In areas of very cold temperatures (-20°C and below), it is not uncommon for small streams to freeze completely (as in, from top to bottom) and larger rivers to develop thick ice covers. However, there is usually still water flowing beneath the ice cover of the larger rivers. This situation can easily persist for weeks or months.
Which leads me to my question - where does the water in the big river come from? Big rivers are fed by smaller rivers, which are fed by large streams, which in turn are fed by small streams. At some point, we get to streams that are frozen solid, and consequently will no longer feed water into the system. Yet, as the big rivers keep flowing into the sea, water is still removed from the system. At this point, I would expect that, after a few days, all water is drained from the system, and the flow stops, but that doesn't seem to be what happens. Where does the water that keeps flowing into the sea come from?
(This is obviously not a big logical issue in rivers flowing towards the poles, but rivers flowing towards the equator, or parallel to it, should have this problem.)