B737 CAPT REPORTS GETTING STUCK IN SNOW DURING TAXI AT CLE DUE TO POOR VISIBILITY REQUIRING TUG FOR EXTRACTION.

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I HOPE AFTER THIS RPT IS READ THAT THERE WILL BE A DISCUSSION WITH THE CLE ARPT MGR. FOR AN ARPT THAT RECEIVES SIGNIFICANT SNOW; BOTH LAKE EFFECT AND OTHERWISE; THE SNOW REMOVAL AND GND OPS ON CLE RAMP WERE THE WORST I HAVE SEEN IN 12 YRS AT MY ACR AND 10 YRS IN THE MIL. WE HAD 2 RETURN TO GATE EVENTS DUE TO HVY SNOW; RWY CLOSURE; NEED FOR ADDITIONAL DE-ICE; RAMP AREAS SO SLIPPERY THAT THE TUGS WERE UNABLE TO PUSH THE ACFT; ETC. OUR MAJOR ISSUE OCCURRED WHEN AFTER DE-ICE WE MOVED FROM TXWY M2; R ON TXWY L; R ON TXWY R; THEN R ON TXWY J. THE R ON TXWY J FROM TXWY R RESULTED IN OUR R ENG INGESTING SNOW PILED ALMOST 2 FT HIGH THAT ROLLED THE ENG BACK AND CAUSED THE GENERATOR TO DROP OFF LINE. THE ENG DID NOT FLAME OUT; AND RECOVERED ONCE THE SNOW PASSED THROUGH. THE PROB I WOULD LIKE TO BE NOTED IS THAT THIS ARPT WAS OPERATING IN REDUCED VISIBILITY WITH SNOW CAUSING ZERO DEPTH PERCEPTION RESULTING IN AN INABILITY TO DISCERN WHERE AND HOW HIGH SNOW HAD BEEN PLOWED. THE SNOW REMOVAL EFFORT WAS FRACTIOUS AND HAPHAZARD RESULTING IN SNOW BERMS ALL OVER THE PLACE THAT FLT CREWS COULDN'T SEE UNTIL GETTING STUCK; AS WE DID SHORTLY AFTER OUR ENG ROLLBACK TURNING ONTO TXWY J. WHEN OUR TUG PUSHED US BACK FROM THE SNOW DAM WE HAD CREATED IN FRONT OF OUR R MAIN GEAR JUST FROM TAXIING; THE SNOW WAS NEARLY 3 FT HIGH. WHEN WE LEFT; ALL TXWY SIGNS WERE STILL UNDER SNOW; AS THEY HAD BEEN THE DAY BEFORE. NONE HAD BEEN CLRED FOR PLTS TO BE ABLE TO USE TO DETERMINE POS AND MERELY SHOVELING SNOW OUT FROM AROUND THEM WOULD HAVE MADE THEM PERFECTLY USEABLE AND VISIBLE. THERE WERE NO TAXI LINES THAT COULD BE SEEN. NO DYE OR PAINT HAD BEEN USED TO MARK SNOWDRIFTS AND BERMS; SOME OF WHICH WERE MORE THAN HIGH ENOUGH TO CLIP A B737 WING OR AN ENG POD. IN SPITE OF THE SEVERELY REDUCED VISIBILITY CONDITIONS AND THE SNOW PILES; LACK OF MARKINGS AND LACK OF SIGNAGE; GND CTL CONTINUED TO USE 'NORMAL' TAXI CLRNC TERMINOLOGY REQUIRING AN AD HOC AND DANGEROUS GAME OF INTERP AND GUESSWORK AS WE TRIED TO DETERMINE IF WHERE WE THOUGHT WE WERE; WHERE WE WERE SUPPOSED TO BE; AND WHERE WE ACTUALLY WERE IN RELATION TO THE ACTUAL TXWY MARRIED UP. RECOMMENDATIONS: 1) CLE NEEDS A MUCH BETTER SNOW REMOVAL PLAN THAT INCORPORATES A PLANNED FLOW OF ACFT TO THE OPERATING RWY AND FOCUSES THE AVAILABLE SNOW REMOVAL EQUIP ON MAKING THAT HAPPEN INSTEAD OF PLOWING IN CIRCLES MAKING SNOWDRIFTS THAT ACFT CAN'T TAXI THROUGH. 2) IF OVERWHELMED (AS OCCURRED HERE) THE ARPT NEEDS TO MAKE AN EARLIER CESSATION OF OPS DECISION. IT ALSO NEEDS TO REALIZE THAT THEY HAD ATIS UP FOR 3 HRS WITHOUT NOTING IN THE ATIS THAT THE ARPT WAS OTS UFN. 3) ONCE THE EVENT IS OVER; THE ARPT NEEDS TO MAKE SIGNAGE VIEWABLE AND SNOW DRIFTS/BERMS/AREAS OBVIOUS WITH DYE OR PAINT. SAME FOR MAKING TAXI RTES MORE OBVIOUS.