Gregg Henyey

Turbulence Production

Internal Wave Cascade to Small Scales: Gregg-Henyey Parameterization: The best way to estimate mixing of a property T is to measure its microscale variability distribution or wavenumber spectra S[T](k) on scales of 0.1-1 cm. This is technically challenging so microstructure oceanographers often instead use turbulence theory to relate the distribution of microscale shear u_z [the kinetic energy dissipation rate ε = (15/2)υu_z² where υ is the kinematic molecular viscosity 10^–6 m²/s] on similar scales to mixing though the dissipation rate does not have to be resolved on as small a scale as properties such as temperature or salinity. The property diffusivity K = 0.2ε/N² for much turbulence in the ocean though the mixing efficiency (0.2) can be smaller.

The dissipation rate ε can also be inferred from
(i) the density overturn lengthscale L_T (ε = < L_T²>N³) which is about 1 m,
(ii) unstable shear v_z > 2N which is also on 1-m scales (Fig. 2)

Fig. 2: Time-series of 1-m shear v_z and stratification N (upper panel) and stratification-normalized shear v_z/N (lower panel) from a neutrally-buoyant float (Kunze et al. 1990 JGR). When v_z/N > 2, shear instability will produce turbulence.

(iii) direct measurement of property fluxes (Fig 3) which is extremely challenging and open to misinterpretation,

 

Fig. 3: Variance-preserving co-spectra of vertical velocity w and temperature T from a neutrally-buoyant float. At low frequencies (lengthscales), the flux w’T’ is negative consistent with heavy water being lifted up or light water pushed down. At high frequencies, the flux is positive (warm water rising, cool water sinking), consistent with water settling back to its level after breaking (from Sun et al. 1996 JPO). Molecular diffusion prevents complete resettling.

(iv) from internal-wave shear and strain on lengthscales of 10’s of meters (Kunze and Gregg 1991 JGR; Kunze et al. 1992; Kunze and Sanford 1996; Sun and Kunze 1999 Kunze et al. 2006 JPO; Nash et al. 2007 GRL). The last parameterization is known as the Gregg-Henyey scaling. It is based on internal wave/wave interaction theory predictions of a cascade of energy from large to small scales. It assumes certain properties of the internal wave field that do not appear to hold on continental shelves, in canyons (Kunze et al. 2002 JPO) or near internal wave generation sites.

The Gregg-Henyey parameterization has been used to show that the diapycnal diffusivity K is closer to 0.1 × 10^–4 than 10^–4 m² s^–1 in the abyssal Sargasso Sea (Fig. 4) but is elevated over seamounts (Fig. 5).

Fig. 4: Dissipation rates ε and diapycnal diffusivities K as a function of depth in the Sargasso Sea (from Kunze and Sanford 1996 JPO). Dots are individual estimates, the black and white curves are averages.

Fig. 5: Depth profiles of rms vertical shear normalized by stratification v_z/N (left panel) and normalized by the GM shear (right panel) in the California coastal margin showing elevated internal wave shear over Pioneer Seamount (squares) (from Kunze et al. 1992 JPO).

An attempt to assess the global deep-ocean turbulence by applying the Gregg-Henyey scaling to about 3500 lowered ADCP and CTD profiles collected on hydrography cruises revealed elevated turbulence levels over rough topography (Fig. 6-8). But average profiles (Fig. 9) show diffusivities of at most a few 0.1 × 10^–4 m² s^–1 in contrast to the 10^–4 m² s^–1 argued to be necessary from a 1-D vertical advective-diffusive balance. The diffusivity K shows a tendency to increase with latitude and with depth for latitudes less than 30º (Fig. 10). Diapycnal velocities w* inferred from these profiles show upwelling through density surfaces of about 0.5 cm/day at high latitudes below 3500- and above 2000-m depth, weakening toward the equator. This is consistent with a 2-cell circulation with one cell circulating between bottom and deep waters and the other between intermediate and upper waters.

Fig. 6: Maps of vertically integrated dissipation rate ∫_ε in the North Atlantic, Pacific and Indian Oceans reveal weak turbulence close to the equator and elevated turbulence over rough bottom topography (from Kunze et al. 2006 JPO).

Fig. 7: Meridional sections in the east Indian and Southern Oceans (Fig. 6) of barotropic (depth-averaged) and bottom 500-m current speed, vertically-integrated dissipation rate ∫_ε, topographic roughness [H²], wind and tidal current speeds eddy diffusivity K. Diffusivities O(10^–4 m² s^–1) extend through the entire water column northeast of the Kerguelen plateau. High diffusivities are also seen at depth over other rough topography such as the Ninety-East Ridge and Austral-Antarctic Rise (from Kunze et al. 2006 JPO).

 

Fig. 8: Zonal sections along 30ºS in the Indian Ocean (Fig. 6) of vertically-integrated dissipation rate ∫_ε, topographic roughness [H²], wind and tidal current speeds eddy diffusivity K. Elevated diffusivities extend through the entire water column above the Southwest Indian Ridge and Madagascar Plateau. High diffusivities are also seen in the bottom 500-1000 m over the generally rough topography (from Kunze et al. 2006 JPO).

Fig. 9: Globally-averaged profiles of number of segments (leftmost), shear v_z and stratification N, shear-and-strain and strain-only Gregg-Henyey diffusivity K, shear-and-strain and strain-only dissipation rate ε, shear/strain ratio R, and ratio of down/upgoing rotary shear as a function of depth z, height above bottom h, neutral density γ_nand stratification N. Diffusivities range from 0.1 × 10^–4 m² s^–1 in the upper few km to a few 0.1 × 10^–4 m² s^–1 near the bottom. A comparison with an inverse bulk budget using hydrographic data (black curve in diffusivity vs. neutral density panel) shows good agreement in the intermediate water and upper deep water but higher values in lower deep and bottom waters either due to overestimation by the bulk budget inverse or missed (non-internal wave) mixing by the Gregg-Henyey method (from Kunze et al. 2006 JPO).

Fig. 10: Diffusivity profiles K as a function of depth z and latitude (left colored number ranges) show a tendency for diffusivity to increase with latitude and increase with depth for latitudes less than 30º (upper panel). Diapycnal velocity w* = (γ/N²)(∂ε/∂z) inferred from cubic fits to the diffusivity and stratification profiles with mixing efficiency γ = 0.2 (lower panel) show strong upwelling in abyssal waters (below 3500 m) and in the upper 2000 m at high latitudes, weakening toward the equator, and a minimum between 2000 and 4000 m (from Kunze et al. 2006 JPO).

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