We consider optimal transport on the circle $\mathbb{T}= \mathbb{R}/(2\pi\mathbb{Z})$ equipped with the metric $d(x,y) = \min_{k\in\mathbb{Z}}|x-y-2\pi k|$. For two probability measures $\mu,\nu$ on $\mathbb{T}$, the (squared) Wasserstein distance between two probability measures $\mu,\nu$ on $\mathbb{T}$ is given by $$W_2^2(\mu,\nu) = \min_{\alpha \in \mathbb{R}}\int_{0}^{1}|\tilde{F}_\mu^{-1}(x)-(\tilde{F}_\nu^{-1}-\alpha)^{-1}(x)|^2\,\text{d}x.$$ Here, $\tilde{F}_\mu$ is the (extended) distribution function of $\mu$ defined as $\tilde{F}_\mu(x)= \mu([0,x])$ for $x\in [0,2\pi]$ and $\tilde{F}_\mu(x+2\pi) = \tilde{F}_\mu(x)+1$ otherwise. Its inverse is given by $\tilde{F}_\mu(x)^{-1}(x) = \min\{x\in\mathbb{R} \colon \tilde{F}_\mu(x)\geq r\}, r\in \mathbb{R}$.
Analogous to McCann's interpolation on $\mathbb{R}^d$ with the Euclidean distance, I would like to find an analytical formula of the interpolant between $\mu$ and $\nu$, i.e. for given $t \in (0,1)$ the minimizer of $$\min_{\gamma} (1-t)\cdot W_2^2(\mu,\gamma)+t\cdot W_2^2(\gamma,\nu).$$ We may assume that $\mu$ is absolutely continuous with respect to the Lebesgue measure. My goal is to find an expression depending only on $t$, the (extended) CDFs and the minimizer $\alpha_{\mu,\nu}$ in $W_2^2(\mu,\nu)$. I'm already stuck in the very beginning and don't know how to deal with the minimization over $\alpha$ in the formula for the Wasserstein distance. Any ideas?