We investigate the expression of non-Kolmogorov turbulence in terms of Zernike polynomials. Increasing the power-law exponent of the three-dimensional phase power spectrum from 2 to 4 results in a higher proportion of wavefront energy being contained in the tilt components. Closed-form expressions are given for the variances of the Zernike coefficients in this range. For exponents greater than 4, a von Karman spectrum is used to numerically compute the variances as a function of exponent for different outer-scale lengths. We find in this range that the Zernike-coefficient variances depend more strongly on outer scale than on exponent, and that longer outer-scale lengths lead to more energy in the tilt terms. The scaling of Zernike- coefficient variances with pupil diameter is an explicit function of the exponent.