- Email: [email protected]
Off-resonant amplification of finite internal wave packets
Dynamics of Atmospheres and Oceans, 2 (1977) 83--105 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
83
OFF-RESONANT AMPLIFICATION OF FINITE INTERNAL WAVE PACKETS
A.D. McEWAN and R.A. PLUMB C. S. I. R. 0., Division o f Atmospheric Physics, Aspendale, Vic. (Australia)
(Received March 4, 1977; revised June 10, 1977; accepted July 15, 1977)
ABSTRACT McEwan, A.D. and Plumb, R.A., 1977. Off-resonant amplification of finite internal wave packets. Dyn. Atmos. Oceans, 2: 83--105. It is shown by experiment and theory that finite fine-scale internal wave packets in a continuous stratification can be amplified by the passage through them of a longer wave of higher frequency. In general the process is a detuned resonant interaction, but the finiteness of the packet provides sidebands which allow enhancement of those components of the packet lying most closely to exact second-order triad resonance with the forcing wave. The amplitude of the forcing wave required for amplification of the packet against viscous dissipation is therefore that appropriate to the nearest resonant mode, and may be significantly less than that for a monochromatic mode with the same central wavenumber. The tendency in an evolving internal wave field is for the packets to develop as beams extending along the characteristic directions of free wave modes.
1. INTRODUCTION E v e n at m o d e s t d i s t a n c e s f r o m i d e n t i f i a b l e sources, t h e average i n t e r n a l w a v e s p e c t r u m o f t h e d e e p o c e a n is n o t a b l y u n i f o r m (Wunsch, 1 9 7 6 ) i m p l y ing an e f f i c i e n t a n d universal p r o c e s s f o r c a s c a d i n g energy. R e s o n a n t w a v e i n t e r a c t i o n (q.v., Phillips, 1 9 6 6 ) is t h e p r i m e s u s p e c t , a n d b y n u m e r i c a l exp e r i m e n t s t h r e e classes o f such i n t e r a c t i o n h a v e b e e n i d e n t i f i e d as d o m i n a n t (McComas and Bretherton, 1977) within the representative spectrum (Garrett and Munk, 1972, 1975). N o w while this s p e c t r u m is b y all e v i d e n c e a g o o d r e p r e s e n t a t i o n o f t h e t i m e - a v e r a g e d i n t e r n a l w a v e m o t i o n field it c a n n o t b e e x p e c t e d t h a t at o n e p a r t i c u l a r t i m e a n d p l a c e in t h e o c e a n t h e m o t i o n e x p e r i e n c e d is c o m p o s e d o f a c o n t i n u u m o f w a v e n u m b e r s a n d f r e q u e n c i e s , b u t is m u c h m o r e likely t o c o m p r i s e o n l y a f e w m o d e s w h i c h are passing t h r o u g h in m o v i n g p a c k e t s o f finite size. I f t h e s p e c t r u m w e r e c o n t i n u o u s t h e w h o l e c o n t i n u u m o f m o d e s c a p a b l e o f s e c o n d - o r d e r triad i n t e r a c t i o n w o u l d b e p r e s e n t s i m u l t a n e o u s l y , b u t in reality t h e c o i n c i d e n t arrival o f even o n e t r i a d o f m o d e s c a p a b l e o f i n t e r a c t i n g r e s o n a n t l y is a rare event. A l m o s t i n v a r i a b l y t h e initial interac-
84 tions will be independent ones between each of the modes instantaneously present. In view of the fundamentally non-linear nature of the problem it is therefore by no means clear a priori that continuum triad wave interaction analyses such as those of MiJller and Olbers (1975) and Olbers (1976) formulated in general by Hasselmann (1967) can be used to evaluate the mean spectral transfers in terms of the representative spectrum. The evolution of the spectral wave action density a(k) is governed by an equation of the form :
Da(_k) Dk
:ffs(k_, .k' . k")a(k')a(k")d3k ' d3k '' . . .
(e.g. Olbers, 1976) where S is the transfer coefficient. If ( > denotes an "ensemble" (time or space) average, then the continuum approach gives a transfer of the form:
ffs(k_, k_',_k")d3k ' d3k '' While such an approach is clearly valid for a stationary spectrum, the extent to which it is affected by variations in spectral density in space and time is not known. It is worthwhile instead to consider in detail the nature of an individual interaction event. The interaction of a pair of free wave packets forces, at second order, waves composed of sum and difference wavenumbers and frequencies. If it happens that either of these approximate a resonant free mode defined by the dispersion relation, that free mode may amplify to form the third member of an interacting triad, and significant energy from the highest-frequency member may proceed so long as the p a c k e t s share c o m m o n space. If the medium is u n b o u n d e d the waves are able to disperse at their respective group velocities. It might then seem that the modes most readily forced by this kind of 'occasional packet' interaction would be those of finest scale, since they have the lowest group velocities, and so would remain longest in interactive proximity. Furthermore, a fine wave interacting with a long wave will force another fine wave of nearly equal and opposite wavenumber to the first, and so will disperse in a nearly equal and opposite direction. The two waves together can then constitute a 'beam' growing both upwards and downwards along this direction. In order that these waves be amplified resonantly the low-wavenumber packets encountering them are required to have a frequency roughly twice that of their own natural frequency. In the limit of zero wavenumber the growth of the fine waves reduces to a 'parametric' instability, as shown by McEwan and Robinson (1975), on which basis it was suggested t h a t even the finest observed microscales in the ocean could be sustained directly by the average action of the large-scale internal wave field. A similar p h e n o m e n o n was suggested by Orlanski (1973) as a mechanism by which internal waves may be generated in the lower layers of the atmosphere.
85 Although the finest-scale waves may, due to their slow dispersion, be those most favoured for quasi-resonant parametric forcing, in a real fluid viscous damping may negate this advantage. In general therefore it can be seen that the capacity for interactive energy cascade between a pair of discrete internal wave packets is determined b y three things: the degree to which they approximate members of a resonant triad, the size of the packets, and their natural 'viscous' attenuation rate. In a b o u n d e d medium the wave modes are discrete, affording a means of quantifying the degree of detuning from precise triad resonance. Plumb (1977) has dealt with the constraints on interaction arising purely from the detuning, while Mander (1974) has defined instability thresholds and evolution of viscously damped, b o u n d e d standing waves under restricted conditions. In the following investigation the criterion for amplification of u n b o u n d e d packets of damped internal waves under the action of a stronger higher-frequency wave are derived and compared with predictions of the simpler parametric instability theory. A laboratory experiment is described which confirms the major predictions and gives emphasis to the importance of internal wave interaction in distributing internal motion within the ocean. For simplicity, the study is restricted to uniform fluids with, e.g., constant b u o y a n c y f r e q u e n c y
2. THEORY The suggestion made in McEwan and Robinson (1975) was that packets of the finer, lower-frequency internal waves in the ocean were energized continually by the passage through them of larger-scale internal waves of approximately double frequency. In conformity with this idea and the conditions of the present experiment we seek the amplitude of a uniform large-scale wave above which a given small-scale wave packet would be amplified, and an understanding of the way the packet would evolve. 2.1. P a c k e t s f o r m i n g e x a c t triad resonance
Consider for convenience a two-dimensional wave field in a medium of constant b u o y a n c y frequency N. In terms of the'stream function • and b u o y a n c y perturbation o:
V21~tt + N 2 ~ x x = J(¢2, V2 ~J)t -- J ( ~ , a t - - ~ x N 2 = J ( ~ , o) + VsV2a
o)x
--
vsV20~ + vV 4 ~t
(2.1) (2.2)
where V2 = ~2/~X2 + ~2/~Z2, a -~ - - g P / P O , N2 = --gPoz/Po, P is density, v is kinematic viscosity, vs is diffusivity of the stratifying constituent, (~/~z, a / ~ x ) = (u, --v) and subscripts denote partial differentiation by horizontal distance x and time t. Motion is taken to be defined in terms of t w o discretely separable scales of
86
space and time, written: T) + O(e 2)
¢2 = e p ( r , t ) ~ ( R ,
a = es(r, t ) Z ( R , T) + O(e 2)
(2.3)
where e is some dimensionless measure of amplitude, r and t are real position vector and time respectively over which a periodic wave-like structure is identified and R and T are expanded scales defined by : R = (X, Z) - er = (ex, ez)
(2.4) T =- e t
The space scale for a packet of internal waves embedded in an extensive fiela of plane waves of higher frequency, and the time scale for the evolution of this packet are characterized by these expanded scales. With gradients in (2.1) and (2.2) defined: a a a -
a
at a
ax
ax
Ot
+ C -
aT a aX
V = V+eV then to leading order, providing ~, v~ ~< O(e): V2 ~ t t + N 2 ~ x x = 0
Ot - - N 2 ~ x = 0
which has the form of free internal waves varying weakly in space and time: = ~ q]~(R, T) exp[i(K, - r - - wnt)] n
(2.5)
o = ~ Z~(R, T) e x p [ i ( ~ . r - -
~nt)]
n
and their complex conjugates. Wave frequency co and wavenumber ~ are connected through the dispersion relation: ~92 = N 2 k 2 / i ~ i
2
(2.6)
in which ~n = (hn, mn) where kn and m , are horizontal and vertical wavenumbers *. Also, by (2.2): ~'n = - - N 2 hn xI~n/ COn
(2.7)
* As a matter of notation K will be used for wavenumbers of order unity (on a scale ILl ) i n the unstretched coordinate x, while K will be used to represent wavenumber increments of order unity i n t h e "slow" coordinatOR.
87 To next order in e:
272 ~tT
+ 2~xXtt + 2t~zZtt + 2N2~ x
j(@,
=
2
(Res)
V ~)t
-- g(¢,
(/)(Res)
VV4~t --Vsq2ax
(2.8)
+
where (Res) denotes resonant forcing terms, such that Z~i = 2:¢o(~i) = 0, where j denotes the (three) modes participating at second order in the interaction and ¢o(~i ) is given by (2.6). Non-resonant terms appear at higher order. Then with (2.5), (2.6) and (2.7):
+ C.. ~ + e-iX.
~I,~ = iS.~I,;~I,~
(2.9)
Here:
(2.10)
X,, = (t, + r,~)1~1/2 denotes a damping coefficient and: _
¢Omn
(2.11)
is the group velocity of mode n. Sn is an interaction coefficient for the forcing of mode n by modes p and q:
Sn = (2E2C~n)-l(kq mp -- kp mq)[w,(Kqz -- ~2) + knN2(kq ¢oql __ kp COp1)] (2.12) In the c o n t e x t of the present experiments where the long wave field has the form: ~)1 = A1 sin(klX -- COxt) sin m l z the appropriate interaction coefficient is quartered in magnitude, viz.:
S(: ) = (SEn2 G)n) - 1 (]~q m l - - k l m q ) [ ¢ o , ( ~ 2 - - ~21) + k , N U ( k q (~71 - - k l co11)] (2.12a) where n = 2 or 3 and q = 3 or 2. If @2 and @3 are small perturbations to an extensive background wave field @1 for which O@ I / a X = O ~ / 1 / a T ~ O, then from (2.9):
(~V + C2" ~ + X2/e)¢2 = iS2 ¢~ ¢~ (2.13)
)
88 or combining:
~-~+ C 2 " ~ + ~
~--~ + C3" ~ +
~2 = 82S3 ~]2l~ll 2
(2.14)
Now supposing that at T = 0:
~ ( R , 0) ; f ~(K) e ' ~ ~ dK
(2~15)
and thereafter that:
a(K, T) = a(K) exp[AQK) -- iFt(K)IT
(2.16)
for each K within the spectrum defining the profile of the wavepacket 2. Here A and ~ are real; A is the growth rate of the disturbance on the long time scale T, while ~ is an O(e) correction to the wave frequency. Then:
C2"K-- ~ ) - - i
A+-e
Cs ' K - - g t ) - - i
+
+$2S31~112=0 (2.17)
or close to marginal stability where I Ae/Xl < < 1: Ft=\~--(X2+XS+2A)-l[(Xs+a-e- ~C2_ + ( X C 2 s+ AI )' K T ~
(2.18)
and the growth rate: A = e 8 2 8 3 ( 1 ~ 1 1 2 - 1 ~ c f 2) (X2 + Xs) - 1
where:
e2[ ~IJc[2 = ($283)-1~,2~3 [ 1 + [(~C2 --~C3)" eg[ 2 (~,2 + ~,3)- 2 ]
(2.19)
Thus the instability is sustained if the amplitude ~1 of the 'background' wave exceeds ~¢. Contributions to ko¢ arise both from the magnitude of the viscous attenuation against which the interaction has to work, and the effect of wave dispersion, since the disturbance waves diverge at a speed [C 2 - - C3[, so the time for cooperative interaction is limited by the size of the packet to IC2 -
C3i" le~
.
2. 2. Detuned packets
If the central wavenumber of a pre-existing packet 2 is not in exact triad resonance interaction with the background wave, the resultant wavenumber detuning may be absorbed into the definition of K in (2.15) and thereafter. Assuming that wave 3 grows from the interaction between the background field wave 1 and the packet 2, wavenumber closure is guaranteed, i.e. : t p NK3 = - - - - ~ 1 - -
t ~2
89 but t t C0(51) + 50(52)
+ G 3 ( 5 3 ) =/= 0
where primes denote the pre-existing, non resonant values. These must lie close to the resonant values such that: c°(51) + co(52) + co(--51 - - 5 2 ) = 0
(2.20)
for significant energy exchange to take place. Writing: t
~2 = ~2 + e~K2
(2.21)
the .wavenumber detuning may be regarded as a long-wave modulation upon a wave of wavenumber ~2. If at T = 0:
t
t
where ¢o2 = ¢o2(52) then if this is equivalent to a suitably modulated wave:
resonant
xIJ2 = a2(R , O) exp[i(52 • r - - ¢02t)] a2 and a 2 become related in K. Specifically, from (2.15): a2(R , 0) = f , ' (/() e 'K .n ch~' then, by (2.4): a2(R , 0)
=fo~'(K')
e'(~'+~K~2)'~ dK' (2.22)
= f a ( K ) eiK~"X dK' where: a(K)
(2.23)
a' K - - 6 K ° )
A non-resonant packet with a given spectrum would therefore be directly equivalent to a resonant packet whose spectrum is the same shape but displaced in wavenumber K an a m o u n t 5K2. A difficulty in applying this result is that the appropriate resonant m o d e cannot be defined a priori in an unbounded system, and it is useful to consider the interaction in wavenumber space 5 = (k, m). For any given wave 1 a locus L(51, 5) may be drawn in 5, as depicted in Fig. l a defining a continuum of waves capable of exact triad resonance satisfying (2.20). If a given resonant wave number 52 is increased a small a m o u n t along this locus: co(~2 + 7 ) + c o ( ~ 3 - - 7 ) + ~ ( ~ )
= 0
90
(a)
K~ K2J/ ~Porometric //~!t~X~' Asymptote --
Fig. 1.a. Resonant locus L(K1, g) in wavenumber space. A specified detuned wave ~ defines a resonant ~2 by the condition that e~K 2 is normal to L.
so as i~i -~ O:
(c2 -
C s ) . ~ ~ o~
Therefore (C 2 - - C3) is always normal to L. For a given non-resonant wave packet with central wavenumber ~ , contours of a'(K') can be constructed in S as shown in Fig. l b . By defining e~K2 = K~ -- ~2 as being normal to L, ~2 becomes specified, and it is then possible to construct contours of A(S2, e~K2, ~ 1 ) , from the complete roots of (2.14) with (2.15) wherein eK = ~ -- ~2. These contours resemble elongated parabolae centred along L. The evolution of a for a given 5 is determined b y A(~) and the most rapid amplification occurs where A is a maximum. Because of this differential amplification, the spectral maximum of a evolves with time towards increasing growth rate, i.e. in the direction of aA/a~. In the direction normal to L this evolution is always towards L. If i~21 >> i~zl, the interaction coefficients $2 and $3 are independent of f~21 and so because of viscous effects, A decreases with increasing 1~2i. Hence along the direction of L, evolution is downward in wavenumber magnitude. The tendency is therefore for the a bontours to become distorted, shrinking in the direction normal to L and for the location of the maximum to move towards and down L. Thus in real space the packet takes the form of a beam, aligned with and gradually extending along the direction of the characteristic ray (arctan(k2/m2) to the horizontal) of the nearest resonant wave.
91 (Xs(K') =c o n s t a n t S ; (b)
(K2, ESK ,q/I )= constant
Fig. 1.b. Contours of constant ~'(~') for a detuned wavepacket centred at ~ and contours of constant growth rate A(~2, e ~ , xP1 ). Evolution of the central wavenumber is along the path marked with a dashed line and arrow.
These effects can be illustrated by a numerical calculation in which the growth (or decay) rates are given b y the complex roots of (2.14). Using the relations N = 1, I~zl = 1 to define scales of length and time, Fig. 2 shows an evolutionary sequence with ~1 (0.841, 0.540), I~11 = 1, e 2 = 0.1, v = 10 - 6 and an initial spectrum: ----
a(K1,0)=expl
(K,,--150)2 25
(K~--4)2 1 25
where KIL and ~± are the components of ~ along and normal to the locus L, which in this example of large 1~2l is very close to being a straight line. Growth rate contours and the locus of Is Imax are also shown. 2. 3. Fine-scale waves and parametric forcing
From (2.6) and (2.11), wavenumbers and group velocities of free modes are orthogonal, i.e.: C , - ~ , = 0~
(2.24)
For wavenumber closure as $2 and $3 b e c o m e large along the resonant locus 1 L, ~2 ~ --~s and C 2 ~ --C a. Also for instability ¢o2 - ¢~a ~ --~¢°z b y (2.20) Thus if detuned an amount e6K2 normal to ~2 : I(C2
-
-
2c02 m2 Ca)" e6K21 ~ i-~-21i-~2 1 le6K2]
(2.25)
92
'501
I
/
/'1
-------7 t .5o
I/I
I'<, '°o
'°°1-/t'/~V/-I
M2
130
,~ol/H/Ill 50
60
~o, I K2
I oCl max
L
_
_
~
_
_
_
_
_
~
_
70
I oC I max
/
(c)
/
(d)
Fig. 2. C o m p u t e d e v o l u t i o n o f a d e t u n e d w a v e p a c k e t w i t h i n i t i a l l y c i r c u l a r a m p l i t u d e c o n t o u r s in ~ ; E1 = ( 0 . 8 4 1 , 0 . 5 4 0 ) , N ffi 1, [XI/l[ = 1 . 0 , e 2 = 0 . 1 , v = 1 0 - 6 . S o l i d c o n t o u r s a r e in i n t e r v a l s o f 0 . 2 I~lmax. D a s h e d l i n e s a r e g r o w t h r a t e c o n t o u r s . (a) t = 0, [0~lmax = 1.0; ( b ) t = 5 0 , I~lmax = 1 . 1 6 ; (c) t = 2 5 0 , 10~lmax = 3 . 7 7 ; ( d ) t = 1 0 0 0 , 10tlmax = 1 8 2 0 ; . . . . marks the locus of amplitude maximum.
93 If wave 2 is in the form of a packet whose.wavenumber is eK, and noting that by virtue of (2.24):
(C2 - C3)" ~2 = C3" ~1 then: ~ a ma [Eli l e K . K2l
c021m21
"~'k2[E212_ l~1[ [e~K°~21
(2.26)
By (2.10), k2 + k3 ~ 7Is 2i so, with (2.25), the detuning term in (2.19) diminishes like i~2i-6 and the interaction rapidly becomes insensitive to detuning for high wavenumbers. Nevertheless, interaction favoursthe lower wavenumberssince for I~21 -~ o o : t~ll~/cl~
82 P[~2[ 2 "~2 ~ [KI[2 " f ( l l ' 1 2 )
(2.27)
where : f = 18(1 +/2)(/2 - - / 1 ) - 1 ( 3 + 12 + 2/1/2)-11
(2.28)
12 = m 2 / k 2 , 1 + l 2 = 4(1 + 12). This can be compared with McEwan and Robinson's (1975) simple parametric instability result * on the basis of the spatially averaged maximum isopycnal slope: a n d 11 = m l / h l ,
I'I~2 f~
(2.28a)
i 212
(2.28) differs from (2.28a) b y a factor between 0.98 as 11 -~ 0 and 1.47 as 11 -~ ~ . Note that, since X2 = X3 in this limit, (2.18) gives ~ = 0. Further, it is straightforward to show that the exponentially growing solution to (2.13) has [~2i = ]~3l. Hence, in this limit, the fine-scale disturbance has the form of a growing standing wave. If at the start of interaction the disturbance amplitudes are unequal, the evolution rapidly brings them to similar levels (q.v. McEwan, Mander and Smith (1972)). To summarize: Compared with large-scale waves, fine-scale wavepackets experience less modification and can be excited by a greater range of large*
Eq. 2.14 o f that paper c o n t a i n s printing errors and s h o u l d read:
s = ~Oot,1 A = 4[--I~2k4(1 + 12)3 + N2](1 + 12)--1~002 Q = 2N2lO~M~.002(1 + /2)--1
94 scale 'background' waves. However, near resonance lower wavenumbers are amplified more strongly, and revolution is generally towards larger, more 'beam-like' packets and lower wavenumbers. 3. THE EXPERIMENT (Fig. 3) A rectangular plate glass channel 5.5 m long and 22.9 cm in width was filled to about 30.5 cm with a salt solution which, apart from the top and b o t t o m diffusion-influenced layers O(v~/2 Tlie ) thick, was linearly stratified in depth and had a b u o y a n c y frequency N of about 0.9 s-1. Here vs is the molecular diffusivity of salt (1.1 • 10 - 5 cm 2 s-1 ) and V is the time elapsed after filling (never greater than 1.8" 105 s). A sheet of thin polythene (Saran wrap) was laid over the water surface after filling to inhibit evaporation. At one end of the channel a vertical paddle occupying the full width and depth was placed. This could be oscillated through an angle of up to -+20 ° to the vertical about a horizontal axis at mid-depth to transmit along the channel a strong internal wave whose vertical wavelength was fundamentally twice the water depth. To sustain this wave in amplitude against viscous dissipation (predominantly due to viscous friction on the side of the channel) a vertical barrier of 0.3 cm thick Plexiglass sheet was placed ahead of the paddle for practically the full length of the channel. This confined the wave generated at the paddle to a region which tapered in width from 22.9 cm at the paddle to 5.0 cm at the far end. This barrier terminated about 10 cm from the far end so that there was a path connecting the opposite sides of the barrier. In practice it was established that due to the absorption along this channel and around the far end, back reflexion of wave energy towards the paddle was negligible. At 3 m from the paddle a cylinder 6 cm in diameter and about 1 cm narrower than the tapering section was suspended laterally and horizontally above the water surface. This could be lowered through a gap in the polythene covering to lie half immersed in the water. By connecting the suspension cords to a motor-driven crank the cylinder could be oscillated vertically through 1 cm at a controlled frequency ¢o~. By oscillating the cylinder at a frequency less than N a pair of discrete internal wave beams at - arcsln(cou]N ) to the horizontal were radiated in both directions along the channel. The beams, being of fine scale, weakened rapidly with distance from the generating cylinder and likewise decayed rapidly when the cylinder was withdrawn from the water. They were the realization of a discrete, spatially non uniform fine-scale internal wavepacket. The waves generated by the main wavemaker represented the large-scale perturbing wavefield. The idea of the experiment was to introduce first the fine-scale wave structure into the previously quiescent medium, then remove its source of excitation (the cylinder) and find the amplitude and frequency conditions -b
"
t
g
c
d
Slit light_~_~ sourc~,~v~ o
Fig. 3. Perspective sketch of experiment, illustrating the f o r m o f a large scale wave and fine scale wave beams in a linear stratification. (a) Crank; (b) fine-scale wave generator; (c) converging sidewall; (d) large-scale wave at mid-plane; (e) large-scale wavemaker; (f) finescale wave 'beams'.
Shodowgroph screen
•
/b
e,D
96
for the large-scale wave to sustain the fine structure against dissipation. To reveal the waves a shadowgraph technique was used in which light from a slit source of 0.2 cm × 1 cm aligned parallel with the internal wave beam was cast on to a projector screen 3.5 m in front of the central region of the channel (see Fig. 3), making it possible to detect the fine-scale wave under typical conditions down to a vertical wave displacement amplitude of 2.5 • 10 - 3 cm. Apart from its simplicity this technique was particularly appropriate since its sensitivity is proportional to the square of the disturbance wavenumber; viscous damping has a similar wavenumber dependence. In principle therefore the amplitude of the large-scale wave at the threshold of detection of a fine-scale instability should not depend on the wavenumber of the instability. The amplitude of the large-scale wave was measured directly at the viewing beam location by tracing the trajectory of neutrally b u o y a n t polystyrene beads (0.2 cm diameter) adrift in the fluid. The wavenumber of the fine-scale disturbance (either the most pronounced structure in the case of a growing disturbance or the initial wave beam in the case of a decaying one) was measured from high-contrast photographic enlargements of the shadowgraph picture. Knowing ¢~, N was found by measuring the angle of the fine wave. 4. R E S U L T S
4.1. Qualitative observations 4.1.1. Continuous forcing near resonance. A shadowgraph sequence typical of a developing instability is shown in Fig. 4. Here, w~ = 0.314 s- I in fluid 30.7 cm deep, with N = 0.94 s- 1 . The cylinder was withdrawn 5 s before the arrival of the large-scale wavefront whose frequency ¢ol was 0.644 s-1. Frames are numbered in cycles of this wave after cylinder withdrawal. While the fine waves were being excited by the cylinder, they defined a beam a couple of wavelengths wide whose phase velocity was directed transvarse upwards, consistent with downward energy propagation. If instability was sustained after cylinder withdrawal the beam remained discrete, widening slowly, but rapidly assumed the appearance of a standing wave with no obvious mean phase propagation. This is consistent with the emergence of a third wave o f nearly equal and opposite wavenumber, forming triad resonance, and with the narrowing of the wavenumber c o n t o u r ~(K) as discussed in § 2.1. The photograph sequence of Fig. 4 does not show the transformation to a standing waveform clearly, since each frame was taken at the same phase position of the forcing wave. It is however plainly evident in cinema records of the same experiments. When the background wave was strong enough, the instability would proceed till 'breaking' occurred as Fig. 4.5 shows: otherwise the beam remained visible, spreading slowly with or w i t h o u t the gradual emergence of
h~ ¢,D
I::u
4a.
e,D
t,D
100
Fig. 4. j,k,l.
101
other instabilities. If the background wave was forced below a threshold level the fine wave beam would decay monotonically and finally disappear.
4.1.2. Off-resonant forcing. If the forced fine-scale wave was appreciably away from resonance with the background wave, instabilities would evolve by the development of the packet towards resonance. For example if the frequency were t o o high ( m 2 / k 2 < m 2 / k 2 resonant) the original beam would appear to decay on its lower side and develop on its upper side to delineate a beam of roughly the same wavenumber magnitude b u t with a shallower characteristic angle. This is entirely consistent with the prediction in § 2.2 that wavenumber evolution is directed towards the nearest part of the resonant locus, the appropriate sideband of the packet forming the nucleus for the instability which subsequently develops.
4.1.3. Irregular forcing. Some attempts were made t o simulate the action of incoherent packets of large-scale waves or of simultaneous large-scale waves separated in frequency. In the first procedure, forcing of the fine-scale wave was as before, but the large wavemaker was arrested after five or six cycles of oscillation and started in oscillation again at the same frequency after a period of one half the wave period. The effect at the position of the fine wave packet resembled that of a pair of interfering large-scale waves separated fractionally in frequency b y 1/10--1/12. Forcing amplitudes required to sustain instability were found to be substantially greater than with continuous forcing, and the instabilities as they developed differed in that there appeared in addition to the original fine-scale beam a set of waves mclined at an equal and opposite direction to the horizontal. In the second procedure, large-scale wave forcing was maintained continuously b u t the frequency was changed cyclicly between three levels of roughly 2.7¢o~, 1.5¢o~, 2.0¢o~ at intervals of six periods of the fine-scale wave (i.e., 121r/co~ s).'In this case the instability when it emerged bore little resemblance to the pre-existing fine-scale wave. From these tests it would appear that the coherency and intermodulation of large-scale waves have an important bearing on the evolution of the instability, not accounted for in the present theory.
Fig. 4. Shadowgraph sequence of developing instability. The previously imposed f i n e - s c a l e disturbance enters from the upper right-hand corner, and the large-scale wave travels f r o m l e f t t o r i g h t . V e r t i c a l s t r i a t i o n s a r e imperfections in the glass sidewalls. Frames are numbered in cycles ( 2 ~ / ~ 1 ) o f t h e l a r g e wave after termination o f e x t e r n a l f o r c i n g o f t h e f i n e s c a l e w a v e . H e r e N = 0 . 9 4 s - 1 , depth is 30.7 cm, ~ 1 = 0 . 6 4 4 s - 1 a n d 0J~ = 0 . 3 1 4 s - 1 .
102
4. 2. Quantitative stability thresholds A range of tests was performed to establish the conditions under which the imposed fine disturbance (wave 2') decayed or could be sustained by the large wave (wave 1). This wave was forced at various levels and its frequency c~1 was varied between 0.52N and 0.88N. The disturbance frequency co~ was imposed at values of between 0.8 COl and 0.38 col. All results are presented on Fig. 5, each being represented by a dashed or solid bar depending on whether the disturbance decayed or remained visible. The abscissa value of the bar is the actual measured stream function amplitude of the large
I
I
ioo
I
I
I
I
I
--
II I
Stable
~'c) E~ measured
I I
~'Fc |
I
i
,
Jti I
IIIIll ~
I
/ .
t
0
/
Unstable
I
5
I
I0
I
I~ ~lJ meosured
15
20
Fig. 5. Measured a m p l i t u d e of background wave, Je~JlJ measured versus critical amplitude ekT'c predicted using eq. 2.19. The upper limit of the bar is the value p r e d i c t e d for the initial d e t u n e d wave, as imposed b y the external wavemaker. The lower l i m i t is the value for the nearest wave on the resonant locus (q.v. Fig. l a ) . The d o t marking this limit is o p e n or closed indicating that the wave was observed to decay or to be sustained. The solid line delineates the c o n d i t i o n e~ c = I e ~ l Imeasured-
103
wave, levi I. The upper and lower limits of each bar on the ordinate are, respectively (a) the critical amplitude of the large wave e~c as predicted using eq. 2.19 for the initially imposed, and usually non-resonant fine disturbance; and (b) the critical amplitude predicted for the nearest resonant fine disturbance. The length of the bar is thus an indication of the level of detuning. In calculating these amplitudes leSK21 = ~ -- h:2 as indicated by (2.23) et seq. I ~ 1 was directly measured from shadowgraphs and its angle was calculated from co'2/N. ~_z was taken as the point on the ~, resonant locus, calculated iteratively, whose normal in wavenumber space passes through ~ . To account for sidewall viscous dissipation the damping coefficients as given by (2.10) were increased by a factor of [1 + ( 2 ~ , / v ) l 1 2 m , ( k , + m,)/WI ~, 14], where W is channel width, vs, being O(10 - a v) in this experiment, was neglected. Also drawn on the figure is a line for the condition e ~ c = l e ~ l I measured, results to the left of which are predicted to show decay and vice versa. It is quite apparent that the lower limit of the bars ( the resonant critical amplitudes) are those which conform most closely to this condition. This is emphasized b y marking these limits b y circles, open for decaying disturbances and closed for those which were sustained. The significance level of the correlation is greater than 99%. Maximum significance, as indicated b y a maximum in f,(a) -- fs (a) where f. and f8 are, respectively, the proportions of 'sustained' and 'decaying' results for which e ~ (resonant)/I e~,L (measured) is less than a, occurs when a is a b o u t 1.15. In other words the results suggest that the disturbances became unstable at a background wave amplitude some 13% less than the minimum predicted b y the present theory. It is seen from the figure that there is no noticeable dependence u p o n the length of the detuning 'bar' and that for the unstable results with only t w o exceptions the predicted critical amplitude for the original detuned wave lies above the measured amplitude. 5. CONCLUDING DISCUSSION
In § 2 it was shown that detuning and limited packet size have equivalent effects on the propensity of an internal wave disturbance to be amplified b y a stronger and more extensive wavefield of higher frequency. In each case in addition to viscous attenuation there occurs a dispersion of the disturbance relative to its partner in triad interaction with the stronger wave. Nevertheless the results of § 4.2 firmly suggest that the amplitude of the stronger wave required for amplification to take place is defined b y stability conditions for the nearest resonant disturbance. This is consistent with the predictions of § 2.3 and the observations of § 4.2. that the disturbance, if limited in space, will have wave number sidebands, the appropriate members o f which are selectively amplified so that the disturbance rapidly evolves towards resonance with the strong wave. Even at resonance the necessary amplitude for
104
instability seems to be slightly overestimated on average, and this supports the suggestion in § 2.3 that the disturbance, if confined to a packet, evolves gradually downwards in wavenumber magnitude, and therefore requires less forcing for instability. This should not be taken to imply a contradiction to McComas and Bretherton's (1977) prediction of cascade to higher vertical wavenumbers since it refers to the process of a single interaction event. Normally at least one of the interaction products is higher in vertical wavenumber than the highest frequency m e m b e r yielding its energy, and each product in turn is susceptible to further interactive cascade until a scale is reached where damping through viscosity, diffusion and wave breaking prevents further evolution except by the present means. Apart from an apparent sensitivity to frequency modulation in the forcing wave as noted in § 4.1.3, the results substantially support assumptions that the oceanic spectrum, although composed of discrete packets, can be treated for the purpose of calculating energy cascade as a continuum of wavenumbers and frequencies. The critical amplitude for unstable growth as predicted by simple parametric instability theory differs by a numerical factor of order unity depending on wavenumber angle from the present theory; nevertheless in view of the tendency towards resonance for wave packet sidebands, the predictions of McEwan and Robinson (1975} concerning parametric reinforcement of oceanic fine-scales should remain reasonably accurate. Finally, it is n o t e w o r t h y that the foregoing theory gives a rough idea of the likely form of the contours of amplitude in wavenumber space of a cascading wave field. For each wave present of frequency cot the resonance locus lies above, and asymptotically approaches, the line at arcos ( ¢ o J 2 N ) to the k-axis. Each interacting packet will tend to align with one or more of these loci and so as evolution proceeds the amplitude contours will progressively b e c o m e lenticular with their major axes directly roughly towards the origin. In real space the packets or at least the finer scale members would become increasingly beam-like, with interaction and breaking events occurring most commonly at the intersection regions. ACKNOWLEDGEMENT
The experimental part of this study was done at Woods Hole Oceanographic Institution, supported b y a Rossby Fellowship and National Science Foundation (Oceanography Section) grant DES 72-01562. These and the assistance of Mr. R. Frazel are gratefully acknowledged. REFERENCES Garrett, C. and Munk, W., 1972. Space--time scales for internal waves. Geophys. Fluid Dyn., 3: 225--264. Garrett, C. and Munk, W., 1975. Space--time scales for internal waves: a progress report. J. Geophys. Res., 80: 191--297.
105 Hasselmann, K., 1967. A criterion for nonlinear wave instability. J. Fluid Mech., 30: 737-739. Mander, D.W., 1974. Thesis, Monash University, Melbourne, Australia. McComas, C.H. and Bretherton, F.P., 1977. Resonant interaction of oceanic internal waves. Submitted to J. Fluid Mech. MeEwan, A.D. and Robinson, R.M., 1975. Parametric instability of internal gravity waves. J. Fluid Mech., 67(4): 667--687. McEwan, A.D., Mander, D.W. and Smith, R.K., 1972. Forced resonant second-order interaction between damped internal waves. J. Fluid Mech., 55(4): 589--608. Miiller. P. and Olbers, D.J., 1975. On the dynamics of internal waves in the deep ocean. J. Geophys. Res., 80: 3848--3860. Olbers, D.J., 1976. The internal wave field in the deep ocean. J. Fluid Mech., 74: 375-399. Orlanski, I., 1973. Trapeze instability as a source of internal gravity waves. Part I. J. Atmos. Sci., 30: 1007--1016. Phillips, O.M., 1966. The Dynamics of the Upper Ocean. Cambridge University Press. Plumb, R.A., 1977. Wave interaction in a bounded fluid. (Unpublished manuscript). Wunsch, C., 1976. Geographic variability of the internal wave field -- a search for sources and sinks. J. Phys. Oceanog., 6(4): 471--485.
83
OFF-RESONANT AMPLIFICATION OF FINITE INTERNAL WAVE PACKETS
A.D. McEWAN and R.A. PLUMB C. S. I. R. 0., Division o f Atmospheric Physics, Aspendale, Vic. (Australia)
(Received March 4, 1977; revised June 10, 1977; accepted July 15, 1977)
ABSTRACT McEwan, A.D. and Plumb, R.A., 1977. Off-resonant amplification of finite internal wave packets. Dyn. Atmos. Oceans, 2: 83--105. It is shown by experiment and theory that finite fine-scale internal wave packets in a continuous stratification can be amplified by the passage through them of a longer wave of higher frequency. In general the process is a detuned resonant interaction, but the finiteness of the packet provides sidebands which allow enhancement of those components of the packet lying most closely to exact second-order triad resonance with the forcing wave. The amplitude of the forcing wave required for amplification of the packet against viscous dissipation is therefore that appropriate to the nearest resonant mode, and may be significantly less than that for a monochromatic mode with the same central wavenumber. The tendency in an evolving internal wave field is for the packets to develop as beams extending along the characteristic directions of free wave modes.
1. INTRODUCTION E v e n at m o d e s t d i s t a n c e s f r o m i d e n t i f i a b l e sources, t h e average i n t e r n a l w a v e s p e c t r u m o f t h e d e e p o c e a n is n o t a b l y u n i f o r m (Wunsch, 1 9 7 6 ) i m p l y ing an e f f i c i e n t a n d universal p r o c e s s f o r c a s c a d i n g energy. R e s o n a n t w a v e i n t e r a c t i o n (q.v., Phillips, 1 9 6 6 ) is t h e p r i m e s u s p e c t , a n d b y n u m e r i c a l exp e r i m e n t s t h r e e classes o f such i n t e r a c t i o n h a v e b e e n i d e n t i f i e d as d o m i n a n t (McComas and Bretherton, 1977) within the representative spectrum (Garrett and Munk, 1972, 1975). N o w while this s p e c t r u m is b y all e v i d e n c e a g o o d r e p r e s e n t a t i o n o f t h e t i m e - a v e r a g e d i n t e r n a l w a v e m o t i o n field it c a n n o t b e e x p e c t e d t h a t at o n e p a r t i c u l a r t i m e a n d p l a c e in t h e o c e a n t h e m o t i o n e x p e r i e n c e d is c o m p o s e d o f a c o n t i n u u m o f w a v e n u m b e r s a n d f r e q u e n c i e s , b u t is m u c h m o r e likely t o c o m p r i s e o n l y a f e w m o d e s w h i c h are passing t h r o u g h in m o v i n g p a c k e t s o f finite size. I f t h e s p e c t r u m w e r e c o n t i n u o u s t h e w h o l e c o n t i n u u m o f m o d e s c a p a b l e o f s e c o n d - o r d e r triad i n t e r a c t i o n w o u l d b e p r e s e n t s i m u l t a n e o u s l y , b u t in reality t h e c o i n c i d e n t arrival o f even o n e t r i a d o f m o d e s c a p a b l e o f i n t e r a c t i n g r e s o n a n t l y is a rare event. A l m o s t i n v a r i a b l y t h e initial interac-
84 tions will be independent ones between each of the modes instantaneously present. In view of the fundamentally non-linear nature of the problem it is therefore by no means clear a priori that continuum triad wave interaction analyses such as those of MiJller and Olbers (1975) and Olbers (1976) formulated in general by Hasselmann (1967) can be used to evaluate the mean spectral transfers in terms of the representative spectrum. The evolution of the spectral wave action density a(k) is governed by an equation of the form :
Da(_k) Dk
:ffs(k_, .k' . k")a(k')a(k")d3k ' d3k '' . . .
(e.g. Olbers, 1976) where S is the transfer coefficient. If ( > denotes an "ensemble" (time or space) average, then the continuum approach gives a transfer of the form:
ffs(k_, k_',_k")
85 Although the finest-scale waves may, due to their slow dispersion, be those most favoured for quasi-resonant parametric forcing, in a real fluid viscous damping may negate this advantage. In general therefore it can be seen that the capacity for interactive energy cascade between a pair of discrete internal wave packets is determined b y three things: the degree to which they approximate members of a resonant triad, the size of the packets, and their natural 'viscous' attenuation rate. In a b o u n d e d medium the wave modes are discrete, affording a means of quantifying the degree of detuning from precise triad resonance. Plumb (1977) has dealt with the constraints on interaction arising purely from the detuning, while Mander (1974) has defined instability thresholds and evolution of viscously damped, b o u n d e d standing waves under restricted conditions. In the following investigation the criterion for amplification of u n b o u n d e d packets of damped internal waves under the action of a stronger higher-frequency wave are derived and compared with predictions of the simpler parametric instability theory. A laboratory experiment is described which confirms the major predictions and gives emphasis to the importance of internal wave interaction in distributing internal motion within the ocean. For simplicity, the study is restricted to uniform fluids with, e.g., constant b u o y a n c y f r e q u e n c y
2. THEORY The suggestion made in McEwan and Robinson (1975) was that packets of the finer, lower-frequency internal waves in the ocean were energized continually by the passage through them of larger-scale internal waves of approximately double frequency. In conformity with this idea and the conditions of the present experiment we seek the amplitude of a uniform large-scale wave above which a given small-scale wave packet would be amplified, and an understanding of the way the packet would evolve. 2.1. P a c k e t s f o r m i n g e x a c t triad resonance
Consider for convenience a two-dimensional wave field in a medium of constant b u o y a n c y frequency N. In terms of the'stream function • and b u o y a n c y perturbation o:
V21~tt + N 2 ~ x x = J(¢2, V2 ~J)t -- J ( ~ , a t - - ~ x N 2 = J ( ~ , o) + VsV2a
o)x
--
vsV20~ + vV 4 ~t
(2.1) (2.2)
where V2 = ~2/~X2 + ~2/~Z2, a -~ - - g P / P O , N2 = --gPoz/Po, P is density, v is kinematic viscosity, vs is diffusivity of the stratifying constituent, (~/~z, a / ~ x ) = (u, --v) and subscripts denote partial differentiation by horizontal distance x and time t. Motion is taken to be defined in terms of t w o discretely separable scales of
86
space and time, written: T) + O(e 2)
¢2 = e p ( r , t ) ~ ( R ,
a = es(r, t ) Z ( R , T) + O(e 2)
(2.3)
where e is some dimensionless measure of amplitude, r and t are real position vector and time respectively over which a periodic wave-like structure is identified and R and T are expanded scales defined by : R = (X, Z) - er = (ex, ez)
(2.4) T =- e t
The space scale for a packet of internal waves embedded in an extensive fiela of plane waves of higher frequency, and the time scale for the evolution of this packet are characterized by these expanded scales. With gradients in (2.1) and (2.2) defined: a a a -
a
at a
ax
ax
Ot
+ C -
aT a aX
V = V+eV then to leading order, providing ~, v~ ~< O(e): V2 ~ t t + N 2 ~ x x = 0
Ot - - N 2 ~ x = 0
which has the form of free internal waves varying weakly in space and time: = ~ q]~(R, T) exp[i(K, - r - - wnt)] n
(2.5)
o = ~ Z~(R, T) e x p [ i ( ~ . r - -
~nt)]
n
and their complex conjugates. Wave frequency co and wavenumber ~ are connected through the dispersion relation: ~92 = N 2 k 2 / i ~ i
2
(2.6)
in which ~n = (hn, mn) where kn and m , are horizontal and vertical wavenumbers *. Also, by (2.2): ~'n = - - N 2 hn xI~n/ COn
(2.7)
* As a matter of notation K will be used for wavenumbers of order unity (on a scale ILl ) i n the unstretched coordinate x, while K will be used to represent wavenumber increments of order unity i n t h e "slow" coordinatOR.
87 To next order in e:
272 ~tT
+ 2~xXtt + 2t~zZtt + 2N2~ x
j(@,
=
2
(Res)
V ~)t
-- g(¢,
(/)(Res)
VV4~t --Vsq2ax
(2.8)
+
where (Res) denotes resonant forcing terms, such that Z~i = 2:¢o(~i) = 0, where j denotes the (three) modes participating at second order in the interaction and ¢o(~i ) is given by (2.6). Non-resonant terms appear at higher order. Then with (2.5), (2.6) and (2.7):
+ C.. ~ + e-iX.
~I,~ = iS.~I,;~I,~
(2.9)
Here:
(2.10)
X,, = (t, + r,~)1~1/2 denotes a damping coefficient and: _
¢Omn
(2.11)
is the group velocity of mode n. Sn is an interaction coefficient for the forcing of mode n by modes p and q:
Sn = (2E2C~n)-l(kq mp -- kp mq)[w,(Kqz -- ~2) + knN2(kq ¢oql __ kp COp1)] (2.12) In the c o n t e x t of the present experiments where the long wave field has the form: ~)1 = A1 sin(klX -- COxt) sin m l z the appropriate interaction coefficient is quartered in magnitude, viz.:
S(: ) = (SEn2 G)n) - 1 (]~q m l - - k l m q ) [ ¢ o , ( ~ 2 - - ~21) + k , N U ( k q (~71 - - k l co11)] (2.12a) where n = 2 or 3 and q = 3 or 2. If @2 and @3 are small perturbations to an extensive background wave field @1 for which O@ I / a X = O ~ / 1 / a T ~ O, then from (2.9):
(~V + C2" ~ + X2/e)¢2 = iS2 ¢~ ¢~ (2.13)
)
88 or combining:
~-~+ C 2 " ~ + ~
~--~ + C3" ~ +
~2 = 82S3 ~]2l~ll 2
(2.14)
Now supposing that at T = 0:
~ ( R , 0) ; f ~(K) e ' ~ ~ dK
(2~15)
and thereafter that:
a(K, T) = a(K) exp[AQK) -- iFt(K)IT
(2.16)
for each K within the spectrum defining the profile of the wavepacket 2. Here A and ~ are real; A is the growth rate of the disturbance on the long time scale T, while ~ is an O(e) correction to the wave frequency. Then:
C2"K-- ~ ) - - i
A+-e
Cs ' K - - g t ) - - i
+
+$2S31~112=0 (2.17)
or close to marginal stability where I Ae/Xl < < 1: Ft=\~--(X2+XS+2A)-l[(Xs+a-e- ~C2_ + ( X C 2 s+ AI )' K T ~
(2.18)
and the growth rate: A = e 8 2 8 3 ( 1 ~ 1 1 2 - 1 ~ c f 2) (X2 + Xs) - 1
where:
e2[ ~IJc[2 = ($283)-1~,2~3 [ 1 + [(~C2 --~C3)" eg[ 2 (~,2 + ~,3)- 2 ]
(2.19)
Thus the instability is sustained if the amplitude ~1 of the 'background' wave exceeds ~¢. Contributions to ko¢ arise both from the magnitude of the viscous attenuation against which the interaction has to work, and the effect of wave dispersion, since the disturbance waves diverge at a speed [C 2 - - C3[, so the time for cooperative interaction is limited by the size of the packet to IC2 -
C3i" le~
.
2. 2. Detuned packets
If the central wavenumber of a pre-existing packet 2 is not in exact triad resonance interaction with the background wave, the resultant wavenumber detuning may be absorbed into the definition of K in (2.15) and thereafter. Assuming that wave 3 grows from the interaction between the background field wave 1 and the packet 2, wavenumber closure is guaranteed, i.e. : t p NK3 = - - - - ~ 1 - -
t ~2
89 but t t C0(51) + 50(52)
+ G 3 ( 5 3 ) =/= 0
where primes denote the pre-existing, non resonant values. These must lie close to the resonant values such that: c°(51) + co(52) + co(--51 - - 5 2 ) = 0
(2.20)
for significant energy exchange to take place. Writing: t
~2 = ~2 + e~K2
(2.21)
the .wavenumber detuning may be regarded as a long-wave modulation upon a wave of wavenumber ~2. If at T = 0:
t
t
where ¢o2 = ¢o2(52) then if this is equivalent to a suitably modulated wave:
resonant
xIJ2 = a2(R , O) exp[i(52 • r - - ¢02t)] a2 and a 2 become related in K. Specifically, from (2.15): a2(R , 0) = f , ' (/() e 'K .n ch~' then, by (2.4): a2(R , 0)
=fo~'(K')
e'(~'+~K~2)'~ dK' (2.22)
= f a ( K ) eiK~"X dK' where: a(K)
(2.23)
a' K - - 6 K ° )
A non-resonant packet with a given spectrum would therefore be directly equivalent to a resonant packet whose spectrum is the same shape but displaced in wavenumber K an a m o u n t 5K2. A difficulty in applying this result is that the appropriate resonant m o d e cannot be defined a priori in an unbounded system, and it is useful to consider the interaction in wavenumber space 5 = (k, m). For any given wave 1 a locus L(51, 5) may be drawn in 5, as depicted in Fig. l a defining a continuum of waves capable of exact triad resonance satisfying (2.20). If a given resonant wave number 52 is increased a small a m o u n t along this locus: co(~2 + 7 ) + c o ( ~ 3 - - 7 ) + ~ ( ~ )
= 0
90
(a)
K~ K2J/ ~Porometric //~!t~X~' Asymptote --
Fig. 1.a. Resonant locus L(K1, g) in wavenumber space. A specified detuned wave ~ defines a resonant ~2 by the condition that e~K 2 is normal to L.
so as i~i -~ O:
(c2 -
C s ) . ~ ~ o~
Therefore (C 2 - - C3) is always normal to L. For a given non-resonant wave packet with central wavenumber ~ , contours of a'(K') can be constructed in S as shown in Fig. l b . By defining e~K2 = K~ -- ~2 as being normal to L, ~2 becomes specified, and it is then possible to construct contours of A(S2, e~K2, ~ 1 ) , from the complete roots of (2.14) with (2.15) wherein eK = ~ -- ~2. These contours resemble elongated parabolae centred along L. The evolution of a for a given 5 is determined b y A(~) and the most rapid amplification occurs where A is a maximum. Because of this differential amplification, the spectral maximum of a evolves with time towards increasing growth rate, i.e. in the direction of aA/a~. In the direction normal to L this evolution is always towards L. If i~21 >> i~zl, the interaction coefficients $2 and $3 are independent of f~21 and so because of viscous effects, A decreases with increasing 1~2i. Hence along the direction of L, evolution is downward in wavenumber magnitude. The tendency is therefore for the a bontours to become distorted, shrinking in the direction normal to L and for the location of the maximum to move towards and down L. Thus in real space the packet takes the form of a beam, aligned with and gradually extending along the direction of the characteristic ray (arctan(k2/m2) to the horizontal) of the nearest resonant wave.
91 (Xs(K') =c o n s t a n t S ; (b)
(K2, ESK ,q/I )= constant
Fig. 1.b. Contours of constant ~'(~') for a detuned wavepacket centred at ~ and contours of constant growth rate A(~2, e ~ , xP1 ). Evolution of the central wavenumber is along the path marked with a dashed line and arrow.
These effects can be illustrated by a numerical calculation in which the growth (or decay) rates are given b y the complex roots of (2.14). Using the relations N = 1, I~zl = 1 to define scales of length and time, Fig. 2 shows an evolutionary sequence with ~1 (0.841, 0.540), I~11 = 1, e 2 = 0.1, v = 10 - 6 and an initial spectrum: ----
a(K1,0)=expl
(K,,--150)2 25
(K~--4)2 1 25
where KIL and ~± are the components of ~ along and normal to the locus L, which in this example of large 1~2l is very close to being a straight line. Growth rate contours and the locus of Is Imax are also shown. 2. 3. Fine-scale waves and parametric forcing
From (2.6) and (2.11), wavenumbers and group velocities of free modes are orthogonal, i.e.: C , - ~ , = 0~
(2.24)
For wavenumber closure as $2 and $3 b e c o m e large along the resonant locus 1 L, ~2 ~ --~s and C 2 ~ --C a. Also for instability ¢o2 - ¢~a ~ --~¢°z b y (2.20) Thus if detuned an amount e6K2 normal to ~2 : I(C2
-
-
2c02 m2 Ca)" e6K21 ~ i-~-21i-~2 1 le6K2]
(2.25)
92
'501
I
/
/'1
-------7 t .5o
I/I
I'<, '°o
'°°1-/t'/~V/-I
M2
130
,~ol/H/Ill 50
60
~o, I K2
I oCl max
L
_
_
~
_
_
_
_
_
~
_
70
I oC I max
/
(c)
/
(d)
Fig. 2. C o m p u t e d e v o l u t i o n o f a d e t u n e d w a v e p a c k e t w i t h i n i t i a l l y c i r c u l a r a m p l i t u d e c o n t o u r s in ~ ; E1 = ( 0 . 8 4 1 , 0 . 5 4 0 ) , N ffi 1, [XI/l[ = 1 . 0 , e 2 = 0 . 1 , v = 1 0 - 6 . S o l i d c o n t o u r s a r e in i n t e r v a l s o f 0 . 2 I~lmax. D a s h e d l i n e s a r e g r o w t h r a t e c o n t o u r s . (a) t = 0, [0~lmax = 1.0; ( b ) t = 5 0 , I~lmax = 1 . 1 6 ; (c) t = 2 5 0 , 10~lmax = 3 . 7 7 ; ( d ) t = 1 0 0 0 , 10tlmax = 1 8 2 0 ; . . . . marks the locus of amplitude maximum.
93 If wave 2 is in the form of a packet whose.wavenumber is eK, and noting that by virtue of (2.24):
(C2 - C3)" ~2 = C3" ~1 then: ~ a ma [Eli l e K . K2l
c021m21
"~'k2[E212_ l~1[ [e~K°~21
(2.26)
By (2.10), k2 + k3 ~ 7Is 2i so, with (2.25), the detuning term in (2.19) diminishes like i~2i-6 and the interaction rapidly becomes insensitive to detuning for high wavenumbers. Nevertheless, interaction favoursthe lower wavenumberssince for I~21 -~ o o : t~ll~/cl~
82 P[~2[ 2 "~2 ~ [KI[2 " f ( l l ' 1 2 )
(2.27)
where : f = 18(1 +/2)(/2 - - / 1 ) - 1 ( 3 + 12 + 2/1/2)-11
(2.28)
12 = m 2 / k 2 , 1 + l 2 = 4(1 + 12). This can be compared with McEwan and Robinson's (1975) simple parametric instability result * on the basis of the spatially averaged maximum isopycnal slope: a n d 11 = m l / h l ,
I'I~2 f~
(2.28a)
i 212
(2.28) differs from (2.28a) b y a factor between 0.98 as 11 -~ 0 and 1.47 as 11 -~ ~ . Note that, since X2 = X3 in this limit, (2.18) gives ~ = 0. Further, it is straightforward to show that the exponentially growing solution to (2.13) has [~2i = ]~3l. Hence, in this limit, the fine-scale disturbance has the form of a growing standing wave. If at the start of interaction the disturbance amplitudes are unequal, the evolution rapidly brings them to similar levels (q.v. McEwan, Mander and Smith (1972)). To summarize: Compared with large-scale waves, fine-scale wavepackets experience less modification and can be excited by a greater range of large*
Eq. 2.14 o f that paper c o n t a i n s printing errors and s h o u l d read:
s = ~Oot,1 A = 4[--I~2k4(1 + 12)3 + N2](1 + 12)--1~002 Q = 2N2lO~M~.002(1 + /2)--1
94 scale 'background' waves. However, near resonance lower wavenumbers are amplified more strongly, and revolution is generally towards larger, more 'beam-like' packets and lower wavenumbers. 3. THE EXPERIMENT (Fig. 3) A rectangular plate glass channel 5.5 m long and 22.9 cm in width was filled to about 30.5 cm with a salt solution which, apart from the top and b o t t o m diffusion-influenced layers O(v~/2 Tlie ) thick, was linearly stratified in depth and had a b u o y a n c y frequency N of about 0.9 s-1. Here vs is the molecular diffusivity of salt (1.1 • 10 - 5 cm 2 s-1 ) and V is the time elapsed after filling (never greater than 1.8" 105 s). A sheet of thin polythene (Saran wrap) was laid over the water surface after filling to inhibit evaporation. At one end of the channel a vertical paddle occupying the full width and depth was placed. This could be oscillated through an angle of up to -+20 ° to the vertical about a horizontal axis at mid-depth to transmit along the channel a strong internal wave whose vertical wavelength was fundamentally twice the water depth. To sustain this wave in amplitude against viscous dissipation (predominantly due to viscous friction on the side of the channel) a vertical barrier of 0.3 cm thick Plexiglass sheet was placed ahead of the paddle for practically the full length of the channel. This confined the wave generated at the paddle to a region which tapered in width from 22.9 cm at the paddle to 5.0 cm at the far end. This barrier terminated about 10 cm from the far end so that there was a path connecting the opposite sides of the barrier. In practice it was established that due to the absorption along this channel and around the far end, back reflexion of wave energy towards the paddle was negligible. At 3 m from the paddle a cylinder 6 cm in diameter and about 1 cm narrower than the tapering section was suspended laterally and horizontally above the water surface. This could be lowered through a gap in the polythene covering to lie half immersed in the water. By connecting the suspension cords to a motor-driven crank the cylinder could be oscillated vertically through 1 cm at a controlled frequency ¢o~. By oscillating the cylinder at a frequency less than N a pair of discrete internal wave beams at - arcsln(cou]N ) to the horizontal were radiated in both directions along the channel. The beams, being of fine scale, weakened rapidly with distance from the generating cylinder and likewise decayed rapidly when the cylinder was withdrawn from the water. They were the realization of a discrete, spatially non uniform fine-scale internal wavepacket. The waves generated by the main wavemaker represented the large-scale perturbing wavefield. The idea of the experiment was to introduce first the fine-scale wave structure into the previously quiescent medium, then remove its source of excitation (the cylinder) and find the amplitude and frequency conditions -b
"
t
g
c
d
Slit light_~_~ sourc~,~v~ o
Fig. 3. Perspective sketch of experiment, illustrating the f o r m o f a large scale wave and fine scale wave beams in a linear stratification. (a) Crank; (b) fine-scale wave generator; (c) converging sidewall; (d) large-scale wave at mid-plane; (e) large-scale wavemaker; (f) finescale wave 'beams'.
Shodowgroph screen
•
/b
e,D
96
for the large-scale wave to sustain the fine structure against dissipation. To reveal the waves a shadowgraph technique was used in which light from a slit source of 0.2 cm × 1 cm aligned parallel with the internal wave beam was cast on to a projector screen 3.5 m in front of the central region of the channel (see Fig. 3), making it possible to detect the fine-scale wave under typical conditions down to a vertical wave displacement amplitude of 2.5 • 10 - 3 cm. Apart from its simplicity this technique was particularly appropriate since its sensitivity is proportional to the square of the disturbance wavenumber; viscous damping has a similar wavenumber dependence. In principle therefore the amplitude of the large-scale wave at the threshold of detection of a fine-scale instability should not depend on the wavenumber of the instability. The amplitude of the large-scale wave was measured directly at the viewing beam location by tracing the trajectory of neutrally b u o y a n t polystyrene beads (0.2 cm diameter) adrift in the fluid. The wavenumber of the fine-scale disturbance (either the most pronounced structure in the case of a growing disturbance or the initial wave beam in the case of a decaying one) was measured from high-contrast photographic enlargements of the shadowgraph picture. Knowing ¢~, N was found by measuring the angle of the fine wave. 4. R E S U L T S
4.1. Qualitative observations 4.1.1. Continuous forcing near resonance. A shadowgraph sequence typical of a developing instability is shown in Fig. 4. Here, w~ = 0.314 s- I in fluid 30.7 cm deep, with N = 0.94 s- 1 . The cylinder was withdrawn 5 s before the arrival of the large-scale wavefront whose frequency ¢ol was 0.644 s-1. Frames are numbered in cycles of this wave after cylinder withdrawal. While the fine waves were being excited by the cylinder, they defined a beam a couple of wavelengths wide whose phase velocity was directed transvarse upwards, consistent with downward energy propagation. If instability was sustained after cylinder withdrawal the beam remained discrete, widening slowly, but rapidly assumed the appearance of a standing wave with no obvious mean phase propagation. This is consistent with the emergence of a third wave o f nearly equal and opposite wavenumber, forming triad resonance, and with the narrowing of the wavenumber c o n t o u r ~(K) as discussed in § 2.1. The photograph sequence of Fig. 4 does not show the transformation to a standing waveform clearly, since each frame was taken at the same phase position of the forcing wave. It is however plainly evident in cinema records of the same experiments. When the background wave was strong enough, the instability would proceed till 'breaking' occurred as Fig. 4.5 shows: otherwise the beam remained visible, spreading slowly with or w i t h o u t the gradual emergence of
h~ ¢,D
I::u
4a.
e,D
t,D
100
Fig. 4. j,k,l.
101
other instabilities. If the background wave was forced below a threshold level the fine wave beam would decay monotonically and finally disappear.
4.1.2. Off-resonant forcing. If the forced fine-scale wave was appreciably away from resonance with the background wave, instabilities would evolve by the development of the packet towards resonance. For example if the frequency were t o o high ( m 2 / k 2 < m 2 / k 2 resonant) the original beam would appear to decay on its lower side and develop on its upper side to delineate a beam of roughly the same wavenumber magnitude b u t with a shallower characteristic angle. This is entirely consistent with the prediction in § 2.2 that wavenumber evolution is directed towards the nearest part of the resonant locus, the appropriate sideband of the packet forming the nucleus for the instability which subsequently develops.
4.1.3. Irregular forcing. Some attempts were made t o simulate the action of incoherent packets of large-scale waves or of simultaneous large-scale waves separated in frequency. In the first procedure, forcing of the fine-scale wave was as before, but the large wavemaker was arrested after five or six cycles of oscillation and started in oscillation again at the same frequency after a period of one half the wave period. The effect at the position of the fine wave packet resembled that of a pair of interfering large-scale waves separated fractionally in frequency b y 1/10--1/12. Forcing amplitudes required to sustain instability were found to be substantially greater than with continuous forcing, and the instabilities as they developed differed in that there appeared in addition to the original fine-scale beam a set of waves mclined at an equal and opposite direction to the horizontal. In the second procedure, large-scale wave forcing was maintained continuously b u t the frequency was changed cyclicly between three levels of roughly 2.7¢o~, 1.5¢o~, 2.0¢o~ at intervals of six periods of the fine-scale wave (i.e., 121r/co~ s).'In this case the instability when it emerged bore little resemblance to the pre-existing fine-scale wave. From these tests it would appear that the coherency and intermodulation of large-scale waves have an important bearing on the evolution of the instability, not accounted for in the present theory.
Fig. 4. Shadowgraph sequence of developing instability. The previously imposed f i n e - s c a l e disturbance enters from the upper right-hand corner, and the large-scale wave travels f r o m l e f t t o r i g h t . V e r t i c a l s t r i a t i o n s a r e imperfections in the glass sidewalls. Frames are numbered in cycles ( 2 ~ / ~ 1 ) o f t h e l a r g e wave after termination o f e x t e r n a l f o r c i n g o f t h e f i n e s c a l e w a v e . H e r e N = 0 . 9 4 s - 1 , depth is 30.7 cm, ~ 1 = 0 . 6 4 4 s - 1 a n d 0J~ = 0 . 3 1 4 s - 1 .
102
4. 2. Quantitative stability thresholds A range of tests was performed to establish the conditions under which the imposed fine disturbance (wave 2') decayed or could be sustained by the large wave (wave 1). This wave was forced at various levels and its frequency c~1 was varied between 0.52N and 0.88N. The disturbance frequency co~ was imposed at values of between 0.8 COl and 0.38 col. All results are presented on Fig. 5, each being represented by a dashed or solid bar depending on whether the disturbance decayed or remained visible. The abscissa value of the bar is the actual measured stream function amplitude of the large
I
I
ioo
I
I
I
I
I
--
II I
Stable
~'c) E~ measured
I I
~'Fc |
I
i
,
Jti I
IIIIll ~
I
/ .
t
0
/
Unstable
I
5
I
I0
I
I~ ~lJ meosured
15
20
Fig. 5. Measured a m p l i t u d e of background wave, Je~JlJ measured versus critical amplitude ekT'c predicted using eq. 2.19. The upper limit of the bar is the value p r e d i c t e d for the initial d e t u n e d wave, as imposed b y the external wavemaker. The lower l i m i t is the value for the nearest wave on the resonant locus (q.v. Fig. l a ) . The d o t marking this limit is o p e n or closed indicating that the wave was observed to decay or to be sustained. The solid line delineates the c o n d i t i o n e~ c = I e ~ l Imeasured-
103
wave, levi I. The upper and lower limits of each bar on the ordinate are, respectively (a) the critical amplitude of the large wave e~c as predicted using eq. 2.19 for the initially imposed, and usually non-resonant fine disturbance; and (b) the critical amplitude predicted for the nearest resonant fine disturbance. The length of the bar is thus an indication of the level of detuning. In calculating these amplitudes leSK21 = ~ -- h:2 as indicated by (2.23) et seq. I ~ 1 was directly measured from shadowgraphs and its angle was calculated from co'2/N. ~_z was taken as the point on the ~, resonant locus, calculated iteratively, whose normal in wavenumber space passes through ~ . To account for sidewall viscous dissipation the damping coefficients as given by (2.10) were increased by a factor of [1 + ( 2 ~ , / v ) l 1 2 m , ( k , + m,)/WI ~, 14], where W is channel width, vs, being O(10 - a v) in this experiment, was neglected. Also drawn on the figure is a line for the condition e ~ c = l e ~ l I measured, results to the left of which are predicted to show decay and vice versa. It is quite apparent that the lower limit of the bars ( the resonant critical amplitudes) are those which conform most closely to this condition. This is emphasized b y marking these limits b y circles, open for decaying disturbances and closed for those which were sustained. The significance level of the correlation is greater than 99%. Maximum significance, as indicated b y a maximum in f,(a) -- fs (a) where f. and f8 are, respectively, the proportions of 'sustained' and 'decaying' results for which e ~ (resonant)/I e~,L (measured) is less than a, occurs when a is a b o u t 1.15. In other words the results suggest that the disturbances became unstable at a background wave amplitude some 13% less than the minimum predicted b y the present theory. It is seen from the figure that there is no noticeable dependence u p o n the length of the detuning 'bar' and that for the unstable results with only t w o exceptions the predicted critical amplitude for the original detuned wave lies above the measured amplitude. 5. CONCLUDING DISCUSSION
In § 2 it was shown that detuning and limited packet size have equivalent effects on the propensity of an internal wave disturbance to be amplified b y a stronger and more extensive wavefield of higher frequency. In each case in addition to viscous attenuation there occurs a dispersion of the disturbance relative to its partner in triad interaction with the stronger wave. Nevertheless the results of § 4.2 firmly suggest that the amplitude of the stronger wave required for amplification to take place is defined b y stability conditions for the nearest resonant disturbance. This is consistent with the predictions of § 2.3 and the observations of § 4.2. that the disturbance, if limited in space, will have wave number sidebands, the appropriate members o f which are selectively amplified so that the disturbance rapidly evolves towards resonance with the strong wave. Even at resonance the necessary amplitude for
104
instability seems to be slightly overestimated on average, and this supports the suggestion in § 2.3 that the disturbance, if confined to a packet, evolves gradually downwards in wavenumber magnitude, and therefore requires less forcing for instability. This should not be taken to imply a contradiction to McComas and Bretherton's (1977) prediction of cascade to higher vertical wavenumbers since it refers to the process of a single interaction event. Normally at least one of the interaction products is higher in vertical wavenumber than the highest frequency m e m b e r yielding its energy, and each product in turn is susceptible to further interactive cascade until a scale is reached where damping through viscosity, diffusion and wave breaking prevents further evolution except by the present means. Apart from an apparent sensitivity to frequency modulation in the forcing wave as noted in § 4.1.3, the results substantially support assumptions that the oceanic spectrum, although composed of discrete packets, can be treated for the purpose of calculating energy cascade as a continuum of wavenumbers and frequencies. The critical amplitude for unstable growth as predicted by simple parametric instability theory differs by a numerical factor of order unity depending on wavenumber angle from the present theory; nevertheless in view of the tendency towards resonance for wave packet sidebands, the predictions of McEwan and Robinson (1975} concerning parametric reinforcement of oceanic fine-scales should remain reasonably accurate. Finally, it is n o t e w o r t h y that the foregoing theory gives a rough idea of the likely form of the contours of amplitude in wavenumber space of a cascading wave field. For each wave present of frequency cot the resonance locus lies above, and asymptotically approaches, the line at arcos ( ¢ o J 2 N ) to the k-axis. Each interacting packet will tend to align with one or more of these loci and so as evolution proceeds the amplitude contours will progressively b e c o m e lenticular with their major axes directly roughly towards the origin. In real space the packets or at least the finer scale members would become increasingly beam-like, with interaction and breaking events occurring most commonly at the intersection regions. ACKNOWLEDGEMENT
The experimental part of this study was done at Woods Hole Oceanographic Institution, supported b y a Rossby Fellowship and National Science Foundation (Oceanography Section) grant DES 72-01562. These and the assistance of Mr. R. Frazel are gratefully acknowledged. REFERENCES Garrett, C. and Munk, W., 1972. Space--time scales for internal waves. Geophys. Fluid Dyn., 3: 225--264. Garrett, C. and Munk, W., 1975. Space--time scales for internal waves: a progress report. J. Geophys. Res., 80: 191--297.
105 Hasselmann, K., 1967. A criterion for nonlinear wave instability. J. Fluid Mech., 30: 737-739. Mander, D.W., 1974. Thesis, Monash University, Melbourne, Australia. McComas, C.H. and Bretherton, F.P., 1977. Resonant interaction of oceanic internal waves. Submitted to J. Fluid Mech. MeEwan, A.D. and Robinson, R.M., 1975. Parametric instability of internal gravity waves. J. Fluid Mech., 67(4): 667--687. McEwan, A.D., Mander, D.W. and Smith, R.K., 1972. Forced resonant second-order interaction between damped internal waves. J. Fluid Mech., 55(4): 589--608. Miiller. P. and Olbers, D.J., 1975. On the dynamics of internal waves in the deep ocean. J. Geophys. Res., 80: 3848--3860. Olbers, D.J., 1976. The internal wave field in the deep ocean. J. Fluid Mech., 74: 375-399. Orlanski, I., 1973. Trapeze instability as a source of internal gravity waves. Part I. J. Atmos. Sci., 30: 1007--1016. Phillips, O.M., 1966. The Dynamics of the Upper Ocean. Cambridge University Press. Plumb, R.A., 1977. Wave interaction in a bounded fluid. (Unpublished manuscript). Wunsch, C., 1976. Geographic variability of the internal wave field -- a search for sources and sinks. J. Phys. Oceanog., 6(4): 471--485.