Integrated Models in Hydrology and Meteorology - Transport in the Atmosphere-Vegetation-Soil Continuum - page 316

SWAP solves this equation numerically, with an explicit, central inite difference
scheme:

θ

j ++ 11

c

j

+ ρ Q

j + 1

− − θ ρ

j

c

j

Q

j

i i b i i i b i

=

∆ t

j

( )
+

( )
+

 

j j j j j j j j

θ Dc c

z

θ Dcc −

j j j j

qc − qc 1

i − ½½ i − i − 1 i i + ½½ i + i i + 1

i −− ++ ½ i ½ i ½ i ½

+ 

 ) −




−

(9.10)

( ( )

∆ z ∆ z ½ ∆ ∆ z ½ ∆∆ z z

 

i i i − 1 i i i + 1

( ) −

µθ ρ b

j j

c

j

+ QKSc

j j j

i i i i ri i

where the superscript j denotes the time level, subscript i the node number and sub-
scripts i - 1/2 and i + 1/2 refer to linearly interpolated values at the upper and lower
compartment boundary, respectively. The scheme is explicit, as the unknown solute
concentration c i j +1 is a function of known variables at the former time level j . To
ensure stability of the explicit scheme, the time step Δt j should meet the criterion (Van
Genuchten and Wierenga, 1974 ):

2 j

∆ z θ j

∆ t ≤

i i

(9.11)

j

2 D

i

This stability criterion applies to nonsorbing substances and is therefore also safe for
sorbing substances.
In general convective water luxes dominate solute transport. To simulate solute
transport, the input data in addition to water low should specify the solute concentra-
tions in rain-, irrigation-, canal-, and groundwater and the initial solute concentrations.

9.1.7 Heat Flow
Soil temperature affects many physical, chemical and biological processes in the top
soil. Examples are the surface energy balance, soil hydraulic properties, decomposi-
tion rate of organic compounds and growth rate of roots. SWAP calculates the soil
temperatures either analytically or numerically. In Chapter 2 a general analytical
solution is discussed; here we explain a general numerical method based on De Vries
( 1963 ). The general low equation for soil heat can be written as ( Chapter 2 ):

∂
∂

T
z


 


 

∂ λ

s

∂
∂ =

T
t

C (9.12)

s

∂ z

where C s is the volumetric soil heat capacity (J m -3 K -1 ), T is the soil temperature (K),
and λ s is the thermal conductivity (J m -1 K -1 s -1 ). SWAP employs a fully implicit inite
difference numerical scheme to solve Eq. ( 9.12 ):