I can find this term stated both ways in different literature.

Are they equivalent?

It's weird because the dot is a dot product in (u⋅∇), but ∇u being a gradient of a vector field, would (presumably) produce a (jacobian?) matrix which would turn that dot-product dot into a regular vector-matrix multiply where I have to swap the matrix and the vector (because as it's written, the vector will be on the left side).