I feel bad answering this one without any research reference, but I can tell you that what you are seeing in the OP's picture is a 'man overboard drill.' In that drill, the objective is to rotate the ship's rear section away from a man who many have fallen overboard and then circle the ship around (in an exact circle) to come up alongside the fallen man.
In order to do this with a ship, you throw the rudder over hard and fire up the engines ahead full. With many warships, the engine's toque is enough to produce massive cavitation (bubble formation) for the duration of the turn. As you end the turn you will usually throw a 2/3 backing bell, which will produce even more cavitation. That is the 'trail' you see behind the ship.
Specifically, the ship you see there is HMS Echo, which has a diesel-electric azimuth drive mounted below the ship, which will provide a very high torque.
Lots of cavitation (and you can feel a 10,000 ton ship vibrate during these maneuvers) means lots of bubbles are formed up to 15m below the surface (depending on the size of the ship; only about 6m for the pictured Echo). As the bubble propagate to the surface they disrupt the passage of surface wavelets and form the calm area shown.
One more factor is that navies train to put the man on the leeward side of the ship during recovery. Thus, in this picture the stern was swung to port (as we can see from the 'trail') so the man must have been on starboard, so the man is on the 'starboard' side of the ship during this recovery. That means the wind should be on the port side, so the calm area is being protected from the wind by the mass of the ship.