Deformation and Axes of Dilatation

Resultant Deformation: In a Cartesian coordinate system, there are two types of deformation: stretching (DEFstr) and shearing (DEFshr) deformation. But these properties of the airflow depend on the orientation of the coordinate system. By combining them into resultant deformation (DEFres) according to (DEFres)^2 = (DEFstr)^2 + (DEFshr)^2, the magnitude of the resultant deformation is independent of the coordinate system - it is an invariant property of the airflow. The orientation angle (THETA) of the axis of dilatation is found by solving the equation: tan (2*THETA) = DEFshr/DEFstr for THETA. In regions of strong DEFres, contraction is likely (unless it is being offset by divergence - see instantaneous contraction rate), which may lead to frontogenesis.

Axis of Dilatation: A line along which deformation of the airflow is causing the maximum stretching of air parcels. The axis of contraction is perpendicular to the axis of dilatation, so in a situation involving frontogenesis, the contours of some property of the air, such as temperature of mixing ratio, will tend to line up roughly parallel to the axis of dilatation. When the axis of dilatation is aligned with the contours, frontogenesis is likely. The larger the length of the axis of dilatation, the larger the magnitude of the resultant deformation.

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.