OK. So let us not call it a job description. So we have nothing now.
Lets go back to the standard iso 9001:2008.
At 5.5.1 you will see this small sentence ~~~
5.5.1 Responsibility and authority
Top management shall ensure that responsibilities and authorities are defined and communicated within the organization.
This is a planning aspect where the top management first defines the R and A and then communicates that within the organization.
Also further in the standard at 5.5.3, you will see this simple sentence ~~~
5.5.3 Internal communication
Top management shall ensure that appropriate communication processes are established within the organization and that communication takes place regarding the effectiveness of the quality management system.
The organization will have to decide how best this can be done to demonstrate both defining and communicating within the organization.
Perhaps by a document ... ? aaaah yes.
A little earlier in the standard at 4.2.1 d) this is said ~~~
The quality management system documentation shall include:
d) documents, including records, determined by the organization to be necessary to ensure the effective planning, operation and control of its processes.
You can call this by any name .... also as Job Description.