Scattering of internal gravity waves

Transcription

1 Scattering of internal gravity waves S.B. Dalziel*, B.R. Sutherland * DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, ENGLAND Department of Mathematical & Statistical Sciences, University of Alberta EGS AGU EUG Joint Assembly Nice April 2003

2 Motivation Internal waves reflecting from boundaries with geometric features comparable with wavelength of incident waves. DAMTP, University of Cambridge EGS, Nice, April 2003

3 Apparatus ω ρ Tank: mm (fill to ~400mm) Salt stratification: N ~ 1.96 rad/s Cylinder: 49 mm Bottom: flat and 90 saw-tooth (10mm and 20mm amplitudes)

4 Synthetic schlieren ξ B W

5 2 2 2 d ξ dξ dζ 1 n = 1+ 2 dy + dy dy n x d ζ dξ dζ 1 n = 1+ 2 dy + dy dy n z 1 1 n' ξ = 2WW ( + 2B) n x 1 1 n' ζ = WW ( + 2B 2 ) n z 0 0 In salt water β ρ 0 dn = n dρ

6 Qualitative mode Look for differences between current image and reference image: diff = γ P ij (t) Q ij,

7 Quantitative mode Interpolation Dot tracking Pattern matching

8 interpolation Lines or 2D features Pixel gives integral of light over area Mid-point rule quadratic interpolation P = Q 0 + ½(Q 1 Q -1 ) ζ + ½(Q 1 2Q 0 + Q -1 ) 2 ζ = 0 Solve for ζ Binomial expansion with consistent order = ( P Q0 )( P Q 1) ( Q Q )( Q Q ) Shift image if out of ζ exceeds one pixel ζ ( P Q )( P Q ) 0 1 ( Q Q )( Q Q ) s

9 Interpolation x gradient z gradient

10 pattern matching Similar to PIV δ Difference between window contents F(P ij, Q ij,δ)

11 Many choices i+ w j+ w ( P, Q, δ ) = P( k + δ, l + δ ) Q( k l) ij ij F, k = i w l = j w PIV normally uses cross-correlation function. Find δ = (integer) that minimises F(..). Fit surface near δ = to improve estimate. Noisey, especially if << pixel. i j A problem in nonlinear optimisation. Aim: to calculate the distortion to the image pair. F( δ ; x) = W ( x x, δ) f ( P( x ), Q( x ), δ) dx f A { } ( x) = δ : min F( δ; x) δ ( P, Q, δ) = diff ( P( x + αδ), Q( x ( 1 α ) δ) ) The measure of the difference, diff, could take many forms.

12 BUT, we do not know P(x + αδ), etc., except at integer spacing. We also do not know δ except at discrete locations. Solution: Interpolate δ Distort P, Q or both Use interpolation when calculating diff(..) for non-integer δ x gradient z gradient

13 Waves from oscillating cylinder

14 Cross-beam structure at r/r = Perturbation in density gradient Key Circular Square Theory Cross-wave position

15 Flat bottom Movie

16 Large amplitude saw-tooth Movie Linear reflection possibilities

17 Nonlinear reflection Also effect of boundary layers and shedding from crests As superposition of waves

18 RMS amplitude Flat bottom 20mm saw-tooth

19 Cross-beam structure x Gradient x Gradient Cross beam Flat bottom Cross beam 20mm saw-tooth Difference movie

20 Spectral development Expt 22: Flat bottom Expt 21: Saw tooth (20mm) Power Incident: 150 mm 140 mm 130 mm 120 mm 110 mm 100 mm 90 mm 80 mm 70 mm 60 mm Reflected: 60 mm 70 mm 80 mm 90 mm 100 mm 110 mm 120 mm Power Incident: 150 mm 140 mm 130 mm 120 mm 110 mm 100 mm 90 mm 80 mm 70 mm 60 mm 50 mm Reflected: 50 mm 60 mm 70 mm 70 mm 90 mm 100 mm 110 mm 120 mm Wave number (k) Wave number (k)

21 Amplitude dependence Spectra Expt 22: Flat bottom Expt 18: Saw tooth (10mm) Key: Incident x gradient Reflected x gradient Incident y gradient Reflected y gradient Key: Incident x gradient Reflected x gradient Incident y gradient Reflected y gradient Power 0.50 Power Wave number (k) Flat Wave number (k) 10mm saw-tooth

22 Expt 22: Flat bottom Expt 21: Saw tooth (20mm) Key: Incident x gradient Reflected x gradient Incident y gradient Reflected y gradient Key: Incident x gradient Reflected x gradient Incident y gradient Reflected y gradient Power 0.50 Power Wave number (k) Flat Wave number (k) 20mm saw-tooth

23 Conclusions Preliminary experiments Increased energy at high wavenumbers Reflection back along incident wave beam Enhanced dissipation & diffusion Back reflection steepening Other nonlinear effects frequency doubling, mixing Synthetic schlieren Benefit from higher resolution digital video