Statistical Parameterization of Heterogeneous Oceanic Convection · Statistical Parameterization of...

Statistical Parameterization of Heterogeneous Oceanic Convection

CLAUDIA PASQUERO

Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California

ELI TZIPERMAN

Earth and Planetary Sciences and Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts

(Manuscript received 17 October 2005, in final form 24 January 2006)

ABSTRACT

A statistical convective adjustment scheme is proposed that attempts to account for the effects of me-soscale and submesoscale variability of temperature and salinity typically observed in the oceanic convec-tive regions. Temperature and salinity in each model grid box are defined in terms of their mean, variance,and mutual correlations. Subgrid-scale instabilities lead to partial mixing between different layers in thewater column. This allows for a smooth transition between the only two states (convection on and convec-tion off) allowed in standard convective adjustment schemes. The advantage of the statistical parameter-ization is that possible instabilities associated with the sharp transition between the two states, which areknown to occasionally affect the large-scale model solution, are eliminated. The procedure also predicts thegeneration of correlations between temperature and salinity and the presence of convectively inducedupgradient fluxes that have been obtained in numerical simulations of heterogeneous convection and thatcannot be represented by standard convective adjustment schemes.

1. Introduction

Ocean general circulation model (GCM) equationsare written for the gridbox-averaged quantities, and thesubgrid-scale variability of temperature, salinity, andvelocities is often parameterized in the form of eddyviscosity and diffusivity. Observations in regions ofdeep oceanic convection, such as the Labrador Sea,show that temperature T and salinity S fields havesmall- and mesoscale variability (Lilly et al. 2003),which is believed to play an important role in the con-vective process. Mesoscale cyclonic eddies are consid-ered key ingredients in the preconditioning process,acting to reduce the vertical stability in the water col-umn and set the location of the convective plumes (Di-Battista et al. 2002). Convective mixing is associatedwith vertical plumes within the preconditioned region,and with nearly isopycnal slantwise motion associatedwith the baroclinic instability of the convectively in-

duced geostrophic rim current around the convectingchimney (Jones and Marshall 1993; Legg and Marshall1993). The final state of a convective event is charac-terized by a rich horizontal structure of mixed andstratified water, with patches on scales of tens of kilo-meters and less [observed in polar convective regions,see Schott et al. (1994) and Lilly et al. (1999); studied innumerical models, see Jones and Marshall (1993), Leggand Marshall (1993), and DiBattista and Majda (2000)].

Convection is therefore both affected by small-scaleprocesses and also creates small-scale anomalies, whosescales are significantly smaller than a grid cell in a typi-cal GCM. Given that temperature and salinity varyhorizontally within the area of a given model grid cell,it is possible for the stratification to become unstable atsome locations within the area of a model grid cell, evenif the gridcell average is stably stratified. Yet, oceanconvection parameterizations (which are required, asthe resolution of models is usually too coarse to resolveconvection) do not normally take this into account ex-plicitly. The effect of vertical plumes results mainly inthe homogenization of the water column, probablywithout a vertical transport of mass (Send and Marshall1995). Current convection parameterizations in GCMsare based on the gridcell mean static stability of the

Corresponding author address: Claudia Pasquero, Departmentof Earth System Science, University of California, 3224 CroulHall, Irvine, CA 92697-3100.E-mail: [email protected]

214 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 37

DOI: 10.1175/JPO3008.1

© 2007 American Meteorological Society

JPO3008

vertical column, and remove static instability either in-stantaneously (Bryan 1969; Marotzke 1991; Rahmstorf1993) or on a finite adjustment time scale (Klinger et al.1996). The differences between mixing adjustmentschemes and enhanced vertical diffusivity schemes arenot significant (Klinger et al. 1996; Marshall and Schott1999), indicating that the instantaneous removal of in-stabilities is a reasonable assumption.

Alternatively, a number of vertical mixing param-eterizations based on turbulence closure theories havebeen introduced, with the aim of representing thedynamics of the surface boundary layer in presenceof both stable and unstable stratifications. Two com-monly used turbulent closures in ocean models are theMellor–Yamada 2.5-level scheme (MY2.5; Mellor andYamada 1982), which includes a prognostic equationfor the kinetic energy of the unresolved scales, andthe K-profile parameterization (KPP; Large et al.1994), which includes a nonlocal term in the expres-sion for the turbulent diffusivity in the presence of un-stable stratification (convective regime) and is meantto represent the effects of plumes or eddies thattraverse large vertical distances. KPP seems to be theonly parameterization used in ocean models that al-lows for upgradient fluxes. A recently introduced tur-bulence closure model (Canuto et al. 2001, 2002) wastested under deep convection conditions (Canuto et al.2004).

In this paper, we suggest a statistical convection pa-rameterization that accounts for possible unresolvedstatic instabilities on a subgrid scale. Specifically, ifthe gridbox-average density at a given location is �,then the actual density variations within the area rep-resented by the grid box is described by a probabilitydensity function (PDF). This allows us to calculate theprobability that there are subgrid locations that arestatically unstable even if the density profile in themodel, based on �, is stable. This probability, in turn, isused to derive the effects of the subgrid-scale con-vection on the resolved grid-averaged temperature, sa-linity, and density. This approach is similar in its phi-losophy (although not in its general outline nor imple-mentation) to that used for deriving fractional cloudcover from an appropriate PDF in state-of-the-art sta-tistical cloud parameterizations (e.g., Bony and Eman-uel 2001; Jakob and Miller 2004; Randall 1989; Tomp-kins 2002).

Standard convection schemes mix the water verti-cally only when the vertical stratification based on thegridcell averages becomes unstable. This is therefore adiscontinuous parameterization as a function of thegridcell-averaged temperature and salinity. The statis-tical parameterization is motivated by two different

needs, both arising from the discontinuous nature ofthe standard convection parameterizations. As ex-plained below, the statistical convection parameteriza-tion is smooth in the gridcell-averaged temperature andsalinity. It may therefore prevent the spatially and tem-porally intermittent behavior of convection in OGCMsthat is normally seen in GCM runs and seems to resultfrom the discontinuous nature of standard convectionschemes. The on–off switch of the convective adjust-ment process is known to create unphysical grid-scaleinstabilities (Cessi 1996), and to lead to temporal andgrid-scale spatial variability (e.g., Hirschi et al. 1999;Lenderink and Haarsma 1994; Marotzke 1991; Rahm-storf 1995) that are strongly sensitive to numerical er-rors, initial conditions, and spatial and temporal reso-lution (Cessi and Young 1996; Titz et al. 2004). Thesecond motivation arises from the difficulties caused bythe standard switch convection parameterizations tothe use of an adjoint of the ocean GCM (e.g., Tziper-man and Thacker 1989). Because switch convection pa-rameterizations are not differentiable, the adjoint maybe ill-defined in convection areas and this may preventits use for both sensitivity and data assimilation pur-poses. The nondifferentiability of the convection ap-plies to both convection schemes that are based on theuse of instantaneous mixing of unstable stratificationand those based on the use of implicit vertical diffusionsolution of the equations, with the vertical mixing co-efficients made large when the density profile is un-stable.

The development of the statistical convection schemeis only one of the two main objectives of this paper. Thesecond objective is to investigate the role of convectionin creating correlations between temperature and salin-ity anomalies. Density-compensated temperature andsalinity anomalies have been observed in the mixedlayer (e.g., Rudnick and Ferrari 1999), and their exis-tence has been attributed to the selective elimination ofuncompensated anomalies by processes that depend onthe buoyancy gradients (Chen and Young 1995; Ferrariand Young 1997). Recently, Legg and McWilliams(2000) observed the selective generation of compen-sated temperature and salinity anomalies in a numericalstudy of convection in the presence of a temperatureand salinity stratification.

Here, we examine how the generation of correlatedT and S anomalies is obtained by mixing in heteroge-neous conditions, and we find a significant appearanceof compensated anomalies. In some specific conditions,negative correlations, corresponding to temperatureand salinity anomalies that enhance each other to cre-ate maximal density anomalies, are obtained. We alsoinvestigate the mean effect of subgrid-scale convection

FEBRUARY 2007 P A S Q U E R O A N D T Z I P E R M A N 215

on the vertical fluxes of heat and salt, obtaining upgra-dient salinity fluxes in polarlike convection. Upgradientfluxes have been observed in numerical simulations ofheterogeneous convection (Legg and McWilliams2000), but they cannot be parameterized in term ofstandard convective adjustments, as perfect homogeni-zation (reduction of any gradient) is the end state of thestandard convective mixing parameterizations. The pa-rameterization proposed here can produce such upgra-dient fluxes.

In the following sections we introduce the statisticalrepresentation of subgrid-scale static instabilities (sec-tion 2) and show how to calculate the effects of thissubgrid-scale convection on the resolved gridbox-averaged temperature and salinity. We then discuss thetemporal evolution of the PDFs for the temperatureand salinity anomalies within a grid box after a con-vection event (section 3). Next, we demonstrate thescheme using a column model (section 4), performsome sensitivity tests (section 5), and conclude in sec-tion 6.

2. Statistical convective instability

Consider two vertically adjacent ocean model levels,with thicknesses zt for the top level and zb for the bot-tom one. The density in each layer is assumed homo-geneous in the vertical direction, but lateral anomaliesmay be present, and they are characterized in terms ofthe PDFs Pt(�) and Pb(�), whose mean values are thegridbox-averaged densities �t and �b. The stratificationis usually considered statically stable if �t � �b. How-ever, some portions of the grid box may be staticallyunstable if the density anomalies in the two layers aresuch that the local density in the top layer is larger thanthe local density in the bottom layer. A parcel in the toplayer with density �t has a probability Punstable(�t) ��t

�� d�b Pb(�b) of being denser than the water below,assuming that the anomalies in the two layers are un-correlated (this assumption is further discussed below).Overall, the probability of having the density in the toplayer larger than the density in the bottom layer is givenby the sum of Punstable(�t) over any density �t in the toplayer, weighted by the probability Pt(�t). This, by defi-nition, is the probability of convection in the two-layergridbox system:

Pconv � P��t � �b�

� ��

�

d�t Pt��t��

�t

d�b Pb��b�

� ��

�

d�t��

�

d�b H��t � �b�Pt��t�Pb��b�. �1�

Here, the Heaviside function depends on the differencein density between the two levels: H(�t � �b) is zero for�b �t and one otherwise, and therefore selects onlythe convectively unstable values of the top and bottomdensities in Eq. (1). Note that the probability Pconv canbe different from zero even when the mean density inthe upper layer is smaller than the mean density in thelower layer. We refer to the cases in which the convec-tion probability Pconv is different from zero as statisti-cally unstable.

a. Probability distribution functions of temperatureand salinity

Statistical stability depends on density stratificationonly, and not on potential temperature and salinitystratifications individually. However, ocean models useequations for salinity and potential temperature, andthe convective parameterization scheme thereforeneeds to be given in terms of salinity and temperaturedistributions. In the following, we shall use a linearizedequation of state to express density as a function oftemperature and salinity:

� � �01 � �� 0� � ��S � S0��. �2�

This choice allows us to find analytic expressions for theeffects of the statistical convective scheme. The linear-ization may be done about the local gridbox-averagedtemperature and salinity, so that the error introducedby not considering the full nonlinear equation of state isminimal.

The temperature and salinity distributions are de-scribed in terms of their mean values, and S, and theirstandard deviations, � and �S. Considering that sub-grid-scale anomalies are created by many diverse andindependent processes such as mesoscale oceanic vari-ability and wind bursts, we assume that temperatureand salinity distributions are described by Gaussianfunctions. We further allow for cross correlation be-tween temperature and salinity:

� �cov��, S�

��S�

�S � � S

��S. �3�

Positive correlations between temperature and salinityindicate that anomalously warm water is anomalouslysalty, and vice versa. The variance of density, �2 ��2

0(�2�2 � �2�2

S � 2�� S), is largest for given tem-perature and salinity distributions, when and S areperfectly anticorrelated (� � �1).

The distribution of temperature and salinity in eachlayer is therefore written as


P��, S; ��, �S, �, �, S� � P��, S� �1

2��S�1 � �2exp��

1

1 � �2 �� 2

2��2 �

�S � S�2

2�S2 � �

�� S � S�

��S��,

�4�

and subscripts t and b will be used to characterize thetop and bottom distributions. Two example of distribu-tions are shown in Fig. 1a.

The probability of convection [Eq. (1)] can now beexpressed in terms of double integrals over potentialtemperature and salinity for each layer, and the equa-tion of state, Eq. (2), is used to calculate the argumentof the Heaviside function:

Pconv � ��

��

�

d�t dSt Pt��t, St�

� ��

��

�

d�b dSb Pb��b, Sb�H��t � �b�.

�5�

b. Convective mixing

Given the above probability of convective mixing, wecan calculate the mean temperature and salinity overthe convecting regions within a grid cell as

�Xt� � Pconv�1 �

��

� ��

�

dSt d�t

� ��

� ��

�

dSb d�b H��t � �b�XtPt��t, St�Pb��b, Sb�,

�6�

where X is either or S and angular brackets indicatean average over the convecting fraction of the layersurface area (see notation in Table 1). The correspond-ing second-order moments are

�XtYt� � Pconv� 1 �

��

� ��

�

dSt d�t

� ��

� ��

�

dSb d�b H��t � �b�XtYtPt��t, St�

� Pb��b, Sb�, �7�

FIG. 1. An example of using the procedure described in section 2b. (a) Shown are contour plots of the two PDFs for two adjacentvertical levels before a convection event, as a function of the potential temperature and salinity in each level. Density contours areshown by the dotted lines. The mean stratification is stable, as seen by the fact that the top PDF (dashed lines) is centered about a lighterdensity than the bottom PDF (solid lines; mean values indicated by asterisks). However, convection occurs between the two levels ona subgrid scale. The probability of convection here is Pconv � 0.254. (b) The PDFs for the convecting portion of the top (dashed) andbottom (solid) layers. Mean values and second-order moments have been obtained from Eqs. (6) and (7). The PDF for the mixedconvecting region, as a result of the mixing of the top and bottom portions, is shown by the shaded area [first- and second-ordermoments of this distributions given by Eqs. (8) and (9)]. (c) The PDFs for the nonconvecting regions, defined by the mean values andvariances in Eqs. (10) and (11). (d) The PDFs for each layer after the first iteration of the convective scheme, defined by the first- andsecond-order moments of Eqs. (12) and (13). Note that the vertical stratification has been stabilized, by increasing (decreasing) thesalinity of the bottom (top) layer; convection in this case has slightly warmed (cooled) the bottom (top) layer. The probability forconvection has not been completely removed, but it has reduced to Pconv � 0.18, corresponding to about a 25% reduction in the staticinstability. (e) The PDFs after the completion of the entire convective scheme (repetition of steps 1–4 in section 2b until the probabilityof convection has been reduced below the threshold Pconv � 0.01). The average stability of the two-layer system has increased, and thedistributions of and S have been modified (smaller variances and positive correlations) so that the probability of convection becamevery small. Here, the initial conditions are t � 3°C, b � 2°C, St � Sb � 33.5 psu, � ,t � � ,b � 0.75°C, �S,t � �S,b � 0.1 psu (so that�� S), and �t � �b � 0. The final conditions are t � 3.1°C, b � 1.9°C, St � 33.45, Sb � 33.55 psu, � ,t � � ,b � 0.5°C, �S,t ��S,b � 0.05 psu, and �t � �b � 0.31.


where X and Y are either or S, allowing us to calculate� 2

t �, �S2t �, and � tSt� over the convecting region. The

mean and second-order moments of convecting waterfrom the bottom layer are calculated similarly.

We next assume that convecting water from the two

layers becomes vertically homogenized (only withinthe convecting areas of a grid cell). Salt content andpotential temperature are assumed to be conserved bythe convective mixing [ignoring the small effect due tothe fact that turbulent mixing actually conserves enthal-py rather than potential temperature; see Fofonoff(1962); McDougall (2003)]. Mixing between top waterwith temperature t and salinity St and bottom waterwith temperature b and salinity Sb creates homog-enized water with temperature (zt t � zb b)/(zt � zb)and salinity (ztSt � zbSb)/(zt � zb). The averaged prop-erties over the convective portion of the gridbox sur-face area after the mixing between the convecting frac-tions of top- and bottom-layer distributions are there-fore

�X� � Pconv�1 �

��

� ��

�

dSt d�t��

� ��

�

dSb d�b H��t � �b�ztXt � zbXb

zt � zbPt��t, St�Pb��b, Sb� and �8�

�XY� � Pconv�1 �

��

� ��

�

dSt d�t��

� ��

�

dSb d�b H��t � �b�ztXt � zbXb

zt � zb

ztYt � zbYb

zt � zbPt��t, St�Pb��b, Sb�. �9�

These first and second moments define the assumedGaussian PDFs of the temperature and salinity, andFig. 1b shows these PDFs for the water in the convect-ing portion of each grid cell as well as that of the mixedwater mass.

Similarly, the statistical properties of the water in thenonconvecting areas of each layer (indicated by thetilde) are

Xt�b� � �1 � Pconv��1��

� ��

�

dSt d�t��

� ��

�

dSb d�b H��b � �t�Xt�b�Pt��t, St�Pb��b, Sb� and �10�

XYt�b� � �1 � Pconv��1��

� ��

�

dSt d�t��

� ��

�

dSb d�b H��b � �t�Xt�b�Yt�b�Pt��t, St�Pb��b, Sb�. �11�

Note that the sign of the argument in the Heavisidefunction in Eqs. (10) and (11) has been reversed. ThePDFs for water in the nonconvecting region of eachlayer, obtained under the hypothesis of Gaussianityfrom the knowledge of the corresponding first- and sec-ond-order statistics, are shown in Fig. 1c.

Last, the new first and second moments after theconvection (denoted by primes) are calculated for bothlayers by averaging over the convecting and noncon-vecting regions in each layer:

Xt�b� � Pconv�X� � �1 � Pconv�Xt�b� and �12�

XYt�b� � Pconv�XY� � �1 � Pconv�XYt�b�. �13�

Given this weighted mean of the first and second mo-ments of the temperature and salinity after the convec-tion, we can calculate the new PDF by assuming againthat it is Gaussian. Figure 1d shows the PDFs afterpartial mixing of waters initially characterized by thePDFs of Fig. 1a.

At the end of the procedure described so far, thePDFs for the top and bottom layers may still overlap(Fig. 1d), indicating that the instability has not beencompletely removed. The procedure is then iterateduntil the probability of convection is reduced below agiven threshold (here, the threshold Pconv � 1% is used).The results after these iterations are shown in Fig. 1e.

In summary, the statistical convection parameteriza-tion is composed of the following steps:

TABLE 1. Notation for different averages used within a grid cell.

Notation Meaning Equation

() Gridcell average() Average over nonconvecting area

within grid cell10

�()� Average over convection areawithin grid cell

6

()� Value after one iteration ofconvective mixing

12


1) compute the probability of convection Pconv be-tween two nearby levels [Eq. (5)];

2) compute the gridbox mean, variance, and correla-tion of the temperature and salinity over the con-vecting and nonconvecting fractions in each layerbefore the convection [Eqs. (6), (7), (10), and (11)];

3) vertically homogenize temperature and salinity overthe convective regions [Eqs. (8) and (9)];

4) compute the modified mean, variance, and correla-tion for the temperature and salinity after the con-vection and over the entire gridbox area for eachlayer [Eqs. (12) and (13)]; and

5) repeat steps 1–4 until Pconv becomes smaller thanthe specified threshold.

The appendix provides the analytic solution to the in-tegrals for the mean and variance required for theabove steps.

The effect of convection occurring in a portion of thegrid area is in general to increase the mean density dif-ference between the two layers, and to reduce the den-sity variance in each layer. This occurs because convec-tion selects the largest (smallest) density values in thetop (bottom) layer, and replaces them with the mixedwater that has smaller (larger) density. The net effect isto reduce density anomalies within each layer. In termsof temperature and salinity, a reduction of the densityvariance can happen in two main ways: by a reductionof the variances of and S individually, and by a modi-fication of the correlations between and S. Even in thecase of no correlation between and S before the con-vective event, the convection introduces some correla-tion, as it selects in each layer those values of the andS anomalies that sum to give a particularly large (orsmall) density. The positive correlations created in eachlayer in the convecting regions are then modified by themixing process. Depending on the water characteristicsof the nearby level, the whole process might result inthe creation of either positive or negative correlations,as further discussed in section 4a below.

3. Evolution of temperature and salinitydistributions

The variance and correlation of temperature and sa-linity within a given grid cell are clearly importantquantities in the proposed scheme. Ocean GCMs donot normally predict the value of the second-order mo-ments of the temperature and salinity distributions ineach grid box. In the absence of a suitable closuretheory that relates the values of � and � to the first-order moments of the distributions and/or to other vari-ables of the model, we attempt in this section to relatethem to observations, and discuss their temporal evo-lution after the end of a convection event.

Local temperature and salinity anomalies in theoceans are created by air–sea interactions, by oceaneddies, and by other ocean instabilities. Typically, vari-ances are large at the surface and decrease with depth.Figure 2a shows typical potential temperature and sa-linity standard deviations (� ,ob and �S,ob) found in theNorth Atlantic Ocean region, at latitudes north of 50°Nand longitudes between 80°W and 20°E. This area in-cludes the Labrador Sea and the Greenland Sea, wheredeep convection is known to occur in wintertime. Theprofiles of the annual mean values of � ,ob and �S,ob

have been obtained by averaging the monthly root-mean-square (rms) values on 1° � 1° boxes [WorldOcean Atlas 2001, WOA01; Conkright et al. (2002)].

It is interesting to observe that the salinity anomaliesin this area are responsible for most of the densityanomalies, as ��S,ob is more than 10 times the �� ,ob atthe surface. The seasonal variations of the surface tem-perature and salinity standard deviations, for the samearea, are shown in Fig. 2b. The surface anomalies in-crease in the summertime and are minimal in the wintermonths. The temperature variance increases in thesummer months at the surface, but is constant below100 m from the surface (not shown). The salinity vari-ability is particularly large during the summer months(Fig. 2b), as sea ice melts and creates pockets of fresh-water.

The observed rms values shown in Fig. 2 representthe temporal variability of the temperature and salinitywithin a 1° � 1° square rather than the spatial variancerequired and used for the above parameterization.Given no other source of information about the spatialvariances, we assume that the spatial variance within a1° � 1° square has the same vertical structure, but asmaller amplitude, which will be specified below. Thereare several reasons to assume that these observed vari-ances are different from, and most likely larger than,the spatial variances � and �S required by the param-eterization. A large part of the horizontal anomalies

FIG. 2. High-latitude North Atlantic averaged vertical profilesof temperature and salinity standard deviations [from WOA011° � 1° grid monthly dataset; Conkright et al. (2002)]. (a) Annualmean values of rms temperature (solid line) and salinity (dashedline). (b) Seasonal variation of the surface temperature (solidline) and salinity (dashed line) standard deviation.


has coherent structure in the vertical direction. In otherwords, a region of particularly dense water at the sur-face is most likely located above a dense water column.We expect only a fraction of the total variance to ac-tively affect the probability of convection. Since theconvective parameterization presented here assumesthat anomalies in different levels before the convectionare not correlated, the values of the model restoringtemperature and salinity standard deviations, �*X, aresignificantly smaller than the observational values. Theactual fraction (� � �*X/�X,ob) is considered a free pa-rameter of the model and the sensitivity to � will beexamined in section 5. The value of � is constrained bythe requirement of obtaining a realistic frequency andintensity for the convective events. In the absence ofappropriate data, the reference temperature–salinitycorrelation, �ob, is assumed to be zero, but can clearlybe considered different from zero when the procedureis used to parameterize a specific observed convectiveevent.

After a convective event modifies the variance of thetemperature and salinity anomalies, we assume thatthese anomalies are regenerated by the different pro-cesses mentioned above and gradually recover their ini-tial values �X,ob, �ob. We assume that the processesresponsible for the gradual recovery of the horizontaltemperature and salinity anomalies are slower than theprocess of vertical convection and therefore happenover many model time steps. To represent this process,we use a gradual relaxation of the rms and correlationvalues to prespecified values (indicated by asterisks),which can depend on location, depth, and/or season:

d�X

dt� � �*X�s, t� � �X� and �14�

d�

dt� ��* � ��. �15�

Here, 1/� is the relaxation time scale and s is the posi-tion vector that defines the geographical location anddepth of the grid box under consideration.

The recovery of the horizontal distribution of anoma-lies about the grid-averaged temperature and salinityoccurs in all vertical levels and may therefore result inthe regeneration of convective instabilities.

4. Testing the parameterization using a columnmodel

To test the above parameterization in a frameworkthat more closely resembles an ocean GCM, yet issimple enough to allow extensive testing, we now use aone-dimensional convection–diffusion model of a ver-tical water column that obeys the following equations:

��

�t� ��* � �� k

�2�

�z2 , �16�

�S

�t� ��S* � S� � k

�2S

�z2 , �17�

��

�t� ��*

� � ��, �18�

��S

�t� ��*S � �S�, and �19�

��

�t� ��* � ��. �20�

The variables (z, t) and S(z, t) represent the gridboxmean potential temperature and salinity at a specifiedhorizontal location. The first terms on the right-handsides of Eqs. (16) and (17) are a relaxation to prescribedvertical profiles of temperature and salinity, meant torepresent the effects of horizontal advection and diffu-sion. The prescribed profiles are chosen as the monthlymean potential temperature and salinity averaged overa 1° � 1° box in the Labrador Sea at 51°W and 56°N.The monthly profiles are interpolated in time to thetemporal resolution desired for the integration of Eqs.(16) and (17). The restoring rms values are �*X � ��X,ob,with �X,ob being the monthly rms typical of the NorthAtlantic region, interpolated in time, and � � 0.05 (seethe discussion in the previous section). The relaxationcorrelation is chosen to vanish: �* � 0. At the surface,temperature is relaxed to an “atmospheric” mean tem-perature (see Fig. 3a), which has the same mean valueas the observed SST, but a seasonal variation 4 times aslarge. This creates a stronger surface temperature vari-ability that triggers strong convection during the wintermonths and allows testing the parameterizations inboth stable and unstable surface forcing conditions.

Convection is parameterized as described in section

FIG. 3. (a) Restoring temperature for the surface box. (b) Air–sea heat flux for the statistical convection parameterizationscheme, with � � 0.05 (solid line), and for the standard convectiveparameterization scheme (dashed line).


2: at every time step a check for statistical instability isperformed between each two adjacent levels, and theconvection scheme is applied if Pconv is larger than thethreshold Pconv � 0.01.

The model is run for 1 yr, with k� � 1 cm2 s�1 and� � 0.1 day�1, with 25 vertical levels from the surface to1500 m. Vertical resolution varies from 10 m at thesurface to 100 m at the bottom of the column. Thegridcell mean temperature, mean salinity, and meanstatic stability, �1/��/�z, in the upper 250 m are shownin Fig. 4. Thermal stratification is unstable in the wintermonths and stable in the summer months, while salinitystratification is almost always stable. This leads tostrong convection during the winter months and veryweak convection below the surface in the summermonths, when the mean stratification is stable (Fig. 4c)but the anomalies can lead to weak (small Pconv) con-vective instability (Fig. 5c). In comparison with the con-vection patterns associated with a standard convectionadjustment scheme shown in Fig. 5a (using the schemeof Rahmstorf 1993), the transitions between states ofconvection off and convection on (and vice versa) aremuch smoother. Note also that while the final stateafter the mixing associated with the standard convec-tion adjustment is inevitably marginally stable from thestatistical point of view, the gridbox-averaged stratifi-cation of the vertical column is statically stable after thestatistical convection parameterization is used (Fig. 4c).The gridbox-averaged stratification is the only part ofthe convection parameterization seen by the modelequations, and so these equations are not expected tosuffer any artifacts due to residual instabilities after theconvection event as parameterized by the statistical ap-proach.

Heat and salt fluxes

During strong convection in wintertime, both grid-cell mean salinity and potential temperature increase

with depth (Fig. 6a), having opposite effects on the den-sity stratification: salinity stratification is stable whiletemperature stratification is unstable. Salinity anoma-lies are very large, and they have a larger impact on thedensity variability than the temperature anomalies.Given that the salinity distribution in any given layer isscattered around the mean value, anomalously saltywater can be found overlying anomalously freshwater,even if the mean S profile is stable. When this happens,the consequent convective vertical mixing is associatedwith a downward salt flux (Fig. 6b), which acts to sta-bilize the water column at any depth, and is up themean salinity gradient. Note that the usual convectiveadjustment schemes cannot reproduce upgradientfluxes, as any gradient is removed by the mixing processthat completely homogenizes the water. It will be in-teresting to look for observational evidence of upgra-dient fluxes in open-ocean convection; such fluxes havebeen obtained in numerical simulations of heteroge-neous convection (Legg and McWilliams 2000).

Heat flux is directed upward in the top 300 m (Fig.6b), as surface cold water is mixed with warmer waterbelow by the mixing, while the heat flux is downwardfarther down the column, where the mean temperaturestratification is slightly unstable. So the heat flux is al-ways downgradient, leading to overall stabilization inthe upper layer and destabilization at large depth.Overall, the density flux is always directed downward(not shown), as denser water from the upper layers ismixed with lighter water from the lower layers, increas-ing the stability of the vertical stratification. It is note-worthy that these results are consistent with the de-tailed simulations of three-dimensional heterogeneousconvection by Legg and McWilliams (2000), who ob-tained downward salinity flux associated with baroclinicmixing for a monotonically increasing mean salinitywith depth (Fig. 10 in Legg and McWilliams 2000). Thisbehavior is thought to reflect the presence of a pre-

FIG. 4. Gridbox-average (a) temperature (°C), (b) salinity (psu), and (c) static stability (s�2) in the upper 250 m, as function of depthand time, from the integration of Eqs. (16)–(20). In (c) black regions indicate static instability, and the numerical labels on contour linesindicate the exponent (e.g., �6 indicates 10�6 s�2). Here, � � 0.05.


conditioned mesoscale eddy region, where salinity islarger than in the surrounding environment. Convec-tive mixing in this case carries salt downward from thepreconditioned area into the surrounding water.

In the context of the statistical representation ofocean convection, upgradient tracer flux can occurwhen the horizontal tracer variability is large relative tothe difference of the mean tracer within the verticalmixing scale, h; that is, |�X/�z|h � �X. Under such cir-cumstances, water in the unstable region of a givenlevel is replaced with mixed water whose characteristicsare closer to the mean value of that level. In most cases,this results in a reduction of the tracer variabilityaround the mean value of each layer. The oppositeholds true if the mean vertical gradient is large, |�X/�z|h �X, where vertical mixing results in the generation ofhorizontal tracer variability. In this situation, the tracerflux is downgradient. For these reasons, convectivemixing mostly increases the temperature variability(Fig. 7a) (apart from the upper layer in wintertime,when stratification is very weak and �* is large), andreduces the salinity variances throughout the year (Fig.

7b). Density anomalies, as anticipated earlier, are al-ways reduced by the convective mixing (Fig. 7c).

During convective mixing, new temperature salinitycorrelations are created. In general, in a layer with agiven average temperature and salinity, mixing of aportion of it with warmer and saltier (or colder andfresher) water creates positive correlations, while mix-ing with warmer and fresher (or colder and saltier) wa-ter creates negative correlations. Large positive corre-lations are created during the strong convection of thewinter months below the surface (Fig. 8), as the meanvertical gradients of temperature and salinity are bothnegative ( and S increase with depth), while negativecorrelations can appear when thermal and salinity ver-tical gradients have opposite signs (as during the sum-mer months below the mixed layer; see Fig. 8).

We show in Fig. 9 the formation of positive and nega-tive correlations, due to vertical convective mixing be-tween two adjacent vertical levels at 125-m depth, attwo different times. During March (Figs. 9a and 9b) theaverage properties in the two-layer system are margin-ally stable, with warmer and saltier water below colder

FIG. 5. Probability of convection in the upper 250 m, as a function of depth and time: (a) classic convectionadjustment, implemented as in Rahmstorf (1993); (b) �* � 0.01�ob; (c) �* � 0.03�ob; and (d) �* � 0.05�ob. Linesindicate the boundaries between nonconvective and convective regions. Minimum Pconv for convecting is 0.01.


and fresher water, and Pconv � 48%. The upper layer ismixed with warmer and saltier water, introducingstrong positive correlations. During September (Figs.9c and 9d), the column is relatively well stratified, re-sulting in Pconv � 8.5%, for � � 0.05. The upper layer isthen mixed with deeper water that is colder and saltierthan the averaged properties at the upper level, result-ing in the creation of negative correlations. Note thatthe density variability in each layer is reduced by theconvective mixing despite the creation of negative cor-relations. We believe that the creation of negative cor-relations is an artifact of the convective mixing scheme,which for large enough variances predicts partial con-vective mixing in the summertime below the mixed

layer. There is neither observational evidence nor theo-retical support for such a convective mixing, suggestingthat the variances of temperature and salinity that havebeen used are too large. We nevertheless show thiscase, with the aim of presenting the sensitivity of theparameterization to different parameter regimes.

5. Sensitivity study

In the previous section, the results of the applicationof the statistical convective parameterization to a col-umn model have been described. The aim of this sec-tion is to analyze how these results differ from the re-sults obtained using standard convective adjustment,based on the stability of the gridbox mean density (e.g.,

FIG. 7. Gridbox (�X � �*X) for (a) temperature (°C), (b) salinity (psu), and (c) density (kg m�3) in the upper 250 m, as function ofdepth and time, from the integration of (16)–(20). Shading indicates the absolute value, |�X � �*X | , with each tone of gray correspondingto a variation of 0.005 (in the respective units), starting from 0 (white). Negative regions, where the modeled variance is smaller thanthe restoring value, are marked with contour lines, while positive regions are only shaded. Convection mostly increases the temperaturevariances and reduces the salinity variances. Overall, the density anomalies are reduced by the convective mixing.

FIG. 6. February mean (a) potential temperature (solid line) and salinity (dashed line)vertical profiles, from the model output, and (b) heat (solid line) and salt (dashed line)convective fluxes. Positive values indicate downward fluxes. Heat flux is downgradient. Saltflux is upgradient. At any time step, fluxes of heat and salt are calculated from the convectivemixing: the heat flux associated with the partial mixing between two nearby levels is computedas �0cPconvzt(� � � � t�)/�t, where c is the specific heat of water and �t is the time step. Theflux is summed for all the iterations of the convective mixing computed at any time step.Similarly, the salt flux for each convective mixing iteration is Pconvzt(�S� � �St�)/�t.


Marotzke 1991; Rahmstorf 1993). Moreover, we alsowant to discuss the sensitivity of the results to the freeparameter used in the statistical convective parameter-ization (�; that is, the magnitude of the variances of and S), to the threshold value of Pconv below which nomixing is applied, and to the relaxation time ��1.

Consider first the effect of the convective scheme onthe air–sea heat fluxes, when restoring temperatureboundary conditions are used. The heat fluxes from thestandard convective adjustment and the statistical con-vection parameterization are very similar (Figs. 3b and3c). During summertime the surface layers are verystable, so that the air–sea heat fluxes are not affected bythe convective parameterization chosen (no convectiontakes place). In the winter months, the heat loss to theatmosphere is larger (by about 25%) in the case of thestandard convective adjustment scheme. This happensbecause the statistical convective scheme allows for apartial mixing (Pconv � 100%) even when the mean

state is statically stable. The mixing between the warmsubsurface water and the surface water is smaller thanin the standard convective parameterization scheme, assome portions of the grid box are not statically unstableand are not affected by the convective mixing. This is anencouraging result, as one of the weak points of stan-dard convective adjustment schemes is the sometimesexcessive convection that upwells too much heat (Kimand Stossel 2001).

The probability of convection is shown in Fig. 5 forthe upper 250 m, for different values of �. For � � 0 thePDFs of temperature and salinity in each layer are deltafunctions, centered around the gridbox mean value.The probability of convection is either 0 or 1. As �increases, the transition between states of convectionon and states of convection off becomes smoother.Eventually, for large enough �, partial mixing is foundat any time and depth, resulting in an unrealisticallystrong mixing.

A quantitative measure of the degree of convectionas a function of � is reported in Fig. 10a. For � � 0 onlyprofiles whose gridbox mean density profile is unstableconvect. The number of convecting cells sharply in-creases for larger �. The time and space average of theprobability of convection increases with increased andS variability. The effect of increasing the specified andS variance on the value of Pconv in convecting cellsdepends on the stability of the mean profile: wheneverthe mean profile is stable, a larger value of � allows forpartial overlapping of the density distributions in thevertically adjacent cells, increasing the probability ofconvection. This effect can be seen in Fig. 10b, wherebelow the first 100 m from the surface the annual meanconvection probability increases with �. On the otherhand, when the mean profile is unstable, a larger vari-ability of temperature and salinity reduces the probabil-ity of convection, as it becomes more likely to haveanomalously light water on top of anomalously densewater (see upper layers in Fig. 10b). For this reason, as� is increased, the mean probability of convection perconvecting cells can be nonmonotonic. In particular, inFig. 10 the mean value of Pconv over the convecting cells(i.e., 1 for � � 0) at first decreases, reaches a minimum,and then increases again as a large number of cellcouples with a stable mean profile increase their un-stable fraction.

As a first guess for the free parameter �, we suggesta criterion based on the sensitivity of the mean value ofPconv per convecting cell: we use a value of � in therange for which the average convection probabilityover convecting cells is relatively insensitive to the ex-act value of �, that is, around its minimum. The specificvalue of �, however, has to be refined by checking that

FIG. 8. Temperature–salinity correlation, �, created by the con-vective mixing. Length of arrow indicates density rms. Directionindicates the �S correlation: vertical means � � 0, horizontalrefers to perfect correlation, either positive (arrow pointing right)or negative (arrow pointing left). (a) For � � 0.01, convectioncreates positive correlations; (b), for � � 0.05, the larger variabil-ity predicts weak convective mixing below the mixed layer in thesummer months (see Fig. 5d), and negative correlations are herecreated.


the maximum convection depth and the occurrence ofconvective mixing are realistic. More work is needed toimprove upon or better justify this choice. In particular,a way of explicitly accounting for vertical correlationsin the temperature and salinity anomalies would greatlyimprove the statistical convection parameterization.

The dependence of the annual mean convecting frac-tion on the relaxation time ��1 and to the specificthreshold value chosen to apply convective mixing areshown in Figs. 10c and 10d. The procedure does notshow a strong sensitivity to these parameters. In gen-eral, a larger threshold value for the initiation of con-vection is associated with a slightly smaller annualmean convecting fraction, as sometimes convection isnot initiated. The dependence is however weak, as usu-ally this results only in a small delay in the initiation ofmixing.

The relaxation time, ��1, is important only for fastrestoring, in which case the annual mean probability ofconvection is large, as the water column tends to becharacterized by the restoring profiles, which are un-

stable during wintertime. For longer relaxation times,convective mixing becomes insensitive to the specificrestoring time.

6. Discussion and conclusions

A parameterization of oceanic convection in generalcirculation models with a typical grid size of hundredsof kilometers needs to account for the small horizontalscale of the ocean convection, as well as for the lateralinhomogeneities associated with the vertical convec-tion, such as those due to the baroclinic instabilities ofthe geostrophic rim current (see Marshall and Schott1999). The statistical scheme described in this paperrepresents a crude attempt along these lines as an al-ternative to standard ocean convection schemes.

The proposed scheme calculates the probability ofconvection in a given grid cell and mixes only the un-stable portion of the cell rather than the entire cell area.The scheme also calculates the effect of the convectionon the variance of small-scale temperature and salinity

FIG. 9. Temperature and salinity properties in two nearby levels at 125 m below the surface(a), (c) before and (b), (d) after convection. Solid lines indicate the properties in the lowerlayer; dashed lines indicate the properties in the upper layer. Top row is for 15 Mar; bottomrow is for 15 Sep. In the former case, convective mixing creates positive correlations; in thelatter case convective mixing creates negative correlations. Here, � � 0.05.


distributions within a grid cell. The net effect of thestatistical convective parameterization is, as expected,an increase of the static stability, via an increase in thedifference between the mean densities in the top andbottom layers, as well as via a reduced anomaly vari-ance in each layer, and the creation of –S correlations.

Convectively induced vertical fluxes of heat and saltand the creation of temperature and salinity correla-tions that are predicted by the statistical scheme seemconsistent with numerical simulations of heterogeneousconvection (Legg and McWilliams 2000). In particular,upgradient salt fluxes are reproduced by the statisticalconvective parameterization while they cannot be ob-tained by standard convective adjustment schemes,whose effect is always the homogenization of any prop-erty, eliminating any gradient. Positive correlations be-tween temperature and salinity (density-compensatedanomalies) arise directly from the mixing scheme.

Unfortunately, the parameterization proposed herestill requires the specification of the second-order sta-tistics of temperature and salinity. These must be speci-fied somehow artificially based upon profiles of the rmsvariability. The parameterization is, in fact, very sensi-tive to the choice of these profiles, as is evident in thesensitivity to the parameter � show in Fig. 10. Further-more, we assume a relaxation to these specified profilesafter a convection event, and this is not likely to be aphysically realistic description of the development ofthese moments. Another limitation is that temperatureand salinity are allowed to be correlated in the hori-zontal directions, but not in the vertical: we do notaccount for vertical coherence of the anomalies. Forthis reason, the actual values of the temperature andsalinity variances that must be used in the parameter-ization are a small fraction, �, of the observed values.The value of � is a tunable parameter, and the sensitiv-

FIG. 10. (a) Fraction of the convective gridbox area as a function of �. Thick solid line is the mean value of Pconv over the wholetemporal duration of the integration and over any depth in the column. Dotted line is the mean value of Pconv over converting cells.Dashed line is the fraction of cells that are convecting. Horizontal line shows the fraction of convecting cells for the standard convectiveparameterization. As � is decreased, the variance of the temperature and salinity distributions goes to zero. (b) Annual mean value ofthe fractional convecting area as a function of depth and �. (c) Annual mean value of the fractional convective area as a function ofdepth and restoring time, ��1. (d) Annual mean value of the fractional convective area as a function of depth and threshold value inPconv, below which no mixing is applied.


ity of the parameterization to its actual value has beeninvestigated. For large values of �, unrealistically largevertical mixing is introduced, and weak convection ap-pears below the pycnocline in summer months. As thishas no observational nor physical base, its appearanceprovides an upper value for the choice of the specifiedtemperature and salinity variances. Further work andmore physically sound criteria for the expressions ofsecond-order moments may significantly improve theparameterization. One possibility is to estimate the spa-tial variability of the static stability from the observa-tional data, so that vertical correlations would be takeninto account. Another possible improvement is tocouple of the stochastic parameterization to a turbu-lence closure model (Canuto et al. 2001, 2002). Last,one may envision combining the idea of a statisticalconvection parameterization with a physically basedparameterization such as KPP (Large et al. 1994) byallowing some of the key parameters in KPP to be rep-resented by a statistical PDF rather than by a singlevalue.

The proposed parameterization may eliminate someof the problems found in standard convective adjust-ment schemes, which are based on the mean strati-fication of the water column. Cessi (1996) showed thatconvective adjustment schemes lead to instabilities ofthe smallest resolved horizontal scale, because mixingoccurs in the vertical direction irrespective of the hori-zontal distributions of any given water property, withthe effect of an amplification of the horizontal spa-tial gradients. Such instabilities may be eliminated bya statistical convective adjustment, as vertical mixingmodifies gridbox mean properties more smoothly intime, and the lateral advection or diffusion has time toreduce the convectively formed spatial gradients. Itmay also be possible that the sensitivity to temporaland spatial model discretization (Cessi and Young1996; Titz et al. 2004) may be eliminated. More researchis needed to address these issues in detail and toexamine how the stability of large-scale circulation maybe affected by the statistical convective parameteriza-tion.

Acknowledgments. The trouble with Carl (accordingto author Tziperman): After my first year as a graduate

student, during which Carl was my assigned adviser, Icame into his office and announced that I would behappy to work with him but that I was not interested indata analysis or inverse methods. Carl looked amusedand let me know that was just fine. A few months later,Carl carefully edited a terribly written draft manuscriptI had given him, deleting every other word and rewrit-ing the rest; he then deleted his name from the authorlist, telling me that an adviser does not necessarily needto be a coauthor on his students’ papers. Some twoyears later, I was getting suspicious. With all these in-verters around, and with Carl still never suggesting thatI try working with inverse models, I became worried.Was I missing something? I ended up wanting to do aninverse model of something—anything. I proceeded,using Carl’s generous help, suggestions, and ideas, andCarl still did not wish to be a coauthor. I have beenworking with inverse models ever since; this clearlywould not have been the case had Carl ever suggestedthat I try this. It took me many years to appreciate justhow unique Carl’s ethics are. The trouble with Carl isthat he is an impossible role model to follow. He isdefinitely an inspiration, though. Thank you, Carl.

Maria Rita d’Orsogna is gratefully acknowledged forhaving suggested the trick reported in Eq. (26). Wethank two anonymous reviewers for their useful sug-gestions. Author Pasquero was supported by theDavidow’s Discovery Fund, and author Tzipermanwas supported by the James McDonnell Foundationand by the NSF’s Climate Dynamics Program (GrantATM-0351123).

APPENDIX

Solution of the Integrals from Section 2

This appendix is devoted to the analytic solution ofthe integrals described in section 2. A FORTRAN sub-routine that uses these integrals and applies the statis-tical convective parameterization to a column modelwas available online (http://www.ess.uci.edu/�cpasquer/research/stat_conv_param/stat_conv_param.F) at thetime of writing.

The probability of convection [Eq. (1)] is

pconv � ��

� ��

�

d�t dSt Pt��t, St��

� ��

�

d�b dSb H��t � �b�Pb��b, Sb� �12�1 � erf� �t � �b

�2��2b � �2

t �� .

�A1�


The average temperature and salinity over the con-vecting regions are given by

�Xi� � Xi � Pconv�1

�X,i�0��X,i� ��,i � �X,i

S ��S,i�

�2��2t � �2

b�e��2

,

�A2�

where X is either temperature or salinity S, the sub-script i stands for either the top or bottom layers, and�

X,i � �i when X � S and is 1 otherwise; similarly,

�SX,i � �i when X � and is 1 otherwise; and ��2 �

(�t � �b)2/2(�2t � �2

b).Similarly, mean properties in the nonconvecting ar-

eas are

Xi � Xi �1

1 � Pconv

�X,i�0��X,i� ��,i � �X,i

S ��S,i�

�2��2t � �2

b�e��2

.

�A3�

Here, ct � �1 and cb � �1.The second-order moments are

�Xi2� � Xi

2� �X,i

2 � ci

1Pconv

�X,i�0��X,i� ��,i � �X,i

S ��S,i�

�2��2t � �2

b��2Xi �

��t � �b��X,i�0��X,i� ��,i � �X,i

S ��S,i�

�2t � �2

b�e��2

,

Xi2˜ � Xi

2� �X,i

2 � ci

11 � Pconv

�X,i�0��X,i� ��,i � �X,i

S ��S,i�

�2��2t � �2

b��2Xi �

��t � �b��X,i�0��X,i� ��,i � �X,i

S ��S,i�

�2t � �2

b�e��2

,

�XiYj� � �Xi�Yj � Xi�Yj� � XiYj � �i�X,i�X, j�ij � cicj

1Pconv

��t � �b��X,i�Y, j

��2t � �2

b��2��2t � �2

b�

� �02��X,i

� ��,t � �X,iS ��S,t��Y, j

� ��,t � �Y, jS ��S,t�e

��2, and

�iSi � �iSi � �iSi � �i��,i�S,i �1

1 � Pconv

��t � �b��,i�S,i

��2t � �2

b��2��2t � �2

b��0

2��i��,i � ��S,i��,i � �i��S,i�e��2

.

�A4�

From the knowledge of those integrals, variances andcovariances are easily calculated.

The solution of the integrals given above is tediousbut fairly simple if the following integrals are known:

f1�a, b� � ��

�

exp��x2� erf�a � bx� dx � � erf� a

�b2 � 1�,

f2�a, b� � ��

�

x exp��x2� erf�a � bx� dx �b

�b2 � 1exp�a2��b2 � 1��, and

f3�a, b� � ��

�

exp��x2�x2 erf�a � bx� dx ��

2erf� a

�b2 � 1� �

ab2

�b2 � 1�3�2 exp�a2��b2 � 1��. �A5�

To solve these integrals [which are not given in Grad-shteyn and Ryzhik (2000)] consider, for example, the

derivative of f1 derivative with respect to a, which con-verts it into a simple double Gaussian whose solution is


�f1

�a� �

��

�

exp��x2� dx�

�aerf�a � bx�

�2

�b2 � 1exp�a2��b2 � 1��. �A6�

Integrating now over a and using the boundary condi-tion f1(0, b) � 0, as when a � 0, the integral in Eq. (A5)is an integral of an odd function over a symmetric in-terval, and we find the above solution.

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