Conventions for writing multiple indices

Are there any conventions as to how indices should be ordered when refering to an element of a multidimensional array?

To take two examples:

1. A 3-dimensional array: I have $J$ objects (data.frames) each with $N$ rows and $K$ columns. How to refer to elements of that array?
2. I have a matrix that contains (theoretical) covariances between each of the elements $a_{ij;1}$ (where this index order is also subject to my question) of some 2-dimensional array $A_1$ with each of the elements $a_{ij;k}$ of other 2-dimensional arrays (including itself) $A_1, A_2, \dots, A_K$. How would I name such an element. Is there a convention? Something like this? $$s_{ij1;ijk'}$$

For the first one, typical convention would either be $x_{jnk}$ or $x_{nkj}$. For the second one, your notation seems fine, it is an extension of the (common) row-column-third dimension convention.
So for example, for your second example, one would say : $s_{ij1;ijk'}$, is the covariance between the element at row $i$ and column $j$ of $A_1$ and the element at the same row and column of $A_k'$. That'll make your intent clear.