Quadratic advection of a velocity field

Using the transport operator, the quadratic advection term is

\mathcal N(u)=(u\cdot\nabla)u, \qquad \mathcal N(u)_i=\sum_j u_j\partial_j u_i.

On smooth velocity fields it is a quadratic map, built from the bilinear expression $B(v,w)=(v\cdot\nabla)w$ .

Expansion about a background

For $u=U+v$ , bilinearity gives the exact identity

\mathcal N(U+v)=\mathcal N(U) +(U\cdot\nabla)v+(v\cdot\nabla)U+(v\cdot\nabla)v.

The two middle terms are linear in the perturbation $v$ , and the last is quadratic. The bilinear expression is generally not symmetric: $(v\cdot\nabla)w$ need not equal $(w\cdot\nabla)v$ .

This identity is the algebraic starting point for linearized fluid equations and estimates of their nonlinear residual.