Flow / Topography Interacctions

Seamounts     |    Ridges     |    Canyons     |    Internal Tide Generation     |     Enhanced Mixing

Flow/Topography Interactions: My own interest in flow/topography interactions is in their possible role in enhancing average ocean mixing. Assuming a 1-D vertical advective-diffusive balance for the ocean’s stratification, Munk (1966 DSR) deduced an average eddy diffusivity K = 10–4 m2 s–1 from the observed stratification and estimates of bottom-water production rates (Fig. 1). Inferred turbulence in the ocean interior routinely find diffusivities an order of magnitude smaller (Fig. 2). However, the mixing might be dominated by elevated turbulence either at the surface or bottom boundaries (Fig. 1).

Fig. 1: Cartoon illustrating possible roles of mixing on density stratification. Basin-average mixing in abyssal waters has been inferred to be 10–4 m2 s–1. This could be contributed from mixing in the interior (A), at high latitudes where density surfaces outcrop at the surface and are exposed to atmosphere forcing (B) or where density surfaces impinge the bottom (C). It would then be stirred along density surfaces by eddies. Elevated mixing is observed in the surface mixed layer and in stratified turbulent boundary layers over rough topography (from Kunze and Llewellyn Smith 2004 Oceanography).

Fig. 2: Profiles of turbulent dissipation rate ε (left panel) and diffusivity K (right panel) inferred from finescale velocity profiles in the Sargasso Sea using the shear-only Gregg-Henyey parameterization. Dots represent individual estimates while the barred lines correspond to the averages. Average diffusivities are 0.1 ×10–4 m2 s–1 at all depths (from Kunze and Sanford 1996 JPO).

Vortex Caps and Shed Vortices: Stratified flow in the ocean can interact with bottom topography in many ways depending on the flow speed U, the flow’s frequency ω, stratification N, topographic height h, the water depth H and topographic horizontal lengthscales L. Well-studied problems include the formation of an anticyclonic vortex (or Taylor column) over a seamount to conserve potential vorticity. A vortex cap can also be created by rectification of fluctuating (tidal) flow over a seamount (Figs. 3-4). Separation of flow around topography can produce a turbulent wake of 2-D vortices with the lengthscales of finescale internal waves. Though measurements in the wake of Ampere Seamount were inadequate to resolve the fully 3-D finestructure, the wave and vortex properties could be distinguished using the relationships of the ratio of kinetic to available potential energy to the aspect ratio (Fig. 5). Examining the vertical spectra of potential vorticity components (Kunze 1993 JPO) led to similar conclusions. Hydraulically-controlled flow through straits and over topography can produce very strong turbulent mixing.

 

 

Fig. 3: Radial and azimuthal velocity profiles as function of radius and depth over the summit of Fieberling Seamount. There is strong anticyclonic (negative vorticity) flow over the summit and upper flanks (from Kunze and Toole1997 JPO).

Fig. 4: Time-series of radial and azimuthal velocity on the summit of Fieberling Seamount revealing fortnightly beating of the azimuthal velocity amplitude as well as longer timescale fluctuations (from Kunze and Toole 1997 JPO).

Fig. 5: Ratio of available potential energy to horizontal kinetic energy PE/KE plotted against the dynamic aspect ratio (Nkh)2/(fkz)2 from 2 surveys in the wake of Ampere Seamount. Dots resolved horizontal scales. Gray horizontal lines either do not constrain the horizontal lengthscales either from above (left) or below (right). While most points lie along the internal wave curve (dotted line) or are ambiguous, some of the large horizontal lengthscale flows appear to be unambiguously geostrophic so potential vorticity carrying (from Kunze and Sanford 1993 JPO).

Enhanced Internal Waves: My own focus is in how interactions with topography can modify the internal wave field and enhance turbulence. Some of my early measurements suggested that seamounts and sloping topography were sites of both enhanced internal wave shear and turbulence (Figs. 6-7) (Gregg and Kunze 1991 JGR).

 

 

Fig. 7: Profiles of gradient Froude number Vz/N and normalized shear in the California continental margin and over Pioneer Seamount (▪) (from Kunze et al. 1992 JPO).

Internal Lee Waves: Subinertial flows interacting with bottom topography can generate internal lee waves provided f > kU > N where f is the Coriolis frequency, k = 2π/λ the topographic wavenumber (spatial frequency), λ the topographic wavelength and N the buoyancy frequency. For typical values of U, f and N, internal waves can only be generated by topography with wavelengths λ < O(1 km) (Kunze and Llewellyn Smith 2004 Oceanography).

Internal Tide Generation: The interaction of barotropic tidal currents with topography on wavelengths O(100 km) and smaller can generate internal waves at the tidal frequency or harmonics (Garrett and Kunze 2008 Ann. Rev. Fluid Mech.). This interaction has been implicated in elevating abyssal mixing. Measurements at Mendocino Escarpment (Figs. 8-9) and along the Hawaiian Ridge (Figs. 10-11) have confirmed that these are sites of strong internal generation. Most of the barotropic-to-baroclinic transfer goes into larger vertical scales (low modes) which radiate away to be dissipated elsewhere with only a small fraction lost to turbulence locally (Klymak et al. 2006 JPO).

 

Fig. 8: Meridional (north-south) and vertical semidiurnal energy-flux profiles (black and red) in a meridional sectional across Mendocino Escarpment. Fluxes appear to be radiating along semidiurnal ray paths (blue lines) from the crest of the ridge running along the escarpment. Depth and fluxes have been WKB-normalized to account for the depth-varying stratification (from Althaus et al. 2003 JPO).

 

Fig. 9: Map of vertically-integrated semidiurnal energy-fluxes over Mendocino Escarpment. Fluxes are northward to the north of the escarpment and southward to the south (from Althaus et al. 2003 JPO).

Fig. 10: Vertically-integrated semidiurnal energy-fluxes along the Hawaiian Island Chain revealing strong fluxes from Kauai Channel west of Oahu and from French Frigate Shoals (from Rudnick et al. 2003 Science; Lee et al. 2006 JPO)

 

Fig. 11: Energy-flux profiles across the southern side of Kauai Ridge showing up- and downgoing beams originating from the north side of the ridge (upper panel). A time-series on top of the ridge (middle panel) shows downward increasing phase, a signature of upward energy propagation. A time-series from the crest rim (lower panel) shows downward phase (upward energy) propagation above 700-m depth and upward phase (downward energy) propagation below 700-m depth (from Nash et al. 2006 JPO).

These internal tides are known to radiate 1000’s of kilometers. Their sink is presently unknown. Possibilities include an internal wave/wave interaction mechanism known as parametric subharmonic instability (Frajka Williams et al. 2009 in prep) and breaking on continental slopes as appears to be happening off Virginia (Fig. 12). The Oregon continental slope also has turbulence hotspots (Nash et al 2007 GRL) but the energy source is less clear. Both scattering and critical reflection (Fig. 13) will transfer energy to small vertical scales and turbulence.

Fig. 12: On the corrugated Virginia continental slope, the onshore energy-flux has larger vertical scales (left panels) while the critically-reflected offshore flux has small vertical scales and is confined to a narrow beam radiating from the critical part of the slope (middle panels). The beam coincides with a layer of elevated turbulent diffusivity (right panels) (from Nash et al. 2004 JPO).

Fig. 13: Illustrating of internal wave reflection when the bottom slope is gentler than the wave’s ray path (subcritical, B), the same (critical, C) and steeper (supercritical, D). Upon reflection from gentle slopes, an internal wave will reflect vertically and, if propagating from deeper to shallower water, will have shrinking wavelength and elevated energy density (A). Upon reflection from steep slopes, an internal wave is sent back into the deep ocean (D). Reflection from near-critical slopes shifts the reflected wave’s wavelength to very small values while its amplitude dramatically increases, both effects likely to cause to the wave to become unstable and break down into turbulence (from Kunze and Llewellyn Smith 2004 Oceanography).

Submarine Canyons: The topography of many shelf canyons acts as a conduit for internal tides, with supercritical canyon walls and near-critical axis slopes, forcing them to follow the canyon to its head (Fig. 14) and forming along-bottom beams of energy-flux (Fig. 15). Energy-fluxes weaken upcanyon. The results energy-flux convergence is lost to turbulence and mixing.

 

 

Fig 14: Depth-integrated semidiurnal internal tide energy-fluxes in Monterey Submarine Canyon. Solid arrows are based on 4 or more profiles over a tidal cycle. Open arrows are based on single profiles and should be discounted. Energy-fluxes are upcanyon and appear to be steered along the canyon meanders. The fluxes decrease toward the head of the canyon (from Kunze et al. 2002 JPO).

 

 

Fig. 15: Profiles of energy-flux magnitude (b) and direction (c) along the axis of Monterey Submarine Canyon. Fluxes are directed up or down the canyon axis (a), particularly near the bottom (red) (from Kunze et al. 2002 JPO).

Topography and Turbulence: Stratified turbulent boundary layers are frequently inferred near seamounts (Fig. 16) and in submarine canyons (Kunze et al. 2002 JPO). The maintenance of this stratification in the presence of turbulence suggests that the water in the bottom boundary layer is rapidly exchanging with that in the ocean interior (McPhee-Shaw et al. 2002 JGR) so that elevated boundary mixing could contribute to mixing of density in the global ocean by spreading laterally. A census of seamounts suggested that this process was unlikely to be important in the N. Pacific for water shallower than 3000 m (Toole et al. 1997 JGR; Kunze et al. 1997 JPO).

 

Fig. 16: Eddy diffusivity K is elevated on the flanks of Fieberling Seamount in a stratified turbulent boundary layer (from Toole et al. 1997 JGR).

Global Internal-Wave-Driven Mixing: A global assessment reached similar conclusions with elevated vertically-integrated dissipation rates ∫ε tending to be elevated over turbulence (Fig. 17). Elevated mixing is inferred over rough topography accompanied by strong tidal or mean currents most dramatically in the abyssal 1 km but sometimes extending through the entire water column (Figs. 18-19). A crude relationship between near-bottom diffusivity and topographic roughness and tidal current speed can be inferred (Fig. 20).

 

 

 

Fig. 17: Vertically integrated dissipation rates ∫ε inferred from application of the shear-and-strain Gregg-Henyey parameterization to 3500 lowered ADCP and CTD profiles in the Indian, Pacific and N. Atlantic. Turquoise indicates the level expected for typical internal wave levels. Dissipation rates are reduced near the equator and elevated over mid-ocean ridges and northeast of the Kerguelen Plateau in the Indian sector of the Southern Ocean (from Kunze et al. 2006 JPO)

 

 

Fig. 18: Zonal (east-west) sections along 30S in the Indian Ocean of vertically-integrated dissipation rate ∫ε (red dots), variance in the bottom topographic height <H2> (black dots), wind speed (blue lines), barotropic tidal speed (red lines) and turbulent eddy diffusivity K (bottom panel). In the bottom panel, white corresponds to diffusivities expected for typical internal wave levels, blue is lower and red higher. Elevated diffusivities are found throughout this section in the bottom 500-1000 m because of the rough topography. High diffusivities extend throughout the water column over the Southwest Indian Ridge and Madagascar Plateau (from Kunze et al. 2006 JPO).

 

 

Fig. 19: Meridional (east-west) sections in the eastern Indian Ocean extending into the Southern Ocean SW of Australia of barotropic flow (BT, blue), bottom 500 m flow (500 mab, red), vertically-integrated dissipation rate ∫ε (red dots), variance in the bottom topographic height <H2> (black dots), wind speed (blue lines), barotropic tidal speed (red lines) and turbulent eddy diffusivity K (bottom panel). In the bottom panel, white corresponds to diffusivities expected for typical internal wave levels, blue is lower and red higher. Mixing is weak in the Bay of Bengal and straddling the equator. Elevated diffusivities are found over the Ninety East Ridge and Austral-Antarctic Rise. Very high diffusivities extend throughout the water column northeast of the Kerguelen Plateau in a region of strong nearly barotropic flows associated with the Antarctic Circumpolar Current (from Kunze et al. 2006 JPO).

 

 

Fig. 20: Near-bottom diffusivity Kb as a function of topographic roughness <H2> and semidiurnal tidal speed variance. Diffusivities appear to be to be higher for higher topographic roughness and tidal speeds but there is considerable scatter (from Kunze et al. 2006 JPO).

The global coverage from these data allows examination of the variability of diffusivity as a function of depth and latitude (Fig. 21, top). Diffusivity K increases with latitude and with depth for latitudes equatorward of 30º. At higher latitudes, diffusivity is independent of depth. A near-bottom peak in diffusivity equatorward of 30 may be a signature of a cascade of energy from largescale semidiurnal internal tides to smallscale near-inertial waves of half the frequency by a nonlinear wave/wave interaction mechanism known as parametric subharmonic instability.

From a vertical advective-diffusive balance (Fig. 1), one can also infer the vertical flow w* through density surfaces due to mixing (Fig. 21, bottom). This is upward everywhere consistent with basin-scale upwelling. It is largest at high latitudes below 3000- and above 2000-m depth with values of 0.5 cm/day. Diapycnal velocities are weak at mid-depth. This suggests 2 meridional overturning circulations, one transforming water from bottom to deep waters and the other transforming water from intermediate to upper ocean waters.

Global-average diffusivities K are 0.1 × 10–4 m2 s–1 in the upper 3000 m, increasing to a few times that in abyssal waters (Fig. 22). Diffusivities inferred from the internal wave parameterization are consistent with those inferred from a global inverse of hydrography on density surfaces shallower than Lower Deep Water but falls short in Lower Deep Antarctic Circumpolar Current and Antarctic Bottom Waters. This may signify sampling problems with the hydrography or turbulence production mechanisms other than internal waves in abyssal waters such as hydraulically controlled flow through passages and internal wave generation, both of which are likely to be important.

 

 

Fig. 21: Average diffusivity K as a function of depth and latitude (upper panel, first column of numbers shows latitude range). Diapycnal vertical velocities w* across density surfaces as a function of depth and latitude (from Kunze et al. 2006 JPO).

 

 

Fig. 22: Profiles of number of estimates i, averaged shear Vz (red) and buoyancy frequency N (blue), eddy diffusivity K, dissipation rate ε, internal wave shear/strain variance ratio Rω and ratio of down/upgoing energy as functions of depth z, height above bottom h, neutral density γn and buoyancy frequency N. Diffusivity and dissipation rate are estimated from both shear-and-strain (blue) and strain-only (red). The black curve and gray stippling in the neutral density diffusivity panel is from an inverse of global hydrography (Lumpkin and Speer 2003 JPO). Global-averaged diffusivities are 0.1 × 10–4 m2 s–1 in the upper 3000 m, increasing to a few times that in the abyss (from Kunze et al. 2006 JPO).

 

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