Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following.

Hydrostatic Correction for Terrain-Following Ocean

Models

Peter C Chu and Chenwu Fan

Naval Postgraduate School

Monterey, California

2003 Terrain-Following Ocean Models Users Workshop  4-6 August, 2003, PMEL/NOAA, Seattle WA

How to effectively reduce the sigma error is a challenge task

for TOMS.

Pressure Gradient Error

Pressure Gradient Error

Original Thought: High-Order Schemes

• Ordinary Five-Point Sixth-Order Scheme (Chu and Fan, 1997 JPO)

• Three-Point Sixth-Order Combined Compact Difference (CCD) Scheme (Chu and Fan, 1998 JCP)

• Three-Point Sixth-Order Nonuniform CCD Scheme (Chu and Fan, 1999, JCP)

• Three-Point Sixth-Order Staggered CCD Scheme (Chu and Fan, 2000, Math. & Comp. Modeling)

• Accuracy Progressive Sixth-Order Scheme (Chu and Fan, 2001, JTECH)

Verification of Current Schemes by Ezer, Arango, Shchepetkin

(OCEMOD, 2002)

Verification of Current Schemes by Ezer, Arango, Shchepetkin

(OCEMOD, 2002)

Verification of Current Schemes by Ezer, Arango, Shchepetkin

(OCEMOD, 2002)

How to reduce errors with low computer cost?

Hydrostatic Correction

• Chu, P.C., and C.W. Fan, 2003: Hydrostatic correction for reducing horizontal pressure gradient errors in sigma coordinate models. Journal of Geophysical Research, J. Geophys. Res.  Vol. 108, No. C6, 3206, 10.1029/2002JC001668.

Finite Volume Consideration

y

C

L p ds F n

Pressure Gradient

2 3 4 1

1 2 3 4

1xl e u w

y

p Fp dz p dz p dz p dz

x L S S

2 4

1 3

z y l uF L p dx p dx

Horizontal Pressure Gradient

2 3

1, 1 , 1 1, 1, 1

1 2

( ), ( ),l l i k i k e e i k i kp dz p z z p dz p z z 4 1

, 1, , 1 ,

3 4

( ), ( ),l u i k i k w w i k i kp dz p z z p dz p z z

1, 1 , 1 1, 1, 1

, 1, , 1 ,

( ) ( )1

( ) ( )l i k i k e i k i k

u i k i k w i k i k

p z z p z zp

p z z p z zx S

Pressure is a linear function of z(or density constant

versus z)

, , 1

2i k i k

w

p pp

1, 1, 1

2i k i k

e

p pp

, 1 1, 1

2i k i k

l

p pp

, 1,

2i k i k

u

p pp

Second-Order Scheme

1, 1 , 1 1 1, , 1 1 1

, 1

i k i k i k i k i k i k i k i k

i k i k i i

p p H H p p H Hp

x x H H

Based on the assumption of linear dependence of P on z.

Steep Topography – Hermit Polynomial Integration

12 1,

l ll l l

l

2

1

1

0

l

lp l dl l d

Hermit Polynomial Integration

2

1

21

1 201 2

( )2 12

l

l

l l p ppdl d p p

l l

2 3 2 31 21 3 2 , 3 2 ,

2 3 3 23 42 ,

1 2

1 2

1 2 3 4l ll l

p pp p l l

l l

Hydrostatic Correction

1, 1 , 1, , 1 1, , 1 , 1, 1

, 1 , 1, , 1 1, 1

i k i k i k i k i k i k i k i k

iki k i i i k i k i k i k

p p z z p p z zp

x x x z z z z

1 , 1, , 1 1, 1

,6

ikik

i i i k i k i k i k

g

x x z z z z

2 2

1 1 1 1, 1, 1 1 , , 1[ik i k i k i k i k i k i k i k i kH H H H

2 2

1 1 1 1, 1 , 1 1 1, , ]i k i k i k i k i k i k i k i kH H H H

Steep Topography – Hermit Polynomial Integration

12 1,

l ll l l

l

2

1

1

0

l

lp l dl l d

Hermit Polynomial Integration

2

1

21

1 201 2

( )2 12

l

l

l l p ppdl d p p

l l

2 3 2 31 21 3 2 , 3 2 ,

2 3 3 23 42 ,

1 2

1 2

1 2 3 4l ll l

p pp p l l

l l

Hydrostatic Correction

1, 1 , 1, , 1 1, , 1 , 1, 1

, 1 , 1, , 1 1, 1

i k i k i k i k i k i k i k i k

iki k i i i k i k i k i k

p p z z p p z zp

x x x z z z z

1 , 1, , 1 1, 1

,6

ikik

i i i k i k i k i k

g

x x z z z z

2 2

1 1 1 1, 1, 1 1 , , 1[ik i k i k i k i k i k i k i k i kH H H H

2 2

1 1 1 1, 1 , 1 1 1, , ]i k i k i k i k i k i k i k i kH H H H

Verification of Current Schemes by Ezer, Arango, Shchepetkin

(OCEMOD, 2002)

HC has the same accuracy as CCD Scheme

• HC has the same accuracy as the sixth-order CCD scheme

CPU time (minute) at the end of 5 day runs of the POM seamount test case using the SD and

HC schemes

Scheme Second-order

HC

CPU Time 171.51 176.92

Ratio 1 1.03

Conclusions

• Hydrostatic correction scheme is a simple scheme with high accuracy and efficiency.

• The accuracy of the HC scheme is similar to the sixth-order CCD scheme.

• The CPU cost is similar to the ordinary second-order scheme.