The internal magnetic field distribution in a mixed state of a cuprate superconductor, Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$ ($T_{rm c}simeq28.5$ K, near the optimal doping), was measured by muon spin rotation ($mu$SR) technique up to 60 kOe. The $mu$SR linewidth $Lambda(B)$ which exhibits excess broadening at higher fields ($B>5$ kOe) due to field-induced magnetism (FIM), is described by a relation, $Lambda(B)proptosqrt{B}$. This suggests that the orbital current and associated quasiparticle excitation plays predominant roles in stabilizing the quasistatic correlation. Moreover, a slowing down of the vortex fluctuation sets in well above $T_{rm c}$, as inferred from the trace of FIM observed up to $sim80$ K, and develops continuously without a singularity at $T_{rm c}$ as the temperature decreases.
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