How Rossby Waves Break. Results from POLYMODE and the End of the Enstrophy Cascade

How Rossby Waves Break. Results from POLYMODE and the End of the Enstrophy Cascade KURT L. POLZIN Woods Hole Oceanographic Institution Submitted, Journal of Physical Oceanography, May 31, 2005 Corresponding author address: Kurt L. Polzin, MS#21 WHOI Woods Hole MA, 02543 E-mail: [email protected] 1 2 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME ABSTRACT The issue of internal wave–mesoscale eddy interactions in Physical Oceanogra- phy is revisited. Previous theoretical work identified the mesoscale eddy field as a possible source of internal wave energy and pseudo-momentum. Adiabatic pseudo- momentum flux divergences, in turn, serve as a sink of eddy potential vorticity. The exchange of internal wave momentum with potential vorticity in the eddy field is an intrinsic component of linear wave propagation in geostrophically balanced mean fields. Previous estimates of internal wave - eddy coupling using data from the POLY- MODE field program are assessed. Characterization of the coupling as a diffusive process provides somewhat smaller transfer coefficients than previously obtained, ∼ 2 −1 ∼ −3 2 −1 ∼ with νh = 50 m s and νv = 2.5 × 10 m s , in contrast to νh = 200 − 400 2 −1 2 −1 m s and νv =∼ 0.01 m s . Current meter data from the Local Dynamics Experiment of the Polymode field program indicate (i) this internal wave – mesoscale eddy coupling through horizontal interactions plays an O(1) role in the energy budget of the internal wavefield and is a significant sink of eddy energy, and (ii) eddy-internal wave interactions play an O(1) role in the eddy potential enstrophy budget. MAY 2005 3 1. Introduction a. Preliminaries Winds and air-sea exchanges of heat and fresh water are ultimately responsible for the basin-scale currents, or general circulation of the oceans. In order to achieve a state where the energy and enstrophy (vorticity squared) of the ocean is not continuously increasing, some form of energy dissipation is required to balance this forcing. While the above statement may seem obvious, little is known about how and where this dissipation occurs. Early theories of the wind driven circulation [Stommel (1948), Munk (1950)] view the western boundary as a region where energy and vorticity input by winds in mid-gyre could be dissipated. Those theories predict Gulf Stream transports that are approximately equal to the interior Sverdrup transport [about 30 Sv, Schmitz et al. (1992)] and that are much smaller than observed Gulf Stream transports after the Stream separates from the coast [about 150 Sv, Johns et al. (1995)]. Subsequent theories of the wind driven circulation have attempted to address the role of nonlinearity and baroclinicity in increasing Gulf Stream transports above that given by the Sverdrup relation. Hogg (1983) proposed the existence of two relatively barotropic recirculation gyres on either side of the Stream that combine to increase the total transport. It is now generally ac- cepted that the recirculation gyres result from the absorption of Planetary and Topographic Rossby Waves generated by the meandering of a baroclinicly unstable Gulf Stream. The absorption process is uncertain. In the atmospheric context, radiative damping [Stone (1972)] is typically invoked to provide damping of planetary scale motions. Much of the atmosphere is sufficeintly close to radiative equilibrium, in which heating associated with incoming solar radiation is balanced by cooling associated with black body radiation, that thermal perturbations associated with eddy motions are damped. Such a process does not operate within the ocean and we are left searching for other mechanisms by which planetary waves could be damped. If one assumes the ocean interior is adiabatic, the energy sinks would neccesarily be located at the boundaries. The oceanic eddy field could be closed by some combination of bottom Ekman layers, eddy-mixed layer interactions, and an interior enstrophy cascade. Despite the intellectual predjudice that views the oceanic interior as adiabatic, or perhaps more precisely because, it seems prudent to enquire whether interior processes could provide a damping of both enstrophy and energy. b. Eddy-Wave Coupling as an Eddy Dissipation Mechanism A convenient starting place to examine eddy-wave coupling is to invoke a decomposi- tion of the velocity [u = (u,v,w)], buoyancy [b = −gρ/ρo with gravitational constant g and density ρ] and pressure [π] fields into a quasi-geostrophic ‘mean’ ( ) and small amplitude internal wave (′′) perturbations on the basis of a time scale separation: φ = φ+φ′′ with −1 τ φ = τ 0 φ dt in which τ is much longer than the internal wave time scale but smaller R 4 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME than the eddy time scale. Employing this averaging process to the equations of motion returns the result that the right-hand-side of the mean momentum and buoyancy equations represents the flux divergence of psuedo-momentum. Following Müller (1976), the pseudo-momentum flux associated with a wave-packet as: CjP i = d3k n(k, x, t) ki Cj (1) g Z g with 3-D wave action spectrum n ≡ E/ω, intrinsic frequency ω = σ −k·u, group velocity 2 2 1/2 Cg, wave vector k = (k,l,m) having horizontal magnitude kh = (k + l ) and energy density E = Ek + Ep. Superscripted indices indicate particular vector/tensor components and repeated indices imply summation. This formulation assumes a slowly varying plane wave solution and manipulations involving the algebraic factors relating (u,v,w,b, and π) for linear internal waves. Psuedo-momentum is not, by itself, conserved. For small amplitude waves in a slowly varying background, it is the action flux which is nondivergent, ∇ · Cgn = 0. If the background velocity field contains horizontal gradients, the horizontal wavenumber of an internal wave packet evolves following a ray trajectory and thus the pseudo-momentum flux is, in general, divergent. The most familiar conservation statement is for a zonal back- z x ground flow, in which case Cg P is nondivergent, Eliassen and Palm (1961); Jones (1967). The zonal problem is, in this respect, trivial. Bühler and McIntyre (2004) point to an analogy between internal wave propagation and the problem of particle pair separation in 2-D turbulence. In this relative dispersion problem, particle pairs undergo exponential separation when the rate of strain exceeds relative vorticity: 2 2 2 Ss + Sn >ζ (2) with Ss ≡ vx + uy the shear component of strain, Sn ≡ ux − vy the normal component and ζ ≡ vx − uy relative vorticity. Equation (2) is simply the Okubo-Weiss criterion [e.g., Provenzale (1999)]. Bühler and McIntyre (2004) argue that the problem of small amplitude waves in a larger scale flow field is kinematically similar to particle pair separation under the hypothesis of action conservation. In this case the ray equations lead to exponential increase/decrease in the density of phase lines, i.e., an exponential increase/decrease in 2 2 1/2 horizontal wavenumber, kh =(k + l ) , Fig. 1. Vorticity simply tends to rotate the horizontal wavevector in physical space. This provides a simple picture of pseudo-momentum flux divergence associated with an internal wave packet interacting with an eddy strain field. Bühler and McIntyre (2004) further argue for a scenario which they term ‘wave cap- ture’. Simply put, exponential growth of the horizontal wave number implies exponential growth or decay of vertical wave number in the presence of geostrophic shear. Those waves with growing vertical wavenumber will tend to be trapped (captured) within the extensive FIG.1 regions of the eddy strain field. MAY 2005 5 The ray-tracing arguments about internal wave packets interacting with mesoscale ed- dies return useful pictures, but they do not provide information about the net transfer rates for the observations. This requires consideration of energy and potential vorticity balances. 1) INTERNAL WAVE ENERGY The internal wave energy equation is [Müller (1976)]: ∂ ( + u · ∇ )(E + E )+ ∇· π′′u′′ = ∂t h k p ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ −2 ′′ ′′ −2 ′′ ′′ −u u ux − u v uy − u w uz − v u vx − v v vy − v w vz − N b u bx − N b v by (3) ′′2 ′′2 ′′2 −2 ′′2 with kinetic [Ek =(u + v + w )/2] and potential [Ep =(N b )/2] energies. Tem- poral variability and advection of internal wave energy by the geostrophic velocity field are balanced by wave propagation and energy exchanges between the quasi-geostrophic and internal wave fields. Nonlinearity and dissipation are assumed to be higher order ef- fects. With the exception of vertical buoyancy fluxes, (b′′w′′), which are negligable for linear waves, the energy exchanges are adiabatic. Similarly, wz does not appear as it is small [order Rossby number squared] in the quasigeostrophic approximation. The thermal wind relation can be invoked to cast vertical Reynold’s and horizontal buoyancy flux as the rate of work by an effective vertical stress acting on the vertical grandient of horizontal momentum: f u′′w′′u + N −2b′′v′′ b = [u′′w′′ − b′′v′′]u z y N 2 z f v′′w′′v + N −2b′′u′′ b = [v′′w′′ + b′′u′′]v z x N 2 z The character of the coupling mechanism can be made more apparent by expressing the right-hand-side of (3) as: i j ij i 3 i3 rhs (1.3) −→ k Cg ns |i,j=1,2 +k Cg ns |i,j=1,2 The factor 1 sij = [∂j ui + ∂i uj] 2 x x represents the rate of strain tensor. Two points are to be made. First, the energy exchange represents the rate at which the pseudo-momentum flux CgP, is doing work against mean gradients. Equation (3) is thus a description of how internal wave energy is altered through radiative processes (work and an energy flux divergence).

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