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The internal magnetic field distribution in a mixed state of a cupratesuperconductor, Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$ ($T_{\rm c}\simeq28.5$ K, nearthe optimal doping), was measured by muon spin rotation ($\mu$SR) technique upto 60 kOe. The $\mu$SR linewidth $\Lambda(B)$ which exhibits excess broadeningat higher fields ($B>5$ kOe) due to field-induced magnetism (FIM), is describedby a relation, $\Lambda(B)\propto\sqrt{B}$. This suggests that the orbitalcurrent and associated quasiparticle excitation plays predominant roles instabilizing the quasistatic correlation. Moreover, a slowing down of the vortexfluctuation sets in well above $T_{\rm c}$, as inferred from the trace of FIMobserved up to $\sim80$ K, and develops continuously without a singularity at$T_{\rm c}$ as the temperature decreases.

Field-induced magnetism in the high-$T_c$ superconductor Ca$_{2-x}$Na$_x$CuO$_2$Cl$_2$ with flat CuO$_2$ planes