Knowlpedia

Definition

Radial-tangential velocity covariance

The pair of averaged products transporting azimuthal and axial momentum in the radial direction.

For a real velocity field expressed in cylindrical components and square-integrable in the averaging variables, define its radial-tangential covariance pair by

C(w)=(⟨wrwθ⟩,⟨wrwz⟩).C(w)=\bigl(\langle w_rw_\theta\rangle,\langle w_rw_z\rangle\bigr).

These are two off-diagonal entries of the . The average may include the angular variable and independent auxiliary torus variables, at fixed slow coordinates. Centering is not part of this definition.

Cross term and amplitude weights

Writing wtan=(wθ,wz)w_{\rm tan}=(w_\theta,w_z), expansion gives

C(w+v)=C(w)+B(w,v)+C(v),B(w,v)=⟨wrvtan+vrwtan⟩.C(w+v)=C(w)+B(w,v)+C(v),\qquad B(w,v)=\langle w_rv_{\rm tan}+v_rw_{\rm tan}\rangle.

Thus DC(w)[v]=B(w,v)DC(w)[v]=B(w,v). If fields bjb_j have vanishing cross products and real coefficients aja_j are independent of every averaging variable, then C(∑jajbj)=∑jaj2C(bj)C(\sum_j a_jb_j)=\sum_j a_j^2C(b_j). Coefficients that depend on an averaging variable cannot in general be taken outside its integral.