  Let $\lambda>0$ and $\triangle_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x}
\frac d{dx}$ be the Bessel operator on $\mathbb R_+:=(0,\infty)$. We show that
the oscillation operator $\mathcal{O}(R_{\Delta_{\lambda},\ast})$ and variation
operator $\mathcal{V}_{\rho}(R_{\Delta_{\lambda},\ast})$ of the Riesz transform
$R_{\Delta_{\lambda}}$ associated with
  $\Delta_\lambda$ are both bounded on $L^p(\mathbb R_+, dm_{\lambda})$ for
$p\in(1,\,\infty)$, from $L^1(\mathbb{R}_{+},dm_{\lambda})$ to
$L^{1,\,\infty}(\mathbb{R}_{+},dm_{\lambda})$, and from
$L^{\infty}(\mathbb{R}_{+},dm_{\lambda})$ to
$BMO(\mathbb{R}_{+},dm_{\lambda})$, where $\rho\in (2,\infty)$ and
$dm_{\lambda}(x):=x^{2\lambda}dx$. As an application, we give the corresponding
$L^p$-estimates for $\beta$-jump operators and the number of up-crossing.
