LONG RANGE PROPAGATION OF FINITE AMPLITUDE ACOUSTIC WAVES IN AN OCEAN WAVEGUIDE Kaelig Castor, Peter Gerstoft, Philippe Roux, W.A. Kuperman,, Marine Physical Laboratory, Scripps Institution of Oceanography B. Edward McDonald,, Naval Research Laboratory, Acoustics Division
0.5 dB/λ, 1800 kg/m3, 1550 m/s
Linear
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Non-linear
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1500 1495 1490
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r=100km, source near bottom interface (zs=4.5km)
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Non-linear 4
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Non-linear Deep water
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Nonlinear coupling sea bottom
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Parametric low frequency
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Frequency-mode coupling
Time series, deep water ocean waveguide Linear, range=100km
WK31 Dec 99
McDonald & Kuperman, J. Acoust. Soc. Am. 81, 1406-1417, 1987.
∫
+∞
∇ ⊥2 p dx +
δ 2
∂ 2x p + 0.02 α ( dB / λ )
c0 2
∫
+∞ 0
p ( x ') dx' x − x'
Deep water time series (10 Hz) 50 km Nonlinear Linear
Bottom attenuation
Diffraction : Crank-Nicholson
Linear, range=3000km
Integration region (moving spatial frame) : x
min
< x < x
- more uniform modal distribution - self-refraction
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Time (s)
The nonlinear NPE code and a time domain finite difference code (Cabrillo) and are used to generate movies. NPE shallow water simulation NPE deep water simulation
MOVIES! Depth (km)
Time series, shallow to Deep water
x 10
2000 km
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reduced time (s)
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Shallow to deep water (10 Hz) Linear, range=100km
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200 m 4 km
Nonlinear, range=100km
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In Depth FDTD modeling: Using Cabrillo Linear
•Staggered Fourier pseudo spectral method •Fourier spectral methods requires less grid points than classical FD (l/2 vs l/10) •Can model both acoustic, elastic and poroelastic (Biot) media. •FD method can better model variations in sound speed (including bathymetry) than classical ocean acoustic propagation codes. Any grid point can have different properties! •The wrap around is destroyed by tapering of the grid (last 30). Shallow to deep water Shallow water to T-phase station.
50 km Nonlinear
500 km Linear, range=3000km
Nonlinear, range=3000km Depth (km)
Nonlinear steepening Multivalued waveform no physical sense Shock dissipation Shock wave formation
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- harmonic generation (2f, 3f, 4f,...)
- shock dissipation
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max
- parametric interaction (f1 ± f2)
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Broader frequency spectrum Structure changes during propagation
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Nonlinear effects Additional frequencies
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1000 km
Important for an ocean waveguide: - Refraction included. - Porous medium attenuation in the bottom
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moving frame time incremental step ∆ t = ∆ x/c0
c0
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Nonlinear, range=3000km
Thermoviscous dissipation : Finite difference scheme Refraction + Nonlinear steepening : Second order upwind flux corrected transport scheme [B. E. McDonald, J. Comp. Phys. 56, 448-460, (1984)] NPE moving frame
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Frequency (Hz)
For both shallow and deep water the NPE is propagating the field the first 20 km where nonlinearities are strong. An adiabatic normal mode code is used for propagating the field to longer ranges. 4 km
Nonlinear Progressive wave Equation (NPE) x
Increased sea-bottom coupling (nonlinearities can be candidate for T-wave formation)
But, we also work with real data!
Non-linear, range=100km
Long range propagation
⎡ ⎤ c β ∂ t p = −∂ x ⎢c1 p + p2 ⎥ − 0 2 ρ0c0 ⎦ 2 ⎣
Nonlinear
Linear
Deep water, r=100km
Nonlinear, r=100km
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Non-linear Shallow water
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1480 1475
Shallow water fondamental frequency first harmonic
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Depth-averaged spectrum
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Depth-averaged spectrum
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Depth-averaged spectrum
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Depth (m )
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Linear, r=100km
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Depth-averaged spectrum
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Linear
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Depth (m)
Nonlinearities can give effects similar to long-range propagation in random media: → modal distribution + arrival time structure → increased sea-bottom coupling (NL candidate for T-phase generation)
dB
Linear
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Nonlinear
Linear
Shallow water, r=100km
2nd
1st mode
1000 km
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1st mode
2000 km -1 reduced time (s) 1 -1 1 Only few modes excited: little time spread, energy close to sound speed axis. The Nonlinear propagation excite higher order modes
Depth (km)
Frequency-Mode Coupling
- frequency distribution → parametric and harmonic generation - modal distribution → larger number of modes excited → tendency toward modal equipartition → changes in the arrival time structure
Parametric low frequency might not appear in a shallow water waveguide
Depth-averaged spectrum
Range=100km
r=100km, source on sofar axis (zs=800m)
Normalized depth-averaged spectrum Shallow water, r=0km
Depth (m )
What is the main difference between shallow and deep water ? In shallow water, => lower number of modes (bottom interaction), faster time-separation of modes. => lower geometrical spreading, higher amplitudes, stronger nonlinear effects How can we identify at long ranges a nonlinear acoustic propagation path ? Redistribution of the energy during the propagation
Source spectrum
Source 50Hz narrowband, depth 100m Source over density Rm=3.5 10-3.
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Conclusion
Deep water ocean waveguide: influence of parametric mode conversion on sea-bottom coupling
Source time series 1500 m/s
group speed (m /s)
Abstract The Nonlinear Progressive Wave Equation (NPE) [McDonald and Kuperman, 1987] computer code was coupled with a linear normal mode code in order to study propagation from a high intensity source in either shallow or deep water. Simulations using the coupled NPE/linear code are used to study both harmonic (high frequency) and parametric (low frequency) generation and propagation in shallow or deep water with long-range propagation paths. Included in the modeling are both shock dissipation and linear attenuation in the bottom.
Shallow water Pekeris waveguide 200 m
Depth-averaged spectrum
Background: The nonlinear progressive wave equation (NPE) [McDonald and Kuperman, JASA, 1987] was developed to obtain accurate and affordable simulations of shock propagation in the deep ocean out to convergence zone ranges.
Sponsored by Defense Threat Reduction Agency Contract No. DTRA01-00-C-0084
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•Cair=0 m/s, Cwater=1500 m/s, Cbot=2500 m/s •Source at 70 m depth •Field tapered outside red box
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