  We introduce $\Psi\mathrm{ec}$, a local spectral exterior calculus for the
two-sphere $S^2$. $\Psi\mathrm{ec}$ provides a discretization of Cartan's
exterior calculus on $S^2$ formed by spherical differential $r$-form wavelets.
These are well localized in space and frequency and provide (Stevenson) frames
for the homogeneous Sobolev spaces $\dot{H}^{-r+1}( \Omega_{\nu}^{r} , S^2 )$
of differential $r$-forms. At the same time, they satisfy important properties
of the exterior calculus, such as the de Rahm complex and the Hodge-Helmholtz
decomposition. Through this, $\Psi\mathrm{ec}$ is tailored towards structure
preserving discretizations that can adapt to solutions with varying regularity.
The construction of $\Psi\mathrm{ec}$ is based on a novel spherical wavelet
frame for $L_2(S^2)$ that we obtain by introducing scalable reproducing kernel
frames. These extend scalable frames to weighted sampling expansions and
provide an alternative to quadrature rules for the discretization of
needlet-like scale-discrete wavelets. We verify the practicality of
$\Psi\mathrm{ec}$ for numerical computations using the rotating shallow water
equations. Our numerical results demonstrate that a $\Psi\mathrm{ec}$-based
discretization of the equations attains accuracy comparable to those of
spectral methods while using a representation that is well localized in space
and frequency.
