We study the generalised 2D surface quasi-geostrophic (SQG) equation, where the active scalar is given by a fractional power alpha of Laplacian applied to the stream function. This includes the 2D SQG and Euler equations as special cases. Using Poincare's successive approximation to higher alpha-derivatives of the active scalar, we derive a variational equation for describing perturbations in the generalized SQG equation. In particular, in the limit alpha -> 0, an asymptotic equation is derived on a stretched time variable tau = alpha t, which unifies equations in the family near alpha = 0. The successive approximation is also discussed at the other extreme of the 2D Euler limit alpha = 2-0. Numerical experiments are presented for both limits. We consider whether the solution behaves in a more singular fashion, with more effective nonlinearity, when alpha is increased. Two competing effects are identified: the regularizing effect of a fractional inverse Laplacian (control by conservation) and cancellation by symmetry (nonlinearity depletion). Near alpha = 0 (complete depletion), the solution behaves in a more singular fashion as alpha increases. Near alpha = 2 (maximal control by conservation), the solution behave in a more singular fashion, as alpha decreases, suggesting that there may be some alpha in [0, 2] at which the solution behaves in the most singular manner. We also present some numerical results of the family for alpha = 0.5, 1, and 1.5. On the original time t, the H-1 norm of theta generally grows more rapidly with increasing alpha. However, on the new time tau, this order is reversed. On the other hand, contour patterns for different alpha appear to be similar at fixed tau, even though the norms are markedly different in magnitude. Finally, point-vortex systems for the generalized SQG family are discussed to shed light on the above problems of time scale. (C) 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4748350]