Loud noise in engine

You seem to be reporting two issues.

1) "Jet Engine Sound" could be the CHRS (Coolant Heat Recovery Storage) pump. This pump is known to make a screeching sound (intermittent for some, regularly for others), but will still work. Replacement will make the sound go away; if you can live w/ the screech, great. This pump is activated only twice. When you make the car READY and when you turn OFF the car. The pump won't leave you stranded so that's good news. When it fails, you just won't pass smog b/c a CEL will be triggered. ~$160 part, 2007 Toyota Prius Parts - Camelback Toyota Parts - Genuine OEM Parts - Free Shipping

2) The other less loud 'rhythmic' sound could be the 3-way coolant control valve, cycling back and forth, trying to get the valve in the correct position. There should be a DTC P1121 (usually), indicating there is a problem w/ this valve. Some values stay stuck intermittently (thus a CEL going away, but the code would be logged unless cleared). ~$80 part, 2007 Toyota Prius Parts - Camelback Toyota Parts - Genuine OEM Parts - Free Shipping

The inverter coolant pump is SILENT; about as a loud as a home aquarium pump. DTC P0A93 usually. They seem to work or not work; no screeching noise, just a silent hum. When it stops working you'll get a CEL and possibly the master warning light. With car OFF, open inverter coolant overflow, see the stillness of the coolant. Now make the car READY, you should see gentle movement of the coolant (envision gentle lake water flow); IG-ON might activate the pump too (I forget if it does, it's been awhile). When you don't see movement, the pump is dead and needs to be replaced ASAP. The OEM inverter pumps seem to last LONG time. Inverter pump life? | PriusChat

 

Agreeing with Exstudent on his point #2.
That became a very familiar sound for me, because I ran it for quite a while that way. The pulsing sound was intermittent at first and then was always present. It did throw the P1121 code.
When I had the coolant control valve replaced, I noticed the engine water temp stayed at a much more constant temp. Where before the temp would drop quickly when the ICE was not running.
Also my mileage went up, because of that change.
My understanding is that the pump gets stuck and constantly tries to set the proper position, making the pulsing sound you recorded.