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In this work we study the generalization of the problem, considered in [{\itPhys. Rev. E} {\bf 91}, 013002 (2015)], to the case of {\it finite} correlationtime of the environment (velocity) field. The model describes a vector (e.g.,magnetic) field, passively advected by a strongly anisotropic turbulent flow.Inertial-range asymptotic behavior is studied by means of the field theoreticrenormalization group and the operator product expansion. The advectingvelocity field is Gaussian, with finite correlation time and preassigned paircorrelation function. Due to the presence of distinguished direction ${\bf n}$,all the multiloop diagrams in this model are vanish, so that the resultsobtained are exact. The inertial-range behavior of the model is described bytwo regimes (the limits of vanishing or infinite correlation time) thatcorrespond to the two nontrivial fixed points of the RG equations. Theirstability depends on the relation between the exponents in the energy spectrum${\cal E} \propto k_{\bot}^{1-\xi}$ and the dispersion law $\omega \proptok_{\bot}^{2-\eta}$. In contrast to the well known isotropic Kraichnan's model,where various correlation functions exhibit anomalous scaling behavior withinfinite sets of anomalous exponents, here the corrections to ordinary scalingare polynomials of logarithms of the integral turbulence scale $L$.

Passive advection of a vector field: Anisotropy, finite correlation time, exact solution and logarithmic corrections to ordinary scaling