  Breaking waves entrain gas beneath the surface. The wave-breaking process
energizes turbulent fluctuations that break bubbles in quick succession to
generate a wide range of bubble sizes. Understanding this generation mechanism
paves the way towards the development of predictive models for large-scale
maritime and climate simulations. Garrett et al. (2000) suggested that
super-Hinze-scale turbulent breakup transfers entrained gas from large to small
bubble sizes in the manner of a cascade. We provide a theoretical basis for
this bubble-mass cascade by appealing to how energy is transferred from large
to small scales in the energy cascade central to single-phase turbulence
theories. A bubble break-up cascade requires that break-up events predominantly
transfer bubble mass from a certain bubble size to a slightly smaller size on
average. This property is called locality. In this paper, we analytically
quantify locality by extending the population balance equation in conservative
form to derive the bubble-mass transfer rate from large to small sizes. Using
our proposed measures of locality, we show that scalings relevant to turbulent
bubbly flows, including those postulated by Garrett et al. (2000) and observed
in breaking-wave experiments and simulations, are consistent with a strongly
local transfer rate, where the influence of non-local contributions decays in a
power-law fashion. These theoretical predictions are confirmed using numerical
simulations in Part 2, revealing key physical aspects of the bubble break-up
cascade phenomenology. Locality supports the universality of turbulent
small-bubble break-up, which simplifies the development of subgrid-scale models
to predict oceanic small-bubble statistics of practical importance.
