Derivation of the scalar stream function equation Distributing the curl: ∂ ∂ t ( ∇ × ( ∇ × ψ ) ) + ∇ × ( ( ∇ × ψ ) ⋅ ∇ ( ∇ × ψ ) ) = ν ∇ × ( ∇ 2 ( ∇ × ψ ) ) {\displaystyle {\frac {\partial }{\partial t}}(\nabla \times (\nabla \times {\boldsymbol {\psi }}))+\nabla \times {\bigl (}(\nabla \times {\boldsymbol {\psi }})\cdot \nabla (\nabla \times {\boldsymbol {\psi }}){\bigr )}=\nu \nabla \times \left(\nabla ^{2}(\nabla \times {\boldsymbol {\psi }})\right)} Replacing curl of the curl with the Laplacian and expanding convection and viscosity: − ∂ ∂ t ( ∇ 2 ψ ) + ∇ × ( ∇ ( ( ∇ × ψ ) ⋅ ( ∇ × ψ ) 2 ) + ( ∇ × ( ∇ × ψ ) ) × ( ∇ × ψ ) ) = ν ( ∇ 2 ( ∇ ( ∇ ⋅ ψ ) ) − ∇ 4 ψ ) {\displaystyle -{\frac {\partial }{\partial t}}\left(\nabla ^{2}{\boldsymbol {\psi }}\right)+\nabla \times \left(\nabla \left({\frac {(\nabla \times {\boldsymbol {\psi }})\cdot (\nabla \times {\boldsymbol {\psi }})}{2}}\right)+\left(\nabla \times (\nabla \times {\boldsymbol {\psi }})\right)\times (\nabla \times {\boldsymbol {\psi }})\right)=\nu \left(\nabla ^{2}(\nabla (\nabla \cdot {\boldsymbol {\psi }}))-\nabla ^{4}{\boldsymbol {\psi }}\right)} Above, the curl of a gradient is zero, and the divergence of ψ is zero. Negating: ∂ ∂ t ( ∇ 2 ψ ) + ∇ × ( ∇ 2 ψ × ( ∇ × ψ ) ) = ν ∇ 4 ψ {\displaystyle {\frac {\partial }{\partial t}}\left(\nabla ^{2}{\boldsymbol {\psi }}\right)+\nabla \times \left(\nabla ^{2}{\boldsymbol {\psi }}\times (\nabla \times {\boldsymbol {\psi }})\right)=\nu \nabla ^{4}{\boldsymbol {\psi }}} Expanding the curl of the cross product into four terms: ∂ ∂ t ( ∇ 2 ψ ) + ( ∇ × ψ ) ⋅ ∇ ( ∇ 2 ψ ) − ( ∇ 2 ψ ) ⋅ ∇ ( ∇ × ψ ) + ( ∇ 2 ψ ) ( ∇ ⋅ ( ∇ × ψ ) ) − ( ∇ × ψ ) ( ∇ ⋅ ( ∇ 2 ψ ) ) = ν ∇ 4 ψ {\displaystyle {\frac {\partial }{\partial t}}\left(\nabla ^{2}{\boldsymbol {\psi }}\right)+(\nabla \times {\boldsymbol {\psi }})\cdot \nabla \left(\nabla ^{2}{\boldsymbol {\psi }}\right)-\left(\nabla ^{2}{\boldsymbol {\psi }}\right)\cdot \nabla (\nabla \times {\boldsymbol {\psi }})+\left(\nabla ^{2}{\boldsymbol {\psi }}\right)(\nabla \cdot (\nabla \times {\boldsymbol {\psi }}))-(\nabla \times {\boldsymbol {\psi }})\left(\nabla \cdot \left(\nabla ^{2}{\boldsymbol {\psi }}\right)\right)=\nu \nabla ^{4}{\boldsymbol {\psi }}} Only one of four terms of the expanded curl is nonzero. The second is zero because it is the dot product of orthogonal vectors, the third is zero because it contains the divergence of flow velocity, and the fourth is zero because the divergence of a vector with only component three is zero (since it is assumed that nothing (except maybe h3) depends on component three). ∂ ∂ t ( ∇ 2 ψ ) + ( ∇ × ψ ) ⋅ ∇ ( ∇ 2 ψ ) = ν ∇ 4 ψ {\displaystyle {\frac {\partial }{\partial t}}\left(\nabla ^{2}{\boldsymbol {\psi }}\right)+(\nabla \times {\boldsymbol {\psi }})\cdot \nabla \left(\nabla ^{2}{\boldsymbol {\psi }}\right)=\nu \nabla ^{4}{\boldsymbol {\psi }}} This vector equation is one meaningful scalar equation and two 0 = 0 equations.