  In this work we analyze the longitudinal instabilities of propagating
acceleration structures that are driven by a relativistically intense laser at
the moving plasma critical layer [1]. These instabilities affect the
energy-spectra of the accelerated ion-beams in propagating critical layer
acceleration schemes [2][3]. Specifically, using analytical theory and PIC
simulations we look into three fundamental physical processes and their
interplay that are crucial to the understanding of energy spectral control by
making the laser-plasma ion accelerators stable. The interacting processes are
(i) Doppler-shifted ponderomotive bunching [1][4] (ii) potential quenching by
beam-loading [2] and (iii) two-stream instabilities. These phenomenon have been
observed in simulations analyzing these acceleration processes [5][6][7]. From
the preliminary models and results we present in this work, we can infer
measures by which these instabilities can be controlled [8] for improving the
energy-spread of the beams.
