An energy-constrained profile parameterization of shear-driven turbulence in the interior ocean

This paper proposes an energy-constrained profile parameterization of both turbulent kinetic energy dissipation rate ( ϵ ) and vertical diffusivity ( κ ), for shear instability-induced turbulence that is initiated in an initial unstable layer (IUL) where the gradient Richardson number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="im1"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>i</mml:mi> <mml:mo> </mml:mo> <mml:mo>∈</mml:mo> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo> </mml:mo> <mml:mn>0.25</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . Large-eddy simulation (LES) experiments provide the data of turbulent processes originating from Kelvin-Helmholtz instability of varied initial shear conditions. The energy-constrained framework posits ϵ and κ as proportional to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="im2"> <mml:mrow> <mml:mo> </mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mtext>a</mml:mtext> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="im3"> <mml:mrow> <mml:msup> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mo>–</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="im4"> <mml:mrow> <mml:mo> </mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mtext>a</mml:mtext> </mml:msub> </mml:mrow> </mml:math> represents available kinetic energy, measuring the released kinetic energy, τ denotes turbulence evolution timescale. Both are determinable by the thickness of IUL ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="im5"> <mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> ), buoyancy frequency ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="im6"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> ), vertical shear ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="im7"> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> ), and Richardson number ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="im8"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>i</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> ) of the IUL. Notably, unlike conventional schemes that parameterize turbulent mixing for single model grid point layer by layer, the present scheme parameterizes the turbulent mixing not only for the grid point(s) of IUL, but also for all the model grid points that are within a determined vertical turbulent penetration layer, by providing a profile of diffusivity. Therefore, the scheme is termed the energy-constrained profile parameterization (EPP). EPP aligns well with the LES results and direct microstructure measurements, outperforming existing parameterizations.