Resonant excitation of trapped coastal waves by free inertia-gravity waves

Transcription

1 Resonant excitation of trapped coastal waves by free inertia-gravity waves V. Zeitlin 1 Institut Universitaire de France 2 Laboratory of Dynamical Meteorology, University P. and M. Curie, Paris, France "PDEs and GFD", INI Cambridge, December 2013

2 Plan 1 2 Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations 3 4

3 Preliminary remarks Topographic/coastal waveguides in rotating fluids Boundaries (coasts) and localized topography (shelves, ridges) are special in rotating fluids, serving as waveguides for specific wave motions. Waveguides: semi-transparent, as free inertia-gravity and Rossby waves may travel through and be refracted/reflected. Nonlinear wave interactions: Considered usually in the infinite/doubly periodic domain (rarely on the sphere) for free waves. Wave turbulence, used e.g. for explanation of Garret-Munk spectrum. Nonlinear interactions of coastal/topographic waves rarely studied.

4 Ideology Trapped waves in the coastal waveguide interact with free inertia-gravity waves Cubic nonlinearity interaction triadic Interaction may be resonant resonant triads If so, resonant excitation of trapped waves by free waves, with two possible mechanisms: 1 F T + T : Free wave Trapped wave + Trapped wave (or (Trapped wave) 2 ) 2 F + F T : Free wave + Free wave Trapped wave Resonant growth of the trapped wave + nonlinearity/dispersion/dissipation coherent structures Twist: infinitely long trapped wave coastal current.

5 Workflow Consider a (straight) coast in a GFD model (below: one- or two-layer rotating shallow water) Obtain the dispersion relations for free and trapped waves and check for resonances among them Apply asymptotic expansions in nonlinearity parameter to the flow consisting of resonating waves, "eliminate" resonances by slow-time modulation, get nonlinear amplitude equations (Landau-type for dispersive waves), look for slow-time growth and nonlinear saturation. Introduce spatial modulation and obtain Ginzburg-Landau (or nonlinear Schrödinger) type (dispersive waves), or Burgers-type (non-dispersive waves) evolution equations for wave amplitudes. Look for pattern formation in modulated amplitudes.

6 Setup: one- or two-layer rotating shallow water model z g f/2 H shelf x

7 Steep vs gentle bathymetry Gentle slope Characteristic horizontal scale of bathymetry O (R d ) Trapped waves: dispersive shelf and edge waves. Details of bathymetry unimportant: spectrum of coastal waves universal, Huthnance, Steep slope Characteristic horizontal scale of bathymetry << R d Trapped waves (leading order): (almost) non-dispersive Kelvin waves

8 RSW equations with coast Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations Equations of motion: u t + uu x + vu y fv + gη x = 0, v t + uv x + vv y + fu + gη y = 0, η t + ((η + h)u) x + ((η + h)v) y = 0. (1) Water depth h(x) - 1d topography. Coast: h x=0 = 0, h x H = const. (2) Boundary conditions: η x 0 regular, ηu x 0 0. (3)

9 Linearized system Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations Linearized non-dimensional equations: Trapped waves: u t fv + η x = 0, v t + fu + η y = 0, η t + (hu) x + (hv) y = 0, (4) (η, u, v) x 0. (5) Free (incident + reflected) wave: ( (η, u, v) x e i(ly σt) Re A + e ikx + A e ikx), (6)

10 Linear waves Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations Wave equation: Solutions of linearized system: (u, v, η) = (iu, V, Z ) (x)e i(ly σt), (7) Hence U = flz σz σlz fz σ 2, V = f 2 σ 2 f 2, d dx. (8) ( hz ) + ( σ 2 f 2 l 2 h fl σ h ) Z = 0, (9) an eigenproblem for eigenfunctions Z n and eigenfrequencies σ n. Solution below: Ball s model h = 1 e x (Ball, 1968); similar for any monotonous profile (Huthnance, 1975).

11 Dispersion diagram Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations 6 f = 1 5 n = 1 4 n = 1 S I G M A 3 n = 0 n = n =

12 Weakly nonlinear system Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations Nonlinear equations: u t fv + η x = ɛ(uu x + vu y ), v t + fu + η y = ɛ(uv x + vv y ), η t + (hu) x + (hv) y = ɛ ((ηu) x + (ηv) y ), (10) ɛ = U << 1 - Froude number. gh

13 Resonant triads Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations If lowest-order solution (u 0, v 0, η 0 ) is in the form: and W 0 + W 1 + W 2, where W i = A i (iu i, V i, Z i )e iθ i, θ i = l i y σ i t, i = 0, 1, 2 - a pair of trapped waves, i = 1, 2, and an incident/reflected wave i = 0, synchronism conditions hold: l 0 = l 1 + l 2, σ 0 = σ 1 + σ 2. (11) the amplitudes A 1,2 grow exponentially in slow time T 2 = ɛ 1 t.

14 Graphical solution for resonances Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations f = 2 6 P o i n c a r e w a v e s σ = σ 1 + σ s i g m a 3 σ 1 n = 0 2 n = 0 σ w a v e n u m b e r l

15 Modulation equations Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations Coupled generalized Landau equations: T2 A 1 + im 11 A 1 2 A 1 + im 12 A 2 2 A 1 + im 02 A 0 A 2 = 0 T2 A 2 + im 21 A 1 2 A 2 + im 22 A 2 2 A 2 + im 01 A 0 A 1 = 0 M ij - convolutions of zeroth and first orders for waves i and j, M 02 M 01 > 0 (condition for resonant growth). A 0 - free wave. General property: Saturation existence of attracting limit cycle, or attracting fixed point in A 1, A 2 - space (Details depend on the parameters of trapped waves).

16 Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations Amplitude equations with spatial modulation Slow space-time variables Modulation equations A i = A i (Y, T 1, T 2 ), i = 1, 2 T1 A i + c gi Y A i = 0, c gi group velocities of waves T2 A 1 + c g1 Y A 1 i 2 c g 1 YY A 1 + im 11 A 1 2 A 1 + im 12 A 2 2 A 1 + im 02 A 0 A 2 = 0 T2 A 2 + c g2 Y A 2 i 2 c g 2 YY A 2 + im 21 A 1 2 A 2 + im 22 A 2 2 A 2 + im 01 A 0 A 1 = 0

17 Comments Linear wave spectrum of the RSW model with a gentle coastal slope Resonant excitation of trapped waves by free waves Modulation equations Details: Reznik & Zeitlin, JFM, 2011 Previous work Resonant excitation of edge waves (Minzoni and Whitham, 1977; Akylas,1983), and of shelf waves (Miles, 1990) - known, although only particular cases analysed.

18 Non-dimensional RSW equations with idealized coast u t v + h x = ɛ(uu x + vu y ) v t + u + h y = ɛ(uv x + vv y ) h t + u x + v y = ɛ ((hu) x + (hv) y ). (12) Boundary condition: x 0, u x=0 = 0.

19 Wave spectrum of the 1-layer model Linear waves (u, v, h) e i(σt kx ly) : Free inertia-gravity waves (IGW) with dispersion σ = 1 + k 2 + l 2 Trapped non-dispersive Kelvin waves (KW) with σ = l, k = i

20 Dispersion diagram Σ l k V. Zeitlin 1 Free Wave - Trapped Wave resonances

21 IGW - KW interaction k1,l1 k2, l2 l

22 Conditions of IGW - KW resonance A pair of IGW with frequencies σ 1,2 and along-coast wavenumbers l 1,2 is in resonance with a KW with wavenumber l if ("difference" resonance) For l < 0: l = σ 1 σ 2 = σ K = l, l 1 l 2 = l, l 0. (13) 1 + k 21 + l k l2 2, l 2 = l 1 + l, (14) and 1 + k 21 + l21 l = 1 + k (l 1 + l ) 2. (15)

23 Algorithm for finding resonances: 1 Take any l and l 1, then l 2 = l 1 + l, ) 2 Take arbitrary k 1 satisfying k1 ( l + l2 2 + l 2, 3 Define k 2 from ) k2 2 = k 1 ( l + k1 2 + l2 1 + l 1. Therefore, a KW with wavenumber l may be resonantly excited by a continuum of incident IGW with wavenumbers l 1 and ( ) k 1 > 2 l 1 + (l1 + l ) 2 + l 1 + l interacting with another incident wave with k 2, l 2 : ) k2 2 = k 1 ( l + k1 2 + l2 1 + l 1, l 2 = l 1 + l. (16)

24 Evolution equation for KW Condition of absence of resonances, R h,v - r.h.s. of h- and v - equations: 0 Evolution equation for the Kelvin wave where η = y + t, S = 0 dx e x (R h R v ) = 0, (17) K T + KK η = Se ilη + S e ilη, (18) dx e x [ (H 1 U 2 + U 1H 2 ) x U 1 V 2 x V 1x U 2 + il(h 1 V2 + V 1H2 V 1V2 )] (19) and (U i, V i, H i ), i = 1, 2 are amplitudes of two IGW.

25 Final form of the evolution equation From the polarisation relations get: and hence S = ia 1 A 2 s, Im(s) = 0 (20) K T + KK η = 2sA 1 A 2 sin lη, (21) - harmonically forced Hopf equation s = 4l (k )(k )[1 + (k 1 + k 2 ) 2 ][1 + (k 1 k 2 ) 2 ] [ (σ 1 l 2 + σ 2 l 1 l 1 l 2 )(1 + k k 2 2 ) +σ 2 l 1 k 1 (1 + k 2 1 k 2 2 ) + σ 1l 2 k 2 (1 + k 2 2 k 2 1 ) +2k 1 k 2 (l 1 l 2 (1 + k 2 1 )(1 + k 2 2 )) ] (22)

26 Isopleths of the interaction coefficient s(l, k 1, l 1 ) for l = 1 at the interval l k 1

27 Integrability of the KW evolution equation Forced simple-wave equation (after renormalizations): Lagrangian (characteristics) approach: K τ + KK χ = sin χ. (23) K = U = X ; (...) = τ + U χ (...) (24) Ẍ + sin X = 0 (25) Pendulum equation: integrable. Shock formation Lagrangian clustering (known in statistical physics: mean-field limit of the kinetics of particles with repulsive long-range interaction on the circle) Implications for transport and mixing.

28 Comments Main property of solutions: breaking and formation of Kelvin fronts a route to dissipation in the ocean The mechanism produces coastal wave "from nothing" Small deviations from infinite slope - weak dispersion forced Hopf equation forced KdV equation, also integrable Details in Reznik & Zeitlin, Phys. Letters A, 2009.

29 2-layer RSW with idealized coast f ρ 1 h 1 H 1 g ρ 2 h 2 H 2

30 Equations of motion Equations in natural variables D 1 Dt v 1 + f ẑ v 1 + g (h 1 + h 2 ) = 0, t h 1 + (h 1 v 1 ) = 0, D 2 Dt v 2 + f ẑ v 2 + g (rh 1 + h 2 ) = 0, t h 2 + (h 2 v 2 ) = 0. (26) Baroclinic/barotropic decomposition, non-dimensional v ± = r v 1 ± v 2, η ± = 2 ( rη 1 ± η 2 ), (27) Here H 1 = H 2, r = ρ 1 ρ 2, η 1,2 - non-dimensional thickness deviations.

31 Baroclinic-barotropic equations t v + + ẑ v r η + = ɛ [( ) 1 ( 2 4 r + 1 v + v + + v v ) ( ) 1 ( + r 1 v + v + v v +)], t η + + v + = ɛ [( ) 1 4 r + 1 (η + v + + η v ) ( ) 1 + r 1 (η + v + η v +)], t v + ẑ v + 1 r η = ɛ [( ) 1 ( 2 4 r 1 v + v + + v v ) ( ) 1 ( + r + 1 v + v + v v +)], t η + v = ɛ [( ) 1 4 r 1 (η + v + + η v ) ( ) 1 + r + 1 (η + v + η v +)].

32 Dispersion relation for 2-layer RSW in a half-plane Wave spectrum: barotropic (BT) (faster) and baroclinic (BC) (slower) versions of each kind of waves: IGWqand KW: σ 2 ± = 1 + c 2 ~k 2 and σ 2 ± = c 2 l 2, with c± = 1± r ± IG K 2 ± -2-1 k Σ l V. Zeitlin Free Wave - Trapped Wave resonances 2

33 What is new due to baroclinicity? Possibility of new IGW+IGW KW resonances: 1 BT KW may be excited by BT - BT, BT + BC, and BC + BC IGW ("difference" and "sum" resonances) 2 BC KW may be excited by BT - BT, BT - BC, and BC - BC IGW (only "difference" resonances) Possibility of the new resonance BC IGW + Mean Current BT KW due to the intersection of dispersion surfaces for BT KW and BC IGW.

34 Sketch of the IG- Mean resonance Coastal curent Solution of both linearized and full systems, v 1,2 0 at x : u ± M = 0, v ± M = 1 ± r x η ± 2 M, Conditions of resonance σ IG = σ K +, l IG = l K + l, c 2 +l 2 = 1 + c 2 (k 2 + l 2 ), ( ) c k 2 2 = + c 2 1 l 2 1 c 2.

35 Comments Evolution equations: harmonically forced Hopf equations for the amplitude of KW - same conclusions as in the 1-layer case. Details: Zeitlin, Nonlin. Proc. Geoph., 2013.

36 Mechanisms of resonant excitation of trapped coastal waves exist, leading to formation of coherent/dissipative structures in the coastal zones. Resonant pairs of free inertia-gravity waves produce coastal waves "from nothing". Important for: Energy and momentum transport from the open ocean to the coast, and subsequent dissipation. A poorly explored route to dissipation in the ocean. Transport and mixing in the coastal zones. Generalization to continuously stratified case straightforward (vertical modes);