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1. To surround something with snow, rendering it impassable, immovable, or inoperable. Often used in passive constructions. A noun or pronoun can be used between "snow" and "under." Traffic on the interstate was so bad that the snowstorm that blew through actually ended up snowing many of cars under, causing even more traffic problems as a result. I just got a call from the ski lodge. Apparently they were snowed under by the blizzard last night, and there's no way to get in or out of the place at the moment.
2. To cause someone or something to be unable to leave a building or area due to the snow. Often used in passive constructions. A noun or pronoun can be used between "snow" and "under." We were snowed under for several days, with only a small cupboard's worth of food to subsist on. The blizzard snowed us under, so we spent a lovely weekend inside reading.
3. By extension, to overwhelm or overwork someone, especially with work. Often used in passive constructions. A noun or pronoun can be used between "snow" and "under." I'm sorry I haven't returned your calls. I've just been so snowed under at work lately that I've barely had any time to myself. The boss keeps snowing me under with these huge overlapping projects.
Very busy or overwhelmed with something. This phrase evokes the image of being buried under an avalanche. I'd love to go out to dinner tonight, but I'm totally snowed under at the office right now. Kate's not coming tonight because she's snowed under with research for her thesis.
overworked; exceptionally busy. Look, I'm really snowed under at the moment. Can this wait? He really has been snowed under with work.