A simple model, parametrized by the Reynolds stress anisotropy, is proposed for the spectrum of weakly anisotropic turbulence. It contains a model constant that affects its region of realizability. This spectrum model is used to derive a one point closure to the rapid pressure‐strain term. The derived pressure‐strain model is linear in the Reynolds stress anisotropy and is of the same form as the closure model of Launder, Reece, and Rodi (LRR) [J. Fluid Mech.68, 537 (1975)]. The spectrum model becomes unrealizable in some regions of wave space for sufficiently large anisotropy of the Reynolds stress, and this is used to infer the region of validity of the linear closure model. It is found that the extent of the valid region is very small when the model constant is set to match rapid distortion theory, and largest for a model constant set close to the value suggested by LRR. However, even the largest valid domain does not extend very far from isotropy, suggesting inherent weakness in the linear pressure‐strain models.