  Given a compact 3-manifold $Y$ and a $\mathbb Z_2$-harmonic spinor $(\mathcal
Z_0, A_0,\Phi_0)$ with singular set $\mathcal Z_0$, this article constructs a
family of local solutions to the two-spinor Seiberg-Witten equations
parameterized by $\epsilon \in (0,\epsilon_0)$ on tubular neighborhoods of
$\mathcal Z_0$. These solutions concentrate in the sense that the $L^2$-norm of
the curvature near $\mathcal Z_0$ diverges as $\epsilon\to 0$, and after
renormalization they converge locally to the original $\mathbb Z_2$-harmonic
spinor. In a sequel to this article, these model solutions are used in a gluing
construction showing that any $\mathbb Z_2$-harmonic spinor satisfying some
mild assumptions arises as the limit of a family of two-spinor Seiberg-Witten
solutions on $Y$.
