A state of fluid projected onto “translational” basis

u(t) is an instantaneous state of the fluid, represented by a 10^{5}dimensional vector.
e_{3} is a 10^{5}dimensional vector, constructed from another state of the fluid, the u_{UB} equilibrium. a_{3}(t) =u(t).e_{3} is the projection on the e_{3} axis defined by the . 