Jia-Fong Fan and Hann-Ming Henry Juang Environmental M odeling C enter NCEP/NWS/NOAA

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

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.

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

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

Initial condition and after 400s

Q(x) 1.0 0.01exp x xm

5

2

The initial acoustic spread

After 800s with different CFL

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

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

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

2D tests in x-y (non-forcing) – Q’(t=30s)

2D tests in x-y (non-forcing) – Q’(t=60s)

2D tests in x-y (non-forcing) – Q’(t=120s)

2D tests in x-y (non-forcing) – U(t=30s)

2D tests in x-y (non-forcing) – U(t=60s)

2D tests in x-y (non-forcing) – U(t=120s)

2D tests in x-y (non-forcing) – V(t=30s)

2D tests in x-y (non-forcing) – V(t=60s)

2D tests in x-y (non-forcing) – V(t=120s)

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 '

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

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

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)

Warm Bubble Case

Time-step=0.01s, CFL~0.7

Results Comparison with lecture by Andre Robert 1992

U speedtime step=0.01 CFL~0.7

W speedtime step=0.01 CFL~0.7

Cascade interpolation

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.

Future work

Applied cascade interpolation with NDSL-RICE on warm bubble test

Improve processes for forcing term

-Thank You-