Skip to main content Accessibility help
Hostname: page-component-cf9d5c678-9z9qw Total loading time: 0.276 Render date: 2021-08-05T05:23:40.541Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Mixing and entrainment are suppressed in inclined gravity currents

Published online by Cambridge University Press:  28 June 2019

Maarten van Reeuwijk
Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
Markus Holzner
Institute of Environmental Engineering, ETH Zürich, CH-8039 Zürich, Switzerland
C. P. Caulfield
BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK


We explore the dynamics of inclined temporal gravity currents using direct numerical simulation, and find that the current creates an environment in which the flux Richardson number $\mathit{Ri}_{f}$ , gradient Richardson number $\mathit{Ri}_{g}$ and turbulent flux coefficient $\unicode[STIX]{x1D6E4}$ are constant across a large portion of the depth. Changing the slope angle $\unicode[STIX]{x1D6FC}$ modifies these mixing parameters, and the flow approaches a maximum Richardson number $\mathit{Ri}_{max}\approx 0.15$ as $\unicode[STIX]{x1D6FC}\rightarrow 0$ at which the entrainment coefficient $E\rightarrow 0$ . The turbulent Prandtl number remains $O(1)$ for all slope angles, demonstrating that $E\rightarrow 0$ is not caused by a switch-off of the turbulent buoyancy flux as conjectured by Ellison (J. Fluid Mech., vol. 2, 1957, pp. 456–466). Instead, $E\rightarrow 0$ occurs as the result of the turbulence intensity going to zero as $\unicode[STIX]{x1D6FC}\rightarrow 0$ , due to the flow requiring larger and larger shear to maintain the same level of turbulence. We develop an approximate model valid for small $\unicode[STIX]{x1D6FC}$ which is able to predict accurately $\mathit{Ri}_{f}$ , $\mathit{Ri}_{g}$ and $\unicode[STIX]{x1D6E4}$ as a function of $\unicode[STIX]{x1D6FC}$ and their maximum attainable values. The model predicts an entrainment law of the form $E=0.31(\mathit{Ri}_{max}-\mathit{Ri})$ , which is in good agreement with the simulation data. The simulations and model presented here contribute to a growing body of evidence that an approach to a marginally or critically stable, relatively weakly stratified equilibrium for stratified shear flows may well be a generic property of turbulent stratified flows.

JFM Papers
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Bak, P., Tang, C. & Wiesenfeld, K. 1987 Self-organized criticality – an explanation of 1/f noise. Phys. Rev. Lett. 59, 381384.CrossRefGoogle ScholarPubMed
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 16451651.CrossRefGoogle Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343368.CrossRefGoogle Scholar
Caulfield, C. C. P. & Woods, A. W. 1995 Plumes with nonmonotonic mixing behaviour. Geophys. Astrophys. Fluid Dyn. 79, 173199.CrossRefGoogle Scholar
Caulfield, C. P. & Woods, A. W. 1998 Turbulent gravitational convection from a point source in a non-uniformly stratified environment. J. Fluid Mech. 360, 229248.CrossRefGoogle Scholar
Da Silva, C. B., Hunt, J. C. R., Eames, I. & Westerweel, J. 2014 Interfacial layers between regions of different turbulence intensity. Annu. Rev. Fluid Mech. 46, 567590.CrossRefGoogle Scholar
Deusebio, E., Caulfield, C. P. & Taylor, J. R. 2015 The intermittency boundary in stratified plane Couette flow. J. Fluid Mech. 781, 298329.CrossRefGoogle Scholar
Ellison, T. H. 1957 Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech. 2, 456466.CrossRefGoogle Scholar
Ellison, T. H. & Turner, J. S. 1959 Turbulent entrainment in stratified flows. J. Fluid Mech. 6, 423448.CrossRefGoogle Scholar
Falder, M., White, N. J. & Caulfield, C. P. 2016 Seismic imaging of rapid onset of stratified turbulence in the south Atlantic Ocean. J. Phys. Oceanogr. 46 (4), 10231044.CrossRefGoogle Scholar
Fedorovich, E. & Shapiro, A. 2009 Structure of numerically simulated katabatic and anabatic flows along steep slopes. Acta Geophys. 57 (4), 9811010.CrossRefGoogle Scholar
Ferrari, R., Mashayek, A., McDougall, T. J., Nikurashin, M. & Campin, J. M. 2016 Turning ocean mixing upside down. J. Phys. Oceanogr. 46, 22392261.CrossRefGoogle Scholar
Ferrari, R. & Wunsch, C. 2009 The distribution of eddy kinetic and potential energies in the global ocean. Annu. Rev. Fluid Mech. 41, 253282.CrossRefGoogle Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10 (4), 509512.CrossRefGoogle Scholar
Hunt, G. R. & Kaye, N. G. 2001 Virtual origin correction for lazy plumes. J. Fluid Mech. 435, 377396.CrossRefGoogle Scholar
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169184.CrossRefGoogle Scholar
Krug, D., Holzner, M., Lüthi, B., Wolf, M., Kinzelbach, W. & Tsinober, A. 2013 Experimental study of entrainment and interface dynamics in a gravity current. Exp. Fluids 54, 1530.CrossRefGoogle Scholar
Krug, D., Holzner, M., Lüthi, B., Wolf, M., Kinzelbach, W. & Tsinober, A. 2015 The turbulent/non-turbulent interface in an inclined dense gravity current. J. Fluid Mech. 765, 303324.CrossRefGoogle Scholar
Krug, D., Holzner, M., Marusic, I. & van Reeuwijk, M. 2017 Fractal scaling of the turbulence interface in gravity currents. J. Fluid Mech. 820, R3.CrossRefGoogle Scholar
Lighthill, M. 1950 Contributions to the theory of heat transfer through a laminar boundary layer. Proc. R. Soc. Lond. A 202, 350377.Google Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207242.CrossRefGoogle Scholar
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 323.CrossRefGoogle Scholar
Mater, B. D. & Venayagamoorthy, S. K. 2014 A unifying framework for parameterizing stably stratified shear-flow turbulence. Phys. Fluids 26, 036601.CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10 (4), 496508.CrossRefGoogle Scholar
Monin, A. S. & Obukhov, A. M. 1954 Basic laws of turbulent mixing in the surface layer of the atmosphere. Contrib. Geophys. Inst. Acad. Sci. USSR 151, 163187.Google Scholar
Morton, B. R. & Middleton, J. 1973 Scale diagrams for forced plumes. J. Fluid Mech. 58, 165176.CrossRefGoogle Scholar
Nieuwstadt, F. T. M. 1984 The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci. 41 (14), 22022216.2.0.CO;2>CrossRefGoogle Scholar
Obukhov, A. M. 1971 Turbulence in an atmosphere with a non-uniform temperature. Boundary-Layer Meterol. 2, 729.CrossRefGoogle Scholar
Odier, P., Chen, J. & Ecke, R. E. 2014 Entrainment and mixing in a laboratory model of oceanic overflow. J. Fluid Mech. 746, 498535.CrossRefGoogle Scholar
Odier, P., Chen, J., Rivera, M. K. & Ecke, R. E. 2009 Fluid mixing in stratified gravity currents: The Prandtl mixing length. Phys. Rev. Lett. 102, 134504.CrossRefGoogle ScholarPubMed
Osborn, T. R. 1980 Estimates of the local-rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 8389.2.0.CO;2>CrossRefGoogle Scholar
van Reeuwijk, M., Jonker, H. J. J. & Hanjalić, K. 2008 Wind and boundary layers in Rayleigh–Bénard convection. I. Analysis and modelling. Phys. Rev. E 77, 036311.Google Scholar
van Reeuwijk, M., Krug, D. & Holzner, M. 2018 Small-scale entrainment in inclined gravity currents. Environ. Fluid Mech. 18 (1), 225239.CrossRefGoogle ScholarPubMed
Salehipour, H. & Peltier, W. R. 2015 Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence. J. Fluid Mech. 775, 464500.CrossRefGoogle Scholar
Salehipour, H., Peltier, W. R. & Caulfield, C. P. 2018 Self-organised criticality of turbulence in strongly stratified mixing layers. J. Fluid Mech. 856, 228256.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory. Springer.CrossRefGoogle Scholar
Scotti, A. 2015 Biases in Thorpe-scale estimates of turbulence dissipation. Part II. Energetics arguments and turbulence simulations. J. Geophys. Res. 45, 25222543.Google Scholar
Sher, D. & Woods, A. W. 2015 Gravity currents: entrainment, stratification and self-similarity. J. Fluid Mech. 784, 130162.CrossRefGoogle Scholar
Shih, L. H., Koseff, J. R., Ivey, G. N. & Ferziger, J. H. 2005 Parameterization of turbulent fluxes and scales using homogenous sheared stably stratified turbulence simulations. J. Fluid Mech. 525, 193214.CrossRefGoogle Scholar
Simpson, J. E. 1999 Gravity Currents. Cambridge University Press.Google Scholar
Smyth, W. D. & Moum, J. N. 2013 Marginal instability and deep cycle turbulence in the eastern equatorial pacific occean. Geophys. Res. Lett. 40, 61816185.CrossRefGoogle Scholar
Spalding, D. 1954 Mass transfer in laminar flow. Proc. R. Soc. Lond. A 221, 7899.Google Scholar
Tailleux, R. 2013 Available potential energy and exergy in stratified fluids. Annu. Rev. Fluid Mech. 45, 3558.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Thorpe, S. A. & Liu, Z. 2009 Marginal instability? J. Phys. Oceanogr. 39 (9), 23732381.CrossRefGoogle Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431471.CrossRefGoogle Scholar
Verstappen, R. W. C. P. & Veldman, A. E. P. 2003 Symmetry-preserving discretization of turbulent flow. J. Comput. Phys. 187 (1), 343368.CrossRefGoogle Scholar
Wahlin, A. K. & Cenedese, C. 2006 How entraining density currents influence the stratification in a one-dimensional ocean basin. Deep Sea Res. II 53, 172193.CrossRefGoogle Scholar
Wells, M., Cenedese, C. & Caulfield, C. P. 2010 The relationship between flux coefficient and entrainment ratio in density currents. J. Phys. Oceanogr. 40 (12), 27132727.CrossRefGoogle Scholar
Wells, M. G. & Wettlaufer, J. S. 2005 Two-dimensional density currents in a confined basin. Geophys. Astrophys. Fluid Mech. 99, 199218.CrossRefGoogle Scholar
Zhou, Q., Taylor, J. R. & Caulfield, C. P. 2017 Self-similar mixing in stratified plane Couette flow for varying Prandtl number. J. Fluid Mech. 820, 86120.CrossRefGoogle Scholar
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Mixing and entrainment are suppressed in inclined gravity currents
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Mixing and entrainment are suppressed in inclined gravity currents
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Mixing and entrainment are suppressed in inclined gravity currents
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *