Kelvin Waves in the Nonlinear Shallow Water Equations on the Sphere: Nonlinear Traveling Waves and the Corner Wave Bifurcation

Transcription

1 Kelvin Waves in the Nonlinear Shallow Water Equations on the Sphere: Nonlinear Traveling Waves and the Corner Wave Bifurcation John P. Boyd Cheng Zhou University of Michigan June, 29 1

2 Two Topics 1. New asymptotic approximation for LINEAR Kelvin waves on the sphere 2. Corner wave bifurcation for NONLINEAR Kelvin waves on the sphere Restrictions 1. Nonlinear shallow water equations 2. No mean currents Two Parameters 1. Integer zonal wavenumber s s> 2. Lamb s Parameter ɛ 4Ω2 a 2 gh Ω = 2π/84, 6 s, g = 9.8m/s a = earth s radius H = equivalent depth 2

3 Parametric range is SEMI-INFINITE in BOTH s & ɛ Table 1: Lamb s Parameter ɛ Description Source.12 External mode: Venus Lindzen (197) 6.5 External mode: Mars Zurek (1976) 12. External mode: Earth (7.5 km equivalent depth) Lindzen (197) 2.6 Jupiter: simulate Galileo data Williams (1996) 21.5 Jupiter Williams (1996) 43. Jupiter Williams (1996) 26 Jupiter Williams (1996) 26 Jupiter Williams and Wilson (1988) 87, ocean: first baroclinic mode (1 m equiv. depth) Moore & Philander (1977) > 1, ocean: higher baroclinic modes Moore & Philander (1977) LINEAR EIGENFUNCTIONS are HOUGH functions (S. S. Hough (1896, 1898) We regard Mr. Hough s work as the most important contribution to the dynamical theory of the the tides since the time of Laplace. Sir George Darwin, Encyclopedia Brit., 11th ed. Figure 1: Sydney Samuel Hough, FRS, ( ) 3

4 u = cos s (latitude) Kelvin Parameter Space & LINEAR Asymptotic Regimes Small ɛ: u = P s s (cos(θ)) exp(isλ iσt) (Longuet-Higgins, 1968)) Large ɛ: u exp( ɛθ 2 ) exp(isλ iσt) s??????? Equatorial beta-plane ε _????? Global Equatorial 4

5 New asymptotic approximation is (µ = sin(latitude)) { } φ (1 µ 2 ) s/2 exp( (1/2) ɛ + s 2 s µ 2 ) Approximation (thick solid curve) and exact (thin solid curve) are GRAPHICALLY INDIS- TINGUISHABLE for s =5,ɛ =5 ε =5 s=5 φ Kelvin exp( 1/2*sqrt(ε)*µ 2 ) exp( 1/2*sqrt(ε+s 2 )*µ 2 ) (1 µ 2 ) s/2 exp( 1/2[sqrt( ε +s 2 ) s] µ 2 ) colatitude 5

6 u = cos s (latitude) Uniform Validity New approximation is uniformly valid for s 2 + ɛ>>1 (shaded in figure) Though not strictly valid when both s and ɛ are O(1), it is not a bad approximations s Equatorial beta-plane Global Equatorial ε _ 6

7 Barotropic (ɛ = )Kelvin Waves Equatorial trapping is not just due to ɛ High zonal wavenumber Kelvin are equatorial modes even for ɛ = Barotropic Kelvin, ε=, s=2 7

8 Barotropic (ɛ = )Kelvin Waves Half-width is inside the tropics for s 5 1 Barotropic Kelvin waves: s=1, 2,..., 1.8 u or φ.6.4 s=1 s=1 s=1 s= colatitude Figure 2: 8

9 Weakly Nonlinear Wave Theory, A Science Fiction Story Perturbative theory yields approximations that have the structure of LINEAR Kelvin waves in LATITUDE & DEPTH multiplied by a function A(x, t) that solves a nonlinear PDE u = h = A(x, t) exp(.5y 2 )φ(z); v On equatorial beta-plane without currents, Kelvin is dispersionless: A t + A x AA x = Breaking and frontogenesis; no steady propagation velocity scale is about 2 m/s length scale is about 3 km [1D Advection Eq] (1) 9

10 Kelvin Fronts & Breaking Boyd (J. Phys. Oceangr., 198) Ripa (J. Phys. Oceangr., 198) Front curvature due to Kelvin-gravity wave resonance Boyd (Dyn. Atmos. Oceans, 1998) Fedorov & Melville (J. Phys. Oceangr., 21) Microbreaking Boyd (Phys. Lett. A, 25) 1 A t + A A x = (1/1) A xx, A(x,)= - sin(x).5 A x 1

11 Shear Currents: Dispersion Equatorial ocean has strong currents: South Equatorial Current (westward), North Equatorial Countercurrent (eastward), etc. Currents induce dispersion in Kelvin wave (Boyd, Dyn. Atmos. Oceans, 1984) A t AA x + da xxx = [KdV Eq] KdV Predictions: 1. Solitons & cnoidal waves of arbitrarily large amplitude 2. No breaking or frontogenesis 11

12 Difficulties with KdV Picture for Kelvin Wave KdV dispersion relation predicts a PARABOLA for the group velocity: true for k< 1 only As k, c g c phase constant Kelvin is WEAKLY DISPERSIVE for large zonal wavenumber (Boyd, JPO, 25) KdV Theory FAILS at LARGE AMPLITUDE because shear-induced dispersion is TOO WEAK to balance NONLINEAR STEEPENING 1.2 Kelvin group velocity in shear flow extrapolated KdV dispersion asymptotically flat zonal wavenumber k 12

13 Solitons on the Union Canal Solitary waves were discovered observationally by John Scott Russell in the 183 s. Below is a modern recreation on the same canal. Figure 3: 13

14 What Really Happens: CCB Scenario CCB Scenario: Cnoidal/Corner/Breaking Breaking Amplitude Corner Cnoidal 14

15 Nonlinear Kelvin Waves on the Sphere (Boyd and Zhou, J. Fluid Mech., 29) NO SHEAR: ALL DISPERSION from SPHERICAL GEOMETRY Small amplitude/small ɛ double perturbation series Spectral-Galerkin model Newton/continuation Kepler change-of-coordinate to resolve the discontinuous slope of the corner wave Zoom plots to identify the corner wave as tallest, non-wiggly solution Cnoidal/Corner/Breaking Scenario: All waves above an ɛ-dependent maximum amplitude break, as confirmed by initial-value time-dependent computations. 15

16 Kelvin Wave on the Sphere-2 With no mean currents, corner wave occurs at SMALLER and SMALLER amplitude as ɛ increases On the equatorial beta-plane (ɛ = ), ALL KELVIN WAVES BREAK φ in the corner wave limit 1 s=1 s=2 φ Figure 4: Maximum of φ(x, y) for the corner wave versus ɛ. The maximum always occurs at the crest of the wave (X =)and right at the equator (y = ). φ is the height at the peak, x =and y = s=east-west wavenumber ε 16

17 Kelvin Wave on the Sphere-3 As ɛ decreases, the corner wave profile becomes narrower and narrower in longitude more soliton-like. φ(x,y=) normalized by φ s=2 φ(x,y=) Longitude ε=.1 ε=1 ε=5 ε=3 Figure 5: An equatorial cross-section of the height/pressure field φ, φ(x, y = ), for several ɛ, normalized by φ(, ). 17

18 Kelvin Wave on the Sphere-4 CONE, CREASE or ONE-SIDED POINT SINGULARITY? Numerical evidence suggests the latter dφ/dy does NOT have a discontinuity: φ x 1 3 s=2 ε=3 φ(x,y=) φ(x=,y) Latitude or longitude Figure 6: Longitudinal (solid) and latitudinal (dashed) cross-sections through φ(x, y) for s = 2 and ɛ = 3 18

19 Kelvin Wave on the Sphere-5 Two views of the same corner wave below: s=1 ε=1 φ =.195 φ Latitude Longitude.15.1 φ Latitude Longitude 19

20 Kelvin Wave on the Sphere-6 dφ/dx has a JUMP DISCONTINUITY at the EQUATOR ONLLY (Insofar as one can judge singularities from numerical computations.) s=2 ε=.1 φ x 5 lat= lat=π/64 lat=π/32 lat=π/16 lat=π/8 lat=π/4 φ x 5 lat= lat=π/64 lat=π/32 lat=π/16 lat=π/8 lat=π/ Longitude Longitude Figure 7: dφ/dx for s =2and ɛ =1/1. The right panel is a zoom plot of the left panel, showing only 1/15 the range in longitude. 2

21 Summary After 35 years of intermittent studies of the Kelvin wave and one thesis chapter and 14 articles (out of 195) from 1976 to present, still are UNRESOLVED QUESTIONS Why does the travelling wave branch, for Kelvin and so many other wave species, terminate in a corner wave? Complete classification of generic & non-generic features in corner wave bifuraction. How does microbreaking promote mixing in the ocean? How does nonlinearity reshape the Kelvin mode s role in El Nińo? Does the breaking Kelvin wave overturn or stay single-valued? Kelvin corner waves in SHEAR flow Critical latitude & resonance issues 21

22 Before I came here I was confused about this subject. Having listened to your lecture I am still confused, but on a higher level. Enrico Fermi 22

23 Equatorial Beta-Plane: ɛ limit of tidal equations Sines & cosines of latitude y and 1. Hough functions become Hermite functions (v, orsums of two Hermite functions (u, φ). Tropical is well-approximated, even NONLIN- EAR, by one-and-a-half-layer model Linear dynamics is beta-plane form Laplace s Tidal Equations with actual depth of upper layer (average: 1 m) replaced by equivalent depth (.4 m). Warm, Light, Moving Layer Cold, Heavy, Infinitely Deep Motionless Layer Figure 8: Schematic of one-and-a-half layer model. equations. The active layer is described by the shallow water 23

24 Kelvin & Rossby waves are disturbances on the main thermocline, which is the interface between the warm layer and the cold layer. Approximation is not too bad for ocean. In one-and-a-half-layer model, wave is only a function of latitude y and longitude x and time t. 24

25 Soliton-Machine at Snibston Discovery Park (England) Recreating solitons in a channel is literally child s play. Figure 9: Soliton-making machine at Snibston Discovery Park, England. 25

26 Kelvin Solitary Waves Initial-value numerical solutions of shallow water equations confirm solitons in a shear flow. u: Kelvin mode du/dx: Kelvin mode t= & t= x 1.5 Mean wind & height Kelvin soliton at t=1 6 y 4 2 U Φ -2 2 x Figure 1: KELVIN SOLITON: left panels: Initial and final amplitude of the Kelvin mode. Upper right: mean flow & height. Lower right: contours of pressure/height and vector arrows. u(x, y, ) = φ(x, y, ) =.12sech 2 (.7x) constant. 26

27 Planetary Waves and the Semiannual Wind Oscillation in the Tropical Upper Stratosphere, Ph. D. Thesis, Harvard (1976) 5. The Effects of Latitudinal Shear on Equatorial Waves, Part I: Theory and Methods, J. Atmos. Sci., 35, (1978). 6. The Effects of Latitudinal Shear on Equatorial Waves, Part II: Applications to the Atmosphere, J. Atmos. Sci., 35, (1978). 7. The Nonlinear Equatorial Kelvin Wave, J. Phys. Oceangr., 1, 1-11 (198). 18. Low Wavenumber Instability on the Equatorial Beta-Plane, with Z. D. Christidis, Geophys. Res. Lett., 9, (1982). 22. Instability on the Equatorial Beta-Plane, with Z. D. Christidis, in Hydrodynamics of the Equatorial Ocean, ed. by J. Nihoul, Elsevier, Amsterdam, (1983). 25. Equatorial Solitary Waves, Part 4: Kelvin Solitons in a Shear Flow, Dyn. Atmos. Oceans, 8, (1984). 6. Nonlinear equatorial Waves. In Nonlinear Topics of Ocean Physics, Proceedings of the Fermi Summer School, Course 19, ed. by A. R. Osborne and L. Bergamasco. North-Holland, Amsterdam, (1991). 98. High Order Models for the Nonlinear Shallow Water Wave Equations on the Equatorial Beta-plane with Application to Kelvin Wave Frontogenesis, Dyn. Atmos. Oceans., 28, no. 2, (1998) 16. A Sturm-Liouville Eigenproblem of the Fourth Kind: A Critical Latitude with Equatorial Trapping, with A. Natarov, Stud. Appl. Math., 11, (1998). 11. Propagation of Nonlinear Kelvin Wave Packet in the Equatorial Ocean, with G.-Y. Chen, Geophys. Astrophys. Fluid Dynamics., 96, no. 5, (22) Beyond-all-orders Instability in the Equatorial Kelvin Wave, with Andrei Natarov, Dyn. Atmos. Oceans., 33, no. 3, (21) Shafer (Hermite-Pade) Approximants for Functions with Exponentially Small Imaginary Part with Application to Equatorial Waves with Critical Latitude with Andrei Natarov, Appl. Math. Comput., 125, (22) Fourier Pseudospectral Method with Kepler Mapping for Travelling Waves with Discontinuous Slope: Application to Corner Waves of the Ostrovsky-Hunter Equation and Equatorial Kelvin Waves in the Four-Mode Approximation, Appl. Math. Comput., 177, no. 1, (26) The Short-Wave Limit of Linear Equatorial Kelvin Waves in a Shear Flow, J. Phys. Oceangr., 35, no. 6 (June), (25). 27