How to install window screen without screen track
It does look like that a screen could fit where you're pointing. 

I've included a picture of some screen hangers and a lock. The hangers would go on the top of the screen and trim allowing the screen to be lifted into your trim track. The locking device would be screwed into the center of the  sill and mounted to hold the screen securely into the trim. Any water that accumulated during rain would be able to run under the screen. The hardware would be permanently attacked to the upper frame, screen and sill so it could be caulked and sealed to prevent any water from migrating into the wood.