Let $K$ be a number field, $\mathcal O _K^\times$ its group of integral units and $\mathcal O _{K,+} ^\times$ its group of totally positive units. Denote further by $\widehat{\mathcal O }_K^\times$ the units of the finite, integral adeles with its idelic topology.

Is it possible to describe the closures of $\mathcal O _K^\times$ and $\mathcal O _{K,+} ^\times$, respectively, in $\widehat{\mathcal O}_K^\times$ (explicitly)? Is there a natural/useful characterization? I think for the rational field and imaginary quadratic fields the integral units are discrete using a theorem saying that the rational points of the Serre torus (attached to these number fields) are discrete in its adelic points. What about, for example, real quadratic fields? In my applications I sometimes encounter these closures but don't have a good intuition about their size, unfortunately.

Thank you very much in advance for your help.

Edit: One (abstract) characterization of the closure $\overline{\mathcal O _{K,+}^\times}$ of the totally positive units is given in terms of class field theory, namely the closure is equal to the kernel of the (restricted) Artin homomorphism $rec : \widehat{\mathcal O}_K^\times \to Gal(K^{ab}/K)$.