Desalination and rinsing methods and systems are provided, which use a liquid column to efficiently perform freeze desalination and enable recyclable washing of produced minerals or combustible material. The liquid column comprises alternating layers of water immiscible liquids and water solutions which form a vertical density gradient (discrete or continuous) and exhibit a lower freezing zone from which ice floats to an upper melting zone. Cooling of introduced upwards flowing salt solutions is carried out by a countering downwards flow of cold water immiscible liquid. The process is cyclical, involves few if any mechanical moving parts and is easily controllable and adaptable to varying desalination circumstances. Rinsing of minerals or combustible material may be integrated in the heat and matter flows of the desalination system to allow effective rinsing, desalination and recycling of the used water.