FLT INSTRUCTOR PLT DESCRIBED CONFLICT WHILE ON CROSSWIND AT GFK WITH TFC ENTERING PATTERN; INDICATING WX AND OR VISIBILITY CONTRIBUTED TO THE INCIDENT.

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DUAL LCL PVT PLT TRAINING FLT: I WAS THE FLT INSTRUCTOR CONDUCTING THE FLT AND PRIOR TO TKOF ATIS WAS RPTING A SCATTERED LAYER AT 2000 FT. UPON CLBOUT FROM GFK; THIS SCATTERED LAYER HAD TURNED TO A BROKEN CEILING. THE DETERIORATING WX CAUSED US TO DISCONTINUE THE FLT TO THE PRACTICE AREA AND PRACTICE TFC PATTERNS AND LNDGS AT GFK. OUR ACFT WAS GIVEN RWY 17R AND TOLD TO MAKE R CLOSED TFC. THE FIRST 3 TFC PATTERNS WENT AS NORMAL AND SEPARATION FROM ALL OTHER TFC WAS MAINTAINED. ON OUR 4TH TKOF WE CLBED TO 1300 FT MSL (500 FT AGL) AND STARTED A R XWIND TURN ONCE WE STARTED TO ROLL WINGS LEVEL; TWR STATED KEEP THE R TURN TO THE N BECAUSE WE WERE CUTTING OFF TFC. I LOOKED TO MY L (SW) AND SAW AN ARROW ON A R DOWNWIND FOR RWY 17R. I TOOK CTL OF THE ACFT AND EXPEDITED OUR TURN TO THE N. WHILE OUR ACFT WAS ON UPWIND I DID NOT SEE ANY ACFT ON DOWNWIND OR ENTERING THE PATTERN AND THOUGHT THE AREA WAS CLR. TO MY KNOWLEDGE THE ONLY TFC ALERT THAT WAS ISSUED TO US FOR THAT ARROW WAS THE ONE TELLING US TO KEEP OUR R TURN IN. PRIOR TO THIS TFC ALERT; I DID NOT RECEIVE NOR VERIFY ANY INSTRUCTION TO FOLLOW THE ARROW OR EXTEND MY UPWIND IN THE PATTERN WITH ATC. I BELIEVE THAT THIS OCCURRED DUE TO THE REDUCED VISIBILITY AND LOWER CEILINGS WHICH MADE IT DIFFICULT TO SEE THE ARROW ON THE WHITE BACKGROUND; AS WELL AS ATC NOT COMMUNICATING INSTRUCTIONS EFFECTIVELY ON HOW THEY WANTED US TO FLY THE TFC PATTERN.