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Local versus volume-integrated turbulence and mixing in breaking internal waves on slopes

  • Robert S. Arthur (a1) (a2), Jeffrey R. Koseff (a2) and Oliver B. Fringer (a2)

Using direct numerical simulations (DNS), we explore local and volume-integrated measures of turbulence and mixing in breaking internal waves on slopes. We consider eight breaking wave cases with a range of normalized pycnocline thicknesses $k\unicode[STIX]{x1D6FF}$ , where $k$ is the horizontal wavenumber and $\unicode[STIX]{x1D6FF}$ is the pycnocline thickness, but with similar incoming wave properties. The energetics of wave breaking is quantified in terms of local turbulent dissipation and irreversible mixing using the method of Scotti & White (J. Fluid Mech., vol. 740, 2014, pp. 114–135). Local turbulent mixing efficiencies are calculated using the irreversible flux Richardson number $R_{f}^{\ast }$ and are found to be a function of the turbulent Froude number $Fr_{k}$ . Volume-integrated measures of the turbulent mixing efficiency during wave breaking are also made, and are found to be functions of $k\unicode[STIX]{x1D6FF}$ . The bulk turbulent mixing efficiency ranges from 0.25 to 0.37 and is maximized when $k\unicode[STIX]{x1D6FF}\approx 1$ . In order to connect local and bulk mixing efficiency measures, the variation in the bulk turbulent mixing efficiency with $k\unicode[STIX]{x1D6FF}$ is related to the turbulent Froude number at which the maximum total mixing occurs over the course of the breaking event, $Fr_{k}^{max}$ . We find that physically, $Fr_{k}^{max}$ is controlled by the vertical length scale of billows at the interface during wave breaking.

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P. Aghsaee , L. Boegman  & K. G. Lamb 2010 Breaking of shoaling internal solitary waves. J. Fluid Mech. 659, 289317.

R. S. Arthur  & O. B. Fringer 2014 The dynamics of breaking internal solitary waves on slopes. J. Fluid Mech. 761, 360398.

R. S. Arthur  & O. B. Fringer 2016 Transport by breaking internal gravity waves on slopes. J. Fluid Mech. 789, 93126.

C. E. Bluteau , N. L. Jones  & G. N. Ivey 2013 Turbulent mixing efficiency at an energetic ocean site. J. Geophys. Res. 118 (9), 46624672.

L. Boegman , G. N. Ivey  & J. Imberger 2005 The degeneration of internal waves in lakes with sloping topography. Limnol. Oceanogr. 50 (5), 16201637.

D. Bogucki , T. Dickey  & L. G. Redekopp 1997 Sediment resuspension and mixing by resonantly generated internal solitary waves. J. Phys. Oceanogr. 27 (7), 11811196.

G. S. Carter , M. C. Gregg  & R. Lien 2005 Internal waves, solitary-like waves, and mixing on the Monterey Bay shelf. Cont. Shelf Res. 25 (12), 14991520.

Y. J. Chou  & O. B. Fringer 2010 A model for the simulation of coupled flow-bed form evolution in turbulent flows. J. Geophys. Res. 115, C10041.

K. A. Davis  & S. G. Monismith 2011 The modification of bottom boundary layer turbulence and mixing by internal waves shoaling on a barrier reef. J. Phys. Oceanogr. 41 (11), 22232241.

A. Dörnbrack 1998 Turbulent mixing by breaking gravity waves. J. Fluid Mech. 375, 113141.

J. F. Dunckley , J. R. Koseff , J. V. Steinbuck , S. G. Monismith  & A. Genin 2012 Comparison of mixing efficiency and vertical diffusivity models from temperature microstructure. J. Geophys. Res. 117, C10.

J. H. Ferziger  & M. Perić 2002 Solution of the Navier–Stokes equations. In Computational Methods for Fluid Dynamics, pp. 157216. Springer.

O. B. Fringer  & R. L. Street 2003 The dynamics of breaking progressive interfacial waves. J. Fluid Mech. 494, 319353.

A. E. Gargett  & J. N. Moum 1995 Mixing efficiencies in turbulent tidal fronts: results from direct and indirect measurements of density flux. J. Phys. Oceanogr. 25 (11), 25832608.

C. Härtel , F. Carlsson  & M. Thunblom 2000 Analysis and direct numerical simulation of the flow at a gravity-current head. Part 2. The lobe-and-cleft instability. J. Fluid Mech. 418, 213229.

P. Hosegood , J. Bonnin  & H. van Haren 2004 Solibore-induced sediment resuspension in the Faeroe-Shetland channel. Geophys. Res. Lett. 31, L09301.

P. Hosegood  & H. van Haren 2004 Near-bed solibores over the continental slope in the Faeroe-Shetland channel. Deep-Sea Res. II 51 (25), 29432971.

E. L. Hult , C. D. Troy  & J. R. Koseff 2011 The mixing efficiency of interfacial waves breaking at a ridge: 2. Local mixing processes. J. Geophys. Res. 116, C02004.

G. N. Ivey  & J. Imberger 1991 On the nature of turbulence in a stratified fluid. Part i: the energetics of mixing. J. Phys. Oceanogr. 21 (5), 650658.

G. N. Ivey , K. B. Winters  & J. R. Koseff 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40 (1), 169184.

J. M. Klymak  & J. N. Moum 2003 Internal solitary waves of elevation advancing on a shoaling shelf. Geophys. Res. Lett. 30 (20), 2045.

K. G. Lamb 2002 A numerical investigation of solitary internal waves with trapped cores formed via shoaling. J. Fluid Mech. 451, 109144.

K. G. Lamb 2014 Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231254.

J. J. Leichter , S. R. Wing , S. L. Miller  & M. W. Denny 1996 Pulsed delivery of subthermocline water to Conch Reef (Florida Keys) by internal tidal bores. Limnol. Oceanogr. 41 (7), 14901501.

B. P. Leonard 1979 A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Meth. Appl. Mech. Engng 19 (1), 5998.

C. L. Lin , J. H. Ferziger , J. R. Koseff  & S. G. Monismith 1993 Simulation and stability of two-dimensional internal gravity waves in a stratified shear flow. Dyn. Atmos. Oceans 19 (1), 325366.

P. F. Linden  & J. M. Redondo 1991 Molecular mixing in Rayleigh–Taylor instability. Part i: global mixing. Phys. Fluids A 3 (5), 12691277.

B. D. Mater  & S. K. Venayagamoorthy 2014 The quest for an unambiguous parameterization of mixing efficiency in stably stratified geophysical flows. Geophys. Res. Lett. 41 (13), 46464653.

A. D. McEwan 1983a Internal mixing in stratified fluids. J. Fluid Mech. 128, 5980.

A. D. McEwan 1983b The kinematics of stratified mixing through internal wave breaking. J. Fluid Mech. 128, 4757.

H. Michallet  & G. N. Ivey 1999 Experiments on mixing due to internal solitary waves breaking on uniform slopes. J. Geophys. Res. 104 (C6), 1346713477.

C. D. Moore , J. R. Koseff  & E. L. Hult 2016 Characteristics of bolus formation and propagation from breaking internal waves on shelf slopes. J. Fluid Mech. 791, 260283.

J. N. Moum 1996 Efficiency of mixing in the main thermocline. J. Geophys. Res. 101 (C5), 1205712069.

W. Munk  & C. Wunsch 1998 Abyssal recipes ii: energetics of tidal and wind mixing. Deep-Sea Res. 45 (12), 19772010.

N. S. Oakey 1982 Determination of the rate of dissipation of turbulent energy from simultaneous temperature and velocity shear microstructure measurements. J. Phys. Oceanogr. 12 (3), 256271.

M. M. Omand , J. J. Leichter , P. J. Franks , R. T. Guza , A. J. Lucas  & F. Feddersen 2011 Physical and biological processes underlying the sudden appearance of a red-tide surface patch in the nearshore. Limnol. Oceanogr. 56 (3), 787801.

T. R. Osborn 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10 (1), 8389.

T. R. Osborn  & C. S. Cox 1972 Oceanic fine structure. Geophys. Fluid Dyn. 3 (1), 321345.

W. R. Peltier  & C. P. Caulfield 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35 (1), 135167.

J. Pineda 1994 Internal tidal bores in the nearshore: warm-water fronts, seaward gravity currents and the onshore transport of neustonic larvae. J. Mar. Res. 52 (3), 427458.

L. S. Quaresma , J. Vitorino , A. Oliveira  & J. da Silva 2007 Evidence of sediment resuspension by nonlinear internal waves on the western Portuguese mid-shelf. Mar. Geol. 246 (2), 123143.

C. R. Rehmann  & J. R. Koseff 2004 Mean potential energy change in stratified grid turbulence. Dyn. Atmos. Oceans 37 (4), 271294.

A. Scotti  & B. White 2014 Diagnosing mixing in stratified turbulent flows with a locally defined available potential energy. J. Fluid Mech. 740, 114135.

H. E. Seim  & M. C. Gregg 1995 Energetics of a naturally occurring shear instability. J. Geophys. Res. 100 (C3), 49434958.

L. H. Shih , J. R. Koseff , G. N. Ivey  & J. H. Ferziger 2005 Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations. J. Fluid Mech. 525, 193214.

J. E. Simpson 1972 Effects of the lower boundary on the head of a gravity current. J. Fluid Mech. 53 (4), 759768.

D. N. Slinn  & J. J. Riley 1998 Turbulent dynamics of a critically reflecting internal gravity wave. Theor. Comput. Fluid Dyn. 11 (3–4), 281303.

W. D. Smyth , J. N. Moum  & D. R. Caldwell 2001 The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31 (8), 19691992.

C. D. Troy  & J. R. Koseff 2005 The instability and breaking of long internal waves. J. Fluid Mech. 543, 107136.

S. K. Venayagamoorthy  & O. B. Fringer 2007 On the formation and propagation of nonlinear internal boluses across a shelf break. J. Fluid Mech. 577, 137159.

S. K. Venayagamoorthy  & J. R. Koseff 2016 On the flux Richardson number in stably stratified turbulence. J. Fluid Mech. 798, R1.

R. K. Walter , M. E. Squibb , C. B. Woodson , J. R. Koseff  & S. G. Monismith 2014a Stratified turbulence in the nearshore coastal ocean: dynamics and evolution in the presence of internal bores. J. Geophys. Res. 119 (12), 87098730.

R. K. Walter , C. B. Woodson , R. S. Arthur , O. B. Fringer  & S. G. Monismith 2012 Nearshore internal bores and turbulent mixing in southern Monterey Bay. J. Geophys. Res. 117, C07017.

K. B. Winters , P. N. Lombard , J. J. Riley  & E. A. D’Asaro 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115128.

C. Wunsch  & R. Ferrari 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281314.

H. Yamazaki  & T. R. Osborn 1993 Direct estimation of heat flux in a seasonal thermocline. J. Phys. Oceanogr. 23 (3), 503516.

Y. Zang , R. L. Street  & J. R. Koseff 1994 A non-staggered grid, fractional step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates. J. Comput. Phys. 114, 1833.

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