  Deep-water surface wave breaking affects the transfer of mass, momentum,
energy and heat between the air and sea. Understanding when and how the onset
of wave breaking will occur remains a challenge. The mechanisms that form steep
waves, i.e. nonlinearity or dispersion, are thought to have a strong influence
on the onset of wave breaking. In two-dimensions and on deep-water, spectral
bandwidth is the main factor that affects the roles these mechanism play.
Existing studies, in which the relationship between spectral bandwidth and wave
breaking onset is investigated, present varied and sometimes conflicting
results. We perform numerical simulations of two-dimensional focused wave
groups on deep-water to better understand this relationship, with the aim
reconciling existing studies. We show that the way in which steepness is
defined, may be the main source confusion in the literature. At breaking onset,
locally defined steepness reduces as a function of bandwidth, and globally
defined steepness increases. The relationship between global breaking onset
steepness and spectral shape (using the parameters bandwidth and spectral
skewness) is too complex to parameterise in a general sense. However, we find
that the local surface slope of maximally steep non breaking waves, of all
spectral bandwidths and shapes (constant-steepness, constant-amplitude, and
JONSWAP), approaches a limit of $1/\tan(\pi/3)\approx0.5774$. This slope based
threshold, is simple to measure and may be used as an alternative to existing
kinematic breaking onset thresholds. There is a potential link between slope
based and kinematic breaking onset thresholds which future work should seek to
better understand.
