Let ${\mathscr{X}}$ be an algebraic ${K^{\circ}}$-model of $X$. A vertical Cartier divisor on ${\mathscr{X}}$ is a Cartier divisor $D$ on ${\mathscr{X}}$ which is supported on the special fiber ${\mathscr{X}}_{s}$. A vertical Cartier divisor $D$ on ${\mathscr{X}}$ determines a model ${\mathcal{O}}(D)$ of ${\mathcal{O}}_{X}$ hence an associated model function

$$\varphi_{D}\coloneqq-\log\|1\|_{{\mathcal{O}}(D)}:X^{\rm an}\to{\mathbb{R}}$$

Note that every ${\mathbb{Z}}$-model function has this form. Indeed, if ${\mathscr{L}}$ is an algebraic model of ${\mathcal{O}}_{X}$ with $\varphi=-\log{\|\hskip 4.30554pt\|}_{\mathscr{L}}$, then the section $1$ of ${\mathcal{O}}_{X}$ extends to a meromorphic section $s$ of ${\mathscr{L}}$ and the vertical Cartier divisor $D:={\rm div}(s)$ satisfies $\varphi=\varphi_{D}$.