Let $(M,\nabla)$ be a smooth manifold with an affine connection $\nabla$ on $TM$. Very often we find the following operation being performed in a coordinate basis $\nabla_\alpha T^{\alpha\beta}$. One example is the conservation condition in General Relativity, which is expressed as
$$\nabla_\alpha T^{\alpha\beta}=0.$$
Here $\nabla_\alpha$ denotes the covariant derivative with respect to $\frac{\partial}{\partial x^\alpha}$ as it is usually done.
Now, is there any other standard notation to mean this operation without involving components?
I mean, there's nothing wrong with this one, I'm quite fine with it. It is just curiosity.
One notation I thought was $\nabla\cdot T$, but this has a problem, since it is not clear whether we are contracting with the first or second index of $T$.