Abstract
Extending earlier studies by the authors, novel aspects of some geometric invariants associated with the Cartesian 2D velocity gradient tensor are explored without restriction to non-divergent flow. Four quadratic invariants are considered in particular: the determinant of the tensor, the square of its Frobenius norm, the square of the resultant deformation, and the sum of the squares of divergence and vorticity. The determinant is the Jacobian of the flow components with respect to the corresponding Cartesian coordinates. For non-divergent flow, this Jacobian is proportional to the Okubo-Weiss parameter of dynamical oceanography (but of opposite sign). Even for 2D flow having divergence as well as vorticity, the Jacobian is the divergence of a vector field (in our Cartesian geometry), and we show that it can be written compactly in terms of isotach and isogon gradients. Any two of the above four invariants can be expressed in terms of the other two. Under formal interchange of velocity components, the Frobenius norm is unchanged, the sign of the Jacobian is reversed, and the other two invariants transform into each other. Various ways of establishing geometric invariance are discussed. We observe that vorticity is not an invariant of the 2D velocity gradient tensor in the usual mathematical sense, but of a tensor obtained from it by a simple orthogonal tensor multiplication.