The presence of isolated acoustic noise sources in the ocean produces an anisotropic noise field. Hence, a hypothesis test for detection of these sources can be formulated as a test of H0 (isotropic noise field) versus H1 (anisotropic noise field). The structured approach taken in this paper is to form M beams from N hydrophone elements and then measure the power in each beam. The test for isotropy reduces to acceptance of H0 if the spatial variance of the measured power is less than a predetermined threshold, and acceptance of H1 otherwise. This so-called spatial test is shown to be asymptotically uniformly most powerful invariant for a very natural group of invariance requirements, provided the measured powers are independent through design or suitable transformation. Strengths of the spatial test are that (1) no detailed specification of the anisotropic condition is required, and (2) the predetermined threshold is easily set without prior knowledge of noise levels. Performance curves for the spatial test are presented for specific models of anisotropy and are compared with performance curves of a conventional temporal test.