B737-300 CREW HAD MULTIPLE TCASII RA'S WITH A CIVILIAN L-39 ALBATROSS FOREIGN MIL ACFT IN BNA CLASS E AIRSPACE.

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WE PUSHED FROM THE GATE IN BNA. AFTER TAXIING TO RWY 2R; WE RECEIVED TKOF CLRNC AND DEPARTED. FO WAS THE PF. WHILE IN THE TURN TO JOIN THE TITAN SID; WE WERE HANDED OFF TO DEP CTL. WE WERE GIVEN A HDG AND TOLD TO MAINTAIN 15000 FT. SHORTLY AFTER; WE WERE TOLD TO TURN R TO 090 DEGS AND MAINTAIN 10000 FT FOR VFR TFC AT OUR 10 O'CLOCK POS AT 10500 FT. I NOTED A TARGET IN THAT AREA ON THE TCASII AND ATTEMPTED TO LOCATE IT VISUALLY. WITHOUT SIGHTING IT; I VERIFIED THAT WE WERE LEVEL AT 10000 FT AND NOTED THE TCASII DISPLAY SHOWED THE TARGET RAPIDLY CLOSING AT OUR 10 O'CLOCK POS AT 10300 FT DSNDING (+3). ALMOST IMMEDIATELY; WE RECEIVED A 'DSND' AND THEN AN 'INCREASE DSCNT' WARNING FROM THE TCASII AS FO DSNDED THE ACFT. I RPTED THIS TO ATC AND GOT NO REPLY DUE TO A LARGE VOLUME OF RADIO TFC. ON MY SECOND RPT; THE CTLR REPLIED 'HE'S SUPPOSED TO BE AT 10500 FT.' AT 9500 FT WE RECEIVED A 'CLB; CLB NOW' TCASII ALERT. AS FO INITIATED THE CLB; I LOCATED THE TARGET; A SMALL MIL TYPE JET; AT OUR 11 O'CLOCK POS; BELOW US APPROX 150-200 YARDS CLOSING AT ABOUT AN 80 DEG ANGLE. I TOOK CTL OF OUR ACFT AND INCREASED OUR CLB RATE/ANGLE AS THE TARGET ACFT PASSED APPROX 100 FT BENEATH THE NOSE OF OUR ACFT. I RPTED THAT TO ATC AND ASKED THAT THEY FORWARD ANY INFO REGARDING THE TARGET ACFT'S IDENT TO BNA OPS. I BRIEFED THE FLT ATTENDANT ON THE SIT VIA THE INTERCOM AND THEY ADVISED THAT NONE OF THE PAX WERE UPSET. SINCE I DIDN'T CONSIDER ANY OF OUR MANEUVERS ABRUPT OR EXTREME; I ELECTED NOT TO MAKE A PUBLIC ANNOUNCEMENT. THE REMAINDER OF THE FLT WAS UNEVENTFUL. HAD IT NOT BEEN FOR TCASII ALERTING; THIS SIT WOULD HAVE BEEN MORE CRITICAL. THE OTHER ACFT WAS IDENTED BY ATC SUPVR AT BNA RAPCON; AS AN L-39 ALBATROSS; FOREIGN MIL JET TRAINER; IMPORTED INTO THE UNITED STATES FOR GA USE. IT HAD FAA REGISTRATION NUMBER. SUPPLEMENTAL INFO FROM ACN 592446: I WAS THE PF. TCASII SHOWED THE TARGET DSNDING AND WE RECEIVED A 'DSND' COMMAND. I DISENGAGED THE AUTOPLT AND PUSHED THE NOSE OVER. THE TCASII THEN STATED; 'INCREASE DSCNT;' WHICH I DID. THE CAPT TRIED TO CALL ATC; BUT THE RADIO TFC WAS HVY AND HE COULDN'T GET THROUGH. AT ABOUT 9500 FT HE GOT THROUGH AND THE CTLR STATED THAT THE TFC HAD BEEN TOLD TO MAINTAIN 10500 FT. WE IMMEDIATELY RECEIVED A 'CLB NOW' FROM THE TCASII. I STARTED PULLING UP AS THE CAPT SAW THE TFC. THE CAPT TOOK THE PLANE AND PULLED IT UP MORE AGGRESSIVELY. I THEN SAW IT PASS UNDERNEATH US TOWARDS OUR 4 O'CLOCK POS. IT WAS ABOUT 100 FT BENEATH US WHEN IT PASSED. THE CAPT CONTACTED ATC AND REQUESTED MORE INFO. CONTRIBUTING FACTORS: ATC DIDN'T HAVE A GOOD HDOF BTWN SECTORS THAT THE VFR TFC WAS IN. WE FOUND OUT LATER THAT HE DIDN'T GET THE 'MAINTAIN 10500 FT' CALL. CORRECTIVE ACTIONS: TRUST THE TCASII. IT WAS CONFUSING TO NOT SEE THE TFC AND HEAR 'DSND' FOLLOWED BY 'CLB NOW.'