When an ideal capacitor is charged by a DC battery

When an ideal capacitor is charged by a DC battery, no current flows. However, when an AC source is used, the current flows continuously. How does one explain this, based on the concept of displacement current?

If an ideal capacitor is charged by DC battery, current flows momentarily till capacitor gets fully charged after that no current flows, however when an AC source is connected, then conduction current
${ I}_{ c }$ = dq/dt
starts flowing in the connecting wire. As charging polarity of AC current changes, the capacitor is alternatively charged and discharged with time. This causes change in electric field between plates of the capacitor which causes electric flux to change and gives rise to displacement current in the region between plates of capacitor as displacement current.