Alfv\'{e}n Cascade modes at high $\beta_{e}$ in the National Spherical Torus Experiment--structure and suppression*

Beam ions and/or fusion alphas are expected to excite Alfv\'{e}n Cascade (AC) modes (i.e. reversed-shear Alfv\'{e}n eigenmodes) in ITER reversed-shear advanced scenarios. The National Spherical Torus eXperiment (NSTX), where fast-ions with comparable v/v$_{Alfv\mbox{\'{e}}n}$ ($\sim $ 2 -- 4) excite ACs, is an ideal device in which to observe ACs and their impact. Its wide range of \textit{$\beta $}$_{e }$(ratio of electron to magnetic pressure) enables tests of AC theory up to, and beyond, a critical \textit{$\beta $}$_{e}$ where suppression is predicted. A value for critical \textit{$\beta $}$_{e}$, $\sim $ [4$q_{min}^{2}{\rm o}$1+(7/4)(T$_{i}$/T$_{e}))$]$^{-1}$, may be derived from the theory of Breizman, et al. [\textit{Phys. Plasmas }\textbf{\textit{12}}\textit{ (2005) 112506}]. Observations of suppression and frequency evolution in NSTX, including onset and saturation, agree well with this theory and calculations by the NOVA-K linear stability code. The dependence of AC frequency on minimum safety factor ($q_{min})$ enables a sensitive determination of $q_{min}$ from the AC spectrum that agrees well with the minimum of the $q$ profile measured using the motional Stark effect. AC structure measurements near critical \textit{$\beta $}$_{e}$ from three fixed frequency (i.e. spatially localized) reflectometers and three tangential interferometers show a structure consistent with predicted localization near the $q_{min}$ radius. Magnetic measurements indicate shear-wave polarization at $q_{min}$. Fast-ion response is monitored with neutral particle analyzers, a fast lost ion probe and neutron detectors. Profile measurements of $q$, density, electron and ion temperature, and rotation are used by NOVA-K to predict mode structure and frequency, or suppression, for direct comparison with the mode measurements. These novel observations of ACs near critical \textit{$\beta $}$_{e}$ are well explained by theory, allowing us to extrapolate our understanding of this physics with confidence.