M easurements of Turbulence Param eters and Observations of
Multipath. Arrivals in Two Contrasting Coastal Environm ents
Using Acoustical Scintillation Analysis
by
Daniela Di Iorio
B.Sc., University of Victoria, 1988A D issertation subm itted in P artial Fulfillment of the Requirem ents for the Degree of
D O CTO R OF PH ILOSOPHY
in the D epartm ent of Physics and A stronom y
We accept tills thhsis as conforming to the required standard
Dr. David M. Farm er, supervisor (Dept, of Physics)
Dr. A rthur W atton, supervisor (Dept, of Physics)
Dr. R obert W. Stew art, member (SEOS, D ept, of Physics)
Dr. Chris J. R. G arrett, m em ber (Dept, of Physics, SEOS)
Dr. Acla^flMfteliiyski, outside m em ber (Dept. Elec. Eng.) ,v
y
1,
1---Dr. Valerian I. Tatarskii, external (N O A A /E T L , Boulder Co.)
© D A N IELA Di IORIO, 1994 University of Victoria
All rights reserved. D issertation m ay not be reproduced in whole or in p a rt, by photocopying or other means, w ithout permission of th e author.
ii
Supervisor: Dr. David M. Farmer
A b str a ct
The purpose of this research is to explore the potential of acoustical propagation as a probe of oceanographic processes in the coastal ocean. This is done by exam in ing acoustical scintillation in two contrasting regimes: a tu rb u len t tidal flow, and a quiescent, weakly stratified environm ent.
In one experim ent, two-dimensional arrays of acoustic projectors and receivers are used to exam ine the refractive index variability in a high Reynolds num ber, low Richardson num ber tidal flow through a channel. Our acoustic d a ta are compared w ith predictions of the weak scattering theory of Tatarskii assuming a Kolmogorov turbulence model, and interpreted in term s of available oceanographic d ata. Using th e acoustic propagation and turbulence models we are able to m easure some of the oceanographic processes characteristic of a tu rb u len t channel flow.
It is found th a t th e dom inant com ponent of the observed acoustical scintillations is due to tu rb u lent velocity fluctuations, leading to estim ates of the p ath averaged tu rb u lent kinetic energy dissipation (e) which rises and falls with the tidal current (e ~ 10~' to 10- 5 m 2s~3). Analysis of th e low frequency variability in the two dim en sional angle of arrival shows th a t advection of the acoustic waveform by current and refraction by stratification can be modelled by a ray approach. The two-dimensional angle of arrival distribution for rapidly fluctuating acoustic signals is correlated du r ing strongly sheared flow. B oth these characteristics confirm th a t the turbulence is anisotropic over scales less than 32 m. This anisotropy is discussed in term s of the horizontal (.t) and vertical (2) cross channel velocity (v ) gradients: d v / d x and d v / d z . Based on th e dissipation m easurem ent the root mean square velocity gradient, cal culated over the tu rb u len t wavenumber scales, does explain the to tal m ean square arrival angle fluctuations w ithin a factor of 3. An a tte m p t to model the anisotropy in term s of th e refractivity spectrum and the Reynolds stress both proved to be im practicable. These m easurem ents provide m otivation for fu rther studies com paring
C o n te n ts
A b str a c t ii
T able o f C o n te n ts iv
L ist o f T a b les v iii
L ist o f F ig u res ix
G lo ssa r y o f N o ta tio n x v iii A c k n o w le d g e m e n ts x x ii D e d ic a tio n x x iv 1 In tr o d u c tio n 1 1.1 M o tiv a tio n ... 2 1.2 E xperim ental a p p r o a c h ... 4 1.3 Thesis o u t l i n e ... 6
2 S ta tis tic a l D e fin itio n s 8
2.1 Covariance f u n c t i o n s ... 8 2.2 S tructure f u n c ti o n s ... 9 2.3 Spectral fu n c tio n s ... 10
V
3 A c o u stic P r o p a g a tio n T h ro u g h R a n d o m M ed ia 13
3.1 T urbulent Random M e d i a ... 14
3.2 Acoustic P ro p a g a tio n ... 17
3.2.1 Acoustic flu c tu a tio n s ... 17
3.2.2 Weak scattering t h e o r y ... 20
3.2.3 M ultiple scattering t h e o r y ... 26
4 A c o u stic I n str u m e n ta tio n 27 4.1 Cordova C h a n n e l ... 27
4.2 Saanich Inlet ... 32
5 M e a su r e m e n ts o f T u rb u len ce P a r a m e te rs in C ord ova C h a n n e l 35 5.1 I n tr o d u c tio n ... 35
5.2 Oceanographic C h a ra c te ris tic s ... 38
5.2.1 C urrent m eter observations ... 38
5.2.2 CTD o b serv atio n s... 40
5.3 Acoustic C h a r a c te r is tic s ... 45
5.3.1 A m plitude and phase c a lc u la tio n s ... 45
5.3.2 Covariance s c a l e s ... 52
5.3.3 S tructure f u n c tio n s ... 53
5.3.4 Spectral d e n s it ie s ... 55
5.4 P a th Averaged Oceanographic M easurem ents Assuming Isotropic T ur bulence ... 6 6 5.4.1 Turbulent outer scale (L 0) e s t i m a t e ... 6 6 5.4.2 C urrent speed . 69 5.4.3 v a r ia b ility ... 76
5.4.4 Turbulent kinetic energy dissipation ( e ) ... 82
5.5 T and X Param eters - Comparison W ith O ther A coustic E xperim ents 8 8 5.6 Two-Dimensional Angle of Arrival D istribution ... 92
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5.6.2 High frequency fluctuations ... 98
5.7 Anisotropy M o d e l s ... 114
5.7.1 Separable refractivity s p e c t r u m ... 114
5.7.2 Turbulent velocity c o rre la tio n s ... 119
5.8 Sum m ary and C o n c lu sio n s ... 1 2 2 6 M u ltip a th A c o u stic P r o p a g a tio n C h a r a c ter istic s in S aan ich I n le t 125 6.1 In tr o d u c tio n ... 125
6.2 O ceanographic C h a ra c te ris tic s ... 128
6.2.1 C urrent m eter observations ... 128
6.2.2 CTD o b serv atio n s... 128
6.2.3 Therm istor chain o b s e rv a tio n s ... 131
6.3 Acoustic C h a r a c te r is tic s ... 137
6.3.1 M ultipath a n a l y s i s ... 137
6.3.2 M ultipath separation algorithm - am plitude and phase calcula tions ... 139
6.3.3 Correlation a n a ly s is ... 147
6.3.4 Probability d is tr ib u tio n s ... 148
6.3.5 Spectral c h a ra c te ris tic s ... 151
6 . 4 Sum m ary and C o n c lu sio n s ... 154
7 C o n clu sio n s o f th e T h e sis 156 7.1 Recom m endations for future w o r k ... 157
B ib lio g r a p h y 159 A B e s s e l I d e n titie s for th e P h a se D iffer en ce S p e c tr u m 164 B A c o u stic a l C u rren t M e a su r e m e n ts 166 B .l Log-am plitude cross correlation technique ... 166
v i i B.3 Delay to p e a k ... 172
C A c o u stic a l M ea su r e m en ts 173
C.l Log-am plitude variance m e t h o d ... 173 C.2 Wave stru ctu re function m e th o d ... i 76
L ist o f T ab les
4.1 Acoustic param eters lor Cordova Channel scintillation experim ent. . . 29 4.2 Acoustic param eters for Saanich Inlet scintillation system for deploy
m ent 1... 33
5.1 Observed phase difference resulting from changes in tem p eratu re alone and changes in both tem perature and parallel current com ponent. . . 48 5.2 C alibrating factor /?. lor current speed determ ination based on diverg
ing acoustic paths (d) and parallel acoustic paths (p )... 70 5.3 A sum m ary of the param eters used for m easurem ents of P and .V for
strong and weak ebb flow... 89
6.1 D istance between therm istor m oorings... 132 6.2 Probability distributions under satu rated (Saanich Inlet experim ent)
and un satu rated conditions (Cordova Channel experim ent) for norm al ized am plitude, intensity, log-am plitude and log-intensity... 151
B .l Inverse of th e calibrating factor R for current speed determ ination based on diverging acoustic paths (d) and parallel acoustic paths (p). 167
L ist o f F ig u r e s
3.1 Diagram to show the scale size which produces destructive interference at the receiving plane (L). This size is known as the radius of the first Fresnel zone... 18
4.1 D eploym ent scheme for the square and linear acoustic array in Cordova Channel. Linear array has 0.61 m separation between transducers. . . 28 4.2 T he in-phase and quadrature components giving the am plitude and
phase as a function of arrival tim e (in samples) for a given receiver listening to four tran sm itters (21 samples = 1.211 m s)... 31 4.3 T ran sm itter and receiver array configurations for Saanich Inlet. Linear
array ( —): T / R 1,2,3,4; Square array (X): T /R 1,4,6,7; L-shaped array (L): T /R 1,4,5, 6... 32 4.4 T he in-phase and q u ad ratu re com ponents giving am plitude and phase
as a function of arrival tim e (in samples) for T 1 /R 7 (73 samples = 3.24 m s)... 34
5.1 Cordova Channel showing instrum ent locations, together with the tra n s m itte r array (T) and hydrophone array (R). Recording current m eters were moored a t stations 1, 2 and 3 (•); CTD profiles and tim e series were obtained at station A ( ■ ) ... 37
5.2 Observed current m eter d a ta (15 m etre) at station 2 (•) for 2.75 days. T he current velocity is shown as two components: along channel cu r rent com ponent (_L) at 4° True, and the cross channel current com po nent (||) a t 94° T rue... 39 5.3 (a) C urrent speed perpendicular to the acoustic path. Sam ple pro
files and tim e series of tem perature taken through the tidal cycle are shown as P and T respectively. Sample acoustic tim e series (Figure 5.6) are taken at A t. A sample phase difference spectrum (Figure 5.18) and one-dimensional spectrum for refractive index fluctuations due to scalars (Figure 5.23) are taken at /l s. (b) Tim e series and (c) profiles of tem p eratu re during the tidal cycle... 42 5.4 Exam ple of phase (<f>) and tim e of arrival (T) plotted to the same scale.
All times are relative to the receiver ping tim ing m ark (R PT M ). . . . 46 5.5 Exam ple of am plitude and phase tim e series recorded during a 12 hour
period on O ctober 24. D ata were averaged for 10 seconds (5 ; = 0.06). 48 5.6 Exam ple of log-am plitude (y), phase (</>) and phase difference {S(j>)
tim e series during (a) strong ebb flow ( Sf = 0.1 1) and (b) slack water.
(S] = 0.05). These tim es correspond to A t in Figure 5 .3(a)... 50 5.7 Twelve (a) log-am plitude and (b) phase spectra taken during a 12
hour m easurem ent period in order to show m easurem ent related noise a t the highest frequencies. The low pass filter cutoff point is shown as
f c — 6.3 H z for log-am plitude and f c = 4.5 H z for phase... 51 5.8 T he norm alized spatial cross covariance function for log-am plitude fluc
tuations for parallel paths (solid curve) and diverging paths (dashed curve). E xperim ental m easurem ents are shown as • (parallel paths) and A (diverging p a th s)... 53 5.9 The norm alized tem poral cross covariance function for log-am plitude
fluctuations for parallel (solid curve) and diverging (dashed curve) paths taken at a tim e of strong ebb and close to slack w a ter... 54
x i 5.10 T he normalized tem poral cross covariance function for log-am plitude
com puted for different receiver spacings using parallel paths. D ata are taken during strong ebb flow... 54 5.11 T he log - am plitude, phase and wave tem poral stru ctu re functions
(D x {px = 1.18m, r) , Dj,{px = 1.18m, r ) and D( px = 1.18m, r ) respec tively) computed for diverging paths. D ata were taken during strong ebb flow... 55 5.12 Theoretical (solid curves) log-amplitude spectra showing th e effects of
aperture averaging and tw enty experim ental (dots) log-am plitude spec tra taken through a tidal cycle superimposed. Normalizing frequency is defined as / 0 = U/ { XL ) 1/2... 60 5.12 Variance preserving plot of the theoretical (solid curve) log-am plitude
spectra and twenty experim ental (dots) spectra taken through a tidal cycle superimposed. Normalizing frequency is defined as /o = U f ( X L ) 1^'2 for (AL) I / 2 = 3.8m ... 60 5.14 Theoretical (solid curves) phase spectra showing the effects of ap ertu re
averaging and twenty experim ental (dots) phase spectra taken through a tidal cycle superimposed. Normalizing frequency is defined as /o =
U / ( X L) ' / 2... 61 5.15 Theoretical (solid curve) phase difference spectra for horizontally spaced
receivers showing th e effects of aperture averaging and tw enty exper im ental (dots) phase difference spectra taken through a tidal cycle superim posed. Normalizing frequency is defined as f ix — U/ px where
px = 0.61m ... 64 5.16 Variance preserving plot of th e theoretical (solid curve) phase difference
spectrum for horizontally spaced receivers and tw enty experim ental (dots) spectra taken through a tidal cycle superim posed. Normalizing frequency is defined as f \ x — U/ px for px = 0.61m ... 64
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5.17 Theoretical curves for the phase difference spectra with ap ertu re aver aging (R = 4.5 crn) as a function of L 0. The Kolmogorov spectrum is identified as L 0 = oo and the receiver spacing is px = 0.61m ... 6 8 5.18 Phase difference spectrum during a tim e of stratification when an outer
scale of 16 m etres is inferred (see A s on Figure 5 .3 ( a '\ The level of th e theoretical spectrum is cidjusted by a factor of two and th e receiver spacing is px — 0.61 rn... 6 8 5.19 Comparison of th e along channel current component observed by a
16 rn current m eter at station 2 (solid curve) with acoustical scin tillation techniques (dots) using (a) the slope of th e normalized log- am plitude cross covariance function (5.28), (b) th e slope of the nor malized wave stru ctu re function (5.29) and (c) the delay to peak (5.30) using 5 m inute averages... 71 5.20 Theoretical (solid curve) and measured wave stru ctu re function during
a strong ebb of —0.85 m ,s_ 1 on 24/10/86 OOOOh. Low pass filtered d a ta are shown as a dashed curve and band pass filtered d a ta are shown as a dash-dot curve... 73 5.21 (a)T he current speed obtained from a 15 in current m eter at station
2 (solid curve) a,rid from acoustic delay to peak m ethod for parallel paths (dots). The mean vertical shear d U / d z determ ined by (b) two vertically spaced current m eters and by (c) the acoustic data. D otted line represents 7.5 m inute averages and solid line represents 15 m inute averages obtained from a moving average filter over 30 m inutes. . . . 74 5.22 (top) C urrent speed for 30 hours of data, (bottom ) S tru ctu re p aram
eter (•) calculated from (a) th e acoustic log-am plitude variance, cr£ (5.33) and (b) the wave stru ctu re function, D ( p , r = 0) (5.34). (c) S tru cture param eter C% (-) calculated from th e level of th e one dim ensional spectrum of th e refractive index fluctuations based on CTD and current m eter m easurem ents, $ i ns(/ci) (5.36)... 78
5.23 T h e one-dimensional spectral density (two-sided) of th e scalar refrac tive index fluctuations from CTD d a ta when an outer scale of 16 m could be inferred (see A s in Figure 5.3(a)). Solid curve is based on 1024 point FFT and broken curve is based on 2048 point FF T . A -5/3 slope is shown for com parison... 5.24 P a th averaged turbulent energy dissipation (per unit mass) e calculated
through th e tidal cycle (+ ) superimposed w ith theoretical estim ates (solid curve) based on a balance between production and dissipation of tu rb u len t energy... 5.25 T he T, X param eter plane. D ata are taken from: Ew art and Reynolds
[22] Cobb71-MATE77 experim ent, Reynold’s et.al. [47] AFAR ex perim ent, Uscinski et.al. [60] Napoli85 experim ent and Di Iorio and Farm er’s [18] presei m easurem ents in Cordova C hannel... 5.26 Schem atic showing th e horizontal arrival angle as it is advected by the
current (left) and th e vertical arrival angle (right) as it is refracted by stratification. Indicated angles are: 0a = m easured acoustic arrival angle relative to the line perpendicular to th e receiver separation (T ),
@x,z = arrival angle relative to the y-axis (H), £ = angle of the receiver
separation relative to H, £ = angle of receiver array relative to the tra n sm itte r array in th e x-direction... 5.27 Schem atic showing th e receiver a.rray viewed from th e center of th e
channel. Transducers are labelled 1 to 4. Horizontal directions make use of th e top and bottom receivers; vertical directions m ake use of th e left and right receivers... 5.28 (a) Horizontal angle of arrival determ ined acoustically (dots) by (5.63)
together with the angle measured by th e Mach num ber (equation (5.67)) (solid curve), (b) Vertical arrival angle determ ined acoustically (dots) by (5.64), together with th e angle m easured by th e sound speed gradi en t (equation (5.68)) (solid curve)...
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5.29 Variance preserving plot of the theoretical (solid curve) and experim en ta l (dots) phase difference spectra for (a) horizontally and (b) vertically spaced receivers. Experim ental spectra are taken through a tidal cy cle. Normalizing frequency is defined as f i x = U/ px and f ix = U/ pz for px = 1.18m and pz = 1.03m... 100 5.30 Norm alized phase difference spectra for (a) horizontally spaced re
ceivers (J \ x — U/ p x , px = 1.18?n) and (b) vertically spaced receivers
( f i z = U / p z , pz = 1.03m). Theoretical spectra are shown as a solid
curve, d a ta with no filtering are shown as a dashed curve and filtered d a ta are shown as a do tted curve... 1 0 1 5.31 Normalized angle of arrival correlations during ebb (left) and flood
(right) flow, (a) Correlations from specific horizontally and vertically spaced receivers. Solid curve is < (r ) > an<^ dashed curve is < #;r3_2#22_ j(r) > . (b) Correlations averaged over the square array. Solid curve is < 0x6z( t ) > a and dashed curve is < 0x6z ( t ) > a0 - ■ ■ ■ 103
5.32 Two dimensional angle of arrival distribution through th e tidal cycle shown in Figure 5.28. Each plot represents 20 m inutes of high passed filtered d a ta and th e colour tabulates the num ber of tim es th e acoustic signal comes from a specific direction... 106 5.33 (a) Structure param eter C% (•) calculated from th e acoustic log - am pli
tu d e variance, (b) M easured tu rb u len t kinetic energy dissipation (+ ) and th e estim ated dissipation for a drag coefficient of Co = 3 x 1 0 - 3 (solid curve)... 107 5.34 T im e series showing the stan d ard deviation of th e scatter along the m ajo r and minor axes, th e angle of th e m ajor axis relative to the x- axis, and the acoustic arrival angle correlation coefficient averaged over th e array... 109
XV 5.35 Theoretical functions Zhvo, £hv2n an^ A v22 plotted against tim e lag.
T h e experim ental wave structure function D ^ is shown as a dashed curve and th e best fit Dn = —2.057 + D^o — 1.5 9 3 7 ) ^ 2 1 + 0.3517)/V22 is shown as a solid curve... 118 5.36 Anisotropy param eters showing (a) the m agnitude « 2 and (b) direction
02 as a function of time. True isotropy occurs for a2 = 0... 118
6.1 Saanich Inlet showing instrum ent locations, together with th e tra n s m itte r array (T) and receiver array (R). Recording cu rren t m eters were moored a t stations 1, 2 and 3 (•). Therm istor chains were moored at station 1, 2, and 3 (A ). CTD profiles were done in term itten tly at statio n 3A and station 2«... 126 6.2 Observed current m eter d a ta (60 m etre) at statio n 3 (•) for a three
day period. Horizontal line coincides with the acoustic and therm is to r chain observations described. The _L com ponent is relative to 120° T r u e and the || com ponent is relative to 30° T r u e ... 129 6.3 T em p eratu re - salinity diagram showing density ( at) in kg m ~ 3 and
sound speed in m s- 1 contours... 130 6 . 4 Average profiles obtained from 6 CTD casts of (a) sound speed and
(b) Brunt-V aisala angular frequency ( N( z ) ) w ith an exponential least squares fit superim posed... 132 6.5 (a) T herm istor tim e series as a function of dep th taken a t statio n 3
(A ) for a three day period, (b) T herm istor chain profiles for an 8 hour period on April 4. (c) Tem perature fluctuations where th e m ean te m p e ratu re profile in (b) is subtracted from each of th e profiles. . . . 134 6 . 6 M ean sound speed profile (solid curve) from th erm isto r chains, together
w ith the m inim um and m axim um profiles (dashed curve). D ata are shown as d o ts ... 136
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6.7 (a) Acoustic am plitude measured as a function of relative arrival tim e and as a function of elapsed time. M easurem ents are from T 4 /R 4 transducers, (b) Averaged sound speed profile using CTD d a ta and th e acoustic eigenrays for a source and receiver a t 35 m d e p th .... 138 6 . 8 Correlation of the received signal with a tem plate of th e tran sm itted
pseudo random noise (PRN ) code (3 samples = 1 b it = 133 fis). . . . 140 6.9 Integration of the noise for an assumed num ber of paths N. The inte
grated signal level is shown for N = 0. Averages are shown as an * on th e far right side... 142 6.10 Acoustic am plitude (*) and phase ( x ) m easured as a function of arrival tim e when (a.) the signal level is high and (b) the signal level is low. Superim posed are the resolved paths together with the modelled signal (dashed curve)... 144 6.11 Expanded view of the second arrival shown in Figure 6.7(a). Superim
posed is a trace of the three separated p a th s... 145 6.12 T h e phase a.nd the tim e of arrival for th e second p ath shown in Figure
6.11 plotted to the same scale... 146 6.13 T im e series of am plitude and phase over a 9 hour period ... 147 6.14 T he normalized cross covariance of log-am plitude for (a) diverging paths and (b) parallel paths. Solid curve is for horizontally spaced receivers and dashed curve is for vertically spaced receivers... 149 6.15 Saanich Inlet probability distributions for (a) w ith an exponential distribution superimposed and (b) In (7 7 5-)... 150 6.16 (a) Power spectral density for f ~ 3 dependence is due to fine
structure, (b) Power spectral density for <j>. f ~ 2'5 dependence is due to internal waves... 152
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B .l W eighting function Wc, (s, /?) for different values of (3 showing the rela tive contribution from different parts of the acoustic path to the current speed m easurem ent, (a) Parallel paths, (b) Diverging p a th s ... 168 B.2 W eighting function showing the relative contribution from dif
ferent p arts of the acoustic path to the current speed m easurem ent for parallel paths (dotted curve) and diverging paths (solid curve)... 171
C .l W eighting function M/c0(iS) showing the relative contribution from dif ferent parts of the acoustic path to C„... 174 C.2 W eighting function H/d0('S) showing the relative contribution from dif
ferent p arts of the acoustic path to for parallel paths (dashed curve) and diverging paths (solid curve)... 177
G lo ssa r y o f N o ta tio n
A c o u s tic p a r a m e te r s
fa, u a = 2irfa acoustic frequency [Hz], and angular frequency [rad s _1],
A, k = 2tt/ X acoustic wavelength [nr] and wavenumber [rad m -1 ],
f>, t spatial [m] and tem poral [s] separations,
L acoustic path length [???.],
XL Fresnel radius [m],
X, Q in-phase and q u ad ratu re com ponents,
A am plitude,
I = A 2 intensity,
X = ln(;4 / < A > ) log-amplitude,
T , <j) tim e of arrival [s] and phase [rad],
6<j) phase difference [rad],
0ax, 0a_ horizontal and vertical acoustic arrival angle [rad],
R i ( r ) intensity auto covariance,
s ] = R , ( o y < I > 2 scintillation index,
C x (p,T) two dimensional space - tim e log-am plitude cross covari
ance function,
x i x
a 2 = Cx(0,0) log-am plitude variance,
D ( p . r ) = D x { p, r ) + D^ (p,t) wave structure function defined as a sum of
log-am piitude and phase stru ctu re functions,
frequency spectrum for log-am plitude [ Hz - 1], phase
[rad? H z -1], or phase difference [rad2 H z -1],
f0 — U/ y / XL characteristic frequency induced by Fresnel scale
sizes advected by the m ean flow [Hz],
f l = U/ p characteristic frequency induced by receiver separa
tion scale sizes advected by th e mean flow [Hz],
T, X scattering strength of the refractive index fluctua
tions, and normalized propagation range.
M e d iu m p a r a m e te r s
x, y, z along stream , cross stream and vertical cartesian coordinates,
c0, ^ mean sound speed [m s -1 ] and sound speed gradient [s-1 ] a.t
th e depth of acoustic propagation,
p0, j : mean density [kg m ~ 3] and density gradient [kg m ~ 4] at the
depth of acoustic propagation,
N ( z ) Brunt-V aisala angular frequency [rad s _1],
n — n 0 + ij refractive index in term s of a mean and fluctuating com ponent,
K , k = |K | = y - 3D refractive index wave num ber and the m agnitude of th e
refractive index wavenumber vector [rad m _1],
t 0, L 0 inner (dissipating) and outer (energy containing) scale of tu r
bulence [m],
Cd drag coefficient a t the depth of the acoustic paths,
K von K arm an ’s constant,
XX
2 path averaged distance from th e boundary to th e acoustic
propagation depth [m],
D average depth of channel or inlet [m],
U, U point and path averaged m easurem ent of th e current perpen
dicular to the acoustic propagation direction [m s -1 ],
d u dz
d u d x
vertical along channel mean velocity gradient [s_1],
velocity fluctuations along the x, y, and z directions [771 s -1 ], turbulent velocity gradient [s-1],
( f f | f ) correlation of horizontal and vertical cross channel velocity gradient [s-1 ],
60x « U/ c 0 horizontal arrival angle due to advection of th e acoustic wave
form [rad],
6Z « vertical arrival angle due to refraction [rad],
t, I point and p ath averaged m easurem ent of th e tu rb u len t kinetic
energy dissipation [77i2s -3 ],
^V<(r ) = refractive index structure function for scalar variables,
oc one dimensional spectral density for isotropic and homoge neous refractive index fluctuations due to scalars [771],
C 2 = C l + 11/6C l refractive index stru ctu re p aram eter in term s of scalar and vector properties jm~2/3],
$ n (« ) oc CnK 1 1 t hr ee dimensional spectral density for isotropic and homoge neous refractive index fluctuations [7n3],
l i n( K , w ) three dimensional space - tim e spectral density for refractive index fluctuations [m3.s],
E ( k ) oc e2/,3K-5 /3 three dim ensional spectral density for isotropic and hom oge neous velocity fluctuations [7n3s -2 ].
XXI
D im e n s io n le s s p a r a m e te r s
Re = ^ Reynolds num ber,
A c k n o w le d g e m e n ts
I would like to thank Dr. David Fanner for giving me the opportunity to work on an extrem ely challenging and interesting research project. 1 am grateful for his enthusiasm , guidance, support and encouragem ent at every stage of my research. 1 would also like to thank Dr. Stew art for many helpful and stim ulating discussions on turbulence and channel flow properties. I thank Dr. G arrett for questioning th e basic physics of acoustic propagation a.nd for careful analysis of the thesis. I thank Dr. W atton for his support as a university supervisor and also for his detailed analysis of the theoretical w rite up. Finally, I would like to thank Dr. Zielinski for his interest in the instrum entation and seiving as a com m ittee member.
Thanks are due to David Lemon and Rene Chave and their colleagues at ASL E nvironm ental Sciences for designing, building and deploying the acoustical scin tillation system . I thank them for providing continuous support in analysing the raw data. T hanks are due to Tom Juhasz, Reg Bigham, John Love, Les Spearing, Andrew Lee and th e crew of the CSS V EC TO R and PARIZEAU for designing, de ploying and recovering all the oceanographic instrum entation. Thanks are also due to Mike Dem psey and David Spear for their support in servicing th e oceanographic instrum ents.
I am very grateful to Grace K am itakahara-K ing for her continuous com puter sup port and to Ron Teichrob for his technical support. I would like to also th an k my Ocean Acoustic friends and colleagues for stim ulating discussions and support through the years, m aking my life a t the In stitu te of Ocean Sciences interesting and enjoyable. In particular, I th an k Grace K am itakahara and David King for th eir wonderful d in
x x n i
ner parties, Len and Liz Zedel for m any wonderful hikes, Yunbo Xie for all the music lessons and duets, Dimitris Menemenlis for the wonderful philosophical argum ents on politics and science and for organising IOS soccer, Li Ding for stim ulating discussions on tim e series analysis, Craig McNiel for all the laughs and the pub nights, Mark Trevorrow for the soccer games ending at the pub, Willi W eichselbaumer for his help on com puter system debugging, Zhen Ye for his ideas on two-dimensional structu re functions, Anna-Lea Rantalainan and A1 Adrian for the bike rides, Donald Booth for his outspoken and entertaining views, P eter Hallschmid for processing the Saanich Inlet d ata, and Terry Russell for her C EO R adm inistrative help.
Last, bu t riot least, I would like to thank Raymond Clary for giving me th e love and support during my undergraduate years and encouragem ent for startin g the Ph.D. Also, I thank Don Newman for giving me love and support during the final stages of the Ph.D . and for his helpful ideas on the design of the frontispiece. Finally, I thank my m other, Livia Di Iorio and sister, M afalda Di Iorio for th eir continued patience and dedicate this thesis to my nephew, Nicholas Edward Benn and to my parents.
This work received support from the Canadian Panel on Energy Research and D evelopment, and the U.S. Office of Naval Research. The author was funded by scholarships from the N atural Sciences and Engineering Research Council of C anada and the Science Council of British Columbia.
I dedicate this thesis to m y nephew
N icholas E dw ard Benn,
an d to iny p arents,
M atth ew an d Livia Di Iorio
XXV
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T w inkle, tw inkle little sta r, how I w onder w h at you are.
U p above the world so high, like a diam o n d in th e sky.
T w inkle, tw inkle little sta r, how I w onder w h at you are.
C h a p te r 1
I n tr o d u c tio n
Much work has been done on the propagation of waves through m edia containing both spatial and tem poral fluctuations in th e refractive index. This type of wave propaga tion occurs in many branches of experim ental physics including radio propagation in the ionosphere, light propagation in th e atm osphere, and acoustic propagation in the ocean (see T atarskii, Ishima.ru, and Zavorotny [54]). The intensity and phase of the signal at th e receiving station show variations th a t evolve with tim e defining a scintil lation p a tte rn . This scintillation p a tte rn contains information about the intervening m edium.
The theory of wave propagation through random m edia (W PRM ) provides the basis for inferring path averaged properties of the propagation m edium from the m easured signal. A theory for wave propagation through a m edium characterized by Kolmogorov’s model of homogeneous, isotropic turbulence, which results in th e characteristic ‘-5 /3 ’ power-law spectrum for velocity and scalar properties of th e fluid, has been developed for the case of weak fluctuations, prim arily by Soviet scientists. This work is sum m arized by Tatarskii [52], and atm ospheric experim ents show good agreem ent w ith th e theory. In m any coastal waters, th e interaction of tid al flow w ith b o tto m topography results in turbulence and hence this theory was used by Farm er et.al. [24] as a tool to m easure current and tu rb u len t stru ctu re in a coastal
environment;.
In th e open ocean, and in more quiescent coastal inlets, however, th e assum p tion of homogeneous, isotropic turbulence and weak acoustic fluctuations cannot be made. Most ocean acoustic experim ents are com plicated by internal waves as the source of random ness, or the path length and refractive index variability are such th a t T atarskii’s weak scattering theory, based on R ytov’s approxim ation, is no longer applicable. Recent theoretical work by Uscinski [57] describes b oth large and small acoustic intensity fluctuations. In 1977 Ewart and Reynolds [22] m ade sim ultaneous m easurem ents of acoustic and refractive index fluctuations. T he physical processes responsible for the refractive index fluctuations are tides, internal waves and fine structure. Their acoustic m easurem ents reveal these effects.
We apply these techniques to active and quiet coastal ocean environm ents by first giving m o ii, ,tion for studying the coastal waters and then describing the experim en tal approach.
1.1
M o tiv a tio n
The coastal ocean is very diverse. The circulation is set by th e local bathym etry, meteorological forcing and the interaction with river run off and deeper oceanic waters. In order for num erical models of the circulation to be accurate, p aram eterization of the turbulent processes is required. To do this, d a ta describing th e stru c tu re of
rbulence are needed.
Turbulence and mixing plays a strong role in th e active coastal environm ent. Since th e currents are strong and the w ater is shallow com pared to th e deep ocean, energy m ust be dissipated in less volume and so th e dissipation ra te and thus th e level of the turbulence is generally higher. This is because th e interaction of th e tidal cu rren t with channel boundaries results in turbulence which occupies a large p a rt and som etim es all of th e w ater column. The turbulence in a channel flow contains stru ctu res over a broad range of scales. The most im p o rtan t source of m ixing in th e coastal ocean is
often turbulence generated by tidally driven currents. Moreover, mixing in a channel flow is a complex and in term itten t process.
In deep coastal inlets and fjords, currents a.re small and stratification changes ?• nally. The circulation in a fjord is complicated because of a sill th a t exists at the entrance of th e fjord. This sill cuts off m ost of the deeper water from interacting with th e adjacent coastal ocean and so circulation is prim arily controlled by density differences. In order to study mixing in this environm ent it is necessary to identify the source of tu rb u len t energy. Is it driven by wind stress, internal waves, directly by current shear or is it driven by bottom stress? Answering this question provides th e necessary background for interpreting and modelling d a ta in this kind of environm ent.
A b e tte r understanding of turbulence, mixing and circulation is vital to progress in coastal oceanography. This is because th e coastal ocean is home to most food fish and shell fish relying on a n u trien t rich environm ent. T he coastal ocean is also subject to th e dilution and dispersion of a great variety of wastes, both liquid and solid. C ontam ination from waste chemicals, paper mill wastes and sewage wastes are often discharged in the coastal ocean and concentrations are diluted through mixing w ith large volumes of ocean waters. T he coastal circulation then disperses th e water masses.
M easurem ent of th e turbulence param eters have traditionally been carried out using in situ sensors of tem p eratu re, conductivity and th ree orthogonal com ponents of current velocity, or using high-resolution vertical profilers. The drawback of these m easurem ents is th a t they are essentially local in character and may not reflect the average properties of the turbulence. P a th averaged m easurem ents of the coastal ocean can be a desirable way to obtain param eterization of turbulence properties in a channel flow for numerical modellers. For many purposes such as tran sp o rt calculations, a p a th averaged result is m ore useful th an a point m easurem ent. P a th averaging has an additional benefit: the effect of sm aller scale processes is filtered and th e scale of greatest sensitivity is controlled by th e p a th length and th e acoustic wavelength. O ur experim ental approach describes a novel way of obtaining p a th
4 averaged m easurem ents of current and turbulent structure.
1.2
E x p e r im e n ta l a p p ro a ch
Acoustical scintillation experim ents described in this thesis were carried ou t in two com pletely different coastal environm ents (Cordova Channel (1986) and Saanich Inlet (1989)) in order to exam ine and contrast the corresponding acoustic characteristics and where possible relate these to th e oceanographic environm ents. T h e prim ary study was carried out in Cordova Channel, and the analysis explores some of th e ways in which m ean and tu rb u len t properties can be m easured acoustically, m aking detailed com parisons with available oceanographic data. T h e contrasting scintillation d a ta from Saanich Inlet are much less completely supported by oceanographic d ata, but nevertheless provides an opportunity to examine che analytical procedures, in particular m u ltip ath separation, th a t m ust be used to derive an in terp retab le signal.
For Cordova Channel we first test th e applicability of th e models within th e lim ita tions of our experim ent, and then use the results to derive p ath averaged turbulence characteristics of th e flow. One quite surprising result of this work is th a t acous tic propagation in a turbulent coastal environm ent leads to a signal dom inated by velocity fluctuations rath er th an tem perature fluctuations, as will be shown. This allows us to evaluate directly the tu rb u len t kinetic energy dissipation. P a th averaged dissipation m easurem ents have been carried out by Menemenlis [41] using reciprocal acoustic transm ission in the boundary layer under A rctic ice to directly m easure th e tu rb u len t kinetic energy spectrum . O ur m easurem ents of p a th averaged dissipation described in this thesis make use of th e forward scattered signal together w ith th e isotropic turbulence model.
Oceanographic m easurem ents from tim e series and profiles together w ith acous tic m easurem ents from square tran sm ittin g and receiving arrays provide a b e tte r understanding of th e two dimensional determ inistic and tu rb u len t effects. T h e d eter m inistic effects are explained in term s of th e ray approach which shows advection and
5
refraction of the acoustic waveform. The turbulent effects are discussed by analysing correlated vertical and horizontal (along-channel) com ponents of th e acoustic angle of arrival fluctuations. These observed correlations imply anisotropic turbulence which is then discussed in term s of the turbulent velocity fluctuations and gradients. An a tte m p t to model the anisotropy is then done.
Propagation of sound over greater path lengths tends to increase the opportunity for m u ltip ath effects. When there is a strong dom inant path the coherence is high; when th ere are m any m ultipaths the coherence is low (Urick [56]). Thus, application of propagation techniques for probing oceanographic processes will invariably confront th e m u ltip ath problem and separation of m ultipaths rem ains one of the outstanding challenges for this approach. Ehrenberg et.al. [20] and Ew art el.al. [21] developed a m axim um likelihood algorithm for separating EM slide and pulsed-tone signals. T he Saanich Inlet d a ta show m ultipath arrivals and we use the m axim um likelihood estim ation algorithm for separating signals processed w ith a m atched filter.
The oceanographic variability in Saanich Inlet was not m easured as com pletely as in Cordova Channel, where tem perature and salinity tim e series m easurem ents in the tidal flow allowed estim ates of the spatial variability over th e inertial subrange. It does however, show an interesting contrast to the tu rb u len t environm ent in Cordova. Channel. T h e analysis of d a ta from Saanich Inlet, which is in some ways a m ore com plex environm ent, is lim ited to a comparison of (1) characteristic oceanographic and acoustic variability and (2) the determ inistic aspects of propagation with m easured sound speed profiles. Some potential explanations for th e form er are provided. Al though th e determ inistic propagation is very well described, th e random com ponents are not, and present a very challenging analysis problem.
This thesis gives further insights into th e way in which tu rb u len t and other fluc tuations in th e ocean contribute to th e received signal, and these in tu rn contribute to our ab ility to tackle the inverse problem of using th e propagation characteristics to resolve oceanic stru ctu re. There are several im p o rtan t features of these experim ents th a t distinguish it from many other acoustic experim ents. These include, th e use of
6
a high acoustic frequency, the use of two-dimensional rigid tran sm itter and receiver arrays, th e use of a coherent scheme allowing precise phase m easurem ents and high enough sam pling rate and signal-to-noise ratio to allow resolution of phase am biguity in the phase tim e series.
An understanding of acoustic propagation and the scintillation p attern is essential in attem p tin g to use the acoustics for measuring oceanographic variables, as well as for other applications such as underw ater comm unication system s, positioning devices and fish stock assessment. ASL Environm ental Sciences L td., have m ade scintillation flow m easurem ents in the Fraser River (see Lemon [37]) and in hydroelectric dam s for efficiency calculations (see Birch and Lemon [2]). The understanding th a t results from this work provides the necessary scientific basis for commercial application of the scintillation technique.
1.3
T h e s is o u tlin e
An overview of statistical functions and their notation i given in C hapter 2. C hap ter 3 provides some of th e background theory for acousti* propagation in random m edia. Following is a description of the random m edium in terms of isotropic and homoge neous turbulence. Finally, a description of the acoustic propagation models for weak and m ultiple scattering is presented. This chapter is t review chapter and th e only original contribution to acoustical oceanography is the discussion of th e refractive index stru ctu re param eter in term s of both velocity and scalar contributions.
C hapter 4 describes the novel acoustic instrum entation used to study th e tu rb u lence in Cordova Channel (1986) and th e m ultipath acoustic signals in Saanich Inlet (1989). T he acoustic system was developed by ASL E nvironm ental Sciences under contract to the In stitu te of Ocean Sciences. I had th e opportunity to p articip ate in th e Cordova Channel experim ent as a co-op student and for th e Saanich Inlet experim ent I helped organize and collect all th e d ata.
7 obtained in Cordova Channel. An overview of the in sii.u oceanograph' m easurem ents are described and the channel is quantified in term s of th e Reynolds and Richardson num bers as a first step in defining the turbulence. Acoustic m easurem ents of am pli tude and phase are described together with a comparison with th e weak scattering theory. Since th e acoustic scintillations can be modelled, we can use our observa tions to obtain path averaged oceanographic m easurem ents. A new technique for the acoustical rem ote sensing of the turbulence outer scale, current, m ean shear, refrac tive index stru c tu re param eter and the turbulent kinetic energy dissipation is the m ost significant original contribution of the thesis. We com pare our acoustic propa gation experim ent to other experim ents and show th a t this is th e first tim e acoustical scintillation m easurem ents have been reported in a coastal environm ent characterized by fully developed turbulence.
The two dim ensional angle of arrival distribution is analyzed to reveal both de term inistic and tu rb u len t effects. The determ inistic effects are described in term s of refraction and advection of the acoustic waveform and th e tu rbu len t effects show a degree of anisotropy. For this reason an anisotropic model for the refractivity spec tru m is exam ined. Also, the angle of arrival correlations are discussed in term s of the cross channel velocity gradients. This section of the thesis is also a significant original contribution.
Chapter 6 describes the acoustic multipath propagation characteristics observed in Saanich Inlet. The most original contribution in this chapter is primarily the multipath separation technique developed for match filtered signals. The results from this analysis together with the oceanographic analysis formulates the “forward” problem of acoustic propagation. This kind of environment demonstrates the difficulty in using acoustical scintillation analysis to solve the “inverse” problem, that is, to measure the oceanographic characteristics.
Chapter 7 is the conclusions of the thesis and outlines all the oceanographic mea surements that can be made with a relatively simple square and linear acoustic array. Some recommendations for future work is also suggested.
C h a p te r 2
S ta tis tic a l D e fin itio n s
A review of the statistical functions m easured and described throughout th e thesis will be sum m arized in this chapter. Although the notation varies from au th o r to author, I have tried to use th e same notation as Tatarskii [5*2].
2.1
C o v a ria n ce fu n c tio n s
R andom processes can be described by statistical quantities. The sim plest statistical quan tity of a random function g(r, t) is its m ean < g(r, t) > , where th e angle brackets
< > denote an average over tim e a t th e location r. T he next statistical q u an tity of im portance is the space-tim e cross covariance function,
Cg( r u t l , r 2, t 2) = < 5'(ri > * i y ( r2,*2)* > i C2-1)
where g' ( v, t ) = g ( r , t ) — < g ( r , t ) > is a zero m ean process and * denotes complex
conjugate in cases where th e function g represents propagating wave fields. If the
random process is a stationary random variable, then th e statistical m om ents rem ain constant w ith tim e and space, and th e covariance function depends only on th e spatial and tem poral separation (p = iq — r2 and r = t\ — t 2 respectively). T h a t is,
< g ( r 1, t i ) n > = < g { T 2, t 2)n > n = 1 ,2 ,3 , ...
and Cg{ r x, t x, r 2, t 2) = C3( p ,r ) . (2.2)
9
Stationarity in space, defines homogeneity. Statistical isotropy implies th a t Cg(p, r ) depends only on th e m agnitude, p = \p\ and not on direction. The spatial covari ance and tem poral covariance function can be recovered by the space-tim e covariance function using th e following notation,
B g(p) = CA P , T = 0), (2.3)
R g(T) = C s ( p = 0,t ). (2.4)
2.2
S tr u c tu r e fu n c tio n s
In oceanographic (or meteorological) conditions, random processes cannot be regarded as statio n ary since mean values change with both space and tim e. To avoid this difficulty, the difference function,
= g'(ri + r , t j + t) - g'{r l5ii), (2.5)
is considered, where ' denotes zero m ean processes. This function removes the effects of large scale inhomogeneities th a t are common to both g(r^ + r, ti + /,) and </(rj, <i). For values of r and t which are not too large, the function G can be considered stationary.
Following Tatarskii [52], the covariance for the function G can be w ritten as the sum,
CG{r!, t u r2, t 2) = < G( r t , t i ) G ( r 2, t 2) > = (2.6)
2 ( < + r, <i + <) - g' ( r2, t 2)}2 > + < ^ ' ( n , <j) - g' (r2 + r, t 2 + t)]2 >
- < ri + M i + 0 ~ 9 ' { r 2 + r , t 2
+
t)}2 > - < ^ ( r i . i i ) -flf'(r2,«2)]2 > ) , by making use of the identity, (a — b) ( c—d) = l/2 [(a — d) 2 + ( b—c)2 — (a — c)2 — ( b—d) 2].Thus the covariance of a stationary random process is expressed as a sum of structure functions,
CG( r i , h , r 2, t 2) = ^ ( D g f a + r , h + t , r 2, t 2) + D g( r 1, t l , r 2 + r , t 2 + t)
1 0 The stru ctu re function can be used for stationary and non-stationary random pro cesses. In order for C g(Ti, Z i,r2,1.2) to depend on th e differences p = ri — 1*2 and r = i } — t 2 it is necessary th a t each structure function depend on the differences in space and time. T h a t is,
C o {p, r ) = ~ D g(p + r, r + t ) + ^ Dg(p - r , T - t ) ~ 2D g(p, r ) (2.8)
Thus th e stru ctu re function characterizes the intensity of those inhomogeneities hav ing periods of order r or smaller and having scale sizes of order p or less. Essentially the stru ctu re function acts as a high pass filter. The spatial stru ctu re function is recovered from th e space-time stru ctu re function by,
V , ( p ) = D , ( p , r = 0). (2.9)
If g'(r, t) is a stationary isotropic random field, then the stru ctu re function can be expressed in term s of the covariance function,
Dg{p, t ) = 2C5 ( 0 , 0) — 2Cg(p, t ) . (2.10)
2 .3
S p e c tr a l fu n c tio n s
The three dimensional spatial spectral density, $ S(K) of a random process g is defined as the Fourier transform of its spatial covariance function (Arfken [1]),
$»(K)= ( 2 ^ / / / e" KPB»(p)^
<211)
where B g(p) characterizes the statistical dependence between fluctuations spatially separated by p. T he spatial covariance function B g and th e spatial spectral density
are Fourier transform pairs so,
S , ( p ) = / / / «iK-p «MK)<OC. (2.12)
Assuming isotropy ($<,(K) = $<?(«0 and B g(p) - B g(p)) and integrating both (2.11) and (2.12) over spherical coordinates (dp = p2 sin 6dpd0d(j> and
11
d K = k2 sin OdndOd(f)) gives,
■r j \ 1 n „ , , sin Kf) , ‘M 'O = 7T~i / P Bg(p) dp, (2.13) 2tt ./(j and. , f ° ° , , s i n w f l , (p) = 4-TT k 4>3(k ) dK. (2.14) J 0 K P
These results are obtained by taking into account the even property of both the covariance function and the spectral density. The variance of the function g K 2) is then defined as,
/»co
,(0) = 4tt / K2$ 9(/c)<k. (2.15)
Jo
= 2 r <*>:ig{K)dK, (2.16)
Jo
where $33(k) = 2nK2(bg(K) is a,n extension of the three-dim ensional spectral density $ fl(/c). T h e spatial stru ctu re function can also be expressed in term s of the spectral density,
© » = 8 t (2.17)
Jo \ K p J
In space, these spectrum functions will have a three-dim ensional character. How ever, w hat can be m easured is the one-dimensional spectrum (see Jlinze [31]),
* ' ’ <■«') = - t J <2-18)
where th e covariance function,
B g(x) =
J
eiK^ g(Kl)dKu (2.19)is m easured along some straight line x in space. The Fourier transform relations are w ritten in this way so th a t the definition of th e variance,
roo to o
2 / $!,(«,)< /«! = 2 / $39(«)d/e, (2.20)
Jo Jo
is consistent. D ifferentiating (2.18) w ith respect to K\ and using (2.13) together w ith th e isotropy assum ption gives,
1 2 The angular frequency spectrum , ^ ( w ) of a random process g is defined as the Fourier transform of its tem poral covariance function (Arfken [1]),
W, M = ^ / (2. 22)
where R g{r) characterizes the statistical dependence between fluctuations tem porally separated by r . The tem poral covariance function R g and th e tem poral spectrum Wg are Fourier transform pairs so,
R g{ r ) =
J
eiu,TWg(u>)dw. (2.23)The variance is then defined as
Rg(0) = 2 r w g{u)du. (2.24)
Jo
The frequency spectrum is recovered from the angular frequency spectrum by
Wg( f ) = 2trWg(u = 27r /) . (2.25)
For three dimensional space and tim e varying random fields, the spectra! density is,
W»(K ’“' ) = v t y J I S dTI i r (2.26)
The spatial (tem poral) spectral density is recovered by integrating the 4D spectral density over all frequencies (wavenumbers). The three dim ensional space-tim e cross covariance is then defined as the inverse Fourier transform ,
C h a p te r 3
A c o u s tic P r o p a g a tio n T h r o u g h
R a n d o m M e d ia
The theory of wave propagation through random m edia (W PR M ) is coupled closely to a detailed knowledge of the random medium. A statistical description of the refractive index fluctuations and the acoustic fluctuations is essential. The acoustic fluctuations considered are those th a t result from random inhomogeneities in the m edium ’s refractive index. These inhomogeneities are produced when the m edium characteristics (e.g. tem p eratu re, salinity or current velocity) deviate from their mean values. T he term ‘turb u len ce’ is defined as this random process and can be caused by m ixing, strong tidal flow or internal waves.
The refractive index fluctuations due to Kolmogorov’s isotropic and homogeneous turbulence is described in term s of some of the statistical param eters described p re viously. Also included in this chapter is a discussion of the weak scattering theory of Tatarskii [51], [52] and the m ultiple scattering techniques of Uscinski [57] and Dashen [11] in describing acoustic fluctuations. T he acoustic fluctuations are then described by statistical functions.
3.1
T u rb u len t R a n d o m M e d ia
1 4
In classical turbulence theory, Kolmogorov hypothesized th a t for sufficiently high Reynolds num ber there exists an equilibrium range where th e statistical properties for tu rb u len t velocity fluctuations are determ ined by the tu rb u len t kinetic energy dis sipation e and wavenumber k. The energy enters the turbulence a t large scales (the outer scale) and cascades to smaller and smaller scales until th e velocity gradients become large enough for viscosity to dissipate the energy (th e inner scale). T h e stru c ture function of the velocity fluctuations is determ ined m ainly by tu rb u len t structures having sizes r ~ )rj — r2|. If these sizes are within the Kolmogorov inertial subrange bounded by the inner (C0) and outer (L0) scale of turbulence (i.e. « r < L 0) then the stru ctu re function becomes dependent on r and e (see Tatarskii [52]). Dimensional analysis techniques lead to the following formula,
V vr(r) oc e2/3r 2/3
- C V /3
4 < r < L 0, (3.1)
where T>Vr(r) =< [i\.(ri + r ) — ur(rj)]2 > is the longitudinal com ponent of the stru ctu re fun l.ion tensor, th e velocity vT is the projection of the velocity v along the direction r and C 2 is called the stru ctu re constant for velocity fluctuations and represents the intensity of th e turbulence. The longitudinal form is used since the transverse stru ctu re function V Vt and the tensor can be determ ined by this function (see Tatarskii [51]).
T he fluctuations of a passive scalar s (for example tem p eratu re and salinity) have statistical properties dependent on c and Afs which is th e ra te of dissipation of taking into account diffusion. The structure function of scalar fluctuations is also determ ined m ainly by structures having sizes r ~ |rj — r2| and if these sizes are w ithin the Kolmogorov inertia) subrange (<?„ C r < L 0) then th e stru ctu re function becomes dependent on r, e and N s (see Tatarskii [52]). Dimensional analysis techniques lead
1 5 to the following formula,
T>s( r ) oc jVsc ‘/-h’2/3
=
c y /3
(3.2)
where C 2 is called the stru ctu re constant for scalar fluctuations and is a m easure of th e intensity of th e fluctuations.
The refractive index (n) fluctuations are dependent on both scalar and velocity fluctuations. Following the development of Tatarskii [52] which assumes zero
scat-fluctuations. The stru ctu re function for velocity fluctuations is dependent on e and the
trom agnetic wave propagation through the atm osphere, scalar fluctuations dom inate over velocity fluctuations in producing refractive index fluctuations dnce the wave speeds are very m uch greater th an the wind speeds (Ishim aru [32]). For acoustic propagation, however, both scalar and velocity fluctuations m ay affect the refractive index (Di Iorio and Farm er [18]).
The three-dim ensional spectral density of the refractive index fluctuations, $ n(«) can be expressed in term s of the refractive index stru ctu re function, T>n(r). Equation (2.17) m ust be inverted in order to obtain $ n(«) in term s of T>n(r) (see Clifford [7]). T h e m athem atical steps have been elim inated, but it is easy to show th at,
tering angle between incident and scattered energy, the refractive index stru ctu re function can be w ritten as,
where cQ is the wave speed and converts velocity fluctuations to refractive index
stru ctu re function for scalar fluctuations is dependent on both c and .Vn«. Therefore,
V n{r) = (Ae2y - 2 +
= ( c i + c i y n = c y p
C « r « L„. (3.4)
where the constants A and B are dimensionless proportionality constants. For elec
16
Inserting the stru ctu re function form ula (3.4) and carrying out th e indicated opera tions, one obtains,
/ r~ 1/3sin(«:r)dr. (3.6)
l07rz J(0
At scale sizes C0 and smaller (typical sizes less th an 1 cm in coastal w aters), viscosity and diffusion effects dom inate to dissipate th e tu rb u len t kinetic energy and the refractive index inhom ogeneit'es respectively. Thus, it can be assum ed th a t the contribution to th e spectral density, a t these scales, is negligible. At scale sizes L 0 and larger the energy is presumed to be introduced into the turbulence. B oth the channel dimensions and the stratification are expected to pu t bounds on th e outer scale lim it. For scales beyond L 0, statistical isotropy and hom ogeneity cannot be assumed. To oboain an analytical expression for the spectral density we let th e lim its on the integral go from 0 to oo giving,
$ „ (« ) = 0.033C^K~n /3, (3.7)
where th e constant 0.033 takes into account factors of tt and th e constant of integra tion. The wavenumber k theoretically ranges from 0 to oo bu t physically,
27r /T 0 <C k <C
27r/4-The m ean current U produces tim e changes by transporting spatial variations of the refractive index past th e point of observation. U nder Taylor’s hypothesis these spatial variations are frozen as they are advected past th e observation area, th a t is, the translation is not accom panied by any mixing. This assum ption implies th a t the three dim ensional wavenumber - frequency spectrum for refractive index fluctuations is (see Tatarskii [52]),
3 .2
A c o u s tic P r o p a g a tio n
1 7
3 .2 .1
A c o u s t ic flu c tu a tio n s
W hen waves propagate through random m edia they are distorted by a num ber of mechanisms. Three such mechanisms are absorption, refraction and diffraction. A b sorption atten u ates th e signal and will be ignored since it is the fluctuations due to th e random m edium th a t are of interest. Refraction causes phase fluctuations since sound velocity changes as a ray passes through regions of different refractive index variations. Refractive index inhomogeneities due to turbulence act as lenses and so because of refraction, lenses will focus or defocus incident rays thus causing am plitude fluctuations. Diffraction (scattering) takes into account the Huygens-Fresnel princi ple which states th a t every point on th e wavefront becomes a source of spherical secondary wavelets. Random scattering due to random fluctuations in the refrac tive index causes am plitude (and phase) fluctuations because the received signal is a superposition of all the wavelets (see Chernov [5]).
If diffraction effects are negligible, then Chernov [5] and Tatarskii [51] used pure ray theory to find expressions for the phase and am plitude fluctuations. However, if diffraction effects a,re im p o rtan t, then random scattering of acoustic signals plays a role in th e observed am plitude (and phase) fluctuations. Strohbehn [50] makes an excellent com parison between th e approxim ations used for ray theory and scattering theory, some of which will be described below.
Consider Figure 3.1, where a turbulent lens of refractive index (S) w ith siz- I (having any distribution) is centrally located between a tra n sm itte r (T) and receiver (R). According to optical theory, the s< ottered p ath T S R should differ from the ray p ath T R by integral m ultiples of one-half wavelength in order for th ere to be destructive interference a t R. Using the simple geometry, it is easy to show th a t
i = VTL, (3.9)
