The Galerkin vertical representation of temperature in terms of basis functions is:
where are coefficients of the expansion in p chosen to minimize square error under an inner product that is just a weighted pressure average. We split off a reference profile that does not vary in space because this can be specified or treated in a separate calculation--only gradients of T affect the circulation. Note that is not assumed to be a solution of the system, and that the spatial average of the model output differs from .
The first trick that aids both numerics and interpretation is that basis functions for the wind fields are constructed from approximate analytic solutions. Consider the leading temperature basis function, . Temporarily neglecting and advection by baroclinic wind, the solution to the momentum equation (3) for circulations driven by this baroclinic pressure gradient, satisfying simple lower and upper boundary conditions, has the form
Using such analytically constructed basis functions to derive nonlinear Galerkin equations can be highly accurate at low truncation, and provides a physical association of and T components.
The approach (6)-(7) could be used generally but larger gains occur when the temperature variations that are typically encountered are restricted by convection. In our approach, we tailor the temperature basis functions to typical quasi-equilibrium profiles. One of the main effects of convection on the large scales is to constrain the vertical temperature profile through the depth of the convecting region--highly unstable profiles with large convective available potential energy tend to be modified by convection (e.g., Xu and Emanuel 1989; Randall and Wang 1992). For many convection schemes one can explicitly or implicitly define, for a given set of large scale variables, a quasi-equilibrium temperature profile that convection brings the large scale temperature towards (e.g., Manabe et al 1965; Arakawa and Schubert 1974; Betts 1986). Thus the variations in quasi-equilibrium temperature profile can be efficiently expressed in the form (6) but with only N coefficients , where is small. The coefficients depend on the large scale T and q variables. If is typical of temperature variations near deep convective regions, associated velocity variations will be largely carried in . Similarly for shallow convection, and so on.