CONSERVATIVE SCHEMES FOR THE SHALLOW-WATER MODEL … · 2006. 10. 17. · CONSERVATIVE SCHEMES FOR...
Transcript of CONSERVATIVE SCHEMES FOR THE SHALLOW-WATER MODEL … · 2006. 10. 17. · CONSERVATIVE SCHEMES FOR...
CONSERVATIVE SCHEMES FOR THE SHALLOW-WATER MODEL
Yuri N. Skiba and Denis M. Filatov
E-mails: [email protected]@mail.ru
Universidad Nacional Autónoma de MéxicoCentro de Ciencias de la Atmósfera
BASIC VARIABLES
( ),u v
x
z
( ), ,H x y t
y
( ),Th x y
- velocity field( , )u v
- free surface height( , , ) Th x y t H h= −
- topography( , )Th x y
( , , )H x y t - fluid depth
( ), ,h x y t
b) periodic channel in x
u u u hu v fv gt x y x
∂ ∂ ∂ ∂+ + − = −
∂ ∂ ∂ ∂
v v v hu v fu gt x y y
∂ ∂ ∂ ∂+ + + = −
∂ ∂ ∂ ∂
0H Hu Hvt x y
∂ ∂ ∂+ + =
∂ ∂ ∂
Initial conditions:
Periodic conditions in x:
Lateral boundary conditions :
( ) ( ) { }0, ,0 , , , ,R x y R x y R u v H= =where
( ) ( ) { }0, , , , , , ,R y t R L y t R u v H= =where
( ) ( ),0, , , 0v x t v x M t= =
SHALLOW WATER MODEL IN PERIODIC DOMAINS ON A PLANE
a) periodic domain in both x and y
( ) 0d HAdt
=Column volume conservation:
Relative hightconservation:
Potential vorticityconservation:
LOCAL CONSERVATION LAWS FOR SHALLOW-WATER MODEL
1 divdA uA dt
=r1 divdH u
H dt= −
r
Since
and
{ } 0Tz hddt H
−=
{ } 0 , (rot ) zd f udt H
ζ ζ+= =
r
( ) constD
M t Hdxdy= =∫
( )2 2
2 2( ) const2 2 T
D
u v gE t H h h dxdy⎛ ⎞+
= + − =⎜ ⎟⎝ ⎠∫
Mass conservation:
Total energy conservation:
The classic schemes by Arakawa & Lamb (1981), Sadourny (1975), Takano & Wurtele (1982), Kim (1984) as well as recent schemes by Ringler &
Randall (2002) and Salmon (2004), conserve the energy and/or the potential enstrophy only if the model is still continuous in time,while the discretization in time destroys all these laws.
Potential enstrophyconservation: ( )
2
2
1 const2D
v uJ t f dxdyx y H
⎛ ⎞∂ ∂= − + =⎜ ⎟∂ ∂⎝ ⎠∫
INTEGRAL CONSERVATION LAWS FOR THE SHALLOW WATER MODEL
( ) 0 (0, )
(0)
d A t Tdt
g
ϕ ϕ
ϕ
+ =
=
inr
r
r r, 0Aϕ ϕ =r r 0ϕ∀ ≠
rLet A be antisymmetric:
( )( ) (0) , 0,t g t Tϕ ϕ= = ∈r r r
11
12
(1 ) 0
( , 0 1)
j jj jAϕ ϕ αϕ α ϕ
τα α
++− ⎡ ⎤+ + − =⎣ ⎦
≠ ≤ ≤
r rr r
111 1
2 2
12
0
( )
j jj jAϕ ϕ ϕ ϕ
τα
++− ⎡ ⎤+ + =⎣ ⎦
=
r rr r
1j jϕ ϕ+ ≠r r
1j jϕ ϕ+ =r r
CONSERVATION LAW AND TIME DISCRETIZATION
Discretization effect:
DIVERGENT FORM OF THE MODELUsing the change of variables (Skiba, 1995)
1 12 2
U uU U vU U hu v fV gzt x x y y x
⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤+ + + + − = −⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦
1 12 2
V uV V vV V hu v fU gzt x x y y y
⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤+ + + + + = −⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦
0H zU zVt x y
∂ ∂ ∂+ + =
∂ ∂ ∂which is essential for constructing conservative difference schemes.
z H= U zu= V zv=
the equations of shallow-water model are transformed to a divergent form
The discrete divergent forms
1 1 1 1 1 1i i i i i i i i i ii
a R a R R R a R a RaR Ra al l l l l
+ + − + + −− − −∂ ∂+ ⋅ ≈ + =
∂ ∂ Δ Δ Δ
Similarly, the discrete divergent forms of advective terms guaranteethe conservation of total energy, since
( )1
1 1 11 0 0 1
0
1 0I
i i i ii I I I
i
a R a RR a R R a R R
l l
−+ + −
− −=
−= − =
Δ Δ∑
for periodic conditions as well as for conditionsat the lateral boundary of channel.
BASIC PROPERTY OF THE DIVERGENT FORM
0 0Ia a= =
1 1i i i ia R a RaRl l
+ + −∂≈
∂ Δ
( )1
1 10 0
0
1 0I
i i i iI I
i
a R a Ra R a R
l l
−+ +
=
−= − =
Δ Δ∑
and
guarantee the conservation of mass, since
( ) ( )cos1 1 1 coscos 2 2
tan ,cos
uU vUU U Uu vt a
u gz hf Va a
ϕϕ
ϕ λ λ ϕ ϕ
ϕϕ λ
⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂ ∂+ + + + −⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
∂⎛ ⎞− + = −⎜ ⎟ ∂⎝ ⎠
( ) ( )cos1 1 1 coscos 2 2
tan ,
uV vVV V Vu vt a
u gz hf Ua a
ϕϕ
ϕ λ λ ϕ ϕ
ϕϕ
⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂ ∂+ + + + +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
∂⎛ ⎞+ + = −⎜ ⎟ ∂⎝ ⎠
( ) ( )cos1 0cos
zU zVht a
ϕϕ λ ϕ
∂ ∂⎡ ⎤∂+ + =⎢ ⎥∂ ∂ ∂⎣ ⎦
SHALLOW WATER MODEL ON A SPHERE
SPLITTING METHOD
0U fVt
∂− =
∂0V fU
t∂
+ =∂
Splitting step 3 (sphere rotation):
Splitting steps 1 (in x) and 2 (in y):
12
U uU U hu gzt x x x
∂ ∂ ∂ ∂⎡ ⎤+ + = −⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦
1 02
V uV Vut x x
∂ ∂ ∂⎡ ⎤+ + =⎢ ⎥∂ ∂ ∂⎣ ⎦
0H zUt x
∂ ∂+ =
∂ ∂
1 02
U vU Uvt y y
⎡ ⎤∂ ∂ ∂+ + =⎢ ⎥∂ ∂ ∂⎣ ⎦
12
V vV V hv gzt y y y
⎡ ⎤∂ ∂ ∂ ∂+ + = −⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦
0H zVt y
∂ ∂+ =
∂ ∂
x
yu t⋅Δr
TWO MAPS USED IN THE SPLITTING METHOD
THE SPLITTING ALONG THE LONGITUDINAL DIRECTION λ
11 1 1 1 1 1
1 1
1 1cos 2 2 2
,cos 2
n nk k k k k k k k
kl
k k k
l
U U u U u U U Uua
gz h ha
τ ϕ λ λ
ϕ λ
++ + − − + −
+ −
− − −⎛ ⎞+ + =⎜ ⎟Δ Δ⎝ ⎠−
= −Δ
11 1 1 1 1 11 1 0
cos 2 2 2
n nk k k k k k k k
kl
V V u V u V V Vuaτ ϕ λ λ
++ + − − + −− − −⎛ ⎞+ + =⎜ ⎟Δ Δ⎝ ⎠
11 1 1 11 0
cos 2
n nk k k k k k
l
H H z U z Uaτ ϕ λ
++ + − −− −
+ =Δ
PROBLEM 1: THE SPLIT PROBLEM IN
11 ( )2
n nR R R+= +
λ
THE SPLITTING ALONG THE LATITUDINAL DIRECTION ϕ
11 1 1 1 1 1
1 1
cos cos1 1cos 2 2
cos 0,2
n nl l l l l l l l
l
l ll l
U U v U v Ua
U Uv
ϕ ϕτ ϕ ϕ
ϕϕ
++ + + − − −
+ −
− ⎛ −+ +⎜ Δ⎝
− ⎞+ =⎟Δ ⎠
11 1 1 1 1 1
1 1 1 1
cos cos1 1cos 2 2
cos ,2 2
n nl l l l l l l l
l
l l l l ll l
V V v V v Va
V V gz h hva
ϕ ϕτ ϕ ϕ
ϕϕ ϕ
++ + + − − −
+ − + −
− ⎛ −+ +⎜ Δ⎝
− ⎞ −+ = −⎟Δ Δ⎠
11 1 1 1 1 1cos cos1 0
cos 2
n nl l l l l l l l
l
H H z V z Va
ϕ ϕτ ϕ ϕ
++ + + − − −− −
+ =Δ
PROBLEM 2: THE SPLIT PROBLEM IN
11 ( )2
n nR R R+= +
ϕ
1
tan 0n nkl kl kl
l l klU U uf V
aϕ
τ
+ − ⎛ ⎞− + =⎜ ⎟⎝ ⎠
1
tan 0n n
kl kl kll l kl
V V uf Ua
ϕτ
+ − ⎛ ⎞+ + =⎜ ⎟⎝ ⎠
PROBLEM 3: A SPHERE ROTATION
MASS AND TOTAL ENERGY CONSERVATION LAWS
The mass and total energy conservation laws
( )2 2
21 const
2 2D
u v gE t H h dxdy⎛ ⎞+
= + =⎜ ⎟⎝ ⎠∫
( ) constD
M t Hdxdy= =∫
( )2 2
2 2( ) const2 2 T
D
u v gE t H h h dxdy⎛ ⎞+
= + − =⎜ ⎟⎝ ⎠∫
Evidently, it is sufficient to conserve only the variable part of total energy
CONSERVATION LAWS FOR DISCRETE MODELS
In discrete SW system, the mass and total energy conservation laws
1 2 cosn n nl kl
l k
M M a Hλ ϕ ϕ+ = ≡ Δ Δ ∑ ∑
11 1n nE E+ =
( )2 2 22 11 2cosn n n n
l kl kl kll k
E a U V g hλ ϕ ϕ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= Δ Δ + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦∑ ∑
where
accept the forms
1n nM M+ = and
EXPERIMENT 1: PLANE. THE HEIGHT h
T=1,3,4,6,7,14 days 4th approximation order
EXPERIMENT 1: PLANE. VELOCITY T=1,3,4,5,6,7 days 4th approximation order
EXPERIMENT 1: PLANE. POTENTIAL VORTICITY
T=1,3,4,5,6,7 days 4th approximation order
INITIALCONDITIONS
T=0
U
h
EXPERIMENT 2: SPHERE.
ROSSBY WAVE
SPHERE
0 50 100 150 200 250 300 350
−80
−60
−40
−20
0
20
40
60
80
λ
ϕ
Velocity at t = 3.5 days
0 50 100 150 200 250 300 350
−80
−60
−40
−20
0
20
40
60
80
λ
ϕ
Velocity at t = 3.5 days
2-order scheme 4-order scheme
0 50 100 150 200 250 300 350
−80
−60
−40
−20
0
20
40
60
80
λ
ϕ
Velocity at t = 7 days
0 50 100 150 200 250 300 350
−80
−60
−40
−20
0
20
40
60
80
λ
ϕ
Velocity at t = 7 days
T=3.5 days
T=7 days
EXPERIMENT 2: SPHERE. VELOCITY
GRID 6º X 6º GRID 3º X 3º
T=7 days
h
T=3.5 days
EXPERIMENT 2: SPHERE. HEIGHT
4-order scheme
0 1 2 3 4 5 6 72.852
2.854
2.856
2.858
2.86
2.862
2.864
2.866
2.868
2.87
2.872x 10
4 Potential Enstrophy
Time (days)
J(t)
0 1 2 3 4 5 6 72.852
2.854
2.856
2.858
2.86
2.862
2.864
2.866
2.868
2.87
2.872x 10
4 Potential Enstrophy
Time (days)
J(t)
0 1 2 3 4 5 6 72.852
2.854
2.856
2.858
2.86
2.862
2.864
2.866
2.868
2.87
2.872x 10
4 Potential Enstrophy
Time (days)
J(t)
0 1 2 3 4 5 6 72.852
2.854
2.856
2.858
2.86
2.862
2.864
2.866
2.868
2.87
2.872x 10
4 Potential Enstrophy
Time (days)J(
t)
EXPERIMENT 2: SPHERE. POTENTIAL ENSTROPHY
1-2 order scheme 3-4 order scheme
4 order scheme2 order scheme
MAXIMUM VARIATION OF POTENTIAL ENSTROPHY
FOR DIFFERENT GRIDSin %
EXPERIMENT 3: SPHERE.TIME BEHAVIOR OF POTENTIAL ENSTROPHY
WITH RANDOM INITIAL PERTURBATIONS
10( ) (0)log
(0)J t J
J−
2nd order scheme 4th order scheme
Yuri N. Skiba and Denis M. Filatov
E-mails: [email protected]@mail.ru
Universidad Nacional Autónoma de MéxicoCentro de Ciencias de la Atmósfera
CONSERVATIVE SCHEMES FOR THE SHALLOW-WATER MODEL