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A generic system of diffusion equations is considered, in which the diffusivities depend on the solutions and their gradients. The stability of the solutions is investigated, showing that small-scale space perturbations can be unstable. The associated stability criterion is established, indicating that instabilities, if any, are due to the diffusivities depending in an inappropriate way on the gradient of the solutions. Numerical simulations of Kato-Phillips flows are performed, which may be regarded as a highly idealised representation of the upper layers of the ocean. Several versions of the Munk-Anderson and Mellor-Yamada turbulence closure schemes are implemented. 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