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Having lived in the city, this author has personally seen snow accumulate on some of the higher hills but melt in lower areas because of just a couple of degree difference in temperature. One day at my apartment complex, on the side of a steep hill in the west side of town, snow was accumulating and turning the ground white by the buildings near the top. Near the bottom at my own building the snow did not accumulate.
Although Asheville is in a valley, it is still in the overall Appalachian highlands and experiences some enhancement of snow from upslope winds from both the north to west and east to south directions. Much of the enhanced precipitation is blocked by higher elevations, especially off to the northwest. In fact, snow events from northwest wind flow and moisture of the off the Great Lakes are frequently reduced to nothing but flurries.
Freezing rain events sometimes occur due to cold air damming. Cold air damming occurs best when a high pressure center over the Northeast or Middle Atlantic states pushes cold air against the mountains. The air is then forced to slide south and west along the adjacent piedmont to the east. Some of the air can spill over into the valleys. When this happens in Asheville and warmer air is running over the colder air, freezing rain results. Depending upon the vertical profile of the air above the cold air near the surface, sleet or snow can also fall. Cold air damming events are generally not as long lasting in Asheville as they are for places like Winston-Salem or Greensboro in the western piedmont. This case is especially true for very shallow events.
What is intriguing about during cold air damming events in Asheville is a strange looking south wind that occurs at the airport. You see the winds in the observations, for example, blowing from the south or south-southeast at 15 to 25 mph. At the same time the temperature is falling. The surrounding topography appears to be the main culprit by forcing the wind into that direction. Two other possible influences are worth considering. First, Asheville is typically situated to the west of the high pressure ridge axis prevalent during cold air damming events. The winds to the west of the axis tend to turn from the east or even southeast. Second, the winds veer with height as one goes from the northeast wind at the bottom of the cold air to the south component (SE to SW) wind at the interface of the cold and warm air. The situation is worth taking another look at.