Comparison of analytical and test-field results for the flow Eq. (C.1) showing the mean-field coefficients as a function of x. Red solid: numerical, blue dashed: analytical. v₀ = w₀ = 1, k₀ = 2, η = 0.05 in arbitrary units. The discrepancies in b₂₁₁,b₁₂₁, and b₃₂₁ near the x boundaries are systematic errors due to the use of periodic boundary conditions for the solution of Eq. (5), which conflicts with the test fields, being linear in x and z.