Let $\mathscr{X}$ be a model of $X$. As $\mathscr{X}$ is flat over $\mathfrak{o}_{k}$, the natural homomorphism $\mathscr{O}_{\mathscr{X}}\to\mathscr{O}_{X}$ is injective. Let $Y$ be a closed subscheme of $X$ and $I_{Y}\subseteq\mathscr{O}_{X}$ the defining ideal sheaf of $Y$. Let $\mathscr{I}_{\mathscr{Y}}$ be the kernel of $\mathscr{O}_{\mathscr{X}}\to\mathscr{O}_{X}/I_{Y}$, that is, $\mathscr{I}_{\mathscr{Y}}:=\mathscr{I}_{Y}\cap\mathscr{O}_{\mathscr{X}}$. Obviously $\mathscr{I}_{\mathscr{Y}}\otimes_{\mathfrak{o}_{k}}k=I_{Y}$, so that if we set $\mathscr{Y}=\operatorname{Spec}(\mathscr{O}_{\mathscr{X}}/\mathscr{I}_{\mathscr{Y}})$, then $\mathscr{Y}\times_{\operatorname{Spec}(\mathfrak{o}_{k})}\operatorname{Spec}(k)=Y$. Moreover, $\mathscr{Y}$ is flat over $\mathfrak{o}_{k}$ because $\mathscr{O}_{\mathscr{Y}}\to\mathscr{O}_{Y}$ is injective. Therefore, $\mathscr{Y}$ is a model of $Y$. We say that $\mathscr{Y}$ is the Zariski closure of $Y$ in $\mathscr{X}$.