Atmospheric convection is the result of a parcel-environment instability, or temperature difference, layer in the atmosphere. Different lapse rates within dry and moist air lead to instability. Mixing of air during the day which expands the height of the planetary boundary layer leads to increased winds, cumulus cloud development, and decreased surface dew points. Moist convection leads to thunderstorm development, which is often responsible for severe weather throughout the world. Special threats from thunderstorms include hail, downbursts, and tornadoes.
There are a few general archetypes of atmospheric instability that correspond to convection and lack thereof. Steeper and/or positive lapse rates (environmental air cools quickly with height) suggests atmospheric convection is more likely, while weaker and/or negative environmental lapse rates suggest it is less likely. This is because any displaced air parcels will become more (less) buoyant, given their sign of adiabatic temperature change, in the steep (weak) lapse rate environments.
Convection begins at the level of free convection (LFC), where it begins its ascent through the free convective layer (FCL), and then stops at the equilibrium level (EL). The rising parcel, if having enough momentum, will continue to rise to the maximum parcel level (MPL) until negative buoyancy decelerates the parcel to a stop.
Acceleration is of little relevance to convection. Drag produced by the updraft creates an opposite force to counter that from the buoyancy [1]. This could be thought of as similar to the terminal velocity of a falling object. This force from buoyancy can be measured by Convective Available Potential Energy (CAPE), or the joules of energy available per kilogram of potentially buoyant air. A theoretical updraft velocity can be derived from this value via substitution into the kinetic energy equation, although this value will be an underestimation given the aforementioned drag or entrainment effects holding back further acceleration at some point. See the CAPE, buoyancy, and parcel links for a more in depth mathematical explanation of these processes.