Does the tracer gradient vector align with the strain eigenvectors in 2D turbulence?

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Abstract
This paper investigates the dynamics of tracer gradient for a two-dimensional flow. More precisely, the alignment of the tracer gradient vector with the eigenvectors of the strain-rate tensor is studied theoretically and numerically. We show that the basic mechanism of the gradient dynamics is the competition between the effects due to strain and an effective rotation due to both the vorticity and to the rotation of the principal axes of the strain-rate tensor. A nondimensional criterion is derived to partition the flow into different regimes: In the strain dominated regions, the tracer gradient vector aligns with a direction different from the strain axes and the gradient magnitude grows exponentially in time. In the strain-effective rotation compensated regions, the tracer gradient vector aligns with the bisector of the strain axes and its growth is only algebraic in time. In the effective rotation dominated regions, the tracer gradient vector is rotating but is often close to the bisector of the strain axes. A numerical simulation of 2D (two-dimensional) turbulence clearly confirms the theoretical preferential directions in strain and effective rotation dominated regions. Effective rotation can be dominated by the rotation rate of the strain axes, and moreover, proves to be larger than strain rate on the periphery of vortices. Taking into account this term allows us to improve significantly the Okubo–Weiss criterion. Our criterion gives the correct behavior of the growth of the tracer gradient norm for the case of axisymmetric vortices for which the Okubo–Weiss criterion fails.