Sound scattering from oceanic turbulence

(1)

Sound scattering from oceanic turbulence

by

Tetjana Ross

B.Sc, University of Manitoba, 1998

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

in the Department of Physics and Astronomy We accept this dissertation as conforming

to the required standard

Dr. R. G. Lueck, Co-Supervisor (School of Earth and Ocean Sciences)

Dr. C. J. R. Garrett, Co-Supervisor (Department of Physics and Astronomy)

Dr. A. Babul, Member (Department of Physics and Astronomy)

Dr. K. Denman, Outside Member (School of Earth and Ocean Sciences)

Dr. D. M. Farmer, Outside Member (School of Earth and Ocean Sciences)

Dr. L. Goodman, External Examiner

(School for Marine Science and Technology, University of Massachusetts)

c

! Tetjana Ross, 2003 University of Victoria

(2)

Supervisors: Drs. R. G. Lueck and C. J. R. Garrett

Abstract

Co-located measurements of acoustic backscatter and temperature/velocity microstruc-ture are used to confirm theoretical predictions of sound scatter from oceanic turbulence. The data were collected with a torpedo-shaped vehicle carrying four shear probes and two thermistors on its nose, and forward-looking 44.7 and 307 kilohertz echosounders (mounted 20 centimetres below the turbulence sensors). The vehicle was towed through the stratified turbulence that forms tidally over the lee side of a sill in a British Columbia fjord. Conven-tional downward-looking echosounder measurements were also made with a 100 kilohertz sounder mounted in the ship’s hull. Populations of amphipods, euphausiids, copepods and gastropods were present in the fjord (sampled with 335-micrometre mesh vertical net-hauls) and could be seen in the sounder data.

These plankton net-hauls indicated that there were too few zooplankton in the turbulent regions to account for the scattering intensity. At both 44.7 and 307 kilohertz, scatter that is unambiguously correlated with turbulence was observed. Turbulent scatter is much stronger at the higher frequency, illustrating the importance of salinity microstructure— long neglected in turbulent scattering models—and shedding some light on the form of the turbulent temperature-salinity co-spectrum.

The turbulent temperature-salinity co-spectrum has never been measured directly. Al-though several models have been proposed for the form of the co-spectrum, they all produce unsatisfactory results when applied to the turbulent scattering equations (either predicting negative scattering cross-sections in some density regimes or predicting implausible levels of correlation between temperature and salinity at some scales). A new co-spectrum model is proposed and shown to be not only physically plausible in all density regimes, but also in reasonable agreement with the scattering data.

At 307 kilohertz, the backscatter is mostly from salinity microstructure and, depending on the strength of the stratification, can be as strong as—or stronger than—the signal from a zooplankton scattering layer. This could easily confound zooplankton biomass estimates in turbulent regions. The two targets’ different natures (discrete targets versus a volume effect) often allow them to be distinguished even when they occur simultaneously. The key is sampling the same targets at multiple ranges. At long-range, discrete targets have a constant volume scattering strength proportional to their number density. The sampling volume, however, decreases as the targets approach the sounder. At some range there will

(3)

be only one (or no) target in the sampling volume and the volume scattering strength will increase (or disappear) as the target continues to near the sounder. Turbulence, as a volume scattering effect, has no range dependence to its volume scattering strength. Thus, by examining the scattering nature at close range we can distinguish discrete targets (like zooplankton) from turbulence.

Examiners:

Dr. R. G. Lueck, Co-Supervisor (School of Earth and Ocean Sciences)

Dr. C. J. R. Garrett, Co-Supervisor (Department of Physics and Astronomy)

Dr. A. Babul, Member (Department of Physics and Astronomy)

Dr. K. Denman, Outside Member (School of Earth and Ocean Sciences)

Dr. D. M. Farmer, Outside Member (School of Earth and Ocean Sciences)

Dr. L. Goodman, External Examiner

(4)

List of Tables

3.1 Summary of data collected in Knight Inlet. Sounders are listed by frequency in kHz. S represents the 4 shear probes and T the 2 thermistors. . . 35 3.2 Summary of sounders: frequencies and -3dB beamwidths. . . 38 3.3 Summary of net-tow data collected in Knight Inlet. . . 39 4.1 Summary of measured mixing efficiencies. [Modified from Ruddick et al.

(1997) and Macoun (2002).] . . . 61 5.1 Average data from two 0-30 metre depth net-hauls collected near the sill

in Knight Inlet (one was taken 1.8 km east and the other 1.8 km west of the sill crest). Only the species that were either numerically significant (i.e. Copepods) or strong scatterers are included in this table. N is the average number density for the entire 0-30 metre water column (± stan-dard deviation). The T S are the species target strengths at 307.2 kHz. (Non-spherical animals were modelled at broadside incidence to maximise target strength.) Sv = T S + 10log10(N ) are the species volume scattering

strengths. Sv(2m) = T S + 10log10(15N ) are what the species volume

scat-tering strengths would be if all the animals collected in the net-hauls were concentrated into a 2-m thick layer. . . 69 A.1 Summary of target sphere strengths. . . 90 A.2 Summary of calibration coefficients. . . 92

(8)

List of Figures

1.1 Comparison of turbulent and zooplankton scattering as a function of acoustic frequency. Modified from Stanton et al. (1994b) by adding salinity only (S!

only) lines. The amplitude of salinity fluctuations was chosen to give the same level of scatter as the temperature fluctuations (T!_{) at low frequencies.} ₄

2.1 Sketch of incident and scattered wavenumbers. . . 13 2.2 Comparison of scattering models for different co-spectrum models when!

!A

Bδ

! ! is close to one (i.e. salinity and temperature contribute roughly equally to sound scattering from turbulence). Both scattering cross-sections (σ) and wavenumbers (k) have been non-dimensionalised, such that the shape of the scattering curve is determined by A_Bδ, κS

ν = 1.5×10

−9_m2_/s

1.3×10−6_m2_/s, and κ_νT =

1.5×10−7_m2_/s

1.3×10−6_m2_/s. The missing portions of the curves are where the models

pre-dicted negative scattering cross-sections. Note that the co-spectrum models are indistinguishable at high q1/2k/kν for these values of _BAδ. . . 25

2.3 Comparison of scattering models for different co-spectrum models for large values of !!A

Bδ

!

! (i.e. scattering from temperature microstructure is domi-nant). Both scattering cross-sections (σ) and wavenumbers (k) have been non-dimensionalised. The missing portions of the curves are where the mod-els predicted negative scattering cross-sections. . . 26 2.4 Examples of zooplankton, taken from Parsons and Takahashi (1973). 1)

ctenophore Pleurobrachia; 2) mollusc Limacina; 3) mollusc Clione; 4) eu-phausiid Thysanoessa; 5) amphipod Parathemisto; 6) copepod Calanus; 7) chaetognaph Sagitta. . . 31 3.1 Location of observations, Knight Inlet, British Columbia, Canada. . . 34 3.2 Sketch of towing configuration. Sketch is not to scale (the towed vehicle fits

easily onto aft deck of ship). Because TOMI was towed 100-200 m behind the ship, the ship-acoustics and the vehicle-mounted acoustics never measured the same water at the same time. . . 36

(9)

3.3 Sketch of the Towed Ocean Microstructure Instrument (TOMI). Not pictured are the pressure sensor, accelerometers, gyros and compass, which are all located inside the forward pressure case in the vehicle. [Sketch courtesy of UVic Ocean Turbulence Lab.] . . . 37 3.4 Picture of TOMI’s nose and part of forward pressure case. [Photo courtesy

of Isabelle Gaboury.] . . . 37 3.5 Detail of port sounder fairing. From 1967 sketch of CCGS Vector. [Sketch

courtesy of crew of CCGS Vector.] . . . 38 4.1 a) An echogram from the 100 kHz ship-board sounder. The white line is

the approximate path of the towed vehicle (the ship-board sounder data are lagged to line up with vehicle data) and the overlying coloured circles are predicted backscatter at 100 kHz as estimated from microstructure data. b) Echogram from the 307.2 kHz vehicle-mounted sounder for time and depth of the pink box in panel (a). Each horizontal line shows the echo from one ping. As time progresses downwards and the range is the distance ahead of the vehicle, parcels of water travel diagonally across the figure, from right to left. The temperature gradient microstructure is shown on the left. The colour scale on the right applies to both images. . . 44 4.2 Range dependence of correlation between 307.2 kHz Sv and !

!dT_dx !

!. . . 45 4.3 a) An echogram from the 100 kHz ship-board sounder. The white line is

the approximate path of the towed vehicle (the ship-board sounder data are lagged to line up with vehicle data) and the overlying coloured circles are predicted backscatter at 100 kHz as estimated from microstructure data. b) Echogram from the 44.7 kHz vehicle-mounted sounder for time and depth of the pink box in panel (a). Each horizontal line shows the echo from one ping. As time progresses downwards and the range is the distance ahead of the vehicle, parcels of water travel diagonally across the figure, from right to left. The temperature gradient microstructure is shown on the left. . . 46 4.4 Range dependence of correlation between 44.7 kHz Sv and !!dT

dx

!

(10)

4.5 Scatterplot of predicted versus measured volume scattering strengths for 307.2 kHz. The x-axis is Svp calculated from 10-second averages of !, dS_dz,

dT

dz, and N . The y-axis is the measured acoustic scatter at one metre range

averaged over 10 seconds (10 pings) with each ping averaged over 22.2 cm of range (equivalent to the 300µs pulse length). Data are compiled from times when turbulent scatter was strong enough to be seen above the -95 dB noise floor of the sounder. The grey line represents one-to-one correspon-dence. The thickness of the grey line shows the uncertainty in the sounder’s calibration coefficient K, and hence, in measured Sv. . . 48 4.6 Histogram of measured minus predicted volume scattering strengths for 307.2

kHz. Solid lines are normal distributions ( √ 1

2πσ2e−(x−µ) 2_/2σ2

) with mean (µ) and standard deviation (σ) for all 454 points (thick line) and with outliers removed (431 points, thin line). . . 49 4.7 Scatterplot of predicted versus measured volume scattering strengths for 44.7

kHz. The x-axis is Svp calculated from 10-second averages of !, dS_dz, dT_dz,

and N . The y-axis is the acoustic scatter, measured at three metres range, averaged over 10 seconds (10 pings) with each ping averaged over 37 cm of range (equivalent to the 500µs pulse length). The data are compiled from times when the turbulent scatter was strong enough to be seen above the noise floor of the sounder. The grey lines represent one-to-one correspondence, with calibration coefficients K = −122 ± 2 dB (dark grey) and K = −120 ± 5 dB (light grey). . . 52 4.8 Histogram of measured minus predicted volume scattering strengths for 44.7

kHz. The solid lines are normal distributions ( √1

2πσ2e−(x−µ) 2_/2σ2

) with mean (µ) and standard deviation (σ) for all 203 points (thick line) and with outliers removed (191 points, thin line). . . 53 4.9 Scatterplot of temperature-only (calculated using (4.3)) volume scattering

strengths versus 307.2 kHz measured acoustic scatter at one metre range. The grey line represents one-to-one correspondence. . . 55

(11)

4.10 Scatterplots of the percentage of turbulent scattering cross-section (σturb)

that comes from salinity microstructure (i.e. σturb(S)/σturb(T &S) × 100%)

for 307.2 (top) and 44.7 kHz (bottom) sound. The percentages are calculated using (2.37), σturb(S) without terms involving T , and the same turbulent

parameters (i.e. !, dS_dz, dT_dz, and N ) as the data plotted in Figures 4.5 (top) and 4.7 (bottom). The solid lines are the percentage of σturbthat is predicted

to come from salinity microstructure for different density ratios (Rρ). Note

that, at 307.2 kHz, all the scatter is from salinity microstructure when ! < 10−7_{W/kg, even for quite large magnitudes of R}_ρ_{. . . .} ₅₆

4.11 Scatterplot of predicted versus measured volume scattering strengths for 44.7 kHz for all four TS co-spectrum models. The x-axis is the Sv calculated from 10-second averages of !, dS_dz, dT_dz, and N for the co-spectrum models discussed in section 2.3.2. The y-axis is measured acoustic scatter at three metres range averaged over 10 seconds (10 pings) with each ping averaged over 37 cm of range (equivalent to the 500µs pulse length). Data are compiled from times when turbulent scatter was strong enough to be seen above the noise floor of the sounder. Pink lines are one-to-one correspondence for K = −122 ± 2 dB (dark pink) and K = −120 ± 5 dB (light pink). . . 57 4.12 Wavenumber spectra of scaled turbulent scattering cross-sections for

dif-ferent co-spectrum models. The pink band is the no co-spectrum or no-correlation model. The light green band is the 3-dimensional upper-bound model. The light blue band is Stern’s model. The points are measured scat-tering cross-section (dark green 44.7 kHz, dark blue 307.2 kHz) scaled by

kν χT = N2 2Γ('ν)3/4 " dT dz #−2

. The thickness of the spectra allow for the range of δ measured when there were acceptable echoes at 44.7 kHz (δ = −0.5 ± 0.15). 59 4.13 Wavenumber spectra of scaled turbulent scattering sections models including

and excluding density fluctuation terms. Pink line is for A = aµand B = bµ

(i.e. density is excluded). Light green line is for A = aµ− α and B = bµ+ β

(i.e. density is included). The points are measured scattering cross-section (dark green 44.7 kHz, dark blue 307.2 kHz) scaled by kν

χT = N2 2Γ('ν)3/4 " dT dz #−2 . 60 4.14 Histogram of mixing efficiencies (Γ) estimated from the difference between

measured and predicted volume scattering strengths at 307.2 kHz. The solid line is a log-normal distribution ( 1

Γ√2πσ2 lnΓ

e−(lnΓ−µlnΓ)2/2σ2lnΓ) with mean in

logspace (µlnΓ) and standard deviation in logspace (σlnΓ) (for the 431 points

(12)

4.15 Turbulent scattering cross-section calculated from (4.14) plotted as a function of dissipation rate (! = x2ν) for the background stratifications of two of the measured 307.2 kHz scattering cross-sections (σm, plotted as horizontal lines).

Asterisks mark the calculated match. Note that, for the black line, a small uncertainty near σm= 6 × 10−9 m would lead to a large uncertainty in fitted !. 65

4.16 Scatterplot of measured dissipation rates (y-axis) versus those calculated using the inverse model (x-axis). The grey dots are the 23 outliers excluded in the earlier analysis. . . 66 4.17 Turbulent scattering cross-section calculated from (4.14) and plotted as a

function of dissipation rate for several values of Rcρ. All curves assume

"

dT dz

#

= 0.16 oC/m. . . 67 5.1 Number densities of the strongest zooplankton scattering targets present near

the sill. The measured number densities (N , white bars) are a taxon-wide sum of the number densities listed in Table 5.1. The errorbars show the uncertainty in the measured N . The next two bars show the number densities needed for each taxon to scatter with the same volume scattering strengths as the maximum (-64.5 dB, dark gray) and minimum (-90 dB, black) levels correlated with turbulence in Figure 4.1, based on the (species abundance weighted) average target strength for that taxon. The light gray background shows the minimum number density necessary for any group to retain a continuous appearance at close range. . . 70 5.2 Near-range 307.2 kHz echogram collected in a zooplankton scattering layer.

The black lines are the paths of parcels of water. Note the absence of temper-ature microstructure (left). The colour scale is in units of volume scattering strength [dB]. . . 72 5.3 A vehicle-mounted echogram from a turbulent layer with zooplankton present.

The black line is the path of a parcel of water. The background scatter is correlated with the temperature microstructure shown on the left, but the discrete target is not. . . 73

(13)

5.4 Plot of scattering strength of the discrete target in Figure 5.3. The solid black line is the maximum Sv between 18.025 and 18.03 hours for each range and the peaks thus correspond to the red dots seen in Figure 5.3. The solid grey lines are the estimated scattering volumes (using (2.90)) for a single 2.8 cm long bent-cylindrical zooplankter positioned with long axis 90o (broadside, thickest line), 82.3o (medium line) and 78.5o (thinnest line) with respect to the sounder beam axis. Also shown are the saturation level of the sounder (dashed black line) and the approximate mean level of turbulent scatter in Figure 5.3 (dashed grey line). . . 74 5.5 Comparison of turbulent and zooplankton volume scattering strengths as a

function of acoustic frequency for turbulence levels, density gradients, and some plankton species found in Knight Inlet. dT_dz = 0.41 oC/m and dS_dz = −0.82 psu/m were used for the ! = 1 × 10−4 _{W/kg line (the gradients were}

generally less for ! ≈ 1 × 10−4 _{W/kg, but this combination of high ! and high}

gradients did occur). dT_dz = 0.21 oC/m and dS_dz _{= −0.42 psu/m were used for} ! = 1 × 10−6 _{W/kg line (these are the average gradients when ! ≈ 1 × 10}−6

W/kg and roughly correspond to the moderate turbulence in Figure 5.3). . 75 5.6 Sketch of the method used to “straighten” the sounder data. U (i) is the

vehicle speed at the time the ping was sampled, and ∆t = 1/fS, where fS is

the frequency at which the vehicle speeds were sampled. . . 77 5.7 Example of the counting method, applied to the 307.2 kHz sounder. The

bugs that were “found” are shown on the left. The “straightened” echogram is shown on the right. “Parcels” 90 to 162 are the same data as displayed in Figure 5.3 (hours 18.015 to 18.035). . . 78 5.8 Example of the counting method, applied to the 44.7 kHz sounder. The bugs

that were “found” are shown on the left. The “straightened” echogram is shown on the right. “Parcels” 90 to 162 are the same data as displayed in Figure 5.3 (hours 18.015 to 18.035). The strong scatter in the upper right hand portion of the echogram is scatter from the surface that is being detected by the sidelobes (at about 55o _{from the beam axis) of the 44.7 kHz sounder.} ₇₉

A.1 Target strengths of steel spheres as a function of acoustic frequency. The grey line is the weighting function sinc2_{(πτ (f − f}0)). . . 91

(14)

Acknowledgements

I’d like to thank everyone who helped me complete this thesis. You are too numerous to name, so I apologise in advance for my omissions.

I’d like to thank my supervisors, Chris Garrett and Rolf Lueck; Chris, for giving me the support and freedom to do whatever I liked (even if it was acoustics); Rolf, for helping me direct my vague ideas into a doable field project and making it happen.

A big thank you to all of you that made the field trip to Knight Inlet a success, no data could have been collected without your technical expertise. In addition to their work during the cruise, Paul Macoun, Roland and Isabelle Gaboury got TOMI in the water, with several new sensors, in no time at all. ASL Environmental Sciences Ltd. generously loaned us the two echosounders that Roland and Isabelle mounted on TOMI. Dave Mackas loaned us the electronics to operate the ship board sounders on the CCGS Vector. The captain and crew of the CCGS Vector, as well as Richard Dewey and Lou St. Laurent, were a great help in Knight Inlet. The Acoustics Group at the Institute of Ocean Sciences (particularly Mark Trevorrow, Svein Vagle, David Farmer, Nick Hall-Patch, Ron Teichrob) were both generous with advice and patient with me as I learned how to calibrate the echosounders using their equipment. Also, thanks to Helen Johnson for her careful proofreading of this thesis. Thanks to Harvey Seim and Andone Lavery for advice and discussion about the turbulent scattering models.

Thanks to Koit and Alan for all the oceanography I’ve learned aboard Imagine. Thanks to Holly—my first Victorian friend—meeting you on the ferry opened the door to adventure and joie de vivre in my new home.

Thanks to my family, David, Rita, Konrad; your support and belief in me has carried me this far.

Last, but not least, a huge thanks to my husband, Jeff Friesen. Your support is invaluable. You keep me balanced, and your proofreading skills keep my writing legible.

(15)

Chapter 1

Introduction

All my life through, the new sights of Nature made me rejoice like a child.

- Marie Curie

1.1 Motivation

Because of light’s strong attenuation in water, acoustics has established itself as a vital tool for exploring the ocean depths. Oceanographers use high-frequency acoustics for everything from tracing oceanic phenomena (e.g. overflows (Farmer and Smith 1980b, Wesson and Gregg 1994), internal waves (Haury et al. 1979, Farmer and Armi 1999a), and buoyant plumes (Hay 1984, Rona et al. 1991)) to measuring currents using Doppler frequency shifts to mapping fine-scale distributions of biomass of different size classes of plankton (Pieper et al. 1990). As long as scatterers are passively following the flow, knowledge of what is actually scattering the sound is generally overlooked in the former examples, but is critical to making quantitative estimates of plankton or fish abundance.

Strong scatter is often observed in regions of the water column expected to be turbulent, but it has never been clear if this is because turbulence is scattering sound or because there are higher plankton concentrations in turbulent regions. Models and laboratory experiments show that zooplankton encounter food more frequently in turbulent regions (Rothschild and Osborn 1988, Peters and Marras´e 2000) and therefore they may be seeking out these ‘high food zones’ (Mackas et al. 1993, Dower et al. 1997). Then again, the elevated scattering in these regions could be due to scattering from turbulence. In that case, acoustic backscatter is potentially a powerful tool for remotely sensing turbulence, possibly replacing the difficult and costly methods currently being used.

(16)

1.2 Previous studies

1.2.1 Sound scattering from turbulent microstructure

The basic physics of sound scattering from turbulence is that the turbulent motions act on the ambient gradients in sound speed and density (in the ocean these are prescribed by the temperature and salinity gradients) to create fluctuations in sound speed and density on the same scale as the incident acoustic wave. When the incident acoustic wave encounters a parcel of water with the same density, but a different sound speed (which can be viewed as a change in density weighted compressibility, as sound speed squared is inversely proportional to density times compressibility), the parcel of water compresses either more or less than the surrounding water and then rebounds isotropically creating a new “scattered” sound wave. On the other hand, if the incident sound wave encounters a parcel of water with a different density, but the same sound speed, the difference in inertia between the parcel of water and the surrounding water will cause it to oscillate in the direction of the passing wave. This dipole moment creates a directional “scattered” sound wave. A sound wave encountering a parcel of water with different sound speed and density will create a “scattered” wave that is a combination of the above. As the sound speed and density of seawater is determined by its temperature and salinity, these fluctuations in sound speed and density are often expressed in terms of fluctuations in temperature and salinity. A full theoretical development of sound scattering from turbulence can be found in Chapter 2.

The theoretical framework for sound scattering from turbulence has been developed over many years. As far back as the early 1950’s, researchers started looking into the possibly of sound scattering from turbulence (see Batchelor (1957) for a review). Early work (up to and including Goodman’s 1990 synthesis of scattering models with classical and empirical turbulence models) considered only fluctuations in sound speed due to fluctuations in temperature. In 1995, Seim et al. first applied the effect of salinity fluctuations on sound speed to predicting scatter from turbulence. In 1999, Seim expanded his salinity scattering theory and made the first attempt at adding temperature and salinity co-variations to the scattering model. Lavery et al. (2003) showed that when significant salinity gradients exist, fluctuations in density are no longer negligible and can in fact increase the turbulent scattering by as much as 6 dB.

Several carefully controlled experiments, with artificially generated turbulence, have shown that temperature fluctuations do scatter sound. Pelech et al. (1983) showed that the wake of an object towed in the freshwater thermocline of a flooded quarry scattered 75 kHz sound. They observed the wake from several angles, but there was too much

(17)

vari-ability within returns from the same angle to make more than a qualitative comparison with models. Thorpe and Brubaker (1983) observed backscatter (at 102 kHz) from the wake of objects towed in a freshwater temperature gradient, and also noted that there was no scatter when the same object was towed in homogeneous water. While their sounder was not calibrated, they went on to compare backscatter to some profiles of temperature microstructure, showing that regions of elevated scatter often correspond to regions of nat-urally occurring temperature microstructure. Oeschger and Goodman (1996) and Oeschger and Goodman (2003) observed broadband (250-750 kHz) scatter at multiple angles from thermally driven buoyant plumes created in a tank.

Although there have been several studies where it is probable that authors are observing sound scatter from oceanic turbulence (e.g. Sandstrom et al. 1989, Trevorrow 1998, Orr et al. 2000, Warren 2001), Seim et al. (1995) and Moum et al. (2003) are the only studies that have compared direct measurements of oceanic turbulence with calibrated scattering data. Seim et al. compared vertical profiles of shear and temperature microstructure through a series of billows in a salt-stratified channel near Puget Sound, Washington with ship-board 120 and 200 kHz sounder data. In spots—particularly near the crest of the billows—turbulent scattering models agreed with acoustic observations, but throughout most of the water column, the measured scatter was many orders of magnitude larger than one would predict from turbulence. This suggests that there were strong sources of biotic scatter present. Seim et al. collected no biological data, so they could not refute the suspicion that the agreement near the billows crest was mere coincidence. The more recent measurements made by Moum et al., comparing scatter in the trough of internal solitary waves (again measured with ship-board sounders) observed off the coast of Oregon to vertical profiles of temperature and velocity microstructure, had the advantage that they were collected in relatively clear water. This makes the correlation between turbulence and acoustic scatter more evident, but they still lacked the biological measurements to explain regions where there was scatter, but no turbulence. Also, while both studies took care to line up the acoustic and microstructure data in space, poor temporal coincidence and large differences in the sampling volumes of the two types of measurements rendered model comparisons—even where the scatter was likely from turbulence—less than convincing.

1.2.2 Distinguishing zooplankton from turbulence

Zooplankton are small enough that they do not scatter low frequency sound (the frequency at which they become significant scatterers depends on the ratio of their size to the wave-length of the sound wave). Turbulence, on the other hand, is expected to scatter sound at

(18)

low frequency, but not at very high frequency (once the wavelength of the sound is smaller than the diffusion scale of salt or heat, the variations in temperature and salinity that cause sound scattering will be damped out through diffusion). Stanton et al. (1994b) suggested that scattering from zooplankton and from turbulence could be distinguished by exploiting their different frequency dependence (Figure 1.1). Using multiple frequencies, one could assume that layers where the scattering intensity dropped with increasing frequency were turbulent, and those where the opposite happened contained zooplankton.

Figure 1.1: Comparison of turbulent and zooplankton scattering as a function of acoustic frequency. Modified from Stanton et al. (1994b) by adding salinity only (S! _{only) lines.}

The amplitude of salinity fluctuations was chosen to give the same level of scatter as the temperature fluctuations (T!_{) at low frequencies.}

Stanton et al. used a temperature only model for turbulent scattering. Adding the possibility of salinity scattering alters their result in that much higher frequencies must be reached before the scatter from salinity microstructure disappears. Unfortunately, as the attenuation of sound increases with frequency, the practical range of a sounder with f > 1 MHz is less than 30 metres, reducing the “remote sensing” capabilities of this method. Sound scattering from turbulence must be better understood before this multi-frequency approach will be truly viable.

(19)

Chapter 2

Scattering theory

Think left and think right and think low and think high. Oh, the thinks you can think up if only you try!

- Dr. Seuss

This Chapter gives an overview of the scattering theory used throughout this study. Section 2.1 derives a very general equation for acoustic waves in a fluid, and although I have yet to find a reference where the acoustic wave equation is derived so generally (most authors either assume a homogeneous, moving medium or a stationary, variable medium), the underlying physics has been long studied. Section 2.2 derives the equation for scattering from turbulence, following the method laid out by Batchelor (1959) and described in Morse and Ingard (1968) and Ishimaru (1978). The scattering from both variability in the medium and turbulent motions are derived simultaneously and, like Goodman and Kemp (1978) and Ostashev (1997), explicit justification is given for ignoring Doppler terms from the scattering equation. Section 2.3 uses classical turbulence theory to predict the form of turbulent temperature and salinity spectra required by the scattering theory. It is here that I have added a new bit of theory, with a new model for the (yet unknown) form of the turbulent temperature-salinity co-spectrum. The effect of this co-spectrum model on the scattering equation is compared with previous co-spectrum models. Finally, section 2.4 first derives the well-established scattering equation for the simplest model of a zooplankter (a fluid sphere) and then outlines some of the more realistic models commonly employed.

2.1 General equations for acoustic waves in a fluid

The derivation of equations for the propagation of acoustic waves in a fluid starts with the equation of continuity for density (2.1),

∂ρ

∂t + u · ∇ρ = − ρ∇ · u, (2.1)

where ρ is the fluid density, ∇·(ρu) = u·∇ρ+ρ∇·u represents the convergence or divergence of ρ due to advection, and it is assumed that there are no sources or sinks of density.

(20)

Next, the density equation is combined with the equation of motion as applied to a fluid, −∇p + ρg + µ∇2u = ρDu

Dt = ρ ∂u

∂t + ρu · ∇u, (2.2)

where ∇p is the force (per unit volume) due to a pressure gradient, g is acceleration due to gravity, and µ is viscosity. (2.2) is already simplified by assuming the fluid is incompressible and has constant viscosity. For the case of acoustic waves, (2.2) is generally further simplified by neglecting both viscosity and weight (ρg). Viscosity causes only a very small amount of damping to acoustic waves. Weight is negligible when the wavenumber of sound is much larger than N2_{/g (Lighthill 1978) (where N}2 _{is buoyancy frequency squared). This is}

generally the case for underwater sound applications (for acoustics frequencies from 1 to 1000 kHz, this is true for N2 << 40 to 40000 s−1_{, and N}2 _{is typically around 10}−4 _s−2_).

Finally, density and pressure are related through the equation of state for an inhomo-geneous medium, ρ =ρ(p, S), Dρ Dt = $ ∂ρ ∂p % S Dp Dt + $ ∂ρ ∂S % p DS Dt, (2.3)

where S is entropy. We must use total derivatives in the above equation because the equation of state holds for a point in the medium, not a fixed point in space (Chernov 1960). If the flow is assumed to be adiabatic (DS_Dt _{≡ 0) and compressibility is defined as κ =} 1_ρ(∂ρ_∂p)S,

equation (2.3) becomes Dρ Dt = ρκ Dp Dt ∂ρ ∂t + u · ∇ρ = 1 c2 $ ∂p ∂t + u · ∇p % , (2.4)

where c = 1/√ρκ is the sound speed. Combining (2.1) and (2.4) gives

∂p

∂t + u · ∇p = −c

2

ρ∇ · u. (2.5)

This, along with the simplified equation of motion, −∇p = ρDu

Dt = ρ ∂u

∂t + ρu · ∇u, (2.6)

(21)

2.1.1 Acoustic perturbation

The pressure, density, and velocity of the fluid are assumed to have two independent parts; a slowly varying ambient part (subscript 1) and a perturbation due to the acoustic wave (subscript 2),

p = p1(r) + δp2(r, t),

ρ = ρ1(r) + δρ2(r, t),

c = c1(r) + δc2(r, t),

u = u1(r) + δu2(r, t), (2.7)

where δ is a small parameter, r is a position vector (x, y, z), and t is time.

Acoustic frequencies used in most underwater applications are of high frequency, so it is reasonable to assume that on the timescale of a few acoustic periods, the environment through which the sound waves are travelling is steady. Hence, in (2.7) the ambient quanti-ties are taken to vary spatially, but not temporally. In addition to only varying spatially, the medium is assumed to be incompressible on scales associated with the ambient velocities, thus ∇ · u1 = 0.

Substituting (2.7) into equations (2.5) and (2.6), and retaining only the O(1) terms, ∂p1 ∂t + u1· ∇p1 = − c 2 1ρ1∇ · u1 (2.8) −∇p1= ρ1 ∂u1 ∂t + ρ1u1· ∇u1 (2.9)

which simply show (as expected) that equations (2.5) and (2.6) are satisfied before the introduction of an acoustic wave.

The O(δ) equations will yield a description of the acoustic waves, ∂p2 ∂t + u1· ∇p2− ρ1u2· (u1· ∇u1) = − ρ1c 2 1∇ · u2, (2.10) −∇p2 = ρ1 ∂u2

∂t + ρ1(u1· ∇u2+ u2· ∇u1) + ρ2u1· ∇u1, (2.11) where in (2.10) it has again been assumed that ∂u1

∂t = 0, ∇ · u1 = 0, and (2.9) was used to

eliminate the ∇p1 term.

Combining _c12

1∂/∂t (2.10) and ρ1{∇ · [(2.11)/ρ1]} gives the single equation for acoustic

(22)

∇2p2− 1 c2 1 ∂2p2 ∂t2 = 1 ρ1∇ρ1· ∇p2 + 1 c2 1 u1· ∇ ∂p2 ∂t − ρ1 c2 1 ∂u2 ∂t · (u1· ∇u1) − ρ1∇ · (u1· ∇u2+ u2· ∇u1) − ρ₁∇ · $ ρ2 ρ1 u1· ∇u1 %. (2.12)

Equation (2.12) is the general equation for the acoustic waves in a non-uniform, slowly moving, quasi-steady medium. This equation is more general that those found in the liter-ature, but, as we will see, it reduces to the correct forms for different limiting cases.

2.1.2 Free-wave equation

If c1 is set to a constant c0, ρ1 set to a constant ρ0, and u1≡ 0, (2.12) becomes the familiar

free wave equation for acoustic waves, ∇2p2− 1 c2 0 ∂2p2 ∂t2 = 0. (2.13) 2.1.3 Doppler shift

If the ambient quantities are assumed to be constant, but u1(r) = u0 (a constant), (2.12)

becomes ∇2p2− 1 c2₀ ∂2p2 ∂t2 = 1 c2₀u0· ∇ ∂p2 ∂t − ρ0∇ · (u0· ∇u2). (2.14) Adding _c12 0u

0· ∇(2.10) to (2.14), we get the Doppler equation in a more familiar form

(Morse and Ingard 1968), & ∇2− 1 c2 0 $ ∂ ∂t+ u0· ∇ %2' p2 = 0, (2.15)

which, assuming an incident plain wave p2 = P eik·r−iωt, gives the dispersion relation ω =

u0· k ± c₀|k|.

2.1.4 Waves in a stationary, inhomogeneous medium

If the ambient velocity (u1) is zero, (2.12) becomes

∇2p2− 1 c2 ∂2p2 ∂t2 = 1 ρ∇ρ · ∇p2, (2.16)

(23)

which is the standard equation for waves in an inhomogeneous medium (Chernov 1960). This equation describes how sound can be reflected from density gradients in the ocean. Analytical solutions to this equation are only possible for the simplest density and sound speed gradients (as described in Tolstoy (1965), Robins (1990)).

2.2 Scattering from turbulence

2.2.1 Turbulent perturbation

To look at the effect of turbulence on the acoustic wave, assume that within a region R the ambient quantities are made up of an undisturbed background and a turbulent fluctuation,

ρ1(r) = ρ0+ ρ!(r),

c1(r) = c0+ c!(r),

u1(r) = u0+ u!(r). (2.17)

The turbulent fluctuations (primed quantities) are assumed to be much smaller than the background quantities, and to be identically zero outside of R. While the turbulent fluctu-ations can vary on the scale of a few acoustic wavelengths, the background does not, and can therefore be assumed to be quasi-constant, as in (2.17).

Substituting (2.17) into (2.12) and neglecting terms that are quadratic in turbulent quantities gives ∇2p2− 1 c2 0 ∂2_p 2 ∂t2 − 1 c2 0 u0· ∇ ∂p2 ∂t + ρ0∇ · (u0· ∇u2) = −2c! c3₀u0· ∇ ∂p2 ∂t − 2c! c3₀ ∂2_p 2 ∂t2 − ρ!∇ · (u0· ∇u2) +∇ρ ! ρ0 · ∇p2 −ρ_c02 0 ∂u2 ∂t · (u0· ∇u !_{) +} 1 c2 0 u!_{· ∇}∂p2 ∂t

− ∇ · (ρ2u0· ∇u!) − ρ₀∇ · (u!· ∇u2+ u2· ∇u!).

(2.18)

The terms on the left hand side of equation (2.18) are simply the Doppler equation for a homogeneous medium (2.14). Many previous authors (Batchelor 1957, Morse and Ingard 1968, Ishimaru 1978) have assumed zero ambient velocity (u0 = 0). This eliminates the

(24)

Doppler terms, but also the first terms in each of the following lines in (2.18) which have a combination of ambient and turbulent quantities. The ratios of similar terms (terms mixing u0 and turbulent sound speed, density, and velocity fluctuations over terms with only the

turbulent fluctuations) are

2c" c3₀u0·∇ ∂p2 ∂t 2c" c3₀ ∂2p2 ∂t2 ∼ u0kωp2 ω2_p₂ ∼ u0 c0 u0 c0+1 ρ"_∇·(u0·∇u2) ∇ρ" ρ0 ·∇p2 ∼ ρ "_ku₀_k" p2 ρ0c0 # kρ"/ρ0kp2 ∼ u0 c0 ρ0 c2₀ ∂u2

∂t ·(u0·∇u")+∇·(ρ2u0·∇u") 1

c2₀

u"_·∇∂p2

∂t−ρ0∇·(u"·∇u2+u2·∇u")

∼ ρ0 c2₀ω " p2 ρ0c0 # u0ku"+k $ p2 c2₀ % u0ku" 1 c2₀u"kωp2+ρ0ku"k " p2 ρ0c0 # ∼ u0 c0, (2.19)

assuming the Doppler dispersion relation (ω = u0· k + c₀k ∼ (u₀+ c₀)k) along with the

approximations (ρ2 ∼ p_c22

0 and u2 ∼

p2

ρ0c0, which are equivalent to the Born approximation,

which assumes the scattered wave has a much smaller amplitude than the incident wave). As Goodman and Kemp (1978) and Ostashev (1997) have pointed out, most oceanographic flows have very low Mach number (M = u0

c0 ≤ 3 × 10

−3_{). Therefore, all the above ratios}

are small and the first term on each line of the right-hand side of (2.18) is negligible. Thus the assumption of u0 = 0 is justified, and gives the correct turbulent scattering equation.

Neglecting all terms containing u0 and dropping the subscript 2 (i.e. p₂ = p, u2 = u), we

get ∇2p −_c12 0 ∂2p ∂t2 = − 2c! c3 0 ∂2p ∂t2 + ∇ρ! ρ0 · ∇p + 1 c2 0 u!_{· ∇}∂p ∂t − ρ0∇ · (u !_{· ∇u + u · ∇u}!_). (2.20)

Echosounders excite acoustic waves of a single frequency, so the acoustic motions will have a simple frequency dependence (i.e. p = Re[p (r) exp(−iωt)] and u = Re[u(r) exp(−iωt)]). Equation (2.20) then becomes

∇2p +ω 2 c2 0 p =2ω 2_c! c3 0 p +∇ρ! ρ0 · ∇p + −iω c2 0 u!_{· ∇p − ρ}0∇ · (u!· ∇u + u · ∇u!). (2.21)

(25)

With k = ω/c0, equation (2.21) is of the form ∇2p(r) + k2p(r) = −f(r), which can be

solved using a standard Green’s function method (Batchelor 1957, Morse and Ingard 1968). This method gives the integral equation for p (r),

p(r) = ∞ ( ( −∞ ) g(r|r0) ∂ ∂n0 p(r0) − p(r0) ∂ ∂n0g(r|r 0) * dS0+ ∞ ( ( ( −∞ f (r0)g(r|r0)d3r₀ g(r|r0) = exp(ik|r − r 0|) 4π|r − r0| , (2.22)

where dS0 is an element of surface area and _∂n∂₀ is the derivative along a direction normal

to the element of surface area.

Applying (2.22) to (2.21) and integrating over the entire (infinite) medium gives

p(r) = pi(r) + ∞ ( ( ( −∞ { − k22c_c! 0p − ∇ρ !_·∇p ρ0 + ik∇p · $ u! c0 %

+ ρ0∇ · (u!· ∇u + u · ∇u!)}exp(ik|r − r 0|)

4π|r − r0|

d3r0.

(2.23)

The volume integral over the entire medium is really just an integral over the region R, because outside R, the turbulent (primed) quantities are zero. The integral over the surface of an infinite sphere produces the incident wave pi(r) which is incident from infinity (Morse

and Ingard 1968).

The Born approximation states that p (r, t) ≈ pi(r, t) + ps(r, t) where the scattered wave

ps(r, t) is much smaller than the incident wave pi(r, t), so that the p’s in the scattering

source terms (in the integrand of (2.23)) can be replaced with pi. Assuming that the

incident wave is a plane wave pi(r, t) = P0eiki·r−iωt, ui and p_i are related by equation (2.11)

as −∂ui/∂t = ∇p_i/ρ₀ −→ ui = p_ie/c₀ρ₀, where e = ki/k. Therefore, by applying the

Born approximation and assuming an incident plane wave (i.e. pi(r0) = P₀eike·r0), (2.23)

becomes ps(r) = ∞ ( ( ( −∞ + −k22c_c ! 0 − ∇ρ !_·(ike) ρ0 + ik(ike) · $ u! c0 % (2.24) −k2(e · u_c! 0) + 2ike · ∇(e · u! c0) − e · ∇(∇ · u! c0 ) , pi(r0)g(r|r0)dr0

(26)

= ∞ ( ( ( −∞ + −2k2$ c! c0 % − ike · ∇$ ρ! ρ0 % − 2k(k − ie · ∇) $ e ·u! c0 %, P0eike·r0g(r|r0)d3r₀.

Assuming that ps(r) is being observed in the far-field, r >> r0 and g(r|r0) → exp(ikr −

iks· r0)/4πr, where ks= kr/r is the scattered wave number. Equation (2.24) then simplifies

to ps(r) = − P0k2eikr 2πr ∞ ( ( ( −∞ +$c! c0 % +ie · ∇ 2k $ρ! ρ0 % +(1−ie · ∇_k )(e ·u_c! 0 ) , ei(ki−ks)·r0 dr0 (2.25) = −P0k 2_eikr 2πr ∞ ( ( ( −∞ +$ aµ− iαe · ∇ 2k % T!(r0) + $ bµ+ iβe · ∇ 2k % S!(r0) +(1−ie · ∇ k )(e · u!_(r₀₎ c0 ) , ei(ki−ks)·r0 dr0. (2.26)

Note that, as described in the introduction, (2.25) has an isotropic relative sound speed (or index of refraction) scattering term and a dipole relative density scattering term. There is also a velocity fluctuation scattering term with a much more complicated directionality. As we will shortly see, this term is identically zero for backscatter, and, hence, is not a consideration when attempting to remotely sense turbulence or zooplankton using ship-board sounders.

In (2.26), turbulent fluctuations in index of refraction (c!_/c₀_{) and relative density (ρ}!_/ρ₀₎

are expressed explicitly in terms of temperature and salinity fluctuations (T!_{, S}!_{). The}

coef-ficients aµ and bµare, respectively, the fractional changes in sound speed from temperature

and salinity changes, while α and β are the coefficients of thermal expansion and saline contraction.

Equation (2.26) can be interpreted as an expression of ps(r) in terms of the Fourier

transforms of the turbulent fluctuations of temperature, salinity, and, velocity along the direction of the incident acoustic wave (e · u!_(r₀_{) =} 1

kkiu!i(r0)):,      γT(K) γS(K) γui(K)      = 1 (2π)3 ∞ ( ( ( −∞ e−iK·r0      T!_(r₀₎ S!_(r₀₎ u! i(r0)      dr0, (2.27)

where K = ks− ki is the Bragg wavenumber.

(27)

in (2.26) are related to the Fourier transforms in (2.27) by ∞ ( ( ( −∞         (aµ− iαe·∇_2k ) (bµ+ iβe·∇_2k ) ki c0k(1 − i e·∇ k )           T!_(r₀₎ S!_(r₀₎ u! i(r0)      e−iK·r0   dr0 = ∞ ( ( ( −∞         (aµ− iαe·∇_2k ) (bµ+ iβe·∇_2k ) ki c0k(1 − i e·∇ k )         ∞ ( ( ( −∞ eik1·r0      γT(k1) γS(k1) γui(k1)      dk1   e −iK·r0   dr0 = (2π)3 ∞ ( ( ( −∞         (aµ− iαe·ik_2k1) (bµ+ iβe·ik_2k1) ki c0k(1 − i e·ik1 k )           γT(k1) γS(k1) γui(k1)        1 (2π)3 ∞ ( ( ( −∞ e−i(K−k1)·r0 dr0     dk1 = (2π)3      (aµ+ αe·K_2k ) (bµ− βe·K_2k ) ki c0k(1 + e·K k )           γT(K) γS(K) γui(K)      = (2π)3      (aµ− α sinθ₂)γT(K) (bµ+ β sinθ₂)γS(K) kicos θ c0k γui(K)      , (2.28) where θ is the angle between the incident and scattered acoustic waves (cos θ = e · r

r, see

Figure 2.1).

θ

Figure 2.1: Sketch of incident and scattered wavenumbers. Thus, equation (2.26) becomes

ps(r) = − (2π)3_P 0k2eikr 2πr ) A(θ)γT(K) + B(θ)γS(K) + kicos θ c0k γui(K) * , (2.29)

(28)

2.2.2 Turbulent scattering cross-section

The scattering cross-section per unit solid angle is defined as (Batchelor 1957, Morse and Ingard 1968, Goodman 1990, Medwin and Clay 1998),

σs = r2_+psp∗s, +pip∗i, (2.30) = (2π)4k4 @$ A(θ)γT(K) + B(θ)γS(K) + ki c0k (cos θ)γui(K) % · $ A(θ)γ_T∗(K) + B(θ)γ_S∗(K) + ki c0k cos θγ_u∗_i(K) %A , (2.31) where the subscript ∗ _{indicates the complex conjugate of that quantity.}

Equation (2.31) contains many terms of the form +γy(K)γz∗(K),, which are related to

the spectra and co-spectra of T!_{, S}!_{, u}! i by +γy(K)γz∗(K), = 1 (2π)6 B ∞ ( ( ( −∞ y!(r1)e−iK·r1dr1 ∞ ( ( ( −∞ z!(r2)eiK·r2dr2 C = 1 (2π)3 ∞ ( ( ( −∞   1 (2π)3 ∞ ( ( ( −∞ Dy!_(r₂_{+ x)z}!_(r₂_{)E e}−iK·x_dx  dr2 = 1 (2π)3 ∞ ( ( ( −∞ Φyz(K)dr2= 1 (2π)3Φyz(K) ( ( ( R dr2 = VR (2π)3Φyz(K), (2.32)

where +y!_(r₂_{+ x)z}!_(r₂_{), and hence spectra (Φ}

yz(K)) are only non-zero inside the turbulent

region R, and VR is the volume of that region.

Assuming that co-spectra between velocity fluctuations and scalar fluctuations are neg-ligible and expressing the spectrum of turbulent velocity (Φuiuj(K) in above notation) as

Φij(K), the turbulent scattering cross-section per unit volume can be expressed as

σturb= 2πk4

$

A(θ)2ΦT(K)+B(θ)2ΦS(K)+2A(θ)B(θ)ΦT S(K)+cos2θ

kikj k2_c2 0 Φij(K) % . (2.33) Excluding terms dependent on salinity and substituting A(θ) = aµ = 0.8, a similar

expression can be found in Batchelor (1959). He derived the scattering from a stationary medium with variable properties (the first two terms in (2.25), which he then argued would

(29)

be dominated by fluctuations in sound speed due to temperature) separately from the scattering from a uniform medium in turbulent motion, and never combined them as in (2.33).

i) Isotropic turbulence

If the turbulence is isotropic, then Φ(K) = Φ(K), and the three-dimensional velocity spec-trum is given by (Batchelor 1953)

Φij(K) = $ δij − KiKj K2 % E(K) 4πK2, (2.34) and, therefore, kikjΦij(K) = (1 + cos θ) k2E(K) 8πK2 , (2.35)

where E(K) is the kinetic energy wavenumber spectrum evaluated at the Bragg wavenumber (K).

The isotropic version of (2.33) is then, σturb= 2πk4

$

A(θ)2ΦT(K)+B(θ)2ΦS(K)+2A(θ)B(θ)ΦT S(K)+(1+cosθ)

cos2θE(K) 8πc2₀K2

% . (2.36) The velocity term in equation (2.36) disappears for backscatter, since θ = 180o (and 1 + cos(180o_{) = 0). The turbulent backscatter equation is therefore}

σbs = 2πk4FA2ΦT(K) + B2ΦS(K) + 2ABΦT S(K)G , (2.37)

where now A = aµ− α, B = bµ+ β and K = 2k.

ii) One-dimensional measurements

Typically temperature, salinity or velocity variance is measured along a single direction, yielding a one dimensional spectrum (e.g. φT(kx) or φS(kz)), not the three-dimensional

spectrum (Φ(K)) of (2.37). These one-dimensional scalar spectra are related to the three-dimensional scalar spectra by φ(kx) = H dkydkzΦ(k). Again assuming isotropy, the

three-dimensional scalar spectra can be expressed as (Panchev 1971) Φ(K) = − 1 2π $ 1 kx dφ(kx) dkx % kx=K . (2.38)

(30)

Thus, (2.36) can be expressed in terms of one-dimensional spectra as σturb = −k 4 K $ A(θ)2dφT(kx) dkx + B(θ)2dφS(kx) dkx + 2A(θ)B(θ)dφT S(kx) dkx % kx=K +(1+cos θ)k 4_cos2_θ 4c2₀K2 E(K). (2.39)

For backscatter, (2.39) becomes, σbs= −k 3 2 $ A2dφT(kx) dkx + B2dφS(kx) dkx + 2ABdφT S(kx) dkx % kx=2k , (2.40)

where, as before, A = aµ− α and B = bµ+ β.

In its full form (2.40) is found only in Lavery et al. (2003). If the substitutions A = aµ

and B = bµ are made, (2.40) is found in Seim et al. (1995) and Seim (1999). Goodman

(1990) used (2.40) without either of the salinity terms and with A = aµ. Goodman also

discussed a more general form for the scattering equation, which included the possibility of anisotropy in the turbulence (while still neglecting scattering from velocity fluctuations).

2.3 Classical oceanic turbulence model

When there are no direct measurements of temperature and salinity spectra at the Bragg wavenumber (K), it is necessary to use an oceanic turbulence model to evaluate (2.37) or (2.40), in order to predict acoustic scattering returns from a region of turbulence.

2.3.1 Scalar spectra

Classically, turbulence spectra have been derived using scaling arguments for different sub-ranges of the total wavenumber k. The inertial-convective subrange, which lies between the buoyancy wavenumber

kb= (N3/!)1/2 (2.41)

and the Kolmogorov wavenumber

kν = (!/ν3)1/4, (2.42)

contains scales where the turbulent eddies are too small to feel the effect of the density gradient (represented by the buoyancy frequency, N ) yet too big to feel the effect of molec-ular viscosity (ν). Both kb and kν are derived dimensionally, by looking at the scale at

(31)

which an eddy with energy dissipation rate ! will feel the effect of either buoyancy (kb) or

viscosity (kν). In this subrange, the one-dimensional spectrum for temperature (or salinity)

is expected to be isotropic, and, can only be dependent on the temperature (or salinity) and velocity variance dissipation rates (χ_(T,S)and !), and the scale of the turbulence (1/K). So, dimensionally (Batchelor 1959),

E(T,S)∝ χ(T,S)!−1/3K−5/3 for kb< K < kν, (2.43)

where E_(T,S), the total-wavenumber spectrum of temperature or salinity variance, is related to the isotropic 3-d spectrum by

E_(T,S)= 4πK2Φ_(T,S). (2.44) When ν >> κ_(T,S) _{(which, given that ν ≈ 1.3 × 10}−6_m/s2_{, κ}

T ≈ 1.5 × 10−7m/s2, and,

κS ≈ 1.5 × 10−9m/s2, is true for seawater), there is a viscous-convective subrange, between

the Kolmogorov wavenumber and the Batchelor wavenumber,

k_B(T,S)= (!/(νκ2_(T,S)))1/4, (2.45) where κ(T,S) is the molecular diffusivity of heat or salt. In this subrange eddies feel the effect

of viscosity, but are still too big for molecular diffusion to be important. An expected form for the spectrum can be constructed by assuming that temperature (or salinity) fluctuations are governed only by the viscous strain rate ((!/ν)1/2), χ(T,S) and K. Then, dimensionally,

E_(T,S)_{∝ χ}_(T,S)(!/ν)1/2K−1 for kν < K < kB(T,S). (2.46)

Batchelor (1959) rigourously derived the high-wavenumber three-dimensional spectrum for a scalar in isotropic turbulence by arguing that beyond the dissipative cut-off, kν =

(!/ν3)1/4, where all velocity variance is destroyed by viscosity, the distribution of a scalar θ (which can be temperature or salinity) will be affected only by the residual pure straining distortion. In a Lagrangian frame, with axes fixed to the direction of constant principal rates of strain (α, β, γ), this can be expressed as,

∂θ ∂t + αx ∂θ ∂x+ βy ∂θ ∂y + γz ∂θ ∂z = κ∇ 2_θ, _(2.47)

(32)

θ0(t)eik(t)·x, with the initial condition θ(t=0) = Θ0eil·x, we get dk1 dt = −αk1 → k1(t) = l1e −αt dk2 dt = −βk2 → k2(t) = l2e −βt dk3 dt = −γk3 → k3(t) = l3e −γt _(2.48) dθ0 dt = −κk 2_θ 0 → θ0(t) = Θ0exp) κ 2α(k 2 1 − l21) + κ 2β(k 2 2 − l22) + κ 2γ(k 2 3− l23) * . (2.49) After a relatively short time, the θ-planes will become perpendicular to the direction of the greatest rate of contraction, which in this case is γ, since α > β > γ and α + β + γ = 0. Thus,

θ → Θ0e

κ(k2−l2)

2γ _eik·x_, _(2.50)

for each spectral component of θ. The contraction of each spectral component of θ means that θ-variance is being transferred from lower wavenumbers (l) to higher wavenumbers (k), such that Eθ(k)dk = Eθ(k) k ldl = Eθ(l)dle κ(k2−l2) γ _. _(2.51)

This combined with the requirement that the total level of θ variance is described by χ (i.e. H∞

0

k2_E

θ(k)dk = _2κχ), leads to a total-wavenumber θ-spectrum of

Eθ(k) = −

χ γke

κk2

γ _. _(2.52)

The principle rate of contraction (γ) is not necessarily constant throughout the fluid, however, so the true form of Eθ(k) is a superposition of exponentials with different

expo-nents. Nevertheless, it is standard in the literature to approximate this superposition of exponentials as a single exponential with γ ∝F_'

ν

G1/2

, whereF_'

ν

G1/2

is the representative vis-cous strain rate of the turbulence. As we know γ < 0, this proportionality can be expressed with the universal constant q as

γ = −1_q "_ν!#1/2. (2.53)

For most modelling purposes—including ours—the fact a steep roll-off occurs at the cor-rect wavenumber is all that is important. The fact that a single exponential cannot really represent a superposition of exponentials is therefore unimportant. The empirically mea-sured value of q can be thought of as a factor that somehow “condenses” the superposition

(33)

into a single term, so that the roll-off occurs at the correct spot.

Combining (2.53) with (2.52) and (2.44), gives the isotropic, high-wavenumber, three-dimensional spectrum for temperature or salinity,

Φ_(T,S)(K) = qχ(T,S) 4πK3 "ν ! #1/2 e−qK2/k2B(T,S)_, _(2.54)

which can be applied directly to (2.37). Note that (2.54) agrees with the dimensional argument (2.46) for kν < K < kB(T,S) and includes a dissipative roll-off for wavenumbers

greater than k_B(T,S).

The one-dimensional Batchelor spectrum can be obtained by integrating (2.38) using (2.54) (Gibson and Schwartz 1963),

φ_(T,S)= qχ(T,S) 2kx "ν ! #1₂      e−qk2x/k2B(T,S)₋ √ 2qkx kB(T,S) ∞ ( √ 2qkx kB(T,S) e−x2/2_dx      for kx > kν. (2.55)

Applying (2.38) to (2.43) for low wavenumbers and using (2.54) for high wavenumbers, the temperature and salinity spectra required in (2.37) are

Φ_(T,S)(K) =      C'_2π2/3_NΓ2 " d(T,S) dz #2 K−11/3 _{for K ≤}F ₅ 12 G3₂ kν q(ν')1/2 2πK3 _NΓ2 " d(T,S) dz #2 e−qK2/kB(T,S)2 _{for K >}F 5 12 G3₂ kν, (2.56)

where the semi-empirical relationships

χT = 2Γ! N2 $ dT dz %2 and χS = 2Γ! N2 $ dS dz %2 (2.57) for the mixing efficiency (Γ) were used to express χ(T,S) in terms of large-scale quantities

and !. The parameter C is a fundamental constant (measured, for example, by Gibson and Schwartz (1963) to be 0.58 and by Gargett (1985) to be 2.7), but in practice C is often chosen to match the pieces of the spectrum (Seim 1999), at the peak of the Panchev and Kesich (1969) theoretical transverse shear spectrum (kx = kν/8). This means the pieces of

three-dimensional spectrum meet at K = k_∗ =F₅

12 G3₂ kν and C = q(ν')1/2 2πk3 ∗ Γ N2 '2/3 2π NΓ2k∗−11/3 = 5 12q = 1.542, (2.58)

(34)

for straining constant q = 3.7 (Oakey 1982).

While this ad hoc way of determining the universal constant C means that we have to accept that there is a large degree of uncertainty in C, it turns out that a correct estimate of C is not vital to the analysis of the data collected in Knight Inlet. The data were of high enough frequency that they all have K >F₅

12

G3₂

kν, and the models are therefore evaluated

using the second part of (2.56) which does not depend on C.

2.3.2 Scalar co-spectra

As there are no measurements of scalar co-spectra in the ocean, all theoretical models are highly speculative.

i) No-correlation model

The simplest model for the co-spectrum is to assume that there is no correlation between temperature and salinity at the scale of the acoustic wavelength employed. With the no-correlation model, ΦT S = 0 and the backscattering cross-section per unit solid angle per

unit volume (σturb, from 2.37) is given by

σturb =        CΓk1/3 8N2 F_' 2 G2₃ + A2"dT_dz#2+ B2"dS_dz#2 , for k ≤F ₅ 12 G3₂ _k_ν 2 q(ν')1/2_Γk 8N2 + A2"dT_dz#2e−4qk2_/k2 BT + B2 " dS dz #2 e−4qk2_/k2 BS , for k >F ₅ 12 G3₂ _k_ν 2 , (2.59) where the substitution K = 2k sin(θ/2) = 2k (for backscatter) has been made.

The no-correlation model for the co-spectrum is unsatisfactory because in many regions (including Knight Inlet) there is a strong correlation between temperature and salinity at large scales. So, although no covariance is expected for very high wavenumbers (certainly for wavenumbers higher than the diffusive cut-off for temperature), this model is not physically plausible at low wavenumbers and tells us nothing about the expected transition from covariance at small wavenumbers to no-covariance for k >> kBT.

ii) Stern’s model

Recently Lavery et al. (2003) proposed a co-spectrum model based on Stern (1968)’s theory. Stern combined the equations for the rates of change of temperature and salinity fluctuations (i.e. dT!_{/dt + u · ∇T = κ}

T∇2T!+ κT∇2T and dS!/dt + u · ∇S = κS∇2S!+ κS∇2S) and,

(35)

fluctuations in an infinitesimally small control volume about a point due to diffusion across the boundary), he derived the relation

1

2∇ · u(T!− δS!)2= −κT(∇T!)2− δ

2_κ

S(∇S!)2+ δ(κS+ κT)∇T!· ∇S!, (2.60)

where δ = ∇T /∇S can be assumed to be equivalent to δ = dTdz/dSdz.

If the T-S relation of the fluctuations is sufficiently tight, the “temperature anomaly” term (on the left-hand side of 2.60) is negligible compared to the other terms. Neglecting the “temperature anomaly” term, recognising that κT + κS ∼= κT, and transforming into

frequency space, (2.60) becomes

ΦT S = 1 δΦT + κS κT δΦS. (2.61)

The linearity of (2.61) means that the relationship is the same for the one-dimensional spectra, φT S = 1 δφT + κS κT δφS, (2.62)

as expressed in Lavery et al. (2003).

Thus, combining (2.56) and (2.61) the backscattering cross section per unit solid angle per unit volume (σturb, from 2.37) is:

σturb =                CΓk1/3 8N2 F_' 2 G2₃ + A2"dT dz #2 + B2"dS dz #2 + 2AB"1+κS κT # " dT dz #" dS dz #, for k ≤F₅ 12 G3₂ _k_ν 2 q(ν')1/2_Γk 8N2 +) A2"dT_dz#2+ 2AB"dT_dz#"dS_dz# * e−4qk2/k2 BT + ) B2"dS_dz#2+ 2ABκS κT " dT dz #" dS dz #* e−4qk2/k2 BS , for k > F₅ 12 G3₂ _k_ν 2 , (2.63) where, again, the substitution K = 2k sin(θ/2) = 2k (for backscatter) has been made.

There are several problems with this co-spectrum model. First, when k >> kBT we

expect no covariance between temperature and salinity (all temperature fluctuations have been damped) and, therefore, the co-spectrum must be zero. This model has a small (since κS/κT ≈ 1/100), but finite co-spectrum for kBS > k > kBT. Second, at low wavenumber

(i.e. k << kBT), the curly brackets in (2.63) are

I ...J= B2"dS dz #2I K_A Bδ + 1 L2 + 2κS κT A Bδ J → −2B2 κS κT " dS dz #2 , as δ → −BA. (2.64)

(36)

scattering cross-sections have no physical meaning. In Knight Inlet, −B/A = −0.67, which is close enough to the mean δ ≈ −0.5 to cause the occasional problem applying this co-spectrum model.

iii) One-dimensional upper-bound model

Washburn et al. (1996) and Seim (1999) applied the mathematical identity for the upper limit of a co-spectrum (Bendat and Piersol 1986) to the one-dimensional spectra,

φT S ≤ (φTφS)

1

2, (2.65)

to evaluate the maximum effect the co-spectrum could have. Using this upper-bound model, the derivative of the 1-d co-spectrum is

dφT S(kx) dkx = 1 2 $ φS φT %1₂ dφT dkx +1 2 $ φT φS %1₂ dφS dkx . (2.66)

Thus, applying (2.66) to (2.43) and (2.55), the backscattering cross section per unit solid angle per unit volume (σturb, from 2.40) is

σturb =                CΓk1/3 8N2 F_' 2 G2₃ + A2"dT dz #2 + B2"dS dz #2 + 2AB"dT_dz#"dS_dz# , for k ≤F ₅ 12 G3₂ _k_ν 2 q(ν')1/2_Γk 8N2 +) A2"dT_dz#2+ AB"dT_dz#"dS_dz# Mf (k,q,kBS) f (k,q,kBT) N1₂* e−4qk2/k2 BT + ) B2"dS_dz#2+ AB"dT_dz#"dS_dz# Mf (k,q,kBT) f (k,q,kBS) N1₂* e−4qk2/k2 BS , for k > F ₅ 12 G3₂ _k_ν 2, (2.67) where f (k, q, k_B(T,S)) = _2k1 e−4qk2/kB(T,S)₋ (qπ)1/2 kB(T,S) erf c(2q 1/2_k/k B(T,S)).

This model predicts non-zero co-spectrum at low wavenumbers and zero co-spectrum at higher wavenumbers (where φT = 0), and is in that way more physically plausible than

either of the two above. But, (2.67) predicts negative scattering cross-sections far more frequently than (2.63). It fails around the point where φT → 0, for most δ < 0.

iv) New co-spectrum model: three-dimensional upper-bound

As all the co-spectrum models proposed by previous authors give unreasonable results in some part of the parameter space, I wondered if there is a better way to model the co-spectrum. One idea is to apply the upper-bound model to the three-dimensional spectra rather than the one-dimensional spectra. This makes more sense physically, as the

(37)

smooth-ing out of temperature fluctuations due to diffusion (and thus the disappearance of the co-spectrum) is more likely to be dependent on the total wavenumber than the wavenum-ber in any particular direction (any kx could be made up of components with large or small

total wavenumber). Thus, applying the upper-bound theorem in 3-d, we get ΦT S(K) = (ΦT(K)ΦS(K)) 1 2 = q 4πK3 "ν ! #1/2 O χTχSe −qK2 $ 1 k2_BT+ 1 k2_BS %_P1 2 = q (χTχS) 1 2 4πK3 "ν ! #1/2 e−qK2/k2 BT S, (2.68) where _k21 BT S = 1 2 " 1 k2 BT + 1 k2 BS # = 1₂F_ν ' G1₂ (κT + κS), and thus, kBT S = I !/MνF_κ_T_+κ_S 2 G2NJ1/4 is the “Batchelor wavenumber” of the co-spectrum.

Looking back to the derivation of the scalar spectrum (2.54), we can see that (2.68) is more than just an upper limit for the co-spectrum, but an exact solution for isotropic turbulence. Note, in (2.48), that the wavenumber depends only on the background straining rate γ = −1q

F_'

ν

G1/2

, which will be the same for all scalars. So, analogously to Batchelor’s (1959) argument, we can write

ET S(k) k ldl = ET S(l)dle κT (k2−l2) 2γ _eκS (k 2−l2) 2γ _, _(2.69) and, assuming H∞ 0 k2ET S(k)dk = " χT 2κT χS 2κS #1/2 , derive (2.68) directly.

Another way to look at it is that, because the time dependence of the wavenumber is not scalar-dependent, T and S will remain phase locked as they evolve to higher wavenumbers and, although the co-spectrum will eventually disappear as T fluctuations get diffused away, the coherency between T and S will remain 1. With perfect coherency, ΦT S(K) =

(ΦT(K)ΦS(K))

1

2 is exact.

Combining (2.68) with (2.56), the backscattering cross section per unit solid angle per unit volume (σturb, from 2.37) is

σturb =                CΓk1/3 8N2 F_' 2 G2₃ + A2"dT_dz#2+ B2"dS_dz#2+ 2AB"dT_dz#"dS_dz# , for k ≤F₅ 12 G3₂ _k_ν 2 q(ν')1/2_Γk 8N2 + A2"dT dz #2 e−4qk2_/k2 BT + B2 " dS dz #2 e−4qk2_/k2 BS +2AB"dT_dz#"dS_dz#e−4qk2/k2 BT S ,for k > F₅ 12 G3₂ _k_ν 2 , (2.70)

Sound scattering from oceanic turbulence