  The thermal quasiparticles in a clean type-II superconductor with line nodes
give rise to a quadratic low-temperature change of the penetration depth,
$\Delta \lambda \sim T^2$, as first shown by Kosztin and Leggett [I. Kosztin
and A. J. Leggett, Phys. Rev. Lett. 79, 135 (1997)]. Here, we generalize this
result to multiple nodes and compare it to numerically exact evaluations of the
temperature-dependent penetration depth in Sr$_2$RuO$_4$ using a high-precision
tight-binding model. We compare the calculations to recent penetration depth
measurements in high purity single crystals of Sr$_2$RuO$_4$ [J. F. Landaeta et
al., arXiv:2312.05129]. When assuming the order parameter to have
$\mathrm{B}_{1\mathrm{g}}$ symmetry, we find that both a simple
$d_{x^2-y^2}$-wave and complicated gap structures with contributions from
higher harmonics and accidental nodes can accommodate the experimental data.
