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Behavior and mixing of a cold intermediate layer near a sloping boundary


As in many other subarctic basins, a cold intermediate layer (CIL) is found during ice-free months in the Lower St. Lawrence Estuary (LSLE), Canada. This study examines the behavior of the CIL above the sloping bottom using a high-resolution mooring deployed on the northern side of the estuary. Observations show successive swashes/backwashes of the CIL on the slope at a semi-diurnal frequency. It is shown that these upslope and downslope motions are likely caused by internal tides generated at the nearby channel head sill. Quantification of mixing from 322 turbulence casts reveals that in the bottom 10 m of the water column, the time-average dissipation rate of turbulent kinetic energy is 𝜖 10 m = 1.6×10−7Wkg−1, an order of magnitude greater than found in the interior of the basin, far from boundaries. Near-bottom dissipation during the flood phase of the M2 tide cycle (upslope flow) is about four times greater than during the ebb phase (downslope flow). Bottom shear stress, shear instabilities, and internal wave scattering are considered as potential boundary mixing mechanisms near the seabed. In the interior of the water column, far from the bottom, increasing dissipation rates are observed with both increasing stratification and shear, which suggests some control of the dissipation by the internal wave field. However, poor fits with a parametrization for large-scale wave-wave interactions suggests that the mixing is partly driven by more complex non-linear and/or smaller scale waves.

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  • Baker MA, Gibson CH (1987) Sampling turbulence in the stratified ocean: statistical consequence of strong intermittency. J Phys Oceanogr 17:1817–1838

    Article  Google Scholar 

  • Becherer JK, Umlauf L (2011) Boundary mixing in lakes: 1. Modeling the effect of shear-induced convection. J Geophys Res 116(C10):1–15. doi:10.1029/2011JC007119

    Article  Google Scholar 

  • Bouffard D, Boegman L, Rao YR (2012) Poincaré wave-induced mixing in a large lake. Limnol Oceanogr 57(4):1201–1216. doi:10.4319/lo.2012.57.4.1201

    Article  Google Scholar 

  • Bourgault D, Cyr F, Galbraith PS, Pelletier E (2012) Relative importance of pelagic and sediment respiration in causing hypoxia in a deep estuary. J Geophys Res 117 (C08033). doi:10.1029/2012JC007902

  • Burchard H (2009) Combined effects of wind, tide, and horizontal density gradients on stratification in estuaries and coastal seas. J Phys Oceanogr 39(9):2117–2136. doi:10.1175/2009JPO4142.1

    Article  Google Scholar 

  • Carter GS, Gregg MC, Lien RC (2005) Internal waves, solitary-like waves, and mixing on the Monterey Bay shelf. Cont Shelf Res 25(12-13):1499–1520. doi:10.1016/j.csr.2005.04.011

    Article  Google Scholar 

  • Cushman-Roisin B, Beckers JM (2011) Introduction to geophysical fluid dynamics: physical and numerical aspects. Academic Press, Amsterdam

    Google Scholar 

  • Cyr F, Bourgault D, Galbraith PS (2011) Interior versus boundary mixing of a cold intermediate layer. J Geophys Res 116(C12029). doi:10.1029/2011JC007359

  • El-Sabh MI (1979) The Lower St. Lawrence Estuary as a physical oceanographic system. Le Naturaliste Canadien 106:55–73

    Google Scholar 

  • Fer I, Widell K (2007) Early spring turbulent mixing in an ice-covered Arctic fjord during transition to melting. Cont Shelf Res 27(15):1980–1999. doi:10.1016/j.csr.2007.04.003

    Article  Google Scholar 

  • Forrester WD (1970) Geostrophic approximation in the St. Lawrence Estuary. Tellus XXII(1):53–65

    Article  Google Scholar 

  • Forrester WD (1974) Internal tides in the St. Lawrence Estuary. J Mar Res 32(1):55–66

    Google Scholar 

  • Galbraith PS, Chassé J, Gilbert D, Larouche P, Caverhill C, Lefaivre D, Brickman D, Pettigrew B, Devine L, Lafleur C (2014) Physical oceanographic conditions in the Gulf of St. Lawrence in 2013. DFO Canadian Science Advisory Secretariat 2014/062:vi +:84

  • Galbraith PS (1992) Relating overturns to mixing and buoyancy flux. PhD thesis. Dalhousie University

  • Galbraith PS (2006) Winter water masses in the Gulf of St. Lawrence. J Geophys Res 111(C6). doi:10.1029/2005JC003159

  • Garrett C, MacCready P, Rhines P (1993) Boundary mixing and arrested Ekman layers: rotating stratified flow near a sloping boundary. Ann Rev Fluid Mech 25 (1):291–323. doi:10.1146/annurev.fl.25.010193.001451

    Article  Google Scholar 

  • Geyer WR, Smith JD (1987) Shear instability in a highly stratified estuary. J Phys Oceanogr 17:1668–1679

    Article  Google Scholar 

  • Geyer WR, Trowbridge JH, Bowen MM (2000) The dynamics of a partially mixed estuary. J Phys Oceanogr 30(2000):2035– 2048

    Article  Google Scholar 

  • Geyer WR, Scully ME, Ralston DK (2008) Quantifying vertical mixing in estuaries. Env Fluid Mech 8:495–509. doi:10.1007/s10652-008-9107-2

    Article  Google Scholar 

  • Geyer WR, Lavery AC, Scully ME, Trowbridge JH (2010) Mixing by shear instability at high Reynolds number. Geophys Res Lett 37(22):L22,607. doi:10.1029/2010GL045272

    Google Scholar 

  • Gilbert D, Pettigrew B (1997) Interannual variability (1948–1994) of the CIL core temperature in the Gulf of St. Lawrence. Can J Fish Aquat Sci 54(S1):57–67. doi:10.1139/cjfas-54-S1-57

    Google Scholar 

  • Godin G (1979) La maree dans le golfe et l’estuaire du Saint-Laurent́. Le Naturaliste Canadien 106:105–121

    Google Scholar 

  • Gregg M, Alford M, Kontoyiannis H, Zervakis V, Winkel D (2012) Mixing over the steep side of the Cycladic Plateau in the Aegean Sea. J Mar Syst 89(1):30–47. doi:10.1016/j.jmarsys.2011.07.009

    Article  Google Scholar 

  • Holtermann PL, Umlauf L, Tanhua T, Schmale O, Rehder G, Waniek JJ (2012) The Baltic Sea tracer release experiment: 1. Mixing rates. J Geophys Res 117(C1):1–17. doi:10.1029/2011JC007445

    Google Scholar 

  • Howard LN (1961) Note on a paper of John W. Miles. J Fluid Mech 10(4):509–512

    Article  Google Scholar 

  • Ingram RG (1979) Internal wave observation off Isle Verte. J Mar Res 36:715–724

    Google Scholar 

  • Kay DJ (2003) Interfacial mixing in a highly stratified estuary 1. Characteristics of mixing. J Geophys Res 108(C3):3072. doi:10.1029/2000JC000252

    Article  Google Scholar 

  • Koutitonsky VG, Bugden GL (1991) The physical oceanography of the Gulf of St. Lawrence: a review with emphasis on the synoptic variability of the motion. In: Therriault JC (ed) The Gulf of St. Lawrence: small ocean or big estuary? Canadian Special Publication of Fisheries and Aquatic Sciences, vol 113, pp 57–90

  • van der Lee EM, Umlauf L (2011) Internal wave mixing in the Baltic Sea: Near-inertial waves in the absence of tides. J Geophys Res 116(C10):1–16. doi:10.1029/2011JC007072

    Google Scholar 

  • Lie HJ, El-Sabh MI (1983) Formation of eddies and transverse currents in a two-layer channel of variable bottom with application to the Lower St. Lawrence Estuary. J Phys Oceanogr 13:1063–1075

    Article  Google Scholar 

  • MacKinnon JA, Gregg MC (2003) Mixing on the late-summer New England shelf—solibores, shear, and stratification. J Phys Oceanogr 33(7):1476–1492. doi:10.1175/1520-0485(2003)033<1476:MOTLNE>2.0.CO;2

    Article  Google Scholar 

  • MacKinnon JA, Gregg MC (2005) Spring mixing: turbulence and internal waves during restratification on the New England shelf. J Phys Oceanogr 35:2425–2443

    Article  Google Scholar 

  • Mertz G, Gratton Y (1990) Topographic waves and topographically induced motions in the St. Lawrence estuary. In: El-Sabh M I, Silverberg N (eds) Oceanography of a large-scale estuarine system: The St. Lawrence. chap, 5, vol 39. Springer, New-York , pp 94–108

    Google Scholar 

  • Miles JW (1961) On the stability of heterogeneous shear flows. J Fluid Mech 10(4):496–508

    Article  Google Scholar 

  • Moum JN, Caldwell DR, Nash JD, Gunderson GD (2002) Observations of boundary mixing over the continental slope. J Phys Oceanogr 32(7):2113–2130. doi:10.1175/1520-0485(2002)032<2113:OOBMOT>2.0.CO;2

    Article  Google Scholar 

  • Moum JN, Perlin A, Klymak JM, Levine MD, Boyd T, Kosro P (2004) Convectively driven mixing in the bottom boundary layer. J Phys Oceanogr 34:2189–2202

    Article  Google Scholar 

  • Nepf HM, Geyer WR (1996) Intratidal variations in stratification and mixing in the Hudson estuary. J Geophys Res 101(C5):12,079–12,086

    Article  Google Scholar 

  • Osborn T (1980) Estimates of the local rate of vertical diffusion from dissipation measurements. J Phys Oceanogr 10:83–89

    Article  Google Scholar 

  • Palmer MR, Rippeth TP, Simpson JH (2008) An investigation of internal mixing in a seasonally stratified shelf sea. J Geophys Res 113(C12). doi:10.1029/2007JC004531

  • Perlin A, Moum JN, Klymak JM, Levine MD, Boyd T, Kosro PM (2005) A modified law-of-the-wall applied to oceanic bottom boundary layers. J Geophys Res 110(C10):C10S10. doi:10.1029/2004JC002310

    Google Scholar 

  • Peters H (1997) Observations of stratified turbulent mixing in an estuary: Neap-to-spring variations during high river flow. Estuar Coast Shelf Sci 45(1):69–88. doi:10.1006/ecss.1996.0180

    Article  Google Scholar 

  • Saucier FJ, Chassé J (2000) Tidal circulation and buoyancy effects in the St. Lawrence Estuary. Atmosphere-Ocean 38(4):505–556

    Article  Google Scholar 

  • Schafstall J, Dengler M, Brandt P, Bange H (2010) Tidal-induced mixing and diapycnal nutrient fluxes in the Mauritanian upwelling region. J Geophys Res 115(C10):C10,014. doi:10.1029/2009JC005940

    Article  Google Scholar 

  • Shih LH, Koseff JR, Ivey GN, Ferziger JH (2005) Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations. J Fluid Mech 525:193–214. doi:10.1017/S0022112004002587

    Article  Google Scholar 

  • Slinn DN, Riley JJ (1996) Turbulent mixing in the oceanic boundary layer caused by internal wave reflection from sloping terrain. Dynamics of Atmospheres and Oceans 24:51–62

    Article  Google Scholar 

  • Smith GC, Saucier FJ, Straub D (2006) Formation and circulation of the cold intermediate layer in the Gulf of Saint Lawrence. J Geophys Res 111 (C06011). doi:10.1029/2005JC003017

  • Smyth WD, Moum JN, Caldwell DR (2001) The efficiency of mixing in turbulent patches: Inferences from direct simulations and microstructure observations. J Phys Oceanogr 31:1969–1992

    Article  Google Scholar 

  • Soulsby R (1997) Dynamics of marine sands: a manual for practical applications. Thomas Telford

  • St Laurent LC (2002) Estimating tidally driven mixing in the deep ocean. Geophys Res Lett 29(23):2106. doi:10.1029/2002GL015633

    Article  Google Scholar 

  • Taylor GI (1921) Tidal oscillations in gulfs and rectangular basins. Proc Lond Math Soc 20:148–181

    Google Scholar 

  • Toole JM, Polzin KL, Schmitt RW (1994) Estimates of diapycnal mixing in the abyssal ocean. Science 264(5162):1120–1123

    Article  Google Scholar 

  • Umlauf L, Burchard H (2011) Diapycnal transport and mixing efficiency in stratified boundary layers near sloping topography. J Phys Oceanogr 41:329–346. doi:10.1175/2010JPO4438.1

    Article  Google Scholar 

  • Walter RK, Woodson CB, Arthur RS, Fringer OB, Monismith SG (2012) Nearshore internal bores and turbulent mixing in southern Monterey Bay. J Geophys Res 117(C7):C07,017. doi:10.1029/2012JC008115

    Article  Google Scholar 

  • Wang J, Ingram RG, Mysak LA (1991) Variability of internal tides in the Laurentian Channel. J Geophys Res 96(C9):16,859–16,875. doi:10.1029/91JC01580

    Article  Google Scholar 

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This work was funded by “Le Fonds de recherche du Québec - Nature et technologies,” the Natural Sciences and Engineering Research Council of Canada, the Canada Foundation for Innovation and Fisheries and Oceans Canada and is a contribution to the scientific program of Québec-Océan. The authors would like to thank Rémi Desmarais and Paul Nicot who were frequent crew members during our summer sampling campaigns, Leo Maas for his help with the derivation of the Poincaré wave equations, and Cédric Chavanne, Luc Rainville and two anonymous reviewers who provided valuable comments to improve this manuscript.

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Correspondence to Frédéric Cyr.

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Appendix: Analytical solutions for the vertical modal structure of isopycnal displacements

Appendix: Analytical solutions for the vertical modal structure of isopycnal displacements

In this Appendix, we re-derive the analytical expression for the vertical modal structure of the Poincaré wave for the isopycnal displacements η 0(z) given a analytical density profile (Eq. 6).

1.1 A1 Problem formulation

For a certain vertical mode n, η 0 is the solution of the ordinary differential equation (ODE) given by (e.g., Cushman-Roisin and Beckers, 2011):

$$ \frac{d^{2}\eta_{0}}{dz^{2}} + \left(\frac{N^{2}-\sigma^{2}}{{c_{n}^{2}}}\right)\eta_{0}=0, $$

where \(N^{2}=\frac {g}{\rho }\frac {\partial \rho }{\partial z} = \frac {gd}{(z+h)^{2}}\) given the density profile imposed by Eq. 6, and c n the phase velocity that, as we will show later, depends on the vertical mode n. If we let \(\nu ^{2} = \frac {gd}{{c_{n}^{2}}}\), and neglecting σ 2 because N 2>>σ 2, Eq. A1 can be re-written:

$$ \frac{d^{2}\eta_{0}}{dz^{2}} + \frac{\nu^{2}}{(z+h)^{2}}\eta_{0}=0. $$

1.2 A2 Simplifications and general solution of the problem

To resolve the problem, we start by introducing two successive variable changes in order to scale the equation. We first introduce the non-dimensional parameter ξ such as z=ξ h (\(d\xi = \frac {1}{h}dz\)). Then, if we let ζ=ξ+1 (d ζ=d ξ), the ODE becomes:

$$ \frac{d^{2}\eta(z)}{d\zeta^{2}} + \frac{\nu^{2}}{\zeta^{2}}\eta=0. $$

We now assume that the solution has the form η 0(ζ)=A(ζ)B(ζ). After applying the derivative rules, the ODE is now:

$$\begin{array}{@{}rcl@{}} &A\frac{d^{2}B}{d\zeta^{2}} + 2A_{\zeta}\frac{dB}{d\zeta} + A_{\zeta\zeta}B + \frac{\nu^{2}}{\zeta^{2}}AB = 0\\ &\frac{d^{2}B}{d\zeta^{2}} + \frac{2A_{\zeta}}{A}\frac{dB}{d\zeta} + B\left[\frac{A_{\zeta\zeta}}{A} + \frac{\nu^{2}}{\zeta^{2}}\right] = 0. \end{array} $$

where subscripts to variable A stand for derivative relative to ζ. The passage from the first to the second line of Eq. A4 was made by multiplying by 1/A.

We now choose A=ζ 1/2. Knowing that:

$$\begin{array}{@{}rcl@{}} A_{\zeta} &= &\frac{1}{2\zeta^{1/2}}, \\ A_{\zeta\zeta} &=& \frac{-1}{4\zeta^{3/2}}, \\ \frac{A_{\zeta}}{A} &=& \frac{1}{2\zeta} ~~\text{and} \\ \frac{A_{\zeta\zeta}}{A} &=& \frac{-1}{4\zeta^{2}}, \end{array} $$

and by multiplying Eq. A4 by ζ 2, the ODE becomes:

$$ \zeta^{2}\frac{d^{2}B}{d\zeta^{2}} + \zeta\frac{dB}{d\zeta} + B\left[\nu^{2}-\frac{1}{4}\right] = 0. $$

We now introduce a new variable change s= ln(ζ) (thus ζ=e S) which results in the following derivation rules:

$$\begin{array}{@{}rcl@{}} \frac{ds}{d\zeta} &=& \frac{1}{\zeta} = e^{-s}, \\ \frac{d}{d\zeta} &=& e^{-s}\frac{d}{ds}~~ \text{and} \\ \frac{d^{2}}{d\zeta^{2}} &=& e^{-2s}\left[\frac{d^{2}}{ds^{2}} - \frac{d}{ds}\right]. \end{array} $$

With this variable change, the equation is:

$$ \frac{d^{2}B}{ds^{2}} + \left[\nu^{2}-\frac{1}{4}\right]B = 0. $$

This last expression is a second-order homogeneous differential equation with constant coefficients that can be resolved with standard procedures (see any ODE textbook). The characteristic equation for this equation is λ 2 + μ2=0, where \({\upmu }^{2} = \nu ^{2}-\frac {1}{4}\). Since μ2 is positive by construction (this can be easily verified later), the characteristic equation leads to complex roots λiμ. In this case, the analytical solution for the variable B has the form:

$$ B = C_{1}\cos(\upmu s) + C_{2} \sin(\upmu s), $$

where C 1 and C 2 are constant coefficients to be determined.

1.3 A3 Specific solutions given the boundary conditions

Recalling that η 0(ζ)=A(ζ)B(ζ) and A(ζ)=ζ 1/2, we can now use the boundary conditions on η 0 to find the specific solution of our problem. Because displacements are vertically limited by the seafloor and the surface, the boundary conditions are η 0(z=0)=0 and η 0(z=H) = 0, where H is the total depth. Given the variables changes made in the preceding, these boundary conditions imply B(s=0)=0 (i.e., z=0→ζ=1→s=0) and \(Bs=\ln\left({H}/{h}+1\right)=0\) (i.e., \(z = H \rightarrow \zeta =\frac {H}{h}+1 \rightarrow s=\ln \left (\frac {H}{h}+1\right )\)).

The first boundary condition (at the surface) implies that C 1 = 0, while the second condition implies that the argument under the sine must be of the form μs=n π, with n=1,2,..., an integer corresponding to the nth vertical mode. Substituting \(s=\ln {\left (\frac {H}{h}+1\right )}\) in the preceding, a necessary condition is that:

$$ \mu = \frac{n\pi}{\ln{\left(\frac{H}{h}+1\right)}}. $$

After replacing all terms by their expressions in z and taking into consideration these boundary conditions, Eq. A9 becomes:

$$ \eta_{0}(z) = \eta_{n}\left(\frac{z}{h}+1\right)^{1/2}\sin \left[\frac{\ln{\left(\frac{z}{h}+1\right)}}{\ln{\left(\frac{H}{h}+1\right)}}n\pi \right]. $$

Note that in the last expression, C 2 has been replaced by η n for clarity and is the parameter that carries the dimension (m) of η 0. This constant parameter is determined by fitting Eq. A11 to observations. Note also that Eq. A11 is slightly different than the one presented in Forrester (1974) (its Eq. 10), but by carefully adjusting the constant parameters in Forrester’s equation, we can show that both equations give the same structure.

Another consequence of the second boundary condition (Eq. A10) is that it gives conditions on the phase velocity (c n ) of the admissible Poincaré waves. Replacing μ by its expression in z this equation leads to:

$$ {c_{n}^{2}} = \frac{gd}{\left( \frac{n\pi}{\ln\left( \frac{H}{h}+1 \right)} \right)^{2} + \frac{1}{4}}. $$

This expression is necessary for the dispersion relation presented in Eq. 5.

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Cyr, F., Bourgault, D. & Galbraith, P.S. Behavior and mixing of a cold intermediate layer near a sloping boundary. Ocean Dynamics 65, 357–374 (2015).

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  • Turbulence
  • Boundary mixing
  • Cold intermediate layer
  • Lower St. Lawrence Estuary
  • Internal tides
  • Internal wave
  • Shear instabilities
  • Bottom shear stress