December
1974
Mean
M.
Zonal
Flows
Induced
Rossby
Uryu
by
Wave
By Michiya
481
a Vertically
Propagating
Packet
Uryu
Department of Physics, Faculty of Science, Kyushu University, Fukuoka
(Manuscript received 1 July 1974, in revised form 17 September 1974)
Abstract
The second order mean motion induced around an internal Rossby wave packet, having an
infinite zonal length and propagating vertically in an inviscid Boussinesq fluid at rest in a channel,
is discussed, and the validity of photon analogy to such a wave packet is examined.
It is shown that the second order mean zonal momentum averaged in the meridional direction is just equal to the wave momentum E/C (where E and C are the wave energy and the
phase velocity in the zonal direction respectively). This guarantees the validity of photon analogy
to the wave packet, and also implies that the treatment done rather intuitively in the previous
paper by the author (Uryu, 1974) is essentially correct.
It is shown that, to the first order in * (where * is a small parameter characterizing the slowness of variation of wave amplitude), the vertical component of Lagrangian mean velocity is zero,
as a result of the cancellation between the Stokes drift and the Eulerian mean velocity.
The case of propagation in a shear flow is also treated, and it is shown that the absorption
of wave at the critical level occurs as a result that the wave momentum is stored up in the mean
zonal flow there.
It is also shown that the results obtained in case of Rossby wave packet can be obtained
without any essential change also in case of internal gravity wave packet under the same situation.
the second order mean motions in it, and he has
concluded that the effect of the wave packet on
In general, when a wave packet propagates in the ambient field can be summarized by the wave
a fluid medium, it exerts forces on the ambient momentum E/C, although he has not shown
mean basic flow and induces changes of the exactly the equality of the magnitude of the insecond order with respect to the wave amplitude duced mean momentum and the wave momentum.
In case of a vertically propagating Rossby
in it. In case of water wave, Longuet-Higgins
and Stewart (1962, for example) has shown that wave, the author (Uryu, 1974) has shown that
such an interaction between wave and flow can the easterly acceleration by the wave can be
be expressed by the concept of radiation stress interpreted by the photon analogy (his treatment
associated with the wave packet.
This suggests has a few insufficiencies, though they are trivial,
that the concept of momentum radiation by wave and it will be complemented in this paper).
However, in his recent paper, McIntyre (1973)
(photon analogy) could be applied generally to
wave packet in a material medium.
That is, a has pointed out that the photon analogy cannot
wave might posess a well defined momentum E/C be applied to such an internal gravity wave that
(where E and C are the wave energy and the is guided in a channel bounded at the top and
phase velocity respectively) and give it to the the bottom.
ambient flow.
On the other hand, several works which treat
In case of internal gravity wave, by studying the so-called critical level absorption, such as
the mean motion induced around a wave packet, Booker and Bretherton (1967), Lindzen and
Bretherton (1969) has shown that the wave packet Holton's model (1968) for the quasi-biennial
and
propagates like a moving "pressure dipole ", oscillation in the equatorial stratosphere
exerting forces on the ambient field and inducing Matsuno's model (1971) for the stratospheric
1.
Introduction
482
Journal of the Meteorological Society of Japan
sudden
warming,
image
and
the
The
the
seem
to internal
analogy
main
origin
included
seems
purposes
of
the
implicitly
to complement
as mentioned
to
gravity
the
conceive
wave
to
and
lead
correct
of this paper
validity
in
the
of
these
works
paper
order
wave,
results.
ing
which
slow
of
is a small
variation
Rossby
and
gravity
Further,
in
order
to
absorption
of
wave,
our
and
to
the
case
it
is
shown
the
also
author
above.
occurs
as
of
parameter
of wave
analogy
above
by
in *
the
cases
are to examine
photon
previous
photon
Rossby
Vol. 52, No. 6
a result
the
that
the
critical
treatments
are
in a shear
critical
the
in both
waves.
discuss
propagation
that
characteriz-
amplitude,
wave
level
level
extended
flow,
and
absorption
momentum
is
We shall treat two kinds of wave packet, stored up in the mean zonal flow.
internal Rossby wave and internal gravity wave.
The wave packet is assumed (1) to have an 2. Problem and its solutions
In this section,
we shall
treat
two
cases;
an
infinite zonal length; (2) to have a slowly varying
envelope; (3) to propagate freely in the vertical internal
Rossby
wave
packet
and
an internal
direction in an inviscid Boussinesq fluid channel gravity wave packet.
But our discussions
will be
bounded by vertical walls at two latitudes. The developed
mainly
in the former
case, because
the
assumption (1) is usually made in meteorological result in the latter
case
is essentially
similar
to
works, such as Lindzen and Holton (1968), that in the former.
Matsuno (1971) and Uryu (1974), and also Booker
2.1. Case of internal Rossby wave packet
and Bretherton (1967) has treated such a wave
As shown schematically in Fig. 1, we shall
packet. It seems that this assumption is essential
to the justification of photon analogy.
consider an internal Rossby wave packet propagatOur discussions will be proceeded in two steps. ing in an inviscid and continuously stratified
First, it is shown that E/C is conserved during Boussinesq fluid channel bounded by vertica
the propagation
of wave packet.
Next, we walls at two latitudes.
We assume the quasicalculate the second order mean flows around the geostrophic and the quasi-hydrostatic approximawave packet by combining the conservation law tions and that the local Cartesian expression
of E/C with the zonally averaged second order is possible.
equation of motion. The treatment is based upon
Then, using the Coriolis parameter at a reference
the so-called two timing method. The first step
discussions are parallel to those done so far by
several authors (Bretherton, 1967., for example),
but it seems that there are few works, except
those by Longuet-Higgins and Stewart in early
stages in the 60's, McIntyre (1973) and Uryu
(1974), which have treated the second order mean
motions in connection with the conservation law
of E/C as in our second step. Such a treatment
is a key point of our discussions.
In the following sections, it will be shown that,
in both cases of internal Rossby wave and internal
gravity wave, the induced mean zonal momentum
averaged in the meridional direction in the channel
is just equal to the wave momentum E/C.
This
implies that the treatment in the previous paper
by the author (Uryu, 1974) is essentially correct.
It will be also shown that the vertical component
of Lagrangian mean velocity is zero up to 1st
* According
to his personal
author,
McIntyre
the
International
Melbourne,
communication
has reported
Conference
Australia,
Jan.
to
the
on this problem
of
IAMAP
at
at
1974.
Fig.
1.
Schematic
illustration
propagation
the second
induced
of vertical
of wave packet and
order mean motions
around
it.
December
1974
M.
latitude f0, the width of the
channel
D and
f0D/N (N is the Brunt-Vaisala frequency) as the
units of time, horizontal and vertical scales, we
can deduce the following
non-dimensional
potential vorticity equation under the assumptions
above;
where p is pressure and x, y and z are the
Cartesian coordinates in the zonal, meridional and
vertical directions respectively, q and *
are
defined as follows;
Uryu
as
483
the
pressure,
the third
The
ing
term
is
the
basic
the second
is the
is the wave-induced
wave
packet
Wave packet solution
time-independent
wave packet
part
mean
pressure.
is assumed
i.e.,
Substituting (2-3) and picking up terms of 1st
order in a, we obtain the following linearized
potential vorticity equation;
We separate p into three parts as follows;
first
envelope,
We further assume that P is of second order in a.
In the following discussions, we assume that n
and * are constant as well as k. This assumption
is allowable because the medium considered here
is a Boussinesq fluid, i.e., homogeneous. In case
of propagation in a shear flow (see * 3), the
constancy of n(and therefore *) does not hold.
P and P are expanded as follows;
(a)
The
wave
to have
and
the follow-
Substitution of the expansion (2-7) into (2-8)
gives, to 0-th order in *, the dispersion relation
of a free Rossby wave;
form;
or
where k and n are the wave numbers in the zonal
and the vertical directions respectively, *
is the
wave frequency, a is a small parameter characterizing the wave amplitude, Re( ) means the real part
of the quantity in the bracket and T and Z are
the slow variables defined as
To
1st
order
in *,
we have
where Cg is the vertical group
wave packet which is given by
where * is a small parameter characterizing the
slowness of the variation of the wave envelope
P(Z, T). Then, the wave packet thus assumed
has an envelope varying slowly in time and in
the vertical direction and also has an infinite
length in the zonal direction.
It is noted that
the wave packet part vanishes if it is averaged
with respect to the fast variables x, z and t, and
also noted that the meridional structure sin *y is
assumed so as to satisfy the boundary condition
that normal velocity vanishes at the vertical wall.
The wave-induced mean pressure is assumed to
vary slowly in the same time and vertical scales
Eq. (2-10) states that the
change its shape during
equation is rewritten in
Since the wave energy E
where
direction,
, the
the
bar
we
following
means
obtain,
result;
the
to
velocity
of
the
wave envelope does not
the propagation.
This
terms of wave energy.
can be defined as
mean
leading
value
orders
in the
in
x-
a and
*
484
Journal of the Meteorological Society of Japan
This
is equivalent
Vol. 52, No. 6
to
the
following
set
of equa-
tions;
where E0 is the leading order term of E in *expansion.
Thus, combining (2-10) and (2-13), we have, to
leading order in *,
This implies that the wave energy packed in the
wave packet is conserved in the propagation
process.
It is noted that, as will be shown in
the next section, the result (2-14) is a special case
of the law of conservation of wave momentum
E/C which holds generally when a wave packet
propagates in a slowly varying medium (Bretherton
and Garrett, 1969., for example).
The present
result originates from the assumption of constancy
of k, n and * which means to assume the homo-
where U0, V1 and W0 are the induced Eulerian
mean zonal, meridional and vertical velocities
respectively, *1
is the induced zonal mean
buoyancy and the suffices 0 and 1 mean the
leading order term of each quantity in *-expansion.
geneous property of medium.
The wave packet solutions become as, up to
Eq. (2-17-d) is the zonal mean adiabatic equation of 0-th order in *, and it means that the
1st order in *,
induced mean vertical motion W0 is in balance
with the convergence (divergence) of the meridional
buoyancy flux due to the wave motion.
It is
noted here that such a balance holds up to 1st
order in *. i.e.,
where
of
u',
velocity
v' and
w'
disturbance
are
x, y and
respectively, *'
z
components
is
density
we can
obtain
disturbance.
Making
that,
use
to leading
of these
order
solutions,
in *,
Thus, we see that the meridional buoyancy flux
is related with the vertical flux of wave momentum. This result is the extension of that
obtained in case of stationary wave by the present
author (Uryu, 1973) to transient case.
(b) Second order mean motion
Sustituting (2-3) into (2-1) and averaging the
result zonally, we obtain the following equation
of second order in a which is correct to 1st order
in *;
This is attributed to the assumption that P is a
function of slow variables only; that is, the
induced mean buoyancy *1 is one order in E*
smaller than P0 due to the hydrostatic balance,
and hence the time change of *1 in the zonal
mean adiabatic equation is two order smaller
than other two terms, buoyancy flux and mean
vertical motion.
In order to obtain the mean
vertical motion correct to 1st order in *, we must
solve a zonal mean potential vorticity equation
of higher order than (2-16). However, as will
be shown in the next sub-section, we can obtain
without solving such a higher order equation but
by making use of (2-17-d') that the vertical component of Lagrangian mean velocity vanishes up
to 1st order in *. In this sense, eq. (2-17-d') is
an important result.
In the previous paper by the author (Uryu,
December
1974
M. Uryu
485
1974), the balance between the buoyancy flux and
the mean vertical motion has been assumed (see
eq. (3-5) in the paper) to obtain the zonal mean
momentum equation, and now the assumption is
proved to be essentially correct. It is further
noted that his resulting equation for the zonal
mean momentum is the one which can be obtained
by integrating eq. (2-16) with respect to y and
setting y =1/2.
This is due to omitting the
(c) Lagrangian mean velocity
meridional structure of induced mean meridional
Using the Eulerian mean quantities obtained so
flow V and evaluating it at the maximum value. far, we can calculate the Lagrangian mean (mass
Making use of (2-14) and (2-10) and taking
transport) velocities which show what mean mointo account the boundary condition that V=0 tions of fluid particles are produced by a vertically
at the vertical walls, we can easily integrate eq. propagating internal Rossby wave packet.
(2-16);
First, we shall examine the vertical component of
Lagrangian mean velocity WL, which is averaged
over one wave length or equivalently one period
of oscillation. Let *' be the meridional displacement of a fluid particle from its mean position
and hence,
due to the wave packet. Then, we can write the
Stokes drift Ws, in the quasi-geostrophic
approximation, as follows, to 1st order in *.
Then, averaging (2-18) and (2-19) with respect to
y, we have
and
where
order
use is made
of
the
solution,
correct
to
1st
in *,
where < > means the meridional mean value.
Thus, the change in Eulerian zonal mean momentum is induced as result of convergence
(divergence) of wave momentum flux, and the which is obtained
from the relation
that
induced momentum is just equal to the wave
momentum E/C.
These results show that the
concept of momentum radiation by wave or the
photon analogy can be correctly applied to the Thus, by making use of (2-17-d'), it is shown that
present case in which the wave packet is assumed the Lagrangian mean velocity WL becomes zero,
to be of infinite length in the zonal direction and up to 1st order in *;
to propagate freely in the vertical direction (cf.
McIntyre, 1973). Further, we note that (2-21)
gives a good agreement with the computational
* An alternative deduction of the result (2-24) is as
result in the previous paper (Uryu, 1974). The
follows. Let *' be the vertical displacement.
As
discrepancy of factor 2 mentioned in the paper
seems to be caused by evaluating U at y=1/2.
The other Eulerian zonal mean quantities are,
to leading order in *, as follows;
The right-hand
side of
higher order than those
WL vanishes
up
this includes
of 1st order
to 1st order
in *.
only
in *.
terms
Thus,
486
Journal of the Meteorological Society of Japan
That is, fluid particles cannot be moved in the
vertical by the wave packet. This has been first
pointed out to the author by Matsuno (personal
communication).
Further, it is suggested by this
result that the circulation of ozone, aerosols and
so on in the stratosphere must explained by any
other mechanism than this type of wave.
The zonal component of the Lagrangian mean
velocity UL becomes as follows, to leading order
in *.
Similarly
section,
wave
and
to the
we
packet
the
we put
Vol. 52, No. 6
treatment
separate
part
induced
all
with
zonal
in the
field
a slowly
mean
previous
variables
varying
part;
for
sub-
into
the
envelope
example
as follows;
where x is the phase function defined by (2-4),
= */d (d is the width of the channel scaled
l by
/N2) and Z and T are the slow variables defined
g
by (2-5).
U and U (and so on) are expanded in *-series
as in the previous sub-section.
(a) Wave packet solution
Thus, we readily obtain <UL> <U0>.
To the first order in a, we obtain the following
It is noted that the meridional component VL
linearized equation for internal gravity wave;
is one order smaller than UL with respect to *.
2.2.
Case of internal gravity wave packet
The situation treated here is similar to that in
where * = aRe(U(Z, T )eix),
the previous sub-section, except that in the present
Then, from (2-28), we obtain, to 0-th order in *,
case the effect of rotation is omitted and that the
the following dispersion relation;
quasi-hydrostatic
approximation is not made.
Then, the basic equations in non-dimensional form
are as follows;
Hence, the vertical
follows;
group
velocity
Cg becomes
as
From eq. (2-28), we obtain, to 1st order in *,
where we have used N-1 and g/N2 (g is acceleration of gravity) as time and length scale units*.
The coordinate system is similar to that used in
the previous sub-section and the notations are
same too unless otherwise mentioned.
* The length scale unit g/N2 is adopted only for the
sake of convenience to remove the coefficient N
from the density equation. The "slowness" of
vertical variation of the wave envelope is defined
in reference to the wave length and not to the
length scale adopted here.
where use is made of (2-30).
The wave packet solutions which satisfy the
linearized equation (1st order in a) to 1st order
in * are written as follows;
December
1974
M.
Uryu
487
(b) Second order mean motion
The zonal mean momentum equation is obtained
by averaging eq. (2-26-a) zonally, and we see that
only the Reynolds stress u'w' contributes to the
zonal mean momentum.
Then, making use of
(2-35-b),
Then, making use of these solutions,
rewrite eq. (2-31) as
we can
tion,
where E0 is the leading order term of wave
energy E (in *-expansion) which is defined as
and E0 becomes
Here,
ponents
in *.
we write
and
the
as
down
buoyancy
the
Reynolds
fluxes
stress
to leading
Averaging
comorder
and
we can obtain, to 1st in *, that
this
equation
in the
meridional
direc-
we have
hence
Thus, we see that the photon analogy can be correctly applied also to the present case.
Since, as already mentioned, U
is determined
by u'w' alone and independently of the mean
motion in the meridional plane (this is an essential
difference from the case of Rossby wave packet),
the result above holds also for a two-dimensional
case (x-z plane).
As to the mean motion in the meridional plane,
we see by averaging eq. (2-26-b) zonally that, to
leading order in *, the Reynolds stress v'2 is
balanced with the meridional gradient of zonal
mean pressure, and hence the mean meridional
motion is induced as a result that the mean
vertical motion is forced by the convergence
(divergence) of meridional buoyancy flux due to
the wave; from the zonal mean adiabatic equation
of 0-th order in *, we obtain that
and
hence
It should
the
mean
meridional
It should be noted that v'w', v'2 and v'*' do not
vanish because the meridional structure of the
wave packet, for example, u'*cos ly, is assumed.
In a two dimensional case in x-z
plane, i.e.,
=0, these terms vanish. Further, we note that
l
(2-35-a) holds in an arbitrary order in * because
there is no phase change in y-direction under the
assumption (2-27).
be noted
vertical
here
motion
buoyancy
in *
also
case
of Rossby
in the
present
wave
flux
that
the balance
and
the
holds
case,
packet;
up
as
that
between
convergence
to
already
is,
1st
of
order
seen
in
488
Journal of the Meteorological Society of Japan
Vol. 52, No. 6
In two dimensional case, the induced mean
The zonal
component
UL becomes,
to leading
motion is only zonal, because the mean vertical order in *, as follows;
flow is required to vanish from the continuity
equation.
This conclusion can be readily obtained also
by setting l=0
in eqs. (2-39) (or (2-39')) and
(2-40). Inversely speaking, the results (2-39) and
(2-40) originate from the assumption of the Thus, we obtain that <UL>= <U0>.
meridional structure of wave packet, such as
The meridional component VL is one order
(2-27). It is further noted that in two dimensional smaller than UL with respect to *.
case the Reynolds stress w12 is balanced by the
vertical gradient of zonal mean pressure (cf. 3. Propagation in a shear flow and critical level
Bretherton, 1969).
absorption
In the discussions in the previous section, we
have assumed only the slow variation of wave
Let *' and *' be the meridional and vertical
envelope, and the other characteristic quantities
components of displacement of a fluid particle due of the wave, such as k, n and a have been
to the wave packet.
Then, in the present case,
assumed to be constant. However, when a wave
the vertical component of Stokes drift Ws is
packet propagates in a shear flow U(z), the wave
given by
number n and therefore * cannot be regarded as
constant.
In case of internal gravity wave, by assuming
that the basic shear flow U(z) is a slowly varying
However, taking into account that *' is equival- function, Bretherton (1967) has shown that
ent to *', we see from (2-35-g) that the second
term on the right of (2-41) is at most of second
order in *. Therefore, Ws which is correct up
to 1st order in * is determined by the first term,
This holds also in case of internal Rossby wave;
i.e.,
the proof is not reproduced here because it is
similar to that used by Bretherton (1967). Further,
Bretherton and Garrett (1969) have shown that
the above result (3-1) holds for a wide class of
physical system. It is the law of conservation
of wave action. That is, when a wave packet
propagates in a slowly varying shear flow, the
Combining (2-42) with (2-39'), we see also in wave energy E is not conserved but the wave
the present case that the vertical component of action E/(*+ k U) is conserved.
Since, in the
Lagrangian mean velocity WL vanishes up to 1st present, we regard the zonal wave number k
order in *;
constant, is readily replaced by E/(C-U),
the
wave momentum.
On the other hand, equation for the second
order
mean flow induced by the wave packet can
It should be noted here that (2-43) holds also
for two dimensional case, because *' is identically be obtained only by replacing C to (C- U) in
zero; that is, both of W and Ws are zero for eq. (2-18), because forms of all the second order
=0. As already mentioned, non-zero values l of quantities necessary for the deduction of (2-18)
W and Ws in the present case are attributed to remains unchanged except C (or *), so far as a
wave train is assumed
to
the assumed meridional structure of the wave quasi-sinusoidal
propagate in a slowly varying shear flow. Namely,
packet.
(c) Lagrangian mean velocity
December
1974
M.
Uryu
489
Although the conservation law (3-1) breaks (2) The bases from which the conclusion (1) is
down at a level where C= U, we can discuss
drawn are different in R and G. In the
former, it is essential that the mean vertical
qualitatively the so-called critical level interaction
between wave and flow by eqs. (3-1) and (3-2).
motion is in balance with the convergence
As a wave packet reaches near and near the
(divergence) of the meridional buoyancy flux
critical level, the vertical group velocity becomes
due to wave, to leading order in * which is
smaller and smaller (In case of internal gravity
a small parameter characterizing the slowness
wave, see Bretherton (1967); in case of internal
of wave envelope. This means that the treatRossby wave, this is readily seen from (2-11) only
ment in the previous paper by the author
by replacing *
to * + k U ), while the wave
(Uryu, 1974) is essentially correct. It is noted
momentum is conserved in the propagation of the
that the existence of the walls (or of the poles
wave packet.
Thus, the wave momentum is
in the actual atmosphere) is one of the most
"stored up" in the mean zonal flow as seen from
important factor in the induction of mean
eq. (3-2). We can say that the acceleration of
zonal motion in R.
mean zonal flow at the critical level by wave is
In G, the decisive factor to the induction
a result of the accumulation of wave momentum
process is the Reynolds stress (correlation
there.
between the zonal and the vertical components
of velocity in wave motion), and hence the
In the treatment described so far, we have
assumed that the wave packet is of infinite length
conclusion (1) holds also in two dimensional
case.
in the zonal direction.
Such an assumption is
often made; such as, in the work of Booker and (3) In both of R and G, the vertical component
Bretherton (1967) which has first pointed out the
of Lagrangian
mean velocity is zero up to
critical level absorption of internal gravity wave,
1st order in *, as a result of the cancellation
in Lindzen and Holton's (1968) model for the
between the Eulerian mean velocity and the
Stokes drift.
This suggests that the circulaquasi-biennial oscillation and Matsuno's (1971)
model for the stratospheric sudden warming both
tion of ozone, aerosols and so on must be
of which include the critical level absorption of
explained by any other mechanism
than these
wave as the essential mechanism bringing about
waves.
such phenomena. According to the present result, (4) In both of R and G, if C is replaced by
the photon analogy included implicitly in these
C- U (where U is the basic zonal flow with
works is valid.
shear), the conclusion (1) holds also in case
of the propagation in a shear flow under the
4. Conclusions
assumption that U is a slowly varying function. Thus, the so-called critical level absorpIn this paper, the second order mean motions
tion of wave can be interpreted as follows.
induced around an internal Rossby wave packet
When a wave packet reaches near and near
have been discussed by a simple model situation.
the critical level, the group velocity becomes
The wave packet is assumed to have an infinite
smaller and smaller, while the wave momenzonal length, to have a slowly varying envelope
tum is conserved.
Therefore, at the critical
and to propagate vertically in an inviscid
level, the wave momentum is stored up in
Boussinesq fluid channel with two vertical walls
the mean zonal flow. The essential mechanism
at two latitudes. We have treated also the case
bringing about the quasi-biennial oscillation
of an internal gravity wave packet under the same
in Lindzen and Holton's model (1968) and
situation (for brevity, we call these two cases as
the
sudden
warming
phenomenon
in
R and G respectively), and our discussions have
Matsuno's
model
(1971)
is
the
accumulation
been extended to the case of propagation in a
of wave momentum at the critical level.
shear flow. The conclusions are summarized as
Finally, it is remarked that the assumption of
follows.
(1) In both of R and G, the induced mean zonal an infinite zonal length of wave packet seems
momentum averaged in the meridional direc- essential to justify the photon analogy.
tion is just equal to the wave momentum
Acknowledgement
E/C. Thus, the photon analogy to such
waves is valid in the present situation.
The author would like to express his sincere
490
Journal
thanks
for
to
his
Prof.
T. Matsuno,
stimulating
critical
also
concerns
of the
this
Prof.
the
for their
by
for
his
author
Sawada,
for
Kyushu
his critical
are due
figure,
and
valuable
comments.
Funds
for
and
Lindzen, R. S. and J. R. Holton,
the quasi-biennial
oscillation.
1095-1107.
Longuet-Higgins,
reading
to Miss
also
Scientific
is
This
Radiation
Sato
due
to
work
waves,
M. S.
stress
and
and
with application
spheric
1494.
of Education.
McIntyre,
sudden
Mean
a guided
internal
gravity
Mech., 60, 801-811.
References
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M. E., 1973:
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R. W. Stewart,
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Matsuno,
T., 1971: A dynamical
Research
Booker, J. R. and F. P. Bretherton, 1967: The critical
layer for internal gravity waves in a shear flow.
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Bretherton, F. P., 1967: The propagation of groups of
internal gravity waves in a shear flow. Quart. J.
Roy. Meteor. Soc., 92, 466-480.
1969: On the mean motion induced
by internal gravity waves. J. Fluid Mech., 36, 785802.
Garrett,
trains in inhomogeneous
moving
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encouragements
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drawing
Ministry
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The
R.
his continuing
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and C. J. R.
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波 束 の まわ りに 誘 導 され る 平 均 流
生
道
也
九州大学理学部物理学教室
経 度 方 向 に は 地 球 を1回
りす る程 度 の長 さを も ち,鉛 直 上 方 に 伝 播 す る,ロ ス ビ ー波 束(又 は,重 力 波 束)の
りに 誘 導 され る平 均 帯 状 流 を,簡 単 な,い わ ゆ る チ ャン ネ ル ・モ デ ル を用 い か つ2-Variables Methodに
まわ
よ って議 論
し,あ わ せ て そ の種 の波 束 に関 す る 「フ ォ トン ・ア ナ ロジ ー」(従 来,波 と流 れ の相 互 作 用 を と りあつ か った 論 文,た
とえ ばMatsuno(1971)の
突 然 昇温 の モ デ ル,Lindzen-Holton(1968)の
準2年 振 動 の モデ ルな どに,暗
に 含 まれ
て い る考 え 方)の 正 当性 を も論 じる.
ロス ビー波 束(又
平 均 す れば,波
は,重 力波 束)の
束 の もつ 運 動 量E/C(Eは
る.こ の こ とは,フ
まわ りに誘 導 され る平 均 帯 状 流 の もつ 運 動 量 は,チ
波 の エネ ル ギ ー,Cは
東 西 位 相 速 度)に
ャン ネ ル の中 で 南 北 方 向 に
ち ょ うどひ と しい こ とが 示 され
ォ トン ・ア ナ ロ ジー の 正 し さを 保証 す る.
ロス ビー波 や 重 力波 の鉛 直 伝 播 に と もな って,流 体 粒 子 も動 くが,そ の運 動 は水 平 的 で あ る こ とも示 され る.つ ま
り,ラ グ ラン ジ ュ的 平均 速 度 の 鉛直 成 分 は,オ イ ラー的 平 均 速 度 とス トー クス ・ ドリフ トが うちけ しあ った 結 果,0
と な る.
な お,鉛 直 シ ャー流 中 に波 が 伝播 す る場 合 も と りあ つ か い,い わ ゆ る ク リテ ィカ ル ・レベ ル で の波 の吸 収 は,波 の
運動 量E/Cが
平 均 流 に 向 って 「蓄 積 」 され る こ と と同 義 で あ る こ とを 示 す.