ACR HAS NMAC WITH SMT ON INITIAL CLB. TCASII RA.

Narrative

FOLLOWING TKOF ON RWY 19 AT SDF; THE COPLT; WHO WAS FLYING; BEGAN A TURN TO 310 DEG PER ATC INSTRUCTION. AS HE WAS ROLLING OUT; WE RECEIVED A TA FROM THE TCASII. I LOOKED AT THE SCREEN; SAW A YELLOW TARGET AT 10-11 O'CLOCK; AND SWITCHED THE UNIT FROM TA; WHERE IT HAD BEEN FOR TKOF; TO RA; THEN LOOKED OUTSIDE BUT DID NOT SEE THE TFC. SOON THEREAFTER WE RECEIVED A 'CLB NOW' COMMAND FROM TCASII. I ALTERNATED LOOKING FOR THE TFC AND INSURING THAT THE COPLT WAS EXECUTING A RAPID CLB. HE; IN FACT; PUT THE ACFT INTO A 4000 FPM CLB WITHIN A FEW SECONDS. I VISUALLY ACQUIRED A DRAB COLORED TWIN PROP ACFT AT 9 O'CLOCK ABOUT 1-2 SECONDS BEFORE IT DISAPPEARED UNDER OUR ACFT. WITH SUCH A SHORT VIEW OF THE OTHER ACFT; I AM UNCERTAIN AS TO THE VERT DISTANCE; BUT WOULD GUESS 300- 400 FT. WHAT IS CERTAIN IS THAT; HAD WE NOT BEEN IN A STEEP CLB; WE WOULD HAVE BEEN MUCH CLOSER. WE RECEIVED NO TA FROM ATC; EITHER TWR OR DEP. WHEN I QUESTIONED THE CTLR ABOUT THE OTHER ACFT; HE STATED THAT 'THE TWR HAD HIM VISUALLY'. I DID NOT TAKE TIME TO QUESTION WHAT THAT MEANT AT THE TIME; BUT CALLED THE DEP SUPVR LATER IN THE DAY. HE STATED THAT IT PROBABLY MEANT THAT SDF TWR HAD CLRED THE OTHER ACFT THROUGH THEIR AIRSPACE VFR TO LAND AT BOWMAN FIELD (A GA ARPT A FEW MI NE OF SDF). WARNING FROM ATC OR PREFERABLY A CLRING VECTOR WOULD HAVE PREVENTED THE CONFLICT. I WOULD HAVE HOPED THAT EITHER THE COPLT OR I WOULD HAVE SEEN THE TFC; BUT THE HAZE AND DULL COLOR OF THE ACFT PREVENTED MY SEEING IT UNTIL VERY CLOSE. THIS WAS TRUE EVEN THOUGH THE TCASII HAD PROVIDED RELATIVE BEARING INFO.