Jia-Fong Fan and Hann-Ming Henry Juang Environmental M odeling C enter NCEP/NWS/NOAA
description
Transcript of Jia-Fong Fan and Hann-Ming Henry Juang Environmental M odeling C enter NCEP/NWS/NOAA
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Applied Non-iteration Dimensional-split Semi-Lagrangian Advection
with Riemann Invariant Characteristic Equation in Non-hydrostatic System
Jia-Fong Fan and Hann-Ming Henry JuangEnvironmental Modeling Center
NCEP/NWS/NOAA
11th RSM workshop, NCU, Taiwan
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Introduction
Primitive equations without approximation contains high- frequency sound waves. These waves impose a severe restriction on the size of time step in order to produce stable integrations. The most common practice consists of using approximations that eliminate sound waves from the equations. Large-scale models use the hydrostatic approximation. Convection models generally use the anelastic approximation.
In this study, we combine non-iteration dimensional-split semi-Largrangian (NDSL) advection with Riemann invariant characteristic equation (RICE) to solve acoustic wave explicitly. We test NDSL-RICE method in a non-hydrostatic system.
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Non-hydrostatic equation on xz can be written as
u
t u
u
x w
u
z RT
Q
xw
t u
w
x w
w
z RT
Q
z g
T
t u
T
x w
T
z T
u
x
w
z
Q
t u
Q
x w
Q
z u
x
w
z
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1D test
u
t u
u
x RT
Q
xQ
t u
Q
x u
x
For Riemann solver, we let the above equations be
t
Q
u
u RT u
x
Q
u
0
R1
t c1
R1
xR2
t c2
R2
x
where
R1 RT /Q u
R2 RT /Q u
c1 u RT
c2 u RT
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Initial condition and after 400s
Q(x) 1.0 0.01exp x xm
5
2
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The initial acoustic spread
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After 800s with different CFL
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2D tests in x-z with isotherm
u
t u
u
x w
u
z RT
Q
xw
t u
w
x w
w
z RT
Q
z
Q
t u
Q
x w
Q
z u
x w
z
gw
RT
where
Q
z g
RT
Q Q Q
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2D tests in x-y (non-forcing)
Here, set g=0, it means on a x-y plane, and we ignore all forcing term to make sure Riemann solver perform well in 2D spatial splitting.
For Riemann solver, we write it into
X direction: Z direction:
t
Q'
u
w
u 0
RT u 0
0 0 u
x
Q'
u
w
w 0 0 w 0
RT 0 w
z
Q'
u
w
gw
RT 0
0
R1
t c1
R1
xR2
t c2
R2
xw
t u
x
R1 RT /Q' u
R2 RT /Q' u
c1 u RT
c1 u RT
R1
t c1
R1
zR2
t c2
R2
zu
t w
z
R1 RT /Q' w
R2 RT /Q' w
c1 w RT
c1 w RT
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2D tests in x-y (non-forcing)
Initial Condition:
T (i, j) 303.16
u(i, j) 0
w(i, j) 0
Q'(i, j) 0.01*exp x xm
5
2
y ym
5
2
Domain:
dx dz 400m
grids : 200 *200
int egral time : 60s
Experiment setting
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2D tests in x-y (non-forcing) – Q’(t=30s)
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2D tests in x-y (non-forcing) – Q’(t=60s)
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2D tests in x-y (non-forcing) – Q’(t=120s)
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2D tests in x-y (non-forcing) – U(t=30s)
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2D tests in x-y (non-forcing) – U(t=60s)
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2D tests in x-y (non-forcing) – U(t=120s)
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2D tests in x-y (non-forcing) – V(t=30s)
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2D tests in x-y (non-forcing) – V(t=60s)
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2D tests in x-y (non-forcing) – V(t=120s)
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Warm bubble case
Non-hydrostatic equation on xz can be written as
u
t u
u
x w
u
z RT
Q
xw
t u
w
x w
w
z RT
Q
z g
T
T
T
t u
T
x w
T
z RT
u
x
w
z
Q
t u
Q
x w
Q
z u
x
w
z
gw
RT
T 303.16
p p0
R
C p
, Q lnp p0
z
g
Cp,
Q
z
g
RT
Q Q Q', T T T '
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Warm bubble case
For Riemann solver, we write it into
t
Q
u
w
u 0
RT u 0
0 0 u
x
Q
u
w
w 0 0 w 0
RT 0 w
z
Q
u
w
gw
RT
R T Q
x
R T Q
z g
T
T
t
ux
wz
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Warm bubble case - sequential forcing
For x direction
t
Q'
u
w
u 0
RT u 0
0 0 u
x
Q'
u
w
t
ux
€
∂R1
∂t= −c1
∂R1
∂x+
1
2
gw
γRT − RT ' ∂Q'
∂x
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
∂R2
∂t= −c2
∂R2
∂x+
1
2
gw
γRT + RT ' ∂Q'
∂x
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
∂w
∂t= −u
∂w
∂x+
1
2g
T '
T − RT ' ∂Q'
∂z
⎛
⎝ ⎜
⎞
⎠ ⎟
∂θ
∂t= −u
∂θ
∂x
€
R1 = RT /γQ' + u
R2 = RT /γQ' − u
c1 = u + γRT
c1 = u − γRT
θ =T
p / p0( )k
For z direction:
t
Q'
w
w RT w
z
Q'
w
gwRT 0
u
t w
u
zt
ux
€
∂R1
∂t= −c1
∂R1
∂z+
1
2
gw
γRT 1− κQ'( ) − RT ' ∂Q'
∂z+ g
T '
T − κQ'
⎛
⎝ ⎜
⎞
⎠ ⎟
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
∂R2
∂t= −c2
∂R2
∂z+
1
2
gw
γRT 1− κQ'( ) − RT ' ∂Q'
∂z+ g
T '
T − κQ'
⎛
⎝ ⎜
⎞
⎠ ⎟
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
∂R2
∂t= −c2
∂R2
∂z+
1
2
gw
γRT 1− κQ'( ) + RT ' ∂Q'
∂z− g
T '
T − κQ'
⎛
⎝ ⎜
⎞
⎠ ⎟
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
∂u
∂t= −w
∂u
∂z−
1
2RT ' ∂Q'
∂x∂θ
∂t= −w
∂θ
∂z
R1 RT /Q' w
R2 RT /Q' w
c1 w RT
c1 w RT
T
p / p0 k
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Warm Bubble case
Initial Condition:
T (i, j) 303.16
u(i, j) 0
w(i, j) 0
'(i, j) A
Ae r a 2 / s2
,r a
r2 x x0 2 z z0 2
,a 50,s 100
Domain:
dx dz 10m
grids :101*150
Experiment setting
Bubble center(51,27)
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Warm Bubble Case
Time-step=0.01s, CFL~0.7
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Results Comparison with lecture by Andre Robert 1992
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U speedtime step=0.01 CFL~0.7
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W speedtime step=0.01 CFL~0.7
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Cascade interpolation
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Conclusion
A non-hydrostatic equation system is used to test NDSL-RICE method to resolve acoustic waves explicitly. Since NDSL is unconditionally stable for advection, theoretically we can use as large a time step as possible; however, RICE in two-dimensions shows ill-solution with large time step.
Thus, the preliminary results show that no limitation of time step should be used in one-dimensional tests but there are some limitations in two dimensional tests due to the nature of Riemann solver. Furthermore, the long distance advection due to large acoustic speeds require us to consider entire trajectory as compared to central mean value.
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Future work
Applied cascade interpolation with NDSL-RICE on warm bubble test
Improve processes for forcing term
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-Thank You-