CL65 CREW OFF CVG; IN HIGH RATE OF CLB TO 13000 FT; RECEIVE TCASII RA TO DSND AND AVOID A DO328 IN A CLB.

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WHILE CLBING TO INITIAL ALT OF 6000 FT; CVG DEP CTL GAVE US CLB CLRNC TO 8000 FT. SHORTLY THEREAFTER; THEY CLRED US TO 13000 FT. A FEW SECONDS AFTER GETTING THE CLRNC TO 13000 FT; ATC TOLD US SUDDENLY TO EXPEDITE OUR CLB (NOT TO A PARTICULAR ALT THOUGH). CLBING THROUGH APPROX 6000 FT AT APPROX +3800-4000 FPM; WE RECEIVED A TA ALERTING US TO TFC ABOVE US (APPROX 1000 FT ABOVE). ALMOST INSTANTANEOUSLY THE TA BECAME AN RA COMMANDING US TO DSND. IMMEDIATELY LOWERING THE NOSE AND REDUCING PWR TO IDLE; I PROMPTLY BEGAN TO TRANSITION THE ACFT FROM AN ALMOST 4000 FPM CLB; TO THE COMMANDED -1500 FPM DSCNT. WHILE PASSING THROUGH APPROX 0 FPM VSI IN THE DSCNT TRANSITION; THE RA QUICKLY CHANGED TO SAY; 'INCREASE DSCNT.' AS SUCH; WE INCREASED THE DSCNT RATE MORE; AND SUDDENLY VIEWED THE RA ACFT WITHIN 300 FT OF US; PASSING OFF OUR R SIDE. SEVERAL SECONDS LATER; WE HEARD THE 'CLR OF CONFLICT' AURAL; AND TRANSITIONED FROM OUR DSCNT BACK TO A CLB (NOW AT APPROX 4400 FT MSL). ONLY THEN DID ATC CALL OUT OUR TFC TO US AS IT PASSED OFF OUR R SIDE. WE INFORMED THEM THAT THAT WAS QUITE A CLOSE CALL; THAT WE HAD RESPONDED TO AN RA; AND WERE NOW AT 4400 FT RESUMING THE CLB. ATC RESPONDED SIMPLY; 'ROGER.' I FEEL THAT WHEN SEEING THE POTENTIAL CONFLICT WELL BEFORE WE DID (AS WE WERE IN IMC); ATC SHOULD HAVE VECTORED ONE OR BOTH ACFT TO INCREASE SPACING; OR HAD ONE OF US PERFORM A QUICK LEVELOFF. BY TRYING TO HAVE US EXPEDITE THE CLB; IT BROUGHT OUR 2 ACFT EVEN CLOSER TO EACH OTHER; AND FORCED AN EVASIVE MANEUVER TO ESCAPE. FORTUNATELY OUR TCASII SYS WAS NOT DEFERRED ON THIS FLT; AS IT PROVIDED THE CRITICAL INFO THAT ALLOWED US TO RESPOND APPROPRIATELY AND IN A TIMELY MANNER. SUPPLEMENTAL INFO FROM ACN 477509: THROUGHOUT ALL OF THIS; I'M STRAINING TO CATCH A GLIMPSE OF THE OTHER TFC AND DID FINALLY AS THE CLOUDS BROKE. THE OTHER TFC WAS LESS THAN 1/4 MI IN FRONT OF US CLBING L TO R. UPON REACHING APPROX 4500 FT THE TCASII ANNOUNCED 'CLR OF CONFLICT.' WE PROMPTLY NOTIFIED ATC OF THE RA AND OUR ALTDEV. PAX FELT THIS MANEUVER; BUT NO INJURIES. NO EXPLANATION WAS GIVEN TO US BY ATC AND WE WERE HANDED OFF TO CTR SHORTLY THEREAFTER. WHAT I FEEL COULD HAVE BEEN DONE TO AVOID THIS SIT FROM WHAT I WAS SEEING ON THE TCASII DISPLAY WOULD HAVE BEEN TO HAVE US TURN 15-20 DEGS L OR R OR HAVE ONE OF US LEVELOFF UNTIL THE OTHER ONE HAD PASSED BY. THE CTLR NEVER SHOULD HAVE GAMBLED THE WAY HE DID BY BETTING THAT OUR PLANE COULD OUT PERFORM THE OTHER.