Can Surface-Tide/Eddy Interactions Excite an Internal Tide?

Home , Eddy (fluid dynamics)

M-Pascale Lelong1 Eric Kunze2

(1) NWRA Seattle WA, USA (2) UVic, Victoria BC, Canada

– p.1/21 Outline

Motivation

– p.2/21 Outline

Motivation Weakly nonlinear theory

– p.2/21 Outline

Motivation Weakly nonlinear theory A resonant triad interaction

– p.2/21 Outline

Motivation Weakly nonlinear theory A resonant triad interaction Delineating eddy, barotropic and baroclinic tides

– p.2/21 Outline

Motivation Weakly nonlinear theory A resonant triad interaction Delineating eddy, barotropic and baroclinic tides Multiple-scale analysis

– p.2/21 Outline

Motivation Weakly nonlinear theory A resonant triad interaction Delineating eddy, barotropic and baroclinic tides Multiple-scale analysis Numerical simulations

– p.2/21 Outline

Motivation Weakly nonlinear theory A resonant triad interaction Delineating eddy, barotropic and baroclinic tides Multiple-scale analysis Numerical simulations Validation of weakly nonlinear formulation

– p.2/21 Outline

– p.2/21 Motivation

– p.3/21 Motivation

Numerous studies and observations have demonstrated generation of internal tide through barotropic tide/topography interactions. (e.g. Cox and Sandstrom, 1962; Wunsch, 1975; Simmons et al. 2004, Vlasenko et al., 2003; Garrett and Kunze, 2007 and many others..) Speculation that barotropic diurnal tide interacting with eddies may be responsible for observed tidal-frequency waves in the vicinity of eddy ﬁelds in Oyashio Current. Caveats: limited data did not rule out wind-forcing or subinertial instability; no convincing justiﬁcation was provided, Rogachev et al. (1992, 1996), Rogachev and Carmack (2002).

– p.3/21 Motivation

M2 barotropic tide −1 ω1 = 2π/12hr m1 = 0, l1 (free parameter) geostrophic eddy ﬁeld 0 ≈ ω2 << ω1 −1 k2, l2 ≈ 2π/150km , m2 ≈ 2π/1000m – p.4/21 A resonant triad interaction

2 2 2 2 k2+(l1+l2) 2 If ω1 = f + N ( m2 ) ⇒ resonance ! Internal tide with wavenumbers k3, l3,m3, frequency ω3:

k3 = k2, l3 = l1 + l2

m3 = m2

ω3 = ω1 is excited, (similar to resonant wave-triads; McComas and Bretherton 1977 ; Müller et al, 1986; wave-vortex triads, Warn, 1990; Lelong and Riley, 1991; Bartello, 1995)

This is the mechanism envisioned for internal tide generation.

– p.5/21 Model equations: 3D Boussinesq, f-plane

∂uh +(u · ∇)uh + ∇hp + feˆ × uh = eˆ F (y, t) ∂t 3 1 ∂w ∂p +(u · ∇)w + − b = 0 ∂t ∂z ∂b +(u · ∇)b + N 2w = 0 ∂t ∇ · u = 0

constant N 2 2 2 ω1 − f Tidal forcing: F (y, t) = UT cos l1y cos ω1t ω1 Eddy ﬁeld introduced as an initial condition.

u = Ue cos k2x sin l2y cos m2z k2 v = − l2 sin k2x cos l2y cos m2z m2 b = l2 cos k2x cos l2y sin m2z

– p.6/21 Multiple time-scale formulation

Introduce fast and slow timescales fast (wave) time T0 ∼ 1/f slow (advective) time T1 ∼ L/U T0/T1 = ǫ, a Rossby number. Expand variables as f = f (0) + ǫf (1) + O(ǫ2) Barotropic and internal tides depend on both fast and slow scales. Barotropic tide is O(1). Internal tide is O(ǫ) unless resonant conditions are encountered. Eddy ﬁeld evolves only on slow scale and is O(1).

– p.7/21 Delineating barotropic tide, eddy ﬁeld and baroclinic tide The eddy velocity is the temporally averaged velocity,

1 t0+T ue(x, t1) = u(x, t0, t1) = u(x; τ0, t1) dτ0 T Zt0

where T0 ≪ T ≪ T1. The barotropic tide velocity is the vertically averaged velocity over the domain depth H.

1 H uBT = u(x,y,t0, t1) = u(x; t0, t1) dz H Z0 The baroclinic tide is described by a forced wave equation. It accounts for all the vertical velocity at high frequencies.

– p.8/21 Baroclinic tide evolution

∂2 ∂2 ∂2w ∂2w 2 ∇2 ∇2 3 ǫ 2 hw+ 2 ( 2 ) + hw + 2 = ǫ(F1+F2+F3+F4+F5)+ǫ F6 ∂t0 ∂t0 ∂z ∂z

∂2 F = ∇h · [(uh · ∇h)uh] 1 ∂t∂z 2 ∂ ∂uh F = ∇h · [w ] 2 ∂t∂z ∂z 2 F3 = −∇h[uh · ∇hb]

2 ∂b F = −∇h[w ] 4 ∂z ∂ F = [~e · [∇ × (u · ∇)uh)]] 5 ∂z 3 ∂ 2 F = − ∇h[uh · ∇hw] 6 ∂t RHS source terms must project onto high frequencies to excite

inertia-gravity waves (e.g. Ford et al., 2000). – p.9/21 Lowest-order equations Eddy ﬁeld:

(0) (0) (0) ∇p + i3 × u + b i3 = 0 b w(0)b = 0 ∇ · u(0) = 0

Barotropic tide:

(0) 2 ∂u (ω1 − 1) − v0 = cos l1y cos ω1t0 ∂t0 ω1 ∂v(0) + u(0) = 0 ∂t0 Baroclinic tide: ∂2 ∂2w(0) ∂2w(0) ∇2 (0) 2 ( 2 ) + hw + 2 = 0 ∂t0 ∂z ∂z

– p.10/21 Lowest-order solutions

Eddy ﬁeld (in geostrophic, hydrostatic equilibrium):

(0) ue = Ue(t1)cos k2x sin l2y cos m2z

(0) k2 ve = − Ue(t1) sin k2x cos l2y cos m2z l2

(0) Uem2 cos k2x cos l2y sin m2z be = − l2 Barotropic tide (tidal ellipse + inertial oscillations):

(0) UT cos l1y uBT = − (sin t0 − ω1 sin ω1t0) ω1 (0) UT cos l1y vBT = (cos ω1t0 − cos t0) ω1

Baroclinic tide: w(0) = 0 ⇒ No waves at this order.

– p.11/21 O(ǫ) equations Eddy ﬁeld:

∂u(0) − − (u(0) · ∇)u(0)  ∂t1 (1)  (0) L =  ∂b  − − (u(0) · ∇)b(0) ∂t1  0   Barotropic tide:

(1) (0) ∂u (0) (0) ∂u − v1 = −u · ∇u − ∂t0 ∂t1 ∂v ∂v(0) 1 + u(1) = −u(0) · ∇v(0) − ∂t0 ∂t1 b(1) = 0 (1) ∂b (0) = −u0h · ∇hb ∂t0 – p.12/21 O(ǫ) baroclinic tide

∂2 ∂2w(1) ∂2w(1) ∇2 (1) (0) (0) (0) 2 ( 2 ) + hw + 2 = F1 + F3 + F5 ∂t0 ∂z ∂z RHS terms represent eddy/eddy, tide/tide, eddy/tide interaction terms. Only eddy/tide terms lead to resonant behavior (similar to forced harmonic oscillator). Resonant terms behavior: (0) (0) ± ∼ uBT (t1)ue (t1)g(κ1, κ2,ω1)exp[±iφ ] ± φ ≡±(k2x ± (l1 + l2)y ± m2z ± ω1t0)

– p.13/21 Resonance curves

k2 l l 2 2 2+( 1+ 2) Resonance condition: ω = 1 + 2 1 m2

Loci of resonant tidal scales

1.5

1 + −

0.5 | 2

κ 0 /| 1 l

−0.5

−1

−1.5 0 1 2 3 4 5 6 7 θ (radians)

2 2 k2 + l2 = κ2 cos θ, m2 = κ2 sin θ p

– p.14/21 Coefﬁcients of resonant forcing terms

± ± ± ± RHS resonant forcing terms are, ΣFi = ΣCs sin φ + ΣCc cos φ

(0) (0) ± ue (t1)uBT (t1)k2m2l1 3 Cs (t1) = (±1 ± l1) 4 4l2 (0) (0) 2 2 2 ± ue (t1)uBT (t1)m2l1 k2 l1l2m2 2k2ω1 Cc (t1) = (∓l1 ± ∓ ± ) 16ω1 l2 16ω1 l2

(0) If l1 = 0 (no horizontal variation of tide), then Fi ≡ 0. w(1) ≡ 0. No baroclinic tide. ± ± 1 If l =6 0, Cs and Cc are maximum when l + l = ± , i.e. 1 1 2 3 l1 = O(l2), assuming k2 = l2. Slow-time dependence of forcing terms comes from elimination of secular terms in eddy and barotropic tide O(ǫ) equations.

– p.15/21 Slow-time dependence of O(1) solutions

Coefﬁcients of secular terms can be written, after much algebra (and thanks to Mathematica) as,

∂ue 2 2 = Aue + BuT ∂t1

∂uT 2 2 = Cue + DuT ∂t1 Can rewrite: ∂ue A ∂uT ADB 2 = − uT ∂t1 C ∂t1 C (or similar equation for uT ). ∂ue 1 −1 If = 0, then uT ∼ (or vice versa). ⇒ ue(t1)uT (t1) ∼ t1 . ∂t1 t1 More work needed to solve for general slow-time evolution which is (hopefully) more interesting.

– p.16/21 Summary of weakly nonlinear analysis

For deep-ocean conditions where barotropic-tide wavelength is very large compared to mesoscales, interaction is negligible and cannot result in significant baroclinic tide excitation, even if resonant conditions are met. In coastal regions where tide can vary on the scale of eddies, resonant interaction can be significant (but probably not as important as tide/topography interactions). Tide or eddy fields may both act as catalysts, but this results in the decay of internal wave source terms as t−1. This mechanism is unlikely to be an effective generator of waves. – p.17/21 Numerical Simulations

f-plane, Boussinesq model (Winters et al., 2003) N/f=20 Initial condition: Eddy field in geostrophic, hydrostatic 0−1 equilibrium (U2 ∼ .1ms ) Forcing by barotropic tide ramped over two tidal periods 1 (U1 ∼ 0.06ms ) Barotropic tide: vertically averaged fields Eddy field: temporally averaged fields (over fast timescale) Baroclinic tide is the residual and accounts for all high-frequency vertical velocity.

– p.18/21 Comparison of 3 simulations

1. no tide (RUN A) 2. spatially constant tide (RUN B) 3. spatially varying tide (RUN C)

Initial condition

Quasi-turbulent surface-conﬁned eddy ﬁeld spun up from perturbed Taylor-Green vortices

– p.19/21 Run A: No tide

Left panels: rela- tive vorticity, mid- – p.20/21 dle panels: ver- Run A: No tide

Left panels: rela- tive vorticity, middle – p.21/21 panels: High-pass-