Equations of Atmospheric Flow in the ABL - Terrestrial Hydrometeorology - page 245

where q is the specific humidity of the air, q is the liquid/solid water content per

unit mass, E v is the rate of creation of water vapor by evaporation/sublimation of
liquid/solid water (in kg m −3 sec −1 ), and S q and S q l are possible separate source/
sinks terms (in kg m −3 sec −1 ) corresponding to chemical formation for vapor and
liquid/solid water, respectively. Equations (16.46) and (16.47) can also be written
more concisely in vector format similar to Equation (16.45), see Table 16.1.

′

Conservation of energy

The time rate of change of potential temperature of a parcel of air in the atmos-
phere is equal to the sum of three terms, namely, the (by now familiar) rate of
inflow or outflow of thermal energy by molecular transfer processes, energy
entering or leaving the parcel of air as radiation, and energy that is released or
absorbed as a result of phase changes between water vapor and liquid or solid
water within the parcel. Hence, the required equation can be written (for concise-
ness here in vector format) as follows:

∂θ +∇θ=υ∇θ− 1 λ E

2

v . ∇ − R

v

(16.48)

θ n

∂ t r c r c

ap ap

where E v is the moisture evaporated within the parcel (in kg m −3 sec −1 ), l is the
latent heat associated with the phase change from liquid/solid to water vapor, r a
and c p are respectively the density and specific heat at constant pressure of moist
air. In this equation net radiation has three axial components and R is the net
radiation vector and the second term on the right hand side of Equation (16.48),
the divergence of the net radiation flux , is made up of three terms, one for each axis,
thus;

∂ () R

⎡ () R () R ⎤

1 1 ∂ ∂

ny

R =

nx nz

(16.49)

∇ + +

⎢ ⎥

n

r c r c ∂ x ∂ y ∂ z

⎣ ⎦

apm apm

Among these three terms it is the third, that associated with the vertical flux of
net radiation, which is usually dominant, and in the ABL and over an 'ideal'
surface, it is assumed that there is no change in horizontal net radiation transfer,
see Equation (5.28).

Conservation of a scalar quantity

The conservation equation for any scalar quantity c (e.g., the concentration of
carbon dioxide) can be written by equating the total derivative of the quantity
to two terms one of which represents the divergence of the flux transferred by