the deep water stratification of ocean general circulation models

This article was downloaded by: [University of Auckland Library]On: 25 November 2014, At: 13:01Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK

Atmosphere-OceanPublication details, including instructions for authorsand subscription information:http://www.tandfonline.com/loi/tato20

The deep water stratificationof ocean general circulationmodelsPatrick F. Cummins aa Institute of Ocean Sciences , Sidney, BritishColumbia, V8L 4B2Published online: 19 Nov 2010.

To cite this article: Patrick F. Cummins (1991) The deep water stratificationof ocean general circulation models, Atmosphere-Ocean, 29:3, 563-575, DOI:10.1080/07055900.1991.9649417

To link to this article: http://dx.doi.org/10.1080/07055900.1991.9649417

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of theContent.

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,sub-licensing, systematic supply, or distribution in any form to anyone is

http://www.tandfonline.com/loi/tato20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/07055900.1991.9649417

http://dx.doi.org/10.1080/07055900.1991.9649417

expressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014

http://www.tandfonline.com/page/terms-and-conditions

The Deep Water Stratification of OceanGeneral Circulation Models

Patrick F. CumminsInstitute of Ocean Sciences

Sidney, British Columbia V8L 4B2

[Original manuscript received 2 January 1991; in revised form 6 March 1991]

ABSTRACT A central problem in climate and ocean modelling is the accurate simulation ofthe climatological state of the oceanic density field. A constant vertical diffusivity for heatand salt is frequently employed in ocean general circulation models (OGCMs) and it isusually assigned a value designed to optimize the depth of the pycnocline. One undesiredconsequence of this choice is a poor representation of the deep water, which is usuallyinsufficiently stratified. In contrast to the uniform diffusivity of many models, some observa-tional studies suggest that the vertical diffusivity is not constant but increases with depth,possibly in inverse proportion to the local buoyancy frequency. Numerical experiments withan OGCM are presented that demonstrate that allowing the vertical diffusivity to increasebelow the pycnocline substantially increases the stratification of the abyssal water mass ofthese models without significantly affecting the pycnocline depth, and hence may lead to abetter representation of the vertical density structure.

RÉSUMÉ Une simulation précise de l'état climatologique du champ de densité océanique estl'un des problèmes importants dans le modélisage du climat et de l'océan. On emploie sou-vent, dans les modèles de circulation générale de l'océan (MCGO), une diffusivité verticaleconstante pour la chaleur et le sel; habituellement, à une valeur désignée pour optimiser laprofondeur de la pycnocline. Une conséquence indésirable de ce choix réside dans le faitqu'on obtient une pauvre représentation, habituellement pas assez stratifiée, de l'eau pro-fonde. Contrairement à la diffusivité uniforme de plusieurs modèles, certaines observationssuggèrent que la diffusivité verticale n'est pas constante mais augmente avec la profondeur,possiblement en proportion inverse à la fréquence locale de flottabilité. Des expériencesutilisant un MCGO montrent qu'en augmentant la diffusivité verticale sous la pycnocline,on augmente de façon importante la stratification de la masse d'eau abyssale du modèlesans déranger grandement la profondeur de la pycnocline, ce qui peut donc conduire à unemeilleure représentation de la structure verticale de la densité.

1 IntroductionIn a review of the state of ocean general circulation modelling, Bryan and Sarmiento(1985) compared the climatological vertical density structure of the mid-latitudeNorth Atlantic with the density structure obtained from the GFDL ocean general

ATMOSPHERE-OCEAN 29 (3) 1991, 563-575 0705-5900/91/0O0O-0563S01.25/0© Canadian Meteorological and Oceanographic Society

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014

564 / Patrick F. Cummins

circulation model (OGCM) of Cox and Bryan (1984). Although the model had ahighly idealized basin geometry and surface boundary conditions, the shape andscale depth of the model pycnocline corresponded well with the data. There werealso some interesting and systematic differences between the observations and themodel. In the depth range 200-800 m, which encompasses the main pycnocline,the model was significantly more stratified than the ocean. Conversely, in the depthrange 1100-3600 m the model density structure was much more homogeneousthan the observations: the Brunt-Väisälä frequency computed from the data wasup to six times larger than in the model. The model studied by Bryan (1987)also displayed a similar density structure, with very homogeneous deep water.The problem is not unique to implementations of the GFDL ocean model. Forexample, Colin de Verdière (1989) also remarked that his numerical solutions ofthe planetary geostrophic equations display quasi-homogeneous deep water. Bryanand Sarmiento (1985) suggested that the excessive homogeneity of the simulateddensity field below the pycnocline may be due to the presence of only a singlesource of deep water in their model, compared with the multiple sources in thereal ocean. They also noted that the constant vertical eddy diffusivity (Kv = 0.3cm2 s"1) that was used to represent small-scale turbulent mixing processes may beinadequate for an accurate simulation of the ocean's density structure. The secondof these remarks is taken up in this note.

The specification of appropriate mixing parametrizations and coefficients incoarse-resolution OCGMs is an area of active research in ocean modelling. Gener-ally speaking, the models display considerable sensitivity to mixing parameters thatare poorly known and whose dependence on the resolved flow structures is not wellunderstood. There have been several attempts to advance beyond the frequently in-voked hypothesis of a spatially uniform eddy diffusivity. For example, Pacanowskiand Philander (1981) have shown that Richardson number-dependent vertical mix-ing can be usefully applied to simulate the density structure of equatorial regionsof the ocean. Other works, such as Gerdes et al. (1991), have advocated the useof advection operators with implicit numerical diffusion designed to preserve themonotonicity of the advected field (e.g. the flux-corrected transport algorithm.)Despite these efforts and many others, the appropriate form for the vertical (i.e.diapycnal) diffusivity remains an unresolved matter.

Gargett (1984) synthesized observational and theoretical estimates of the verti-cal diffusivity and proposed that the diffusivity is proportional to N~g, where Νis the Brunt-Väisälä frequency and the exponent, q « O(l). This form is consid-ered particularly plausible if internal wave breaking is the dominant mechanismfor small-scale mixing in the ocean interior (Gargett and Holloway, 1984). Animmediate consequence of incorporating a diffusivity proportional to N~l into anOGCM is that the diffusivity increases below the pycnocline. It is not evidentwhether an increase in the vertical diffusivity at depth can help improve the rep-resentation of the density structure in OGCMs, particularly if the stratification isalready too weak. Nevertheless, in a recent numerical study, Cummins, Hollowayand Gargett (1990, hereinafter referred to as CHG) noticed that incorporating astability-dependent diffusivity in an OGCM led to an increase in the stratification

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014

The Deep Water Stratification of Ocean General Circulation Models / 565

of the deep water. Several ocean modelling and coupled atmosphere-ocean mod-elling studies (e.g. Bryan and Lewis (1979), Toggweiler et al. (1989), Stouffer et al.(1989)) and many others have appeared in the literature in which the diffusivity isincreased at great depth with the intention of incorporating greater realism into themodels. The effects of this putative enhancement have not yet been made explicit.

The purpose of the present note is to clarify the issue of the deep water stratifi-cation and to examine the effects on the structure of the density field of the deepocean produced by allowing the diffusivity to increase below the pycnocline inan OGCM. The numerical solutions confirm that increasing the vertical diffusivitybelow the pycnocline can lead to a substantial increase in the abyssal stratificationwithout significantly affecting the depth of the pycnocline. These numerical resultsare interpreted in terms of a simple advection-diffusion model of the density strat-ification. It is shown that the abyssal stratification increases with decreasing Pécletnumber (the ratio of the advective term to the diffusive term). The results suggestthat OGCM simulations of the density structure of the ocean may be improved byallowing the diffusivity to increase below the pycnocline.

2 Numerical experimentsThe Bryan-Cox ocean general circulation model (Cox, 1984) is employed for thisstudy. The configuration of the model is similar to CHG's except for the specifi-cation of the vertical diffusivity. Briefly, the model domain is an idealized sectorof ocean ranging 45° in longitude and extending from the equator to 66°N, witha uniform depth of 5000 m. The resolution is 3° in both the meridional and zonaldirections; there are 33 levels in the vertical. The horizontal and vertical mixingcoefficients for momentum are set to 109 and 1 cm2 s"1, respectively. The hori-zontal mixing coefficient for heat and salt is set to a uniform value of 107 cm2

s~l. The vertical diffusivity is assigned a value of 104 cm2 s~' in regions wherethe density stratification is unstable, to allow for vigorous mixing of the unstableportions of the water column. A non-linear equation of state (Bryan and Cox, 1972)is used to obtain density from the temperature and salinity fields. At the surface,the temperature and salinity are required to relax on a time-scale of 25 days tothe zonally constant reference values given in Fig. 1 of CHG. These values wereobtained by zonally annually averaging the sea surface temperature and salinitydata from the Levitus (1982) atlas for both hemispheres. The wind stress forcingretains the essential characteristics of the observed zonally averaged stress (Fig. 1of CHG).

Five numerical experiments were conducted to verify the effects on the stratifi-cation of a vertical diffusivity that increases below the pycnocline. The diffusivityis given the form used in the model of Bryan and Lewis (1979):

Kv(z) = Ao+— tan-'[5(zo-z)] (1)κ

where ζ is the (positive upwards) vertical coordinate. The parameters Ao, Cr, 5 andzo are given in Table 1 for each experiment. In addition, the profiles of verticaldiffusivity versus depth are illustrated in Fig. 1. Experiment 0 is a control run with

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014


TABLE 1. Parameters used in Eq. (1) to specify the vertical diffusivity for the different experiments. Ablank entry indicates no value was specified. Also included are the pycnocline scale depthsdefined according to Eq. (2), and the maximum of the meridional overturning streamfunction,denoted by ψον<.Γ, in Sverdrups (Sv; 1 Sv = 106 m3 s"1). In the calculation of the scale depths,the potential density is referenced to the surface.

ExperimentNumber

01234

Ao(cm2 s"1)

0.300.711.392.690.80

CT

(cm2 s"1)

01.332.515.101.05

S(10-3m-')

1.55.0

10.84.5

z0

(m)

-1000-1000-1000-2500

PycnoclineScale Depth

(m)

283278266256283

>lw,(Sv)

8.510.111.914.28.6

1 -

Q 3 -

5-1

• ΕΧΡΤ0

Δ ΕΧΡΤ1

Χ EXPT2

φ ΕΧΡΤ3

Χ ΕΧΡΤ4

I0.0 1.5 4.5 6.03.0

Κ, (cm2 s1)

Fig. 1 The variation of the vertical diffusivity with depth for the five experiments of this study.

the vertical diffusivity held constant at the commonly used value of 0.3 cm2 s~'. Forthree other experiments, the vertical diffusivity is in the range from 0.3 to 0.4 cm2

s~' in the upper 500 m of the water column. Below 500 m, the diffusivity increasesprogressively with depth to values of 1.3, 2.6 and 5.2 cm2 &~l in the deep ocean(Expts 1, 2 and 3, respectively). The transition from the upper ocean to the deep

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014


ocean diffusivity values occurs at a depth determined by zo· This parameter waschosen such that the transition occurs at some point below, but still in proximity to,the pycnocline. The incorporation of a stability-dependent diffusivity as proposedby Gargett (1984) leads to an increase in the vertical diffusivity at a similar pointrelative to the pycnocline (e.g. Fig. 2 in CHG). Finally, in Expt 4, the diffusivityassumes the vertical profile used by Bryan and Lewis (1979) and others cited above,and the transition depth is much deeper than in the other experiments.

Each experiment was integrated from an initial motionless state having a uniformtemperature and salinity of 5°C and 33%e, respectively. The time integrations werecarried out using the asynchronous time-stepping method of Bryan (1984) for about2500 (7500) model years in the upper (lower) ocean. This time is sufficient for theabyssal temperatures and salinities to adjust completely to the forcing so that astatistically steady state is achieved^. At the termination of each experiment, thelevel-averaged temperature and salinities at all depths are without a systematictrend.

Pycnocline scale depths, d, were calculated according to the formula of Bryanand Sarmiento (1985), viz.

d — {ç>s — Pb)~l / (P - Pb)dz (2)J-H

where ρ is the potential density, ps and Pi are the potential densities at ζ = 0 andζ = —Η, i.e. at the ocean surface and bottom, respectively. The scale depths foreach experiment were averaged over the subtropical gyre and the resulting valuesare given in Table 1. For a spatially constant diffusivity, Bryan (1987) has shownthat the pycnocline scale depth is approximately proportional to the cube root ofthe vertical diffusivity. When only the diffusivity at depth is varied, the situationis entirely different. The range of scale depths in Table 1 is less than 30 m, whichindicates that the pycnocline depth is quite insensitive to the specified variations inthe vertical diffusivity. Although the magnitude of the diffusivity in the upper oceangreatly influences the depth of the pycnocline, a similar sensitivity to variations inthe diffusivity below the pycnocline is not obtained. This result is consistent withthe numerical experiments of CGH and Gerdes et al. (1991).

The vertical density gradient was computed from the temperature and salinitydistributions and then area-averaged over the interior of the subtropical gyre at thesteady states. The Brunt-Väisälä frequency, N, was computed according to

Ρ

where g is the gravitational constant. The resulting vertical profiles of Ν are given

t Typically, the only remaining variability in steadily forced, coarse-resolution general circulationmodels after the long adjustment of the deep ocean to equilibrium is caused by small scale intermittentconvective events, confined to localized regions.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014


ι

2-

3 -

4-

5-1

• EXPTO

Δ ΕΧΡΤ1

Χ ΕΧΡΤ2

Ο ΕΧΡΤ3

Κ ΕΧΡΤ4

Ν (rad h 1 )

Fig. 2 The vertical distribution of the Brunt-Väisälä frequency, Ν, averaged over the subtropical gyre

(13.5^»3.5°N, 10.5-34.5°E, with the western boundary located at 0°E), for the lower 4 km of

the basin.

in Fig. 2. As expected, Expt 0 with a uniform constant vertical diffusivity producesa very homogeneous water mass below 2000-m depth. As the diffusivity at depth isprogressively augmented, the density gradients between 1500 and 4000 m increasesignificantly over the constant diffusivity case. For example, in Expt 1 at 3000-mdepth the buoyancy frequency is five times the frequency in Expt 0. In Expt 2, thereis an additional 50% increase over Expt 1 at this depth. A further, more modestincrease in the density gradient below 3000-m depth occurs in Expt 3. The densitygradient in the depth range 1000-1500 m is somewhat reduced in Expt 2 fromthose in Expts 0 and 1, and is further reduced in Expt 3. The stratification in Expt4 differs very little from that in Expt 0, which indicates that, in this experiment,the increase in the diffusivity takes place at a depth that is too great to redistributevertical gradients as in Expts 1-3.

In summary, the profiles of Fig. 2a confirm, at least within the context of an ide-alized sector model, that allowing the diffusivity to increase below the pycnoclineleads to an increase in the stratification of the abyssal water mass. The entire depthof the water column below the pycnocline is affected in this way, not just the re-gion immediately below the pycnocline. These changes occur without a significantvariation in the depth of the pycnocline.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014


TABLE 2. Temperature and salinity averaged overthe subtropical gyre and over the bottomkilometre of the model domain

ExperimentNumber

01234

Temperature(°C)

2.642.782.963.212.65

Salinity(ppt)

33.2833.2933.3133.3433.28

Although in this note we are concerned mainly with the stratification of themodel, other properties of the solutions that are of climatic significance may beaffected by the specified variations in the diffusivity. Two quantities of primaryinterest are the magnitudes of the poleward heat transport and of the meridionaloverturning stream-function. The latter is a measure of the intensity of the ther-mohaline circulation. The poleward heat transport is rather insensitive to changesin the diffusivity below the pycnocline. The latitudinal profiles of the heat fluxare similar, and maximum amplitudes vary by less than 15% over the five experi-ments. This is due to the relative invariance of the pycnocline depth, which is thedetermining factor for the heat flux, given the imposed thermal forcing (Bryan,1987).The overturning streamfunction displays considerably greater sensitivity to the vari-ations in the diffusivity. Whereas the overall pattern of an intense sinking regionalong the poleward boundary and a broad upwelling region over the interior remainsbasically unaffected, the amplitude of this overturning mode varies significantly.Maxima of the overturning streamfunction, which are given in Table 1, increase bynearly 70% between Expts 0 and 3. Variations in the intensity of the overturningmode occur in response to variations in the temperature and salinity in the deepocean. Averages of these quantities in the bottom kilometre of the basin over thesubtropical gyre are given in Table 2. The enhanced deep diffusivities of Expts 1-3augment the heat and salt content of the deep ocean. Increased upwelling velocitiesare required to balance the diapycnal diffusion of heat and salt through the pycn-ocline; as a result, the overturning circulation is enhanced. Alternatively viewed,the overturning mode must be amplified when the deep waters are warmed but thepoleward heat flux is unchanged (CHG).

Allowing the diffusivity to vary as in Expts 2-A introduces modifications to thecirculation and water properties that are similar to those arising from a stability-dependent diffusivity. In fact, all of the effects resulting from the N~l parametriza-tion of vertical diffusivity discussed by CHG appear to be retained in a qualitativesense in the present series of experiments. This confirms that the essential charac-teristic of specifying an N~l parametrization is to increase the diffusivity belowthe pycnocline.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014


3 Discussion

In this section we seek to provide an explanation of the process by which the deepwater stratification of the numerical model increases in response to an increase inthe diffusivity below the pycnocline. A simple vertical advection-diffusion modelis applied for this purpose. The starting point for this discussion is Eq. (7) inMcDougall (1987), which expresses the relation between the diapycnal mass fluxand the small-scale diapycnal mixing through a local isopycnal surface, viz.

4 = τ (Df)+NLT

dz dz V àz)where e is the diapycnal velocity, D is the diapycnal diffusivity and NLT representsall the terms arising from non-linearities in the equation of state, including thoseresponsible for cabbeling and thermobaricity. In (3), derivatives normal to localdiapycnal surfaces are approximated by vertical derivatives; the error involved inthis approximation is O(10~6), and hence is safely ignored (McDougall, 1987). Inthe following we will also neglect the terms represented in NLT, not because theyare particularly small, but because they are not essential to understand the effectson the stratification discussed in Section 2.

The diapycnal velocity, e, in (3) is related to the vertical velocity, w, accordingto

w-u-Vhe = j , (4)

(Bennett, 1986) where V is the horizontal gradient operator, VÄ = — Vp/pz givesthe components of the isopycnal slope, and u = («, v) is the horizontal velocityvector. In the presence of sloping isopycnal surfaces, the horizontal diffusivity aswell as the vertical diffusivity contribute to diapycnal diffusion. For gently slopingsurfaces (V/i · Vh <C 1), we can approximate D by

D fa Kv + KhVh • Vfc (5)

where Kh is the horizontal diffusivity of the model.In addition, the boundary conditions

p(-#o) = Po and p ( - # i ) = pi (6)

are assumed, where Ho is a depth below the base of the pycnocline (say 1000 m)and Hi is the depth of the bottom water (say 4500 m). The vertical advection-diffusion model (3) is to be applied to the thick intermediate region between thetwo major water masses in the subtropical gyre of the sector model, i.e. between apoint situated within the bottom water and a point near the base of the pycnocline.The boundary conditions represent the presence of these water masses. There isa near-surface water mass, formed by interaction with the surface forcing, andassumed to have a density of po. The bottom water mass is assumed to have a

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014


density of plm Typically, in sector models the bottom water mass is formed at thepoleward boundary, is forced to sinks to great depth through intense convectionalong the poleward wall, and is advected and diffused laterally so that it fills thebottom of the basin.

To obtain an expression for the vertical density gradient, (3) is rewritten as

and a non-dimensional density and vertical coordinate, ρ and z, are defined accord-

ing to

ρ = ρΔρ + po

(8)

where Δρ = (p! — p0) and AH = H\ - HQ. In addition, we let

_P(z)' AH

(9)

where P(z) is a non-dimensional function. This function gives the local Pécletnumber of the flow and it is a measure of the relative importance of verticaladvection to vertical diffusion at a point. Substituting (8) and (9) into (7) weobtain

l %with boundary conditions

p(l) = 0andp(0) = 1 (11)

Integrating (10) once we have

f f )

The constant of integration is obtained by applying the boundary conditions;

f 'cHi'expLfIt is possible to deduce the qualitative dependence of the stratification on the

Péclet number from an inspection of (10). Suppose that P(z) = P, where Ρ is apositive constant. In the diffusive limit, Ρ —> 0, (10) reduces to d2p/dz2 & 0 andthe (dimensional) vertical density gradient is uniform and equal to —Ap/AH. Inthe adiabatic or nearly-ideal fluid limit, Ρ is large and dp/dz m 0 over most of the

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014


ι-,

=

«-ι 3-

ïο

4 -

5-J

Δ ΕΧΡΤ1

Χ ΕΧΡΤ2

<!> EXPT3

0.0 0.5

D'(e •

I1.0

dD) (105 cm1)dz

1.5

Fig. 3 The variation with depth of the Péclet function. P(z)/AH = D~\e — dD/dz), computed fromEqs (4) and (5). Model data from a representative point near the centre of the domain (28.5°N,25.5°E) were used to construct the three curves. Refer to Section 3 for definitions of thesymbols.

depth range. In this limit, the density gradient is nearly zero everywhere except inthe neighborhood of ζ = 1, where density gradients are concentrated in a boundarylayer of thickness O(P~l). The latter is necessary to satisfy the boundary conditionat ζ = 1. For positive values of P, it is therefore in the small Péclet number,diffusive limit that density gradients at depth are greatest.

The Péclet function, P(z)/AH, was calculated from (4) and (5) over the depthrange 1300 to 4750 m at a representative point near the centre of the domain. Itis given for Expts 1-3 in Fig. 3, where the three curves illustrate the ordering ofthe experiments from the least diffusive case (Expt 1) to the most diffusive case(Expt 3). This also corresponds to the progression in Fig. 2 from less stratified tomore stratified deep water. The corresponding Ρ(ζ)/ΔΗ curves for Expts 0 and 4(the two are nearly identical) are omitted from Fig. 3 because they had a shapedifferent from those of Expts 1-3, making any comparison difficult.

Approximating the curves of Fig. 3 by the linear form P(z) = PI, we can obtainthe variation of the Brunt-Väisälä frequency with depth from (12) and (13). The

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014


1.0

Ζ 0.5 -

0.75 1.5

Ν (rad h 1 )

Fig. 4 The Brunt-Väisälä frequency, Ν, obtainedfrom Eqs. (12) and (13) with P(z) = Pi,for different values of the global Pécletnumber, P. To obtain dimensional re-sults, values of Δρ = 0.05 kg m~3 andAH = 3500 m were chosen. The value ofΔρ was obtained by integrating the av-erage vertical density gradient between1300- and 4750-m depths. Note the re-duced stratification at depth as Ρ is pro-gressively increased.

global Péclet number, P, is estimated to range from about 5.0 to 1.5 in Expts 1-3.The theoretical curves corresponding to this range of Ρ are given in Fig. 4 forrepresentative values of Δρ and AH. While Ρ is reduced, the vertical gradients areprogressively more uniformly distributed with depth, and the stratification of thewater mass at depth increases in a manner that is consistent with the numericalresults of Fig. 2. This comparison shows that the advection-diffusion model is ableto capture, in both qualitative and quantitive senses, the essential character of theprofiles of Fig. 2 for Expts 1-3. The variation in the stratification of the watercolumn occurs as a response to variations in an overall Péclet number, which takesinto account adjustments in both the diapycnal diffusivity and velocity. For Expts0 and 4, the simple form assumed for P(z) is inappropriate. Assuming a morecomplex, piecewise linear form for P(z) in (12) and (13), it is possible to reproducethe character of the stratification curves of these experiments. To preserve therelation of these curves to those of Expts 1-3 it is necessary to use an appropriatevalue of Δρ that is smaller than the value appropriate for Expts 1-3.

4 Conclusions

The concern addressed in this note is the representation of the climatological state ofthe deep water mass in certain ocean general circulation models. Frequently, thesemodels employ a spatially uniform vertical diffusivity chosen to yield a realisticpycnocline depth. This choice also tends to produce a deep water mass that isvirtually homogeneous. This is in contrast to the real ocean, which is considerablystratified even at great depth (Bryan and Sarmiento, 1985).

A possible reason for the discrepancy between the models and observations maybe the choice of a uniform diffusivity. Some observational estimates of verticaldiffusion in the ocean suggest that the diffusivity increases with depth below thepycnocline, possibly in inverse proportion to the local buoyancy frequency (Gar-

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014


gett, 1984). Several numerical experiments were conducted that demonstrate thatallowing the vertical diffusivity to increase below the pycnocline can lead to a sub-stantial increase in the stratification of the abyssal water mass without significantlyaffecting the pycnocline depth. This suggests that the misrepresentation of the deepwater in the models may, at least in part, be symptomatic of an underestimation ofdiffusion below the pycnocline. Additional sensitivity studies with a more realisti-cally configured model are warranted to examine this possibility further.

It is worth noting that increasing the diffusivity at depth is also advantageousfrom a numerical standpoint. The coarse, irregular vertical resolution typically usedto represent the deep ocean in OGCMs is prone to numerical difficulties, such asinstabilities due to grid Péclet number violations (Weaver and Sarachik, 1990),and artificial negative diffusivities (Yin and Fung, 1991). These problems can beavoided, without recourse to increased resolution, if the diffusivity is enhancedbelow the pycnocline, where the resolution is coarse.

The parametrization of vertical diffusion used first by Bryan and Lewis (1979)and subsequently in several ocean climate simulations has also been examined.In this case, the density structure is virtually unchanged from that of the con-stant diffusivity case. In addition, other properties of the solution, such as merid-ional overturning rates and heat transports, were similarly unaffected. With thisparametrization, the diffusivity increases at a depth that is too great to redistributevertical density gradients or to otherwise alter the solution.

References

BENNETT, s.L. 1986. The relationship between verti-cal, diapycnal and isopycnal velocity and mix-ing in the ocean general circulation. J. Phys.Oceanogr. 16: 167-174.

BRYAN, F. 1987. Parameter sensitivity of primitiveequation ocean general circulation models. J.Phys. Oceanogr. 17: 970-985.

BRYAN, K. 1984. Accelerating the convergence toequilibrium of ocean-climate models. J. Phys.Oceanogr. 14: 666-673.

_____ and M.D. cox. 1972. An approximate equa-tion of state for numerical models of the oceancirculation. J. Phys. Oceanogr. 2: 510-514.

_____ and L.J. LEWIS. 1979. A water mass modelof the world ocean. J. Geophys. Res. 84: 2503-2517.

_____ and J.L. SARMIENTO. 1985. Modeling oceancirculation. Adv. Geophys. 28A: 433-459.

COLIN DE VERDIÈRE, A. 1989. On the interaction of

wind and buoyancy driven gyres. J. Mar. Res.47: 595-633.

cox, M.D. 1984. A primitive equation, 3-dimensionalmodel of the ocean. GFDL Ocean Group Tech.Rep. No. 1, GFDL/Princeton University.

_____ and K. BRYAN. 1984. A numerical model of

the ventilated thermocline. J. Phys. Oceanogr.14: 674-687.

CUMMINS, P.F.; G. HOLLOWAY a n d A.E. GARGETT. 1990.

Sensitivity of the GFDL ocean general circula-tion model to a parametrization of vertical dif-fusion. J Phys. Oceanogr. 20: 817-830.

GARGETT, A.E. 1984, Vertical eddy diffusivity in theocean interior. J. Mar. Res. 42: 359-393.

_____ and G. HOLLOWAY. 1984. Dissipation anddiffusion by internal wave breaking. J. Mar.Res. 42: 15-27.

GERDES, R.; C. KÖBERLE a n d J. WILLEBRAND. 1 9 9 1 .

The influence of numerical advection schemeson the results of ocean general circulation mod-els. Climate Dyn. 5: 211-226.

LEVITUS, s. 1982. Climatological atlas of the worldocean NOAA. Prof. Pap. No. 13, [U.S. Gov.Printing Office] Washington, DC

MCDOUGALL, T.J. 1987. Thermobaricity, cabbeling,and water-mass conversion. J. Geophys. Res.92: 5448-5464.

PACANOWSKi, R.c. and S.G.H. PHILANDER. 1981. Para-metrization of vertical mixing in numericalmodels of the tropical ocean. J. Phys. Oceanogr.11: 1443-1451.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014


STOUFFER, R.J.; s. MANABE and κ. BRYAN. 1989. Inter-hemispheric asymmetry in climate response toa gradual increase of atmospheric CO2. Nature,342: 660-662.

TOGGWEILER, J.R.;K. DIXON a n d K. BRYAN. 1 9 8 9 .

Simulations of radiocarbon in a coarse-resolutionworld ocean model. 1. Steady state prebombdistributions. J. Geophys. Res. 94: 8217-8242.

WEAVER, A.J. and ES. SARACHIK. 1990. On the im-

portance of vertical resolution in certain oceangeneral circulation models. J. Phys. Oceanogr.20: 600-609.

YIN, F.L. and I.Y. FUNG. 1991. On the net diffu-sivity in ocean general circulation models withnon-uniform grids. J. Geophys. Res. 96: 10,773-10,776.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

uckl

and

Lib

rary

] at

13:

01 2

5 N

ovem

ber

2014

the deep water stratification of ocean general circulation models

Documents