There are several synonyms for the phrase "rid on current." Some of the common synonyms include flowing, drifting, moving, sailing, gliding, cruising, and sliding. All these terms describe an object or person that is carried or propelled by a current, whether it is water, wind, or something else. These synonyms can be used interchangeably based on the context of the sentence or the object being described. For example, we can say a boat is sailing downstream because it's being carried by the river's current, or a leaf is drifting aimlessly along the water's surface. Ultimately, the choice of the synonym depends on the writer's preference and the intended meaning.