Spin vectors and rates of change of wind direction

Abstract

AbstractEstablished formulae for various rates of change of horizontal wind direction may be unified by a treatment in terms of the spin vector, S, of a vector field M. The spin vector measures the rate of change of direction of M, in the appropriate osculating plane, and may be defined with respect to any infinitesimal displacement in space and time. When M is a horizontal vector field, the osculating plane is horizontal, and S immediately gives the rate of change of direction relative to a fixed horizontal axis. The formulae involved are those of Hide and of Bryden for the rate of change of geostrophic wind direction with height, and of Burk and Staley and of Lecluyse and Neumann for the Lagrangian rate of change of wind direction. When M is a 3‐dimensional vector field, the orientation of the osculating plane is not in general constant and the Cartesian components of S do not represent the rate of change of direction in the most concise terms. The Lagrangian spin vector of the 3‐dimensional velocity field may nevertheless be considered as a basic kinematic quantity, on a par with the rates of change of kinetic energy, vorticity and divergence.