rates

The rate of water mass change with time can be represented by the inverse of a characteristic time,, the time required for the mass to change by about one third of its value:

Rate = ^-1 ( 1 / M ) dM / dt

sedimentation rate (cloud particles)

For conditions in the lowest 10-15 km of Earth's atmosphere, the sedimentation (or fall) rate of a spherical cloud particle is given as the characteristic time for it to fall a distance of one atmospheric scale height (about 8 km on Earth) by

^-1_fall = (2 _p mg² / 9 kT ) r²_M

where is the atmospheric dynamic viscosity (the resistance of the atmosphere to movement through it), (kT/mg) is the atmospheric scale height (k is Boltzmann's constant, T is atmospheric temperature, m is weight of a gas molecule, g is acceleration by gravity), _p is the mass density of the cloud particle, and r_M is the mass-weighted-average cloud particle radius. Essentially, the sedimentation rate increases proportionally to the particle cross-section (r²_M).

sedimentation rate (precipitation particles)

For precipitation-sized particles, the sedimentation rate formula must be modified. In the case of rainfall, for particles larger than several hundred microns, the shape is distorted so that the particles fall more slowly than a sphere. Thus, their sedimentation rate is smaller than given by the formula above. If the droplets become too large, they actually fall apart:

r_max (hydro) ( / 2) [ / g ( _v - )]^1/2

where is the surface tension of liquid water and _v is the density of water vapor at the droplet's surface. For snowfall, the highly non-spherical shapes and low densities mean that the sedimentation rates are much larger than for a sphere containing the same amount of water. Hence for larger precipitation-sized particles the sedimentation rate increases more slowly with increasing size than for a rigid sphere.

condensation rate

In Earth's lower atmosphere the condensation rate is given by

^-1_cond = ^-1 (4 _s / _p) S_max r_M^-2

where is the atmospheric density, _s is the water vapor density at saturation (100% relative humidity), _p is the density of condensed water, is the atmospheric viscosity and S_max is the supersaturation vapor density (S_max = 0.001 is equivalent to a relative humidity of 100.1%). Essentially the condensation rate increases as the supersaturation, which increases with stronger upward motions, and decreases as the cloud particle grows larger (proportional to the particle surface area, r_M^-2).

collision rate

In Earth's lower atmosphere the collision rate of cloud particles (or the coalescence rate) is given by

^-1_coal ( _p g / 9 ) N r_M⁴

where _p is the particle mass density, g is the acceleration of gravity, is the atmospheric viscosity, and N is the number density of cloud particles. Essentially, the collision rate increases as the number density of falling cloud particles increases and increases very rapidly as the size of the particles increases, both because larger particles fall faster and because the collision cross-section is larger. Note that this expression assumes a distribution of particle sizes; if all the particles are the same size and fall at the same speed. there will be no collisions.

For growth to occur, colliding particle must stick together. If the cloud particles are too small the air layer between two approaching particles prevents actual contact. This is why most clouds with less than the critical amount of water do not produce precipitation. If the particles are large enough, then liquid droplets always combine upon contact, whereas ice crystals and snow flakes do not always combine. At temperatures near freezing, there is usually some liquid on the ice particles that greatly enhances sticking; at much colder temperatures, ice particles do not stick very well.