Nilpotent Orbits for Borel Subgroups of $SO_{5}(k)$

Let $G$ be a quasi-simple algebraic group defined over an algebraically closed field $k$ and $B$ a Borel subgroup of $G$ acting on the nilradical $\mathfrak{n}$ of its Lie algebra $\mathfrak{b}$ via the Adjoint representation. It is known that $B$ has only finitely many orbits in only five cases: when $G$ is of type $A_{n}$ for $n \leq 4$, and when $G$ is type $B_{2}$. In this paper, we elaborate on this work in the case when $G =SO_{5}(k)$ (type $B_{2})$ by finding the polynomial defining equations of each orbit. We use these equations to determine the dimension of the orbits and the closure ordering on the set of orbits. The other four cases, when $G$ is type $A_{n}$, can be approached the same way and are treated in a separate paper.