A property $p$ of identities of a fixed type $\tau$ is said to be 
hereditary if for every set $I$ of identities having the property $p$,
every consequence of $I$ (under the usual derivation rules for
identities) also has the property. A characteristic algebra
for such a hereditary property is an algebra $\cal A$ such that for any
variety $V$ of type $\tau$, we have ${\cal A} \in V$ 
iff every identity satisfied
by $V$ has the property $p$. P\l onka has produced minimal characteristic 
algebras for a number of hereditary properties, including regularity, 
normality, uniformity, biregularity, outermost, and external-compatibility. 
We study characteristic algebras for the hereditary property of $k$-normality,
for $k \geq 1$, which extends the usual normality property. For type $(2)$
and the usual depth valuation of terms, 
we produce minimal characteristic algebras
for $k=1,2,3$ and $4$. \\