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HEMISPHERIC SOLUTION OF THE OMEGA EQUATION INCLUDING TERRAIN AND SURFACE FRICTIONAL EFFECTS by Leo Co Clarke // Commander, United States Naval Reserve and George E. The three-dimensional integration of the resulting equations excludes areas "inside 11 the terrain. To solve (6) numerically, the following finite-difference approxima- tions, based on the adjoining five-point array, were used* 4 © 2 © i © © 0 3 © 4 For arbitrary quantities A and A x = A 3 — A j j A y A = A 2 'A 4 A r A = GM? a =A 2 A 4 -2Ao » R A u p - Letting Cj — First term on right in (12) ^ C A to Second term on right in (12) equation (12) becomes v X - B K X = C T -C& =C K , K = 1,2...

The effects of terrain and surface frictional stress on the vertical velocity at the lower boundary are investigated in some detail and applied at the earth's surface. The grid used is the Fleet Numerical Weather Facility Northern Hemi- sphere octagon of 1977 points centered on the pole and extending to about latitude ION, The omega equation (6) is to be solved at the four levels of 850 mb, 700 mb, 500 mb and 300 mb, with the lower boundary being the surface of the eartho The upper boundary is 100 mb where it is assumed that CCr O , and the lateral boundaries are near 10N where it is also assumed that t O O .

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This substitution is illustrated schematically in figure 3.

The geostrophic vorticity jf^o is also required and is based on v L q Vlo=Kx|(v p d) t • (31) Here it is noted that^Vp D must be determined at terrain pressure.

Thus it is necessary to "reduce" or extrapolate the D-p values at points sur- rounding the central point. The extrapolation will be done on altimetry principles using the previously calculated values for the layer in which the terrain lies. ), the extrapolated value is (pr) p, / — (p t ) i - i / '— oc coco o o ococoooo oo oo o o o o co 00 00 O 00 coco o 00 o cococ - 1 oooooo I | OOOOCO | COCO 00 1 OOOOCO 1 CC co | oooooo ooooco ooooco ooooco 00 0 — — oo cm oooo com N- in CM CD co 00 O CC oo 00 00 CO CD 00 o cou 00 o o 00 00 oo oo aoo — CM oaocooo — oooooo— COCO coo c 3 00 0000 O OCCOOOOOOOOO o a — — o O CC CD 00 O 00 oo 00 O co CD 00 O o c § 383 »: 33835 )' r— , i mea 1 •‘GOCOCOr OOOOOOC 1 — 00 00 oo ro 00 -a- — — — O — — CM 00 CD OOO X* — xxxtnxx c 5 cc oo coo co coco ococoooo o CM'— — — — oco 00 m x* xrnxx c 3 00 00 oo o oo CO 00 O 00 00 o o O — — — o— — ooo 00 o o o oxxxc CCCCC0 1 cooooc 1 | oo — — — ' — ■ — - 00 1 1 1 r-r- i xx - CO CO co coco — — — — CO r^* xx -00 00 00 CM CD 00 o o— — 00 ON m r- m xx- ooco o o o o o O — — o— o oo coo xao coo coco co ococoooc 1 000000 000000 | 00 — — — — — X 1 XXX 1 XXX I XXX 1 000000 000000 ooooco oooo 0C0C0G0 J-OOCOOOOCCCOOOin OOCO — DOOOOCCOOOOOOOOOOOOCCOOOOOOOO D oo oo 00 o 00 00 00 o oo 00 00 O oo 00 00 o ] oooooo | cococo | coco 00 I cocooc 00 00 oo oooooo oooooo oooooo 1 00 00 00 in 00 CO 00 00 CC 00 00 r- 00 CC CO o :ororo ro ro rororo rororo ro— CM — PO— rororo— rororo— cm— cm — ro— rororo— fomro— o cm — rororo- rororo— rororo — ro — rom CM rororo rororo rororo rororo rororo • — cm rororo ro cm cm cm — — o CM — — — ro CM CO CM CM CM CM u — in CM ro— — — CM ro PO— C •— — ocm — ro ro— — — oaococoo — — c p ~ CM rororn ro ro CM— co | i — r— r^- | coco — ro cm- — ro CC — - o- - o O 00 CO CO C 03 oo oc c 1 £8353 4* X^SS88 1 ^ ^ ¥ *~ rwr *$ 8 ^ 1 W89 * 1 1 mm' mm' ^00 Xl CC LOCO CO CO rococo CO CO - 00 CO X o 00 00 CO O 00 CO CO o DCOCOCOOOOOOCOOOOOO o I CO 00 CO I 00 00 00 | 00 00 00 CO 00 CO 00 0 00 CO CO C\ 00 CO CM 0 CO CO coo — o — — m— — — o— 0 00 CO coo 1 00XX4- coco SJ 00 CO ox ox I 00 X vjx ox oxco o o o 4 - o o o o o o o o CNJ o " o — — -O — — in no o- 4 - ' O' CNJ- O' 4 - - O' -no- -o- •N" -CM* -o - 4 - o I I XXX 4* o no o o 'CM -CNJ -o 4* r — XX — ro CM CM C o xxo o o o c — XX| I I I I — X XX — d*— mcoooco N — mu — — — CMXXXOXXX — xxx — XX - —o—o xoxxxo xxxo xxxc — — 4 - X I XXX I XXX I xxx — — xxx xxx xxx — ro ——CM— in XXCMXXXr OXXX.: — ro — — — ro— — XOXXXOXXXC — 0—0*-— o oxxxoxxxc — 4 -— — — 4 -—— 4 - XX I XXX — — — — — — - - x xxx — in— — — X — — ro Jd xoxxxc ——————— — o oxxxc o — — o— o o oxxxc A A A I i c p H rfl CM OXXXOXXX- ro Jt XOXXXC O o o xxc I r- I r^-rs-is- j r-N- N-N-N- no a o o 6C o OL | UJUJUJ I VAJUJUJ I VJULA/UJ I t 'JLA'JL/LAJ I L 000000 CO CO CO CO 00 00 00 CO CO 00 00 CO - oo co oo un cc oo oo — oo co oo o co oo oo — oo oo oo ro coooocooccooooooooooooocoooooco — c CO coco o CO 00 cco CO 00 coo oo co coo o | CO COCO | 00 0000 I 00 00 CO I ao CO CO I cococo cococo cococo ao co 5 00 CO 00 ro 00 CO 00 C\J CO CO coo CO O O ' 1 - 00 CO 00 OCO CO 00 O COCO COO o CM' DCOCOCOOCOCOCOOCOOOCOO O O' 1 S 8 SBS 3 1 SSS 1 EBSB® 1 * -=*= - 3 - “CM -o UJUJ I ULMJUWJ I UUUJUJ i WJU COCO COCOCO CO 00 CO CO 00 CM CO CO CO st 00 CO CC CM 00 OCC CO CO O CO CO CO O o co co CO ooo CO CO o | COCOCO I cocooo 4 00 00 CO COCOOO 00 00 00 N.

CO CO 00 st 00 OXXX — XXX-XXX-XXXCMXXX- OOOOOOOOOOOOOOOOOOOOOOOOCOCOOOOCOCOC XX o — xxu p— O XX- o oxxxc 4 4 XXX | XXX P- XXCMXXX ^ ’S83!

2 * I I C&&S I W I -xcoxoxxx^xoo m cm — — m— — -~o* — OXX o o in — c -XXX ox XX ox — cm— — m— — ro— OXX ro ro inr— r— r— in - OXXXO XXX OX o O— — O— — — o— — OXX O O NNON^O C rxxx | XXX 1 X 4 — 4 — — — 4 — 4 — — I X 1 i^r-i^ I 1 r-r-r- I I s - I XXX XXX — — — “ — — *— «— *—■ X r^r-r- r— r — i N- r- I s - I s - CXXXCMXX st N- — --0 — -'—0 — r-— — r OX aor-r-r-xr-r- *— -^r-r- 4 w— «— p— — ro st — — ro oxxxoxxxc o o— — — o o — o — o xxoxxxc 4 4 — — — 4 4 — 4 — 4 X | XXX .

Note that under this scheme, the Laplacian of CO at ( c, ) involves the uses of 5 which generally more closely approximates the value of than if zero were used.

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