure 31.
The w'-O'
V
correlation coefficient is a high constant value (.765)
throughout the daytime PBL.
The nocturnal (8:00 pm) boundary
layer profile of the correlation coefficient clearly does
not exhibit this behavior.
99
7.
EFFECTS OF THE CORIOLIS TERMS ON TURBULENT
FLUXES IN THE PBL
Most higher order closure models of the PBL contain assumptions to the effect that the Coriolis terms of
the turbulence moment equations are negligible
Mellor and Yamada, 1974, 1975).
(e.g.,
Wyngaard et al. (1974),
however, discuss the importance of the Coriolis terms in the
Reynolds stress equations.
In
importance of these terms in
order to evaluate the
influencing the results in
the simulated boundary layer, runs were made in which the
rotation terms were included
(turned on) and set equal to
zero (turned off).
The inclusion of the Coriolis terms in
the u!u!
1 J
and u!6' equations presented no problems other than to
increase the complexity of the equations.
Computational
problems, however, were encountered with the level 3 moisture
equations with rotation.
A-18)
An examination of equations (A-16 -
for w'r' indicates that the effect of the Coriolis
terms will be to increase the moisture flux if
flux is
in
the same direction
moisture flux.
If
the momentum
(upward or downward)
the momentum is
in
as the
the opposite direction
of the moisture flux, the Coriolis terms tend to decrease
Consider a situation in which 3R/az = 0.
w'r'.
(A-21)
w' r'
=
Equation
reduces to:
Ar'' v
(87)
100
A is defined by equation (A-22).
Equation (48) is
approximately:
- w'r'
(r'6')
-(A
=
r'6
-
v+
q
r'
Usually the second term in
unstable boundary layer.
(88)
parentheses dominates in
the
In the stable region above the
boundary layer, the first term is usually larger.
of the terms of equation (A-22)
An analysis
reveals:
2
A ~
q
(89)
gq
+ 9k2 qfy li
The turbulence in the region above the boundary layer is
weak, indicating q
3.
is a small positive number.
Since fy
is positive, a.region of au/3z < 0 can make A negative.
This occurred in the stable region above h.
Equation (88)
reduces to:
C r'O'
1
v
v
at (r'O')
where C1
-A 3
exponential.
(90)
> 0.
The resulting solution is a growing
101
In order to evaluate the effects of the Coriolis
equations it was necessary to
terms in the u.u'
i v
i 3 and u.''
set fy = fz = 0 in the moisture equations above the PBL
The moisture equations are decoupled from the other
top.
No moisture variable is
equations.
and,
or u.6'
i v
of uu!, 3
i
therefore,
used in
the Reynolds stresses and
heat fluxes are unaffected by this change in
In addition,
equations.
the calculation
the u!r'
the boundary
the area of interest is
layer, and in this region,
no restriction on fy or fz was
necessary.
Figures 32-35 contain u'w',
w'r'
v'w',
w '',v
and
profiles for runs with and without this Coriolis
terms.
There is
no difference in
the linear w'0'v
throughout the boundary layer (Figure 34).
profile
The w'r'
profile also shows little sensitivity to the inclusion of
rotation.
The Reynolds stresscomponents u'w' and v'w',
however, are influenced somewhat (Figures 32-33).
The u'w'
profile retains the same shape as without rotation, but
the v'w' profile is- flattened.
None of the prognostic turbulence equations con2 ~~
Their inclusion affects q , 6'v
tains Coriolis terms.
r'
and r'0'
only through the diagnostically determined
Time-height variation plots
u!u!, u.'', and u.r' equations.
i
i v
i 3
for the mean variables and the prognostic turbulence variables
are almost identical for runs with and without the Coriolis
terms.
The effects of the Coriolis terms, therefore, are
102
.0
0.9 0.80.7
0.6h 0.5 -
040.30.20-1 -
-.03
-. 02
U W' (
Fig. 32.
0
01
-.
2
.01
.02
.03
S2 )
Profiles of the Reynolds stress component
u'w' at 1200 hours, Day 33, for two cases
[Coriolis terms on (---)
and Coriolis
terms off (-)]
as a function of z/h.
103
h
0.5
-.0!
0
.01
0.02
.03
.04
( M2 C-2
Fig. 33.
Profiles of the Reynolds stress component
v'w' at 1200 hours, Day 33, for two cases
[Coriolis terms on (---) and Coriolis
as a function of z/h.
terms off (-)
.05
104
1.0
0.9
0.80.70.6h.
5
04 0.30.20.1-
0
w 1 e"
Fig. 34.
.1
.2
.3
.4
(in s-1 K)
Profiles of the heat flux component w'6'
at 1200 hours, Day 33, for two cases
[Coriolis terms on (---) and Coriolis
. as a function of z/h.
terms off (-)
.5
105
543-
I
0
2
'"""
wr
Fig.
35.
(m S
3
-
4
1-1
kg kg
6
5
)
x 10
-5
Profiles of the moisture flux component,
w r', at 1200 hours, Day 33, for two
cases [Coriolis terms on
Coriolis terms off (-)]
of z/h.
and
function
a
as
(---)
7
106
felt mainly in the details of the uIw' and v'w' profiles.
These terms can therefore usually be safely neglected.
107
EVALUATION OF THE SIMILARITY FUNCTIONS
8.
A, B, C AND D
8.1
Definition of the Scales and Derivation
of A, B, C, and D
The similarity theory proposed by Monin and
Obukhov
(1954)
for a stationary, horizontally
assumes,
homogeneous flow, that the surface layer profiles of all
mean and turbulent variables, when appropriately nondimensionalized,
are universal functions of a small number
of dimensionless parameters.
The theory is applied in
the surface layer where the Reynolds stress, and the heat
and moisture fluxes may be considered constant with height.
It is assumed that the surface layer structure is determined
by the dimensional variables:
z,
z
, g/G
The flow is
,u,,w'
0
, w'r'
(91)
.
assumed to be of sufficiently high Reynolds
number so that the kinematic viscosity, thermal diffusivity,
and water vapor diffusivity need not be included in
(91).
It is possible to form two dimensionless combinations of
these dimensional variables:
z/z 0 , z/L
(92)
108
3
-U,
where,
is
L =
v0
VO
length, and
K
the Monin-Obukhov
)w'e'
K(g/G
v
O
0
is the von Karman constant, which is tra-
ditionally included in the definition of L.
The assumption that (91) contains all the relevant
information needed to specify the structure of the surface
layer implies that the non-dimensional forms of the mean
velocity components, virtual potential temperature, and
mixing ratio must be functions of only z/z0 and z/L.
The
appropriate scaling factor to nondimensionalize the velocity
components is the friction velocity u,.
Scaling factors
for virtual potential temperature and mixing ratio are obtained from (91) and are given by:
v0
=
(93)
K,
R;
=
0 .(4
Ku
*
(94)
Therefore, in the surface layer,
u
=
*
F (z/z0 , z/L)
0
(95)
109
V
-
F 3 (z/z0 , z/L)
O=
(97)
(98)
0 = F 4 (z/z0 , z/L)
In this section, the velocity components u and v
are in surface coordinates, i.e., u is defined to be in the
direction of the surface wind and v is perpendicular to it.
If the non-dimensional vertical gradients of u,
v, 0V, and ~A are assumed to be independent of z/z0 (Monin
and Obukhov, 1954), it is possible to identify the z/z
0
dependence of the mean variables as logarithmic.
Equations (96-98) can be written in the form:
U
[ln (z/z0 ) -
=
m(z/L)]
(99)
= 0
(100)
U*
.
Pr
V0
Pr
K_
K
[ln
(z/z
o)
(z/L)]
-h
(101)
*
R K
R*
Pr [ln (z/z
K
-
0
$(z/L)]
R
(102)
For the remainder of the PBL, above the surface
layer, Monin-Obukhov similarity theory does not apply, and
the parameters in (91) do not suffice to determine the
structure.
The roughness parameter, z0 , is not important
Other
for z >> z0 , and can be eliminated from the list.
effects, such as rotation and baroclinicity, should be
included.
The height of the PBL, h, representing externally
determined influences, such as diurnal heating, subsidence,
etc., must also be included.
The similarity theory applied
to the region of the PBL above the surface layer is called
Ekman layer similarity theory.
Ekman layer similarity theory has evolved through
the years and is based on the work of many people.
and Monin (1960, 1961),
Csanady (1967),
Gill
Kazanski
(1968),
Blackadar and Tennekes (1968), Clarke and Hess (1973),
Deardorff (1972a,b), Hess (1973), Arya and Wyngaard (1975),
and others have contributed to its development.
In an
analogous fashion with the surface layer, the non-dimensional
forms of the mean variables can be represented as universal
functions of non-dimensional combinations.
The Ekman layer
relations are:
u - u
uum
v -
G1 (z/h, h/L, IfIh/u*)
(103)
G2 (z/h, h/L, iflh/u*)
(104)
v
u* m
ill
j
v
-
j
v
_*
m= G3 (z/h, h/L,
R - RK
R* m
(105)
jfIh/u*)
(106)
G4 (z/h, h/L, Iflh/u,)
m'
The variables uM'
av , and Rm used in the
Vm
deficit relations (103-106) are PBL mean values of velocity,
and mixing ratio.
virtual potential temperature,
Xdz,
= 1 h
X =u,
v,
e
(107)
, R
0
The use of the PBL averaged quantities allows the effects
of baroclinicity to be included implicitly, and thus simplifies the analysis (Arya, 1978).
The relations (99-102) apply in the surface layer,
while (103-106) are their counterparts in the rest of
Assuming there is a transition region,
the boundary layer.
or matching layer, in which both sets of relations apply,
it is possible to obtain the following relations:
--
[ln (z/z
S
u*
K
-
0
$
m
(z/L)]
u
m + G (z/h, h/L,
1
u,
jfIh/u,)
(108)
112
Writing ln(z/z ) as ln(z/h) + ln(h/z ), and absorbing the
z/h dependence into the unknown function G1 yields:
KU
ln(h/z)
-
--
fIh/u*)
G (z/h, h/L,
=
(109)
The function G', however, must be independent of z because
the z dependence of G
,
independent of z.
is
the left hand side of (94)
Eliminating
and calling the unknown function A,
we obtain:
u
A(h/L,
Iflh/u*)
= ln(h/z )
-
K
(110)
-
A similar matching argument for the relations
(100-102, 104-106) yields the other universal similarity
functions B, C, and D.
Kv
B(h/L, Ifln/u*) =
-
--
(111)
sign f
-
e.
C(h/L,
IfIh/u*) = ln(h/z 0 )
D(h/L
fI h/u,) = ln(h/z )
+
K[
vo
+
K[
0R
e
(112)
m
(113)
113
Attempts have been made to determine the similarity
functions from an empirical data base.
Clarke and Hess
(1974) evaluated A and B using the geostrophic wind in
their definition instead of um and vm, and assumed
h = u,/If
I.
Melgarejo and Deardorff
(1974)
used the components
of the wind u and v at the PBL top, h, in the deficit relations.
The PBL top was determined by profiles of Ov
and R (unstable conditions) or u and v (stable conditions).
Both studies, however, show a large amount of scatter in
the data points, especially on the stable side.
Arya
(1975) reanalyzed the data of previous studies in an
attempt to reduce the huge amcunts of scatter.
The re-
sults, although somewhat better, still retain considerable
scatter.
The similarity theories discussed assume that a
steady-state, horizontally homogeneous situation exists.
In the real atmosphere neither condition is satisfied.
Diurnal variations and large scale changes in the flow
pattern violate the assumption of a steady-state.
Changes
in the surface characteristics and horizontal advection
are usually present to violate the horizontal homogeneity
assumption.
In addition, it is very difficult to measure
Reynolds stresses or heat and moisture fluxes in the
field.
It is not surprising, therefore, that empirical
determinations of the similarity functions contain a
considerable. amount of scatter.
114
An alternative approach, the use of a model, has
been used to determine the similarity functions (Arya,
1977; Yamada, 1976; Arya and Wyngaard, 1975).
The assump-
tions of horizontal homogeneity and stationarity can be
satisfied using this approach.
There is no "measure-
ment error" as with an empirical determination.
The main
limitation is the ability of the model, with its modeling
assumptions and approximation-s, to faithfully reproduce
nature.
8.2
Results
The similarity functions A, B,
C, and D are
evaluated for five cases and are tabulated with h, u*,
T*, R,, L, RiB, h/zo, h/L, and Iflh/u* (see Tables VI-X).
The bulk Richardson number, RiB, is defined as:
Bgh(e
B
-E
)
-2
-2
u + v
m
m
In each case, a 24-hour simulation is started
at 0900 hours local time,
described in section 6.1.
using the initial
conditions
The results contained in
section 6.2 are from case B.
115
Case
Table
Surface Albedo
Geostrophic Winds
A
VI
0.10
Wangara
B
VII
0.20
Wangara
C
VIII
0.25
Wangara
D
IX
0.30
Wangara
E
X
0.20
constant
As summarized above, cases A-D use the observed
geostrophic winds from the Wangara experiment, as described
by Yamada and Mellor (1975).
Spatially and temporally
constant geostrophic winds are used in case E (u = v
--
4 m/s).
The similarity function A is
of h/L in Figure 36.
are used in
plotted as a function
Hourly values of A (from Tables VI-X)
the construction of Figure 36.
The data for
the 9th and 10th simulated hours (6:00 - 7:00 p.m. local
time) are not used because in this period the boundary
layer height is very rapidly changing.
Similarity theory
cannot be expected to do well under these highly nonstationary conditions.
Figures 37-39 are the same as
Figure 36 except for the similarity functions B, C, and D.
On the unstable side, there is a small amount of
scatter in the similarity functions which is probably
attributable to the fact that all values of h/z
fjh/u, are allowed.
A, B, C, and D on h/z
and
In other words, the dependence of
and jflh/u, is not considered in
116
List of the Variables in Tables VI -
X
t (hours)
simulated time starting at 9:00 a.m. local
time (i.e., time = 1 corresponds to 10:00 a.m.,
time = 2 corresponds to 11:00 a.m., etc.)
h Cm)
Boundary layer height
u,(m/s)
Friction velocity
0* (*K)
Scaling factor for virtual potential temperature (eqn. 93)
R,
Scaling factor for water vapor mixing ratio
(eqn. 94) multiplied by 1000
(gm/gm)
L (m)
Monin-Obukhov length
RiB
Bulk Richardson number (eqn. 114)
A
Similarity function A
B
Similarity function B
C
Similarity function C
D
Similarity function D
h/z
Ratio of boundary layer height to surface
o
roughness parameter
(z
= 0.01 M)
h/L
Ratio of boundary layer height to MoninObukhov length
IfIh/u*
Ratio of the magnitude of the Coriolis parameter
s-1) to u*/h
(f = -8.2 x 10-
ou
0 055.
a
c
au
O
C
0
00
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60
g
r
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5040302010 -
0 -'
A
-10.-
-20 -
-30
-40-50-60-70
1-
-600
.
I
-400
I
-200
I
0
I
200
h/L
Fig. 36.
Similarity function A as a
function of h/L: open circles (o)
- case E; closed circles (-)
- cases A-D.
123
1009080-
706050403020100-10-20-600
-400
-200
0
200
h/L
Fig.
37.
Similarity function B as a function
of h/L:
open circles (o) - case E;
closed circles (-) - cases A-D.
124
60
I
I
I
I
I
I
50
40
30
-%
20
10
C
0
-10
-
-
-20
-30
-40
p
3
I
I
-50
-60
-70
-60 0
-400
-200
200
h/L
Fig. 38.
Similarity function C as a function
open circles (o) - case E;
of h/L:
- cases A-D.
closed circles (-)
125
50
0 -p
-.
-WV
0-
-50-
D
-100-
-150-
-200 -
-250I
-600
I
p
-400
-200
p
0
200
h/L
Fig. 39.
Similarity function D as a function of h/L:
open circles (o) - case E; closed circles
(.)
-
cases A-D.
126
Similarity Function A
h
(a)
3
(x10
-600
-200
-400
h/L
Similarity Function B
h
(b)
(x0
-600
-200
-400
h/L
Fig.
40.
(a)
Similarity function A as a function of h/L
and h/zo for values of
|flh/u*
> 0.1.
(b) Similarity function B as a function of h/L
and h/z 0 for values of IfIh/u* > 0.1.
127
Similarity Function C
150
h
(a)
x1O3
-200
-400
-600
h/L
Similarity Function D
h
"20
(b)
(x 10 3
o1
-600
Fig.
41.
(a)
1
'
-400
h/L
I
'
1
-200
'
0
Similarity function C as a function of h/L
and h/zo for values of IfIh/u, > 0.1.
(b) Similarity function D as a function of h/L
and h/z0 for values of
|f|h/u*
> 0.1.
128
Figures 36-39.
Figures 40 (a,b) and 41 (a,b) show isolines
of A, B, C, and D as functions of h/z
restricted range of values of
and h/L for a
Iflh/u*.
An examinations of Tables VI-X indicates that the
similarity functions C and D are equal (within a few
percent) in the unstable boundary layer.
Under stable
conditions, however, C and D do not appear to be equal.
Brutsaert and Chan (1978),
analyzing experimental
data, evaluated the similarity functions C and D.
With
the height of the inversion as the length scale, 6, they
found D to be about 0.65 C.
Their results, however, are
based on the use of 0 (6) and R (6) in the deficit relations
instead of the vertically averaged variables
As pointed out by Arya
(1977,
em
and R .
1978) the use of 0 (6)
and R (6) in the formulation of C and D is less desirable
than the use of the vertically averaged variables because
the results are more sensitive to baroclinicity and
sampling errors.
129
9.
SUMMARY
A higher order closure turbulence model, using
the full level 3 equations
(Mellor and Yamada, 1974;
Yamada and Mellor, 1975) is described in detail in
section 2.
A new formulation for the length scale Z,
defined by equations (65a - 65c) is used.
This length
scale, appearing in each of the modeling parameterizations
described in section 3, has the same shape in the PBL
as Deardorff's
(1974, 1975) profile of the turbulence
energy dissipation length scale.
Equilibrium boundary conditions for the second
moments are applied at the lower boundary.
The surface
mixing ratio and potential temperature are predicted
with a single layer ground thermodynamics model (Deardorff,
1978).
The results of a simulation of Day 33-34 of the
Wangara boundary layer are examined in section 6.
The
boundary layer grows through the day, reaching a maximum
(1320 m) around 1800 hours (6:00 p.m. local time).
an hour after sunset,
About
the PBL top falls rapidly to a more
or less constant value of 100-150 m throughout the night.
Twenty-four
hour simulations are made both with
ahd without the Coriolis terms.
The results are nearly
identical for all the mean and prognostic turbulence
variables.
The details of the Reynolds stress components
130
u'w' and v'w',
however,
did show some sensitivity to the
inclusion of the Coriolis terms.
Finally, the similarity functions A, B, C, and D
are evaluated in section 8.
Layer-averaged variables are
used in the deficit relations.
The similarity functions C
and D are found to be equal in the unstable boundary layer,
although this appears not to be true in the stable boundary
layer.
131
APPENDIX A
SOLUTION OF THE DIAGNOSTIC REYNOLDS STRESS, HEAT FLUX,
AND WATER VAPOR FLUX EQUATIONS
The level 3 equations for the Reynolds stress and
the heat flux (equations 50-58) represent a closed set of
9 diagnostic equations.
Once the prognostic equations for
the mean variables (u, v,
(q2 ',2)
e,
R) and the turbulence variables
are solved, the individual components of the
V
Reynolds stress tensor and the heat flux vector are determined.
The sclution of the simultaneous equations (50-58),
It is
however, requires a great deal of algebraic work.
convenient to represent equations (50-58) in the matrix
form (equations A-1).
The matrix is solved by performing elementary row
reduction operations until all the non-diagonal elements are
zero and the diagonal elements are equal to one.
If, at
each step in the matrix manipulation, a new variable is
defined in terms of combinations of previously defined
variables, the answer will be of the form u'u= N
where
Nk is defined in terms of M's, and the M's are defined in
terms of L's, etc.
following:
This procedure.will lead to the
132
0
1
0
0
A5
1
0
B5
0
A
3
0
0
B3
B
0
0
0
S1
0
2
E1+E5
0
0
0
0
0
0
C4
0
0
0
u'v
0
u'w
+C5
0
0
1
D
D2+D
0
1
E6
E3
0
1
0
F3
0
1
G
4
3
0
0
o
0
O
0
F1
F
F5
o
0
0
0
G
0
0
0
0
0
o
0
12
0
0
u'2
I-
0
C
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v'w
0
3
0
0
2
+G
3
v
S
0
H2
v'O'
0
1
w'6'
v
Equations A-i
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
Equations
w'
2
VI
u'w'
I
5
N
6
N7
v
(A-2)
N
vi
I
vV
N8
N
9J
133
where,
o
0
~4 1
A 1 = q /3
0
1
A2
z
-21
0
A
al u
q
3
z
*
C
5
=
-6Z1
q
az
D
1
=
--
y
v
3z
1
q
F3
3
-
0
f
q
v
0
F2
-
-3z 1
q
1
A4
-62,
0
A5
*
D2
3 1
q
0
3, Y
D
3
au
azE
--
=
Yg
F4
if
0
F
q
-3Z
=
5
G
ajz
q
0
A
0
6
q
D4 =
Y
0
2
B5 = q /3
q
D5 =
fy
q
fz
G2
G3
q
0
0
B2
q 12Z
az
aii
E
1
q
-31
az
q
E2
0
C a=
E= 3Z 1 Cq a
a
G4
H1
-q
E3
Sg
H2
0
B
6 1
5
-3
0
~
fz
q
0
-3
0
4
0
Y
q
az
32
fh
0
+2Z
0
321
o
4Z
B3
B
fu
0
z
q
g
=-3
0
0
3vz
f
E4
-q
fy
=
q
3z
q
lz
392
32
q
z
q
a
H3
0
0
ci =
0
C2
0
C3
32
0
q
/3
-2.2
3u
-1
q
-22
q
z
1
q
E5
u
au
a
y
-3,
E
32,
S1 F
~~~ v
0
12
q
0
I
v
3v
0
13
-3Z 2
q
y
134
N1 = M 2/M1
N
=
M
M43 (M2 /M
-
N 3 = L8-
N
N
1 l6 ~
M15
1)
M16
K 18
M5 (M2 /M
)
-
=L
-M
M 7 (M /M
2
1)
N
=M
L
N
= M
M 11 (M /M
2
1)
-
Ml3 (M2/M
)
N9 = M 16
-
M1 5 (M /M
2
1)
- L2 L7
M,
L
M2
L3 - L2L8
2 L8
=MK
-
K
F1 - F2K5
L2
K5L
M6
6
M7
E 6
M8
E18
M9
K
3 - F2K6
8
=
1
13
2
3
4
3
12
1 I2
112
1
3
2
3
4
3
L
=
F5 K5
L = FF4 -F5K5
1
I1
5 FSK6
2
2
31
K 1/K2
K1/K2
3
4
3G2
L8
K3 /K2
Jg
I5
3G3
K=
F7 - F8 (11
K2
1
K3=
F
F6
L =
L =
K
3
F8 ( 2 / 3)
-
F8
1
4
3
3)
K5=
G
-
G 2 ( 2 /1
3
K6
G
-
G2 (4 /3
EyL7
17L 7
E17 L8
-
-G2 (
=
K7=
K8L7
K
I
10
K l8
-F5 K
-
9
16
1 3
8
=
I1=
-H4 B32
4
I
H1
=
H4B33
L5L8
K5L8
K
M5
I7
- L5L
M3 =L
N
K 17L8
=
M 9 (M2 /M 1 )
-
6
1 - F2K
L =
N 6 = M10-
K 13
K13=
K15
L7 (M2 /M 1 )
M
17 L7
K8
J /3
213
J 2/J3
I
H
=
2
H3 4HgB34
I5 =
19 H5 - H4B35
I6=
-F15 B32
I =
1 2 F 13
5F 5B33
8 = F14 -
15B34
K8L8
I
K=
M1 = K10 -K K11 LL7
M2
Ml3
M14 =K
3 12
K11 L8
L7
4 13 - K 4K
L
15
K14 L8
K=
K=y
K 11
K=
K1
B32 - B3 4 (1 /3
=
B33
B35
-
-
=
I
Il0
F16 =
-
B 3 4 (J2 /J
3
I
B 3 4 (14 /J
3
Il2
11
-
15B35
A44 B32
A43 - A 44B33
-A44 34
I =
Il3 A45
A 44B35
135
H2
E20
H3
1 -
H
=
E2
D
D 3(D 9/D1)
E5
21F16
=
C2/C
D2
D3 (D10/D 11
D
E 4 =D
F15
22 -
1 -
E3 =D2
21F14
= -E2
H5 =
G
E
9E - E21F13
H1
31/C1
D
D34 = C4/Cy1
C1
- D3 (D 1 2 /D 1 )
1
D
1 - D7 (D1 3 /D 16
=
C /C
5
7
D6
C8
(
6
= D
-
D 7(D
E
= D
-
D7 (D 15/D16
D,
= C
- C6 (C4/C1
7/D16
D
= C10
-
E8
G3 = F12/F
F
E9
= E2/E
D8
= D
E10
E
F2 = E 3 /E 1
/D16
= E6
F5 =
F6
F
E
/E5
E8/E5
= E /E
O
F8 = 1/E 10
F
C6 (C 3/C )
-
F
F 2
=
E1
E
E 5
E
12(E6 /E 5
22
-
B2 4 (D 9 /D
B23
-
B2 4 (D 1 0 /D
12 (E8
Fl3 = -A42A43
F
= A40
F
=5A
F
= -A42A45
- A
42A
5
11)
24 (D12/D 11
Dl8/D20
D
11)
20
-
B34
114 -
35
B 18
= B
D15
E 16 =
E20 =
12 (E7 /E5
D
1 7 /D 1 6
25 -
= Cl
D13
E15 =
19
324 1
-C B33
D12
13/D16
= D15/D16
E18 =
103
F10
D
-B
1 -
D16 =
D1 7
C6 (C5/C )
9
D10
D14/D16
17
= E1 /E10
D7
/D 1
= Dl2/D
E13
5
D7 (D
D O/D
E12 =
F
1
E
= F 10/F
G2 = 1/F 1 1
2
17 (C3/C )
20 -B
B21
7(C4/C )
7(C5
B
D
=B
D
= B28 -
D20
7(C2/C )
-B
l
(C/C
26(C3/C )
B29 -B26
(C4/C )
E 2 1 =D21/D20
D
E 2=
D22 =B31 -B 26(C5/C
)
D2/D20
= B 30
136
Cy
C,1= 1-
5
BB 4 (B12
(B12/B
/B15)
C 2 =B 1
C3 =
C4
B4 (B1 3 /B 1 5
-
B3
-
C5 =B5
= B
C
= B7
C8
13 -
3117
A 4A
B4 (B1 4 /B 1 5
B1
B14 = Al
A15
B4 (B1 6 /B1 5
A1
B1
B15 = Al
A16 - AA44A17
- B 0 (B 2/B15
B 1 0 (B1 3 /B1 5
A8
1 -
B16
B
=7A
B18
B8
A 20
B
C10 =B
B20 =
22 -
B21 =A
24
B22
25
- B10 (B16 /B 1 5
B12 /B15
C12 =
13/B 15
C 1 3 =B
C14
B
/B15
B16/B15
=A
-A
A
B2
A2
B 3=A3
45A17
- A23(A58/A51
CC9 = BB9 - B1
/B1
B10 (B
(B14/B15
C11
45A12
B12 = AA14
Bl
B13
2
C
B11 =
= A21 -
23 (A49/A51
23(A50/A51
-A
23
(A5 2 /A 5 1
A 28 (A48/A51
B23
A2 6 -A
28
(A4 9 /A 5 1
B24
A 2 7 -A
28
(A5 0 /A 51
B25
A29
B26
A 30 -A
B27 = A
31
B28 =
32
A 28
-A
52
51
33
(A4 8 /A51
33
(A4 9 /A 51
B3 =A3
B
= A
- A
A5
B29 =1 -A
B5 =A6 - A45A5
B30 =
B6 = A
B6=A
B7
-A7
B
- A33 (A52/A 5l
= A 48/A5
B8 = A
B33 = A9 /A51
B = A9
B9 = 10
B10
A11
B
A44A12
B35
=
(A50 /A 51
34
B 31 = A35
A8 - A 43A12
33
A 5 0/A 51
A 52
51
137
A1
(-a
A2
27
1/ (1-a16 2
(18
28 =
19
0
A3
A
X2 )/ (1-X16 2
17
33-a 1 8 a2 )/(1-a a 2)
16
=
A5
-a
a2 /
A6
a2
1 6C2 2
A3 1
)
A7 = -16
(20-=l7a22
)/(
16 a2)
Ct6
176
A32
C 2 1 /(l-t
A33
-a
A34
A3 5
= a5
A10
7
19 6
8
A 13
o
0
B1
6E2
18 9
A16
1 9 a9
/(-t17 a 9
A1 8
=
1 1 /(1-
1822
A 19
12
A20 =
13
A
22
A23
17
16 14
17a 4
at
O
1 6 aC
24
/(l-a
* 37
-a17a24/(-
*38
-a 1 8 a
)822
1 9ga 2 4
19 a,24
2 4 /(1-aga
24
C2 4 E2 /(1-a
17 9
at9 )
27
41 =
'28
42
29
A43
1 9g 2 4
=
a
3 0
0
(C 1 -9 E 2
A21
-a
A40
at9 )
10
0
22
F2-22 E 2)/(1-a
=
A39
A15
a
t
0
A 1 4 = -a1 6 a9 /(-17
A 17
22 )
18
23
A36
18 6
A 11
A12
18 22
1 9 a 2 2 /(l-a 1 8
0
A
/ (1-a 1 8 'X2 2
O
(A -2E2
A8
2
30
(1-16a2
a 4 /(1C16
a
29 =
15
-a
a
18 14
19 14
(1-
17
a9 )
A
= a3
A45
1
A46 =
25/
A 47 =
26/(-a
A48
36
A
= A37
l-a1924
19a24
43(A 47 -A42A46
0
A24
A25 =
A
14 2
16
a 7
A50
A38 -
A51
A52 =
40A46
41A46 - A
39 -
(A47-A42A46
A 4 5 (A 4 7-A 42 A46
138
a
= C
2 + A6
(22
=
aL
1 3
23
D5
0
a
= D
(X14
D1
o
24
a24
0
X15
a5 = B5
=GG1
0
0
2 + D4
=
a25
a 6 =B3
16
=E
a 2 6 =G
4
0
0
7X = B2
a17
o
0
+ E5
E
18
27 =
2 + C5
a 19
a 1 0 =3
+ G3
1
28 =
6
3
0
29 =
E3
0
0
2
0
0
0
0
0
0
0
a8 = B
G4
0
0
0
3
=
0
A
a4
Ot4=
=A4
5
0
0
o
X9 =
= F
0
D3
a12
0
a3
2
0
0
0
t2
0
0
0
at = A5
2
0
a 2 0 =F 130
2
31
0
331
2
a%
3 1 =13
The finite difference scheme described in section 5
requires that the turbulent moments u'w', v'w', and w'6'V
be known at time step k in order to solve equations (40-47)
at time step k + 1.
From equations
v'w', and w'6' are N 5 , N
O
N6
and Ng,
(A-2) , we see u'w',
respectively.
Substi-
tution of the A through M into the N's results in a very
long and complicated expression for each N.
The model
calculates the N's by evaluating the intermediate variables
0
(A-M) first.
Matters are simplified considerably when the Coriolis
terms are omitted.
The model has an option to allow the
Coriolis terms to be included or to be set equal to zero.
Mellor and Yamada (1974) evaluated expressions for u'w',
v'w', and w'6'v in the case fy =f z =0.
It is possible to
139
to show that N 5 , N 6 , and N reduce to their expressions
9
in that special case.
With f =f =0,
LJ
= NS
=E
E,, (I-
Eas Aw.)
-
Coo (-
z A%#*)
-
Ev& *- F-t' Avo&Avv)
(A- 3)
- E,
S A,A. t
3Jo
-
3
+Er. A4, A43)
1A
. A,.EL(%QV&
3A, C
t
(CE..
I
a e I,;a
r~
v1AA~
(A-4)
)E
;);z
(A- 5)
1
'e, A
0
3 +411612toa
(,St) 1
4/3
101I I
d.
r
S'f,AR, 't15
-2,
-0
Z
~
4
3
Qv
54F T
3?)
+ '.
2tI
4
*eA
g I
22
(A-6)
6v
t
%
(
(A-7)
140
IE2.-
1
iAxv7=
7 Or
[3.A C. */ 3a
I
I
a
)iuj
+ ~
iT
+
;":
(A- 8)
;
- & 7a
) zGo,,
]
27 .18
3
It 3
s'ij,'g~
i
4,~~C~ C1@,
(;.~)*~ a.,,
a-
4
qAtg
ISS~
(A-9)
+ Ftv Aiz A%43 I =
13.Al zz
z
& 06
; O r;z
(A-10)
Substitution of equations
(A-4 to A-10) into (A-3) will
yield, after considerable manipulation, equation (A-ll).
.4A'A4i
1.,aa)
3. U,
7(00) tit (5-12
7+361
(A-11)
Mellor and Yamada (1974)
represent equation (A-ll)
in the form
a.
-u'w' = Km 3z
(A-12)
141
and
-v''
(A- 13)
= Kom
maz
From equations (A-2)
(E2 2 +E21A42A45
v'w' = N
6
=
E 1 1 (-E
K
9
8
A
)
E 1 0 (l-E2211 A4400 )
-
(E 1 9 +E
2 1 A 4 2 A 4 3)
(A-14)
It
can be easily shown that:
K 9 /(av/az)
=
E 1 8 /(3a/az)
and
K 8 /(aV/az)
=E
7/(au/3z)
so ccmparing equation (A-14) with (A-3) and (A-il)
yields
The expressions for N5 and N 6 , therefore, reduce as expected to Mellor's and Yamada's expressions in the simplified case fy = fz = 0.
142
Prognostic equations (45, 48) require that w'r'
be known at time step k to calculate R and 7'r' at time
V
step k+l.
Once the Reynolds stresses and heat fluxes
(N1 -N9 ) have been calculated, the only unknowns at time
Equations (59-61)
step k will be u'r', v'r', and w'r'.
are three equations for these three unknowns.
The solution
for w'r' is:
ro.
-+
-AA
2
(
Expressing w'r' in the form:
-
=
Kw az
(A-16)
R
and comparing equations (A-15) and (A-16) yields the following expressions for Kw and
R'
(A-17)
9
+
143
( tz , ol-R' 1.2L ) ( 3,9, y3a r'Qv')
(A-18)
CI.A "
2 t
(
eI
+ 4z L +-
-a
a2e
Z.. ) + 7.-1,12.I
z
Once equation (A-16) is solved, u'r' can be
evaluated by:
IVI
A I.
~
I
(A-19)
Finally, v'r' is the only unknown and is determined by:
L
20)
Numerical stability considerations for the r'e'v
equation require w'r' to be expressed as:
w'r' = Ar'O'
v
-
B.
A comparison of equation (A-21) with equations (A-16
- A-18)
yields expressions for A and B:
A-2 1)
144
3-fl p
IG
IL
( tz .
'
%[1?
43
ffi,)'t
3, (
Is
't~j4
L+4
)
t I
(A-22)
al.Z
a
+
4
-)?-]
IA
I
* A 'JL
-a
Z~-L43
t
7
i~~a
-~
-Ar
D-7
(A- 23)
145
APPENDIX B
THE STAGGERED GRID SYSTEM
A 44 point one dimensional staggered grid is used
in the model.
The mean variables are defined at integer
grid points and turbulence energies and fluxes are defined
at half integer grid points (see Figure B-l).
A trans-
formed coordinate system is used (Yamada and Mellor, 1975)
which provides greater resolution near the ground where
gradients are the largest.
3 grid points.
The lowest 26 meters contain
Above this level, the distance between
grid points quickly approaches 50 meters and remains nearly
constant with height thereafter.
The new vertical coordinate,
C, is defined as:
(B-l)
= a 1 z + a 2 ln(z/a 3 )
(B-2)
(a1 ,a2 ,a3 ) = (0.02, 0.25, 0.01)
The z-C values are tabulated in Table (B-l).
Yamada and Mellor evaluate vertical derivatives in the
C coordinate system and that approach is followed here.
The derivative of any quantity, $,
4
-
a
is:
(B-3)
146
a
a = a1 +
2
(B-4)
The lower boundary conditions for the mean variables
are applied at C = 0; those for the turbulence variables
are applied at C = 1/2.
The upper boundary conditions
(turbulence moments vanish, mean gradients constant) are
evaluated at grid point 43-1/2 (2022 m).
147
Table (B-l)
z-G Values
zeta
z(m)
zeta
(Equation B-l)
z(m)
zeta
z(m)
0
0.01
15
612.22
30
1352.32
1/2
0.07
15-1/2
636.73
30-1/2
1377.09
1
0.52
16
661.26
31
1401.87
1-1/2
3.14
16-1/2
685.80
31-1/2
1426.65
2
11.70
17
710.36
32
1451.43
2-1/2
26.48
17-1/2
734.94
32-1/2
1476.22
3
44.88
18
759.53
33
1501.01
3-1/2
65.21
18-1/2
784.13
33-1/2
1525.81
4
86.66
19
808.74
34
1550.61
4-1/2
108.81
19-1/2
833.37
34-1/2
1575.41
5
131.45
20
858.00
35
5-1/2
154.44
20-1/2
882.65
35-1/2
1600.21
1625.02
6
177.69
21
907.30
36
1649.83
6-1/2
201.14
21-1/2
36-1/2
7
224.75
22
931.97
956.64
37
1674.64
1699.46
7-1/2
248.49
22-1/2
981.32
37-1/2
1724.28
8
272.35
23
1006.01
38
1749.10
8-1/2
296.29
23-1/2
1030.71
38-1/2
1773.92
9
320.32
24
1055.41
39
1798.75
9-1/2
10
344.41
368.57
24-1/2
25
1080.12
1104.84
39-1/2
40
1823.58
1848.41
10-1/2
392.77
25-1/2
1129.57
40-1/2
1873.24
11
11-1/2
417.02
441.31
26
1154.29
26-1/2
12
465.64
27
1179.03
1203.77
41
41-1/2
42
1898.08
1922.92
1947.76
12-1/2
490.01
27-1/2
1228.52
42-1/2
1972.60
13
514.40
28
1253.27
43
1997.44
13-1/2
538.82
28-1/2
1278.02
43-1/2
2022.29
14
563.26
29
1302.78
14-1/2
587.73
29-1/2
1327.55
148
- = 43.5
z = 2022 m
=
43
z = 1997 m
- = 42.5
z = 1973 m
z
=
z =
1
{-rVrjv
0.52 mn
g
u'w'
az
C = 0.5
z =
0.07 m
{q
2
i
'
~W
6v
,
u.uT-,
v'w
v
3
,
3z
'
~~.
r
,
u.'',
1
I
av
3z
az
v }zaz
g'
Vgz
r
v'
u.'r',
v
au
u
lia
v
~
3v
~
7
3
liR
~
u , V
g
g
z = 0.01 m
77 7 7-7 /77
Fig. B-1.
Staggered grid system used in the model.
149
APPENDIX C
Read Input Parameters
Starting Run
Yes
)
Values
Read Initial
of Second Moments
From Disk
Read Last Values
of Prognostic Variables
from Disk
Call CORIOL
KES
for fy = fz = 0)
(Call
Calculates Diagnostic Variables
and Derivatives
Enter Main Loop
Produce Plots
Run To Be Restarted
Write Values of
Prognostic Variables
to Disk
-
STOP
Fig. C-i.
Flowchart of the level 3 model.
150
ENTER MAIN LOOP
Call PROG
Calculates Prognostic
Variables at t + 6t
Call BOUNDC
Ground Thermodynamics
Model
Call GEOTV
Calculates New Values of
u
g
,
v
g
Call LZERO
Calculates Z
0
Call CORIOL
(Call KES for fy=fz=0)
Calculates Diagnostic
Variables and Derivatives
EXIT MAIN LOOP'
Fig. C-l (continued)
151
REFERENCES
Arya, S. P. S., and E. J. Plate, 1969: Modeling of
the stably stratified atmospheric boundary layer.
J. Atmos. Sci.,
26,. 656-665.
Arya, S. P. S., 1975: Geostrophic drag and heat transfer
relations for the atmospheric boundary layer.
Quar. J. Roy. Met. Soc., 101, 147-161.
Arya, S. P. S., 1975: Comment on "Similarity theory for the
planetary boundary layer of time-dependent height."
J. Atmos. Sci., 32, 839-840.
Arya, S. P. S., and J. C. Wyngaard, 1975: Effect on
baroclinicity on the wind profiles and the geostrophic
drag laws for the convective planetary boundary layer.
J. Atmos. Sci., 32, 767-778.
Arya, S. P. S., 1977:
Suggested revisions to certain boundary
layer parameterization schemes used in atmospheric
circulation models. Mon. Wea. Rev., 105, 215-227.
Arya, S. P. S., 1978: Comparative effects of stability,
baroclinicity and the scale-height ratio on drag laws
for the atmospheric boundary layer. J. Atmos. Sci.,
35, 40-46.
Blackadar, A. K., 1962: The vertical distribution of wind and
turbulent exchange in a neutral atmosphere. J. Geophys.
Res., 67, 3095-3102.
Blackadar, A. K., and H. Tennekes, 1968: Asymptotic
similarity in neutral barotropic planetary boundary
layers. J. Atmos. Sci., 25, 1015-1020.
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