DTIC ADA535041: A Conserved Minimal Adjustment Scheme for Stabilization of Hydrographic Profiles

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1072 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME27
A Conserved Minimal Adjustment Scheme for Stabilization of Hydrographic Profiles
PETERC.CHUANDCHENWUFAN
DepartmentofOceanography,NavalPostgraduateSchool,Monterey,California
(Manuscriptreceived25September2009,infinalform26January2010)
ABSTRACT
Ocean(T,S)dataanalysis/assimilation,conductedinthethree-dimensionalphysicalspace,isageneralized
averageofpurelyobserveddata(dataanalysis)orofmodeled/observeddata(dataassimilation).Becauseof
thehighnonlinearityoftheequationofthestateoftheseawaterandnonuniformverticaldistributionofthe
observationalprofiledata,falsestaticinstabilitymaybegenerated.Anewanalyticalconservedadjustment
schemehasbeendevelopedonthebaseofconservationofheat,salt,andstaticstabilityforthewholewater
columnwithpredetermined(T,S)adjustmentratios.Asetofwell-posedcombinedlinearandnonlinear
algebraicequationshasbeenestablishedandissolvedusingNewton’smethod.Thisnewschemecanbeused
foroceanhydrographicdataanalysisanddataassimilation.
1. Introduction andr istheinsitudensitytothedepthoftheupperof
thetwo adjacent levels z . The density inversion isde-
Raw and averaged observational hydrographic data k
finedbytheoccurrenceofthenegativevalueofE .The
contain substantial regions with vertical density in- k
minimumstaticstabilityisrepresentedbyE 5E .It
versions. For example, Jackett and McDougall (1995) k min
is not always possible to reach zero exactly due to the
foundthattheannuallyaveragedfieldoftheoceanatlas
precision limitations of the temperature and salinity
of Levitus (1982) hadmore than 44%of thecasts pos-
values used (Locarnini et al. 2006). As a result, the
sessing static instability at least at one level. Here, the
minimumvalueforthestaticstabilityisgivenby
word ‘‘cast’’’ is used to denote a pair of vertical tem-
peratureandsalinityprofiles.Awidelyusedconceptfor
E $ E , k51,2, ... ,K, (2)
staticstabilityEisdefinedbyLynnandReid(1968)as k min
‘‘the individual density gradient by vertical displace-
whereE isthereferencevaluefortheminimumstatic
ment of a water parcel (as opposed to the geometric min
stability,whichisuser-defined.Ifstaticinstabilityoccurs
density gradient).’’ For discrete samples (T , S ) at
k k
inanobservedoraveragedhydrographiccast[i.e.,(2)is
depthz ,k51,2,...,K(kincreasingdownward),the
k
notsatisfied],thisprofileneedstobeadjusted.
densitydifferencebetweentwoadjacentlevelsistaken
The National Oceanographic Data Center (NODC)
after one is adiabatically displaced to the depth of the
usesalocalinteractive(T,S)separatedadjustmentmethod
other.Computationally,E iscalculatedby
k
(Locarnini et al. 2006), which is based on the method
E 5r(S ,T ,z )(cid:2)r(S ,T ,z ), proposedbyJackettandMcDougall(1995)withsome
k k11 k11 k k k k
modifications,tominimallyadjustunstabletemperature
k51,2, ... ,K(cid:2)1, (1)
and salinity profiles (hereafter referred to as the MA
method)
wherer(S ,T ,z )isthelocalpotentialdensityof
k11 k11 k
thelowerofthetwoadjacentlevelsbetweenz andz ,
k k11 x5(T ,T , ... , T , S ,S , ... ,S )
withrespecttotheupperofthetwoadjacentlevelsz , 1 2 K 1 2 K
k
intostableprofiles
Correspondingauthoraddress:Dr.PeterC.Chu,NavalOcean
AnalysisandPrediction(NOAP)Lab,NavalPostgraduateSchool, x1Dx5(T 1DT ,T 1DT , ... ,T 1DT ,
833DyerRd.,Monterey,CA93940. 1 1 2 2 K K
E-mail:[email protected] S 1DS ,S 1DS , ... ,S 1DS ).
1 1 2 2 K K
DOI:10.1175/2010JTECHO742.1
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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
A Conserved Minimal Adjustment Scheme for Stabilization of
5b. GRANT NUMBER
Hydrographic Profiles
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
Naval Postgraduate School,Department of REPORT NUMBER
Oceanography,Monterey,CA,93943
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JUNE2010 CHU AND FAN 1073
AfterassumingTandSarelinear,theadjustmentisto thisexampleisthe18latitude–longitudeboxcentered
solvetheproblem: at53.58S,171.58EfromLevitusetal.(1998).Thisison
theNewZealandPlateau,withabottomdepthbelow
MinimizekDxk subject to A(cid:3)(x1Dx) $ E , (3)
min 1000 mandabove1100 m.ThemonthisOctober,dur-
ingtheearlyaustralsummer.Thereisnotemperatureor
wherethefinite-differenceapproximationofstabilityEk salinitydatawithinthechosen18box.Thus,theobjec-
becomes the inner product of the matrix A and the tivelyanalyzedvaluesinthis18boxwillbedependenton
profilevectorx1Dx.Obviously,matrixAdependson the seasonal objectively analyzed field and the data in
the solution Dx to the minimization problem (3), im- nearby18gridboxes.Thereismuchmoretemperature
plyingthattheconstraintsin(3)arenonlinear.Usually, datathansalinitydataontheNewZealandPlateaufor
aniterationmethodisused. October. This contributes to six small (on the order of
Before deciding which level to change, the values of 1022 kg m23) inversions in the local potential density
›T/›zand›S/›z,thegradientsoftemperatureandsalinity field calculated from objectively analyzed temperature
betweentwoadjacentlevelsinvolvedintheinstability, andsalinityfields(Table1).AfterusingtheMAmethod,
areexamined.Thishelpsdetermineifthetemperatureor theoriginalandadjustedprofilesfT ,S ,k51,2,...,
k k
salinityprofile,orboth,aretobechangedtostabilizethe KgareasshowninFig.1,andtheadjustedtemperature
densityfield.If›T/›z,0,›S/›z,0,onlytemperatureis and salinity profiles are listed in Table 2. Readers are
changed;if›T/›z.0,›S/›z.0,onlysalinityischanged; referred to appendix B of Locarnini et al. (2006) for
andif›T/›z,0,›S/›z.0,bothtemperatureandsalinity detailed information on the stabilization procedures.
fieldsareadjustedwithalocallineartrendtest(Locarnini The relative root-mean adjustment (RRMA) using the
et al. 2006). Here, the z axis points upward. The prin- MAmethodcanberepresentedby
cipleistostabilizethehydrographicprofileswithmini-
mumadjustment. uvffiffiffiffiffiKffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uvffiffiffiffiffiKffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
The benefit of using the MA method can be easily tu1 (cid:2)(DT )2 tu1 (cid:2)(DS )2
identifiedfromcomparisonbetweentwooceanatlases: Kk51 k Kk51 k
RRMA5 1
theocean atlasof Levitus (1982; withoutMA) andthe max(T )(cid:2)min(T ) max(S )(cid:2)min(S )
k k k k
World Ocean Atlas 2005 (Locarnini et al. 2006; with
50.0712. (4)
MA).Bothatlasesconsistofannuallyandmonthlyav-
eraged vertical profiles of temperature and salinity on
RRMA represents the mean adjustment relative to the
aglobal18318gridat33verticallevels.Theoceanatlas
rangeofaprofile.Thetotalheatandsaltchangesofthe
ofLevitus(1982)hasconsiderablecastspossessingstatic
watercolumnwithinthis18318gridboxareestimatedby
instability; however, the World Ocean Atlas 2005 con-
tainsnoprofilepossessingstaticinstability. ð0 ð0
Toeliminatethestaticinstability,theMAmethoddoes DQ5Ar c DTdz, D(salt)5A DSdz,
0 p
notrequiretheconservationofheatandsalt.Becauseone (cid:2)H (cid:2)H
of the ocean’s important roles in the earth’s climate is
where r (51028 kg m23) is the characteristic density,
heattransport,anadjustmentmadewithouttakingheat 0
c (54002 J kg21 K21) is the specific heat for the sea-
conservationintoaccountmayleadtoerrorsinestimat- p
water,H51000 m,andAistheareaofthegridbox,
ingtheocean’simpactonglobalclimatechange.Inthis
study,anewconservedschemeisdevelopedtosimulta- (cid:3)(cid:4)p 2
A5 R cosu,
neouslyadjustthetemperatureandsalinityprofilesfrom 180
(T ,S )to(T 1DT ,S 1DS ).Asetof2Kalgebraic
k k k k k k
(linear and nonlinear) equations are established to get whereR(56370 km)istheearth’sradius,andu(553.58)
(DT , DS ) on the base of heat and salt conservation, isthe latitude ofthe gridbox.The temperatureand sa-
k K
predetermined (DT /DS ) ratios (or called adjustment linityadjustments(DT,DS)areobtainedbycomparison
k K
ratios)foralllevels,andtheremovalofstaticinstability betweenTables1and2,theheatandsaltchangesofthe
byadjustingE toE 1DE withacombinedconserva- watercolumnforthisgridboxarecalculatedby
k k k
tionandnonuniformincrementtreatment.
DQ5(cid:2)7.0411 3 1017J, D(salt)5(cid:2)0.5443 3 1010kg.
2. Unconservedadjustment
Becauseoneoftheocean’simportantrolesintheearth’s
AnexampleasdescribedinappendixBofLocarnini climate is transporting heat fromlow to high latitudes,
etal.(2006)isusedforillustration.Theareachosenfor nontrivialheatandsaltlossesshowthattheunconserved
1074 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME27
TABLE1.Gridbox171.58E,53.58SLevitusetal.(1998)profilesbeforestabilization(fromLocarninietal.2006,TableB1).Here,the
asterisksinthelastcolumnindicatethestaticinstability.
k Depth(m) T(8C) S(ppt) r(S ,T ,z )(kgm23) r(S ,T ,z )(kgm23) E (kgm23)
k11 k11 k k k k k
1 0 7.1667 34.4243 26.9476 26.9423 0.0054
2 10 7.1489 34.4278 26.8982 26.9939 20.0957*
3 20 7.0465 34.2880 26.9529 26.9443 0.0085
4 30 7.0050 34.2914 27.0104 26.9990 0.0114
5 50 6.9686 34.2991 27.0967 27.1028 20.0061*
6 75 7.0604 34.3073 27.2406 27.2120 0.0286
7 100 6.9753 34.3280 27.3892 27.3560 0.0332
8 125 6.9218 34.3604 27.5164 27.5046 0.0117
9 150 6.8919 34.3697 27.6000 27.6316 20.0316*
10 200 6.9363 34.3364 27.8123 27.8302 20.0179*
11 250 7.0962 34.3415 28.0295 28.0421 20.0126*
12 300 7.1622 34.3367 28.2684 28.2593 0.0092
13 400 6.8275 34.2852 28.6664 28.7281 20.0618*
14 500 7.4001 34.3123 29.3699 29.1238 0.2461
15 600 6.2133 34.4022 29.9386 29.8292 0.1094
16 700 5.9186 34.4868 30.5869 30.3978 0.1891
17 800 4.5426 34.4904 31.0754 31.0488 0.0266
18 900 4.1263 34.4558 31.6539 31.5377 0.1162
19 1000 3.3112 34.4755 32.1176
adjustmentmaychangeheattransportandinturnaffect E*5E , (5)
k min
theoverturningthermohalinecirculation. i
thatis,theminimaladjustmentwithincrementof
3. Stabilization
Thestabilizationprocessisdividedintothreeparts:1) DE 5E (cid:2)E .
k min k
stability increasing at unstable levels, 2) stability de- i i
creasing at stable levels, and 3) normalization for con-
servation of stability for the cast. Let static instability Suchanincreaseofstabilitywillbecompensatedbythe
occuratlevelk1,k2,...,ki[i.e.,satisfiestheinequality decreaseofstabilityatneighboringlevelski6m(m51,
(2)], the static stability E is increased to its marginal 2, ...) with skipping the unstable levels until reaching
k
stabilityvalue(E*), i thetopandbottomoftheprofile,
k
i
(E (cid:2)DE /2m11 if E (cid:2)DE /2m11$E
E* 5 ki6m ki ki6m ki min.
ki6m Emin if Ek6m(cid:2)DEk/2m11,Emin
i i
The static stabilities for the whole profile before and r(S 1DS , T 1DT , z )(cid:2)r(S 1DS ,
k11 k11 k11 k11 k k k
aftertheadjustmentarecalculatedby
T 1DT , z )5E**, k51,2, ... , K(cid:2)1. (8)
k k k k
K K
(cid:2) (cid:2)
I5 E , I*5 E*. (6)
k51 k k51 k WhenEminisspecified[seeEq.(5)],theright-handside
of(8)(i.e.,E**)istheknownadjustment,whichiscal-
k
Thenormalizationprocessisconductedby culated through (5)–(7). Equation (8) is used to de-
terminethetemperatureandsalinityadjustmentsateach
I depth for given E**. The difference between (7) (i.e.,
E**5 E* (7) k
k I* k the direct determination of E**) and (8) is that (7)
k
to keep the conservation of the static stability for each showstheminimaldensityadjustmenttoremovestatic
profile. After three stabilization processes, the static instability,and(8)istocalculatethe(T,S)adjustment
stabilityisrepresentedby[seeEq.(1)] ateachdepth.
JUNE2010 CHU AND FAN 1075
FIG.1.Original(dashed)andadjusted(solid)profilestemperatureofTk,salinitySk,andstaticstabilityEkatthegridbox53.58S,171.58E
usingtheMAmethod(Locarninietal.2006).
4. Constraintsfortemperatureandsalinity ð0 ð0
DTdz50, DSdz50, (9)
adjustment
(cid:2)h (cid:2)h
Conservationofheatandsaltfortheadjustmentcan
berepresentedby whichcanbediscretizedby
TABLE2.Gridbox53.58S,171.58EimprovedLevitusetal.(1998)profilesafterstabilizationusingtheMAmethod(fromLocarninietal.
2006,TableB2).
k Depth(m) T(8C) S(ppt) r(S ,T ,z )(kgm23) r(S ,T ,z )(kgm23) E (kgm23)
k11 k11 k k k k k
1 0 7.1667 34.3096 26.8521 26.8519 0.0002
2 10 7.1489 34.3063 26.8982 26.8982 0.0000
3 20 7.0465 34.2880 26.9529 26.9443 0.0085
4 30 7.0050 34.2914 27.0042 26.9990 0.0051
5 50 7.0132 34.2991 27.0967 27.0967 0.0000
6 75 7.0604 34.3073 27.2361 27.2120 0.0240
7 100 6.9796 34.3228 27.3513 27.3513 0.0000
8 125 6.9897 34.3243 27.4667 27.4667 0.0000
9 150 7.0242 34.3301 27.5820 27.5820 0.0000
10 200 7.0628 34.3364 27.8123 27.8123 0.0000
11 250 7.0962 34.3415 28.0422 28.0421 0.0000
12 300 7.0748 34.3367 28.2719 28.2719 0.0001
13 400 6.8275 34.2894 28.7314 28.7314 0.0000
14 500 6.9604 34.3123 29.3699 29.1899 0.1799
15 600 6.2133 34.4022 29.9386 29.8292 0.1094
16 700 5.9186 34.4868 30.5869 30.3978 0.1891
17 800 4.5426 34.4904 31.0754 31.0488 0.0266
18 900 4.1263 34.4558 31.6539 31.5377 0.1162
19 1000 3.3112 34.4755 32.1176
1076 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME27
TABLE3.Changeof(T 1DT(j))(8C)ateachiterationusingtheNewton’smethod.Itisnotedthattheiterationconvergesatthe
k k
thirditeration.
k Depth(m) j50 j51 j52 j53 j54
1 0 7.166700 7.212634 7.212833 7.212833 7.212833
2 10 7.148900 7.289401 7.289072 7.289072 7.289072
3 20 7.046500 6.818173 6.816828 6.816828 6.816828
4 30 7.005000 6.872865 6.872591 6.872591 6.872591
5 50 6.968600 6.888794 6.888861 6.888861 6.888861
6 75 7.060400 7.023494 7.023712 7.023712 7.023712
7 100 6.975300 6.977379 6.977638 6.977638 6.977638
8 125 6.921800 6.965175 6.965378 6.965378 6.965378
9 150 6.891900 6.983992 6.983997 6.983997 6.983997
10 200 6.936300 6.959537 6.959779 6.959779 6.959779
11 250 7.096200 7.125999 7.126229 7.126229 7.126229
12 300 7.162200 7.228075 7.228205 7.228205 7.228205
13 400 6.827500 6.995044 6.994489 6.994488 6.994488
14 500 7.400100 7.229221 7.228652 7.228652 7.228652
15 600 6.213300 6.129374 6.129400 6.129400 6.129400
16 700 5.918600 5.883923 5.884121 5.884121 5.884121
17 800 4.542600 4.542873 4.543127 4.543127 4.543127
18 900 4.126300 4.153784 4.154020 4.154020 4.154020
19 1000 3.311200 3.362894 3.363075 3.363075 3.363075
K(cid:2)1 Equations(10)and(11)canberewrittenby
(cid:2)(DT 1DT )
k k11 (z (cid:2)z )50, (10)
k51 2 k k11 K K
(cid:2) (cid:2)
a DT 50, a DS 50, (12)
k k k k
K(cid:2)1 k51 k51
(cid:2)(DS 1DS )
k k11 (z (cid:2)z )50. (11)
k51 2 k k11 where
z (cid:2)z z (cid:2)z z (cid:2)z z (cid:2)z z (cid:2)z
a 5 1 2 , a 5 1 3 , a 5 2 4 , ... , a 5 K(cid:2)2 K , a 5 K(cid:2)1 K . (13)
1 2(z (cid:2)z ) 2 2(z (cid:2)z ) 3 2(z (cid:2)z ) K(cid:2)1 2(z (cid:2)z ) K 2(z (cid:2)z )
1 K 1 K 1 K 1 K 1 K
Obviously,wehave toillustratethebasicmethodologyofthisanalyticalad-
justmentprocedure.Thisratiomayvarywithdepth.A
K largepartofthepaperbyJackettandMcDougall(1995)
(cid:2)
a 51, a .0 for k51,2, ... , K. (14) wasdevotedto developing amethodtodetermine g .
k k k
k51
Interestedreadersarereferredtotheirpaper.
Theadjustmentratiosg areusedforN21levels, Equations (10), (11), (15), and (8) represent a set of
k
2K algebraic equations for 2K unknowns (DT , DS ),
k k
k51,2,...,K.Thus,theyareclosure.Amongthem,(8)
DT 1g DS 50, k51,2, ... , K(cid:2)1. (15)
k k k isnonlinearand(10),(11),and(15)arelinear.
Becausetemperatureandsalinitycorrectionsaffectthe 5. Example
density differently, that is, the increase (decrease) of
The same example as described in section 2 is used.
temperature(salinity)decreases(increases)thedensity.
SubstitutionoffS ,T ,z gvaluesfromTable1into(8),
This leads to a positive value of g . Here, we use the k k k
k
(10), (11), and (15), and the Newton iteration method
simplestform,
(Kelley1987,seeappendixB)isusedtosolvethesetof
2Kalgebraicequations.Forthehydrographiccastlisted
g 5g [ max(Tk)(cid:2)min(Tk), (16) inTable1,onlythreeiterationsareneededtoeliminate
k max(S )(cid:2)min(S ) the static instability. Tables 3 and 4 list the values of
k k
JUNE2010 CHU AND FAN 1077
TABLE4.Changeof(S 1DS(j))(ppt)ateachiterationusingNewton’smethod.Itisnotedthattheiterationconvergesat
k k
thethirditeration.
k Depth(m) j50 j51 j52 j53 j54
1 0 34.424300 34.421995 34.421985 34.421985 34.421985
2 10 34.427800 34.420749 34.420765 34.420765 34.420765
3 20 34.288000 34.299459 34.299526 34.299526 34.299526
4 30 34.291400 34.298031 34.298045 34.298045 34.298045
5 50 34.299100 34.303105 34.303102 34.303102 34.303102
6 75 34.307300 34.309152 34.309141 34.309141 34.309141
7 100 34.328000 34.327896 34.327883 34.327883 34.327883
8 125 34.360400 34.358223 34.358213 34.358213 34.358213
9 150 34.369600 34.364978 34.364978 34.364978 34.364978
10 200 34.336400 34.335234 34.335222 34.335222 34.335222
11 250 34.341500 34.340005 34.339993 34.339993 34.339993
12 300 34.336700 34.333394 34.333388 34.333388 34.333388
13 400 34.285200 34.276792 34.276820 34.276820 34.276820
14 500 34.312300 34.320875 34.320904 34.320904 34.320904
15 600 34.402200 34.406412 34.406410 34.406410 34.406410
16 700 34.486800 34.488540 34.488530 34.488530 34.488530
17 800 34.490400 34.490386 34.490374 34.490374 34.490374
18 900 34.455800 34.454421 34.454409 34.454409 34.454409
19 1000 34.475500 34.472906 34.472897 34.472897 34.472897
fT ,S gattheeachiteration.Theyshowthehigheffi- The heat and salt are conserved for the whole water
k k
ciencyofthismethodforeliminationofstaticinstability columnwiththerelativeroot-meanadjustment
inthehydrographiccast.Figure2showstheoriginaland
RRMA50.0482. (18)
adjustedprofiles
Comparing(18)to(4),wemayfindthatthisanalytical
conserved adjustment scheme has a smaller RRMA
(cid:5)(cid:6)
Sk, Tk, Ek , k51,2, ... ,K. (17) (0.0482)thantheMAmethod(0.0712).
FIG.2.AsinFig.1,butusingtheanalyticalconservedmethodproposedinthispaper.
1078 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME27
FIG.3.DistributionofstaticallyunstablecastsintheJPL-ECCO10-daydatacenteredon31Dec2008(availableonlineathttp://ecco.jpl.
nasa.gov/external/).Thedatawereproducedbyadataassimilationsystem.
Data assimilation is required in operational ocean (Fig. 3). Here, the NODC’s criterion for flagging out
data access and retrieval (Sun 1999). It is to blend the staticallyunstableprofiles,
modeled variable x with observational data y (e.g.,
m o
Kalnay2003;Chuetal.2004),
8(cid:2)0.03 kgm(cid:2)3 (0 $ z $ (cid:2)30m)
>< k
x 5x 1W(cid:3)[y (cid:2)H(x )], (19) E 5 (cid:2)0.02 kgm(cid:2)3 ((cid:2)30m.z $ (cid:2)400m), (20)
a m o m min k
>:0 kgm(cid:2)3 ((cid:2)400m.z )
k
where x is the assimilated variable, H is an operator
a
thatprovidesthemodel’stheoreticalestimateofwhat
is observed at the observational points, and W is the is used. Because such a false static instability is due to
weight matrix. The difference among various data as- theblendingofobservationaldatawiththemodeldata,
similationschemessuchasoptimalinterpolation(e.g., itisnotarealinstability.Useoftheconvectiveadjust-
Lozano et al. 1996), Kalman filter (e.g., Galanis et al. mentschememayovercorrecttheprofiles.
2006),andvariationmethods(e.g.,TangandKleeman To illustrate this, we discuss the existing convective
2004), is the different ways to determine the weight adjustmentschemesinoceanmodels.Thevariouscon-
matrix W. The data assimilation process (19) can be vective adjustment schemes are based on the same
consideredastheaverage(inageneralizedsense)ofx original idea (e.g., Bryan 1969): whenever a water col-
m
andy .Inocean(T,S)dataassimilation,theobserva- umnisstaticallyunstable,temperatureandsalinityare
o
tional data y may contain several casts, which are verticallyadjustedtomakethewatercolumnneutrally
o
staticallystable.Themodelprofilex isalsostatically stable,withheatandsaltconservedintheprocess.The
m
stable because convective adjustment (Bryan 1969) is adjustment takes an iterative approach. The iteration
usuallyconductedateachtimestep. continues between all adjacent levels until the static
Falsestaticstabilitymaybegeneratedafter(T,S)data instability is removed in the whole water column. Be-
assimilation[i.e.,performing(19)].Forexample,10-day cause the adjustment acts on only neighboring points,
JetPropulsionLaboratory(JPL)EstimatingtheCircu- the number of iterations required to reach the final
lation and Climate of the Ocean (ECCO) (T, S) fields stablestateisinfiniteforagivenunstableprofile(Smith
centered on 31 December 2008 (available online at 1989). In practice, however, the number of iteration is
http://ecco.jpl.nasa.gov/external/)showthataconsider- alwaysfinite,andthisleadstosomeresidualinstability
ableportion(35.32%)ofprofilesarestaticallyunstable (Killworth1989).
JUNE2010 CHU AND FAN 1079
FIG.4.AsinFig.1,butusingthecompleteconvectiveadjustmentmethod(YinandSarachik1994).
Several algorithms were developed to remove these adjuststhetemperatureandsalinityprofilesfDT ,DS ,
k k
residual static instabilities such as the implicit vertical k 5 1, 2, ... , Kg simultaneously and efficiently on the
diffusion scheme (Cox 1984; Killworth 1989) and the basisofthreetypesofconstraints:1)heatandsaltcon-
complete adjustment scheme (Yin and Sarachik 1994). servation,2)predetermined(DT /DS )ratios(orcalled
k k
The former tests the static stability between the verti- adjustment ratios) for all levels, and 3) the removal of
callyadjacentlevelsand,ifunstable, theverticaldiffu- static instability by adjusting the static stability with
sivity is set to a large value (convective diffusivity) to a combined conservation and nonuniform increment
smooth out the instability. The latter determines the treatment.Withtheseconstraints,asetof2Kcombined
upper and lower boundaries of each adjusted region linear/nonlinearalgebraicequationsareestablishedfor
whilekeepingtheinstantaneousadjustmentwithineach fDT , DS g. Among them, (K 1 1)algebraic equations
k k
unstable region. Yin and Sarachik (1994) showed that arelinear,and(K21)equationsarenonlinear.Newton’s
the complete convective adjustment scheme is more method is used to solve this set of equations. The pro-
efficientthantheimplicitverticaldiffusionschemeand posedschemehasverysmallrelativeroot-mean-square
guaranteedacompleteremovalofstaticinstabilityofa adjustment compared to the existing methods. More-
watercolumnateachtimestep.Forthesameexampleas over, it has three features: 1) conservation of heat and
described in section 2, the complete convective adjust- salt, 2) removal of static instabilities with small (T, S)
mentschemeremovesthestaticinstabilities(Fig.4)with adjustments,and3)analyticalform.Withthesefeatures,
therelativeroot-meanadjustment it can be widely used in ocean (T, S) data analysis. Be-
sides, ocean data assimilation may cause false static
RRMA50.2192. (21) instabilities. Because this instability is not real, com-
monlyusedconvectiveadjustmentschemesmayover-
Thisvalueis4.5timeslargerthanthatof(0.0482)using adjust the profiles. Therefore, the proposed analytical
theanalyticaladjustmentmethod. conserved scheme can be used in ocean (T, S) data
assimilation.
6. Conclusions
A new analytical conserved adjustment scheme is Acknowledgments.TheOfficeofNavalResearch,the
developedtoeliminate thestaticinstability ofraw and NavalOceanographicOffice,andtheNavalPostgraduate
averagedobservationalhydrographicdata.Thismethod Schoolsupportedthisstudy.
1080 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME27
APPENDIXA individualobservationss .Thus,theconservationcon-
e
straints(10)and(11)guaranteethat
ValidityoftheConservationConstraints
adj t
(c )5(c)1(c9), (A8)
In ocean modeling, all the convective adjustment
schemes for stabilizing (T, S) profiles require heat and
the same smaller error variance of the vertically in-
saltconservation(e.g.,YinandSarachik1994).Inocean
tegratedobservedvalues.
data analysis, such conservation constraints are also
valid.Afterqualitycontrolprocedures,itisreasonable
toassumethatoceanobservationaldataccontainran- APPENDIXB
domerrorc9,
c5ct1c9, (A1) NewtonMethod
Let the temperature and salinity adjustment be rep-
with population meanhc9i5 0 and standard deviation
s .Here,cdenotes(T,S),andctisthetruevalueatthe resentedbya2K-dimensionalvectorP,
e
samelocationandtimeastheobservationtakenplace.
2 3 2 3
Thepopulationmeanof(A1)gives p DT
1 1
6 p 7 6 DS 7
hci5hcti. (A2) 66 p2 77 66DT1 77
6 3 7 6 2 7
6 7 6 7
6 p 7 6 DS 7
An observational profile (c , k 5 1, 2, ... , K) can be 6 4 7 6 2 7
taken as a sample. Verticalkintegration of the observa- P[66 : 77566 : 77, M52K. (B1)
tional profile is represented by weighted average [see 66 : 77 66 : 77
6 7 6 7
(12)], 6 : 7 6 : 7
6 7 6 7
6p 7 6DT 7
K K K 4 M(cid:2)15 4 K5
(c)5(cid:2)a c , (ct)5(cid:2)a ct, (c9)5(cid:2)a c9. pM DSK
k k k k k k
k51 k51 k51
(A3)
ThealgebraicEqs.(10),(11),(15),and(8)[notethatwe
put(8)atthelast]canberepresentedby
Therandomerrorsatdifferentdepthc9 areconsidered
k
independent. The central limit theorem states that the
F(P)50, (B2)
linearcombination
whereFhasthedimensionof2K.TheclassicalNewton
K
(cid:2) method (Kelley 1987) for approximating a desired so-
Y95 a c9 (A4)
k51 k k lutionPto(B2)isformallydefinedbytheiteration
hasanormaldistributionwithzeromeanandvariance, P(j11)5P(j)(cid:2)J(cid:2)1(P(j))F(P(j)), j50, 1, 2, ... , (B3)
K K where P(j) is the jth approximation to the solution of
(cid:2) (cid:2)
s2Y95 a2ks2e5s2e a2k. (A5) (B2),J(P(j))istheJacobianmatrixofF(P)evaluatedat
k51 k51 P(j).InversionoftheJacobianmatrixisnotperformed
inpractice;rather(B3)isimplementedviasolutionof
From(14),wehave
the following system of linear equations at the each
iteration:
K K
(cid:2) (cid:2)
a2, a 51. (A6)
k51 k k51 k J(P(j))(cid:3)d(j)5b(j), b(j)[(cid:2)F(P(j)), (B4)
Substitutionof(A6)into(A5)leadsto followedbytheupdate
sY9,se, (A7) P(j11)5P(j)1d(j), (B5)
whichindicatesthattheerrorvarianceofthevertically whered(j)iscalledtheNewtondirection.Theiteration
integratedobservedvaluescissmallerthanthatofthe stopsatstepJwhen