FXCTTATTON 0? THE ThERTIO-ORAVTTT WAVES
IN A.
NUMERICAL WEATIWER P'RZDTCTTON EXPERI4ET
i4ST. TECs
DANTFL RaUSSAU
LIBRA
LINDa~
fRUMETED IN -PARTTAL WULPILLMENT
Or THT. REmJTREMNTs FoR THt
DECREE
w MARTER 0r
RCIENCE
AT THIF
MASACHUSETTS IUTITUTE 0lW TECUiMLOO?
June 1966
SIgnature of Author.C.....
.v*
woe
Departmat of Meteorology, 13 June, 1966
Cercifisd
by
accepted by
.
.
.
.
.
.
*
.o
4
.
.
aThsis Advisor
.
.
.
.
.
.
.
a
.
.
Chairman, Departmental CometIttee on
(raduate Studeits
S
A
.
rxcitation of the inertio-gravity waves
in a
numerical weather
redietion emneriment
by
Daniel Rousseau
Submitted to the Department of Meteorology ou
June J, 1966 in partial fulfillmsnt of the requiremats for the degree of Master of Science
AB'RTRACT
Inertio-grwity waves are observed in a numerical weather
prediction experiment. The chataeteristies of the types of inertiogravity waves, which can exist in the wodal, are than studied. Finally, by considering a simpler model, the renartition of renrgr between
mean flow and wave motion is discussed as a function of latitude and
the effect of the spherieity of the earth on the periods of the inertiogravity waves is suggested.
Thesis Supervisor: Jule G. Chernay
Title: Professor of eteorology
ACKNO'LEDCfENT
T an gresatly
indebted to Professor Charney for guidance
and useful sugtestions throughout this studv.
Mr. Tidoro
I wish also to thank
Orlanski who helped me by many fruitful discussis
and particularly Mrs. Jutta Thorn* wbo beautifully typed the manuscript.
TAPALLE lci' CONlfl-TS
1.
ITT,
ITT.
WV.
V.
INTROIUCT70H
WrIIWNAL REISULTP
TWERTTO-GRAVTIY WAVES TN HTNrZ S M*ODEL
PARTITION (W TRE EXER(~v TV AN~ TNCOHPE'.TLE
n'uID ur[tI PFlz qURi'ACE
2
t
25
CONCLUISTONI
41
REVFRE1NCES
43
I.
INTRODUCTION
Experiments have been made to check the limit of predictof the numerical weather prediction models.
-ability
In these experi-
ments the evolution of the simulated atmosphere is compared for two
slightly different initial conditions.
One can say that the limit
of predictability is reached when one gets two atmospheric states as
different as randomly chosen states.
Such experiments have been performed with the model devised
by Mints (1).
A disturbance is introduced initially in the field of
temperature alone.
The general behavior of the root-mean-square (I0MS)
difference of temperature between disturbed and undisturbed states is
the following:
-
A decrease of the RMS difference of temperature which reaches
a minimum after two, three or four days (depending on the initial conditions of the mean flow and of the disturbance
-
Then a nearly exponential increase of the RMS difference,
which finally levels off after about 28 days, limit of predictability
of the model.
The two phases of the evolution of the 104S difference corres-
pond to two different processes. In the first phase, the disturbed
flow, which is initially geostrophically unbalanced, becomes progressively quasi-geostrophic: in this adaptation the inertio-gravity waves
play the essential role.
In the second phase the baroclinic instability
becomes the preponderant factor.
This paper deals with the first phase of the experiment,
when adaptation to the geostrophism occurs.
It.
EXPERIMENTAL RESULTS
The model used to perform the experiments has seven time
dependent variables: the two horisontal components of the velocity
and the temperature at two levels, and surface pressure.
The upper
and lower levels correspond to o a 1/4 sad a a 3/4 where a is the
vertical coordinate defined as a a-=--E
Ps -Pr
surface pressure and
to 200 ab.
p pressure,
Ps
pr pressure at the top of the model and equal
The whole earth is represented with 1000 grid points for
every level.
In the experiment which will be considered in what follow*,
a perturbation of temperature is introduced Into a meteorological
flow.
It has the same expression for both levels and is equal to
AT = . sin 6)Acos |1 1o
'I A longitude, f latitude).
However, the
surface pressure and the velocity field are initially undisturbed.
Figures 1, 2 show the time variation of the difference of
temperature between the meteorological flow initially disturbed by
the sinusoidal perturbation for the upper and lower levels.
The
curves are drawn for different latitudes along the sane meridian.
To
simplify the representation, the absolute value of the initial distur-
Upper Level
77 *N
630
350
140
00
140
35*
t4
9
*
630
77 0 S
0C
Figure 1.
I day
AT3 for longitude X = 162*
2
4
Lower Level
630
4-490
350
14*
00
140
350
4490
630
+770
Figure 2. AT
Iday
2
5
Surface Pressure
66.5*N
45.50
315*
17.50
3.50
3.5
17. 50
31.50
14 5.5*
Figure 3.
Ap
.
I doy
S
f(
2
bance is plotted: the sign is given on the left of the figures.
Figure 3 shows the same thing for the difference of sur-
face pressure.
Because the surface pressure is not computed for the
same grid points as the temperatures, it is not given for the same
longitudes and latitudes as the differences of temperature.
Although the mean flow is not simple and could correspond
to a real meteorological situation, one can nevertheless detect a
characteristic long damped oscillation which corresponds, as will be
seen, to the adaptation to the geostrophism by Inertia gravity waves.
A long oscillation is clearly visible from the equator up
to latitude 35*N or S. At the equator all the initial disturbance
energy seems to go into the wave notion.
If =w goes to high lati-
tude, however, the wave motion becomes less and less important and
it is often hardly visible.
and 63*S for
The behavior of AT, and
AT3 at 49*S
A - 162*W for instance is saply explained by the
transit at the meridian of the initial calls of disturbance, transported by the seen flow.
From Figures 1 and 2 and from some other curves corresponding to other longitudes (not reproduced hre), the pseudo period of
the long-period oscillation is computed as a function of the latitude.
Since the damping is relatively important one san rarely follow more
than two alternations: hence, in Figure 4, twice the time between the
two first extreme is plotted as a point and twice the time between
the second and the third extrema
as a cross.
high latitudes no oscillation is evident:
Often at middle and
this explains the smaller
number of points plotted.
The data were available every two hours,
so that a precision better than 4 hours for the period cannot be expected.
Moreover, more confidence has to be given for latitude les
than 35* where the long oscillation is clearly visible.
For compari-
eon the inertial period corresponding to half a pendulum-day has been
drawn:
except at the equator, where this period becomes infinite,
the pseudo periods and the inertial period are fairly close.
In the eight first hours of the experiment a short oscillation is also visible.
It is very important in the surface pressure,
small in the temperature of the lower level, hardly visible at the
upper level.
The damping is very important and since the half period
seems to be of the same order as the time interval, where the data
were available, one camat compute it more precisely.
One can see that in average the amplitude of the taMperature difference decreases during the three first days.
This is not
surprising, for the disturbanee is initially introduced in the temperature field.
So only available potential energy is given initially
to the system.
Part of this energy is then trasformed into kinetic
energy, so that the available potential energy and consequently the
temperature perturbation decreases.
To explain the behavior of the system as shown by the numerical experiment it is first useful to compute the period of the inertiogravity waves, which can develop in the numerical model.
8
50
x x
-
Figure 4
pseudo period of the first
oscillation
y
40-
x
pseudo period of the second
.oscillation
inertial period
30-
-x
20x
xx
-
10-xX-
0*
10*
200
30*
40 *
500
60*
700
Latitude
800
III.
AVES IN MINTZ'S MODEL
INERTIO-GRAVITY
The numerical model uses the "primitive equation" with
the hydrostatic equation, so that the acoustic waves are suppressed.
Due to the constraints on the vertical profiles of temperature and
wind, only two types of inertio-gravity waves are possible, as will
be shown.
Figure 5 schematizes the vertical levels and the variable
of the system
-
0,
Tt1/4 ~
V 6orizon-qJ 'ind
20o,,,4
400mb
T
fnperaure
600i4
40
3eootenhJ
--
3
So0b
2,.
4
10004
? Fressvre
Fr
where
Fpure 5
If one assumas a linear profile of wind
V=T
NYr .V1.
1- r'
z I
and of temperature, except in the hydrostatic equation, the continows equations of the model given in (I) becone:
Equations of horisontal motion:
+ 2 (Z
AV +V4' + - RT
Vr
0-(F%- Pr) +Yg.
by 1.+ 21Z^ A3+ V+,+
V3 3
VpS
!j R.T.
91 %(S- Pr}+ Pr
F=0
(1)
0 (2)
(7. )3
wbere
3
-
Equatiens of euergy:
)TA+
.. Kl
+ *,
+ IV V'F +
where
3
WC
-_
T-3
(4)
CP
e3[n.. pr)}+ pr
3)T
(3)
.. 0O
mna
bC:R
Pressure tendency equation:
( Vi+V
(5)
,)
Hydrostatie equation:
fkys
I-S
5+
O
2.
.I.-L ( (ar.
2n
FS
I)I RT. 4--1
f
4.-
)
2- PS
F3
(6)
3
L
.. 21+
S
P3,+
-7)
-2{1-.
RT,
7
These last expressions are obtained from the continuous hydrostatic equation by the requirements of total energy conservation.
Continuity equation:
W, = <7, V, VP, .
)P,-
(rV,-
(8)
16
3
3f3
And from the definition of 4-
~
-YS
16
-
.-(PS
,)
Z.-(10).
- FT()10)
-Pr
p,- fy
From equations (1) - (11) and neglecting friction and diabatic heating one gets the following perturbation equations:
+
)\+Y2' 2
-__
+
A.1V
7+
+
,3 +
___3
I
V'
--
(12)
(13)
+
(14)
t (ps.
+
+Pvr
O(pSi-)
(15)
5r / I
s- Fr)
+4
V
(
1
I
Pr
+
_L
.3
-
(16)
V')
--1)RT,' +I kfrCI)IT.
QWAlb
F)
pi
I I IE.: ?
( V+
+
R;,
~((~X.-)
--I R13'+
-
+
P3
Is*3 P'v
+
P3;
(17)
--e-T3
-
( 0-
3)-L
e--
I
+
(18)
P3~~
rV -
Ps6
\G
-3
_ - ?'Pr
s
V(5'-VY)
(
+
(19)
(20)
(21)
V (VsV7)
r',e
(22)
The next step is to eliminate
tioms (12) - (22).
'
cui
so3
F
To simplify let
Moreover
Pow 200 ub ,
800 mb
f w 400 mb
:
O -|
J
and
10oo0
0= w 3/4
di a 1/4,
0i '
in.
equa-
12 Az I = k
h :! 1000 mb
Aie
1 . 0.288
With these values one gets:
-
Lrb.+@(t
-
+
I~.+t
+
,r
-
+
.
(23)
(24)
-
~4
(25)
)x
(26)
3 -+ a
Va-
+ T 43
)T':iI.
)f.
b
3
)hI-
b
+
3 -
. o
0
+
4)
>PSI
yA)
4
)v
1 7
+
(27)
(28)
N 443
3 x
+
0
(29)
where
at, - 0.441R :
e(13
0.0591 ;
S&an 0.532R :
,-
03 a 0.218'R;
ra 0/061 RT + 0.991 R 3
Q,a 0.555, -- 0.37573 :
,
9-
;
The system (23)
- 0.4111,
+ 0.375T
0.876 RT + 0.181RT
0.3757 -0.2137
-
; C
0.4
0.375 , + 0.429 73
(29) has exponential solutions
-
3
-.
V'
A
V'
P' propottional to exp(i( X+6-vt)J if the followins con-
IT7
dition is fulfilled:
.40-
f
0
o
o
0
..ir
-4
its, 'ar,
4
,
ahy
41,
,
.. .
0
0
- .
.. ce
0
?0
- ft(
tna3
-Ld
O
c
G
AMC
(30)
C'-ft, 3
f
' LIita,
4: Fri0
0
0
-. (-
0
6nc
This equation gives the frequencies of the inertio-gravity modes of
It eon be written i
the model.
''(&T3[
a
AOn+e. V) (e.')
*
+(v+
P)'b) -.
a
(31)
where
A
.
,t
5
C_
t 3J +0,48+
+
,C
+ ig
(32)
and
(3~'3
Y3 R3e3
(33)
i)%-1C+()-3
_;--u,
Besides the triple solution I~ i 0 which cerresponds to a geostrophie
notion, there are the two inertio-gravity modes with frequencies:
+
G
ga A + rA
N
8-K
(34)
wad
-+(z-
a-
A-
f)
-
K4
2
(35)
Let us compute first the variation of "',' in function of time.
do this let us write the differential equation that satisfies
if one assuse that the initial conditions are :4 exp[4(fr)%.
(23)
To
1"
From
(29) one deduces:
-.. iM4,,v'.-
ia
-
a, v +
e -4(,,-t
- -
a Is
+
-t I Ct
+
.
dek,
iVI,.%
-'4*a V,. -t
(36)
V3
4v.
S7'.. s'
kti
+
im
1
(37)
(38)
,tC4,
+-
.
V,
-T'
(39)
(40)
where
.3
-.-. (
e,, e }( + )
ue,+
(CA,
(e-*)
-34 U3)
+
S4-.( +
L3
0,;
)
(E.
)
L=!
P
and
CL,&41
cta.
+ a,
,1
+
+ c S3
.
CLg,
na~
3~
13s.
+ C
-. CL
-Lt, + +4n V,
By elimination of
(36),
Lt + S
S
~4t I :0/where
w~re.
c S,
- 4.' +4.hnv
and
between
(38) and (40) one gets:
,.
+
Sr
',
a 4
31
(41)
+
3 t3
..
-
ct I C
Q
Cel
The frequency equation derived from this differential equation is:
4.Q
CL CL3a - 4-L
CIL
I c 3 1 -. ec3 C#*
1
a+
-A-d 3 All
-ticl
The roots of this equation are d'
,t
1
(42)
1i
,
where
(; ' and
s3 are Piven by (34) and (35).
This shows that
T,
c,
+ c,
-Let
e
+ C, e*
*a1
a can be written:
Ga's
- -ig
+
,bq
z +
(*
(43)
IC
I
* ~~ 4
The five constants are determined by the initial conditions.
us consider the case vhere the initial perturbation Is
Let
3
A,
It correspond. to an initial gain of energy in the form of potential
energy
C, +
.Z.4
C+
C'+
C +4-;
C-
C
C, +
T
-
C'
Ct
(4s)
'1
C, - - C
-
y; ,
e,'C
..
g
+ e c,' -+
Fros (45) and (48) C
too
-ac
c
ofC ,
(44)
- ' C'
c
...
+
'c'
, and Ca
-0
40
gg
I-'- o .
s
C
s
T,+
-)
sn
7*
iP)
(48)
.
From (46) and (48)
C
2,
(S1T S00 Ty'T ') (S. 3.+Till +
2 0"
( -. |
o2
(50)
MEMEL-
and from (44)
C.:
s1r;s1?),,1
si, ra+tsP)-4.f+4)(s*.
+(siLT.
so that
T
y)
+ - Ccoura
C,.i 2C,coet
f
(52)
The same computation can be made to determine the variation of
One gets
i'
:
9,t
X,
coo
+
,pat
}
(53)
one has equations analogous to (36) - (40) if one changes
For
the
2
4
L, and
= (4,a
(
2: 3i.a=
So by
4
9,
E,L defined by:
a and
0,),3 (
3s,
0.4)
j
(fa+vw)
0tI
(it , ,+ '3,j W,;) (Vne)
and
b,4.,
-
b,4
:g + a,
+ ,
t
+
ii.
1 -.
b,,, = b,.
+ a Ze
d'be 2mi +b
9, ,y ad 0 are then deterined by
I^
i w&PE.
n (YRr- : -+
T~.4I~
poo
'
When
of
and
Y7"nd
'psi
4,+<)(*
,+ E,
a.4(5
SIT+
are known, one can easily get V
VV' between (27) - (28) - (29).
-
-
c
TS+
Z3-1'
(T,* +
(
o1.
4
o+
i
(54)
*
'.
CL,- C
-.
D-ta
LO4,-A
by elimination
This gives:
(57)
)7(
'T'+
",Tf)(56)
COtb)h
_Wl__C
(58)
b,"Q,... 43 L,
The constant is determined by the initial difference of surface pressure.
The solutions which have been developed above are really
valid only for an infinite rotating plane with sinusoidal perturbation
extending to infinity in the T and Y directions.
Nevertheless it is
useful to cospare these solutions with the result of the numerical experiment.
We will see later qualitatively, what may be the effect of
the sphericity of the earth by considering a simpler model.
Equations (52),
(53) and (58) have been applied for two par-
at the equator and at a middle latitude (49*).
ticular cases:
For
these two eases the coefficients have been computed using values of
and T gven by table I (see further en).
*IV =
and
I
bance was V, - 5
The initial distur0.
The numerical result for the equator is:
,
Ps"
(0.770
(..
0.4
+ 1.0
caws
e;
-
... 1,*;3' cCJ C-,
+F 1 91
(59)
w
f)
(61)
og is the frequenty of the short-period (here 3.3 houts) gravity
the frequency of the long period (here 23.2 hours)
wave and
gravity wave.
The amplitude of the short-period gravity waves agrees, at
least qualitatively, with what is observed in Pig. 1, 2, 3 before the
quick disappearanc, of the short-period oscillation.
However, the be-
havior of the long-period gravity wave is quite different from what
could be expected from (59),
(60),
(61).
It ft believed that it is
due to the negleCt of the heating term in the treatment of the linearised equation.
Indeed, the model contains a convective adjustment pro-
cess whieh acts when the vertical gradient of temperature is less than
the saturated adiabatic gradient:
tion at latitude less than 30.
this is precisely the usual condiSince the long-period oscillation,
which is an internal oscillation, is characterised by a sensible
oscillation of the gradient of temperature, the adjustment has an effect on both the amplitude and the period of internal oscillation by
preventing a rore instable gradient.
The disappearance of the short-
period gravity wave is due to a computational property of the model.
Indeed, the time integration is done with the folloving differentiatIng scheme:
+ At P(4i)
41.- A E(63)
(62)
4%
One can see the effect of this acheme of a differential equation
such as
4,
I1
when
the real solution is
e
But by (62) and (63),
so that
-6
or
.
[
_
4
-At .
ef()
dV), +
: Nat
1'At
The modulus Is always less than one and tends to seTo when
to infinity.
The modulus is reduced to
4~ .-,2At /
I /r when
Lsrr fCrj-4Pt+
cr2 A
t
tends
Taking the values of e- corresponding to the two types
of gravity
waves at the equator, mne gets (atice A1 = 12 minutes):
for
a-
for
;
=5.31
- 10'' s-'
9
. 69,
10-
At other latitudes, c;
the sa
t
*
3 hours
t a 160 hours
s'
and05 are somewhat modified but stay in
order of magnitude.
So this shows that in 10 hours the
short inertio-gravity waves are practically eliminated and that the
long Inertio-gravity waves decrease very slowly.
The modification in the frequency is meall.
computed by the
differeneing
me.Arc hn
a
*'f
For W, one gets
3.0 hours instead of
S
t
scheme is equal to
..
'
w 3.3
se±L
stead of ,-
.3Ilo" 'or T/
instead of a :
4%
gt"lefor T'
s'
hours.
eO0.691 xIrs
For T3 one gets
The frequesy
25.2 hours practically equal to 1
*
So if the computational scheme does not practically modify
the properties of the long-period oscillation, it forbids the developmost of the short-period gravity waves : the consequence is then to modify the mean flow.
At middle and high latitude one has also this effect, but
equations (52),
(53) and (58) are likely to represent better the experi-
mat, sine generally no convective adjustmat takes place.
For
49* latitude the Uumrical values are:
, (0.26S + 0.092 c.amk 4 O.G63
(40. 2
P'-. ( ..
,
1I( ceset + .
22 + 2.13 c.0r,t .
casat)C
(64)
cC &
(65)
.91 c.C
)
'4
(66)
Here there Is a qualitative agreement with Pigs. 1, 2 and 3:
Increasing importance of the short-period waves when one goes to lower
level, decreasing importance of the long-period waves for lower levels.
As ha been said before, the long oscillation (with period nor than
2 days) seen in the temperature profile at 49*N does mot correspond
to the long-period wave, but to the sain flow, since the curwes are for
a fixed point in space.
The fact that the amplitude of the long-oscil-
lation is smaller than could be expected from (64),
(65) and (66) i
possibly due to the important vertical and horisontal gradient of these
latitude.
gradient not being taken into account In these computations.
Let us now compare the period computed from (34) and (35) with
the numeical experient.
The period* computed from (34) and (35) do-
pends on the latitude for three reasons:
- By the presence of the Coriolis parawter f,
- In the particular experiment studied here, by the variation
of
E
with the latitude.
In fact, for a perturbation of the form ... sinGA
tf,
":
i-1
3)_.
CIL
and
z=
.--
, where a is the earth radius.
- By the variation of K1 and K2 in function of the latitude.
Indeed K and K2 are functions of the coefficients of equations (23)
to (29), whicb depends on the average temperature of the upper and
lower level.
ij
and
Using the average along a latitude circle to evaluate
3, the values of K and K2 are given in Table I for 4
different latitudes.
490
770
Latitude
0*
28*
T
251.7
243.4
231
221.4
T3
292.5
274.8
252.1
236.9
Ic
0.739
0.721
0.662
0.638
K2
0.013
0.018
0.022
0.023-
TABLE I
and T3 are in *K and
and K2 in 105,m2a
One ses that the variation of K2 is Important in function
of the latitude, while the variation of X1 is not considerable.
This
is because K, depends principally on the uswn temperature of the troposphere, which does not vary very much from equator to pole.
Ever,
K2 depends principally on the difference of temperature between the
two levels, which varies largely from a stable atmosphere at high latitudes to a less stable atmosphere at low latitudes.
The vagiation in function of the latitude of the periods T
and T2 of the long and short inertie-grawity waves is computed from
(24) and (35) by T a 2tr and is given in vigure 6.
A coparson of Pigure 4 and Figure 6 shows that the observed
oscillation bs a period larger than C2 and particularly Neh larger
at low latitude.
This large discrepancy at these latitudes is po-
sibly due - at least partially - to the convective adjustment.
cannot
Ns
lso neglect the sphericity of the earth and consider the prob-
lam is a tangent plane, with sinusoidal perturbation of constant wave
lengths in both directions extending to infinity.
In fact, waves cos-
Ing from different latitudes with different period interfere to create
the oscillating istion.
Another effect of the spherietty of the earth is that the
energy going into the wave notion depends on the latitude.
To se* this
effect and In order to apply it to Mints's experiment, let us consider
a one level modael, since the computational scheme permits an adjustwaat with only one type of inertie-gravity wave.
TV.
PARTITION OF TRF ENERGY IN AN INCOPRESSTBLE
FLUID 1ITH RE SURFACE
The simplest possible %*del able to exhibit a geoatrophie
wmtion and inertio-gravity waves is an incompressible fluid of height
26
50
Figure 6
I
Period of the external inertiagravity wave
II
Period of the internal inertiagravity wave
40
30
20
10
00
100
200
30 0
40 0
50
60*
70 0
Latitude
80*
90'
H, on a rotating piene with rotation frequency f, aubject to the
If moreover one assume small velocities u and v and
gravity g.
perturbation of height h, and that the initial velocity field i
independent of a, the relvant equatione
governing the motion of
this flow axe:
(67)
+
. +
-40
(68)
.1.+ H ( )..
)-x
).:o
(69)
Looking for a solution proportional to ep(S(t..'y.s)), on
e
that o* WSat3 satisfy:
S0)
or
-ls"( t'.. . g a
4
))
(71)
whicb has solutions 0 a 0 cowresponding to the geostrophic motion
and two opposite solutions correspondiUg to inertio-gravity waves
+stisfying:
This has the soe foi and latitudinal variation as (34) and (33).
Siacs the mdel peraits practieally ony long-perIod waves of type
(35) a one eaa expect that the behaviot of these waves would be the
Sam as that of an incompreseible fluid with free surface, provided
that Om puts gR - 1 into the numetieel applicatiom.
For a gravity
equal to 9.8 a/* the equivalest height varies from 132 a at the equatot to 235 a at latitude 77*.
The ratio wuich will play the impetant tel in the partities of the mory is
kz
Ise where
S*
(d-')13)
afH
*mt.A.)]. the variatio
For a varieties of h proportiosal to ea 14(e
of u and v is
fom (67) and (66):
IL
(74)
(75)
I-.
the
So if the initial conditions are proportimmal to e
solutica is (because of the linearity of equation
,
Aj.+ C
Fr-. V+fe1
(67) to (69):
ee
i*iMJe'
(76)
(77)
where A, B and C are constants (without relation with the values deIf
They are deterind by the initial conditions.
fMned before).
the initial perturbation is only in the height field and is equal to
A. e 4
with
-R,
A
. the equations (76) - (78) aoplied to
"*+.,
ttx+.)
4
-
detrmine A, R and C. One gets:
f
= I,Z.9I
-
Now it is convenient to cons back to real values and consider initial eondltiens proportional to sil em coon,
rather than
ine
C ^**
-::e
Safxa
stig)
--o c
it-txMy
+ e
(iex-.y
-
-n
4
a
-ma)
4(m
the solutas is foand by a linear ecabinatios of solutins of type
(76) to (76).
--
+
-..
&.a(
L -..
This yield.:
(80)
ycta
} R, 5hPxc
- CA rI]st.
.
.
se
eC k ceAx COP'%)
(81)
-0c%
-+
5U.Jj
f3 5h.a
S
(82)
0
Prem these expressious it is easy to columte the kinette energy
(A.P.l) of the motion.
Sergy
and the available potential
(KL)
por a unit &a*.
Y1E'
4 *v0]
t)
H
(83)
A ,?.E .
(84)
ase can divide the time and spatial average of the two knds of *aergy
Into two parts: ae part odag from the coeffiatents of sI
co
t,
t and
ich will be called energy E the oscillation, the other part
beig the energy of the ass flow
CK
eKe)..aa
of time.
dpedt
So ome ba:
(85)
flow
+Now"
-+
(86)
_
S(Ad'*L)asan flow
(87)
"NOW%
-
r
so that the partitioe of the total energy (T.E.)
(88)
beteen eseu flow is
the follovtg:
(T
*%tq"
low
R
(89)
If k is the ratio
(T-E).,a flow
on sees that:
k
This mase that at the equator, where k a 0, all the initial energy
of the perturbation goS
into the oscillation.
Is the experlent, k
Increases from the equator to latitude 50* where it to slightly larger
then one, and thou decreases dow to sero at the pole, because them a
beces infinite. The variation of k as a fumetioe of latitude is
gen a ?igure 7. This is oa of the reasons why the vave sotion is
clearly visible at low latitudes and is the mest perturbed and somemopletely
asked at
times
iddle latitudes.
The intesity of the wind
amd of the gradiat ef wuld at these latitudes is another rssen, sce
the vealoaty of propagation of the isteroal waves (of the order 50 u/s
there) is only of the order of the man wind, which can to loner be
neglected.
It is interesting to see how the partition between oscilleties and amn flow is dome on the form of kinette energy nd available
potential ene2gy:
,,......,,,(92)
I,5
1.0
.5 -
00
10*
200
30 *
Figure 7. Variation of K -
4 0*
50*
60*
7C
g2
2
as a function of latitude
K (12+m 2 )
and
(A. P.E.)".a
go93)
(A.-P. E.)00T
The curves cowrspoding to (91),
(92) and (93) are given in Figure 8.
Vigure 9 shove the variation as a function of k of the ratio
between kinetic energy and available potential energy in the mean flow,
in the oscillation, and for the total notion.
The expressions corres-
ponding to the curves are:
- OE**-
a
ean flow
A.'.B*
.JpKE.
A...
(94)
k
--
scillation
total
*1
-
+2
(95)
$
(96)
+ 242
Figures 8 and 9 show for instance that if the inertial effect is much
more important than the gravitational effect, i.e. if k >' 1, the
energy stays in the man flow principelly in available potential energy.
On the other extrme, if the gravitational effect is preponderant, i.e.
if k <41 (near the equator for instance)
the mejor part of the energy
noes into the oscillation, where kinetic and avilable
potential energy
are nearly balanced.
One has to peint out that these results applied only when the
initial perturbation is in the height field, the velocity field being
undisturbed: them energy is given initially to the system in the form
6Figure 8
5
Ratio
cillation
energy mean flow
total energy
fort
4
a function of k
kinetic energy
S-
- -available energy
3
km 0
k:-0
I
S32
2
3
4
of available potential energy.
Completely onoisIte results are of-
tained if kinetie enetgy is intoduced initially into the Syste,
having initially a rotational perturbatioe
by
of veledity and ne pertur-
bation in the height field.
Pigeres I and 9 een also be interpreted as giving the
partition of energies between nean flow ad oncillatioe, and the partititn of the total energy between kinette enerxv and available potewtial energy in function of the wave length for a fixed latitude.
These reults ar quite analogous to the elansteal problem
first studied by Rossby f2) of the adaptation to the geostrophinm of
an oen uifortiv disturbed on a strip of width a, between two ordinafes of a 0-plane.
It is then found that if the initial perturba-
tien is In the longItudinal velocity, the largest point of the initial
emergy goes into gravity wave if a V3
as
A, radius'of
and stays in the mean motion if a
H
deformation, defined
4
The opposite
is true if the initial perturbation is in the height field, for instease by adding wase 'over the strip.
Our problem is analogous to thfin
latter case, the wave length playing the saae role as the width of the
strip.
The variation with latitude of the anont of energy, which
initially gees into the inertie-gxavity motion eusgests the difference
between the solution relative to an infinite plane and the real ephertcal problem.
'r
instance let as see what hapenas at the equator.
Tf
one considers the solution for the tangent plane, the sinuseidal -elmtion will be neutral and will have the sas period on the whole inf1-
nite plane, since the enerpy of the initial sinusoidal disturbance
which goes into the wave motion is uniformly distributed on the plane.
gowever, if one cOnsiders the sphere, more energy Will be given to
the weve notion at the equator than at higher latitudes, so that a
transfer of energy from the equator toward middle latitudes e
expecteds
be
the oscillation at the equator will then hae a decreas-
ing uplitude.
Anothet aspect of this is that the influence of the
initial disturbance on the otion at the equater will be relatively
more important In the vicinity of the ocuator.
So"e light an the influence of this sereading of energy on
the periodicity of the motion is wo by solving the initial value
problem in a plan., thanks to an influence function.
Let
:
the divergeowe of the horisontal wind.
+ )v
The divergence equation derived from (67) - (69) lo:
3HP.
IV2[)
+~
(97)
=0
If the disturbance is initially only in the height field, the initial
cenditions are:
DIn O
VD =0a
O
4
, a ..
g,
(98)
where to is the initial disturbance of the height field. Let us
sup-
pose that the initial conditions are sinusoidal in the z direction so
that the solution can be written
)z
ACez
2 with:
+ SM
d
a H
) 2') 0.. o
a
0(9)
By taking the characteristies of this equation an ne
im coordi-
sates, I.e. if:
+
t(100)
(101)
,' N
the ouatios (98) beses asply:
..
0$
(102)
*a
vith Initial ooditiOUs
*
,(1
oan the ttaight lina X
vaiatio
(
Ct
0), whtet
(* * ±
&Contains
(103)
onlyuov the
in the y direction.
Equatia (103) to solvWd in closed fort by the Rieman method.
If U is a faaetion %tichsatisfies the equation (102) (equation equal
to its .adjoint) one has:
., I 0~5~
w)
(104)
LX
T O31N 00 -r
(104) is sex.
both members
plan f
By (103) the ntegtral on a doain of the 1-
of
If one applies this to the surface MC where Hi Coe-
ponde to a point ( I,
0 ). B and C to voiui t a 0 and distance
k
T
m (A ye)
nmGURt 10
tj3~
SMC
frm the -point
iSz
ertesponding to N. one gets
CLX.CEY O
1)y
(105)
T
J+
d
But, by Riesein-Green formulei
3Iay
4I
6MC B
tz+
2
...
a
2)
u
Jtc
1
M21
3y
L
:: 0
(106)
X
2.
(107)
d.X
If U is choen equal to 1 mo NB and MVC, one has r
4y
[. 3S
.3 W
(108)
But using the bouadary conditioms (104) one Vets:
U
30.
(, oef1o
r wvith (100) and (101),
(109)
cy
ad siace d? w dy os WC
).+
1,,ai1mmb
(110)
(rkj.YRO ) i
wbere y is the ordinate of the poiat W. where the divweos
is com-
puted. The influence function U =at satisfy
and
be equal to 1 on 1W and Mt, i.e. for X *
ad Y * Y,.
?hre-
fote let
:
X4
X) (a)(Y-0 Y6)
1Witb this vus valable, U maust be equal to I for
+~
lu:
+ 14t) z 0
a 0 4d satisfy:
vanee U is the vessel function JO(A).
Sines
Tc( X.-X)(Y.Y)
. (!**-9t
W
(110) becomes:
lot fpt
(112)
MfOO
This expression showS that at tim. t, the Initial disturbasee h, has
-
influenet
fluaene for points further than
of the waves
'
There is no int-
"
waxnimm at distance eoqul to
since the ns'inum velocity
i
If h0 is a sinusoidal fuaction extending to infinity, the
oolution to almply a sinusoidal variation with frequency deftind by
t ce
{
a
- -.
. However, if one supposes that the initial
simusoidal disturbance is limited around y, to a width 2a, the limit
of the integwal of (112) becomas constant for
t
7 C/
tam of the asymptotto expansion of -1Cz) when x -gao
.
The first
is
so that the variation of D becomes a damped standoidal perturbation
and frequecy defind by
with *Mplitude deereasing as
-
1+
instead of
1e
H
w4=
(113)
(4-+
this process may be a factor in the characteristics of the
long-oscillation observed near the equator in Mits's experient,
sias the influence of higher latitude dereses, due to smaller
amout of enrgy, which initially goes into the oseillation,
The
decreasing of the amplitude of the wave eould be due to this spreading of euergy.
An upper limit of the period is then given by (113)
where gH is replaced by
stead of 25.2 hours,
K. : one finds 51.7h at the equator in-
otre in agreement with the obervations than
this latter value.
If the influence of the initial disturbance is also limited
in longitude, one gets only
'=
for the limiting frequency of
the oscillation, that is the inertial frequency.
This is exactly the
case for an initial localised disturbance.
The Rossby-adjustmsst problem is a. quits aualomous problem,
and it has been shown by Cah [31 that gravity waves oropagate from
the disturbed ocean strip with foreruanr velocity
,
bt
tht' the
oactllation of the flow in the stril %as a decreasing amp.itude and a
period which tends to the inertial period,
CONCLUSION
An experiment of predictability has put in evidence inertiogravity waves, which occur when atmospherie states are initially geo-
etrophically unbalanced.
The vertical profile of the **del permite
the development of Wly one type of internal inertio-gravity Wave.
Moreover, a computatiemel en"straint eliainates quickly the external
gravity wave.
ty lmearising the equations of the modal, negleetiog the
sphericity of the earth, the heating and friction te
, n analytic
solution was found, which deviates however greatly from the computed
solution near the equator.
The differece is believed to be due par-
tially to the neglect of the conveetive adjustmat of the temperature
profile ouch is important at low latitudes.
The difference of the iaportance of the ve, metioAm s a
fmatis of latitude has been studied by omparien with an inempresible fluid with free surface.
Sinm all the mergy mes into
the wave notioe at the equator, this has an importat consequence :
a epreading of the esergy Is possible toward the middle latitude.
This affects the amplitude of the wave
the pat"od, which lengthens.
motion. which decreses.
PEventually the period
and
eehes the iner-
tial period If the Initial disturbame is localised in spa@e.
43
RLTRENICES
[11
fints., v. (1065), Very long-ter slebel integraticn of the
primitive equatioms of staospheric motion. -1(0T~ch:.stsli
no. 66.
(21
Rossby, C. (1438), On the mutual adjustment of pressure
sad velocity distributions in certain simple current
systems, TI., Ejfar.sep.., T (3). pp. 239-263.
[31
Cahn, A, Tr., (1945), An investigation of the free osellations of a simple current system, 3. Het. 2, pp. 113119.