Sorry to switch subject, I have a question I do not know where to ask:
(please note I am not a hardware specialist)

A processor with two layer cache writes a byte to zero, none of the memory is in cache.
So the processor system looks for the address in layer 1 cache, do not find it, looks
in layer 2 cache, do not find it, instruct the layer 2 cache to fetch the memory.
When the layer 1 cache line is filled, the instruction to clear the byte executes.
If at that point both layer cache are no more needed and evicted, what size is written
back to memory: a layer 1 cache line or a layer 2 cache line (which would be a lot bigger).
In other words, is there "dirty bits" for every layer 1 cache line inside the layer 2 cache?

Thanks. Pointers to documentation welcomed.