This is in continuation of this post which explained the meaning of the vols in the vol smile as follows:
Suppose you have a downsloping vol smile. Then, the fact that the vol is higher on the downside means that the magnitude of down moves (if they happen) is higher than the magnitude of up moves. It means traders expect a down move would be more violent (and imply more subsequent volatility) that an up move.
Maybe Dupire's formula that gives local vol as a function of implied vol can help here?