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 $\mathit{\text{Ultrafilters}}\mathit{\text{on}}\omega$, is empty. We translate the result to the language of ${<}_{\mathrm{\infty }}$-sequences under an ultrafilter, investigated by C. Laflamme in $\mathit{\text{A}}\mathit{\text{few}}\mathit{\text{special}}\mathit{\text{ordinal}}\mathit{\text{ultrafilters}}$, to show that if there is an arbitrary long finite ${<}_{\mathrm{\infty }}$-sequence under $u$ than $u$ is at least strict $J_{{\omega }^{\omega +1}}$- ultrafilter.