Thayer WatkinsSilicon Valley & Tornado Alley USA |
---|

Due to Diffusion |

**In cloud physics there is an equation, usually called the growth
equation, which describes how a cloud droplet grows. Rogers and
Yau in their A Short Course in Cloud Physics provide a derivation
of this equation. Although overall Rogers and Yau's text is excellent,
in the matter of the derivation of the growth equation their presentation
is incomplete and points confusing because it misleads the reader. What
follows is an attempt to provide a more comprehensive derivation. The
presentation of the derivation differs from that of Rogers and Yau's in
notation. The generic radial distance variable is denoted by the lower
case symbol r and upper case R is used to denote the particular radius of
the droplet. This is the opposite of Rogers and Yau. The environmental
levels of water vapor density and temperature are identified by a
subscript e, in contrast to Rogers and Yau who denote the environmental
levels by the absence of a subscript. The values of water vapor
density and temperature are denoted here by a subscript R.
**

Cloud droplets may grow by condensation or diminish by evaporation. Both condensation and evaporation are diffusion processes described by the Fick diffusion equation; i.e., the flux of water vapor passing through a plane M is given by:

where D is the diffusion coefficient for water vapor and ρ is the density of water vapor in the space surrounding the droplet. The negative sign in the above equation reflects the fact that the mass flow is in the opposite direction from the gradient of water vapor density.

When water vapor diffuses into the droplet it brings heat energy which raises the temperature of the droplet and that heat energy may diffuse out of the droplet into the surrounding moist air. Such diffusion of heat energy obeys the same form of equation as does the diffusion of the water vapor. The heat energy flux Q passing through a plane is given by

When the net flow of water vapor or heat into or out of an infinitesimal volume is considered they must generate a change in the density and the temperature within that infinitesimal volume; i.e.,

and

c∂T/∂t = K∇

Under steady-state conditions (∂ρ/∂t = 0 and ∂T/∂t = 0) the vapor density and temperature fields in the space surrounding the droplet must satisfy the equations:

and

∇

In spherical coordinates under conditions of spherical symmetry
the Laplacian ∇^{2} is:

where r is the distance from the center of the droplet.

For a droplet of radius R the boundary conditions are:

and

ρ -> ρ

where ρ_{e} and T_{e} are the environmental vapor density and temperature,
respectively.

The solution to the steady-state diffusion equation reduces to integrating

once with respect to r to get

where C_{1} is a constant.

Dividing by r^{2} gives the equation

Integating this with respect to r gives

The integration constants must be chosen to satisfy the boundary conditions.

and

ρ

Subtracting the second equation from the first gives

and thus

C

The solution to the boundary value problem for the steady-state diffusion equation is therefore

This equation reflects the fact that if ρ_{e} is greater
than ρ_{R} then ρ is decreasing smoothly as r decreases
toward R.

Temperature obeys the same equations as water vapor density so

which can be rearranged to

T(r) = T

The latter form reflects the fact that if
T_{R} is greater than T_{e} then T increases smoothly
as r decreases to R.

At the surface of the droplet

and

(∂T/∂r)

Therefore the net amounts of water vapor mass and heat energy diffusing into and out of the droplet surface are:

and

Q = (4πR

The heat energy supplied to the droplet from the condensation of water vapor into the droplet is:

For equilibrium this must match the heat energy Q diffused out of the droplet; i.e.,

4πLDR(ρ

and hence

(T

This above equation may be solved for the unknown ρ_{R}
in terms of the unknown T_{R} as

The water vapor density at the surface of the droplet, ρ_{R},
also obeys the ideal gas law so

where R_{V} is the gas constant for water vapor and
e'_{s}(T_{R}) is the saturated vapor pressure at
the temperature T_{R} taking into account the Kelvin effect
due to the curvature of the droplet.

From the Kelvin Equation and the Raoult effect e'_{s} is given by

where e_{s} is the vapor pressure of water over an
infinite plane.

Combining the two equations for ρ_{R} gives

Dividing by (e_{s}/R_{V}T_{R})
gives:

The RHS of the above equation is a function of the known parameters
and T_{R}; the LHS is a cubic equation in (1/R). Thus for
given ρ_{e} and T_{e} the solution for R for any
value of T_{R} can be found. From the value of T_{R}
the value of ρ_{R} can be found. It is, of course, the
inverse relationships of T_{R} and ρ_{R} as functions
of R that are desired but that is a mere mathematical transformation.

of the solution to the equation

Let ρ_{VS} and e_{S} be the density and vapor pressure
of water vapor under saturated conditions. The Clasius-Clapyeron equation
is usually expressed in the form:

where L is the latent heat of fusion and the α's refer to the specific densities of water vapor and liquid water under conditions of saturation. The specific volumes are just the reciprocals of the densities. Therefore the Clasius-Clapyeron equation can be expressed as:

Since ρ_{VS} in negligible compared to
ρ_{L} the above equation reduces to:

But water vapor satisfies the ideal gas law

and so

de

Equating the two expressions for de_{S}/dT and dividing both sides
by ρ_{VS}R_{V}T gives:

or, in differential form

dρ

The integration of this latter differential equation from T_{e}
to T_{R} yields

= (L/R

At this point Mason makes use of an approximation:

Thus if (y/x) is approximately 1

= ln(1+(y-x)/x) = (y-x)/x

Applying this approximation to the previous equation yields

= (L/R

or after factoring

(ρ

Since (T_{R}-T_{e}) = L(dm/dt)/(4πKR)
the previous equation can be expressed as

= [(L/R

It was previously established that

or, upon division by ρ

(ρ

The two equations above can be added together. The addition of the LHS gives:

(ρ

Under the assumption that ρ_{R}=ρ_{RS} the above
reduces to

where S is called the saturation ratio and (S-1) the supersaturation ratio.

The RHS of the two equations which were added together is

Solving for dm/dt gives

= (4πR(S-1))/[(L/R

Since

and thus

dm/dt = ρ

It thus follows that

= (S-1)/[(L/R

The density ρ_{eS} can be replaced by e_{eS}/R_{V}T_{e} to obtain
the growth equation:

= (S-1)/[(L/R

= (S-1)/[(L/R

the final modification is to assume T_{R} to be equal
to T_{e}.
Thus the final version of the growth equation is:

= (S-1)/[(L/R

Sources:

- R.R. Rodgers and M.K. Yau,
*A Short Course in Cloud physics*, 3rd Edition, Butterworth-Heinemann, Woburn, MA, 1989. - Basil J. Mason,
*The Physics of Clouds*, 2nd Edition, Clarendon Press, Oxford, 1971.