The most common and most likely answer to this crossword clue is the 9 letter word ASCENDANT
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More clues leading to the result ASCENDANT
Buddy explains!

The straight part
We know that "on the rise" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case on the rise = ascendant.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does ASCENDANT mean?
ASCENDANT noun- Being in control; superiority, or commanding influence; ascendency.
- An ancestor (antonym of descendant)
- (usu. followed by to) A royal heir assuming (a place of power)
- Ascent; height; elevation.
- The horoscope, or that degree of the ecliptic which rises above the horizon at the moment of one's birth; supposed to have a commanding influence on a person's life and fortune.
- Rising, moving upward.
- Surpassing or controlling.