AN IFR C210 PILOT REPORTS A NEAR MISS WITH CIRRUS AFTER DEPARTURE AT 2100 FT. TCAS ALERTED THE TFC BUT ATC LCL CTLR DID NOT UNTIL IT PASSED.

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OUR ACFT (A C210D) DEPARTED ZZZ ON AN IFR FLT TO ZZZ1 AND EXPERIENCED WHAT WE WOULD CLASSIFY AS A 'NEAR MISS' WITH AN INBOUND ACFT. THE FLT WAS FLOWN BY AN INST CERTIFIED PVT PLT; WITH A PVT PLT PAX ALSO ON BOARD. OUR IFR CLRNC GIVEN ON CLRNC DELIVERY WAS INITIALLY CLB TO 3000 FT; EXPECTING 6000 FT WITHIN 10 MINS AFTER DEP. EITHER SHORTLY BEFORE (30 SECONDS OR LESS) OR SHORTLY AFTER (15 SECONDS OR LESS) WE WERE GIVEN 'CLRED FOR TKOF RWY 2L; FLY HDG 330 DEGS;' A CIRRUS ACFT CALLED ZZZ TWR AND ADVISED 8 MI N; INBOUND; VFR. WE ROTATED; AND ONCE WE CROSSED THE N BOUNDARY OF THE ARPT; BEGAN A TURN TO A HDG OF 330 DEGS. ON THAT HDG; WE WERE CLBING APPROX 1200 FPM. WITHIN 1 MIN OF BEING AIRBORNE; WE RECEIVED A VISUAL AND AURAL TFC MESSAGE FROM THE ONBOARD GTX330D XPONDER RECEIVING DATA VIA MODE S; CONNECTED TO THE GNS530 GPS DISPLAY. IT WAS A RED TFC ALERT INDICATING IMMEDIATE ACTION REQUIRED. AS SOON AS WE BOTH LOOKED OUT TO THE FRONT R OF THE ACFT; WE OBSERVED A CIRRUS LESS THAN 1 MI AWAY. IMMEDIATELY PULLED BACK ON THE PWR; ACTIVATED ONBOARD SPD BRAKES; AND PUSHED THE NOSE DOWN TO AVOID A COLLISION. SHORTLY THEREAFTER; THE CIRRUS APPEARED TO MAKE A TURN TO AVOID US; HOWEVER NOT SHARPLY. IT IS MY OPINION THAT HAD WE NOT STOPPED THE CLB AND BEGUN A DSCNT; THAT WE WOULD HAVE COLLIDED WITH THE CIRRUS ACFT. JUST AS SOON AS WE BEGAN OUR DSCNT; I NOTIFIED ZZZ TWR THAT WE HAD RECEIVED A TFC ALERT ON BOARD AND WERE DSNDING TO AVOID. THE CTLR ANSWERED BACK WITH SOMETHING TO THE EFFECT OF 'YOU HAVE TFC AT 12-1 O'CLOCK POS; LESS THAN 1 MI; 2100 FT.' I CAN'T SPECIFICALLY RECALL IF WE WERE GIVEN DISTANCE; BUT I RECALL THE DIRECTION AND ALT INFO SPECIFICALLY. BY THE TIME THE CTLR FINISHED THAT CALL; THE CIRRUS PASSED OVERHEAD. AT THAT TIME; WE CLEANED UP; BEGAN A CLB AGAIN. BY THE TIME WE PASSED 2500 FT IN THE CLB; WE WERE TOLD TO CALL DEP. AT THAT TIME WE CONTACTED DEP CTL AND THE FLT PROCEEDED NORMALLY. I DO NOT KNOW IF THE CIRRUS HAD BEEN IN CONTACT WITH DEP OR NOT. I THINK THAT THIS COULD HAVE BEEN AVOIDED IF THE TWR AT ZZZ HAD ISSUED TA'S; ESPECIALLY SINCE WE WERE ON AN ACTIVE IFR FLT PLAN. HOWEVER; I HAVE BEEN TOLD BY CTLR FRIENDS SINCE THE INCIDENT THAT CLASS D TWRS; SUCH AS ZZZ; ARE NOT AUTH TO DO THAT WITH PRECISE LOCATION INFO -- AT LEAST; THAT'S THE WAY I UNDERSTOOD IT. TO MY KNOWLEDGE; THERE ARE NO LASTING EFFECTS FROM THIS INCIDENT; OTHER THAN THOUGHTS OF IT. IT IS ALARMING; AS WE WOULD HAVE LIKELY HIT SINCE WE WERE IN THE PROCESS OF GEAR RETRACTIONS; POSTDEP CHKLISTS; ETC; WHEN THE TFC AURAL WARNING ACTIVATED.