Meltwater transport and mixing-layer growth near the ice–ocean interface

Ice melting into saline water plays a fundamental role in the dynamics near the ice–ocean interface in polar oceans. The physics of meltwater transport involves a non-trivial interplay between the thermodynamics at the interface, hydrodynamic transport in the bulk and the properties of the ambient ocean. The key control parameters are the density ratio upper R Subscript rho <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:msub mml:miR</mml:mi> mml:miρ</mml:mi> </mml:msub> </mml:math> $R_\rho$ , which is proportional to the ambient salinity and measures the balance between the temperature and salinity effects on density, together with the Lewis number italic Le equals kappa Subscript upper T Baseline divided by kappa Subscript upper S Baseline <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:mrow <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Le</mml:mtext> </mml:mrow> mml:mo=</mml:mo> mml:msub mml:miκ</mml:mi> mml:miT</mml:mi> </mml:msub> mml:mrow mml:mo/</mml:mo> </mml:mrow> mml:msub mml:miκ</mml:mi> mml:miS</mml:mi> </mml:msub> </mml:math> $\textit{Le} = \kappa _T/\kappa _S$ , which compares thermal and solutal diffusivities. In quiescent horizontal configurations, increasing the salinity is known to slow down melting, with the melt rate transitioning from subdiffusive to diffusive as upper R Subscript rho <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:msub mml:miR</mml:mi> mml:miρ</mml:mi> </mml:msub> </mml:math> $R_\rho$ increases. Here, we assess the role of turbulence in this transition, using highly resolved numerical simulations of the two-dimensional Boussinesq equations with a slowly melting upper boundary. We analyse the non-stationary growth of the thermal and solutal mixing layers, varying the Lewis number and the density ratio. While meltwater transport is continuously driven by convection within the bulk, we identify a transition from convection to diffusion close to the interface. This transition is reflected by the formation of an interfacial boundary layer that regulates the flux of meltwater pouring into the turbulent bulk. The boundary layer has little effect on thermal transport, but strongly suppresses solutal transport for upper R Subscript rho Baseline greater than or equivalent to 10 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:msub mml:miR</mml:mi> mml:miρ</mml:mi> </mml:msub> mml:mo≳</mml:mo> mml:mn10</mml:mn> </mml:math> $R_\rho \gtrsim 10$ , only allowing a fraction proportional to upper R Subscript rho Superscript negative 1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:mo∝</mml:mo> mml:msubsup mml:miR</mml:mi> mml:mrow mml:miρ</mml:mi> </mml:mrow> mml:mrow mml:mo−</mml:mo> mml:mn1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> $\propto R_{\rho }^{-1}$ of the input flux to reach the bulk. Using mixing-layer diagnostics based on solutal-concentration thresholds, we observe that the turbulent layer grows super-diffusively proportional to t Superscript 1.33 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> mml:mo∝</mml:mo> mml:msup mml:mit</mml:mi> mml:mrow mml:mn1.33</mml:mn> </mml:mrow> </mml:msup> </mml:math> $\propto t^{1.33}$ , while the interfacial boundary layer expands diffusively <jats:inline-graphic xmlns:xlink=