Knowlpedia

Definition

Quadratic self-interaction

The quadratic term involving a perturbation with itself in an exact expansion.

For a BB, set Q(v)=B(v,v)Q(v)=B(v,v). The quadratic self-interaction of an increment ww is B(w,w)B(w,w), as seen in

Q(u+w)=Q(u)+B(u,w)+B(w,u)+B(w,w).Q(u+w)=Q(u)+B(u,w)+B(w,u)+B(w,w).

The two middle terms are interactions with the background. Symmetry of BB is not assumed; symmetrizing it leaves QQ unchanged.

Fluid example

For B(v,w)=(v⋅∇)wB(v,w)=(v\cdot\nabla)w, self-interaction is (w⋅∇)w(w\cdot\nabla)w. If ww is divergence free, this equals ∇⋅(w⊗w)\nabla\cdot(w\otimes w). An oscillatory or geometric construction may cancel selected components or averages; that does not establish cancellation of the full expression without a calculation.