Tumultuously just means a person place or thing is involved in or is becoming a victim violent or overwhelming turbulence or upheaval.
”There was something coming to her and she was waiting for it,fearfully.What was it? she did not know:it was too subtle and elusive to name.But she felt it,creeping out of the sky,reaching toward her through the sounds,the scents,the color that filled the air.”
”now her bosom rose and fell tumultuously.”
(The Story Of An Hour;Kate Chopin)
This word is usually used by people who do not want to use words like violent,rapid,fast or crazily.Whats happening in this instance is the main character is having a mini but violent borderline heart\lung attack.For extra visualization the author threw in the fact that her bosom and or chest area was going up and down.