Looks like they used loess as fill.
As good as that would be to garden on... as fill on a slope... what you are experiencing is the rule rather than the exception.
Really depends on your end goal as to what you should do.
As mentioned, fist-sized rock... would fill in the trenches without washing...
Or... a nice drainage box/catch basin at the top (where the water leaves the impermeable surface) hooked to that drain line...
Or... research swales and rain gardens.
There are plenty of ways to make lemonade from lemons here... I only wish I had all that water...
