In fresh water below the temperature of maximum density, the equation of state for density is a nonlinear function of temperature. Moreover, the maximum possible density differences are less than $0.1\mathrm{kg}{\mathrm{m}}^{-3}$. This implies that in the late winter and early spring most mid to high latitude lakes are in a regime of very weak buoyancy effects. We perform three-dimensional direct numerical simulations (DNS) to ascertain how the nonlinearity of the equation of state in such situations impacts the onset of shear instability and transition. We find that regardless of the smallness of the buoyancy term, accounting for buoyancy is essential in setting the timing and scale of three-dimensionalization. The nonlinearity of the equation of state manifests in setting the details of transition and mixing characteristics at moderate shear-layer based Reynolds number ($\text{Re}=60$), while for weak Reynolds number ($\text{Re}=45$) ignoring nonlinearity can change whether instability occurs at all. At Reynolds numbers greater than about 120, inertia dominates and the effects of both the nonlinear equation of state and the buoyancy term are minimized. We explore the effect of the commonly made approximation of setting the Prandtl Number ($\text{Pr}=\nu /\kappa$) to be 1 instead of the physical value of 12 in cold water. We observe that this common, though unnecessary, approximation also leads to a different onset and time evolution of the instability, and hence should be avoided in simulations of the cold water regime.

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• Kelvin-Helmholtz instability
• Buoyancy
• Direct numerical simulations