Download numerical methods for

Different numerical approaches and numerical
schemes for solving the governing equations.
Special subject: numerical methods for solving
continuity equations
Eigil Kaas, Niels Bohr Institute, University of Copenhagen, Denmark
MUSCATEN / NETFam Young Scientist Summer School – 2011,
Odessa State Environmental University
On-line and off-line atmospheric
chemistry transport (ACT) models
On-line chemistry permits explicit simulation of feedbacks
between atmospheric dynamic and chemical processes.
On different time-scales the following issues are particularly
relevant:
•  CO2 and CH4 cycles (well mixed greenhouse gases)
•  O3 (effects in both the short and longwave domain)
•  Direct effect of aerosols (mainly shortwave)
•  Indirect/semi-direct aerosol forcing (coupling to clouds
and related radiation)
Applications
Air quality predictions (e.g. Enviro HIRLAM)
Earth system (climate) simulations (global and regional
models, e.g. HadCM3, ECHAM6, RegClim).
Governing equations
Repetition from Sergyi’s presentation
Exact governing dynamical equations
Navier-Stokes equation (Unit mass
version of “Newton's second law)
expressed
in
the
accelerated
coordinate system of the Earth
dU
1
= −2Ω × U − ∇p + g + F
dt
ρ
The first equation of thermodynamics
dT
dα
dT
dp
cv
+p
= cp
−α
=Q
dt
dt
dt
dt
Continuity equation for dry air
d ρd
= − ρd ∇ ⋅U
dt
Continuity equation for various tracers
(e.g. water vapour, liquid and solid
water, SO2, 137Cs, particles …)
d ρt
= − ρt ∇ ⋅U + S ρ
t
dt
Equation of state for ideal gases (no
particle density included in ρ).
p = ρ RT
ρ = ρ d + ∑ ρt
t
Continuity equations
Conservation of mass or conservation of mixing ration?
Conservation of dry air mass. The volume density continuity equation:
∂ρd
= −∇ ⋅ ( ρdV )
∂t
d ρd
= − ρd ∇ ⋅V
dt
⎫
Eulerian formulation ⎪ ρ density of
⎪
d
⎬
dry air
⎪
Lagrangian formulation
⎪⎭
Conservation of mass of a chemical species, t:
∂ρt
= −∇ ⋅ ( ρtV ) + Dρ + S ρ
t
t
∂t
d ρt
= − ρt ∇ ⋅V + Dρ + S ρ
t
t
dt
⎫ ρ density of
⎪⎪ t
tracer t
⎬
⎪ S sources/sinks
⎪⎭ ρt
Continuity equations
Chemistry works on mixing ratio - NOT on density!
ρt
qt =
ρd
mixing ratio of chemical tracer t
From the volume density continuity equation for tracer t we have:
dqt ρd
= −qt ρd ∇ ⋅V + Dρ + S ρ
t
t
dt
d ρd
dqt
qt
+ ρd
= −qt ρd ∇ ⋅V + Dρ + S ρ
t
t
dt
dt
Continuity equations
Using the continuity equation for dry air, and re-ordering
terms we get the advection equation for mixing ratio:
⎛ d ρd
⎞
dqt
qt ⎜
+ ρd ∇ ⋅V ⎟ + ρd
= Dρ + S ρ
t
t
dt
⎝ dt
⎠
⇒
dqt
= Dq + Sq
Lagrangian form
t
t
dt
∂qt
or
= −V ∇qt + Dq + Sq
Eulerian form
t
t
∂t
Sρ
Dρ
t
with Sq =
and Dq = t
t
t
ρd
ρd
Desired properties for numerical solutions to the continuity equation
•  Locally mass conserving
•  Shape conserving (positive definite, monotonic and nonoscillatory)
•  Avoid numerical mixing of tracers
•  Transportive and local (solution must follow characteristics)
•  Consistent (avoid mass-wind inconsistency problem)
•  Conserving a constant field in a non-divergent flow
•  Computationally efficient (i.e. high accuracy for a given
computational resource). E.g numerically stable for long time
steps
Semi-Lagrangian integration
d ρt
∂u
= − ρt
dt
∂x
t
U∆t
tn+1
tn
xj‐p‐1
α∆x
xj‐p
xj‐p+1
xj‐p+2
x nj
xj
x
Semi-Lagrangian approximation to the
continuity equation
One-dimensional
dρ
∂u
= −ρ
dt
∂x
"
#$
!
n
∂u
n+1
ρk SL = ρ − 0.5∆t ρ
∂x
∗k
%
'n+1
&
∂u
−0.5∆t ρ
∂x
k
Any-dimensional
dρ
= −ρ∇ · "v
dt
$n+1
!
"n
#
!
ρ∇
· "v
ρn+1
=
ρ
−
0.5∆t
(ρ∇
·
"
v
)
−
0.5∆t
k SL
∗k
k
Second order Adams-Bashforth type
Traditional semi-Lagrangian (SL) scheme. Here solving
the volume density continuity equation as an example:
Explicit forecast in grid point k :
ρ
n+1
k SL
(
= ρ − 0.5Δt ρ∇ ⋅ v
n
*k
K
(
)
n
*k
(
= ∑ wk ,l ρ − 0.5Δt ρ∇ ⋅ v
l
n
l
(

− 0.5Δt ρ
∇⋅v
))
n
l
)
n+1
k
(

− 0.5Δt ρ
∇⋅v
)
n+1
k
k
Grid point/cell index. k = 1, …, K ,
K = nlon*nlat
l
Grid point/cell index. l = 1, …, K.
wk,l Weights on upstream departure grid points representing the polynomial
upstream interpolations. These weights are only different from zero in
Eulerian points l close to the semi-Lagrangian departure point.
Per definition we have
Extrapolated value
Semi-Lagrangian approximation to the continuity
equation
ρn+1
k SL
$n+1
!
"n
#
!
· "v
=
ρ − 0.5∆t (ρ∇ · "v )
− 0.5∆t ρ∇
=
K
%
l=1
&
∗k
n
wk,l ρnl − 0.5∆t (ρ∇ · "v )l
'
k
#
!
· "v
− 0.5∆t ρ∇
E.g. cubic (third order) interpolating scheme for the case of
pure 1-D advection of a tracer:
φ kn+1SL = φ kn
*
= wk ,k − p −1
= wk ,k − p
α (1 − α )(2 − α ) n
(1 − α 2 )(2 − α ) n
=−
φ k − p −1 +
φk − p
6
2
α (1 + α )(2 − α ) n
α (1 − α 2 ) n
+
φ k − p +1 −
φk − p + 2
2
6
= wk ,k − p +1
= wk ,k − p + 2
$n+1
k
Semi
Lagrangian
scheme
for
solving
the
volume
density
con6nuity
equa6on
(two
dimensions)
ρ
n+1
k SL
(
= ρ − 0.5Δt ρ∇ ⋅ v
n
*k
)
n
*k
k
k
(
− 0.5Δt ρ
∇⋅v
)
n+1
k
Bicubic
interpola1on
requires
4×4
=
16
points.
Semi
Lagrangian
scheme
Semi
Lagrangian
scheme