Entrainment and mixed layer dynamics of a surface-stress-driven stratified fluid

Abstract

We consider experimentally a linearly stratified fluid with buoyancy frequency N in a cylinder subject to surface-stress forcing from a disc of radius R spinning at a constant angular velocity Ω. We observe the growth of the disc-adjacent turbulent mixed layer bounded by a sharp primary interface with a constant thickness l I . To a good approximation the depth of the mixed layer scales as h U /R ∼ (N/Ω) −2/3 (Ωt) 2/9 . Generalising the previous arguments and observations of Shravat, Cenedese & Caulfield (2012), we show that such a deepening rate is consistent with three central assumptions that allow us to develop a phenomenological energy balance model for the entrainment dynamics. First, the total kinetic energy of the deepening mixed layer K U ∝ h U u 2 U , where u U is a characteristic velocity scale of the turbulent motions within the upper layer, is essentially independent of time and the buoyancy frequency N . Second, the scaled entrainment parameter E =ḣ U /u U , depends only on the local interfacial Richardson number. Third, the potential energy production is driven by the local energy input at the interface, and hence is proportional to the third power of this characteristic velocity u U . We establish that internal consistency between these assumptions implies that the rate of increase of the potential energy decreases with Ri I , suggesting following Phillips (1972), that the mixing in the vicinity of the primary interface leads to the spontaneous appearance of secondary partially-mixed layers which we indeed observe experimentally below the primary interface.

Statistical state dynamics analysis of buoyancy layer formation via the Phillips mechanism in two-dimensional stratified turbulence

“…The present work provides a theoretical underpinning of PI based on SSD, but is not designed to capture alternative mechanisms that may be important in the case of rod-stirred tanks. That a background shear is required to enable layer formation in our system is consistent with the results of previous studies in which layering occurred in the presence of background shear flows (Manucharyan & Caulfield 2015;Lucas et al 2017;Zhou et al 2017). We note that although vertical shear is used in our study, Lucas et al (2017) observed layer formation in simulations in which a body force was used to drive horizontal shear.…”

“…Horizontal layers of density or buoyancy are commonly identified in observations of stratified turbulence in the ocean (Gregg 1980;Duda & Rehmann 2002), the atmosphere (Cho et al 2003), laboratory studies (Manucharyan & Caulfield 2015) and simulations (Lucas, Caulfield & Kerswell 2017;Fitzgerald & Farrell 2018a). Several physical mechanisms have been advanced to explain layering phenomena, including the inertial instability (Ménesguen et al 2009), double-diffusive instability (Radko 2003) and the Phillips instability (PI) (Phillips 1972;Posmentier J. G. Fitzgerald and B. F. Farrell 1977).…”

The Evolution and Arrest of a Turbulent Stratified Oceanic Bottom Boundary Layer over a Slope: Downslope Regime

“…The exponents fall between two limits. For the smallest initial Bu, the convection is weak, and BML growth follows a 2/9 power law, consistent with stress-driven mixed layer growth (Manucharyan and Caulfield 2015). For larger values of Bu, BML growth follows a 1/2 power law, consistent with a classic upright convection-driven mixed layer development (Deardorff et al 1969).…”

Submesoscale Sea Ice‐Ocean Interactions in Marginal Ice Zones

“…The depth of melt‐water layer, H m , depends on the intensity of mixed layer turbulence and is expected to be smaller than H d because of the suppressing effect of stratification on mixing (Ivey & Imberger, ; Kaneda & Ishida, ; Manucharyan & Caulfield, ). Thus, we fix H d = 35 m and keep constant the sea ice melt m resulting in a varying .…”