Vertical modes

Standard vertical modes, , \({\mathsf{p}}_n (z)\), are the eigenvectors of the “stretching matrix”

\[{\mathsf{S}}\,{\mathsf{p}}_n = -R_n^{-2}\, {\mathsf{p}}_n\,,\]

where the \(R_n\) is by definition the n’th deformation radius (e.g., Flierl 1978). These orthogonal modes \({\mathsf{p}}_n\) are normalized to have unitary \(L2\)-norm

\[\frac{1}{H}\int_{-H}^{0} {\mathsf{p}}_n {\mathsf{p}}_m {{\rm d}}z = \delta_{nm}{\, ,}\]

where \(\delta_{mn}\).