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We establish the vanishing viscosity limit of the zero-mode of the linearized Primitive Equations in a cube. Our method is based on the explicit construction and estimates of the boundary layers. This result, together with that in [12,15], allows us to conclude the vanishing viscosity limit of the linearized Primitive Equations in a cube.

BOUNDARY LAYERS FOR THE 3D PRIMITIVE EQUATIONS IN A CUBE: THE ZERO-MODE