Investigation of Applying Normal Mode
Initialization Method to Planetary
Stationary Wave Disturbances
by
Yang Niu
B.S. in Meteorology,
Peking University, Beijing
(1990)
Submitted to the Department of
Earth, Atmospheric and Planetary Sciences
in partial fulfillment of the requirements
for the degree of
Master of Science in Meteorology
at the
Massachusetts Institute of Technology
May, 1992
@Massachusetts Institute of Technology, 1992
Signature of Author
Center fr Meteorology and Physical Oceanography
Department of Earth, Atmospheric and Planetary Sciences
May, 1992
-. 0
11.7
Certified by
Richard S. Lindzen
Professor of Meteorology
Thesis Supervisor
Accepted by
Thomas Jordan, Head
Department of Earth, Atmospheric and Planetary Sciences
MASSACHUSES IN I
MAY12?1292s
M-RIES
Investigation of applying normal mode initialization method
to planetary stationary wave disturbances
by
Yang Niu
submitted to the Department of Earth, Atmospheric and Planetary Sciences
in May, 1992 in partial fulfillment of the requirements for
the Degree of Master of Science in Meteorology
Abstract
While the prediction of planetary-scale waves has been studied for many
years, skillful prediction has not been achieved yet. Complexity is found
in some serious problems such as data initialization, domain of integration,
model forcing, physical parameterization, etc.. In this paper the problem
of applying normal mode initialization to planetary stationary wave disturbances is investigated.
A linearized baroclinic primitive equation model is explored. The initial
field is a calculated topographically forced and radiating stationary wave on
a beta plane. However, for the purpose of normal mode initialization an
artificial model 'top' is imposed. The emphasis of this paper is to determine
whether normal mode initialization can affect the time behavior of the initial
stationary wave under this circumstance and the dependence of this result
on mean zonal flow. In addition, the effect of the placement of the model
top on the accuracy of forecasting the stationary waves is examined.
It is found that the vertical modes for the streamfunction 0' and v' velocity are good representations of the vertical structures of initial fields varying
in mean zonal flow. However, for u' velocity only in westerlies do they look
like the original ones. Consequently, the time behavior of the initialized and
the noninitialized fields bears resemblance in the regime of westerly mean
zonal flows. On the contrary, in the presence of easterlies, the initialized
field behaves quite differently from the noninitialized field.
Moreover, it is found that in westerlies both of the noninitialized and the
initialized fields behave far from stationary waves. However, in easterlies the
time behavior of the noninitialized fields is near stationary.
Thesis supervisor: Richard S. Lindzen
Title: Professor of Meteorology
Acknowledgements
I wish to thank my advisor, Prof. Richard Lindzen for his guidance,
insight and patience throughout the course of this research. I also wish to
thank Prof. Randy Dole for his helpful discussions and suggestions in the
early course of the work.
I would like to thank many of my fellow students for their friendship,
encouragement and assistance. Special thanks goes to Michael Morgan for
his advice and correcting some parts of the manuscript.
Other students,
Daniel Reilly, Nilton Renno, Dezheng Sun, Shuntai Zhou, Chun-Chieh Wu
and Dennis Boccippio also deserve considerable thanks. Additionally I thank
Jane McNabb at CMPO headquarter and Cindy Hanes at EAPS graduate
office for their kindness and help in my graduate life at M.I.T.. I'd also like
to thank Diana Spiegel for her assistance in using the computational facilities
at CMPO.
And last I gratefully thank my host family Mr. and Mrs. Wollman and
my good friend Lynn Smith for their love, care and support, which make me
feel home, encouraged and joyful.
Contents
1 Introduction
6
2 Initial forced stationary wave solution
8
. . . . . . . . 10
2.1
The horizontal structure of the stationary wave
2.2
The vertical structure of the stationary wave . . . . . . . . . . 12
14
3 Normal mode initialization
...
Normal modes . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Normal mode initialization . . . . . . . . . . . . . . . . . . . . 18
19
4 Model description
5
14
3.1
24
Results
5.1
The comparison of the initial stationary wave solution and the
model stationary wave response . . . . . . . . . . . . . . . . . 24
6
5.2
Normal mode initialization . . . . . . . . . . . . . . . . . . . . 27
5.3
Tim e behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
47
Discussion
6.1
The effect of model lid on initialization . . . . . . . . . . . . . 47
6.2
The sensitivity of initialization to mean zonal flows
6.3
The role of vertical wind shear . . . . . . . . . . . . . . . . . . 53
6.4
The departure of the evolution of the stationary wave from its
initial state
. . . . . . 52
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7
Conclusion
67
8 References
68
1
Introduction
Normal mode initialization(NMI) attempts to suppress forecast noise by
excluding gravity waves from initial data. Normal modes are determined for
the system, and the coefficients in the normal mode expansion are determined
from independent initial values of all variables. Since normal mode initialization was first introduced by Dickinson and Williamson (1972), employing
this approach to exclude gravity waves from initial data has been extensively
studied and applied in short and medium range numerical weather prediction.
However, questions remain about the relevance of normal modes to realistic flows.
One of these concerns is the prediction of planetary waves.
Although the theoretical study of Lorenz (1969) suggests that the planetary
scales are the most predictable, forecasting experiments indicate that the
long waves are predicted less accurately than the synoptic scales. The errors
of forecasting planetary waves are suggested by Lambert and Meriless (1978)
and Baumhefner and Downey (1978) as due to initial data, numerical formulation, domain of integration or physical parameterization. Moreover, Daley
and Williamson (1981) suggested that the above first three reasons can all
result in the excitation of large-scale free Rossby waves. The purpose of the
present paper is to investigate the impact of normal mode initialization on
the prediction of stationary planetary waves.
The initial field is a calculated stationary wave on a beta plane in the
presence of topography. The stationary wave is obtained for an atmosphere
with no top and hence no complete set of discrete normal modes. One of
the critical issues is to evaluate the validity of the application of NMI to
the stationary planetary wave if an artificial top is imposed for the purpose
of obtaining normal modes. For example, are the initialized fields(4', u', v')
good representations of the original ones?
Secondly, the linearized baroclinic primitive model is employed in our
test. As a rigid top is imposed in the model, the topographically forced
stationary wave response of the model is different from the initial calculated
stationary wave with radiation condition. The difference field is expected to
travel with the phase speed of free Rossby wave. It was found by Lindzen,
Farrell and Jacqmin (1982) that interference can occur between coexisting
stationary wave and Rossby wave with the same horizontal wavenumber but
different phase speeds. In addition, in our baroclinic primitive model, the
traveling Rossby wave may propagate upward. Therefore, the focus of the
paper is to test whether the initialization can make a difference in the time
behavior of the initial stationary waves in our model.
Furthermore, attention is paid to how the NMI is affected by mean zonal
flow, which primarily determines the propagation properties of planetary
waves.
It was first found by Charney and Drazin (1961) that energy is
trapped in the region where zonal winds are easterlies or sufficiently strong
westerlies. We propose to examine the sensitivity of NMI to different mean
zonal flows.
In addition, the time behaviors of the initial stationary waves from with a
model lid and without a model lid are compared. One advantage of choosing
stationary waves as initial fields is that stationary waves do not change at all
in nature, and so the solutions from without a lid are already known. When
a model lid is imposed, spurious Rossby waves are excited and so the time
behaviors of the initial fields are expected different from stationary waves.
Therefore, we wish to determine to what extent the spurious Rossby waves
contribute to the errors of forecasting the stationary waves.
Finally, to assess the effect of the more realistic flows of easterlies and
westerlies together, some simple cases of zonal mean flow with shear are
tested.
2
Initial forced stationary wave solution
We are concerned with planetary wave disturbances whose horizontal
wavelength is comparable in size with the earth's radius and whose doppler
shifted period is large compared with the period of the earth's rotation.
Such disturbances are therefore both hydrostatic and quasi-geostrophic. The
initial field is a topographically forced stationary planetary wave on a
#-
plane. The perturbation is assumed to be around the basic state , which is
isothermal and has dry static stability N 2 . The mean zonal flow of the basic
state satisfies the geostrophic relation with the pressure of the basic state.
Let
u = u + u', v = v', w = w',p = p(y,z)+ p',p =
(y,z) + p',T
=
V'+T'
(y, z) exponentially decay
where prime denotes perturbation. P(y, z) and
with z with a scale height H.
Linearized perturbation equations are:
_U'
1 Op'
- -(fo+y)v'=--
-- +(fo+#y)
O-xp
x
-
(1)
--
+ --~) +
/
'vzcy =
(2)
0
(3)
T
U (9p'
9T
g
_
+ w'(--+--) =0
(5)
P
x
op'|_p
T'
az
9T'
U-5
8x
where Z(2
c,
cPPp 8xz
-=- + (6)
p
p
T
+ (-) = N2 is Brunt-Vaisala frequency and c, is heat capacity.
Since P satisfies (aY
=
-fops and T is constant, - satisfies 2ay =
However, we do not include v'lay term in continuity equation (3).
because we compared the magnitude of v'a1 't with that of w&
RTP
This is
and found
that the magnitude of 21'2
ay is about 4 order smaller than the magnitude of
8
, while v' is only about 2 order larger than w'.
Let 0' to be streamfunction:
fop
(7)
We seek perturbations with independent wave form components of the form
of a product of some separable functions of x, y and z, namely:
4" = Re[eikx'(y)(z)]
((8)
So horizontal and vertical equations are separated as:
ikau' - (fo + Py)v' = -ikfoo'
ikiiv'+ (fo +
iku'9v'
i'+
y--
y)u' = -fo a
ay
ifokfz
gh
ghn '
(9)
(10)
(11)
and
iksfoopp; 18p~w'! ikcifo,
-i
+ P- az - ghn #2
gp az
(12)
op2 f= T'
-(13)
az
ikcT' -
T
'i'N 2
iki fo
k /+
w' =0
c,
g
(14)
where hn is the equivalent depth.
The horizontal and vertical structures of the stationary wave are resolved
by solving the horizontal equations(9)-(11) and vertical equations(12)-(14)
respectively satisfying their own boundary conditions.
2.1
The horizontal structure of the stationary wave
Our geometry consists in a channel centered at the latitude of 45'N. The
width of the channel is comparable to the radius of the earth. Using this condition, we can solve the eigenvalue-eigenfunction problem of the stationary
solution.
The eigenvalue is found by solving the horizontal equations(9)-(11). The
boundary conditions are:
at y=0 and y=L:
V= 0
(15)
or equivalently:
01 - -- 0' = 0
(16)
UU
9y
Solving u' and v' from equations(9)-(10) and substituting them into equation(11), yields:
a201
(
2
By2
(k U)2 _
h
ghn
f220
+ u-k
2
)Vi=0
(17)
Let:
12
2
=k
Uk)2 - f2
ghn
0 + -k-2k(18)
u
the solution to the equation(17) is:
0'(y)
= sin 4lY + A, cos lny
(19)
satisfying the restriction(16). So the ln and An are found:
in =
L
, An =
(20)
fo
Correspondingly, from equation(18) the equivalent depth hn is:
ghn =
ki_2
f02-
(21)
n
Then u' and v' are solved from equations(9)-(10):
fo[k 2 5$((y)
u(y) =
-
(fo + 0y)
2
2
(f + #y)2 - (k )2
* ](Y)
(22)
3)
(23)
fo[ik(fo + #y)40(y) - i(2
o
2.2
(y)y
(fo + #)2
2
- (k)
The vertical structure of the stationary wave
Having found the eigenvalue of the problem, we can resolve the vertical
structure of the stationary wave by solving equations(12)-(14). Eliminate T'
and w' from equations(13)-(14) and substitute them into equation(12), yield:
a2#
Oz 2
1
-- (-
H
.
+
Tc,
g
2
N
N24
g
)
az
(
g
-
gH
N
HTc,
2
)#2 2=
ghn)
0
(24)
where h is obtained from the eigenvalue solution equation(21). The vertical
boundary conditions are:
at z=O:
ikifo
a
g
,
-)p =
-
z
I'c,
ikifo(a
T
N2 N2(
,
(25)
at z=top:
,
= 0
(26)
The w' appearing on the right hand side of equation(25) is the surface forcing:
w' = ikii7
(27)
where 9, is the height of topography.
We noticed that the coefficients of 0' and 8
in equation(24) and in the
boundary condition equations(25)-(26) are slightly different from those in the
classical equations of Charney and Drazin(1961). This is because we added
in continuity equation(3) and -U29Z
ax term
f
CPT ax
term in thermal equation(5)
as we considered the stationary equations as compatible as possible to the
time dependent equations used later. We also calculated that the magnitudes
of the additional terms in equations(24),(25) and (26) are about 1 to 2 order
less than those of the original terms in Charney and Drazin. Therefore, the
change we made here does not matter.
The equation(24) can be transformed into canonical form:
d29
djz-2
ap22
+ (a1 -
)
(28)
= 0
by setting a1 and a 2 to be the coefficients of 0' and 9
respectively in
equation(24):
N
gH
a2 =-(-
g
HTcP
1
g
+ -H
c,
N2
gh
N2
g
)
(29)
(30)
and
=(31)
2
It's easy to see that the propagating property of the solution to equation(28)
depends on the sign of (a 1 - "1). Here we define
2
72 =
2
a2
ai-- 4
2
72 is a function of mean zonal flow R. In weak westerlies -
(32)
> 0, and
the stationary wave will propagate, while in easterlies or sufficiently strong
westerlies -2 < 0, and it will be trapped.
Satisfying the boundary conditions(25)-(26), we obtain the vertical structure solution 0': (for simplicity, use - = [7| > 0)
For propagating wave (/2 > 0):
e(
:7
N2
(Z) =
2-)Z
(33)
For trapped wave (-t2 < 0):
e(--#)Z
'(Z) = fo(-2-
(34)
?:-)
Therefore, the initial forced stationary wave solution is found:
For propagating wave (-y2 > 0):
'=
Re[Ape kx(sin lay + A, cos ly)e-iY~#)z]
(35)
where
AP = -(36)
fo(-i-Y - y
- g)
For trapped wave (-y2 < 0):
,' = Re[Ateik (sin ly + A, cos ly)e(~)"]
(37)
where
At = -
"S
fM-7
-
?
-
(38)
c-
Normal mode initialization
3
3.1
Normal modes
First, we are going to find the normal modes of atmosphere. Assume the
atmosphere is restricted between two vertical boundaries with depth of D.
The vertical dependent equations are:
1_ Ipw
iwfo
n /'+ -
io-
z"
P
gh,
aO'
Oz
(39)
0
N2
+
n
=
-- W' = 0
(40)
fo
where w is frequency and hn is the equivalent depth of mode. Eliminating 0'
from equation(39) and substituting it into equation(40), yields:
a2w*
N2
1
N)W AH2
82z + ( gqhn
-
=
0
(41)
where
w* =w'e 2
(42)
To find normal modes is to solve the following eigenvalue-eigenfunction problem:
,
82
2= + m2
=0
(43)
where
2
2
m2 m N
__ _21i
__
ghn 4H 2
(44)
satisfying the boundary conditions:
at z=0 and z=D:
w* = w = 0
(45)
or equivalently
az
= 0
(46)
So ths solution to equation(43) is:
w* = sin mz
(47)
and
0'
(48)
= cos mz
where m = P, p=01,2,...
In order to do normal mode initialization, we then project the initial
stationary wave into normal modes. Set:
k
=
(49)
Re[ E Om cos mz]
M=0
u =
(50)
Re[ E um cos mz]
m=0
v
=
Re[ E
(51)
vm Cos mz]
m=O
the coefficients of normal modes are determined by Fourier expansion of the
stationary wave solutions(35)-(38).
For propagating wave (-y2 > 0):
AP
=
=2
0-(-i7
_(e(-iy-')D
-
- ?)D
-
-
(
(e(
)D
1)o'(y)eikx
(52)
)u/(y)e ikx
(53)
ikx,
-1)vI(y)e
M=
0
(54)
and
2
(-iy
SDm2
(-i
)Ap
((
)Pe(-i-
)D
1)7pi(Y)eikx
(55)
m
(56)
- a, )2
(-ig, - ?)Ap ((
n'=2
-- T2 ) 2 (
"' D m 2 + ( --
?(-i--
D
)
~ '(~
k
(-i7 - g)Ap
D m2
+ (-i2
-27)2
Dm
2
V= 2
m
* ((_1)P e(-i,-)D -
m
w here A ,
=
p = 1 2,3...
D"1=
(57)
(e(-'Y-7
(58)
=
-
M-
-
,
1)v'(y)e
For trapped wave ()j2 < 0):
At
=
(e(-,-)D
A~
-
-
)D
_)
luyeikz
(-- -')
(59)
i)D
At
0
ikx
(e(--
)D
1)v'(y)eikx,
c- )D
(-
M
=
(60)
0
and
m
(D m2 +
u
D m 2 + (-
-
(--
((
e( -
)D
_ 1)0 (y)e ikx
(61)
((_ lPe( --.
)D
-
(62)
2
-
2)
(-y- ')At
D m2+ (
- 2)2
V' = 2
m
)At
2 -
e)2
*(-(-(1
m=
where At
=
~ fM-
-)D
l)u'(y)e kx
_
p= 1 2, 3...
i
kx
(63)
Normal mode initialization
3.2
For normal mode initialization the horizontal dependent equations are
linearized around the basic state of rest. Set the perturbation to be the form
of ei(kx-wt), the linearized equations for each vertical tranformation can be
written in a matrix form:
W
V,
i(fo +
0
U'm
=
i(fo
+ Oy)
#y).
0
k/g-hi
U'
AM
8i
-
(64)
The 3x3 matrix explicitly appearing on the right-hand side of the above
matrix equation(64) is Hermitian, so every eigenvalue w of the matrix is real
value. The three eigenvalues of the 3x3 matrix are Rossby wave frequency
and two gravity wave frequencies.
Since the initial field includes both basic state and perturbation, we need
to initialize both of them by NMI. The basic state is geostrophic and independent of x:
(fo + /y)
= -fo
8#(y)
(5
(65)
ay
i = 0
(66)
so the matrix(64) for basic state becomes:
0 f
o
i(fo +#y)
i(fo+#Oy)
0
0
y
- i V/goaT
(
(67)
The projection of the stationary wave into Rossby wave and gravity waves
is:
r
Ur
V'
X
g
(68)
where r denotes Rosby wave, g and g' denote east and west propagating
gravity waves respectively. The matrix X is composed of the corresponding
three eigen-vectors. Therefore, the r, g, and g' are obtained by inversing the
matrix X:
r
Urm
X-1
g
(69)
V/
For the purpose of normal mode initialization, the gravity wave component of the initial stationary wave is excluded by setting both g and g' to
zero, namely:
Um
VI
=X
r
0
(70)
0
4
Model description
We employ time-dependent linearized baroclinic primitive equation model.
The basic state is the same as we used in the initial stationary wave. Namely,
isothermal, constant mean zonal flow U and static stability N2 . The time-
dependent linearized equations are:
&u'
- +
Bu'
(fo + #y)v'
-
&v'
&v'
at
ax
/
f ap'
ax
ap'
at-
+
18Op'
(71)
_
18&p'
7y
aT' + T'+n -----1 ap'
at
=
2 ,
TN
_--+w'=0
c~Pat c7,Pax
g
apw'
ap' + X(- au' + av'
-- ) +
f-
ay
ax
ax
al
az
=
0
(74)
TT(75)
pt
_l
az
(73)
p
T'
T
(76)
The horizontal and vertical boundary conditions are the same as we used
in obtaining the initial stationary solution except that there is a model 'top'
at the height D. Thus, the vertical boundary conditions in our model are:
at z=D,
w' = 0
(77)
at z=0,
w' = iku-
(78)
The stationary part of the model can be computed numerically by setting all
a part in equations(71)-(74) to zero.
y
As horizontal boundary is satisfied, the solutions to equations(71)-(76)
are set to be the form of the product of function of (z,t) and eikxF(y), where
F(y) is the same as the y-dependent structure of the initial stationary solutions(19), (22) and (23). Therefore, the time-dependent part of our model is
only in vertical modes.
For numerical computation the above time-dependent equations are transformed into prognostic and diagnostic equations:
at
a?)
at
#y)v'
-
ikfop'
(79)
(fo +#y)u'
-
fo-
(80)
=
-ikiu' + (fo +
-
-ikiv'
-
T'RT
-
-(i
-ikiiT'-
at
CV
--(
at Pss/$'-POfy)=
az
w
1
cTN 2 I
'
-)w'
=
(
fo
T'
T
!-($'
ga t
-p')
-
(81)
cVg
H
Dz
-iki(pp'
ay
- iEJ?(iku'+
v'
- )dz
Ty
(82)
(83)
P$)+
-
ikfo
p
g
+--i p(i ku' + --- )dz
P z
p' =
-- )
C.,
RT&(
where
av'
ku' +
ay
Oy
,
p
Pt_
p
(84)
p'/fop and H is the scale height. .The subscripts s and t denote
the surface and the model top respectively. The tendency equation(82) is
obtained by eliminating p' from equations(75)-(76) and substituting it into
continuity equation(74), and then integrating the continuity equation from
surface to top. The diagnostic vertical velocity is obtained by integrating the
same continuity equation from z to top.
The prognostic equations(79)-(82) are integrated in time using mid-point
Euler-backward method. That is: for the first-order differential equation
-=
4t
R
(85)
where A denotes time-dependent variable and R the right-hand side, the A
forwards in one time step after being computed four times:
An+1 = An+ R(An) At
(86)
A**1 - An + R(A*+1 )At
(87)
A**
A*** = An + R( An + An+12
An+1 = An + R(An +
-A
A*** - A
n+12
)A t
(88)
n)At
(89)
where the subscript n denotes time point.
Finite differences are used for vertical direction. Vertical layers and the
displacement of all 5 variables u', v', T', 0', w' in these layers are sketched in
Fig.1.
k=K
U', v', V, W=,
k= K-1 ..................
T' ..................
v', ,',
k=K-2
U',
k=3
I', )'V
k= 2 .....................
k=1
U',I'
k=K
k=K-2
WI
', W'
TI
k=K -1
k=3
................
', W',
k= 2
k=1
Figure 1: Vertical layers k and the displacement of variables on these
layers. k=1: the surface, k=K: the model top
The second-order leap-frog scheme is used to approximate the vertical
spatial derivatives. For y direction the equations(79)-(84) are integrated independently in specified grid points. All the partial differentials 2- appearing
on the right-hand side of equations(79)-(84) are the direct derivatives of ydependent structure
if)a~y
Results
5
In this section we present the major results in our test. First we list the
parameters used in our model:
model top: D = 20.Km
horizontal wavenumbers: k=4.44 * 10-
7
m-1, 1=4.8 * 10- 7 m-
scale height: H = 7.99Km
temperature: T = 273 0 K
static stability: N 2 = 4.0 * 10- 4 sec-2
the height of topography: rj, = 1.Km
fo
=
1.03 * 10- 4 sec-1
#=1.6*10-11sec-m~ 1
vertical resolution: 1.Km
time step: 3min.
5.1
The comparison of the initial stationary wave solution and the model stationary wave response
The initial stationary wave solution is calculated by using radiation condition on the top, while the model stationary wave is computed by imposing
a lid on the top.
The vertical structure of these two waves against dif-
ferent mean zonal flows are plotted in Fig.2. It is shown that the initial
stationary waves under mean flows of ii = 10.m/s and U = 25.m/s are both
vertically propagating. Due to the reflection of the model lid the model sta-
tionary responses are different from the initial stationary waves especially in
i = 10.m/s. At stronger westerly i = 45.m/s the initial stationary wave
is trapped. However, the amplitude of the model stationary wave decreases
with height up to 14Km and then increases up to the top. Moreover, in
easterlies the initial stationary waves are heavily trapped and they look like
the model stationary waves.
a)
=
b) ; = 25.m/s
10.m/s
20
1816
14
12
10
20
18
16
14
12
10
'
'
6
4
2
00
6
4
2
1
2
3
4
5
6
7
8
AMPLITUDE OF STREAM FUN.*1E+7.
4
6
8
10
12
14
16
18
20
AMPLITUDE OF STREAM FUN.*1E+7.
d) ;a = -5.m/s
A MPLITU
C
V;4
44
4U
LA
LA
LA
AMPLITUDE OF STREAM FUN.*lE+7.
0
LSR
E
FN*E7
LA
.2
AMPLITUDE OF STREAM FUN.*1E+7.
.4
.6
.8 1.0 1.2 1.4 1.6
AMPLITUDE OF STREAM FUN.v1E+7.
1.9
Figure 2: The vertical structure of streamfunction 0' in m 2 /s
* 107
of the initial stationary wave with radiation condition (A letter) and
the model stationary wave with a lid (B letter) against different
mean zonal flows: a) z- = 10.m/s, b) U-= 25.m/s, c) U-= 45.m/s, d)
U = -5.m/s, e) U-= -15.m/s.
5.2
Normal mode initialization
We first initialize the basic state of the initial field. It is found that in
varying mean flows ranging from easterlies to westerlies the Rossby wave
frequencies are near zero and the projections of the initial field almost completely go into Rossby waves. This result is reasonable as the basic state is
geostrophic. We do not show the result here.
The coefficients of the normal modes of the original perturbation fields
(V)', u ,v' ) and the initialized fields against different mean flows are plotted
0
in Fig.3-Fig.7. Here we only show the results of initialization at 45 N. In
addition, since in the cases of ii = 10.m/s and i! = 25.m/s the stationary
waves are propagating, and all the variables are complex values, we plot both
of the amplitude and the phase of ', v' and u'. However, in the other cases
the stationary waves are trapped and all the variables are real values, so we
only plot the variables themselves.
u
10.m/s
=
-
4.5
4.5
4.0
* 4.0
3.5
z 3.5
LL
3.0
3.0
c 2.5
2.5
2.0
2.0
1.5
U 1.5
1.0
0
1.0
-
-
I R q Pn rA - 10
8
4
6
2
VERTICAL MODES M
0
2
12
14
0 -.
2
1
20
1s
16
14
>12
0 10
08
j
0
-2
2.2
2.0
1.8
1.6
1.4
- 1.2
w 1.0
0
-j
CL
.8
.6
z.4
.2
0
-
2
A0
-2
FA
4
6
8
10
VERTICAL MODES M
0
2
4
6
B
10
VERTICAL MODES M
6
-j
CL4
Z
0
12
14
1
-2
0
2
4
6
8 10 12
VERTICAL MODES M
14
16
10
4
6
8
VERTICAL MODES M
12
14
16
-2
0
2
4
6
8
10 12
VERTICAL MODES M
14
16
*
107, V'
and u' in m/s versus vertical modes from mode 0 to mode 15 for
field.
1
7!l
2
=
14
S.1
:
Figure 3(a): The amplitude of streamfunction 0' in m 2 /s
U
12
10.m/s.
Left: the noninitialized field, Right: the initialized
ii = 10.m/s
M
200
Li
Li
200
8
150
U.
Li
150
100
100
S50
50
r
0
W
-50
In
L0
0
cc
-50
-100
100
w -150
U,
in
=-200
-2
(L
0
2
8
10
4
6
VERTICAL MODES M
12
14
16
-150
-200
2
0
2
4
6
8
10
VERTICAL MODES M
12
14
14
2
0
2
4
6
8
10
VERTICAL MODES M
12
14
1(
-2
0
2
4
6
8
10
VERTICAL MODES M
12
14
0
200
200
150
150
100
100
50
50
>
0
0
o
-50
Li
Li
3
-50
Li
w -100
-100
-150
-150
-200
-2
0
2
10
6
8
4
VERTICAL MODES M
12
14
-200
1
200
200
150
150
100
CC 100
Li
C3
50
50
0
S 0
o
Li
n
Li-
-50
0
-50
LiJ
in-100
100
(.-150
C-
-200
-2
0
2
4
6
8
10
VERTICAL MODES M
12
Figure 3(b): The phase of
-200
14
4',
-150
v' and u' in degree versus vertical
modes from mode 0 to mode 15 for - = 10.m/s. Left: the noninitialized field, Right: the initialized field.
ii = 25.m/s
12
.
.
12
.
-
11 10
-11
10
Z9
8
7
w
<7
6
5
U-4
I-1
w3
6
4
w3
9
2
I-
B
0
2
0
2
10
8
6
4
VERTICAL MODES M
12
14
20
2:-
1
2
0
2
10
4
6
8
VERTICAL MODES M
12
14
1
2
0
2
10 12
6
8
4
VERTICAL MODES M
14
16
14
16
55
50
'45
55
50
45
40
>35
30
w 25
L' 40
820
820
15
15
> 35
30
w 25
210
0
0
2
0
2
6
8
10
4
VERTICAL MODES H
12
14
1
-A
16
cfl
N
-
14
14
m 12
12
= 10
3 10
LU-
Lu.
08a
w
6
6
4
.
2
2
0
-2
4
0
2
8
10
4
6
VERTICAL MODES M
12
14
16
0-2
H
U
Y~
0
2
-
4
6
8
10 12
VERTICAL HODES M
Figure 4(a): As in Figure 3(a) but for il = 25.m/s
U =
25.m/s
W
200
200
8
150
150
100
100
50
S50
X:
0
0
-50
-50
L'..-100
-100
w -150
-150
cn
0
-200
= -200
-2
w
o
0
2
8
10
6
4
VERTICAL MODES M
12
14
200
200
150
150
o
0
2
6
8
10
4
VERTICAL MODES M
12
14
1
2
0
2
6
8
10
4
VERTICAL MODES M
12
14
11
-2
0
2
10
8
4
6
VERTICAL MODES M
12
14
100
100
o
50
0
Li.
2
1
50
0
Z
Li.
o
-50
-50
-100
C -100
i
C- -150
a. -150
-200
-200
-2
0
2
8
10
6
4
VERTICAL MODES M
12
14
1
200
200
150
-
w
Lii
150
100
w
9
50
50
LLi
50
8D 100
c
100
-50
-100
S-150
n- -150
-200
-200
-2
0
2
8
10
4
6
VERTICAL MODES M
12
14
16
Figure 4(b): As in Figure 3(b) but for U-= 25.m/s
i = 45.m/s
5.0
4.5
+ 4.0
3.5
3.0
2.5
i-
5.0
4.5
+ 4.0
3.5
3.0
z 2.5
2.0
LL 2.0
1.0
0- 1 .0
in
.5
0
2
0
2
4
6
8
10
VERTICAL MODES M
12
14
0
1
2
0
2
4
6
8
10
VERTICAL MODES M
12
14
11
2
0
2
4
6
8
10
VERTICAL MODES M
12
14
1
24
22
20
18
16
14
12
10
8
6
4
2
24
22
20
18
16
14
12
10
8
6
4
2
0-2
0
2
4
6
8
10
VERTICAL MODES M
12
14
1
-2
0
2
4
6
8
10
VERTICAL MODES M
12
14
16
0
-
=r]-1
1
-2
0
2
[
1
4
6
8
10
VERTICAL MODES M
12
Figure 5: Streamfunction ?P' in m 2 /s * 10', v' and u' in rn/s versus
vertical modes from mode 0 to mode 15 for U = 45.m/s
14
16
u
= -5.m/s
.28
.24
.20
.16
.12
.08
.04
0
-2
0
2
10
8
4
6
VERTICAL MODES M
12
14
1
-2
0
2
4
6
8
10
VERTICAL MODES M
12
14
11
0
2
8
10
4
6
VERTICAL MODES M
12
14
1
-2
0
2
4
6
8
10
VERTICAL MODES M
12
14
1
-2
0
2
4
6
8
10
VERTICAL MODES M
12
14
16
1.2
1.0
S.8
-
.6
.4
.2
0
-2
.16
.16
.12
.12
.08
.08
.04
.04
0
0
-. 04
-. 04
-. 08
-. 08
-2
0
2
8
10
6
4
VERTICAL MODES M
12
14
16
Figure 6: As in Figure 5 but for U = -5.m/s
= -15.m/s
.60
.60
-
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
0-2
0
2
6
8
10
4
VERTICAL MODES M
12
14
16
0
2
10
6
8
4
VERTICAL MODES M
12
14
14
-
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.05
0
-2
0
2
4
6
8
10
VERTICAL MODES M
12
14
16
0
2
4
6
8
10
VERTICAL MODES M
12
14
16
2
4
6
B
10
VERTICAL MODES M
12
14
16
2.4
2.0
1.6
1.2
.8
.4
0
2
-2
.5 -
.4
.3
.2
0
-. 2
-. 3
-2
0
2
4
6
8
10
VERTICAL MODES M
12
14
16
-. 4-2
-
0
Figure 7: As in Figure 5 but for U = -15.m/s
From the upper two panels in all Fig.3-Fig.7 we found that for both the
propagating and trapped waves the initialized 0' and the initialized v' are
pretty similar to the original ones in terms of their amplitude and phase/or
sign. This means that the projection of V' and v' into gravity wave is much
smaller than that into Rossby wave.
However, the lowest panels in Fig.3.-Fig.7. show that the initialized u'
is different from the original u' and the difference depends on the mean
flow. In westerlies ranging from weak to strong, the amplitude of the leading
mode, which has the largest amplitude among all other modes, changes less
than twice after initialization except at i = 10.m/s. At ii = 10.m/s the
amplitude of mode zero increases about 3 times after initialization. However,
the phase or the sign of each mode remains unchanged in westerlies. On the
contrary, the most pronounced feature in easterly mean flows is that the sign
of the first several leading modes is reversed. Moreover, in Fig.6 and Fig.7
it is found that the relative importance of the first two modes (m=0,1) is
changed. The first mode (m=O), which is originally comparable in size to
the second mode (m=1), becomes much larger than the second one after
initialization. In addition, the amplitude of the first mode increases about 5
times in U-= -5.m/s.
5.3
Time behavior
We plot the time behavior of the noninitialized and the initialized field 4'
within 20 days after day 0 against different mean zonal flows. First we show
in Fig.8(a)-(e) the behavior of the vertical structure of the streamfunction 0'
fixed at point (00 E, 45 0 N). As reference we also plot the vertical structure
of the initial stationary wave (day 0) and the model stationay wave in the
same picture. It is found that in westerlies ranging from weak to strong the
behaviors of the noninitialized and initialized fields are qualitatively similar.
However, in easterlies about at day 10 the vertical structure of the initialized
field begins apart from that of the noninitialized field. As time increases the
initialized field behaves more and more differently from the noninitialized
one. This feature can be more captured in the case of ii = -15.m/s.
Moreover, we plot in Fig.9(a)-(e) the contour of the noninitialized and initialized 0' at 5 Km within 20 days after day 0 against different mean flows.
It is found that in westerlies ranging from weak to strong the phase speed
as well as the amplitude of the high and low are similar between the noninitialized and the initialized fields. In the case of - = 45.m/s the similarity
is smaller compared with that in U-= 10.m/s and i = 25.m/s. However, in
easterlies it is noticed that at day 5 the initialized field begins to oscillate
in place with time compared with the noninitialized field especially in its
southern part. And then the oscillation becomes more and more pronounced
as time increases.
In addition, we noticed in Fig.8 and Fig.9 that in westerlies neither of the
noninitialized and the initialized fields behaves like the original stationary
waves. However, in easterlies, the time behavior of the noninitialized fields
is near stationary. We will discuss this later.
=
10.m/s
20
20
18
16
14
12
10
8
6
4
2
0
/7
18
16
14
12
y10
-
w8
UT6
'
0
20
18
4
2
0
Lb
*
'
'
12
10
6
8
4
2
AMPLITUDE OF STREAM FUN.*1E+7
'
'
'
'
'
18
16
14
( 1 DAY I
0
12
10
6
8
4
2
AMPLITUDE OF STREAM FUN.*1E+7
18
16
14
( 2 DAYS)
18
0
10
12
6
8
4
2
AMPLITUDE OF STREAM FUN.*1Et7
16
18
14
(10 DAYS)
-rv
16
14
12
10
8
W=6
4
2
0
0
2
4
6
8
12
10
AMPLITUDE OF STREAM FUN.*1E+7
14
I5
16
DAYS)
20
18
16
14
12
10
a
20
18
16
14
12
10
*
/~7~
$
0
12
9
10
6
4
2
AMPLITUDE OF STREAM FUN.*1E+7
Ld6
W6
4
4
2
0
2
0
0
12
10
8
6
4
2
AMPLITUDE OF STREAM FUN.v1E+7
14
16
115 DAYSI
18
16
14
(20 DAYS)
Figure 8(a): The time behavior of the vertical structure of streamfunction 4' in m 2 /s
*
10
from day 1 to day 20 for i = 10.m/s. A: the initial
stationary wave (day 0), B: the model stationary wave, C: the behavior
of the noninitialized field, D: the behavior of the initialized field.
u
= 25.m/s
AMPLITUDE OF STREAM FUN.*1E+7
t I DAY I
AMPLITUDE OF STREAM FUN.*1E+7
( 2 DAYS)
AMPLITUDE OF STREAM FUN.*1E+7
I5
DAYSI
AMPLITUDE OF STREAM FUN.*1E+7
(10 DAYS)
AMPLITUDE OF STREAM FUN.*1E+7
(15 DAYS)
AMPLITUDE OF STREAM FUN.*1E+7
0
ALU STRA
O
Figure 8(b): As in Figure 8(a) but for U = 25.m/s
38
F
N
0
DNS
(20 DAYS)
U
= 45.m/s
18
14
12 10
8
6
4
2
q
(l
N.*1E+
LFU..STE
Lo LA
APLITUDE
0
'
1 1 DAY
AMPLITUDE OF STREAM FUN.*lE+7
"o 441 N1.N
146
u
...L
LA
..
4
4
LA
LA
r0
AMPLITUDE OF STREAM FUN.*IE+7
20
(4)
CM)
V,
LA
LnA
.0
AMPLITUDE OF STREAM FUN.*1E+7
( 2 DAYS)
6
b....
.0
...6..
X0 ... LAhh
rr
1
r
.
r-
AP L
I 1 0
1 5 DAYS)
U .E
L
RAM
F.*
1E
(1
DA
AMPLITUDE OF STREAM FUN.*1E+7
(10 DAYS)
c')
qt
't
LA
LA
10 .0
AMPLITUDE OF STREAM FUN.*lE+7
rrC
(20 DAYS)
-
18
16
14
12
10
8
6
4
2
...
LA.
CV
.
3
.
qt
A
.
.
LA
I..
Ln
I.
,
.0
n 0 in
.0
AMPLITUDE OF STREAM FUN.*1E+7
6 X LA
1- r-
6,
Lq
C6
115 DAYS)
Figure 8(c): As in Figure 8(a) but for U- = 45.m/s.
39
0
O = -5.m/s
20
18
16
14
12
20
18
16
14
12
10
10
6
4
2
0
6
4
2
0
0
4 5
6 7
8
9 10 11 12
1 2
3
AMPLITUDE OF STREAM FUN.*1E+6 ( 1 DAY I
20
18
16
14
12
20
le
16
'14
12
10
10
6
4
6
4
2
0
2
0
0
1 2
3 4
5
6 7
8
9
10 11 12
AMPLITUDE OF STREAM FUN.*1E+6 1 5 DAYS)
1 2
3
4
5 6
7
8 9 10 11 12
AMPLITUDE OF STREAM FUN.*1E+6 (10 DAYS)
0
20
18
16.
20
18
16
14
14
12
12
10
-10
=6
4
2
0
1 2
3
4
5 6
7
8 9 10 11 12
AMPLITUDE OF STREAM FUN.*1E+6 ( 2 DAYS)
0
0
1
2
3 4
5 6
7
8
9 10 11 12
AMPLITUDE OF STREAM FUN.x1E+6 (15 DAYS)
Uaj 6
4
2
0
'
0
1 2
3
4
5
6
7
8 9 10 11 12
AMPLITUDE OF STREAM FUN.x1E+6 (20 DAYS)
Figure 8(d): As in Figure 8(a) but for U-= -5.m/s and the unit is
m 2 /s * 108.
i = -15.m/s
20
18
16
14
12
10
8
64
2
0
'
0
.2
.4
.6
.8
1.0 1.2
AMPLITUDE OF STREAM FUN.*1E+7
1.4 1.6 1.8
1 1 DAY I
0
.2
.4
.6
.8
1.0 1.2
AMPLITUDE OF STREAM FUN.*1E+7
1.4 1.6 1.8
( 2 DAYS)
0
.2
.4
.6
.8 1.0 1.2
AMPLITUDE OF STREAM FUN.*1E+7
1.4 1.6 1.8
(10 DAYS)
.2
.4
.6
.8
1.0 1.2
AMPLITUDE OF STREAM FUN.*1E+7
1.4 1.6 1.8
(20 DAYS)
20
18
16
14
12
10
8
0
0
.2
.4
.6
.8
1.0 1.2
AMPLITUDE OF STREAM FUN.*1E+7
.2
.4
.6
.8
1.0 1.2
AMPLITUDE OF STREAM FUN.*1E+7
6
4
2
0
1.4 1.6 1.8
( 5 DAYS)
1.4 1.6 1.8
(15 DAYS)
20
18
16
14
L 12
10
"8-d 64 2 0
0
Figure 8(e): As in Figure 8(a) but for - = -15.m/s.
1.m/s
=
S5N
j
50*N
50*N
5'N
( 0 Day)
45*N
40'N
40-N
0W
120/
l
80
0*
60*E
*
120W
12*E
80*E
60w
120*E
180
WI*N
55*N
50*N-
46*N
50*N
1 Day)
C"'
35*N-
40*N
-
35*8
04W
1201
60W
0
S0E
180*E
120*E
55'N
180W
120W
60W
0*
60*E
120*E
DE
180
MIN-
50'N
50*N
.
45*N
45*N
.
40*N
40*N-
11e
8'
5801W
rs
( 5 Days)
1201W
601W
0
60*E
120*E
1800*E
Days)
Days)
Days)
Figure 9(a): The time evolution of streamfunction at 5 Km from day
2
0 to day 20 for - = 10.m/s. The contour interval is 2.m /s
*
107 and
negative contours are dashed. Left: the noninitialized field, Right: the
initialized field.
i = 25.m/s
( 0 Day)
(1
Day)
E
( 5 Days)
E
(10 Days)
-E
(15 Days)
I'E
(20 Days)
180*E
Figure 9(b): As in Figure 9(a) except that ii = 25.m/s and the
contour interval is 4.m 2 /s
*
107 .
= 45.m/s
(0 Day
(1 Day
rC2
65'N
50'N -
40*N
60*W
120eW
35*0
'
T80E
120E
I
/
50*N
I
*
45*N
55-N
55*N
-
I
~
*
45*N
~1
*~*
(5 Days)
40*N
'C
I-
35*N
180*E
180oW
120*W
60E
0*
60W
50N
5'N
120*E
18 0Dy
120*E
180'E
-
iC
-
)
40*N
3536'N80 *W
60"W
120'W
0*
120*E
50*E
1SC".E
358(
0*
60*W
120'W
0
60E
55*N
55*N
50*N
45"N
4*N
40N
40"N
35'N
1DC*W
60*W
120v
0*
120E
80E
180 "E
s%
80"v
120*W
60*W
(15 Days)
%
0
60E
120*E
180
55*N
50*N
45-N
40'N
35N
*
-
-
1810-W
I
I
I
I
I
I
I
I
I
I
120-W
I
I
I
I
I
I
~
45*N
I
II
I
I
II
I
I
I
I
0-
(20 Days)
II
40N
I
60-W
50N
I
80-E
I
I
120"E
180*E
180*W
120W
501W
0.
0E
120*E
Figure 9(c): As in Figure 9(a) except that U-= 45.m/s and the
contour interval is 4.m 2 /s
*
107.
180"
i
55*N
3
3
50*N
3
'
OS'S
3
3
3
.
~3
3
3
3
3
3
I
'
45*N
3''
3
3
3
3
-5.m/s
=
3
'3
3
I
33
--.
50'N
3
3
3
3
3
3
45'N
40*N
3
-~
3
3
3
3
I
3
3
3
3
3
3
3
3
I
3
3
3
3
3
3
I
(0 Day )
40'N
MI'N
18 0.W
0*W
SOOT
60*W
120W
120*W
35*N
0*
0E
120*E
ISO
180*W
55'N
120*W
S0W
0'
SOE
120*E
180'S
5501
3
3
r3
50-N
WI'N
45-N
I
SON
3
45'N
3
( 1 Day)
-
40*N
40'N
-
35'N
18
I
'E
35'N
180 *W
120*W
60*W
0*
60*E
120*E
18C'E
55*N
55'N
50'
50*N -
45*N
0'W
120'W
80*W
0*
60*E
1205
180
/
45'N -
40*N
-
( 5 Days)
40*N
35*N
18
*E
36'N
180
*w
120'W
60*W
0*
0*E
120*E
180'E
55'N
5O*N
46'N
(10 Days)
40*N
10 0*W
60*W
120'W
120*E
60*E
180'E
60*W
0
60*E
120'E
180'E
56'N
--
-
--
I
(15 Days)
- 33
40'N
35*N
8NI
1S 0*W
120*W
60*W
0*
80*E
120*E
18(0*E
80OW
0*
SO0E
120*E
180*E
55*N
50mN
-
45*N
(20 Days)
40'N;
35'
180*W
120'W
60'W
0*
60*E
120'E
180"E
Figure 9(d): As in Figure 9(a) except that n = -5.m/s and the
contour interval is 2.m 2 /s * 106.
U
= -15.m/s
I
I
I
--
-
55'N
18*E
I
I
I
I
46N
35I*W
I
I
I
( 0 Day)
I
I
I
'
'
/
/
40*N
35*N
55'N
120W
120'W
60W
60*W
C,
0*
50-N
I
I
I
I
I
I
/
45*N
45N
I
I
I
15(
18 0D
I
I
50N
V
I
120E
120*
BOE
60 E
- -
g
I
*
I
I
1 Day
40'N
40N
180*W
120W
0OW
0*
80*E
1800E
120*E
35N
i80'w
120W
60w
0*
12O*E
6OE
1 30*E
55*N
BOON
5-N
I
( 5 Days)
45'N
40*N
35*N
1800W
I
120-W
60*W
0
60*E
1800E
1200E
55'N
55"N
I
I
I
/
I
~
I
I
/
I
N
SON
--
I
50N
46*N
*
/
40*N
40*N
M"N
1801W
I
(10 Days)
~D<~:'KII
120'W
601W
0
0*
60E
W0E
120E
120"E
180*E
(15 Days)
5-N
40N
"Ta0O
120OW
60W
0*
60*E
120*E
180' E
55*N
50-N-
(20 Days)
40'N
-
50.w
45-N
180"W
Figure 9(e): As in Figure 9(a) except that i = -15.m/s
contour interval is 4.m
2
/s
*
106.
and the
6
Discussion
6.1
The effect of model lid on initialization
In all of our experiments, we impose an artificial top in the model. We
found that the behaviors of the initialized and noninitialized fields are similar
in westerlies and different in easterlies respectively. Thus, we expect that the
initialization is insensitive to the existence of model top. In order to verify
this, we increased the model top to 30Km for the cases of U = 10.m/s and
f = -15.m/s as comparison with the model top at 20Km.
Fig.10(a) and Fig.10(b) show the behavior of the vertical structure of
streamfunction at (00 E, 450N) and the contour of the streamfunction at 5Km
respectively for mean flow of ii = 10.m/s when the model top is at 30Km.
It is found that the vertical structures for the noninitialized and the initialized fields are pretty similar within 20 days. Moreover, the contour of the
initialized field looks like the noninitialized one. Thus the behavior of the
initialized field is generally in agreement with that of the noninitialized one
when the model top is placed at 30Km.
Fig.11(a) and Fig.11(b) show the behaviors of the noninitialized and the
initialized fields for the mean flow of U-= -15.m/s when the model top is at
30Km. It is found that in easterly the initialized field begins apart from the
noninitialized one at day 5. As time increases the initialized field behaves
more differently from the noninitialized one.
U = 10.m/s
28
24
20
16
C12
=8
4
0
30
20
25
15
10
5
AMPLITUDE OF STREAM FUN.*1E+7
45
40
35
I 1 DAY I
0
25
30
20
15
10
5
AMPLITUDE OF STREAM FUN.*1E+7
45
40
35
1 5 DAYS)
35
40
115 DAYS)
0 0
25
30
5
10
15
20
AMPLITUDE OF STREAM FUN.*1E+7
35
40
45
( 2 DAYS)
0
20
25
30
5
10
15
AMPLITUDE OF STREAM FUN.*1E+7
35
40
(10 DAYS)
0
5
10
15
20
25
30
AMPLITUDE OF STREAM FUN.x1E+7
45
40
35
(20 DAYS)
28
24
20
16
5
12
=8
45
/
28
24
20
16
0 12
8
4
0
30
25
15
20
5
10
AMPLITUDE OF STREAM FUN.*1E+7
45
Figure 10(a): The time behavior of the vertical structure of streamfunction in m 2 /s * 10' from day 1 to day 20 for in = 10.mn/.s. The
model top is at 30Km. A: the initial stationary wave(day 0), B: the
model stationary wave, C: the behavior of the noninitialized field, D:
the behavior of the initialized field.
48
1.m/s
=
55*N
56N
50'N
50*N
45*N
II
4'N-
40*N
35'N
120*W
180*W
35*N
N0W
0*
80O%
1200E
1800E
( 0 Day)
120*W
180*W
0*
80E
120*
180
55*N
50*N
I
N
45*N
( 1 Day
40-N
35*N
160*W
120'W
120*W
801!
0'
60*E
120E
180 'E
55"N
45*N-
( 5 Days)
35*Nf
40*NI
I
I
i
0*
80*W
I
60*E
120'E
I
180*E
55'N.
50*N
180 *W
120'W
601W
0*
W0E
120*E
180'E
55"N
-
180*W
50*N
-
2.0
--
120"W
55*N
.
45'N
-O
-0
(10 Days)
'
40*N -
40*N
35*N
60'w
0'
80"E
120*E
18(
36*N
"W
120*W
601W
80'E
0*
120'E
180*E
56*N
50'N
80N
-,
1801W
45'N-
120'W
45*N
40-N
55*N
(15 Days)
40*N
-?
-
35*N
5-N
1
80*W
0*
80E
120*E
1IC
35*N
180 *W
120*W
O0N
601W
0
0E
120*E
180E
0.0
45'N
-
-
I~l
I,
I
.
.
--
(20 Days)
,'
/
/
40*N
//
II--
180N
'V
120*W
801W
0"
6O0E
120"E
Figure 10(b): The time evolution of streamfunction at 5Km from
day 0 to day 20 for
nu
= 10.m/s. The model top is at 30Km. The
contour interval is 2.m 2 /s * 10' and negative contours are dashed.
180E
ii
= -15.m/s
28
28
24
24
20
X 20
16
16
o12
w12
=8
8
4
0
0
.4
.6
.8
1.0 1.2
.2
AMPLITUDE OF STREAM FUN.*1E+7
1.4 1.6 1.8
1 1 DAY I
0
.2
.4
.6
.8 1.0 1.2
AMPLITUDE OF STREAM FUN.*1E+7
1.4 1.6 1.8
( 2 DAYS)
1.0 1.2
.2
.4
.6
.8
AMPLITUDE OF STREAM FUN.*1E+7
1.4 1.6 1.8
I5 DAYS)
0
.2
.4
.6
.8
1.0 1.2
AMPLITUDE OF STREAM FUN.*1E+7
1.4 1.6 1.8
(10 DAYS)
.4
.6
.8
1.0 1.2
.2
AMPLITUDE OF STREAM FUN.x1E+7
1.4 1.6 1.8
(15 DAYS)
0
.2
.4
.6
.8
1.0 1.2
AMPLITUDE OF STREAM FUN.*1E+7
1.4 1.6 1.8
(20 DAYS)
28
24
20
16
o12
8
4
0
28
24
20
16
012
8
4
0
0
Figure 11(a): As in Figure 10(a) but for - = -15.m/s.
-15.
m/s
56*N
--
4'N
( 0 Day)
40rN
1'E
356*N
18 0*W
120*W
60*W
0*
80*E
120*E
180 *E
(1
Day)
40*N
5'E
I
( 5 Days)
55'N
55*N
50'N
50'N
7
2
K~~~!
I.. -
v
45-N
45"N
IV
46I
40*N
35'N
18131W
~;:
I If
(10 Days)
-I
-
120*W
801W
0.
WE
120*E
180*E
180*W
60*W
120*W
0"
0E
120*E
180"E
55*N
50-N
- -I
48*N
(15 Days)
40'N
kV1
55*N
50-N
45'N
I
%
(20 Days)
40-N
35*N
10 "W
120*W
__I
80*W
0*
I
60*E
120'Z
I
I
180*E
35'N
180"W
'.'
z
'-/
80'W
120*W
0"
:50"E
120'
Figure 11(b): As in Figure 10(b) except that n = -15.m/s and the
contour interval is 4.m 2 /s
*
106.
Therefore, the results from the study of model top at 30Km are in agreement with those from the study of model top at 20Km. This confirms that
the existence of model lid does not affect initialization.
6.2
The sensitivity of initialization to mean zonal flows
We have found before that the time behaviors of the initialized fields
look like those of the noninitialized ones in westerlies ranging from weak to
strong. However, this does not hold in easterlies. In order to investigate why
the time behavior of the initialized field depends on the mean flow, we recall
the results from the normal mode initialization.
We have seen before that in westerlies in spite of small changes of the size
of U', the phases or the signs of u' remain unchanged. However, in easterlies
the first two leading modes (m=0,1) are heavily affected by the loss of gravity
part in terms of the sign, the relative importance and the amplitude. Since
the initial field is stationary wave, namely c = 0, according to wave phase
speed:
C =U-(90)
k2 + 12 + gh
f0
where h is the equivalent depth, in the presence of westerly mean flow Rossby
wave propagates westward, while in the presence of easterly mean flow Rossby
wave propagates eastward. However, eastward Rossby wave is not the eigenmode of the 3X3 eigenmatrix in equation(64), and therefore the initial stationary wave can not be corrected projected into the eigen system.
Thus, we conclude that the change of the mode structure after initialization is sensitive to mean flow and the change is crucial to the future behavior
of the stationary wave. However, questions still remain about the relevance
of initializtion to more realistic mean flows such as vertical wind shear.
6.3
The role of vertical wind shear
In order to investigate how the behaviors of the initialized and the noninitialized fields are affected by vertical wind shear, we did some simple experiments as comparison.
The wind shears are chosen as one case of all
westerlies from surface to top and two cases of weaker and stronger easterlies
near surface respectively and westerlies above. These are:
(a) U(z) is from 5.m/s at surface to 40.m/s at top.
(b) ii(z) is from -5.m/s at surface to 25.m/s at top.
(c) ii(z) is from -15.m/s at surface to 25.m/s at top.
The three different U(z) distributions are shown in Fig.12.
In the presence of vertical wind shear, the initial field is still the solution
to equation(24) or equation(28) as we used before, but solved numerically.
The radiation condition in model top is assumed as:
d'P
- =
dz
i(91)
where -y is the same as in equation(32), and Im(/) < 0 is required. The
initial field is then projected into normal modes by using discrete Fourier
transform.
20
16I
121
14
16/
0
14
2:
-15
-10
-5
0
5
15
10
U0
20
25
30
35
40
(M/S).
Figure 12: The vertical distribution of wind shear. a) in long dashed
line: Ufrom 5.m/s to 40.m/s, b) in short dashed line: U-from -5.m/.s
to 25.m/,s, c) in dotted line: U from -15.m/s to 25.m/s. The solid
line is zero wind line.
The results of the time integration of the noninitialized and the initialized
fields are plotted in Fig.13 and Fig.14. Fig.13(a)-(c) shows the behaviors
of the vertical structure of streamnfunction at (00E, 450N) for the cases of
54
shear(a)-(c) respectively. Again as reference the initial stationary wave with
radiation condition and the model stationary wave with lid are plotted in
the same picture. In case(a) the initial stationary wave is propagating from
surface to about 11Km then turns trapped on top, while in both case(b) and
case(c) the initial stationary waves are first trapped from surface and then
propagating upward with the latter trapped less heavily.
Comparison from Fig.13(a)-(c) shows that in all three cases the behaviors
of the initialized and the noninitialized fields generally bear resemblance.
However, it is noticed that there is some slight difference of the behaviors of
the initialized and the noninitialized fields in different time scales except in
case(c), where the behavior of the initialized field is pretty similar to that of
the noninitialized one in all 20 days. In case(a) the initialized field is similar
to the noninitialized one in the first several days, but at day 10 it differs
most especially in the upper part. Finally this difference seems diminishing
with time. However, in case(b) the initialized field differs most from the
noninitialized one in the first several days until at day 10. After day 10
the behaviors of the initialized and the noninitialized field look much more
similar than before.
Fig.14(a)-(c) shows the contour of streamfunction at 5Km for cases(a)-(c)
respectively. It is found that in all three cases the behaviors of the initialized
fields are pretty similar to those of the noninitialized ones not only in size
and phase speed but also the fine structures of contours.
.5
4.0
3.0
3.5
2.5
2.0
1.5
1.0
AMPLITUDE OF STREAM FUN.*1E+7 1 1 DAY
4.5
I
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
.5
AMPLITUDE OF STREAM FUN.*1E+7 ( 2 DAYS)
18
1x6
20
4
2
.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
AMPLITUDE OF STREAM FUN.*1E+7 (10 DAYS)
.5
4.5
4.0
3.5
2.5 3.0
2.0
1.0
1.5
AMPLITUDE OF STREAM FUN.*1E+7 1 5 DAYS)
0
.5
4.0
4.5
3.0 3.5
2.0
2.5
1.5
1.0
AMPLITUDE OF STREAM FUN.*1E+7 (15 DAYS)
.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
AMPLITUDE OF STREAM FUN.*1E+7 (20 DAYS)
Figure 13(a): The time behavior of the vertical structure of streamfunction in m 2 /s * 107 from day 1 to day 20 for shear (a). A: the
initial stationary wave(day 0), B: the model stationary wave, C: the
behavior of the noninitialized field, D: the behavior of the initialized
field.
C
0
.2
.4
.6
.8
1.0 1.2
AMPLITUDE OF STREAM FUN.*1E+7
1.4 1.6
( 1 DAY I
0
.2 .4
.6
.8 1.0 1.2 1.4 1.6 1.8 2.0
AMPLITUDE OF STREAM FUN.*1E+7 ( 2 DAYS)
20
18
16
14
12
10
6
4
-N
2
AMPL
U DE OF STREAM
F.*1E
AMPLITUDE OF STREAM FUN.*lE+7
0
5
Y
( 5 DAYS)
.5
1.0
1.5
2.0
2.5
3.0
3.5 4.0
AMPLITUDE OF STREAM FUN.*1E+7 (15 DAYSI
AMPLITUDE OF STREAM FUN.*1E+7
(10 DAYS)
AMPLITUDE OF STREAM FUN.*1E+7
(20 DAYS)
,N N
Figure 13(b): As in Figure 13(a) but for shear (b).
57
N
20
18
14
12
20
18
16
14
12
10
10
16
8
6
8
6
4
2
0
4
2
0
A I N
AMPLITUDE OF STREAM FUN.*IE-7
N.
N
N
1 1 DAY I
AMPLITUDE OF STREAM FUN.*1E+7
20
18
16
14
12
20
10
10
[ 2 DAYS)
19
16
14
12
=6
=6
4
2
0
4
2
AMPLITUDE OF STREAM FUN.*1E+7
1 5 DAYS)
53 N
CO -T tol 'o
f-
CO a,
M
c
AMPLITUDE OF STREAM FUN.*1E+7
(10 DAYS)
AMPLITUDE OF STREAM FUN.*1E+7
(20 DAYS)
20
18
16
14
12
10
L3 8
:C6
4
2
0.
0
6
8 10 12 14 16 18 20 22
2
4
AMPLITUDE OF STREAM FUN.*1E+7 115 DAYS)
Figure 13(c): As in Figure 13(a) but for shear (c).
55*N
0"N
-0
45'N
1
I
IGI
G
&
I
( 0 Day )
\
40*N
71*N
55"N
It
II
35*8
18~
0W
55*N
50'N
I
40*N
180eW
120W
60W
60*E
0*
120E
18C
35'N
i8
120W
60W
805E
0*
~
13
O
120W
18C0E
120E
I'
NI'
I'
I,
80E
2
120E
( 1 Day )
0*E
50-N-
( 5 Days)
4*N 55*N
5'N
IS
IS
4IN-
0N
40N
36'N
180
55"N
0
50*N
0
120'W
60*W
0'
80"E
120*
-
181
(10 Days)
Z -Z
180 *W
60*W
120"W
0"
60E
120*E
55N
50*N
50*N
180"E
0.
0 0
0.0
45N
(15 Days)
40'N
40*N
18 ow
80*W
120*W
0"
0*E
120"E
180'E
58*N
50*N
I.
45*N
(20 Days)
40N
35'N
180*W
120*W
80"W
0*
8O*E
120E
Figure 14(a): The time evolution of streamfunction at 5Km from
day 0 to day 20 for shear (a). The contour interval is 2.m 2 /s
* 107
and the negative contours are dashed. Left: the noninitialized field,
Right: the initialized field.
180E
55'N
55-N
0N
-N
( 0 Day
120W
35801
50ON
60T
0*
60*E
120'E
180*E
120*W
0OW
15'8
0*
60*W
60'E
120*E
18
50*N
-OE
(1 Day
40'N
40-N
( 5 Days)
0"E
55*N
----
500N
46'N
~
'
(10 Days)
40'N0
86*N
180*W
601W
120'W
0.
80E
120*E
180E
(15 Days)
OE
50'N
W-N
<
(20 Days)
40-N
1
35"N
-
Figure 14(b): As in Figure 14(a) except for shear (b) and the
contour interval is 4.m 2 /s
60
*
106.
55*N
55'N
5Q*N
50'N
45"N
45'N
( 0 Day )
40*N
40*N
38N
Ie0*
120*W
800E
0*
806
180*E
120E
358N
1810*W
55N
55'N
50'N
50*N
35ON
00'
45-N
O
0*
60'!
60W
\\>
\'~\
120E
120*E
80E
W0E
180*E
'I
'I
'.4/
( 1 Day )
,L
40N
120'!
120'W
40'N
120
806E
0*
806
180*E
120*E
35*N 0'!
18 00W
1~o.,w
120O
0'
0*
uu-w
60*W
60'E
80*E
120'E
120*E
18O'E
180*E
50N
55"N
( 5 Days)
45*N
4O*N
35'N
8
0'!
120*W
0*
60!
0"
180*E
1200E
60E
60*E
1200E
180'E
55'N
50ON
(10 Days)
46*N
4ON
55'N
56*N
50*N
50*N
4-N
45'N
-..-..-
-I
40N
35
1
120OW
.
0*
80W
p
.4
I
/
(15 Days)
180E
120*E
80*E
180*W
120*W
a'
8O~W
0*
80eW
8OE
W0E
120E
120*E
550N
0N
~
kI::~
f
II
I
40*N
120'W
501W
I
(20 Days)
%
/
/
II
18(
18 0'E
4O*N
I
I
~
180 .w
(V
4O*N
55ON
3'N
.4
,-
I.
*W
-
-
--
0*
120*E
8OE
180"E
35'N
1801W
120'W
801W
0*
0E
12OE
Figure 14(c): As in Figure 14(a) except for shear (c) and the contour
interval is 2.m 2 /s
*
107.
180*E
The above results show the similarity among all three cases. The resemblance of the time behaviors of the initialized and the noninitialized fields in
both case(b) and case(c) is a little surprising and interesting, as we found
before that in the presence of uniform easterlies the time behaviors of the
initialized and the nontialized fields are different. It seems that the effect of
the easterlies near surface is vanishing in the presence of mean shear. It is
not clear that this is because westerlies are dominant in the domain and easterlies are small. Therefore, it is necessary to further investigate the situation
of mean flows with both easterlies and westerlies.
6.4
The departure of the evolution of the stationary
wave from its initial state
Although we have found that initialization is not affected much by the
existence of model lid, there is some other problem rising from the placement of a model lid. We noticed in Fig.8(a)-(e) and Fig.13(a)-(c) that the
vertical structures of both the initialized and the noninitialized fields behave
differently from those of the initial fields. As planetary stationary waves are
concerned, their time solutions should not change at all in nature. However,
due to the existence of model lid the difference field of the initial stationary
wave and the model stationary wave travels with the phase speed of free
Rossby wave. In order to see how much the time behavior of the stationary wave is different from its initial state, we calculated the change of total
energy with time to estimate this difference.
The energy equations involved in our time-dependent linearized baroclinic
primitive model can be derived from equations(71)-(76):
cp
2 c, P
-
_
p'2 1,
T-u
8x 2 c p
- pV -'
--
(92)
p'V-
12 -V. p'+
-p|
p' pw'
az
where V. denotes the divergence in x, y plane, and
N2
- -pw
(93)
g
}-plv|
2
and
}-2
are
defined as kinetic energy and internal energy respectively. Now integrate
equations(92)-(93) over x, y, z domain:
a jf
|I|2dxdydz = JJp'V - dxdydz
=
- ' dxdydz Jfdxdydz
-
J
(94)
dxdydz
(95)
Np'w'dxdydz
The advection terms ii in equations(92)-(93) and the V - p'v' term in
equation(92) drop out because of the continuity in x direction and the zero
normal velocity at y boundary. Then adding equation(94) and equation(95),
we get the rate of the change of total energy in whole domain:
-NPa
'97t_2IIIN 2'P
V|2+
--
P
)dxdydz =
dxdydz
2
-
p'w'dxdydz
(96)
Due to the periodicity in x direction the integral at right-hand side of equation(96) is zero, and the total energy in whole domain is conserved. However,
the column energy over z between zero and model top in a fixed (x,y) point
changes with time. Therefore we use the variable f(}lv'I
+
}
- )dz to de-
scribe the local difference between the time behavior of the stationary wave
and its initial state.
The cases we chose are both the noninitialized and the initialized fields
of ii = 25.m/s, iR= -5.m/s and for wind shear(b), namely U from -5.m/s
at surface to 25.m/s on the top.
Fig.15 shows the time evolutions of the column kinetic, internal and total
energies at (00 E, 450N) as compared to those at the initial state. It is found
that in the mean flow of ii = 25.m/s the column energy changes around that
at the initial state with the period about 18 days for the noninitialized field
and 16 days for the initialized one. The maximun increase of the column
energy of the stationary wave from its initial state is by 180% for the noninitialized field, while the maximun increase for the initialized field is smaller
than that for the noninitialized one, namely 110%. On the other hand, at
u = -5.m/s for both the noninitialized and the initialized fields the maximum departure of the energy from its initial state is within 5%. However,
in the case of shear(b) for both the noninitialized and the initialized fields
the energy first jumps from the initial state to some level, and then oscillates
around that level with the period of 3 to 4 days. The mean value of the
energy oscillation is as about 4 to 5 times larger than the initial value.
Therefore, the above results suggest that in the presence of moderate
westerlies, in which the planetary stationary waves are propagating, the fore-
casting error is large when a model lid is imposed. However, in the presence
of easterlies the stationary waves are trapped, and the forecasting error is
relatively small. This is because in the presence of westerlies the difference
of the initial stationary waves with radiation condition and the model stationary wave is large, and so spurious Rossby wave is excited. The spurious
Rossby wave travels around with time in the absence of damping. However,
in the presence of easterlies the difference of the initial stationary waves and
the model stationary waves is very small, and so the time behavior of the initial fields is near stationary. Moreover, the comparison of the noninitialized
and the initialized cases indicates that initialization does not tend to change
the major features of the forecasting errors resulted from the placement of
the model lid.
l
I
'
i
I
i
i
i
'
'
.
32
28
. 24
4.
L 20
a)
16
6
z
z
wU
12
8
4
-
---------------
-..----
0
0
2
4
6
8
10 12
DAYS.
14
16
18
20
2.8
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
.8
.6
.4
.2
b) *
0
)
i
c.
zr
2
4
6
8
10 12
DAYS.
14
16
18
2
0
26
24
22
20
18
16
14
26
24
22
20
12
10
12
6
10 12
DAYS.
8
------------ ----------
0
0
20
4
2
4
6
14
16
10 12
DAYS.
,
I
20
-----
-
8
18
14
16
18
22
18
20
16
1
14
18
LU
10
4
' -
0
2
-
-'
4
6
8
-
10 12
DAYS.
--
14
4 ,
-
16
18
0
20
Figure 15: The change of column energy in J/m
.
2
2
4
6
8
,
10 12
DAYS.
-.
14
,
16
from day 0 to day 20. a):
u = 25.m/s, b): ii = -5.m/s, c): with wind shear(b). Solid line-:
total energy, long dashed line-------:
kinetic energy, short dashed line
---------- : internal energy. The horizonal lines are the initial values of energy
as reference. Left: the noninitialied field, right: the initialized field.
66
7
Conclusion
Applying the normal mode initializtion method to planetary stationary
wave disturbances has been studied. It is found that the vertical modes for
streamfunction
4'
and v' are good representations of the vertical structures
of the initial fields varying in mean zonal flows. However, the mode structure of u' is heavily affected by NMI in the presence of easterlies. This is
because eastward Rossby waves are absent in the eigenmodes of the normal
mode initialization. Therefore, only in westerlies is the time behavior of the
initialized field similar to that of the noninitialized one.
Furthermore, it is concluded that initialization is sensitive to mean zonal
flow but not the existence of model lid. The change of mode stucture by
NMI strongly depends on mean zonal flow and this change is essential to the
time behavior of stationary wave. However, the relevance of initialization to
more realistic mean flows needs to be further investigated.
In addition, it is found that in the presence of moderate westerlies the
error of forecasting stationary waves is large due to the placement of a model
lid. Initialization, however, does not tend to change this feature. On the
other hand, in easterlies the noninitialized fields behave like the original
stationary waves and the forecasting errors are small.
8
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