  We obtain a complete description of anisotropic scaling limits and the
existence of scaling transition for nonlinear functions (Appell polynomials) of
stationary linear random fields on $\mathbb{Z}^2$ with moving average
coefficients decaying at possibly different rate in the horizontal and vertical
direction. The paper extends recent results on scaling transition for linear
random fields in Puplinskait\.e and Surgailis (2016), Puplinskait\.e and
Surgailis (2015).
