Environmental Engineering Reference
In-Depth Information
∂
c
r
c
c
modes
∑
(1)
+
div v
(
ρ
)
=
div
(
K
ρ
∇
)
+
( ;
R c T q
,
)
+
Q
−
k
m
(
c
−
c
*
(
a
m
))
c
t
Eq
∂
t
ρ
ρ
m
m
m
m
∂
a
r
a
a
(2)
m
m
m
*
m
+
div v
(
ρ
)
=
div
(
K
ρ
∇
)
+
P
( ;
a T q
,
)
+
Q
+
k
(
c
−
c
(
a
))
a
t
Eq
∂
t
ρ
ρ
for all modes
m
. Here
c
is the vectors of gas phase concentrations. The vectors
a
m
denote the component masses and the particle number of mode
m.
The wind
vector
v
r
, the density ρ, the temperature
T,
and the humidity
q
have been provided
by the meteorological code. The terms
R
,
Q,
and
P
m
stand for the chemical
reactions in the gas phase, the emissions, and the aerosol dynamical transformations
(e.g., coagulation). The parameters
m
k
denote the mass transfer coefficients
t
m
E
c
stands for the equilibrium saturation of the particles in mode
m
. Note that
gas and particle phase are only coupled by the last term which describes the phase
transfer.
After spatial discretization this system can be written as a system of ordinary
differential equations (ODEs) of the form
and
Slow
Fast
c
'
=
f
(,)
tc
+
f
(,)
tc
+
f
(,)
tc
+
f
(,,)
tca
+
f
(,)
tc
(3)
Hor
Hor
Vert
Phase
Chem
af
'
=
Slow
(,)
t a f
+
Fast
(,)
t a f
+
(,)
t af
+
(,,)
t acf
+
(,)
t a
(4)
Hor
Hor
Vert
Phase
Phys
The time step restrictions imposed by the different terms on the right hand side
may differ by several orders of magnitude. In chemistry-transport models,
stiffness is expected especially for the fast chemical and aerosol dynamical terms,
whereas the horizontal transport part is usually characterized by moderate
changing courses. The behaviour of vertical exchange is varying from slow to fast
depending on the dynamics of the atmosphere. Therefore, as extension to the
scheme implemented in MUSCAT (Wolke and Knoth, 2000), the singlerate
Runge-Kutta method applied for the horizontal transport integration is replaced by
multirate schemes where the step sizes are restricted only by the local CFL
numbers. The stiff chemistry, aerosol dynamics and all vertical transport processes
are integrated in a coupled manner. This integration can be performed by higher
order explicit schemes or, again, by an IMEX scheme where the vertical transport
is treated explicitly. In both cases, multirate techniques can be applied again.
Special care must be taken in regard to time step selection and the partitioning for
parallel execution.
3. Numerical Tests
The efficiency of the multirate schemes is analyzed for more “theoretical” advection
problems and one real scenario. Here the model system COSMO-MUSCAT was
applied in a nested hierarchy with the superior control by the global reanalysis
