How is a second material derivative defined?

Question

I am reading a paper and it used a second material derivative written like this: $$ \dfrac{D^2\delta}{Dt^2} $$ I know the first order material derivative operator is defined $$ \dfrac{D\delta}{Dt}=\partial_t\delta + \vec{u}\cdot\nabla\delta $$ and I know that the second material derivative would just be taking the material derivative again. However, if I wanted to write it a form similar to above it would be $$ \dfrac{D^2\delta}{Dt^2}=\partial_t\dfrac{D\delta}{Dt}+\vec{u}\cdot\nabla\dfrac{D\delta}{Dt}. $$ Is there a better way to define/write this? It feels bulky in this form.