A LEARJET 31 CAPT RPTED BUSTING THEIR ASSIGNED ALT DURING CONVECTIVE ACTIVITY AT 8000 FT.

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WHILE CLBING TO 8000 FT MSL AS ASSIGNED BY ATC; WE HIT AN UPDRAFT AT 7800 FT MSL THAT CAUSED THE AIRPLANE TO GO FROM 2000 FPM TO 4000 FPM ON THE VSI. WE WERE FLYING WITH THE AUTOPLT ENGAGED AND ALT SELECT DEPRESSED. WHEN WE HIT THE UPDRAFT; WHILE FLYING THROUGH A SMALL CELL; THE ALT SELECT DISENGAGED BY ITSELF AS THE ACFT ACCELERATED TO 4000 FPM CLB AND THE AIRSPD JUMPED FROM 245 KIAS TO 270 KIAS. I IMMEDIATELY RETARDED THE THROTTLES TO IDLE AND STARTED TRIMMING FULL NOSE DOWN AND ATTEMPTED TO DEPRESS THE PRESELECT ALT KEY TWICE. THE CLB RATE WAS TOO GREAT FOR THIS FUNCTION; SO I HIT THE CAPTURE KEY AS WE WERE CLBING THROUGH 8000 FT MSL; ALL THE WHILE STILL TRIMMING DOWN. WE ARRESTED THE CLB AT 8240 FT MSL AND STARTED BACK DOWN TO 8000 FT MSL. (WE ENDED UP DSNDING BACK TO 7800 FT BEFORE WE LEVELED SINCE WE WERE SO AGGRESSIVE IN TRYING TO STOP THE PLANE FROM CLBING AT THIS TIME.) THE CTLR ASKED US OUR ALT ASSIGNED AND MY FO REPLIED 8000 FT. THE CTLR THEN ASKED OUR CURRENT ALT AND I RESPONDED '8200 FT AND DSNDING TO 8000 FT; WE HIT AN UPDRAFT IN THAT CELL AND IT POPPED US UP.' HIS RESPONSE WAS THAT WASN'T GOOD ENOUGH; THAT WE SHOULD HAVE ALERTED HIM IMMEDIATELY WITH THE PROB. HOWEVER; WE COULDN'T AVOID THE CELL SINCE WE DIDN'T WANT TO DEVIATE WITHOUT PERMISSION AND THE FREQ WAS CONGESTED. THEN; DURING THE ALT CLB THROUGH; I WAS SO BUSY FLYING THE ACFT I DIDN'T HAVE THE TIME TO TAKE TO TALK TO ATC. I WAS BUSY ALERTING MY FO; WHILE I TOOK MEASURES TO CORRECT THE ERROR. THE CTLR CAME BACK WITH A NUMBER WE NEEDED TO CALL FOR A POTENTIAL VIOLATION INTO ANOTHER ACFT'S AIRSPACE THAT WAS FLYING ABOVE US. I BELIEVE; TO PREVENT A RECURRENCE; WE EITHER NEED MORE CTLRS WHEN THE WX IS DOWN SO THERE IS MORE AIR TIME TO COMMUNICATE AVAILABLE; OR PUT A LARGER SEPARATION BTWN ACFT THAN 1000 FT WHEN THERE IS MODERATE RAIN AND CONVECTIVE ACTIVITY IN THE AREA TO BE FLOWN IN.