Acoustic mode coupling induced by nonlinear internal waves: evaluation of the mode coupling matrices and applications.

Acoustic mode coupling induced by nonlinear internal waves: Evaluation of the mode coupling matrices and applications T. C. Yanga) Institute of Applied Marine Physics and Undersea Technology, National Sun Yat-sen University, 70 Lien-Hai Road, Kaohsiung, Taiwan

(Received 6 April 2013; revised 11 December 2013; accepted 16 December 2013) This paper applies the mode coupling equation to calculate the mode-coupling matrix for nonlinear internal waves appearing as a train of solitons. The calculation is applied to an individual soliton up to second order expansion in sound speed perturbation in the Dyson series. The expansion is valid so long as the fractional sound speed change due to a single soliton, integrated over range and depth, times the wavenumber is smaller than unity. Scattering between the solitons are included by coupling the mode coupling matrices between the solitons. Acoustic fields calculated using this mode-coupling matrix formulation are compared with that obtained using a parabolic equation (PE) code. The results agree very well in terms of the depth integrated acoustic energy at the receivers for moving solitary internal waves. The advantages of using the proposed approach are: (1) The effects of mode coupling can be studied as a function of range and time as the solitons travel along the propagation path, and (2) it allows speedy calculations of sound propagation through a packet or packets of solitons saving orders of magnitude computations compared with the PE code. The mode coupling theory is applied to at-sea data to illustrate the underlying physics. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4861253] V PACS number(s): 43.30.Re, 43.30.Bp, 43.30.Pc [HCS]

I. INTRODUCTION

Mode coupling by nonlinear internal waves, also known as solitary internal waves (SIWs) consisting of packets of solitons, was first noted by Zhou et al.—mode coupling effects are most pronounced when the mode wavenumber differences match the energetic wavenumbers of the solitons, causing either abnormal transmission loss or large signal fluctuations.1–3 Motivated by this work, various theoretical and experimental studies have been carried out to investigate the effect of SIWs on sound propagation assuming the direction of (sound) propagation is closely parallel to that of SIWs. For example, Tielbuerger et al.4 and Finette et al.5 studied numerically the effects of mode coupling on transmission loss due to internal waves. Headrick et al.,6,7 Duda and Preisig,8 and Duda9 studied the effects of SIWs on acoustic signals and compared the modeled results with data collected on New Jersey and New England continental shelf, respectively. Katsnelson et al.10 investigated the temporal sound field fluctuations in the presence of SIWs in shallow water. Yoo and Yang11 studied the effects of internal waves on source localization. Preisig and Duda12 studied mode conversion/coupling by a single soliton assuming a sharp interface approximation (i.e., a square wave soliton) to gain theoretical understanding of the coupling mechanism.7 Duda and Preisig8 and Rouseff et al.13 modeled acoustic energy transfer between the modes using a parabolic equation (PE) model. For moving SIWs, mode coupling causes the acoustic intensity to fluctuate as the SIWs travel a mode cycle distance, resulting in intensity fluctuation on the scale of a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

610

J. Acoust. Soc. Am. 135 (2), February 2014

Pages: 610–625

minutes. Duda and Preisig8 observed that the energy transfer between the modes can cause the acoustic intensity envelop at the receiver to change with the position of the (moving) solitons due to different mode attenuation between the modes, referred to as the energy gain or loss factor. This causes the acoustic intensity envelop to vary with time on the scale of hours. Others have studied three-dimensional variations of the sound field, which will not be covered here. For non-square SIWs, most prior work adopted the numerical approach. Recently, Colosi used the coupled mode equation to model the mode intensities at the receiver in the presence of SIWs, assuming single scattering.14 This approach illustrates explicitly the wavenumber matching condition mentioned above. The coupled mode equation approach is adopted in this paper. Instead of modeling the mode intensity, it is used to model the mode-coupling matrix. The mode-coupling matrix formulation allows much more flexibilities as it does not impose any condition on the nature of the incoming field (or its mode content), and has been adopted by others who use the PE to calculate the mode-coupling matrix.8,13 The approach presented in this paper differs from Ref. 14 in several aspects: (1) The calculation is carried out to second order in sound speed perturbation for the case of a single soliton, whereas Ref. 14 included only the first order calculation. (2) Multiple scatterings between the SIWs are included using the coupling matrix formulation whereas Ref. 14 included only single scattering. (3) Mode attenuation factors responsible for energy gain/loss mentioned above are derived explicitly which was implicit in Ref. 14. (4) Using the mode-coupling matrix, one can evaluate the acoustic intensity propagating through multiple packets of SIWs, which would be difficult using the approach presented in Ref. 14.

0001-4966/2014/135(2)/610/16/$30.00

C 2014 Acoustical Society of America V

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 139.184.14.159 On: Wed, 11 Feb 2015 23:18:32

An example is given below involving data where sound propagates through two packets of SIWs, separated by 28 km before reaching the receivers. The validity of the mode-coupling matrix formulation is demonstrated numerically by comparing the received acoustic intensity using the mode-coupling matrix with that obtained using a PE code. The benefits of the normal mode approach compared with the PE code are: First, the physics of mode coupling can be traced as a function of range and time for traveling SIWs. For example, many features of the data can be characterized by the mode content and coupling. In particular, the intensity fluctuation frequencies and energy gain/loss factor can be estimated based on the mode wavenumbers and attenuations. This is demonstrated with both simulated and real data. It might also be possible to assess the locations based on acoustic intensity time series when visual images of the SIWs and/or oceanographic measurements are not available. Second, this approach uses significantly (2 orders of magnitude) less computations than PE. For example, the numerical calculations presented in this paper take 4 to 5 h using PE and only 40 s using the coupled mode approach. For time critical missions (such as tactical decisions), this difference can be critical. This paper is organized as follows. An analytic form of the mode-coupling matrix is given in Sec. II using the coupled mode equation to the second order in sound speed perturbation. Sound propagation through SIWs are simulated numerically in Sec. III to compare with that obtained using the Range-Dependent Acoustic Model (RAM) PE code.15 The 400 Hz data received on the Naval Research Laboratory (NRL) vertical line array (VLA) during the SWARM95 experiment16 are analyzed in Sec. IV. Summary and conclusions are given in Sec. V.

FIG. 1. (Color online) (a) Geometry of the SIWs relative to the source and receivers. (b) Mode propagation and coupling matrices, and mode amplitudes defined for each range grid.

mode attenuation; K(rkþ1,rk) is the mode-coupling matrix due to kth soliton. The objective is to calculate the mode-coupling matrix for which we shall use the coupled mode equation.14,18–21

A. Coupled mode equation

Expressing Am ðrÞ ¼ am ðrÞeilm r , and assuming narrow angle (forward scattering) approximation for the Helmholtz equation, one obtains the following coupled mode equation14,18,19 for the reduced mode amplitude am: M X @an ðrÞ nm ðrÞam ðrÞ; ¼ i @r m¼1

(3)

where II. COUPLED MODE CALCULATION

For a receiver located at range R and depth zj, the pressure field for a narrow band signal can be expressed in terms of normal modes M X Am ðRÞ pffiffiffiffiffiffiffiffi /m ðzj Þ; pj ðzj Þ ¼ km R m¼1

p Y

Kðrkþ1 ; rk ÞTðr1 ; 0ÞAð0Þ;

lnm ¼ ln lm ¼ lmn

(4)

ð k02 dCðr; zÞ nm ðrÞ ¼ pffiffiffiffiffiffiffiffiffi dz /n ðzÞ/m ðzÞ; C0 kn km

(5)

and

(1)

where /m(zj) is the mode depth function for the mth mode, zj is the depth of the jth receiver, and Am(R) is referred to as the (complex) mode amplitude at range R. Consider acoustic propagation through a packet of SIWs as shown in Fig. 1(a). The mode amplitude vector A(R) ¼ [A1(R), A2(R),…,AM(R)]T, where the subscript denotes the mode number and the superscript T denote the transpose, can be expressed as17 AðRÞ ¼ TðR; rpþ1 Þ

nm ¼ nm eilnm r ;

where C0 is the mean sound speed profile, and dC(r,z) is the sound speed perturbation (due to internal waves). The solution to Eq. (3) is given by the Dyson series,14,18,19,22 which can be expressed in two forms: !# " ðR M X R^ exp i dr ðrÞ am ð0Þ an ðRÞ ¼ ¼ an ð0Þ i

(2)

J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014

M ðR X

nm

dr nm ðrÞam ð0Þ

m¼1 0

k¼1

where T(r2,r1) ¼ diag{exp(ilm(r2 r1))}, with m ¼ 1,2,…,M, is the mode propagation matrix assuming range independent or adiabatic modes between range r1 and r2, and lm ¼ km þ iam, where km is the mode wavenumber and am is the

0

m¼1

M X M ðR X m¼1 l¼1 0

ðr

dr

dr 0 nm ðrÞ ml ðr0 Þal ð0Þ þ

0

(6) and T. C. Yang: Mode coupling by internal waves

611


an ðR þ DRÞ ¼

" M X

R^ exp i

ð RþDR

dr ðrÞ

R

m¼1

¼ an ðRÞ i

M ð RþDR X m¼1 R

!#

M X M ð RþDR X m¼1 l¼1

nm

dr nm ðrÞam ðRÞ

ðr dr

R

am ðRÞ

dr 0 nm ðrÞ ml ðr 0 Þal ðRÞ

0

þ ;

(7)

where R^ is a range ordering operator and the boldface symbol is a matrix.19,22 [Equation (6) can be obtained from Eq. (7) by setting R ¼ 0. Equation (7) can also be obtained by taking the difference of Eq. (6) at range R þ DR and R.] The term linear in nm is referred to as the single scattering term, and the terms quadratic and higher power in nm are referred to as multiple scattering terms. B. Mode amplitudes evaluated using the mode-coupling matrix formulation

Equation (7) can be used to calculate the mode-coupling matrix between the reduced mode amplitudes P at range r2 ¼ R þ DR and r1 ¼ R, defined by an ðr2 Þ ¼ M n¼1 Cnm ðr2 ; r1 Þam ðr1 Þ. One can use the series expansion, if the series converges. One finds that the single scattering term in Eq. (7) has an order of magnitude given by (noting that the off-diagonal terms are smaller than the diagonal) b¼ ¼

ð RþDR

nm ðrÞe R ð k02 RþDR kn

k0

R ð RþDR

ilnm r

drdzdCðr; zÞ/2n ðzÞ=C0 (8)

R

Equation (7) converges if b < 1, which requires (approximately) that the fractional sound speed change integrated over the range and depth of the perturbation be less than one. For example, b 0.6 for the fractional sound speed changes induced by a single soliton in previous publications (Refs. 8 and 13); b is approximately the same for all three solitons. Note that if b is greater than, say, 0.3, the single scattering term may not be sufficient and one needs to include the second order (multiple scattering) term in Eq. (7). In this paper, we will calculate the mode-coupling matrix, keeping terms up to the second order. The fractional sound speed change due to internal waves is related, to a good approximation, to the particle displacement g by lðr; y; z; tÞ ¼ dC=C0 ¼ GðzÞN 2 ðzÞgðr; y; z; tÞ, where N(z) is the Brunt-V€ais€al€a buoyancy frequency, and G(z) is a smooth function of depth.4,5,11 Expanding the displacement in terms of the internal wave mode depth functions, and assuming that the first internal-wave mode dominates,4,11 the displacement for a soliton of the hyperbolic secant-square type (based on the KdV equation23) can be expressed as gðr; y; z; tÞ ¼ g0 sech2 ½r r0 ðtÞ=DWðzÞ, where g0 is the amplitude of the soliton, and W(z) is the first mode depth-function of the internal waves; r0 and D denote 612


where H(z) GN2(z)W(z). Using Eq. (7) and expressing the mode-coupling matrix in terms of ð2Þ Cnm ðr2 ; r1 Þ ¼ dnm iSð1Þ nm ðr2 ; r1 Þ Snm ðr2 ; r1 Þ þ ;

(10) one finds Sð1Þ nm ðr2 ; r1 Þ

ð r2

nm ðrÞeilnm r dr ¼ g0 Znm eilnm r0 ;

(11)

r1

where r0 ¼ (r1 þ r2)/2, and Znm given by ð k02 dzHðzÞ/n ðzÞ/m ðzÞ Znm ¼ pffiffiffiffiffiffiffiffiffi kn km ð d=2 q ilnm q e sech2 dq D d=2 ð k02 dzHðzÞ/n ðzÞ/m ðzÞ ’ pffiffiffiffiffiffiffiffiffi kn km ð1 q ilnm q e sech2 dq; D 1

(12)

where d ¼ r2 r1 is the range span of the soliton, assuming d D. [It should be obvious that other waveforms can also be used in Eq. (12) such as when the soliton waveform changes with respect to range.] With some mathematical manipulations (see the Appendix), one finds

drjn¼m

drdzdCðr; zÞ=C0 :

the center and width of the soliton. The fractional sound speed perturbation is then given by dC r r0 ; (9) ¼ g0 HðzÞsech2 C0 D

Sð2Þ nm ðr2 ; r1 Þ

M ð r2 X k¼1

dr nk ðrÞe

ilnk r

r1

ðr

dr 0 km ðr 0 Þeilkm r

0

r1

M X

¼

1 ð1Þ ð1Þ S ðr2 ; r1 ÞSkm ðr2 ; r1 Þ 2 k¼1 nk

¼

M g20 X Znk Zkm eilnm r0 : 2 k¼1

(13)

Recall the definition: Am ðrÞ ¼ P am ðrÞeilm r . The modecoupling matrix defined by An ðr2 Þ ¼ M n¼1 Knm ðr2 ; r1 ÞAm ðr1 Þ is then given by Knm ðr2 ; r1 Þ ¼ Cnm ðr2 ; r1 Þeiðln r2 lm r1 Þ ¼ dnm eiln ðr2 r1 Þ ig0 Znm eiðln þlm Þðr2 r1 Þ=2 ! M g20 X Znk Zkm eiðln þlm Þðr2 r1 Þ=2 þ : 2 k¼1 (14) 1. Single soliton

Based on Fig. 1(a), the mode amplitude at range R propagating through a single soliton located at a range between r1 and r2 is given by T. C. Yang: Mode coupling by internal waves


An ðRÞ ¼ e

iln ðRr2 Þ

(

M X

Knm ðr2 ; r1 Þe

~ 0 ðtÞÞ ’ 2g0 ½Qðr R

Am ð0Þ

m¼1

’ eiln R An ð0Þ ig0

ilm r1

M g20 X Znk Zkm eilnm r0 Am ð0Þ 2 k¼1 )

( M X M 2an R X e kn n¼1 m¼1 )

Zmn eamn r0 ðtÞ

Am ð0ÞAn ð0Þ ;

M X Znm eilnm r0 Am ð0Þ þ ; m¼1

(15) where r0 ¼ (r1 þ r2)/2. One finds that the mode intensity to first order in g0 is given by " M X Zmn sinðkmn r0 Þ jAn ðRÞj2 ¼ e2an R A2n ð0Þ þ 2g0

(19)

where the gain factor, eanm r0 ðtÞ , is a function of the range of the soliton to the source. In the case when mode coupling occurs predominantly between a pair of modes, denoted as the nth and mth modes, one finds from Eq. (19), keeping only the n, m term in the summation, that the intensity envelope changes by the following amount: ~ dðln½QðtÞÞ ’ amn dr0 ¼ amn vdt;

(20)

m¼1

eamn r0 Am ð0ÞAn ð0Þ :

(16)

Equation (16) is similar to the expression obtained by Colosi [Ref. 14, Eq. (17) or (25)] but is different in the (explicit) mode attenuation term: eanm r0 . The difference is addressed numerically in Sec. III B. The mode attenuation factor is related to the energy gain/loss factor discussed by Duda et al.8 for SIWs traveling along the propagation path. The acoustic energy summed over a vertical array of receivers is given by X p ðzj Þpðzj Þ Q¼ j

X A ðRÞAm ðRÞ X n pffiffiffiffiffiffiffiffiffi ¼ /n ðzj Þ/m ðzj Þ kn km R m;n j ¼

M X jAn ðRÞj2 n¼1

kn R

when the soliton moves a distance dr0 ¼ vdt. Equation (20) expresses the energy gain (loss) factor (of the energy envelope) in terms of the difference of the mode attenuations. The result for multiple solitons is more complicated and will be studied using numerical simulations. 2. Multiple solitons

For sound propagation through multiple solitons, one uses the coupled mode solution, Eq. (2), based on the modecoupling matrix derived above, Eq. (14), for each soliton. As an example, consider sound propagation through three solitons. To simplify the discussions, let us take the first order approximation for the coupling matrix. Using Eqs. (2) and (14), one has An ðRÞ ¼ eiln ðRr4 Þ

M X

Kni ðr4 ; r3 ÞKij ðr3 ; r2 ÞKjm ðr2 ; r1 Þ

i;j;m¼1

;

(17)

where we have used the mode orthogonality condition PN / ðzj Þ/m ðzj Þ ¼ dm;n . To keep the equation simple, we n j¼1 shall keep only terms up to first order in g0 for the following discussions. The depth-integrated energy for sound propagating through a single soliton located at r0 is given, using Eq. (16), by ( M 2an R M X M 2an R X 1 X e e A2n ð0Þ þ 2g0 Qðr0 Þ ¼ kn R n¼1 kn n¼1 m¼1 ) Zmn sinðkmn r0 ðtÞÞeamn r0 ðtÞ Am ð0ÞAn ð0Þ ;

(18)

where we assumed r0 ðtÞ ¼ vt þ r0 for a soliton moving at speed v along the propagation direction. From Eq. (18), one observes that the intensity oscillates as a function of time with periods given by T ¼ 2p/vknm. [During this period the soliton has traveled a distance equal to the mode cycle distance vT ¼ jknm j.] Denote the time-varying components of ~ ¼ QðtÞ hQðtÞi, or the the depth-integrated energy by QðtÞ ~ range variation by QðrÞ ¼ QðrÞ hQðrÞi, where hxi denotes ~ 0 ðtÞÞ has an envelope [via the the mean of x. Then Qðr Hilbert transform of Eq. (18)] given by J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014

eilm r1 Am ð0Þ ’ eiln ðRr4 Þ

M X dni eiln ðr4 r3 Þ ig3 Zni i;j;m¼1

iðln þli Þðr4 r3 Þ=2

e h i dij eili ðr3 r2 Þ ig2 Zij eiðli þlj Þðr3 r2 Þ=2 h i djm eilj ðr2 r1 Þ ig1 Zjm eiðlj þlm Þðr2 r1 Þ=2 eilm r1 Am ð0Þ; (21) where Zij is different for different solitons [see Eq. (12)]; the dependence is not explicitly shown. One notes that Eq. (21) contains multiple scattering terms between the (different) solitons, i.e., terms involving products of gi and gj, i 6¼ j. One can likewise write down a similar equation including terms up to second order in g2j , j ¼ 1,2,3 as derived above to demonstrate the “full” effects of multiple scattering between solitons. Our approach is to evaluate the mode-coupling matrix derived above for each soliton numerically, and conduct mode propagation using Eq. (2). Multiple scatterings between the (multiple) solitons are automatically included. T. C. Yang: Mode coupling by internal waves

613


C. Mode amplitudes evaluated using the full range Dyson series

An alternative method is to calculate the mode amplitudes by evaluating the Dyson series at the receiver as given by Eq. (6). Colosi adopted this approach keeping only the first order terms in g0.14 It will be shown that the two approaches are the same for the soliton case, order by order. For multiple solitons, higher orders are required using this approach to account for the multiple scattering effect. The (major) problem with this approach is that the series is not expected to converge when b [Eq. (8)] is not small, as is the case when multiple solitons are included. A secondary problem is that mode attenuation over a long range is difficult to evaluate analytically; it was expressed in a wavenumber representation in Ref. 14. 1. Single soliton

Using Eq. (6), the mode amplitude at the receiver is given by An ðRÞ ¼

M X

Knm ðR; 0ÞAm ð0Þ ¼ eiln R

m¼1

M X

Cnm ðR; 0ÞAm ð0Þ;

m¼1

(22) where one finds ðR Cnm ðR; 0Þ ¼ dnm i nm ðrÞeilnm r dr

M ðR X

0

dr nk ðrÞe

k¼1 0

ilnk r

ðr

dr0 km ðr0 Þeilkm r

0

0

ð r2 ¼ dnm i nm ðrÞeilnm r dr

calculate the depth-integrated intensity at 400 Hz as a function of the SIW position. We are interested in the intensity oscillation as SIWs move along the propagation path between the source and receivers for the purpose of understanding and interpreting the data presented in Sec. IV. Sound propagations through the SIWs were previously modeled using the PE approach.3–11,13 We will use the RAM PE model15 as the bench mark and show that similar results can be obtained using the mode-coupling matrix formulation with orders of magnitude reduction in computation time compared with the RAM code. We will show that many features of the simulated data agree with the mode coupling theory, and are independent of the details of acoustic environments. For this study, we shall assume that the soliton waveform does not change with time. For the numerical simulations, we adapt a shallow water environment with inputs from the SWARM95 experiment16 which provided extensive data about the SIWs. Based on Eq. (9), one estimates the maximum sound speed perturbation to be [dC] ’ C0G[N]2 h0 5 m/s, given, for example, that C0 1500 m/s, G 2.4 s2/m2, g0 14 m for the internal wave displacement, and the maximum buoyancy frequency measured is [N] 0.01 Hz.4,11 For comparison with the RAM model calculation, we shall use a simple analytical expression for dC due to internal waves, often used for benchmark testing of propagation models in the presence of internal waves,24 given by r r0 ; (25) dC ¼ cðz=BÞez=B sech2 D

Since vnm(r) is nonzero only within range r1 and r2, one finds that Eq. (23) yields the same answer as Eq. (10) using Eqs. (11) and (13).

where B is the thermocline depth and c is the amplitude, which is related to the maximum sound speed perturbation by [dC] ¼ ce1. One finds, for example, c 14 m/s for [dC] 5 m/s. Comparing Eq. (25) with Eq. (9), the theoretical expression derived in Sec. II remains unchanged if one replaces g0 by c, and H(z) GN2(z)W(z) by H(z) ¼ (z/B)ez/B/C0. For bench mark testing, we shall assume a range independent environment with a constant water depth of 60 m. The bottom has a sound speed 1650 m/s, density 1.76 g/cm3, and attenuation 0.8 dB per wavelength.8

2. Multiple solitons

A. Single soliton

For the multiple solitons case, using Eq. (6), and taking the first order approximation, one has

For a secant type soliton with a width D ¼ 100 m, the acoustic intensity integrated over depth is shown as a function of the soliton position (the center of the soliton) relative to the source in Fig. 2(b) using the RAM PE code. The source is at a depth of 50 m, exciting primarily the first mode. The receivers are located 30 km from the source and are spaced with 1 m separation in depth covering the entire water column. For a soliton at a fixed range r0 to the source, one estimates the sound speed perturbation using Eq. (25) within r1 ¼ r0 2D, and r2 ¼ r0 þ 2D, and calculates the acoustic intensity at the receiver using the RAM code. The sound speed within r1 and r2 is sampled every 10 m. The acoustic field is calculated using 0.2 m depth increments and 2 m range increments. The RAM calculations are conducted/repeated for a soliton at various ranges to the source, r0 ¼ 1 to 28 km. The results (depth-integrated acoustic energy)

M ð r2 X

r1

dr nk ðrÞeilnk r

ðr

k¼1 r1

0

dr 0 km ðr 0 Þeilkm r :

r1

(23)

P X Cnm ðR; 0Þ ¼ dnm iSnm ¼ dnm i gp Znm eilnm rp :

(24)

p¼1

One notes that Eq. (24) ignores the scattering terms between the solitons, i.e., terms involving products of gi and gj, i 6¼ j, which are present in Eq. (21). To include multiple scattering between the solitons, one must include the second (and higher) order terms in Eq. (6). III. NUMERICAL EXAMPLES: COMPARISON WITH PE CALCULATIONS

Numerical results for sound propagating through SIWs are given in this section to illustrate mode coupling, and 614


T. C. Yang: Mode coupling by internal waves


FIG. 2. (Color online) (a) Sound speed profile. (b) Variation of the depthintegrated acoustic energy at the receiver as a function of the soliton range to the source for three sizes of soliton with amplitudes given by [dC] ’ 5, 10, and 15 m/s.

are displayed as a function of the soliton range as shown in Fig. 2(b). The RAM calculations are conducted for three different sizes of solitons characterized/designated by their maximum sound speed perturbation: [dC] ’ 5, 10, and 15 m/s or their amplitudes c ’ 14, 27, and 40 m/s. The corresponding depth-integrated acoustic energy are shown by the solid, dashed, and dotted lines, respectively, in Fig. 2(b). One notes that the depth-integrated acoustic energy oscillates as the soliton moves. Furthermore, the amplitude of the oscillation decreases as the soliton moves out in range. The decrease of the energy envelope is the gain loss factor discussed in Ref. 8. For a soliton at a fixed range r0, one calculates the mode-coupling matrix between range r1 and r2, and then the mode amplitudes at the receiver using Eq. (2). The depth-integrated intensity is determined from the mode intensities using Eq. (17). We use KRAKEN for the normal mode calculation.25 Note that the mode-coupling matrix (the range ordered exponential operator) in Eq. (6) is unitary since the exponent is skew Hermitian, but the truncated series representation is not. This is corrected by a normalization factor determined by requiring the (total) mode energy to be conserved when there is no mode attenuation. With this correction, one finds that acoustic intensity calculated using the mode-coupling matrix agrees very well (within 0.01 dB) with that obtained using the RAM code for [dC] ¼ 5 m/s (result not shown here, see comparisons below). Without the correction, the two differ by 0.22 dB. However, when the soliton amplitude is increased to c ’ 27 and 40 m/s or [dC] ¼ 10 and 15 m/s, the acoustic intensity using the mode-coupling matrix formulation does not agree with that obtained using the RAM code. The reason is that b 0.6, 1.2, and 1.8 for [dC] ¼ 5, 10, and 15 m/s using Eq. (25). For b ¼ 1.2 and 1.8, the series expansion is not expected to converge, and the mode-coupling matrix so deduced is not expected to work. Methods to solve this (large amplitude soliton) problem will be discussed in Sec. III E. [It is noted that the solitons used in Refs. 8 and 13 yield b 0.6 for g0 ¼ 15 m. Although the maximum sound speed perturbation is much larger in that model compared with this model using c ’ 14 m/s, the sound speed perturbation is confined to a much narrow depth span than this model, Eq. (25). As a result, they yield approximately the same b.] J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014

One notes from Fig. 2(b) that the amplitude of the intensity oscillation (at a given r0) increases monotonically [exponentially by Eq. (6)] with the soliton amplitude c. The oscillation frequency with respect to range remains basically unchanged. Since the coupling physics does not depend on the (particular) soliton amplitude used, one can study the effects of the soliton on sound propagation using the c ’ 14 m/s, or [dC] ¼ 5 m/s soliton. The results can be extrapolated to cases with larger amplitudes based on the system behavior as shown in Fig. 2(b). To compare the details of mode coupling, one places a (virtual) VLA of receivers before and after the soliton to estimate the amplitudes of the incoming and outgoing modes. For a soliton located at r0 ¼ 1 km, one calculates the pressure field at a range of 700 m and 1.3 km using the RAM code, and extracts the mode amplitudes using mode decomposition. The resulting modal spectrum (mode amplitude versus mode number) is shown by the dashed line in Fig. 3(a) for the incoming modes and in Fig. 3(b) for the outgoing modes. The incoming modal spectrum at 700 m estimated using the normal mode program KRAKEN is shown by the solid line in Fig. 3(a). The outgoing mode amplitudes are calculated at 1.3 km using the mode coupling matrix approach, Eq. (2) described above. The result is shown by the solid line in Fig. 3(b). We find that the incoming and outgoing modal spectra agree pretty well within the numerical tolerances between the two (KRAKEN and RAM) programs. While the normal mode and PE code agree very well in terms of depth-integrated acoustic energy, the acoustic fields calculated using the two approaches are known to have different phases (by as much as 10 to 60 in simple range independent environments);26 the larger the depth and range grid the worse it is. Consequently, the modal spectra are not expected to agree as nicely as the total acoustic energy. [The fact that the PE and normal mode calculations agree well for the depth-integrated acoustic energy should not be a surprise. In a lossless medium, they must agree since the total energy is fixed. In a lossy medium, while the energy at a fixed receiver may fluctuate greatly in the presence of moving solitons, the envelope of the total (depth-integrated) energy is expected to decrease with range at a constant rate.] T. C. Yang: Mode coupling by internal waves

615


FIG. 3. (Color online) Comparisons of mode amplitudes estimated using the RAM code (dashed lines) and the mode-coupling matrix formulation (solid lines): (a) Incident mode amplitudes at a range of 700 m. (b) Outgoing mode amplitudes after traveling through a single soliton evaluated at a range of 1300 m. (c) Outgoing mode amplitudes after traveling through a packet of three solitons evaluated at a range of 3000 m.

B. Single packet of SIWs

In real oceans, the SIWs usually appear as a train of solitons. Thus, we will study the effects of multiple solitons on mode coupling and intensity oscillation as the SIW packet moves. For this simulation, we assume three solitons as in Refs. 8 and 13, with widths D ¼ 100, 125, 150 m, amplitudes c ’ 14, 11, 9 m/s (or [dC] ’ 5, 4, 3.3 m/s), and ranges at r0, r0 þ 500, r0 þ 1100 m, where r0 denotes the range of the (center of the) first soliton to the source. Considering the case r0 ¼ 1 km, one calculates the mode amplitudes at a range of 3 km using the mode coupling matrix formulation and the RAM code. The results are shown by the solid and dashed 616


lines, respectively, in Fig. 3(c). The two modal spectra are in good agreement within the numerical uncertainties. Comparing Fig. 3(c) with Fig. 3(b), one finds that the outgoing modal spectra are similar between the case of three solitons and single soliton. This indicates that multiple solitons have a similar effect on sound propagation as a single soliton (for the present case). More quantitative analyses will be given below. Similar to what was done in Sec. III A, we calculate the depth-integrated acoustic intensity for receivers at a range of 30 km for different SIW positions. For the mode coupling approach, the mode-coupling matrix is evaluated for each soliton, and inserted in Eq. (2) to calculate the mode amplitude at the receivers. Denoting r0 as the center position of the first soliton (of the SIW packet), the depth-integrated intensity, obtained using Eq. (17), is shown by the dashed line in Fig. 4(a) as a function of the range r0 to the source (as the SIW packet moves out in range). The depth-integrated acoustic intensity obtained using the RAM code is shown by the solid line in Fig. 4(a). The two results are in good agreement with each other (except when the solitons are close to the receiver, due presumably to a limited number of range grids between the SIWs and receiver). We also calculate the depth-integrated acoustic intensity using only the single scattering term, by extending the second term in Eq. (18) to include all three solitons [see Eqs. (27) and (28) of Ref. 14]. The result is shown in Fig. 4(b) by the solid line to compare with the results using the mode-coupling matrices mentioned above [the dashed line, repeated from Fig. 4(a)]. One finds that the magnitude of the intensity oscillation is about 10% (in dB scale) smaller when multiple scatterings between the solitons are ignored. To understand the intensity oscillation (as the SIW packet moves out in range), we plot the mode intensity spectrum at the receiver as a function of the SIWs position. The modal spectrum for the lowest seven modes is shown in Fig. 4(c) as a function of range r0 to the source. Comparing Fig. 4(c) with Fig. 4(a), one finds that the depth-integrated acoustic intensity is high when the acoustic energy is concentrated in mode 1, and is low when nearly half of the acoustic energy moves into mode 2. As the SIW packet shifts in range, the acoustic energy shifts between mode 1 and mode 2 causing the total (depth-integrated) energy to oscillate. The shift in energy between the modes can also be seen in the depth dependence of the received acoustic intensity as shown in Fig. 4(d). One observes that the intensity peaks at a depth around 45 m when the energy is concentrated in mode 1 (the depth function of the first mode peaks at 45 m depth), and spread more evenly between depth from 40 to 55 m when the energy is split between the first and second modes. The above phenomenon can also be confirmed by plotting the energy of modes 1 and 2 as a function of the SIW packet range r0 as shown by the dashed lines in Figs. 5(a) and 5(b), respectively. Also shown in Fig. 5 is the energy of the first and second modes without mode attenuation (loss). With or without attenuation, mode 1 energy is high when the mode 2 energy is low, and vice versa. It shows that energy is transferred from mode 1 to mode 2 and vice versa every 1.2 km. This agrees with the theoretical expectation given by the mode cycle distance k12 ¼ 2p/(k1 k2), T. C. Yang: Mode coupling by internal waves


FIG. 4. (Color online) (a) Comparison of the depth-integrated acoustic energy traveling through a packet of SIWs using the RAM code (solid line) and modecoupling matrix formulation (dashed line) plotted as a function of the soliton range. (b) Comparison of the depth integrated acoustic energy including only single-scattering (solid line) and including multiple scatterings between the solitons via the mode-coupling matrices (dashed line), plotted as a function of the soliton range. The horizontal (dotted) line denotes the received level without SIWs. (c) Modal spectrum of the received signal as a function of the soliton range. (d) Depth variation of the acoustic energy at the receiver as a function of the soliton range. Range is measured from the center of the first soliton to the source.

where k1 ¼ 1.70 m1 and k2 ¼ 1.695 m1 according to the KRAKEN program. When there is no mode attenuation, one finds that the sum of mode 1 and mode 2 energy remains approximately the same independent of the SIWs position. Consequently, the envelopes of mode 1 and mode 2 time

series [the solid lines in Figs. 5(a) and 5(b)] remain approximately constant with respective to the SIW position. With mode attenuation, the envelope of the mode 1 energy decreases, and at the same time the envelope of the mode 2 energy increases as the SIW packet moves away from the

FIG. 5. (Color online) Variations of mode 1 (a) and mode 2 (b) energy at the receiver after traveling through a packet of SIWs plotted as a function of range from the first soliton to the source. Solid lines: No attenuation. Dashed lines: With attenuation. J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014


617


source [see the dashed lines in Figs. 5(a) and 5(b)]. This indicates that more energy is transferred from mode 1 to mode 2 in the latter (nonzero attenuation) case. The reasons for this are not at all obvious. Note that the source-receiver range remains fixed (at 30 km), and thus in the absence of SIWs, the received mode energies remain at a constant level. In the presence of SIWs, one expects that the total received mode energy envelope will be smaller when the SIWs are closer to the source, since the energy converted from mode 1 to the higher order modes suffers a higher loss due to the longer range to the receiver.4 The numerical results shown above [Figs. 2(b) and 4(a)] do not seem to follow this intuitive argument. Note that the above argument ignores the range (r0) dependence of the mode coupling matrix. What happens for the present (simulated) data is another phenomenon, namely, that more energy is being taken out from mode 1 and transferred to mode 2, when the SIWs are further away from the source. One can numerically confirm this by evaluating the mode conversion term in Eq. (15) as a function of range. The reason for the range dependence of the mode conversion rate lies perhaps in the input modal structure, namely, that the energy ratio between the incoming first and second modes is higher when the SIWs range to the source is greater, causing more mode 1 energy to be transferred to mode 2 and less mode 2 energy to be transferred back to mode 1. Since mode 1 energy is dominant, the (envelope of the) total mode energy decreases as the SIWs move out in range as shown in Fig. 4(a). Thus, whether the energy envelope will gain or lose depends not only on mode attenuation but also on mode energy conversion (rate). Figures 5(a) and 5(b) show that mode attenuations have a significant effect on the received mode amplitudes at 400 Hz. The loss of mode energy (in dB) with and without attenuation in the above figures is much larger than what was shown in Fig. 4 of Ref. 14.

C. Energy fluctuation frequencies and gain/loss factor

Taking a Fourier transform of the energy time series shown in Fig. 4(a) (as an example), one obtains a fluctuation spectrum shown in Fig. 6(a). One finds two spectrum peaks in Fig. 6(a) marked by arrows 1 and 2. Arrow 1 points to a frequency of 0.84 cycle/km, which agrees with the theoretical prediction, namely 1/k12, due to mode coupling between mode 1 and 2 [Eq. (18)]. Arrow 2 points to a frequency of 2.07 cycle/km, which agrees with 1/k13, due to mode coupling between modes 1 and 3. Note that the energy fluctuation frequencies due to mode coupling/conversion are determined by the mode wavenumbers, and are independent of the number of solitons in the packet. It is no surprise that Figs. 2(b) and 4(a) show a similar oscillation time series, since in both case, mode coupling between the first and second modes is dominant. From the energy time series, one can estimate the ~ envelope of energy fluctuation ½QðtÞ defined above in ~ 0 Þ as a funcSec. II B 1. As an example, we calculate ½Qðr tion of the soliton range to the source r0 using the energy time series shown in Fig. 2(b). The results are shown in Fig. 6(b) in dB scale. One finds the energy envelope changes by an amount of 2 6 0.5 dB from 2 to 15 km, which is consistent with a12 13 km ¼ 1.9 dB. [Note that the envelope, Fig. 6(b), has a complicated structure, making it difficult to measure its exact slope or energy change.] While the slope of the envelope change is determined primarily by mode attenuation, its magnitude is proportional to the amplitude of the soliton [Eq. (20)], as shown in Fig. 6(b). The above numerical results [Figs. 2(b) and 4(a)] confirm that the energy fluctuation frequency and slope of the energy change are insensitive to the number of solitons (one or three) in the packet (so long as they are similar). As such, we can apply the above analysis to real data (see Sec. IV). Specifically, given the mode wavenumbers, one can estimate the energy fluctuation frequency and given the mode

FIG. 6. (Color online) (a) Spectrum of the time series shown in Fig. 4(a). (b) Envelope of the energy fluctuation determined from the energy time series shown in Fig. 2(b). 618




attenuation, one can estimate the gain change of the energy envelope. D. Two packets of SIWs

For receivers at long ranges to the source, one may find two packets of SIWs in between the source and receivers at times. With a 12-hour generation cycle, the SIW packets will be separated by a distance d ¼ v 12 h, where v is the average velocity of the SIW packets projected along the propagation path. One finds d 28 km for v ¼ 0.65 m/s, and thus when the source-receiver range is greater than 28 km, one can expect to find two packets of SIWs between the source and receivers at times (such as the data analyzed in Sec. IV). The question is: What are the effects of two packets of SIWs compared with only one packet of SIWs studied in Sec. III B? This will be studied in this section using the mode-coupling matrix formulation. Given a source-receiver range of 41 km (see Sec. IV), and assuming that the SIWs are moving in the direction toward the source, one finds that there is only one packets of SIWs between the source and receiver, during the time period when the SIW packet is within a range of 13 to 28 km from the source. Before and after this period, one will find two packets of SIWs between the source and receivers. Again, denoting r0 to be the position of the first soliton of the first SIW packet, we calculate the depth integrated acoustic energy as a function of the position of the SIW packet r0 using the RAM PE code and the mode-coupling matrix formulation. The result using the RAM code is shown in Fig. 7(a) by the solid line. The result using the mode coupling matrix formulation is shown by the dashed line for r0 < 12 km when there are two packets of SIWs present, and by the dotted line for r0 between 13 and 27 km when there is only one packet of SIW present between the source and receivers. [For r0 between 12 and 13 km, the receiver array is covered by the second packet of SIWs, the calculation is

more tedious and is not shown here.] One observes that the depth-integrated energy agrees rather well using the mode coupling matrix approach and the RAM code when there is only one packet of SIWs present (as in the previous case, Sec. III B). When there are two packets of SIWs in between the source and receiver, the results using the mode-coupling matrix agree well with the RAM results when the energy is high, and deviate from the RAM results by 0.3 dB when the energy is low. (This could be a numerical accuracy problem which remains to be investigated.) We next investigate the difference of intensity fluctuations between the case of two packets of SIWs and one packet of SIWs; for the latter case the second packet of SIWs is not generated. The results using the mode-coupling matrix formulation is shown in Fig. 7(b) using the same convention as in Fig. 7(a), namely, the dashed line, denotes the case of two packets of SIWs and the dotted line denotes the case of one packet of SIWs. One finds that the amplitude of the intensity oscillation is reduced slightly when the second packet of SIWs is present. The fluctuation frequency remains practically speaking unchanged. E. Computational advantages

In order to calculate the depth-integrated acoustic energy due to moving SIWs and the associated modal spectrum using a PE code, such as RAM, the sound must be propagated from the source to the receiver for each given SIW location. This can be a computation intensive process, depending on the accuracy required and the granularity used to represent the sound speed perturbation relative to the acoustic wavelength. The normal mode approach, given a closed form solution for the mode-coupling matrix, has a distinct speed advantage, because mode coupling occurs only in the range spanned by the SIWs and outside of this range, one can apply adiabatic mode propagation, which is simple and straightforward. For the numerical simulations shown above, to calculate the energy time series at a range of 30 km using

FIG. 7. (Color online) (a) Comparison of the depth-integrated acoustic energy for a receiver array at a range of 41 km using the RAM code (solid line) and mode-coupling matrix formulation (dashed line for r0 < 12 km when there are two packets of SIWs present and a dotted line for 13 km < r0 < 27 km when there is only one packet of SIW present between the source and receivers; r0 is the range from the first soliton to the source). (b) Comparison of the depth-integrated acoustic energy traveling through two packets of SIWs (dashed line) versus that traveling through only one packet of SIWs (dotted line). J. Acoust. Soc. Am., Vol. 135, No. 2, February 2014


619


the RAM code with a range increment of 2 m and depth increment of 0.2 m, it takes 3.7 h for 271 positions of the SIW packet on a personal computer. The same time series, calculated using the mode-coupling matrix formulation, takes less than 40 s using an approximate closed form solution for the hyperbolic secant integral Eq. (A5). At a range of 41 km, when two packets of SIWs are present, it takes nearly 5 h versus a slightly increased computation time (

À LA mode atrioventricular mechanical coupling.

Edge Mode Coupling within a Plasmonic Nanoparticle.

Strong polarization mode coupling in microresonators.

Mode Coupling and Nonlinear Resonances of MEMS Arch Resonators for Bandpass Filters.

Dynamic mode coupling in terahertz metamaterials.

Mode coupling at connectors in mode-division multiplexed transmission over few-mode fiber.

Phononic crystal surface mode coupling and its use in acoustic Doppler velocimetry.

Coupling light into graphene plasmons through surface acoustic waves.

Mode conversion and coupling in a slanted grating.

Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers.

Measurement of distributed mode coupling in a few-mode fiber using a reconfigurable Brillouin OTDR.

Measurement of mode coupling distribution along a few-mode fiber using a synchronous multi-channel OTDR.

Mode-coupling analysis of residual stresses in colloidal glasses.

Equalizer tap length requirement for mode group delay-compensated fiber link with weakly random mode coupling.

Magnesium ion-induced changes in the binding mode of adenylates to chloroplast coupling factor 1.

Mode-coupling instability in a fluid two-dimensional complex plasma.

Wave-vortex mode coupling in neutrally stable baroclinic flows.

Shear-induced breaking of cages in colloidal glasses: Scattering experiments and mode coupling theory.

Orbital angular momentum mode division filtering for photon-phonon coupling.

Mode-coupling polarization rotator based on plasmonic waveguide.

Efficient single-mode photon-coupling device utilizing a nanofiber tip.

Optically induced mode coupling and interference in a terahertz parallel plate waveguide.

Nonlinear rheology of glass-forming colloidal dispersions: transient stress-strain relations from anisotropic mode coupling theory and thermosensitive microgels.

Mode control and mode conversion in nonlinear aluminum nitride waveguides.