  In this paper we study the degeneration behavior of the norm of the Riemann
$\theta$-function in a family of principally polarized abelian varieties over
the punctured complex unit disc in terms of the associated polarized real
torus. As an application, we obtain the degeneration behavior of the
Zhang--Kawazumi invariant $\varphi(M_t)$ of a family of Riemann surfaces $M_t$
in terms of Zhang's invariant $\varphi(\Gamma)$ of the associated metrized
reduction graph $\Gamma$. This allows us to deduce a uniform lower bound for
the essential minimum of the N\'eron-Tate height on the tautological cycles of
any Jacobian variety over a number field.
