Cascades, Order and Ultrafilters

We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. Using that relation we prove (ZFC) that the class of strict $J_{\omega^\omega}$-ultrafilters, introduced by J. E. Baumgartner in \textit{Ultrafilters on $\omega$}, is empty. We translate the result to the language of $<_\infty$-sequences under an ultrafilter, investigated by C. Laflamme in \textit{A few special ordinal ultrafilters}, to show that if there is an arbitrary long finite $<_\infty$-sequence under $u$ than $u$ is at least strict $J_{\omega^{\omega+1}}$- ultrafilter.