The turbulent passive advection under the environment (velocity) field with fi-
nite correlation time is studied. Inertial-range asymptotic behavior of a vector (e.g., magnetic)
field, passively advected by a strongly anisotropic turbulent flow, is investigated
by means of the field theoretic renormalization group and the operator product expansion.
The advecting velocity field is Gaussian, with finite correlation time and prescribed
pair correlation function. The inertial-range behavior of the model is described by two
regimes (the limits of vanishing or infinite correlation time) that correspond to nontrivial
fixed points of the RG equations and depend on the relation between the exponents in the
energy spectrum E ∝ k
1−ξ
⊥ and the dispersion law ω ∝ k
2−η
⊥ . The corresponding anomalous
exponents are associated with the critical dimensions of tensor composite operators built
solely of the passive vector field itself. In contrast to the well-known isotropic Kraichnan
model, where various correlation functio