Mixing Efficiency of Inertia-Gravity Wave Breaking: an oceanographic perspective

Transcription

1 Mixing Efficiency of Inertia-Gravity Wave Breaking: an oceanographic perspective Pascale Lelong, NWRA, Seattle USA Pascale Bouruet-Aubertot, U. of Paris VI, France Sylvain Beaufils, ENS-Cachan, France Chapman Conference on Gravity Waves, Honolulu, March, 2 Tuesday, March, 2

2 Outline Background How do inertia-gravity waves break (in the absence of ambient shear?) Potential energy budgets High-frequency vs. low-frequency wave breaking Tuesday, March, 2 2

3 Oceanographic motivation Inertia-gravity wave breaking provides a principal pathway to energy dissipation in the ocean (Wunsch and Ferrari, 29). Estimates of mixing efficiency from ocean microstructure measurements range from.5 to.48, and appear to depend on instability mechanism (Caldwell and Moum, 995). Low-frequency inertia-gravity waves are primarily unstable to shear instability, high-frequency waves to convective instability (Lelong and Dunkerton, 998). Shear instability in a stratified fluid develops where density gradients are maximum. Convective instability develops where gradients are minimum. Do the two types of instability lead to different mixing rates? Need to parameterize wave breaking in larger scale models Tuesday, March, 2 3

4 Instability mechanism as a function of wave frequency a/c TC: Transverse convective TS: Transverse shear HS: Homogeneous shear DS: Diagonal shear Ri l = g ρ ( dρ(z) dz + ρ z ) u z 2 + v z 2 >.25 R f/ω (Lelong & Dunkerton, 998) Tuesday, March, f/ω

5 Numerical simulations 3D Boussinesq equations coupled with density advection-diffusion equation. Pseudo-spectral code, periodic boundary conditions. Isotropic horizontal resolution, comb. of hyper and Laplacian dissipation Initial condition: Monochromatic, propagating wave+3d white noise u = a c cosφ v = a c (f/ω) sinφ w = ack m cosφ φ = const ω = N 2 k 2 + f 2 m 2 k 2 + m 2 b = acn2 m sinφ φ = kx + mz ωt z (k,,m) y x 2π/N 2π/f L/U L 2 /ν 2 Tuesday, March, 2 solve dynamically similar problem with reduced N/f keeping relative separations between all timescales constant.

6 density Instability of a near-inertial wave shear-driven and fast-growing. horizontal velocity is nearly circularly polarized. instability develops nearly simultaneously in all azimuthal directions 6 Tuesday, March, 2

7 Breaking of an intermediate frequency wave, R=.35 hybrid shear/convective instability appearance of mushroom-like formations Tuesday, March, 2 7

8 high frequency waves break down via convective instability Given that the onset of convective and shear instabilities occurs at different positions on the wave phase, is there a difference in the mixing efficiency of wave breaking for waves of different frequencies? 8 Tuesday, March, 2

9 Estimating turbulent diffusivity κ e in the ocean Assumptions: density gradient constant over integral scale of turbulence flow is statistically stationary κ e = < ρ w > dρ/dz buoyancy flux is proportional to mean density gradient < ρ w > dρ dz = κ ρ 2 Osborn-Cox model (972) = κ e = κ ρ 2 (dρ/dz) 2 Based on < ρ w>and, mixingefficiency Γ.2 in the ocean. Tuesday, March, 2 9

10 Splitting potential energy into background & available parts Extension of Lorenz s notion of APE to 3D stratified flows (Winters et al. 995; Caulfied and Peltier, 2, Staquet, 2, Bouruet-Aubertot et al., 2...) total potential energy E p = V ρgzdv background potential energy E b = V ρgz dv available potential energy E a = E p E b z (x, y, z, t) is the position of a fluid parcel located at {x, y, z} at time t, following adiabatic rearrangement to an equilibrium state. Tuesday, March, 2

11 Energy budgets and mixing efficiency de p dt de b dt = Φ a + Φ i + Φ s = Φ d + Φ s Φ a : advective buoyancy flux Φ i : laminar diffusive flux Φ d : diapycnal flux : kinetic energy dissipation Φ s : surface fluxes de a dt = Φ a (Φ d Φ i ) E_k Φ a E_a de k dt = Φ a + E_b Φ d Φ i Φ s E_i Φ i Mixing efficiency: Γ Φ d / Tuesday, March, 2

12 background potential energy equation de b dt = g z ρu nds + κ 2 g z ρ nds +g ρu z dv + κ 6 g z ( 4 ρ) nds κ 2 g z ρdv κ 6 z ( 4 ρ)dv 2 Tuesday, March, 2

13 KE budget (J kg s ) 5 Kinetic energy budget 5 x 3 R= dke/dt KE diss + buoy flux f/ω =.85, a =. Pr = KE budget (J kg s ) 5 5 x dke/dt KE diss buoy flux 3 Tuesday, March, 2

14 background potential energy budget E b budget (J kg s ) x 3 R = de b /dt + adv flux E b budget (J kg s ) x de b /dt adv flux diap flux 4 Tuesday, March, 2

15 Mixing efficiency of a low-frequency wave breaking event (J kg s ) R = / i (J kg s ) no evidence of pre-turbulent mixing characteristic of KH instability, mixing efficiency is low 5 Γ.5 Tuesday, March, 2

16 Kinetic energy budget R =.55,a=. KE budget (J kg s ).5 x 3 R= dke/dt KE diss + buoy flux Pr = KE budget (J kg s ) 2 2 x dke/dt KE diss buoy flux 6 Tuesday, March, 2

17 intermediate-frequency wave 2.5 x 3 R =.55 2 de b /dt + adv flux E b budget (J kg s ) 2.5 x de b /dt adv flux diap flux 7 Tuesday, March, 2

18 Intermediate-frequency wave f/ω =.55,a=.,Pr = (J kg s ) 2.5 x 3 R = i (J kg s ) / Tuesday, March, 2 8

19 Background energy budget, R=.55, Pr= E b budget (J kg s ) 3.5 x 3 R = de b /dt + s E b budget (J kg s ) x de b /dt s 9 Tuesday, March, 2

20 Intermediate frequency wave, R=.55, Pr= (J kg s ) 2.5 x 3 R = i (J kg s ) / Tuesday, March, 2

21 Impact of amplitude on mixing efficiency, R=.75 R=.75, a=.7, Pr= R=.75, a=., Pr= / /.5 / weaker initial amplitude results in longer laminar-diffusive regime. Γ =.35, irrespective of initial wave amplitude 2 / Tuesday, March, 2

22 Mixing efficiency as a function of wave amplitude, R=.55 R=.55, a=,7, Pr= R=.55, a=., Pr= / same story... Tuesday, March, 2 22

23 Prandtl number dependence R=.55, a=., Pr= R=.55, a=., Pr= d.5 / Higher Prandtl number flows lead which leads to reduced mixing efficiencies. Bouruet-Aubertot et al. (2) found Γ /( + Pr) for non-rotating waves. Tuesday, March, 2 23

24 Conclusions Low-frequency wave instabilities do not result in higher mixing efficiencies. Energy expended in overturning isopycnals during shear instability is significant. Initial wave amplitude does not significantly impact the mixing efficiency. Regime dominated by laminar diffusion is longer for smaller-amplitude waves, due to slower growth of instability but resulting mixing efficiency is the same for weak and strong waves. Larger Prandtl numbers result in decreased mixing efficiency. Mixing efficiencies of breaking waves range Tuesday, March, 2 between. and.5. Therefore the typical oceanic value of.2 appears quite reasonable. 24