hydrostatic correction for terrain-following ocean models peter c chu and chenwu fan naval...
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![Page 1: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/1.jpg)
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
![Page 2: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/2.jpg)
How to effectively reduce the sigma error is a challenge task
for TOMS.
![Page 3: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/3.jpg)
Pressure Gradient Error
![Page 4: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/4.jpg)
Pressure Gradient Error
![Page 5: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/5.jpg)
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)
![Page 6: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/6.jpg)
Verification of Current Schemes by Ezer, Arango, Shchepetkin
(OCEMOD, 2002)
![Page 7: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/7.jpg)
Verification of Current Schemes by Ezer, Arango, Shchepetkin
(OCEMOD, 2002)
![Page 8: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/8.jpg)
Verification of Current Schemes by Ezer, Arango, Shchepetkin
(OCEMOD, 2002)
![Page 9: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/9.jpg)
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.
![Page 10: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/10.jpg)
Finite Volume Consideration
y
C
L p ds F n
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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
![Page 12: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/12.jpg)
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
![Page 13: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/13.jpg)
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
![Page 14: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/14.jpg)
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.
![Page 15: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/15.jpg)
Steep Topography – Hermit Polynomial Integration
12 1,
l ll l l
l
2
1
1
0
l
lp l dl l d
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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
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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
![Page 18: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/18.jpg)
Steep Topography – Hermit Polynomial Integration
12 1,
l ll l l
l
2
1
1
0
l
lp l dl l d
![Page 19: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/19.jpg)
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
![Page 20: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/20.jpg)
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
![Page 21: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/21.jpg)
Verification of Current Schemes by Ezer, Arango, Shchepetkin
(OCEMOD, 2002)
![Page 22: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/22.jpg)
HC has the same accuracy as CCD Scheme
![Page 23: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/23.jpg)
![Page 24: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/24.jpg)
• HC has the same accuracy as the sixth-order CCD scheme
![Page 25: Hydrostatic Correction for Terrain-Following Ocean Models Peter C Chu and Chenwu Fan Naval Postgraduate School Monterey, California 2003 Terrain-Following](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d4c5503460f94a2b259/html5/thumbnails/25.jpg)
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
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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.