We study the local asymptotic behaviour of divergence-like functionals of a family of d-dimensional infinitely divisible random fields. Specifically, we derive limit theorems of surface integrals over Lipschitz manifolds for this class of fields when the region of integration shrinks to a single point. We show that in most cases, convergence stably in distribution holds after a proper normalisation. Furthermore, the limit random fields can be described in terms of stochastic integrals with respect to a Lévy basis. We additionally discuss the relationship between our results and the advective kinetic energy flux in a possibly turbulent flow.
