C.4 The Streamfunction

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As we discussed in Chapter 11, for two dimensional fluid flows, we may introduce a streamfunction Ψ(x,z, t), which carries all the information about the fluid flow. The actual fluid velocity components are obtained by taking partial derivatives of the streamfunction:

(We are free to place the minus sign on either of the velocity components. The sign choice made here gives us the conventional signs in the Lorenz model equations.) We now use the streamfunction in the thermal diffusion equation:

in which we have expanded the grad term explicitly in terms of components. (Mathematically experienced readers may recognize the middle two terms on the left-hand side of the previous equation as the Jacobian determinant of the functions Ψ and τ with respect to the variables x and z.)

The fluid flow equations can also be written in terms of the streamfunction. Unfortunately, the equations become algebraically messy before some order emerges. The v_z equation becomes

The v_x equation becomes

If we now take ∂/∂x of Eq. (C.4-3) and subtract from it ∂/∂z of Eq. (C.4-4), the pressure terms drop out, and we have

Eq. (C.4-2) and the rather formidable looking Eq. (C4-5) contain all the information on the fluid flow.