CHINESE JOURNAL OF PHYSICS
JUNE 1999
VOL. 37, NO. 3
A Study of Secondary Waves Generated by
Wave-Wave Interaction in a Wind Field
F. S. Kuol, C. L. Lol, H. Y. Lue2, and C. M. Huangl
‘I nstitute of Space Science, National Central University, Chung-Li, Taiwan 320, R. 0. C.
2Department of Physics, Fu Jen University, Hsin Chuang, Taiwan 242, R.O.C.
(Received December 29, 1998)
Propagation of an atmospheric gravity wave (AGW) in a wind field is studied by
numerical simulation. Two models are applied: the first is a linear model to check
with the gravity wave theory; the second is a quasi-nonlinear model to study the characteristics of the secondary waves produced by the interactions between two gravity
wave packets. The magnitudes and directions of the vertical phase and group velocities
obtained by linear simulation are consistent with the prediction of linear gravity wave
theory. That is, the vertical propagation velocities of the phase and the energy of an
AGW are always opposite to each other. In the quasi-nonlinear model, two primary
waves are forced to oscillate at the ground level with their group velocities upward and
phase velocities downward, which creates two secondary waves by the interaction between the primary waves. The smaller one is found to have group velocity upward and
phase velocity downward just the same as that of the two primary waves. The group
velocity of the larger one is found to be downward always, while its phase velocity is
upward when the primary waves in the source region are still oscillating and becomes
downward after the primary waves stop oscillation. The result reveals that the smaller
one is a production of a resonant interaction; while the larger one is the production of
a non-resonant interaction. Both secondary waves do not follow the dispersion relation
of the linear theory. The result of this simulation confirms our previous observation
that the vertical phase and group velocities of a non-linear AGW are not necessarily in
the opposite direction.
PACS. 02.60.Cb - Numerical simulation.
PACS. 92.6O.Dj - Gravity waves.
I. Introduction
According to linear theory, the vertical phase and group velocities of the gravity waves
are in the opposite direction. Hines [l] applied this principle to point out that the energy
source of the gravity waves observed in the upper atmosphere must generally lie below
the height of observation, because the observed phase progression was almost invariably
downward. Since then, gravity waves have been believed to play an important role in
transporting the disturbance energy from the lower atmosphere to the upper atmosphere.
Also, people tend to follow the same principle to determine the whereabouts of the energy
source of the observed gravity waves in the atmosphere. However, linear theory is valid
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@ 1999 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
VOL. 37
F. S. KUO, C. L. LO, H. Y. LUE, AND C. M. HUANG
241
only when the wave amplitude is small. If the wave amplitude is large enough for the
nonlinear process to become important, the sense of the phase and energy propagation may
be different from that of the linear theory. In fact, a method of analysis had been developed
[2] to derive the vertical group velocity and phase velocity of the wave propagation, and had
been applied to analyze two sets of observation data made by Chung-Li VHF radar under
different conditions. In one case, the mean wind is mild at the lower heights and becomes
violent above the height of 8 km; in another case, strong vertical shear exists in the zonal
component of the mean wind throughout the heights of observation. It was found in the
first case that, the vertical phase and group velocities were mostly in the opposite direction,
and the energy source must lie above the height of 8 km since the group velocities of most of
the events were downwards. While in the second case, with the existence of strong vertical
shear, the sense of the phase and energy propagation were no longer certain: some were
in the same direction while others were in the opposite direction. It was proposed that
nonlinear processes might be responsible for this uncertainty of the propagation sense.
There are a number of papers discussing non-linear wave-wave interactions of gravity
waves in the ocean [3,4,5,6,7] and in the atmosphere [8,9,10,11,12]. Their main subject
of interest is the energy transfer among different wave components. The most efficient processes they had found are called elastic scattering, parametric subharmonic instability, and
induced diffusion. Through elastic scattering an upward-propagating wave is scattered into
a downward wave by interacting resonantly with a vertical shear; parametric subharmonic
instability transfers energy to much smaller scale at near-inertial frequency, and the induced
diffusion diffuses wave action in wave number space. The interactions considered in these
three processes are among three plane waves satisfying conservation of frequency and wave
number (i.e. the so called resonant condition). The phase velocity of the third wave generated by the interaction between the other two waves can be calculated by the equations of
the resonant condition. However, the group velocity cannot be obtained from these studies
since no dispersion relation of these nonlinear waves is given. It seems to be unrealistic
trying to derive the dispersion relation for a nonlinear wave. A simpler way to study the
group velocity of wave propagation is by numerical method to simulate the wave propagation in the mean wind field with strong vertical shear. Because the critical level provides
a natural propagation boundary for the gravity wave, and that simplifies the most difficult
part of the boundary value problem in numerical simulation. A numerical code described in
[l3] was successfully applied to study the saturation spectrum of atmospheric gravity wave
by a nonlinear model. The nonlinear model is a self-consistent numerical model including
both wave-mean flow interaction and wave-wave interaction with mean flow dependent on
time. In addition to the non-linear model, three models can be applied by this code: linear
model, quasi-linear model and quasi-nonlinear model. A linear model, by which wave-wave
interactions are excluded and the mean flow field is kept constant, is suitable to simulate the
wave propagation in the wind field. In the quasi-linear model, only the interaction between
the gravity wave and the vertical shear of the mean wind field is considered, the mean state
is time dependent, and all the wave-wave interactions are excluded. So the quasi-linear
model is a proper one to study elastic scattering. In the quasi-nonlinear model, only the
wave-wave interaction among the primary waves to produce new waves are included, those
interactions among the secondary waves and between the secondary wave and the primary
wave are all neglected. The quasi-nonlinear model is suitable to study induced diffusion
242
A STUDY OF SECONDARY WAVES GENERATED BY ...
VOL. 37
process. Analysis of the group velocity of a single wave packet is simple, but the analysis
of the group velocities of multiple wave packets propagating simultaneously in the same
wind field can become very complicated. Therefore, a delicate method to analyze the group
velocity of the composite wave packet is required. In this paper, we will concentrate on the
results of linear and quasi-nonlinear models with the mean flow state independent of time.
Because wave-shield interaction will cause a partial reflection of the wave [14,15,16,17,18],
and the superposition of the reflected wave packet on the incident wave packet will complicate the analysis of the wave propagation. By keeping the mean flow independent of time
we can completely eliminate the partial reflection to simplify our analysis. The organization
of this paper is as follows: the numerical models and the methods of analysis are presented
in section 2; followed by a test of the method of analysis by a linear model in Section 3.
The results of quai-nonlinear simulation with mean state independent of time is presented
in Section 4, then a short summary of this paper is given in Section 5.
II. Numerical models and methods of data analysis
II-l. Simulation code
In this paper, we will study the oscillation data in the lower atmosphere generated
by numerical simulation using a numerical code modified from the one described in [13].
Following the previous work of [13], we assume that the motion is two dimensional, the
Boussinesq approximation is valid, the mean state is hydrostatic balance (dP,/dz = -p,,g)
and rotational effects are negligible. The equations of motion are
(14
(Id)
In the above equations, 6 is the potential temperature, vh and vL, are the horizontal and
vertical eddy diffusion coefficients; Kh, and KI, are the horizontal and vertical thermal conductivities; u and zu are the total horizontal and vertical velocities, respectively. An overbar
denotes a mean state quantity and primes denote perturbation quantities. This system of
equations had been numerically studied in [13,19,20,21]. The numerical procedures to simulate different problems can be found in these literatures, and the work described in this
paper, unless specifically stated, generally follows the procedures described in the paper of
PI.
There are two types of data to be studied in this paper: the data generated by
linear model and quasi-nonlinear model. The linear model includes only wave-mean flow
interaction while the mean state is kept constant. In addition to the wave-mean flow
243
F. S. KUO, C. L. LO, H. Y. LUE, AND C. M. HUANG
VOL. 37
interaction, the quasi-nonlinear model further includes secondary wave production through
wave-wave interaction, and the mean state is again kept constant to purity the data for
analysis. The initial mean flow velocity profile and potential temperature profile for all the
gravity wave simulations in this paper are shown respectively in the left panel and right
panel of Fig. 1. A wave or several waves of the following form are forced to oscillate at the
lower boundary:
w(Z,) = F(t)
2 w, cos(h - at) = ;F(t) c w, pk-) + e-i(kz-ct)]
k,a
(24
k,o
where Ic = 27r/X,, g = 27r/T are the horizontal wave number and frequency respectively
of the forcing wave, wk is the oscillation amplitude of the vertical velocity, and F(t) is the
amplitude modulation function
t < Tr = NkT
t 2 T, = NkT
(2b)
where T is the period of the forced-mode wave and Nk is an integer to be given case by
case. All the horizontal phase velocities of the forcing waves are 30 m/see, corresponding to
a common critical level at about 10 km height where the horizontal phase velocity is equal
to the mean flow velocity, and the wave energy will propagate between the ground level and
the critical level. The amplitude modulation function F(t) plays a role of the envelope of
a wave packet. Thus it is the wave packets, not the single plane waves, that are generated
at the ground level. That enable us to evaluate both the phase and group velocities. In all
of the simulations we choose a spectral representation for all the physical quantities of the
following form:
I.
0
20
40
U. (m/g=)
60
I..
I..
1
300 320 340 360 360 400 420
To
09
FIG. 1. Initial mean flow velocity profile (left panel) and potential temperature profile (right panel)
of all the simulations of the gravity wave propagation in the lower atmosphere in this study.
244
A STUDY OF SECONDAR17 WAVES GENERATED BY.
VOL. 37
(3a)
L=-M
where $’ represents the physical quantities such as stream function, vorticity function,
potential temperature and velocity, and cr is the horizontal wave number of the fundamental
(forced) mode, and cr is taken to be cr = 27r x (250 km)-l in all the simulations throughout
this study. Among these quantities, 4: and fl_( are each other’s complex conjugate as can be
seen from Eq. (2a), and the mean state is represented by the state functions corresponding
to 1 = 0. All the vertical derivatives and the time derivatives are represented by finite
differences. Because of such spectral representation in the horizontal variation, we are able
to analyze the wave packets with fixed horizontal wave length.
ln each simulation, the time variation of the velocities v(.z,~) contributed by the
specific 1th horizontal component at 600 consecutive heights z and the fixed horizontal
position z = 0 are calculated from Eq. (3a):
where v~(z, t) is the complex velocity amplitude of the .&h horizontal wave number component at height z and time t. It is obvious that v(z,~) includes the contributions of both the
leftward propagating wave and rightward propagating wave of the same horizontal wave
number !cr. These two oppositely propagating waves can be uncoupled by Fourier analysis
of both the real part and the imaginary part of v;(z,~) (see Section 2.3).
11-2. Method of data analysis
The method of obtaining the vertical phase velocity and group velocity of a wave
packet is as follows: Consider the Fourier analysis of the time series of the atmospheric
velocity at the height z {V(z,ti), i = 1,2,3;..,N} with time resolution St and height
resolution 6z,
v(z,~) = Ae + C(Aj COSU~~~ + B, sinajt;).
I=1
(4)
lf the wave packet under study is composed of a single horizontal wavelength, we can express
v as the combination of plane waves by the following form
NJ 2
v(z,ti) = & •I C
Cj COS(ajti - TL~,Z)
(5)
j=l
where ti = (i- 1)St is the time of the ith time step, aj = 2rj/NSt is the observed frequency,
and nj is the vertical wave number of the jth harmonic. Then from equations (4) and (5).
we obtain
njz = tan-‘(Bj/Aj) = @j(Z).
(6)
F. S. KUO, C. L. LO, H. Y. LUE, AND C. M. HUANG
VOL. 37
245
As the same procedure of analysis is applied to the successive heights z - 6z and z + 6z,
the vertical wave number nj and its corresponding wave length Xj, can be readily obtained
by differentiating the phase aj(z) with respect to z:
=
z = d%(z)
Xj,
%
@j(Z
”
dz
+
6Z)
@j(Z
26z
-
-
6Z)
and the vertical phase velocity vPZ of the corresponding plane wave is given by
VP” =
(8)
Ujfnj.
The vertical group velocity of a wave packet with central frequency urn is known to
be
us2
duna
= -
dn, ’
lf the time series under analysis is such a long data set that its frequency resolution 6cr =
27r/N& is fine enough to resolve the frequencies of the composition waves of the wave
packet, i.e., Sa 5 AU, where Aw is the frequency resolution of the wave packet, equation
(9) can be approximated by
We call this procedure the method of long data scheme in contrast to another method
called short data scheme [2], which was developed to analyze the group velocity of a wave
packet when the available data is very limited. Since the resolutions of both the temporal
and spatial sampling rates of the numerical simulation data can be controlled to meet the
requirement of the present method of data analysis (except near the critical level), we will
use the long data scheme to analyze the simulation data throughout this paper.
11-3. Separation of eastward wave and westward wave
Consider a wave with horizontal wave number k, vL(z, t) = Qk(z, t) . eikx, and make
Fourier analysis of the time series of Qk(z,t) at the height z:
ho . e -t((Jt--6~)
e ikz . Qk(z, t) = eikz .
0
= e ikr
’
+
Bko .
et(nt-6B)
1
(11)
[&(z, t) + i&(2, t)]
where Ak. Bk, bA, and Sg are real functions of the height Z; and,
15~ =nA.z+6a,
6~ =nB-z+ SL.
(12)
Obviously, Al;ei(kz-ot+6a) is a wave propagating eastward with horizontal phase velocity
+a/k; and Bket(kz+ot-6E) is a wave propagating westward with horizontal phase velocity
--a/k. Finally, &(z,t) and Ik(z,t) are respectively the real and the imaginary component
of the velocity generated by simulation:
246
A STUDY OF SECONDARY WAVES GENERATED BY . .
VOL. 37
Ik(-z, t) = z[-Ako . sin(crt - 6,) t Bko . sin(at - JB)]
*
Pb)
And by Fourier analysis of the time series of &(z, t) and Ik (z, t) separately, we have:
c
Q
Rk(t, t) =
(Ckol *
(14a)
cos at + CkC2 . sin at)
By comparing Eqs. (13) and (14) after some algebraic manipulation, it is readily obtained:
A ko ’
=
a (ckd’ t cko2’ t Dkd2
Bko2 = + (ck,12
tan6
t
2Ckdkol)
tcko22 t Dko12 tDko22 t2CkolDko2 - 2Ckc72Dkol)
(154
w4
Cko2 t Dkol
_
A
t Dk02~ - 2CkolDko2
-
tan&g =
C
(164
kol-
Dkaa
ck o 2
- Dkal
Ckal
+ Dko2
PW
From the phases JA and ~7~ given in Eq. (16a) and (16b) we are able to obtain the vertical
wave numbers nA, and ng for the eastward wave and westward wave respectively. Thus the
propagations of these two oppositely propagating waves can be completely separated.
III. Test of analysis method by linear model
It is well known that the dispersion relation of the atmospheric gravity waves (AGW)
derived by the linear theory in two dimensional space is exact, so are the vertical phase
velocity and group velocity of an AGW. Therefore, the simulation of the gravity wave
propagating in the wind field by the linear model provides a good check on the accuracy
of the simulation code, and the result of the wave propagation velocities obtained from the
linear simulation data can be used to evaluate the reliability of the method of analysis.
The dispersion equation of the 2-D gravity wave model with vertical shear, Boussinesq
approximation and no earth rotation, is given in [22]:
(17)
with
w = o - kuo,
(18)
F. S. KUO, C. L. LO, H. Y. LUE, AND C. M. HUANG
VOL. 37
247
where wb = 1.2247 x 10-2s-’ is the Brunt-Vaisala frequency taken in the simulations; u. and
ui are the mean flow velocity and its second derivative with respect to height; n and Ic are
the vertical and horizontal wave number; w and g are the intrinsic and observed frequency,
respectively. The vertical phase and group velocities observed in the fixed reference frame
are obtained from (17) as following:
VP” =
u/n,
(19)
These two equations (19) and (20) will be compared with the result of simulations. It is
worth to note that the vertical group velocity observed by an observer tied on earth is same
as that observed by an observer co-moving with the mean wind as shown in Eq. (20), while.
n )ob served by an observer moving at the same speed of
the vertical phase velocity (= w/
the mean wind, is different in amplitude from that (a/n) observed by an observer fixed on
earth such as VHF radar. Also, when the horizontal phase velocity a/k exceeds the mean
wind speed uo, the two vertical phase velocities a/n and w/n are in the same direction (see
Eq. 18 and 19).
In the linear simulation case LS, the wave with horizontal wave length 41.67 km
(corresponding to wave-number !cr with ! = 6) and eastward phase velocity 30 m/s (corresponding to frequency ~7 = 4.52 x 10-3s-’ and wave period of 23.15 min) is forced to
oscillate 24 cycles (nTk = 24) with forcing amplitude wk = 0.001, corresponding to 6 cm/s
of the vertical velocity amplitude. The vertical grid space size and the time step of the simulations are AZ = 33.3 m and At = 2.2 set respectively. The vertical range of simulation is
6OOAz = 20 km. Fig. 2 shows the horizontal velocity profile generated by the simulation,
from which we can see that the wave amplitude is significantly enhanced as it approaches
the critical level (at the height z = 10 km), and almost no oscillation is found above the
critical level. This is a typical phenomenon of dynamic instability. The data from the 1st
to 19035th time steps (total time interval = 41877 see) of the profile of Fig. 2 is analyzed by
long data scheme to obtain the vertical phase and group velocities at each height. Examples of the frequency spectra at each height are shown in Fig. 3, from which we observe the
maximum power at the 3~)‘~ Fourier mode, corresponding to an oscillation period of 1395.9
set (23.27 min). Since the horizontal wave length of the forcing wave is about 41.67 km,
we obtain the horizontal phase velocity of the main oscillation period of the wave packet
to be 29.85 m/set, which is only slightly different from the phase velocity of the forcing
wave. So, we pick the 27th w 33rd Fourier modes to calculate their phase-height relations
(Fig. 4) from which the corresponding vertical wave numbers are obtained by Eq. (7). The
frequency dependence of the vertical wave number for these modes are demonstrated in
Fig. 5. The vertical phase velocity of the 30th Fourier mode at each height can be readily
obtained, and the slopes of these curves at the 30th mode are calculated by the method of
least square fitting to give the vertical group velocities at each height. The results of the
phase velocity profile and group velocity profile of the eastward wave packet are plotted in
the right panel and left panel respectively in Fig. 6 along with the theoretical values (thin
lines) calculated from Eqs. (19) and (20). The agreement between the simulation and the
248
VOL. 37
A STUDY OF SECONDARY WAVES GENERATED BY .
01.
0
y,.
1.0*104
2.0~10~
,
3.0~10~ 4.0X104
Trne (SC)
,,,
5.3~10~
6.0~10~
FIG. 2. Height profile of the perturbation horizontal velocity from z = 0 km to z = 13 km generated
by the linear simulation Case LS. The profile above the height of z = 13 km is not shown
here because the oscillation amplitudes at these heights are negligibly small.
10-6/
0
40
20
Mode
60
10-81
0
20
40
80
Mode
FIG. 3. Examples of the frequency spectra of the oscillation data of Fig. 2 at two heights.
theory is nearly perfect, so both the accuracy of the simulation and the reliability of the
method of long data scheme are justified. In addition, the amplitudes of the westward wave
at each height is found to be three orders of magnitude smaller than that of the eastward
wave as expected. When we raised the forcing amplitude Wk from 0.001 to 0.002 and 0.004,
we obtained the identical results regardless of the forcing amplitude. This is the result of
the assumption of the model that the mean flow state is independent of time.
IV. Quasi-nonlinear simulations on wave-wave interaction
IV-l. Quasi-nonlinear model with mean state time independent
By the convection terms G.‘?z7 in Eqs. (la) and (lb) and z7.G6’ in Eq. (Id), new waves
can be produced through wave-wave interaction. A self-consistent numerical simulation
249
F. S. KUO, C. L. LO, H. Y. LUE, AND C. M. HUANG
VOL. 37
Mode=30
Mode=29
:_/fl
0
2
4
IleqhL
6
6
(Km)
Mode=31
j&J
0
10
4
6
6
2
Helght ( K m )
:;m
10
0
4
6
0
2
Height (Km)
10
FIG. 4. The phase - height relation of three consecutive Fourier modes around the central frequency
( i.e. z 4.5 x 10-3s-1, Mode = 30) of the wave packet generated by the linear simulation
case LS.
2=1.33 Km
Z=2.67 Km
Flip/
- 6
- 2
ii/j1
-6
- 2
K (l&n)
K (l&m)
FIG. 5. Examples of the frequency N vertical wavenumber relation around the central frequency at
two heights for the case L S .
15
? lo
5 5
t
>-
0
- 5 I.
0
6
8
2
4
Height (Km)
10
0
6
2
4
6
Height (Km)
10
FIG. 6. Vertical group velocity (left panel) and vertical phase velocity (right panel) of the gravity
w a v e ( w i t h X, = 41.67 km and r = 1388.8 set) as functions of height for the linear
simulation case LS. The thin lines are obtained from the gravity wave theory Eq. (19) and
(29).
250
A STUDY OF SECONDARY WAVES GENERATED BY
VOL. 37
should include not only the wave-wave interaction and wave-shear interaction, but also
time dependence for the mean state. Any new wave produced by wave-wave interaction
will in turn interact with other waves, and a wave-shear interaction with the time dependent
mean state will result in a partial reflection. A numerical model which includes all these
processes is called a nonlinear model. The merit of a nonlinear model is its ability to
simulate the reality. But it is difficult to study the characteristics of any specific process by
a non-linear model because of the interplay among different processes. So we design a quasinonlinear model to isolate the specific process of interest from all the other processes. By
this model we are able to study the characteristics of the secondary wave packets produced
by wave-wave interaction.
By quasi-nonlinear model, we force two primary wave packets characterized by (pi, k,)
and (02, IcZ) to oscillate at the ground level, where cr and k are understood to stand for the
observed frequency and horizontal wave number respectively. As these two primary wave
packets propagate upward, two secondary wave packets with horizontal wave numbers lia
and k4 will be created according to the convective terms in Eqs. (la) N (Id) such that
k3 = kz - kl
k4 = kz + kl.
(21)
Then all the following interactions are truncated by forcing these new wave’s amplitudes
to be zero at each time step of simulation:
Therefore, only two primary wave packets associating with ICI, k2 and two secondary wave
packets associating with k3, k4 are active throughout the simulation of the quasi-nonlinear
model. The vertical wave numbers n1,n2,n3,n4 and the frequencies o~,o~,o~,~~ must be
obtained by analyzing the propagation data of their corresponding wave packets following
the same procedures described in Section 3.
Since the self-consistent wave-mean state interaction (i.e. with the mean state time
dependent) will result in wave reflection from the wind shield, and the reflected wave packet
will have same frequency and horizontal wave number as the incident wave packet. The
superposition of the reflected wave packet on the incident wave packet will complicate and
even confuse our analysis of the wave propagation. For this reason, we further simplify
our simulation by keeping the mean state time independent. Under such conditions, the
propagations of the two primary wave packets are exactly same as they will behave in
the linear model, i.e. the vertical phase velocities are downward and group velocities are
upward with the amplitudes consistent with the prediction of the linear theory. This latest
statement is repeatedly confirmed by the results of analysis on the primary waves (not
presented in this report) in all the quasi-nonlinear simulations.
IV-2. Propagation velocities of secondary wave packets
In the first case of simulation (Case QNL-l), the horizontal wave numbers of the two
primary waves are set to be:
VOL. 37
;
i
6
._
TfJ
.E
4
F. S. KUO, C. L. LO,
H. Y. LUE,AND C. M. HUANG
251
1
0
2x104 4 x 1 0 4
0
6~10~
T i m e (Set)
6~10~
FIG. 7. Height profile of the perturbation
horizontal velocity from t = 0 to z =
10 km of the larger secondary wave
03, Its) generated by the wave-wave
interaction in the Quasi-Nonlinear
simulation Case QNS-1.
2lr
=7&=7X-.
250 km’
Time (Set)
FIG. 8. Same as Fig. 7 except for the smaller
secondary wave (~4, /cd) in the height
range between z = 0 to z = 6.6 km.
k2=8a=8x&
and the central frequencies of the corresponding wave packets are:
o1 = k1 x 3 0 mlsec;
o2 = k2 x 3 0 mlsec.
Correspondingly, the horizontal wave numbers of the two secondary waves are:
k,=k2-k,=Ix&;
k4 = k2 $ k1 = 15 x &.
The oscillation profiles of the large secondary wave packet (which associates with k3) and the
small wave packet (which associates with k4) are shown in Fig. 7 and Fig. 8 respectively. To
analyze the propagation velocities of the secondary wave packets, we analyze the oscillation
profiles just as we did to the case of linear simulation to obtain,
03 2
27r
1.2 x p.
250 km’
2?T
cT‘$ = 15 x ~
250 km’
The resulting vertical phase and group velocity profiles of the wave packets associated with
k3 and k4 are shown in Fig. 9 and Fig. 10 respectively. Here we observe that the central
frequency of the smaller wave equals to the sum of the central frequencies of the two primary
252
VOL. 37
A STUDY OF SECONDARY WAVES GENERATED BY ...
-4’;
0
1
2
4
6
6
0
Height (Km)
2
4
6
6
Height ( K m )
2
4
6
6
10
Height (Km)
~_~~
Cl
1
- 4 0 I*
‘1 0
~~~~~
10
0
2
4
6
6
Height (Km)
10
(b)
FIG. 9. (a) Vertical group velocities (left panel) and vertical phase velocities (right panel) of the
larger secondary wave packet (03, Jzs) as a function of height at the earlier time period when
the primary waves are still oscillating at the ground level. (b) Same ot Fig. 9a except at
the later time period after the primary waves have stopped oscillating at the ground level.
-51
0
2
4
6
6
Height (Km)
10
- 2 0 I..
0
2
4
6
6
Height (Km)
10
FIG. 10. Vertical group velocities (left panel) and vertical phase velocities (right panel) of the
smaller secondary wave packet (~4, k4) as a function of height.
.ri:_. :
F. S. KUO, C. L. LO, H. Y. LUE, AND C. M. HUANG
VOL. 37
0
-20
-40
__”
,:’
_:’
,:’
..:
253
,.,’
_6o ,..:....
-BO/
0
2
4
6
Height (Km)
B
10
FIG. 11. The vertical phase velocity of the smaller secondary wave packet as a function of height
(black dots) along with two theoretical curves. The solid curve is obtained from the
assumption that the secondary wave is produced by a resonant wave-wave interaction..
The dashed line is obtained from the linear gravity wave theory.
wave packets, i.e., u4 = ~1 + ~2; while the central frequency of the larger wave is slightly
larger than the difference between the central frequencies of the two primary wave packets,
namely, us !Z l.2(g2 - cri). Besides, all the amplitudes of the westward waves are found to
be negligible comparing with that of the corresponding eastward waves.
Fig. 10 reveals that the vertical phase velocity of the small wave packet is downward
at each height (right panel) and the vertical group velocity is upward at almost every
height (left panel). Such behavior is exactly same as that of the two primary wave packets.
To investigate the characteristics of the small wave (u4,k4), we redraw its vertical phase
velocity profile (denoted by dots) along with two theoretical curves in Fig. 11. Where
the dashed line is the vertical phase velocity obtained by substituting g4, and k4 into the
dispersion relation (Eq. 17) to calculate the vertical wave number n4 and the phase velocity
cr4/n4 at each height; and the solid line is the vertical velocity profile of a secondary wave
(gi + cr2, kl + k2,n1 $ n2) produced by a resonance interaction between the two primary
waves (gi, kl, nl) and (c2, k2, n2):
VP"
01t 02
= ____
nl tn2
(22)
Fig. 11 clearly shows that the vertical phase velocity profile of the secondary wave (u4, k4)
is closely matched with the solid line, that means n4 g n1 + n2 at each height, so the
generation of the smaller secondary wave packet charaterized by (u4, k4) is a process of
resonance interaction. It further indicates that, from the large difference between the
dashed line and the solid line, this secondary wave doesn’t follow the dispersion relation of
the linear theory. This is not a surprising matter because the dispersion relation (17) is a
second order function of n and k.
The characteristics of the larger secondary wave packet characterized by (ua, k,) are
quite different from the smaller one. The smaller wave packet oscillates only during the
two primary waves were oscillating at the ground level (see Fig. 8); while the larger wave
packet keeps oscillating even after the two primary waves have stopped oscillating at the
ground level for a long time. This difference is due to the effect of energy dissipation: as
A STUDY OF SECONDARY WAVES GENERATED BY . . .
254
Cl
2x104
6~10~
4x104
Tome (SW)
6X104
FIG. 12. Partial height profile of the perturbation horizontal velocity from t = 9.33
km to z = 10 km of the primary wave
(cl, ki) in the Quasi-Nonlinear simulation Case QNS-1.
4x104
6~10~
8X104
Time (Set)
VOL. 37
1x106
FIG. 13. Partial height profile of the perturbation horizontal velocity from z = 9.33
km to z = 10 km of the larger secondary wave (63, ks) generated by the
wave-wave interaction in the QuasiNonlinear simulation Case QNS-1.
revealed in Eqs. (la), (lb) and (Id), the energy dissipation of the gravity wave is proportional to the square of its wave number, since the wave number of the smaller wave is larger
than the larger wave by approximately one order of magnitude, the energy dissipation of the
smaller wave will be larger than that of the larger wave by about two orders of magnitude.
The gravity wave will gain energy from the wind shield as it propagates upwards in the
wind field, the closer the wave approaches to its critical level, the more energy it will gain
from the wind shield [23]. Fig. 12 shows the oscillation profile of one of the primary wave
(oi, Ici) near the critical level, it reveals that this primary wave oscillates much longer when
it is near the critical level than when it is near the ground level. Fig. 13 shows the later part
of the oscillation profile of the secondary wave (os, ks) near the critical level. We can clearly
see that the phase velocity is upward before the time of 5.5 x lo4 see and become downwards
after the time of 6.6 x lo4 sec. Therefore we divide the oscillation profile into two parts
N earlier part before and later part after the primary waves stop oscillating at the ground
level N to analyze its propagation velocities separately. The propagation velocity profiles
of the earlier part and of the later part of the oscillation data are respectively shown in
Fig. 9a and Fig. 9b. They show that the group velocity is always downward at each height;
while the phase velocity is upward at the earlier time of simulation and become downward
at later time. Conferring with the oscillation profiles of the primary wave such as Fig. 12,
we conclude that when the primary waves are still oscillating at the ground level, the large
secondary wave paclet generated by their interaction has downward group velocity and upward phase velocity, when the primary waves no longer oscillate at the ground level. the
larger secondary wave packet is produced mainly near the critical level with both its group
velocity and phase velocity downward. Consequently, a standing wave-type oscillation
F. S. KUO, C. L. LO, H. Y. LUE, AND C. M. HUANG
VOL. 37
1st Mode
-7th Mode
260
255
0th M o d e
zoo
150
100
0
F I G.
2
4
e
8
Helpht (Km)
10
0
2
4
e
8
Helphl (Km)
10
50
0 L/I
0
2
4
6 8
10
HslphL (Km)
14. Integrated oscillation energy as a function of height of the wave packets. Left panel: of the
larger secondary wave (~3, Its); Middle panel: of the primary wave (~1, kr); Right panel:
of the primary wave (c~2, Icz).
pattern emerges as shown in Fig. 14, which shows the height profiles of the integrated (over
time) oscillation energy of the secondary wave (us, ks) along with that of the two primary
waves. The middle and right panels reveal that the oscillation energy of the primary
waves are increasing with height, and reconfirm the theory of [23] that the wave will gain
energy from the wind shield as it propagate upward. ln contrast to the primary waves,
the oscillation energy profile of the larger secondary wave (left panel) appear to exist two
nodes and one antinode between them, since the ground level is the energy source and the
critical level would supply the maximum energy to the waves, our system of simulation for
the larger secondary wave is similar to an open pipe for the standing wave. Furthermore,
we have obtained similar result by a second case of simulation (Case QNL-2), in which the
two primary waves are given as follows:
27r
k,=3cY=3x--250 km’
k2=4a=4x&
and
crz = kl x 30 mlsec;
g2 = k2 x 30 mlsec.
V. Summary
In this paper we have examined the vertical group velocity of the atmospheric gravity
waves of various origin in a wind field with vertical shear by the method of numerical
simulation. Two numerical models have been applied for this study: one is called linear
model (LS model) which is used to check with the linear theory of gravity wave; the second
one is called quasi-nonlinear model (QNL model) which aims to study the propagation
characteristics of the secondary wave packets produced by wave-wave interaction in the
wind field. ln order to isolate the specific process under study from the feedback effect of
the wind shield motion, we artificially keep the mean state independent of time. Under the
basic assumption that the wave-wind field interaction does not change the horizontal wave
length and the frequency of the wave, we have investigated the group velocities of the wave
packets propagating in the wind field under different conditions.
256
A STUDY OF SECONDARY WAVES GENERATED BY . .
VOL. 37
Prom the result of the linear simulation, we found that the vertical phase velocity and
group velocity of the gravity wave are consistent with the gravity wave theory, implying that
the gravity wave packet will execute no reflection from the stationary wind field. Therefore,
the condition of the time independence of the mean state forced in the simulations has
clarified the wave propagation from the interference by the reflected wave.
By quasi-nonlinear simulation, two primary wave packets are forced to oscillate at
the ground level. As these wave packets propagate upward, two secondary wave packets are
generated: a small wave (small horizontal wave length and short period) and a large wave
(large horizontal wave length and long period). The vertical phase velocity of the small wave
packet is always downward while its group velocity is always upward just the same as the
two primary waves. In contrast to the smaller secondary wave, the vertical group velocity
of the larger one is always downward, opposite to that of the two primary waves; while its
vertical phase velocity is upward (opposite to the vertical group velocity) when the primary
waves are still oscillating at the ground, and becomes downward (same as the vertical group
velocity) after the primary waves stop oscillating at the ground. The smaller wave is found
to be the production of a resonant interaction, while the larger one is a production of a
non-resonant interaction. Both secondary waves do not satisfy the dispersion relation of
the linear gravity wave theory. And consequently, the vertical phase and group velocities of
the secondary waves are not necessarily opposite to each other (which is the prediction of
the linear theory). This result support our earlier conjecture [2] that some events observed
by Chung-Li Radar, with vertical phase and group velocities in the same direction, might
be resulted from non-linear process.
In the quasi-nonlinear simulation, the two primary waves are purposely selected to
have such close frequency and wave length that the difference between their propagation
velocities is small. Because then the two primary wave packets can stay close together
for a long time to interact with each other to produce secondary waves with significant
amplitudes. The most interesting result is the production of the large secondary wave out
of a non-resonant interaction. It has large wavelength and long wave period, the closer
the frequencies and wavelengths of the primary waves are, the longer the period of the
secondary wave is. The confinement of the two primary waves in a finite region (e.g.,
between ground and the critical level) has resulted in the standing wave-type behavior of
this secondary waves N its energy is concentrated in some limited region. In the real world,
the concentration of the oscillation energy of a big wave in a limited area can be very
devastating. It is well-known that Yan-Tze River of China has been flooded by heavy rain
approximately once every ten years. Is this a result of some kind of non-resonant wave-wave
interaction?
Acknowledgment
This work is supported in part by National Science Council under the contract number
NSC 87-2111-M-008-012-AP9.
References
[ 1 ] C. 0. Hines, Can. J. Phys. 38, 1441 (1960).
[ 2 ] F. S. Kuo, et al., J. Atmos. Terr. Phys. 55, 1193 (1993).
VOL. 37
F. S. KUO, C. L. LO, H. Y. LUE,AND C. M. HUANG
257
[ 3 ] P. Muller and D. J. Olbers, J. Geophys. Res. 80, 3848 (1975).
[ 4 ] D. J. Olbers, J. Fluid Mech. 74, 375 (1976).
[ 5 ] C. H. McComas and F. P. Bretherton, J. Geophys. Res. 82, 1397 (1977).
[ 6 ] C. H. h/IcComas and P. Muller, J. Phys. Oceanogr. 11, 139 (1981a).
[ 71 C. H. McComas and P. Muller, J. Phys. Oceanogr. 11, 970 (1981b).
[ 8 ] K. C. Yeh, and C. H. Liu, Radio Science 5, 39 (1970).
[ 9 ] K. C. Yeh and C. H. Liu, J. Geophys. Res. 86, 9722 (1981).
[lo] K. C. Yeh and C. H. Liu, Radio Science, 20, 1279 (1985).
[ll] P. Dong and K. C. Yeh, J. Geophys. Res. 93, 3729 (1988).
[12] K. C. Yeh and B. Dong, J. Atmos. Terr. Phys. 51, 45 (1989).
[13] C. M. Huang, et al., J. Atmos. Terr. Phys. 54, 129 (1992).
[14] W. L. Jones, J. Fluid Mech. 34, 609 (1968).
[15] R. J. Breeding, J. Fluid Mech. 50, 545 (1971).
[16] R. S. Lindzen and K. K. Tung, Mon. Wea. Rev. 104, 1602 (1976).
[17] R. S. Lindzen, Pure Appl. Geophys. 126, 103 (1988).
[18] W. Blumen, J. Atmos. Sci. 42, 2255 (1985).
[19] H. Tanaka, J. Meteor. Sot. Japan, 52, 425 (1975).
[20] M. A. G 11
e er, H. Tanaka, and D. C. Fritts, J. Armos. Sci. 32, 2125 (1975).
[21] D. C. Fritts, J. -4tmos. Sci. 35, 397 (1978).
[22] E. E. Gossard and W. H. Hooke, Waves in The Atmosphere, (Elsevier, New York, 1975),
p.124.
[23] F. S. Kuo, ei al., J. Atmos. Terr. Phys. 54, 31 (1992).