A CLBING B737-300 FLT CREW DISREGARD A ZFW CTLR DSCNT CLRNC AFTER RECEIVING A TCASII CLB COMMAND DUE TO A CONVERGING; DSNDING BE40.

Narrative

NORMAL OP UNTIL SHORTLY AFTER TKOF RWY 16R IN MAF. CLBING ON THE RWY HDG; WE TOOK A HDOF TO DEP CTL AND WERE THEN GIVEN AN INITIAL HDG OF 130 DEGS. APCHING OUR INITIAL CLRNC ALT OF 13000 FT; WE ASKED DEP CTL FOR HIGHER AND WERE HANDED OFF TO ZFW ON FREQ 126.15. WE WERE NOW LEVEL AT 13000 FT. CHKING IN WITH ZFW; WE WERE TOLD TO 'CLB AND MAINTAIN FL180.' WE ACKNOWLEDGED AND IMMEDIATELY STARTED THE CLB. HAVING MADE CONTACT WITH CTR; MY FO NOW WENT OFF TO CALL IN THE TIMES TO MAF. WHILE CLBING AT A HIGH RATE PASSING 14200 FT; I SAW A TARGET ON THE TCASII SCREEN APCHING FROM 1 O'CLOCK POS; AND DSNDING TOWARD US FROM ABOVE. AT ALMOST THE SAME TIME; THE ZFW CTLR CALLED US: 'MAINTAIN 13000 FT.' I STARTED A L 25 DEG BANK TURN AND REPLIED THAT I WAS APCHING 15000 FT. HE RADIOED; 'RETURN TO 13000 FT.' I ROGERED THAT AND ADDED THAT I WAS ALSO TURNING L TO AVOID THE TFC. I PULLED OFF PWR AND PUSHED THE NOSE OVER ABRUPTLY. WITHIN A FEW SECONDS; THE TCASII GAVE ME AN RA; 'CLB.' I IMMEDIATELY PUSHED THE PWR UP AND BEGAN TO CLB. THE FO WAS BACK ON FREQ BY NOW AND HE INFORMED THE CTLR THAT WE HAD RECEIVED A CLB RA AND WERE COMPLYING WITH IT. WHEN THE TFC WAS WELL CLR; WE REQUESTED SOME INFO ON THE ACFT WITH WHOM WE HAD THE CONFLICT. THE CTLR SIMPLY RESPONDED THAT IT WAS A 'BEECH JET.' ALTHOUGH WE HEARD THE CTLR MAKE SEVERAL CALLS TO THE BEECH JET; WE NEVER HEARD ANY OF THE RESPONSES. THE CTLR DID NOT ACCUSE THE BEECH JET OF DSNDING THROUGH AN ASSIGNED ALT. I DO NOT KNOW EXACTLY WHAT CAUSED THE TFC CONFLICT. BECAUSE OF THE UNUSUAL ASPECT OF 'CLB; ABRUPT DSCNT AND ABRUPT CLB' THAT HAD OCCURRED; I MADE A PA TO THE PAX TO EXPLAIN WHAT HAPPENED. NOBODY HAD ANY QUESTIONS OR COMMENTS AFTER THE FLT.