Diagnosis of Kinematic Vertical Velocity in HYCOM. By George Halliwell, 28 November ( ) = z. v (1)
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1 Diagosis of Kiematic Vertical Velocity i HYCOM By George Halliwell 28 ovember 2004 Overview The vertical velocity w i Cartesia coordiates is determied by vertically itegratig the cotiuity equatio dw ( = z v ( dz z dowward from the surface where subscrits deote the variable held costat durig artial differetiatio Model variables i HYCOM are stored o a o-cartesia ( xys coordiate system where the geeralized vertical coordiates are surfaces of costat s tyically desity i the ocea iterior ad fixed ressure levels ear the ocea surface ad i shallow coastal regios To use equatio ( to estimate w rofiles at HYCOM grid oits horizotal velocity comoets must be re-gridded to costat z levels before itegratig ( dowward from the surface This re-griddig must be erformed at high vertical resolutio to rovide accurate vertical rofiles of w Sice this high-resolutio re-griddig is time cosumig a formula is derived here to estimate w rofiles directly from fields stored o the HYCOM geeralized coordiate system This formula is ow icluded i the HYCOM ost-rocessig rogram (hycomroc to estimate w rofiles from fields stored o model archives A differet formula is the derived to calculate w durig HYCOM rus for the urose of advectig three-dimesioal Lagragia floats This is accomlished by itegratig the HYCOM cotiuity (thicess tedecy equatio dowward from the surface ad combiig it with the reviously-derived w rofile equatio A Vertical Velocity Profile Equatio Suitable for HYCOM Post-Processig Sice HYCOM equatios use ressure uits for the vertical coordiate HYCOM vertical velocity is defied as d w = (2 dt By covertig the vertical coordiate i ( from z to we obtai dw = s d v (3 s Sice the HYCOM geeralized coordiate system is ot Cartesia itegratio of (3 dowward from the surface itroduces additioal terms related to the sloig s iterfaces The vertical discretizatio cosists of layers = 2 with each layer bouded by vertical coordiate surfaces located at ressure deths ( x y s above ad + ( x y s below From this oit forward it is uderstood that s is held costat i artial differetiatio ad the subscrit s is droed
2 Assumig w = 0 at the surface vertical velocity at the base of model layer = is 2 u v u v w( 2 = d ( 2 + = + (4 where the itegratio is carried out from the surface dow to a ifiitesimal distace above iterface 2 To obtai the vertical velocity at the to of layer 2 the cotiuity equatio is itegrated across iterface 2 from 2 to 2 + : u v 2 u v w ( 2 = d d (5 If iterface 2 is ot level the a jum coditio arises i the evaluatio of the rightmost itegral i (5 This jum coditio is obtaied by evaluatig the rightmost itegral as illustrated for the x directio i Figure The x derivative of u is give by u 2 δ 2 ( x u δ u + u 2 ( x δu 2 = (6 δx δx The vertical velocity jum across the iterface is umerically evaluated by u2 x u x w( 2 ( 2 2 ( ( + w δ u δ u δ = + + (7 δx where δ = 2 2 I the limit as the box defied by δ x ad δ i Figure shris to zero area δ/ δx / x ad δx /2 0 Thus w( 2 + from (5 becomes after addig the jum coditio i the y directio u v 2 2 w( 2 + =( 2 + ( u2 u ( v2 v x y + + (8 Cotiuig the itegratio dowward the vertical velocity at a ressure level P withi model layer 2 where P = + q( + 0 < q < (9 is u v u v w( P = ( + + q( + + = (0 + ( u u + ( v v = 2 It is easy to show that + + wp ( = w( + q[ w( + w( ] ( Thus w varies liearly i the vertical withi each layer while discotiuities ca exist at model iterfaces Equatio (0 is used to evaluate w i the HYCOM ost rocessig rogram (hycomroc Equatio (0 is validated by showig that it gives the correct bottom vertical velocity whe itegrated from the surface to the bottom If model layer is the layer itersectig the bottom (the deeest layer with ozero thicess the Equatio (0 yields the followig exressio for vertical velocity at the bottom: u v w( b = ( ( u u + ( v v (2 = = 2
3 where b = + is the bottom ressure deth For this bottom velocity to be correct it must equal the bottom velocity derived from the cotiuity equatio for the barotroic velocity Defiig barotroic velocity comoets uv as vertical averages from the surface to the bottom ad assumig that surface vertical velocity is zero the followig bottom vertical velocity is obtaied from the barotroic cotiuity equatio: u v w( b = b + (3 The barotroic vertical velocity comoets are give by u = u( + b = (4 v = v( + b = Equatio (2 is obtaied by substitutig (4 ito (3 which validates (0 A Vertical Velocity Equatio for Use durig HYCOM Rus It is ecessary to estimate w durig HYCOM rus for the urose of vertically advectig sythetic floats This calculatio is made more efficiet by taig advatage of calculatios already made durig model rus secifically the time evolutio of the thicess of model layer calculated by the HYCOM cotiuity (thicess tedecy equatio The thicess tedecy equatio is itegrated dowward from the surface ad combied with equatio (0 to derive the exressio used to estimate w durig HYCOM rus If sub-grid scale rocesses (thicess diffusio are eglected the thicess tedecy is give by (Blec 2002: ( = s ( v ( s + ( s (5 t s s + s where ( s / s is the etraimet velocity i ressure er uit time across iterface ad the subscrits s idicate that the geeralized vertical coordiate is held costat durig artial differetiatio Equatio (5 is summed dowward from the surface assumig that the surface iterface is statioary The vertical motio of iterface 2 with the subscrit s agai droed is 2 u v 2 2 =( 2 u( v ( s t + x y x x y y (6 s 2 Cotiuig to iterface = + ( 2 (7 t t t which results i 3 u v 2 2 =( 2 + u( v t (8 u2 v ( 3 2 u2( v + 2 ( s s 3 Rearragig terms yields
4 3 u v u2 v2 =( 2 + ( t ( ( u2 u + ( v2 v u2 v2 ( s s 3 More geerally the vertical motio of iterface + at the base of layer 2 is + u v = ( + + t = + ( u u + ( v v (20 = u v ( s s + The iterface vertical velocity at ressure deth P withi model layer with P give by (9 is P + = + q ( = t t t t ( s + q ( s ( s s s + s u v u v ( + + q( + = x y + x y (2 + ( u u + ( v v = u + q ( v + q x From (0 the third ad fourth lies of (2 are idetified as the fluid vertical velocity w at ressure deth P As a result (2 becomes + w( P = + q( + ( s + q ( s ( s t t t s s + s ( u + q( + v + q The vertical velocity of model ressure iterfaces ca be searated as follows: = ( s t t s (23 If the etraimet velocity is zero the iterface vertical velocity equals / t which ca therefore be iterreted as the local vertical velocity of a material surface Sice ad surfaces are co-located at the time vertical velocity is evaluated equatio (22 ca be writte as ( + w P = + q t t t ( u + q( + v + q
5 The first term o the right side of (24 is the vertically iterolated material surface vertical velocity (the vertical velocity of s surfaces i the absece of diaycal mass fluxes The other two terms o the right side rereset the vertical comoet of layer flow whe the layer is ot flat It is a fuctio of mometum comoets withi layer ad the sloe of the iterfaces at the to ad bottom of layer the latter vertically iterolated to ressure deth P The vertical velocities at the to ad bottom of layer are obtaied by settig q = 0 ad q = resectively: + w( = + u( + v t ( w( + = + u( + v t Vertical velocity at the cetral deth of layer is obtaied by settig q = /2: ( + u + v + w P = ( t t 2 2 (26 From Equatio (24 the vertical velocity at the ocea bottom reduces to b b w( b = ub + vb (27 where b is bottom ressure ad ub v b are mometum comoets i the deeest model layer with ozero thicess To estimate vertical velocities from Equatios (24 through (26 it is ecessary to estimate / t at all model iterfaces It ca be obtaied by solvig the HYCOM thicess diffusio equatio (5 with etraimet velocity set to zero: ( = s ( v (28 t s The advatage to estimatig w durig model rus is that / t is already calculated by HYCOM i subroutie cuityf It is oly ecessary to calculate the iterface sloe terms ad add them to the iterface vertical velocity calculated i cuityf Validatio The two equatios for estimatig w as a fuctio of ressure P [equatios (0 ad (24] are ow demostrated to be equivalet to each other ad to w estimated by first re-griddig velocity comoets oto level ressure coordiates ad vertically itegratig (3 dowward from the surface Calculatios were made withi a low-resolutio Atlatic simulatio Before itegratig (3 horizotal velocity comoets are re-gridded usig u ( = u ( l (29 v ( = v ( l where l is the umber of the model layer withi which ressure deth is located The regridded velocity comoets uv are the substituted ito (3 The vertical itegratio is erformed at high vertical resolutio (0 m or 000 MPa i ressure uits to reduce trucatio errors ad resolve the velocity jums that exist across model iterfaces The velocity comoets
6 are re-gridded oto ressure deths = MPa ad the umerical itegratio is erformed dowward from the surface usig the traezoidal rule The w rofile resultig from the vertical itegratio of (3 assumig zero vertical velocity at the surface is illustrated i Figure 2 at the model grid oit located o the Equator ear 28W The rofiles obtaied from equatios (0 ad (24 varyig P i icremets of 0 m are also reseted i Figure 2 The rofiles are idetical withi umerical trucatio errors validatig the derivatio of these two equatios Vertical velocity varies liearly withi each layer ad jums ca exist across model iterfaces These jums are most clearly evidet i the uer 300 m The rimary differece amog the rofiles is that the jums i w occur over a fiite deth rage i the rofiles calculated from (3 because of the horizotal grid sacig (Figure This deth rage will decrease toward zero as horizotal grid sacig decreases Referece Blec R 2002: A oceaic geeral circulatio model framed i hybrid isoycic-cartesia coordiates Ocea Modellig Figure Vertical itegratio of the cotiuity equatio (3i a o-cartesia grid across a model iterface that sloes i the x directio
7 Figure 2 Vertical rofile of w i m/day calculated from re-griddig horizotal velocity comoets oto a Cartesia coordiate system ad vertically itegratig (3 (blac lie for the uer 3000 m (to ad the uer 300 m (bottom Also show are rofiles calculated from (0 (red lie ad from (24 (blue lie each dislaced 05 m/day to the right
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