C.5 Fourier Expansion, Galerkin Truncation, and Boundary Conditions

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Obviously, we face a very difficult task in trying to solve the partial differential equations that describe our model system. For partial differential equations, the usual practice is to look for solutions that can be written as products of functions, each of which depends on only one of the independent variables x, z, t. Since we have a rectangular geometry, we expect to be able to find a solution of the form

where the λs are the wavelengths of the various Fourier spatial modes and ω_m,n  are the corresponding frequencies. We would, of course, have a similar equation for τ, the temperature variable. (Appendix A contains a concise introduction to Fourier analysis.)

As we saw in Chapter 11, the standard procedure consists of using this sine and cosine expansion in the original partial differential equations to develop a corresponding ser of (coupled) ordinary differential equations. This procedure will lead to an infinite ser of ordinary differential equations. To make progress, we must somehow reduce this infinite ser to a finite ser of equations. This truncation process is known as the Galerkin procedure.

For the Lorenz model, we look at the boundary conditions that must be satisfied by streamfunction and the temperature deviation function and choose a very limited ser of sine and cosine terms that will satisfy these boundary conditions. It is hard to justify this truncation a priori, but numerical solutions of a larger ser of equations seem to indicate (SAL62) that the truncated form captures most of the dynamics over at least a limited range of parameter values.

The boundary conditions for the temperature deviation function are simple. Since τ represents the deviation from the linear temperature gradient and since the temperatures at the upper and lower surfaces are fixed, we must have

For the streamfunction, we look first at the boundary conditions on the velocity components. We assume that at the top and bottom surfaces the vertical component of the velocity v_z must be 0. We also assume that we can neglect the shear forces at the top and bottom surfaces. As we saw in Chapter 11, these forces are proportional to the gradient of the tangential velocity component; therefore, this condition translates into having ∂v_x/∂z = 0 at z = 0 and z = 1. For the Lorenz model, these conditions are satisfied by the following ansatz for the streamfunction and temperature deviation function:

where the parameter a is to be determined. As we shall see, this choice of functions not only satisfies the boundary conditions, but it also great1y simplifies the resulting equations.

The particular form of the spatial part of the streamfunction Ψ models the convective rolls observed when the fluid begins to convect. You may easily check this by ca1culating the velocity components from Eq. (C.4-l). The form for the temperature deviation function has two parts. The first, T₁, gives the temperature difference between the upward and downward moving parts of a convective cell. The second, T₂, gives the deviation from the linear temperature variation in the center of a convective cell as a function of vertical position z. (The minus sign in front of the T₂ term is chosen so that T₂ is positive: The temperature in the fluid must lie between T_w and T_c.)