Post on 16-Jan-2016
NUMERICAL MODELINGNUMERICAL MODELING OF THE OF THE OCEAN AND MARINE DYNAMICS ON OCEAN AND MARINE DYNAMICS ON THE BASE OF MULTICOMPONENT THE BASE OF MULTICOMPONENT
SPLITTINGSPLITTING
Marchuk G.I., Kordzadze A.A., Tamsalu RMarchuk G.I., Kordzadze A.A., Tamsalu R, ,
Zalesny V.B.Zalesny V.B., , Agoshkov V.I.,Agoshkov V.I.,
Bagno A.V.Bagno A.V., , Gusev A.V.Gusev A.V., , Diansky N.A.Diansky N.A., , Moshonkin S.N.Moshonkin S.N.
MoscowMoscow, 2010 , 2010
Contents
• I. Splitting method is a methodological basis for the construction and treatment of the complicated system
• II. Nonhydrostatic FRESCO model of the Baltic Sea
• III. Numerical model of the Black Sea dynamics
• IV. World Ocean -coordinate splitting model
• V. 4D VAR data assimilation techniques based on splitting and adjoint equation methods
I. Splitting method is a methodological basis for the construction and treatment of the complicated hydro-ecosystem. Key points
• The splitting method can be considered not only as a cost-effective solution of the complex problem but as the basis for the construction of the hierarchical model system as well
• In the framework of the unified approach there can be constructed a particular model of sea/ocean dynamics of a different complexity: from the point of view of its physical completeness, dimension, and spatial resolution
• We need to find a conservation law which holds in the model in the absence of external sources and internal energy sinks
Splitting-up methods(Yanenko, Marchuk, Samarskii et al., 1960 - 2010)
, *A ft
1 1
, 0, 1,2,3,...,I I
i i ii i
A A A f f i I
Let the governing equations are represented in operator form:
To solve (*) we reduce the solution of this complex problem to the solution of a set of problems with simpler operators Ai :
1/1/
1 1
2 / 1/2 / 1/
2 2
1 ( 1) /1 ( 1) /
1 ,
1 ,
..............................................................................
1 .
j I jj I j
j I j Ij I j I
j j I Ij j I I
I I
A f
A f
A f
All these simple tasks may be solved by effective and stable implicit and semi-implicit methods.α = 1 - implicit schemeα = ½ - Crank-Nickolson schemeα = 0 - explicit scheme
Multicomponent splitting• Symmetrized form of governing equations
• Energy conserving space approximations using V.I. Lebedev grids
• Multicomponent splitting into series of nonnegative subproblems
• Separate subproblem has its adjoint analog. The adjoint model consists of the respective subsystems adjoint to the split subsystems of the forward model
• Implicit schemes and exact solutions
II. Nonhydrostatic FRESCO numerical
model (Tamsalu et al.) • The goal of experiments is to simulate the dynamics of the
Baltic Sea in an eddying regime
• Experiments are carried out for four nested regions with a gradual improvement of the spatial resolution: the Baltic Sea (h = 3.7 km), Gulf of Finland ( h = 1.85 km), Tallinn-Helsinki basin ( h=460 m ), Tallinn Bay (h = 93 m). Atmospheric forcing: HIRLAM forecast for August 2003
• The model simulates the processes of enhanced turbulence activity in the near-shore zones
Two-equation (k-) turbulence model
Analytical solutions for the 2nd and 3rd stages
kcS40
222
2 111
w
H
v
H
u
HG
potg
HN
0
2 1
k
c
c
S
uS
u 0
Pru
3
40
2
22
0
1
Pr
1kc
kNG
c
cHk
k
Ht
kH S
S
uS
k
u
3
2402
2
2
32
10
1
Pr
1
S
S
uSu cc
NcGc
c
cH
HtH
FRESCO. Subdomain space resolution: (1) 3*3 nm; (2) 1*1 nm; (3) 1/4*1/4 nm; (4) 1/20*1/20 nm
open boundary
Depth: Gulf of Finland (1.85 km, left), Tallinn Bay (93 m, right)
Tallinn Bay. Zonal section along 59.5 N: a) horizontal velocity (cm/c), b) vertical velocity (cm/c), c) turbulent viscosity coefficient, d) turbulent kinetic energy (12 06.08.2003)
Turbulent kinetic energy at the sea surface. A. Without waves: k(min) = 2.8 cm2/s2, k(max) = 3.7 cm2/s2
B. With waves: k(min) = 40.5 cm2/s2, k(max) = 226 cm2/s2
III. Mathematical modeling of the Black Sea dynamics Institute of Geophysics, Georgia, Tbilisi (A. Kordzadze et al. )
• Primitive equation model. Splitting numerical technique
• Splitting with respect to (x,z) and (y,z) plans
• 5-10 km resolution for the most part of the Black Sea
• 1 km resolution of the Eastern Black Sea (from 39.32 E): 216x347x30
• Forecast duration: 4 days ... initial cond. from MHI (Sebastopol), atmospheric forecast fields at 1 hour intervals from ALADIN atmospheric model
Velocity vectors (cm/s) in the Eastern Black Sea. Day 18.07.2010, 00:00 h (a) z=0 m, (b) z=50 m, (c) z=200 m, (d) z=500 m
Sea surface velocity in the Eastern Black Sea. (a) 24 h, (b) 48 h, (c) 72 h, (d) 96 h
IV. World Ocean -coordinate model• Symmetrized form of the ocean dynamics equations
ux
Zx
Z
r
gZ
gp
xrvu
y
rv
x
r
rrl
dt
duu
xx
xy
yx
00 22
11
vy
Zy
Z
r
gZ
gp
yruu
y
rv
x
r
rrl
dt
dvv
yy
xy
yx
00 22
11
Z
ZgZ
gp
22
01
vDry
uDrxrrt xy
yx
TT
Ty
T
r
DvTDvr
yrrx
T
r
DuTDur
xrrt
DT
t
TD
yx
yxxy
xy
)(
11
2
1
SS
Sy
S
r
DvSDvr
yrrx
S
r
DuSDur
xrrt
DS
t
SD
yx
yxxy
xy
)(
11
2
1
),,( ZST
Splitting by physical processes. Stage I: convection-diffusion
uDdt
duu
vDdt
dvv
TDTvTrZyrr
uTrZxrrt
TZTx
yxy
xy
)(
11
SDSvSrZyrr
uSrZxrrt
SZSx
yxy
xy
)(
11
Splitting by physical processes. Stage II: adaptation of density and velocity fields
xZ
x
ZgZ
gp
xrvl
t
u
x
22
1
0
yZ
y
ZgZ
gp
yrul
t
v
y
22
1
0
Z
ZgZ
gp
22
01
vrZy
urZxrrt xy
yx
0
t
TZ 0
t
SZ gZSTpSSTT 0,,~,,~
Splitting by space coordinates
,,2
1xxxyy
xy
DDx
urZurZxrrt
Z
,,2
1yyyxx
xy
DDy
vrZvrZyrrt
Z
,2
1D
t
Z
V. 4D-var data assimilation technique (Marchuk, Penenko, Le Dimet, Talagrand, Agoshkov, Shutyaev et al., 1978 – 2010)
• 4D-Var data assimilation method is applied in oceanography to solve inverse problems
• It is used to find a set of control variables, which minimize the norm of distance between observations and model predictions (cost function)
• Using adjoint equation method the gradient of the cost function is computed and optimal control method is implemented to solve problems arising in ocean modeling
t°
4D-VAR data assimilation – initialization problem
0
fAt
?0
0L
0*
L
t
data dtJ0
2 min
tt
data dtfAt
dtL0
2*
0
2 min,
Example of optimality system. 1D nonlinear problem
0
pot
T
D
TT
t
TD
t
DT 1
2
1
S
D
SS
t
SD
t
DS 1
2
1
***
*** 1ˆ
2
1
T
T
DTT
TT
t
TD
t
DT potT
***
*** 1ˆ
2
1
S
S
DSS
SS
t
SD
t
DS potS
01 **
*
SSTT
D
Numerical experiments
• Indian Ocean modeling in an eddying regime: 1/8° 1/12°21
• 4D-VAR Indian Ocean initialization problem: 1°1/2°33 50
• 4D-VAR World Ocean initialization problem: 2°2.5°33 30
Observations, January and July (Shankar et al, 2002)
Model, January and July. Monthly mean velocity averaged over 100m (20-120m).
Indian Ocean
4D-VAR Indian Ocean initialization problem. Climatic SST (left), SST observed data (right)
Indian Ocean initialization problem. SST assimilation: optimal solution (left), deviation from data (right)
Indian Ocean initialization problem. Sea level height assimilation: optimal solution (left), climatic data (right)
4D-VAR World Ocean initialization problem
• Two stages for the numerical experiments: climatic run and 4D-VAR initialization of temperature and salinity fields
• First stage: the World Ocean circulation under climatological atmospheric forcing (~ 3000 years)
• Second stage: 4D-VAR initialization of temperature and salinity fields using ARGO data (Zakharova, 2009).
5-day assimilation interval, every month, 1 year.
ARGO floats
Буи АРГО
4D-VAR World Ocean initialization problem. ARGO data assimilation
4D-VAR World Ocean initialization problem. Temperature at 10 м, April 2008: Optimal solution (left), ARGO data (right)
4D-VAR World Ocean initialization problem. Temperature at 100 м, April 2008: Optimal solution (left), ARGO data (right)
4D-VAR World Ocean initialization problem. Temperature at 10 м, October 2008: Optimal solution (left), ARGO data (right)
4D-VAR World Ocean initialization problem. Temperature at 100 м, October 2008: Optimal solution (left), ARGO data (right)
4D-VAR World Ocean initialization problem. Optimal solution. Temperature and currents at 100 м, April 2008:
4D-VAR World Ocean initialization problem. Optimal solution. Temperature and currents at 100 м, October 2008
Conclusions
• Splitting numerical technique for the solution of the prognostic and 4D-VAR ocean data assimilation problem is constructed
• As a result of splitting, a rather simple subsystems of the forward and adjoint equations are solved at each separate stage
• Adjoint model consists of the respective subsystems adjoint to the split subsystems of the forward model
• The method is the constructive basis for the INM modular computing system of simulation and initialization of the World Ocean hydrographic fields
To split or not to split?