CL65 FLT CREW ENCOUNTERS WINDSHEAR DURING A MISSED APCH; RECEIVES A GPWS ALERT OF 'DON'T SINK;' REGAINS A POSITIVE RATE OF CLB AND EXITS THE ADVERSE CONDITIONS.

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DURING MISSED APCH AND GAR; SHORTLY AFTER PF CALLED FOR FLAPS 8 DEGS AND GEAR UP; WE ENCOUNTERED LIGHT TO MODERATE TURB FOLLOWED BY A 'DON'T SINK' AURAL WARNING. A RATE OF DSCNT OF 800-1000 FPM WAS NOTED. FULL PWR WAS CONFIRMED AND THE NOSE ATTITUDE WAS FURTHER ADJUSTED UPWARD BUT WE STILL HAD THE DON'T SINK AURAL WARNING AND A VERT SPD OF 500 FPM DOWN FOR APPROX 5 MORE SECONDS. OUR RESULTING HIGH PITCH ATTITUDE AND EXCESSIVE AIRSPD AND RELUCTANCE TO OVER CTL THE ACFT AT THIS POINT CARRIED US TO 3800 FT MSL; 800 FT ABOVE THE ASSIGNED ALT. A SECOND APCH WAS MADE WITHOUT FURTHER INCIDENT. SUPPLEMENTAL INFO FROM ACN 693008: I WAS THE PF AND EXECUTED A PRE-PLANNED MISSED APCH. I HIT THE TOGA BUTTON; APPLIED GA THRUST; ASKED THE PNF TO SELECT THE FLAPS TO 8 DEGS; ACHIEVED A POSITIVE RATE OF CLB; AND ASKED THE PNF TO SELECT THE LNDG GEAR UP. AS WE CLBED THROUGH ABOUT 1200 FT MSL; I ASKED THE PNF TO SELECT HDG AND SPD MODE ON THE FLT DIRECTOR AND BUG OUR TWR ASSIGNED HDG OF 120 DEGS. AS I ENTERED THE R TURN; WE ENCOUNTERED MODERATE TURB; FOLLOWED BY A DOWNWARD TREND ON THE VSI. THE VSI THEN SHOWED A DSCNT AND THE GPWS GAVE THE ALERT 'DON'T SINK.'