1229
Progress of Theoretical Physics, Vol. 90, No. 6, December 1993
Theory of Eddy Viscosity Coefficient for Two-Dimensional Inviscid
Barotropic Fluid
Takahiro IWA YAMA and Hisao OKAMOTo*>
Department of Physics, Kyushu University, Fukuoka 812
(Received August 25, 1993)
§ 1.
Introduction
It is widely known that the terrestrial atmosphere can be considered as an
inviscid fluid, because the atmospheric waves, e.g., inertio-gravity wave, propagate up
to the great heights almost conserving their energy density. 0 ' 2> On the other hand,
there is an example contradictory to what the atmosphere can be considered as
inviscid. In laboratory experiments, it is known that a condition for generating the
Karman vortices on the leeward of a·cylinder is given by Reynolds' number Re= UDiv
~ 102 , where U is the mean velocity of the fluid, D is the diameter of the cylinder and
v is the kinematic viscosity coefficient of the fluid. 3 > If we apply the Reynolds' law of
similarity to the atmospheric Karman vortices generated on the leeward of Cheju
Island (Cheju-Do) of South Korea, we have v= UDIRe~(10 mls) X (30 km)l102 ~ 103
m2 I s. 13 > This implies that the atmosphere must have an effective viscosity coefficient of
eight orders of magnitude larger than that of the molecular viscosity coefficient of the
air, which is Vmoi ~10- m Is, in order to explain generating the atmospheric Karman
vortices. Since this anomalous viscosity originates from turbulence in the atmosphere, it is called the eddy viscosity. The eddy viscosity has been qualitatively
explained by Reynolds' stress [e.g., see Ref. 4)]. However, no quantitative arguments
on the eddy viscosity, especially elucidation of its anomalousness has been performed
yet.
In a previous paper, 5>we performed a quantitative argument on the eddy viscosity
for a two-dimensional (2-D) inviscid barotropic fluid confined to a square of sides 27r
6
X 27r. According to Mori's theory, >the spectral form of the vorticity equation w~s
reduced to the generalized Langevin equation for the spatial Fourier component of the
vorticity of interest. A damping term in the Langevin equation was represented in
*> Present affiliation: Department of Information Science, Kochi University.
5
2
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According to Mori's theory, the spectral form of the vorticity equation for two-dimensional
inviscid barotropic fluid is reduced to the generalized Langevin equation for the vorticity. The
damping term in the Langevin equation is interpreted as the eddy viscosity damping term. A
formula for the eddy viscosity coefficient is derived by using the fluctuation-dissipation theorem. The
eddy viscosity coefficient thus obtained depends on the length scale of a phenomenon of interest.
Under Cheju Island scale, the theory derives that the eddy viscosity coefficient is about eight orders
of magnitude larger than that of the molecular viscosity coefficient of the air. This value is of the
same order as the eddy viscosity coefficient obtained by applying the Reynolds' law of similarity to
atmospheric Karman vortices generated on the leeward of Cheju Island. It is theoretically derived
that the reason for the enormousness of the atmospheric eddy viscosity coefficient comes from that
of the length scale of interest.
1230
T. Iwayama and H. Okamoto
§ 2.
Theory
For a 2-D flow of an inviscid barotropic fluid confined to a square of sides LX L
and satisfied doubly periodic boundary conditions, the spectral form of the vorticity
equation is expressed as
(1)
where k and p are wave vectors, summation runs over all wave vectors, {(27Cnx/L,
27Cny/L); nx and ny are integers}, l;t(k) is the spatial Fourier component of vorticity
with wave vector k at time t, and D(k, p) is the interaction coefficient of nonlinear
term, which is described as follows:
1
( IPI2
1
D(k,p)=-z(kXp)z
(2)
According to Mori's theory, 6> the generalized Langevin equation for the vorticity
of interest is derived from (1):
(3)
The first term on the right-hand side of (3) is an oscillating term. The frequency
is described by
Q
__ . (Fi;(k), i;(k)*)
k-
z (i;(k), i;(k)*)
Qk
(4)
The real part of the second term represents an effective damping of the vorticity due
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terms of the convolution of a memory function and the vorticity of interest. The
memory function was represented by the correlation function of the random force in
the Langevin equation. This is the fluctuation-dissipation theorem. We interpreted
the damping term as an effective viscosity force due to turbulence acting on the
vorticity, i.e., the eddy viscosity damping term, because the vorticity equation was the
one for the inviscid fluid. Moreover, we estimated a damping coefficient of the
damping term definitely from the fluctuation-dissipation theorem. A preliminary
numerical simulation was performed by using a spectral form of the vorticity equation with 20 independent variables. Agreements between the theory and the simulation were satisfactory and then we obtained that the damping coefficient divided by
square of wave number, i.e., the eddy viscosity coefficient, is about 1010 times as large
as the molecular viscosity coefficient under the terrestrial synoptic scale condition.
In the present paper, we consider the 2-D inviscid barotropic fluid confined to a
square of sides LX L and derive a formula for the eddy viscosity coefficient by taking
the limit as L-HXJ. Then the eddy viscosity coefficient is readily estimated for given
characteristic values of the system. Moreover, the reason for the enormousness of
the atmospheric eddy viscosity coefficient in comparison to the molecular viscosity
coefficient is elucidated.
Eddy Viscosity Coefficient for Two-Dimensional Inviscid Barotropic Fluid 1231
to nonlinear effects from which a turbulence comes out, i.e_, the eddy viscosity
damping. The third term is a randomly fluctuating force and is expressed as
(5)
Rt(k)=etiirQ"'£,D(k, p)i;(p)i;(k- p),
p
where i;(k)= St=o(k) and the Liouville operator
system (1): 7>.s>
a
r= ~q:~D(q, p)i;(p)i;(q- p)} at;(q) .
r
is defined as follows in the present
(6)
PF
cif'h)~~~1*}*) i;(k),
Q=1-P.
(7)
(8)
These projection operators mean that only one variable i;(k) is regarded as the
observed variable and all the other variables, {i;(k'); k'=l=-k}, are regarded as the
projected-out variables. The parenthesis denotes the average over an ensemble p:
(F, G)= 1:·--l:d{i;}pFG,
(9)
where {i;}={···, i;(k); ···} and * denotes the complex conjugate. Equation (5) represents an evolution of the initial value of the nonlinear term in the phase space spanned
by the projected-out variables.
The relation between the random force and the memory function is given by
(Rt(k), R(k)*)
(i;(k), i;(k)*) .
(10)
This is the fluctuation-dissipation theorem.
The equation-of-motion for the auto-correlation function of l;t(k) is derived from
(3):
(11)
';?
...., k
(t)- (l;t(k), i;(k)*)
( i;(k), i;(k)*)
(12)
If the relaxation time of the memory function is short enough in comparison to the
time of interest, then rk(t) behaves as the delta function,
(13)
where o(t) is the Dirac delta function. Then the generalized Langevin equation
becomes the Langevin equation for Brownian motion,
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P and Q are projection operators that project a function onto the phase space spanned
by observed variables and projected-out variables, respectively. In the present work,
P and Q are taken as follows:
1232
T. Iwayama and H. Okamoto
(14)
where
(15)
The eddy viscosity coefficient l/~y is defined by setting the damping term to be of
diffusion type. Equation (15) is the general expression for the eddy viscosity
coefficient. The auto-correlation function is expressed as
(16)
p= Cexp[ -( ttff
= CIIexp[k
+
~) J
e+ 2tt X
Bk
ls(k)l2]
2
'
(17)
where k is the absolute value of wave vector k and parameters B and tt are determined from the energy and enstrophy per unit mass. B corresponds to the temperature parameter in the boson problem, while tt is the chemical potential. 9> C is the
normalization constant. In the limit L~oo, the power spectrum e(k) of the energy
density of the vortices with wave number k is given by
L ) 2 Bk1r
e(k)= ( 27r k2+ tt .
(18)
Then E and Z are given by
E=
(kmax
Jkmin
e(k)dk=(~)2 7rB log[ k~ax+ tt]
27r
2
kmin + tL
(19)
and
2
2
_
(kmax 2
-( L7r ) 1rB { 2
tt]}
ZJkm!n k e(k)dk- 2- (kmax-kmin)-ttlog [ k'!r..ax+
k~ln+tt ,
2
(20)
respectively. Here, kmin=(27r)/L and kmax is the maximum wave number or the
truncation wave number. The power spectrum depends on k as k- 1 for larger wave
·numbers and has a maximum value at f!;..
Using this distribution function p, we calculate the frequency Qk and the memory
function rk(t). The frequency Qk is given by
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We will next determine the memory function definitely from the fluctuationdissipation theorem (10).
In order to express the memory function definitely, we require the distribution
function p of the fluctuation of the vorticity. It has commonly been hypothesized that
the statistical dynamics of classical 2-D turbulence is controlled by two quadratic
invariants of energy and enstrophy. 9>' 10 > Thus in this work, we use a canonical
ensemble which has energy, E=L:kls(k)l 2/(2k 2), and enstrophy, Z=L:kls(k)l 2/2, per
unit mass as the invariants,
Eddy Viscosity Coefficient for Two-Dimensional Inviscid Barotropic Fluid 1233
Q
--z-~D(k p) (l;(p)l;(k-p), t;(k)*)
7 '
(t;(k), t;(k)*)
0
0
k-
(21)
Since the propagator etQr having appeared in the random force (5) is expressed by the
expanded form about time,
(22)
the memory function is also described by the expanded form,
(23)
2
~ 2 D(k p)z <I t;(p)l ><1 t;(k- p)J2>
rk <o>= 7
'
<lt;(k)l2>
(24)
where rkp is a positive quantity. The second and the third term on the right-hand side
of (23) are expressed as
(25)
and
rk<z>= ~2rkprpq+ ~<I t;(p)JZ><I t;(q)lz><l ~(k- p- q)lz>
p,q
p,q
<it;(k)l >
X16D(k,p)D(k-p, q)D(k, q)D(k-q,p)
+~4
p
4
p
Yk YP
k-p
<lt;(k-p)i >
(<lt;(k-p)l2>)2'
(26)
respectively. The second order moment <I t;(k)l 2> and the fourth order moment
<I t;(k)l 4 > are calculated according to (9) and (17):
<lt;(k)l2>=(t;(k), t;(k)*)=
l::!2f.L '
<I t;(k)l 4>=C t;(k) t;(k), t;(k)* t;(k)*) =3{ <I t;(k)l 2 >}2 .
(27)
(28)
In the short-time approximation, we terminate (23) at the third term and express
it in a Gauss function at first, that is; n(t)~rk<o>exp[ -(rk<2>/rk<o>)·(t 2/2)], and then
approximate the Gauss function by an exponential function,
(29)
Here, by making equal the integrated values of both the approximated functions, we
have
(30)
Then the Laplace transformed auto-correlation function is given by
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Here the first term on the right-hand side of (23) is given by
1234
T. Iwayama and H. Okamoto
1
r
s+-ks+ck
(0)
(31)
'
where
(32)
~2D(k p)2 (tt(p), f;(p)*)(f;t(k- p), ?;(k- p)*)
rk (t) = 7
' .
(?;(k), ?;(k)*)
(33)
Equation (33) is consistent with (24) at t =0. We substitute (16) with (21) into (33),
then
Yk ( t )
-~2D(k 'p )2<1?;(p)l2><1t(k-p)l2>
-.o;
<I ?;(k)l2>
exp [ -
{lJeddyp
~ 2+ lJeddy
~ lk - p 12}]
t ' (34)
. (35)
Thus we obtain
(36)
In the limit L -HX),
))~y= l;l?;{1::•xpdp( ;:!~)
}1' 2
sin28(k2-2Pkcos8)2
2
2
2
2
2
2
x Jo de (k +p -2pkcos8)(k +P + f-l-2pkcos8)(k +2P -2Pkcos8)
(
2
"
(37)
where
e is the angle between k
and p,
Ir(P) ={x(P)- (K2- 2!-l)log[ t9(p )l.;(p) + x(P )I]
[I
4
2
+2 3I
t9(p) k 4 +4f-lP2+ K37J(P) . 2 2 . 1
K og -f-l-· k -4f-lP + K37J(P)' K2P - Kr - f-l.;(p)- K3X(P)
IJ
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Under this approximation, the integration (15), V~y, reduces to
j TC(yk<0 >)3/(2rk<2>) /k 2 • However, in the limit L-Hx), it is difficult to express the
summation in (26) by an integration and to integrate it. Thus we estimate V~y
approximately by a self-consistent treatment. The self-consistent treatment in the
following overestimates V~y by about 50 % compared to that in the short time
approximation. 5 >
By an analogy to rk<o>, we consider rk(t) as follows:
Eddy Viscosity Coefficient for Two-Dimensional Inviscid Barotropic Fluid 1235
-8,ulog[I2P2 + TJ(P)IJ + Kllog[,uiKzP2 - K1 2 - K1X(p)IJ},
lz(P)=Kllog[8(p)]-P2 ,
K1=k 2+ ,lL,
t;(p)=p2-k2+ .u'
8(p)=P2+ .u
Kz=k 2-
(38)
(39)
,lL,
(40)
and
(41)
where
I(p)={ (x(P)- P2 ) -(Kz-2.u)log[8(P)it;(p) + x(P)I]
[18(p)
2
4
k 4 +4.uP2+ K37J(P)
+2 31
2 2 1
K og -.u-· k -4,uP +K37J(P) · KzP -K1 -.ui;(p)-K3X(P)
-8,ulog[I2P2 + TJ(P)I] + Kdog[,u8(P)IKzP2- K/- KIX(P)I]}.
IJ
(42)
Equation (41) is the analytical result of our theory of the eddy viscosity coefficient.
§ 3.
Results and discussion
For performing numerical calculation, we take parameter values as follows.
Letting the area of a square domain equal the surface area of a sphere with radius r,
we have the side of the domain L=2;-ir. Under the terrestrial condition, r=6.4
7
6
X 10 m, then L=2.3 X 10 m. The maximum wave number or the truncation wave
number, kmax, is taken to be 105 km;n that corresponds to the minimum wavelength to
be Am;n = 2.3 X 102 m. The parameter .u is determined from the form of the power
spectrum, because j"i; is the wave number at which the power spectrum is maximum.
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Up to now, we considered that the eddy viscosity acting on st(k) comes from the
nonlinear interaction between wave number k and all the other wave numbers, when
we are interested in st(k). Thus not only smaller scale eddies thank but also larger
ones act on st(k) as viscosity. However, it is natural to consider that an eddy
viscosity is an effective viscosity due to an action of small scale eddies upon large
scale eddies/ 1> because the concept of the eddy viscosity is an analogous one of the
molecular viscosity that is an effective viscosity due to thermal agitation of molecules,
i.e., small scale fluctuation. Hence we take into account the contribution of wave
numbers larger than k in (37), if we consider a phenomenon whose wave number is k.
That is,
1236
T. Iwayama and H. Okamoto
(b)
(a)
106
e(k)
MMEIII
N
td
Ul
N
::;;
u
Comparing the power spectrum e(k)
lOll
with observed ones (see Fig. 1), we select
!;;. =3 kmin, for the time being. On the
v
other hand, the parameter B is deterv
1010 mined from (19). Typical wind speed U
!-----is chosen to be 10 m/s so that we have
2
2
/
energy per unit mass E=50 m /s •
Hence four parameter values, L, kmax, fJ109
and U or E, are specified.
Figure 2 shows the numerical result
of the theory. The ordinate indicates
10 8 ~~~~~~~~~
the ratio of the eddy viscosity coefficient
1~
1~
lif
1~
Vectctik) to the molecular viscosity
length[2n/k(m)]
coefficient l/moi of the air. The abscissa
indicates the characteristic length of a
Fig. 2. Numerical result as calculated from formula (41). The ordinate indicates the ratio of
phenomenon. It shows that the eddy
the eddy viscosity coefficient to the molecular
viscosity coefficient depends on the
viscosity coefficient of the air. The abscissa
length of the phenomenon. For Cheju
indicates the characteristic length of a phenomIsland (Cheju-Do) scale, whose mean
enon, 2Jr/k.
diameter is about 30 km, the theory
derives that the eddy viscosity coefficient is about eight orders of magnitude larger
than that of the molecular viscosity coefficient of the air. This value is of the same
order as that of the eddy viscosity coefficient obtained by applying the Reynolds' law
of similarity to the atmospheric Karman vortices generated on the leeward of Cheju
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Fig. 1. (a) Power spectra as calculated from Eq. (18), for ,ffJ.=km1n (thin solid line), 3 km 1n (thick solid
line), 5 km1n (dashed line), 10 km1n (dash-dotted line) and 100 km1n (dotted line). (b) Observed power
spectra after Chamey.12>
Eddy Viscosity Coefficient for Two-Dimensional Inviscid Barotropic Fluid 1237
.....
X
-
I'
-
8
-
•
I=
10-12
,-....
'il
u
6
I
...!:C
;:;;:;
I
7
SxlO
~
4
1!0
...!:C
-IJi / kmin
Fig.
3. Dependence of the eddy viscosity
coefficient on the parameter J1 of the power
spectrum, for Cheju Island scale. The or·
dinate indicates the ratio of the eddy viscosity
coefficient to the molecular viscosity coefficient
of the air. The abscissa indicates the wave
number normalized by km1n at which the power
spectrum is maximum.
2
0+--+--1-l--l+l"'--4-++H-I-+++""f---+-++1-+Hll
10°
10
1
Fig. 4. Dependence of the integrated value in (41)
on the truncation wave number kmax for Cheju
Island scale, whose wave number corresponds
to kc-o=27r/(30 km). The ordinate indicates
the integrated value in (41). The abscissa
indicates the truncation wave number kmax
divided by kc-o.
Island, which was mentioned in § 1.
We examine the dependence of the eddy viscosity coefficient for Cheju Island
scale on the parameter J.L of the power spectrum (see Fig. 3). The ordinate indicates
the ratio of the eddy viscosity coefficient to the molecular viscosity coefficient of the
air. The abscissa indicates the wave number at which the power spectrum is
maximum, which is normalized by km1n. Although jJ;. changes five orders, the ratio
changes by less than two orders at most. Hence enormousness of the eddy viscosity
coefficient is almost independent of J.L.
We also examine the dependence of llectctik) on the truncation wave number kmax.
Figure 4 is the dependence of the integrated value in (41), I(kmax)- I(k), on kmax, for
Cheju Island scale, whose wave number is k=kc-D=ZJC/(30 km). The truncation
wave number used in the previous estimation, kmax=10 5 km1n, corresponds to about 102
ke-D. The integrated values for kmax>5 ke-D do not vary in significant figures of
third-order. Thus we conclude that the eddy viscosity coefficient llectctik) is
dominantly contributed by the modes whose wave numbers are almost equal to k in
which we are now interested and the enormousness of the eddy viscosity coefficient is
independent of the assumed parameters, kmax and J.L. In these respect, the enormousness of the eddy viscosity is independent of the local shape of the spectrum.
Equation (41) implies that eddies which have Cheju Island scale recognize the
extent of earth, because it includes the scale of domain L. What is physics of that?
As we consider the properties of fluid confined within the earth as a basin of a finite
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10° 101 102 103 104 105 ;:;;:;
1238
T. Iwayama and H. Okamoto
lJeddy(k)=/(
-try-q- ~~ {J(kmax)- J(k)}
= j kf+::;z / Kk! 4 {J(kmax)-J(k)}
Y max min Y 4fl.
= Vl(k),
(43)
where
V=
-
Z+f.l.E
k'fnaxk'fn1n '
l(k)=/
·1
4 ~k4 {J(kmax)- J(k)} . ·
(44)
The former part of (44), V, is constant that is independent of k and is determined by
properties of total system, and has the dimension of velocity. For selected parameter
values in this work, we have V =1.5 m/s. The latter part, l(k), is a function of the
wave number k, and has the dimension of length. Since the molecular viscosity
coefficient is expressed in terms of the product of the mean free path l of particles and
the mean value of fluctuating velocity u of them, the function l(k) corresponds to the
mean free path of vortices, or the mixing length. The form of l(k) is the same as that
of the curve in Fig. 2, i.e., l(k) ~ k- 1 • For air at atmospheric pressure (1000 hPa) and
room temperature (300°K) one has approximately u ~ 102 m/s and l ~10- 7 m, hence
JJmol~ul~10- m /s. > In this respect, we see that the enormousness of the eddy
viscosity in comparison to the molecular viscosity is due to that of the function l(k),
whereas l(k) is the function of the length scale of interest. Thus we conclude that the
enormousness of the eddy viscosity coefficient comes from that of the length scale of
interest.
We comment on the application of the present discussion to atmospheric phenomena. We assumed that the system is of two-dimension and has the specific wind speed
to be 10 m/s. Although large scale atmospheric motions are actually three-dimensional phenomena, two-dimensionality is approximately satisfied well. For example,
5
2
15
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size of V, eddies of Cheju Island scale are regarded as a phenomenon in the basin.
Therefore the earth and the eddies correspond to a heat bath and sub-systems contact
with it in the statistical-thermodynamics, respectively. Because the properties of the
sub-systems depend on those of the heat bath, it stands to reason why the eddies
recognize the extent of earth, i.e., the formula for the eddy viscosity coefficient (41)
includes the size of domain L. For conserved systems, dynamical properties of the
systems are completely specified by initial conditions, by which conserved quantities
are also determined. However, properties of systems in statistical mechanical theory
are specified not by individual initial conditions but by conserved quantities. Since
the scale of domain L and the parameters B and fl. determine the total energy and
enstrophy, i.e., conserved quantities, it is also reasonable that these quantities are
included in the formula for the eddy viscosity coefficient. We can rewrite (41)
without including L. By using (19) and (20), Eq. (41) becomes
Eddy Viscosity Coefficient for Two-Dimensional Inviscid Barotropic Fluid 1239
§ 4.
Summary
Applying Mori's theory 6> to the spectral form of the vorticity equation for 2-D
inviscid barotropic fluid, we derived the formula for the eddy viscosity coefficient.
The eddy viscosity coefficient thus obtained depended on the length scale of a phenomenon of interest. For Cheju Island scale, it was ~erived theoretically that the eddy
viscosity coefficient was eight orders of magnitude larger than that of the molecular
viscosity coefficient of the air, whose value was of the same order as that of the eddy
viscosity coefficient obtained by applying the Reynolds' law of similarity to the
atmospheric Karman vortices generated on the leeward of Cheju Island. Moreover it
was theoretically found that the enormousness of the atmospheric eddy viscosity
coefficient was due to that of the length scale of interest.
Acknowledgements
We would like to thank ProfessorS. Miyahara, Professor 0. Morita, Professor T.
Hirooka of Kyushu University and Professor M. Takahashi of University of Tokyo
for their helpful discussions and encouragement. We wish to thank Dr. Y. Ookouchi
of Yatsushiro National College ofTechnology, Dr. H. Akiyoshi of Fukuoka University, and all of the members of our dynamic meteorology group at Kyushu University
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the characteristic length of the synoptic scale disturbances that correspond to the high
or low atmospheric pressure systems is about 3000 km in the horizontal and about 15
km in the vertical. Moreover, for atmospheric Karman vortices generated on the
leeward of Cheju Island, the two-dimensionality is a good approximation. In general,
the atmospheric temperature decreases monotonically with height from the ground to
the tropopause, which is located at an altitude of about 15 km. However a layer in
which the temperature increases with height is locally observed. This layer is called
an inversion layer. The atmospheric Karman vortices appear on the leeward of
Cheju Island when the wind speed is more than 10 knots ( ~5 m/s) and a well-defined
inversion layer exists at about 1 km altitude. 13 > Then the vertical motion of the
atmosphere is suppressed at this altitude. As seen from meteorological satellite
photograph [e.g., see Ref. 13)], the horizontal scale of the atmospheric Karman vortex
is 30~50 km. Thus the two-dimensionality is a good approximation in this case. In
addition, it is seen that the specific wind speed assumed in this study is reasonable.
It is known that the eddy viscosity coefficient of the atmosphere has different
values in both horizontal and vertical directions. The vertical eddy viscosity
coefficient is 1 ~ 10 m 2 / s for the mesosphere, thus 5 ~ 6 orders larger than the molecular
viscosity of the air, which is by two orders less than that of the horizontal one.l),r 4 >
The vertical eddy viscosity originates from unstable breakdown of gravity and tidal
waves, whose vertical wavlength is about 10 km. 14> Although the present theory is
one for horizontal two-dimensional system, we can infer from the theory that the
smallness of the vertical eddy viscosity coefficient in comparison to the horizontal one
comes from that of the vertical scale of atmospheric phenomena. This problem for
the three-dimensional inviscid barotropic fluid will be discussed in a subsequent paper.
1240
T. Iwayama and H. Okamoto
for encouragement and Mr. E. M. P. Ekanayake for his work to clarify the manuscript. We are also grateful to Emeritus Professor K. Gambo of University of Tokyo
for valuable comments.
Thanks are also extended to one of the referees for valuable comments on the
manuscript. He especially introduced Ref. 13) to the authors.
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