Notes on the GSW function gsw_geostrophic_velocity (geo_strf,long,lat,p)

Transcription

1 Notes o gsw_geostrophic_velocity Notes o the GSW fuctio gsw_geostrophic_velocity (geo_strf,log,lat,p) Notes made 7 th October 2, ad updated 8 th April 2. This fuctio gsw_geostrophic_velocity(geo_strf,log,lat,p) calculates the differece betwee the geostrophic velocity at pressure p ad that at the referece pressure (where p p ref dbar ). The geeral expressio for this velocity differece is k f v v, surf where f is the Coriolis parameter ad is the geostrophic streamfuctio i a particular surface. Note that the type of geostrophic streamfuctio should be carefully chose to be appropriate to the surface i which it is beig used. The referece pressure p pref dbar is the value that was selected i the calls to the various geostrophic streamfuctios whose outputs are used as geo_strf i gsw_geostrophic_velocity(geo_strf,log,lat,p). Below we reproduce sectios of the TEOS aual (IOC et al. (2)) which describe four differet types of geostrophic streamfuctio. Dyamic height aomaly is the geostrophic streamfuctio for use i a isobaric surface, while the otgomery geostrophic streamfuctio is desiged for use i a surface of costat specific volume aomaly. The remaiig choices of geostrophic streamfuctio, the Cuigham ad the isopycal geostrophic streamfuctios of cdougall ad Klocker (2) are desiged for use i various types of desity surfaces, ad are ot exact geostrophic streamfuctios i ay particular surface. For these streamfuctios the argumet p i the iput to this fuctio is the pressure o these desity surfaces ad so varies alog each surface. It is emphasized that the choice of geostrophic streamfuctio must be made carefully to match the surface i which it is to be used. Figures 2 ad 3 of cdougall ad Klocker (2) are reproduced below ( Eq.(62) o the figure refers to their isopycal geostrophic streamfuctio). These figures illustrate the errors i geostrophic velocity that are itroduced whe usig a geostrophic streamfuctio i a surface for which it is ot desiged. The correspodig r.m.s. errors for the otgomery geostrophic streamfuctio i each of these six approximately eutral surfaces are ot show i these 3 figures, but are cosiderably larger, at approximately 4 x ms, illustratig that while the otgomery geostrophic streamfuctio is the correct streamfuctio i a specific volume aomaly surface, it is rather iaccurate whe used i surfaces (Klocker et al. (29a,b)), surfaces (Jackett ad cdougall (997)) ad i potetial desity surfaces, such as 2 surfaces. The Cuigham geostrophic streamfuctio is cosiderably more accurate tha the otgomery streamfuctio, but is ot as accurate as the isopycal streamfuctio of cdougall ad Klocker (2) i these six surfaces. Hece it is the geostrophic streamfuctio gsw_geo_strf_isopycal_ct which is recommeded for use i potetial desity surfaces, i surfaces (Klocker et al. (29a,b)) ad i surfaces (Jackett ad cdougall (997)). ref Refereces IOC, SCOR ad IASO, 2: The iteratioal thermodyamic equatio of seawater 2: Calculatio ad use of thermodyamic properties. Itergovermetal Oceaographic Commissio, auals ad Guides No. 56, UNESCO (Eglish), 96 pp. Available from See sectios 3.27, 3.32 ad appedix A.3. Jackett, D. R. ad T. J. cdougall, 997: A eutral desity variable for the world s oceas. Joural of hysical Oceaography, 27,

2 Notes o gsw_geostrophic_velocity 2 Klocker, A., T. J. cdougall ad D. R. Jackett, 29a: A ew method for formig approximately eutral surfaces. Ocea Sci., 5, html Klocker, A., T. J. cdougall ad D. R. Jackett, 29b: Corrigedum to ʺA ew method for formig approximately eutral surfacesʺ published i Ocea Sciece, 5, 55 72, 29, Ocea Sci., 5, sci.et/5/9/29/os html cdougall, T. J. ad A. Klocker, 2: A approximate geostrophic streamfuctio for use i desity surfaces. Ocea odellig, 32, 5 7. Figure 2. Rms errors of the geostrophic velocities calculated from the expressios of various geostrophic streamfuctios o a surface, a surface ad a 2 surface. Each desity surface has a average pressure of 3 dbar. Figure 3. Rms errors of the geostrophic velocities calculated from the expressios of various geostrophic streamfuctios o a surface, a surface ad a 2 surface. Each desity surface has a average pressure of 8 dbar.

3 Notes o gsw_geostrophic_velocity 3 Here follows sectios of the TEOS aual (IOC et al. (2)) Dyamic height aomaly The dyamic height aomaly with respect to the sea surface is give by ˆ SA p, p, p, where ˆS,, p vˆ S,, p vˆ S, C, p d. (3.27.) A A SO This is the geostrophic streamfuctio for the flow at pressure with respect to the flow at the sea surface ad ˆ is the specific volume aomaly. Thus the two dimesioal gradiet of i the pressure surface is simply related to the differece betwee the horizotal geostrophic velocity v at ad at the sea surface v accordig to k fv fv. (3.27.2) Dyamic height aomaly is also commoly called the geopotetial aomaly. The specific volume aomaly, ˆ i the vertical itegral i Eq. (3.27.) ca be replaced with specific volume ˆv without affectig the isobaric gradiet of the resultig streamfuctio. That is, this substitutio does ot affect Eq. (3.27.2) because the additioal term is a fuctio oly of pressure. Traditioally it was importat to use ˆ i preferece to ˆv as it was more accurate with computer code which worked with sigle precisio variables. Sice computers ow regularly employ double precisio, this issue has bee overcome ad cosequetly either ˆ or ˆv ca be used i the itegrad of Eq. (3.27.), so makig it either the dyamic height aomaly or the dyamic height. As i the case of Eq. (3.24.2), so also the dyamic height aomaly Eq. (3.27.) has ot assumed that the gravitatioal acceleratio is costat ad so Eq. (3.27.2) applies eve whe the gravitatioal acceleratio is take to vary i both the vertical ad i the horizotal. 2 2 The dyamic height aomaly should be quoted i uits of m s. These are the uits i which the GSW Toolbox (appedix N) outputs dyamic height aomaly i the fuctio gsw_geo_strf_dy_height(sa,ct,p,p_ref). Whe the last argumet of this fuctio, p_ref, is other tha zero, the fuctio returs the dyamic height aomaly with respect to a (deep) referece pressure p_ref, rather tha with respect to (i.e. zero dbar sea pressure) as i Eq. (3.27.). I this case the lateral isobaric gradiet of the streamfuctio represets the geostrophic velocity differece relative to the (deep) p ref pressure surface, that is, k fv fv ref. (3.27.3) Note that the itegratio i Eq. (3.27.) of specific volume aomaly with pressure must be doe with pressure i a (ot dbar ) i order to have the resultat isobaric gradiet,, i the usual uits, beig the product of the Coriolis parameter (uits of s ) ad the velocity (uits of m s ). The GSW fuctio gsw_steric_height(sa,ct,p,p_ref) returs 2 divided by the costat gravitatioal acceleratio g ms. Hece steric height remais proportioal to a exact geostrophic streamfuctio but the spatial variatio of the gravitatioal acceleratio esures that it caot be exactly equal to the height of a isobaric surface above a geopotetial surface.

4 3.28 otgomery geostrophic streamfuctio Notes o gsw_geostrophic_velocity 4 The otgomery acceleratio potetial (otgomery, 937) ˆ ˆ SA ˆ f f p, p, p defied by d ˆ (3.28.) is the geostrophic streamfuctio for the flow i the specific volume aomaly surface ˆ SA,, p ˆ relative to the flow at (that is, at p dbar). Thus the twodimesioal gradiet of i the ˆ specific volume aomaly surface is simply related to the differece betwee the horizotal geostrophic velocity v i the ˆ ˆ surface ad at the sea surface v accordig to k v v or ˆ k f v f v. (3.28.2) The defiitio, Eq. (3.28.), of the otgomery geostrophic streamfuctio applies to all choices of the referece values SA ad i the defiitio, Eq. (3.7.3), of the specific volume aomaly. By carefully choosig these referece values the specific volume aomaly surface ca be made to closely approximate the eutral taget plae (cdougall ad Jackett (27)). It is ot ucommo to read of authors usig the otgomery geostrophic streamfuctio, Eq. (3.28.), as a geostrophic streamfuctio i surfaces other tha specific volume aomaly surfaces. This icurs errors that should be recogized. For example, the gradiet of the otgomery geostrophic streamfuctio, Eq. (3.28.), i a eutral taget plae becomes (istead of Eq. (3.28.2) i the ˆ ˆ surface) k fv fv ˆ, (3.28.3) where the last term represets a error arisig from usig the otgomery streamfuctio i a surface other tha the surface for which it was derived. Zhag ad Hogg (992) subtracted a arbitrary pressure offset,, from i the first term i Eq. (3.28.), so defiig the modified otgomery streamfuctio Z-H ˆ ˆ S p, p, p d. (3.28.4) Z-H A The gradiet of i a eutral taget plae becomes Z-H k fv fv ˆ, (3.28.5) where the last term ca be made sigificatly smaller tha the correspodig term i Eq. (3.28.3) by choosig the costat pressure to be close to the average pressure o the surface. This term ca be further miimized by suitably choosig the costat referece values SA ad i the defiitio, Eq. (3.7.3), of specific volume aomaly so that this surface more closely approximates the eutral taget plae (cdougall (989)). This improvemet is available because it ca be show that ˆ S A,, p ˆ SA,, p T b The last term i Eq. (3.28.5) is the approximately. (3.28.6) 2 (3.28.7) 2 T b ad hece suitable choices of, SA ad ca reduce the last term i Eq. (3.28.5) that represets the error i iterpretig the otgomery geostrophic streamfuctio, Eq. (3.28.4), as the geostrophic streamfuctio i a surface that is more eutral tha a specific volume aomaly surface. 2 2 The otgomery geostrophic streamfuctio should be quoted i uits of m s. These are the uits i which the GSW Toolbox outputs the otgomery geostrophic

5 Notes o gsw_geostrophic_velocity 5 streamfuctio i the fuctio gsw_geo_strf_otgomery(sa,ct,p,p_ref). Whe the last argumet of this fuctio, p_ref, is other tha zero, the fuctio returs the otgomery geostrophic streamfuctio with respect to a (deep) referece sea pressure p_ref, rather tha with respect to p dbar (i.e. ) as i Eq. (3.28.) Cuigham geostrophic streamfuctio Cuigham (2) ad Alderso ad Killworth (25), followig Sauders (995) ad Killworth (986), suggested that a suitable streamfuctio o a desity surface i a compressible ocea would be the differece betwee the Beroulli fuctio B ad potetial ethalpy h. Sice the kietic eergy per uit mass, uu, is a tiy compoet 2 of the Beroulli fuctio, it was igored ad Cuigham (2) essetially proposed the streamfuctio (see his equatio (2)), where B h uu 2 h h hs ˆ(,, p) hs ˆ(,,) vˆ S ( p), ( p), p d. A A A (3.29.) The last lie of this equatio has used the hydrostatic equatio z g to express gz i terms of the vertical pressure itegral of specific volume ad the height of the sea surface where the geopotetial is. The differece betwee ethalpy ad potetial ethalpy h h i this equatio has bee amed dyamic ethalpy by Youg (2). The defiitio of potetial ethalpy, Eq. (3.2.), is used to rewrite the last lie of Eq. (3.29.), showig that Cuigham s is also equal to ˆv S A ( p ), ( p ), p ĥs SO,C, p ˆvS A,, p d ĥs,, p A c p. (3.29.2) The first lie of this equatios appears very similar to the expressio, Eq. (3.27.), for dyamic height aomaly, the oly differece beig that i Eq. (3.27.) the pressureidepedet values of Absolute Saliity ad Coservative Temperature were S SO ad C whereas here they are the local values o the surface, S A ad. While these local values of Absolute Saliity ad Coservative Temperature are costat durig the pressure itegral i Eq. (3.29.2), they do vary with latitude ad logitude alog ay desity surface. The gradiet of alog the eutral taget plae is 2 T b (3.29.3) z, 2 (from cdougall ad Klocker (2)) so that the error i i usig as the geostrophic streamfuctio is approximately T 2 2 b. Whe usig the Cuigham streamfuctio i a potetial desity surface, the error i is approximately T 2 b 2r. The Cuigham geostrophic 2 2 streamfuctio should be quoted i uits of m s ad is available i the GSW Oceaographic Toolbox as the fuctio gsw_geo_strf_cuigham(sa,ct,p,p_ref). Whe the last argumet of this fuctio, p_ref, is other tha zero, the fuctio returs the Cuigham geostrophic streamfuctio with respect to a (deep) referece sea pressure p_ref, rather tha with respect to p dbar (i.e. ) as i Eq. (3.29.).

6 Notes o gsw_geostrophic_velocity Geostrophic streamfuctio i a approximately eutral surface I order to evaluate a relatively accurate expressio for the geostrophic streamfuctio i a approximately eutral surface a suitable referece seawater parcel SA,, p is selected from the approximately eutral surface that oe is cosiderig, ad the specific volume aomaly is defied as i (3.7.3) above. The approximate geostrophic streamfuctio is give by (from cdougall ad Klocker (2)) 2 SA,, p 2 T b 2 d. (3.3.) This expressio is more accurate tha the otgomery ad Cuigham geostrophic streamfuctios whe used i potetial desity surfaces, i the surfaces of Klocker et al. (29a,b) ad i the Neutral Desity surfaces of Jackett ad cdougall (997). That is, i these surfaces z k f v f v to a very good approximatio. I Eq. (3.3.) T b is take to be the costat value 2.7x K (a) m s. This approximate isopycal geostrophic streamfuctio of cdougall ad Klocker (2) is available as the fuctio gsw_geo_strf_isopycal i the GSW Toolbox. Whe the last argumet of this fuctio, p_ref, is other tha zero, the fuctio returs the isopycal geostrophic streamfuctio with respect to a (deep) referece sea pressure p_ref, rather tha with respect to the sea surface at p dbar (i.e. ) as i Eq. (3.3.).