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We present a study of a two-point spectral turbulence model (Local Wave-Number model or LWN model) for the Rayleigh-Taylor (RT) instability. The model outcomes are compared with statistical quantities extracted from three-dimensional simulation of the RT problem. These simulations are initialized with high wavenumber perturbations at the interface of a heavy fluid placed on top of a light fluid so that the density gradient is in the direction opposite to acceleration due to gravity. We consider flows of low to medium density contrast and compare the LWN model against simulation data using the mix-width evolution as the primary metric. The original model specified physically reasonable but largely \emph{ad hoc} terms to account for the inhomogeneous mechanisms involved in growing the mixing layer. We systematically assess the role of each of the terms in the LWN model equations by comparison with simulation. Two of these, the kinematic source term, introduced to maintain a finite covariance between density and specific volume, and a spectral distortion term, introduced as spectral modifications of the density-specific-volume covariance, both result in severely over-predicting the mix layer growth. A simplified model eliminating those two terms is shown to improve the capture of both mix-width evolution as well as the turbulent mass flux velocity profiles across the mix layer at different times. However, this simplification reveals that fidelity to other metrics such as the density-specific-volume covariance, and the turbulent kinetic energy are somewhat compromised. The implications of this outcome are discussed with respect to the physics of the RT problem, and we provide this study as a guide for the practical use of such a model.