We present a method for detecting and describing features in vector flow fields. Our method models flow fields locally using a linear combination of complex monomials. These monomials form an orthogonal basis for analytic flows with respect to a correlation-based inner product. We investigate the invariance properties of the coefficients of the approximation polynomials under both rotation and scaling operators. We then propose a descriptor for local flow patterns, and developed a method for comparing them invariantly against rigid transformations. Additionally, we propose a SIFT-like detector that can automatically detect singular flow patterns at different scales and orientations. Promising detection results are obtained on different fluid flow data.