Horizontal Velocity Gradient Tensor Magnitude

The kinematic properties of the airflow are those associated with the first order partial derivatives of the horizontal velocity. There are four kinematic properties in Cartesian coordinates because there are two velocity components (u,v) and two coordinate directions (x,y). The magnitude of the tensor formed from these four partial derivatives is a measure of the combined velocity gradients in the airflow. These four derivatives can be combined to produce three scalar quantities that are independent of the coordinate system (i.e., are invariant): the divergence (DIV), the vorticity (VOR), and the resultant deformation (DEFres). The magnitude of the horizontal velocity gradient tensor (MAG) is related to the three invariant kinematic properties according to the following (MAG)^2 is proportional to (DIV)^2 + (VOR)^2 + (DEFres)^2. Strong boundaries in the airflow (wind shift lines) typically are associated with large values for one or more kinematic properties, so this parameter is useful for identifying and tracking significant airflow boundaries. These boundaries may or may not be associated with thermal and/or moisture gradients.

Reference:
Cohen, R. A., and D. M. Schultz, 2005: Contraction rate and its relationship to frontogenesis, the Lyapunov exponent, fluid trapping, and airstream boundaries, Mon. Wea. Rev., 133, 1353-1369.