climb up in WordNet English dictionary

1. appear to be moving upward, as by means of tendrils
   "the vine climbed up the side of the house"
   ascend  climb up 
   
2. go upward with gradual or continuous progress
   "Did you ever climb up the hill behind your house?"
   climb  climb up  mount  go up 
   
3. rise in rank or status
   "Her new novel jumped high on the bestseller list"
   rise  jump  climb up 
   

climb up in English Wiktionary