  We derive explicit expressions for a family of radially symmetric,
non-differentiable, Spartan covariance functions in $\mathbb{R}^2$ that involve
the modified Bessel function of the second kind. In addition to the
characteristic length and the amplitude coefficient, the Spartan covariance
parameters include the rigidity coefficient $\eta_{1}$ which determines the
shape of the covariance function. If $ \eta_{1} >> 1$ Spartan covariance
functions exhibit multiscaling. We also derive a family of radially symmetric,
infinitely differentiable Bessel-Lommel covariance functions valid in
$\mathbb{R}^{d}, d\ge 2$. We investigate the parametric dependence of the
integral range for Spartan and Bessel-Lommel covariance functions using
explicit relations and numerical simulations. Finally, we define a generalized
spectrum of correlation scales $\lambda^{(\alpha)}_{c}$ in terms of the
fractional Laplacian of the covariance function; for $0 \le \alpha \le1$ the
$\lambda^{(\alpha)}_{c}$ extend from the smoothness microscale $(\alpha=1)$ to
the integral range $(\alpha=0)$. The smoothness scale of mean-square continuous
but non-differentiable random fields vanishes; such fields, however, can be
discriminated by means of $\lambda^{(\alpha)}_{c}$ scales obtained for $\alpha
<1$.
