Disk space disappears to 0, then after a while I have free space

If you are saying you have 1GB of free memory then the problem is not what is grabbing that 1GB but the lack of memory in the first place.

This will get worse overtime because you are already hitting 0 bytes and operating system needs space to write out its files.

I strongly suggest you move essential files like photos a video media off to an external drive to free up some memory. These type of files being normally the largest and easiest to move. ( Do not delete any system files unless you know exactly what they are and removing them will not break you system)

Try and give yourself at least 10-15% free space as a minimum and keep at or above that.

kernel_task is doing that when it is paging out contents of RAM to disk.

Based on your virtual memory usage (8GB+ also written to your disk) you are going to want to keep at least 12GB of space free on your startup disk.

OS X uses virtual memory when it runs out of RAM, which means it moves some stuff from RAM to your disk and that is what is filling up the last bit of space.