We show that low-order g-modes with large enough amplitudes to affect significantly the solar neutrino fluxes would produce surface velocities that are 10⁴ times larger than the observed upper limits and hence are ruled out by existing data. We also demonstrate that any large-amplitude, short-period oscillations that grow on a Kelvin-Helmholtz time scale will require, to affect solar neutrino fluxes, a large amount of energy (for g-modes, 10⁹ times the energy in the observed $p-$ mode oscillations) and a tiny amount of dissipation (for g-modes, 10⁻⁸ the dissipation rate of the p-modes).
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Bahcall et al. (1993) studied this question.