Parameterization of Mixed Layer Eddies. I: Theory and...

8 October 2007 FOX-KEMPER, FERRARI, and HALLBERG 1

Parameterization of Mixed Layer Eddies. I: Theory and Diagnosis

Baylor Fox-Kemper

Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge,Massachusetts, 01239∗

Raffaele Ferrari

Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge,Massachusetts, 01239

Robert Hallberg

NOAA Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey

(8 October 2007)

ABSTRACT

Ageostrophic baroclinic instabilities develop within the surface mixed layer of the ocean at horizontal frontsand efficiently restratify the upper ocean. In this paper a parameterization for the restratification driven by finite-amplitude baroclinic instabilities of the mixed layer is proposed in terms of an overturning streamfunction that tiltsisopycnals from the vertical to the horizontal. The streamfunction is proportional to the product of the horizontaldensity gradient, the mixed layer depth squared, and the inertial period. Hence restratification proceeds faster atstrong fronts in deep mixed layers with a weak latitude dependence. In this paper the parameterization is theoreticallymotivated, confirmed to perform well for a wide range of mixed layer depths, rotation rates, and vertical andhorizontal stratifications. It is shown to be superior to alternative extant parameterizations of baroclinic instabilityfor the problem of mixed layer restratification. Two companion papers discuss the numerical implementation andthe climate impacts of this parameterization.

1. Introduction

A typical oceanic stratification and shear allowstwo types of baroclinic instability (Boccaletti et al.,2007, hereafter BFF): deep mesoscale instabilitiesspanning the entire depth and shallow submesoscaleinstabilities trapped in the weakly-stratified surfacemixed layer (ML). The troposphere and its surfaceboundary layer provide two analogous types of insta-bility (Blumen, 1979; Nakamura, 1988). The shallowML instabilities are ageostrophic baroclinic instabil-ities (Stone, 1966, 1970, 1972a; Molemaker et al.,2005) and differ from the deep mesoscale instabili-ties in their fast growth rates of O(1/day) and smallscales of O(1 km). BFF suggest that ML instabili-ties play an important role in restratifying the upperocean after strong mixing events.

Presently ocean models use a variety of bound-ary layer parameterizations to represent the pro-

∗Present address: Cooperative Institute for Research inthe Environmental Sciences (CIRES) and Department of At-mospheric and Oceanic Sciences, University of Colorado atBoulder, Boulder, Colorado 80309

cesses that mix away stratification in response tosurface forcing (e.g., Kraus and Turner, 1967; Price,1981; Price et al., 1986; Large et al., 1994; Thomas,2005), while ML restratification occurs only by sur-face heating. Dynamical restratification by slump-ing of horizontal density gradients within the ML istypically ignored. As a consequence, ocean modelshave a bias towards weak near-surface stratification(e.g., Oschlies, 2002; Hallberg, 2003; Chanut et al.,2005). Large-scale ocean models are beginning toresolve deep mesoscale eddies with O(10km) grids,but resolving restratification by submesoscale insta-bilities requires O(100m) grids. Submesoscale in-stabilities are subgridscale even in ”eddy-resolving”models.

In this paper scalings are developed for restrati-fication by finite-amplitude ML instabilities, hereinreferred to as mixed layer eddies (MLEs). Thesescalings are tested in idealized simulations and for-mulated into a parameterization. Two companionpapers provide more insight into the workings of theparameterization. Fox-Kemper and Ferrari (2007,

2 submitted: JOURNAL OF PHYSICAL OCEANOGRAPHY 8 October 2007

hereafter FF) compare the parameterization withsubmesoscale-resolving simulations and estimate theimportance of MLE restratification from data. Thenumerical implementation of the parameterizationand its effects in realistic global simulations are thesubject of a third paper (Fox-Kemper et al., 2007).

Tandon and Garrett (1994) first proposed thatdynamical restratification occurs at lateral densityfronts in the ML after strong mixing events. How-ever they considered only restratification by Rossbyadjustment of lateral fronts (Rossby, 1937, 1938; Ou,1984; Tandon and Garrett, 1995). Young (1994) andHallberg (2003) derive parameterizations for the re-stratification by Rossby adjustment. However, BFFshow that most dynamical restratification occurs af-ter the initial Rossby adjustment, when ML insta-bilities reach finite amplitude and start releasing thepotential energy (PE) stored in the front.

The parameterization developed here representsthe restratification by ML instabilities. FollowingGent and McWilliams (1990) (henceforth GM), therestratification is cast in terms of an eddy-inducedstreamfunction that adiabatically overturns isopy-cnals from the vertical to the horizontal. Scalingsare derived directly for the overturning streamfunc-tion, in contrast to the traditional approach of usingmixing length arguments to relate fluxes to meangradients via an effective diffusivity. The proposedscaling depends only on finite-amplitude propertiesof MLEs that are confirmed by simulations.

There are many notable studies of baroclinic in-stabilities in the ML. References for the linear anal-ysis are given in BFF. This work is closer in spirit toprevious studies at finite amplitude (Samelson andChapman, 1995; Spall, 1997; Jones and Marshall,1993, 1997; Haine and Marshall, 1998). However,the focus here is a parameterization of ML restrati-fication which does not appear elsewhere. It will beshown that MLEs restratify importantly through anupward buoyancy flux; capturing MLE horizontalfluxes is less important. The strength of the ver-tical fluxes is predicted by the parameterization asa function of the lateral frontal buoyancy gradientand the ML depth. The parameterization applies torestratification by the submesoscale eddies observedthroughout the extratropics (Weller, 1991; Rudnickand Ferrari, 1999; Hosegood et al., 2006). The pa-rameterization also recovers the scaling laws foundby Jones and Marshall (1997) and Haine and Mar-shall (1998) for eddy transport and restratificationduring deep convection at high latitudes.

The paper is structured as follows. Section 2 givesa relevant phenomenology of MLEs through study

of two idealized numerical simulations. Section 3presents the theory behind the parameterization.Section 4 validates the parameterization by diag-nosis of the simulations. Concluding remarks and areview of observational evidence of ML restratifica-tion are presented in section 5.

2. Phenomenology of MLEs

Two numerical simulations are used to gain a senseof the phenomenology of MLEs. The first contrastsand connects mesoscale eddies and submesoscaleMLEs. The second focuses on MLE restratificationat a single front.

a. Simulation with both Mesoscale and Subme-soscale Eddies

The first simulation is configured to produce deepmesoscale eddies extending through the whole wa-ter column and shallow submesoscale eddies trappedin the surface ML. The MITgcm model (Marshallet al., 1997) is configured to simulate a reentrantchannel where a baroclinically unstable jet is main-tained by restoring temperature profiles along theside walls (Fig. 1). The upper 75 m are initially un-stratified, and are subsequently mixed by a diurnalcycle of 200W/m2 cooling compensated by penetrat-ing heating during the day. Nightly cooling thor-oughly mixes the ML to roughly 50m depth. Thesimulation is run at 8km resolution for 900 days,interpolated to 2km resolution, and continued for100 days. At this resolution, the largest MLEs1 arepermitted but only marginally resolved in order topermit mesoscale features as well. Below, dedicatedsimulations of MLEs alone allow better resolution ofsubmesoscale features. Details are in Appendix A.

A vigorous mesoscale eddy field develops through-out the full water column (Fig. 1b), while variabilityin the ML is dominated by small-scale meanderingfronts (Fig. 1a). The tightly packed isotherms resultfrom straining by the mesoscale eddies and frontoge-netic processes compacting outcropping isotherms.The meanders that develop along the fronts areMLEs. The large mesoscale eddies result from baro-clinic instability of the mean jet with growth ratesof O(1 month) and length scales near O(80 km).The smaller MLEs result from ageostrophic baro-clinic instabilities that develop along fronts withinthe ML. Their scales begin near the linear insta-bility scale based on ML depth and stratification,

1MLEs vary in size according to the strength of the frontupon which they grow, see (2).


Figure 1: Contours of temperature at the a) surface and b) below the ML base in a simulation with both mesoscaleeddies and MLEs (0.2C contour intervals). Shading indicates w′b′ (upper panel) and |u′Hb′| (lower panel) at 20mdepth, the depth at which eddy fluxes are largest.

O(1 to 5 km), and enlarge as a result of an in-verse cascade as discussed in BFF. MLEs clusteralong fronts, where frontal vertical shear endows thefastest growth (Stone, 1966). Elsewhere the growthrates are too slow to compete with the damping dueto turbulent mixing.

BFF argue that the ML instability linear growthrates are only weakly affected by large scale strain-ing, yet Spall (1997) shows that a large scale straincan substantially alter finite-amplitude baroclinicinstability. This effect is notable in Fig. 1a near(450, 80) km, where a powerful surface tempera-ture front is pinched between three mesoscale eddies(Fig. 1b). MLEs develop only after the front exitsthe strain field near (400, 140) km. The strain ratesin this idealized simulation are larger than is typicalin the real ocean, yet MLEs are present throughoutthe domain. Thus, while mesoscale straining can oc-casionally suppress MLEs, the effect is confined tothe regions of largest convergence.

Basin-wide restratification can occur only by anet upward transport of buoyancy. That is, on thewhole, the near-surface ML is made more buoy-ant and the deeper ML becomes denser. Thevertical eddy buoyancy flux, w′b′, is shaded in

Fig. 1a. (Primes denote departures from along-channel, x−direction averages. See Table 1). Thefigure shows fluxes near the depth where they arelargest (20m). Two features emerge. First thelargest vertical fluxes are small-scale features clus-tered near fronts. In fact, filtering w′ and b′ indi-cates that 70%(50%) of the basin average, w′b′

xy,

is generated by scales smaller than 12km (8km).(Overbars denote along-channel averages, and su-perscripts indicate additional averaging along othercoordinates, See table 1). Second, w′b′

xyis posi-

tive rather than negative, implying a tendency torestratify the ML. The shading in Fig. 1b indicatesregions where |u′Hb′| is largest. The horizontal fluxesare coherent on scales associated with mesoscale ed-dies, while the vertical fluxes are distinctly subme-soscale. The mesoscale eddies and MLEs have com-parable horizontal velocities, but mesoscale eddiesstir over longer distances and dominate the horizon-tal fluxes. MLEs and fronts have larger Rossby num-ber and thus larger vertical velocities, so they dom-inate the vertical fluxes. In sum, mesoscale eddiesdominate the lateral fluxes while fronts and MLEsdominate vertical fluxes and restratification.

The role of MLEs in restratification is clarified by


comparing the simulation described above with anotherwise identical simulation run without a diurnalcycle from day 900 to day 1000. When the resolutionis increased at day 900, near-surface restratificationincreases as a result of sharper fronts from mesoscalestraining (Spall, 1997; Nurser and Zhang, 2000; Os-chlies, 2002; Lapeyre et al., 2006). But, do MLEsand their associated fronts lead to even more restrat-ification as suggested by BFF? With and without acontinued diurnal cycle, the mesoscale eddies differlittle and fronts of a similar strength develop at thesurface–the averaged surface |∇Hb|2 differs by lessthan 25%. But without a diurnal cycle the ML dis-appears through unchallenged restratification, andsoon MLEs are stabilized. The average flux, w′b′

xy,

in the upper 40m is half (a third) that of the simu-lation with a continued diurnal cycle in ten (forty)days.

Some recent investigations without active remix-ing of the ML underestimate the restratification ef-fect of MLEs (e.g., Mahadevan, 2006; Mahadevanand Tandon, 2006), as MLEs do not occur after theML has already restratified by other mechanisms.Other studies have found few or no MLEs becausethe ML in the regions considered were very shallow,so very high horizontal resolution would be requiredto resolve the instabilities (e.g., Capet et al., 2006;Lapeyre et al., 2006). Of course, these simulationsare not without near-surface restratification: fron-togenesis, winds, and solar heating may all restrat-ify. Ferrari and Thomas (2007) compare the relativecontributions from these mechanisms quantitativelyand conclude that MLEs play an important role innear surface restratification. MLE restratification isthe subject of this paper.

b. Restratification by Spindown of a ML Front

Frontal instabilities–MLEs–develop once the subme-soscales are permitted (Fig. 1). However the compu-tational expense of refining grid resolution to con-vergence for MLEs while retaining properly sizedmesoscale eddies is onerous. Hence, the MLE re-stratification study continues by focusing on thespindown of a single ML front representing the af-termath of a mesoscale straining event as in Fig. 1or the edge of a recent vertical mixing event (Price,1981; Haine and Marshall, 1998). The front is ini-tialized as a horizontal density gradient in a flat-bottom reentrant channel. Vertical stratification isuniform in the interior and weak in a surface ML. Atypical model configuration is shown in Fig. 2a anddetailed in Appendix B. The initial velocity may be

either resting (hereafter ”unbalanced”) or in thermalwind balance (”balanced”). Many other parametersvary across the simulations, and resolution is var-ied to ensure the baroclinic instability is resolved;tripling the resolution does not significantly changethe results.

An unbalanced ML front first slumps gravitation-ally and oscillates inertially about the Rossby ad-justed state (Tandon and Garrett, 1995). Soon af-terward, ML instabilities are detectable. Withina few days they are evident as wavelike distur-bances along the oscillating front (Fig. 2a) that en-large in time (Figs. 2b-c). Five days are requiredto reach finite amplitude because the initial per-turbations away from uniform flow in the along-channel direction are artificially small. In test simu-lations, and presumably also the ocean, larger initialperturbations develop into finite amplitude MLEswithin one day. Balanced simulations do not un-dergo Rossby adjustment, but the development andnonlinear growth of ML instabilities is very simi-lar. In all cases the initial PE is the primary energysource, and the MLEs grow by slumping the frontto extract this energy (BFF).

MLE restratification increases the balancedRichardson number,

Ri ≡ N2

∣∣∣∣∂ug

∂z

∣∣∣∣−2

=N2f2

M4,

where N2 and M2 are volume-average values of bz

and by over the frontal region in the ML. The bal-anced Ri differs from the traditional Ri in that geo-strophic shear replaces the full shear.

Fig. 3 shows the increase in Ri in four simulations.Two of the simulations have no initial velocity. Theother two begin in thermal wind balance. In eachpair, N2 at t = 0 is set to either M4/f2 or 0.2

The unbalanced simulations oscillate inertially nearRi ≈ 1 for about 5 days, while the balanced simula-tion with N2 = 0 develops symmetric instabilities ina few hours that then increase Ri to one. This earlyrestratification is overwhelmed once MLEs are ac-tive (after day 5), and the MLE restratification rate(∂N2

∂t ) is the same once finite amplitude is reached.Only the time to reach finite amplitude differs: thelarger Ri simulations reach finite amplitude later (asthe linear growth rate (3) below predicts).3 In sum-

2The initial Richardson number is Ri0 = 1 or Ri0 = 0,respectively. See Table 1 for notation.

3Symmetric instabilities are strengthened in a tripled res-olution version of the balanced, Ri0 = 0 case. At higherresolution, Ri = 1 is reached a day earlier, but the averagerestratification rate after Ri > 2 differs by less than 3%.


a) d)

b) e)

c) f)

Figure 2: Temperature (oC) during two typical simulations of a ML front spinning down: a-c) no diurnal cycle, d-f)with diurnal cycle and convective adjustment. (Black contour interval=0.01oC, white contour interval=0.1oC.)

mary, there are a variety of instabilities that rapidlybring the front to Ri ≈ 1, but the subsequent MLErestratification is insensitive to the details of these

processes.

Ageostrophic baroclinic instabilities, which ex-tract PE by slumping isopycnals, are the dominant


0 2 4 6 8 10 12 14 160

10

20

30

40

50

time (days)

Ri

unbalanced, Ri0=0unbalanced, Ri0=1balanced, Ri0=0balanced, Ri0=1

Figure 3: Balanced Richardson number for four simu-lations starting from a thermal wind ”balanced” initialcondition or resting initial velocity, i.e.,”unbalanced”.All parameters are identical across simulations and atfront center, but initial N2 may be M4

f /f2 (labeled hereas Ri0 = 1) or 0 (Ri0 = 0). N2 is bz averaged over thecenter of the front (|y − y0| < Lf/4), and M4

f is thelargest value of b2

y in the initial condition (Appendix B).

form of ML instabilities at Ri ≥ 1 (BFF, Haineand Marshall, 1998). Their main characteristics arecaptured by Stone (1970) in his analysis of the Eady(1949) problem. The linear growth rate is

τs(k) =kU

2√

3

[1− 2k2U2

15f2(1 + Ri)

], (1)

and the fastest growing mode has

Ls =2π

ks=

2πU

|f |

√1 + Ri5/2

, (2)

τs(ks) =

√545

√1 + Ri|f |

. (3)

For the simulations shown in Fig. 2, Ls = 3.9 kmand τs(ks) = 16.8hr for Ri = 1. MLEs appear nearthese scales in both the frontal spindown simulations(Fig. 2) and the mesoscale plus submesoscale simu-lation (Fig. 1a)–these values are much smaller andfaster than those of mesoscale eddies.

The scales from linear theory are helpful in deter-mining the numerical resolution necessary to permitMLEs, but they are not useful for parameterizingthe frontal slumping effect of MLEs. Fig. 4 com-pares the power density spectrum of eddy kineticenergy (EKE) at various times from a nonlinear sim-ulation and the linear theory prediction. Linear the-ory tracks the nonlinear spectrum only for the first

10!2

10!1

100

10!6

10!4

10!2

100

102

PS

D o

f horiz. pert

urb

ation v

elo

city (

m3/s

)

Wavelength (cycle/km)

Figure 4: Perturbation power spectral density, E(κ)for a simulation from Fig. 3 (solid). Spectra are plot-ted at two day intervals from day 1.5 to day 29.5. Thelinear prediction of the spectrum (Es(κ), dashed) is setequal to the nonlinear spectrum on day 1.5, and thenevolved at each along-channel wavenumber as predictedby linear theory taking into account the changes in Riand U . That is, Es(κ) is evolved using τs(k) from (1)based on the instantaneous Ri and U from the nonlinearsimulation: Es(κ) = e2t/τs(k)

RE|t=1.5(k, l)dl. The de-

crease in growth rate with cross-channel wavenumber, l,is ignored for simplicity and because low l modes soondominate.

six days. During this period the spectral peak tracksthe most unstable wavenumber predicted by (2) andshifts to larger scales because Ri grows as the strat-ification increases (Fig. 3). However the nonlinearspectrum departs the linear prediction as the insta-bilities reach finite amplitude. EKE is transferred toscales larger than the most unstable mode througha vigorous inverse cascade (Fig. 4).

The inverse cascade complicates the parameteri-zation problem. Eddy mixing length arguments areroutinely used to study baroclinic eddy fluxes (Haineand Marshall, 1998; Spall, 2000; Larichev and Held,1995; Schneider and Walker, 2006). In these argu-ments the lateral transport of tracers is dominatedby the largest energy-containing eddies (e.g., How-ells, 1960). The eddy saturation strength follows asimple scaling: the eddy velocity within the frontsaturates at the initial mean flow velocity as shownin Fig. 5 (Stone, 1972b). The mixing length, how-ever, is not fixed in time in spindown problems suchas this one, nor is it readily estimated from otherhorizontal scales: the most energetic eddies enlargebeyond the most unstable scale (e.g., Cehelsky andTung, 1991) and beyond the initial frontal width


0 5 10 15 20 25 30

10!8

10!6

10!4

10!2

time (days)

kin

etic e

ne

rgy (

m2/s

2)

basin!avg. pert. KE

linear predict. pert. KE.

initial mean KE2: 1/2(M

2 H/f)

2

avg. pert. v2 in front

Figure 5: Kinetic energies and cross-channel pertur-bation velocity variance as a function of time from thesame simulation as Fig. 3 (solid) and Fig. 4. The slightincrease in the basin-average EKE after day 15 is sim-ply a result of the enlarging eddy scale widening thearea of eddy activity into previously motionless fluid (seeFig. 4). That is, the basin-average of perturbation KEcontinues to grow (dashed line) while the average overonly the center of front saturates (solid line).

(Fig. 2c).Another popular approach for parameterizing

baroclinic spindown relies on linear stability the-ory of the basic state (e.g., Stone, 1972b; Killworth,2005). The core assumptions are that eddies andmean state satisfy the same scaling and that finite-amplitude eddies resemble the fastest-growing lin-ear instability. In the MLE problem, not only arelonger lengthscales energized by the inverse cascade,but frontogenesis leads to smaller lengthscales aswell. The mean state is well described by quasi-geostrophic (QG) scaling–perhaps modified to al-low variable background stratification (e.g., Naka-mura and Held, 1989), but the MLE Richardson andRossby numbers approach one as a result of fronto-genesis at the boundaries. This spontaneous loss ofbalance is a distinguishing feature of fronts that out-crop at the ocean surface (Molemaker et al., 2005).

Nakamura and Held (1989) and Nakamura (1994)argue that the nonlinear, frontogenetic developmentof MLEs can be captured by stability analysis in geo-strophic coordinates (Hoskins, 1976). This approachcorrectly predicts frontal development of Richard-son and Rossby numbers of O(1). However, thisapproach also predicts that the ageostrophic sheargenerated through frontogenesis grows as large asthe geostrophic shear and arrests further restratifi-

0 5 10 150

0.2

0.4

0.6

0.8

1

time (days)

!/H

Figure 6: Typical vertical excursion scale, ζ ≡pb′2/N2, scaled by ML depth, H, for initially bal-

anced simulations where the initial ML depth was 200m(solid), balanced simulations where the ML depth was50m (dashed), and unbalanced simulations where theinitial ML depth was 200m (dotted). Other parametersvary as well: Lf/Ls varies by a factor of four and initialN is 0 or 4f . The value of ζ/H shown is the maximumin z, horizontally-averaged over the front center.

cation, as verified in 2d simulations by Nakamura(1994). In three dimensions restratification contin-ues despite the appearance of fronts (Fig. 2); theMLEs twist and fold the front and prevent the fron-togenetic two-dimensional saturation (as in Spall,1997).

Traditional approaches therefore provide littleguidance in developing a parameterization of frontalslumping and spindown by MLEs. There are how-ever aspects of the nonlinear frontal spindown thatcan be used to develop a parameterization. First,the initially vertical isopycnals slump from the ver-tical to the horizontal without spreading much, i.e.,M2 decreases only 10 to 20% while N2 increases byorders of magnitude. Second, the inverse cascadeproceeds to ever increasing scales in the horizontal,but it is arrested by the ML depth in the vertical.The typical vertical excursion scale is a fixed propor-tion of the ML depth across different simulations(Fig. 6). Third, the MLEs release PE by fluxingbuoyancy along a surface at a shallower slope thanthe mean isopycnal surface (i.e.,the flux directionis more horizontal than the isopycnals), a charac-teristic of linear and nonlinear baroclinic instability(Fig. 7). The ratio of the slopes is fixed near two, thevalue yielding the maximum extraction of PE (Eady,1949; Haine and Marshall, 1998). Fourth, the rms


Figure 7: Ratio of the horizontal to vertical eddy fluxesscaled by isopycnal slope for the same simulations as inFig. 6. The z-level shown is the ML midpoint, and allquantities are averaged over the center of the front.

eddy velocities in the middle of the front saturate ata value that scales with the initial mean geostrophicshear (Fig. 5). These four elements constitute thebasic ingredients of the parameterization.

3. Theory for the Parameterization

A parameterization of ML restratification is to bederived based on the phenomenology of MLEs. Aschematic of the slumping process of a ML front isshown in Fig. 8. The vertical eddy buoyancy fluxesare everywhere positive, and the horizontal cross-channel eddy fluxes are everywhere down the meanhorizontal gradient. The fluxes are along a shallowerslope than the mean isopycnal slope to slump thefront and reduce the mean PE.

The ML restratification problem shares many as-pects with the mesoscale restratification consideredby Gent and McWilliams (1990, hereafter GM) andGent et al. (1995). First, restratification proceedsthrough baroclinic instabilities and releases meanPE. Second, isopycnal slumping is largely adia-batic and can be represented through advection byan eddy-driven overturning streamfunction. Third,momentum fluxes are weak compared to Coriolisforces, hence only buoyancy fluxes need to be pa-rameterized. Despite these similarities, the GM pa-rameterization is not optimal for MLE restratifica-tion for two reasons. The MLE vertical structure isdictated by the ML depth (Fig. 6); there is no suchconstraint in the ocean interior or GM. Second, theML is frequently remixed, so M2 is nearly depth-

Figure 8: Schematic of the ML restratification.Thin contours denote along-channel mean isopycnals.Straight arrows denote direction of the eddy buoy-ancy fluxes, and circular contours/arrows indicate eddy-induced streamfunction contours and direction. Thedecorrelation lengths of the eddies ∆y and ∆z are in-dicated. The reader is reminded that after Rossby ad-justment the isopycnals are already flattened to slopesof O(10m per km) despite their near vertical appearancein this figure.

independent. The Rossby adjustment or symmetricinstabilities that follow remixing provide a nearlydepth-independent N2 from the depth-independentM2. Hence, MLIs develop with nearly uniform back-ground M2 and N2, which simplifies the parameter-ization.

It is an open question whether parameterizationof mesoscale restratification should be cast in termsof PV4 or PE budgets. During MLE restratificationthe horizontally-averaged PV outside of frictionallayers is nearly uniform in the vertical. Thus,

PE ≡ −zbxyz ∝ H2bz

xyz ≈ H2PVxyz

/f,

and the two approaches are equivalent. Lapeyreet al. (2006) note that frontogenesis can intensifyPV near the surface without affecting PE. Whilethis effect appears significant for mesoscale eddies,it is secondary for MLEs. We can therefore developthe parameterization using the along-channel meanbuoyancy budget and the volume average PE budget(over a large volume containing the frontal slumpingso that boundary terms vanish),

∂b

∂t+ ∇ · ub +∇ · u′b′ = D, (4)

dPEdt

≡ − d

dtzb

xyz= −wb

xyz. (5)

4Ertel potential vorticity, or PV, is (f + ∇ × u) · ∇b,but when averaged over the meandering front it is well-approximated by f bz .


Overlines denote averaging (Table 1).

a. Magnitude of the Potential Energy Release

A simple scaling for the magnitude of the verti-cal and horizontal eddy buoyancy fluxes begins byconsidering the PE extraction by exchange of fluidparcels over a decorrelation distance (∆y, ∆z) in atime ∆t, as sketched in Fig. 8,

∆PE∆t

∝−∆z

(∆yM2 + ∆zN2

)∆t

. (6)

We may estimate the extraction rate by assuming:

1. The relevant timescale ∆t is advective: the timeit takes for an eddy to traverse the decorrelationlength with typical eddy velocities, V:

∆t ∝ ∆y/V. (7)

2. The horizontal eddy velocity, V, scales as themean thermal wind, U (see Fig. 5):

V ∝ U =M2H

f. (8)

3. The vertical decorrelation length scales with theML depth (see Fig. 6):

∆z ∝ H. (9)

4. Fluid exchange occurs along a shallower slope(i.e., PE extracting) and proportional to themean isopycnal slope (see Fig. 7):

∆z

∆y=

1C

M2

N2, C > 1. (10)

Thus,

∆PE∆t

∝ −C − 1C

M4H2

|f |. (11)

Taking the absolute value of f ensures that PE isextracted in southern and northern hemispheres.

The MLE vertical flux dominates the mean, so

wbxyz ≈ w′b′

xyz ∝ C − 1C

M4H2

|f |. (12)

Assumption 4 implies

v′b′xyz

= −Cw′b′

xyzN2

M2,

∝ −[(C − 1)

N2H2

|f |

]M2. (13)

To conclude, (12) and (13) are consistent with Fig. 8:w′b′ is upward and v′b′ is down the mean buoyancygradient, M2.

b. Magnitude of the Overturning Streamfunction

One might base a parameterization of w′b′ and v′b′

directly on the scalings (12) and (13), but introduc-tion of an overturning streamfunction aids numericalimplementation.

The eddy buoyancy fluxes may be decomposedinto a skew flux generated by a streamfunction(v′sb′ ≡ −Ψbz, w′

sb′ ≡ Ψby) and the remaining resid-

ual flux,

∇ · u′b′ = − ∂

∂y

(Ψbz

)+

∂

∂z

(Ψby

)(14)

+∂(v′b′ − v′sb

′)∂y

+∂(w′b′ − w′

sb′)

∂z,

In an adiabatic statistically-steady setting the resid-ual flux would vanish, so all fluxes would be skewwith a unique streamfunction. In spindown prob-lems, the residual flux does not vanish–the fluxesare more horizontal than the isopycnals–primarilydue to time dependence. Thus a choice of stream-function remains, and this choice should be gov-erned by the ease of parameterization of the residualflux (Plumb and Ferrari, 2005). Traditionally, thestreamfunction is chosen to eliminate the horizontalresidual flux (Andrews and McIntyre, 1978),

Ψtr ≡ −v′b′

bz, (15)

∇ · u′b′ = − ∂

∂y

(Ψtrbz

)+

∂

∂z

(Ψtrby

)(16)

+∂(w′b′ + v′b′by/bz)

∂z.

The residual flux here is the vertical cross-isopycnalflux. It is O(Ro) compared to the skew flux, andthus it can be neglected in the ocean interior whereRo 1 and (15) is useful. In the ML setting, thevertical fluxes are leading order. Producing w′b′ in(12) would require a delicate balance of the decom-posed fluxes implied in (15) and (16) to stably pro-duce the upgradient vertical flux and have it vanishat the surface: a daunting numerical task. The Heldand Schneider (1999) streamfunction, Ψhs, is moreconvenient,

Ψhs ≡ w′b′

by, (17)

∇ · u′b′ = − ∂

∂y

(Ψhsbz

)+

∂

∂z

(Ψhsby

)(18)

+∂(v′b′ − v′sb

′)∂y

.


With this definition, Ψhs is readily given by w′b′

in (12) and vanishes naturally at the ocean surface.Furthermore, the horizontal residual flux is an easilyparameterized downgradient flux, as C > 1.

Care must be taken if the scaling (12) for w′b′xyz

is to be used to estimate Ψhs in the definition (17).The scaling (12) applies to the large scale yz-averageof w′b′ while (17) requires local values in y and z ofw′b′ and by. The simulations suggest that smooth-ing horizontally over an MLE lengthscale–or equiva-lently the resolution of any model where the param-eterization will be used–is sufficient to quell thesesubtle distinctions (see FF). Thus, a local relation-ship in y is presumed,

Ψz ∝ (C−1)H2by

xz

C|f | . (19)

The vertical structure of the parameterization is notlocal and is presented next.

c. Vertical Structure of the Overturning Stream-function

In linear theory, the lengthscale at which the ver-tical velocity and the buoyancy perturbations arecorrelated specifies the vertical structure of w′b′.Fig. 9 shows the dominant lengthscales contribut-ing to the correlations between w′, v′, and b′. Whilethe correlations and autocorrelations of v′ and b′ aredominated by features larger than the most unsta-ble lengthscale, the typical horizontal scale at whichw′ and b′ correlate remains close to Ls. The differ-ence in correlation scales is consistent with a verticalmode saturation and an horizontal mode inverse cas-cade. Thus, the vertical structure of w′b′ from lineartheory persists at finite amplitude (per Branscome,1983a,b).

A vertical structure function, µ(z), is taken fromthe w′b′ of linear theory and implemented as

Ψ =CeH

2byxz

µ(z)|f |

. (20)

Fig. 10 shows how little µ(z) changes as finite am-plitude is attained. Normalization of µ(z) to peakat one collects all remaining constants into an effi-ciency factor Ce.

An accurate approximation of µ(z) is given by asimple extension of the analysis described in Stone(1972a). The vertical fluxes due to ageostrophicbaroclinic instabilites are obtained by expanding thelinear solutions to O(k2U2/f2). The expression forw′b′ is evaluated at the k of the fastest growing modeas suggested by the numerical simulations (Fig. 9).

0 5 10 15 20 250

0.5

1

1.5

2

time (days)

corr

ela

tion length

/Sto

ne length

vb

wb

KE

Figure 9: The horizontal lengthscales typical of thecorrelations v′b′, w′b′, and EKE for the same simula-tion as in Fig. 4 are compared to the most unstablelengthscale. Lengthscales from the v′ and b′ cospec-trum, the w′ and b′ cospectrum, and the EKE spec-trum, E(k), are shown rescaled by the time-evolvingLs. L2 =

R<(S(k))dk/

Rk2<(S(k))dk for a cospec-

trum S(k), and theR<(S(k))dk is the full correlation.

For more detail on cospectra see Emery and Thomson(2001).

Taking the large Ri limit, a µ(z) appropriate soonafter MLEs begin strong restratification is found:

µ(z) =[1−

(2zH + 1

)2] [

1 + 521

(2zH + 1

)2]. (21)

Below the ML base, µ(z) is set to zero. By differen-tiating the buoyancy budget (4) in z and averagingin the horizontal over a region large enough that thefluxes vanish on the boundaries, one finds that thedominant balance observed in the simulations:

∂bzxy

∂t≈ −by

x ∂2Ψ∂z2

y

,

≈ −Ce

H2∣∣∣by

xz∣∣∣2

|f |

y

∂2µ(z)∂z2

. (22)

All of the factors on the right are depth-independentexcept ∂2µ(z)

∂z2 . Hence, µ(z) controls the relative rateof restratification at different depths.

Equations (21) and (22) suggest that restratifica-tion near the surface and base of the ML is nearly 3times faster than in the center, consistent with nu-merical simulations (see FF). A long-wave approxi-mation to µ(z) is easily obtained by neglecting thesecond factor in brackets in (21), as shown by Stone(1972a). This quadratic form is inappropriate for


0 2 4 6 8 10 12 14 16!300

!250

!200

!150

!100

!50

0

de

pth

(m

)

!hs

(normalized+shifted by 1/day)

Figure 10: Daily snapshots of Ψhs from a typical simu-lation without a diurnal cycle. The streamfunctions arerescaled to have maximum of unity for comparison toµ(z) in (21), and they are shifted by 1 each day (dottedlines show the origins, ML depth ≈ 200m).

the ML frontal spindown, because it produces uni-form restratification at all depths contrary to theresult of the simulations.

d. Comparison with other theories

Comparison with other parameterizations is usefulto clarify the implications of (20). In particular,most eddy parameterizations assume a steady statewith constant N2. During ML restratification byfrontal spindown both the stratification and eddylengthscale change dramatically: this time depen-dence must be predicted by the parameterizationrather than ignored.

Stone (1972b) uses linear instability analysis tocompute the correlations v′b′ and w′b′ for small am-plitude linear waves, and then sets the eddy velocityamplitude to be proportional to the mean flow ve-locity, U , as is done here. From Stone’s equations2.22-2.23,

v′b′ = −1.3[

N2H2

f

√1+RiRi

]M2, (23)

w′b′ = 0.09H2M4

f

[1−

(2zH + 1

)2]

1√1+Ri

. (24)

These fluxes differ from the ones proposed here, (12)and (13), by a dependence on Ri that originates fromthe linear theory correlations. The difference canbe traced to the linearized perturbation buoyancybudget,

b′ ∼ (M2v′ + N2w′)τs(k), (25)

where τs is given by the linear growth timescale(3), or just τs ∼

√Ri/f for large Ri. In turbu-

lent flows, such as in Fig. 2, b′ decorrelates on themuch longer advective timescale τa ∼ Ri/f , hencethe

√Ri discrepancy of Stone’s formulae with the

simulation results. Eddy-damped Markovian the-ory nicely demonstrates the transition from fluxesgoverned by linear timescales to fluxes governedby advective timescales as the instabilities reach fi-nite amplitude (e.g., Holloway and Kristmannsson,1984; Salmon, 1998). A symptom of the failure of(25) is that it predicts a vertical excursion scale ofζ =

√b′2/N2 ∝ Hτa/τs, while the simulations in

Fig. 6 demonstrate ζ ∝ H.Haine and Marshall (1998) use a mixing length

argument to advocate v′b′ ∝ −LfUM2, and asadvocated here emphasize an advective timescale:the timescale to transfer buoyancy across the baro-clinic zone, Lf/U . However, they presume the zonewidth, Lf , holds fixed during restratification. Inthe ML, the vertical lengthscale and vertical fluxesare more constrained than the horizontal, leadingto ∆y ∼ N2H/M2. Indeed, the snapshots in Haineand Marshall (1998) reveal eddies that enlarge be-yond the initial baroclinic zone. They analyze fluxscalings at only one time per simulation, τmodel,when ’lateral transfer by eddies has become signifi-cant’, which occurs naturally when ∆y ∼ Lf . Theirforcing provides Lf = |N2H/M2| initially, so

v′b′ ∝ −LfUM2 ∝ −N2H2

fM2. (26)

This expression agrees with (13). However, the workhere uses ∆y instead of Lf , which extends the evo-lution of v′b′ and w′b′ beyond τmodel and applies tosituations where Lf is not equal to |N2H/M2| ini-tially (e.g., Fig. 3).

Green (1970) proposes a scaling based on equat-ing the total difference in PE between an initialbaroclinic zone and an hypothetical one with theminimum PE accessible by adiabatic rearrangement.The PE released is equated to EKE to yield an eddyvelocity scale and–with b′ scaling as the buoyancydifference across the zone–a scale for v′b′. Greenassumes constant N2, but by adapting Green’s ap-proach to allow for large changes in N2 yields,

∆PE ∝ HM2Lf , (27)

v′b′ ∝ −M3L3/2f H1/2 (28)

∝ −M2N2H2

|f |1

Ro3/2Ri1/2. (29)


Where Ro = U/(fLf ). Once the eddy lengthscaleexceeds the front width, one may replace Lf withN2H/M2 (or equivalently Ro with Ri−1), and then(29) becomes,

v′b′ ∝ −N2H2√

Rif

M2. (30)

Different arguments lead Visbeck et al. (1997) andLarichev and Held (1995) to the same expression.While the amount of PE extracted in (27) is thesame as proposed in Section 3a, the results for v′b′

differ by√

Ri. The extraction of mean PE is close toEKE+EPE, but Green assumes that EKE+EPE ∝EKE. Yet, from the numerical simulations (Figs. 5and 6),

EKEEPE

∝ M4H2

f2N2H2∝ Ri−1.

As Ri increases, the mean PE extracted goes in-creasingly to EPE, while EKE saturates near theinitial mean KE. The work here avoids this prob-lem by using the PE budget, (5), to directly relatePE extracted to w′b′

xyz.

Some eddy parameterizations (e.g., Canuto andDubovikov, 2005; Eden, 2006) suppose that thedecorrelation length is approximately the linear in-stability lengthscale for a mixing length theory. Us-ing the linear lengthscale in (2) yields,

v′b′ ∝ −LsUM2 ∝ −[N2H2

|f |

√1 + RiRi

]M2. (31)

Except for an unspecified efficiency factor, this ex-pression is Stone’s (23). This approach fails becausethe linear instability lengthscale during frontal spin-down is smaller than ∆y ∝ N2HM−2 by Ls/∆y ∝√

1 + Ri/Ri.In summary, the scaling here differs from others

in approach and by nondimensional factors. Theparameterization is tested against these alternativesin Section 4.

e. Residual Diffusive Fluxes

The skew flux generates restratification because it ispart of the overturning circulation, but the residualflux,

R = v′b′ + Ψbz = v′b′ + w′b′bz

by, (32)

merely widens the front slightly (FF). In the linearEady model, v′b′, bz, and by are depth-independent

while w′b′ depends on µ(z), so the relationship be-tween residual flux and v′b′ is depth dependent. Inthese simulations v′b′ and bz change as the flow re-stratifies until R ≈ v′b′/2. Perhaps not coinciden-tally, parcel exchange theory indicates that if R isv′b′/2 at all depths then potential energy extractionis maximized. Using the scalings for v′b′ and w′b′

in (13) and (12), R can be parameterized with anonlinear horizontal diffusivity scaling as

v′b′ + Ψbz = −κH by, (33)

κH =Cebz

xzH2µ(z)|f |

. (34)

Given the value of Ce ≈ 0.06 as determined in Sec-tion 4 and typical ML stratifications, κH is onlyO(1 − 100m2/s). This small value confirms thatMLE horizontal fluxes–residual or not–are smallerthan mesoscale horizontal fluxes. FF show that ina forward simulation of parameterized frontal spin-down, adding the residual flux widens the front,but minutely–comparably to changing the buoyancyadvection scheme. They also show that includingresidual fluxes makes the model less stable numeri-cally. In sum, adding the residual fluxes is possible,but the costs outweigh the benefit.

While w′b′ is upward in the ML, it is downwardbelow the ML base, as the reversal of the sign ofΨ just below the ML base in Fig. 10 shows. Thistendency is easily understood: v′ and w′ are con-tinuous, so fluxes roughly along isopycnals in theML overshoot as the mean isopycnal slope flattenssuddenly at the ML base. Below the ML base thevertical buoyancy flux is downward and thus downthe mean vertical buoyancy gradient; a vertical di-apycnal diffusivity κv of O(10−4m2/s) acting onthe mean buoyancy gradient could parameterize thisflux. This magnitude was estimated by diagnos-ing w′b′/bz for the simulations run using the typ-ical MLE parameter values in Table 2, but κv variesstrongly with MLE strength. FF show that usingκv = 3 ·10−5m2/s in a forward model of the param-eterization slightly improves agreement with a com-parable submesoscale-resolving simulation. Turbu-lent mixing parameterizations may already containpenetrating turbulent fluxes of this magnitude (e.g.,Large et al., 1994). The additional diffusivity mightbe important where ML entrainment is critical, buta full study of this secondary effect of MLEs is be-yond the scope of the present work.


a) b)

c) d)

e) f)

Figure 11: Temperature (oC) during one diurnal cycle using convective adjustment. Panels d-e) are afternoonvalues. (Black contour interval=0.01oC, white contour interval=0.1oC.)

f. MLEs under Additional Mixing: Diurnal Cycle

The preceding discussion has paid little attentionto the mechanisms that cause the ML to be mixed

in the first place: turbulent vertical mixing. SomeRossby adjustment simulations were inundated witha diurnal heat flux cycle for a more realistic ML en-


Figure 12: As in Fig. 7, but for a simulation with adiurnal cycle. The afternoon values, when the surface isnot being actively cooled, are shown as diamonds.

vironment. An example is shown in Fig. 11 andFig. 2d-f. With a diurnal cycle, the initial instabil-ity wavelength is slightly smaller during the lineargrowth stage (see BFF), but later the MLEs andtheir nonlinear saturation are remarkably similar tothose in Figs. 2, 4, and 5.

Considering ”afternoon” snapshots suffices to iso-late the effects of MLEs; during the night convec-tion blurs the MLE signal. Fig. 12 shows that theafternoon MLE fluxes are along a slope shallowerthan the isopycnal slope just as without a diurnalcycle. This effect is apparent once the MLEs aresufficiently strong to overcome the noise of the di-urnal cycle (after about day 10). The next sectionshows that the Ψ scaling (20) holds nearly as wellas in the no diurnal mixing case.

The diurnal cycle causes a notable change to thevertical structure of the fluxes. Fig. 13 shows thatthe streamfunction does not vanish at the surface,but at some level below. This is because the MLis capped by large N2 during solar heating (Figs. 2and 11). The streamfunction structure µ(z) maybe trivially altered by translating and rescaling thevertical coordinate in µ(z) so that it vanishes at adepth just below the surface rather than the surface.This shortcut approximates the result from linearinstability analysis for a ML with surface-intensifiedN2.

In conclusion, the scaling for Ψ in (20) holds inthe presence of spatially-uniform intermittent mix-ing due to a strong diurnal cycle, as MLEs are rel-atively unaffected. Haine and Marshall (1998) find

0 2 4 6 8 10 12 14 16!300

!250

!200

!150

!100

!50

0

de

pth

(m

)

!hs

(normalized+shifted by 1/day)

Figure 13: Daily snapshots of afternoon Ψhs from atypical simulation with a diurnal cycle. Compare toFig. 10.

the same scaling with a constant 400W/m2 coolingof the surface to represent strong wintertime cool-ing, so even larger fluxes without daytime restratifi-cation do not halt MLEs. However, MLE restratifi-cation may not overtake convective destratification.Indeed, here the basin-average ML stratification de-creases each night as the MLE w′b′ is dwarfed bythe peak cooling. Yet, the carefully chosen balancebetween nighttime cooling and solar heating in thesesimulations is tipped by the MLE flux so that thelong term tendency is toward a shallower ML. Inthe ocean and in realistic models a balance will ex-ist between long-term-average (e.g., monthly) sur-face forcing and MLE restratification, while duringactive convection the effects of MLEs will be sec-ondary. MLE restratification does not prevent ac-tive convection but immediately initiates restratifi-cation when convection ceases, during the daytimehere and at the edge of the cooling region in Haineand Marshall (1998).

4. Diagnostic Validation

This section validates the scaling argument pre-sented above by diagnosing the magnitude of theoverturning streamfunction in MLE-resolving nu-merical simulations.

The simulations provide instantaneous 3-dimensional fields of buoyancy and buoyancy fluxes.The relevant diagnosed quantity is,

Ψd =1T

∫ t0+T

t0

w′b′xy

byxy µ(z)−1dt, (35)


Where the time averaging in (35) is restricted to aninterval after MLEs have reached finite amplitudeand before lateral boundary effects are important.Appendix C discusses further details of the diagno-sis.

Fig. 14a shows that Ψd scales well with

H2byxz|f |−1

yt

for a set of 37 runs with balanced ini-tial conditions and no diurnal cycle. The slope onthis figure demonstrates the scaling, and the inter-cept amounts to Ce = 0.06 in (20). Fig. 14b showsthat the same scaling holds over the whole set of 241simulations varying front strength, initial stratifica-tion, front width, vertical and horizontal viscosity,rotation rate, etc. Consistent with Fig. 3, balancedand unbalanced simulations obey the same scaling.The diurnal cycle introduces noise in the estimationof Ψd, as can be anticipated from the noisy after-noon results in Fig. 12, and increases the estimateof Ce to near 0.08. The scaling agrees best when Ψd

is large, which is when MLE restratification is mostimportant (Fig. 14b).

The magnitude of Ce may be compared to otherstudies measuring baroclinic eddy horizontal fluxesby using the scaling for the vertical flux (12) andconverting to an horizontal flux with (13). Cenedeseet al. (2004) present a laboratory result for horizon-tal flux scaling approximately equivalent to Ce =0.05, and cite many studies covering a range equiv-alent to 0.02 < Ce < 0.12. The wide range found inthese studies is likely an artifact of fitting inappro-priate scaling laws to v′b′ and consequently foldingparameter variations (of Ri, for example) into themeasured ’constants’ of eddy processes.

A clever intuition might arrive at the scalingΨ ∝ CeM

2H2µ(z)/|f | by pure dimensional analy-sis, but dimensional analysis cannot rule out nondi-mensional parameters. Dependence on Ri is quicklyeliminated. Figs. 14c-d show that the scaling ofStone (1972b) from (24) and (17),

Ψs = CsH2by

xzµ(z)

|f |1√

1 + Ri, (36)

and Green (1970) from (30) and (15),

Ψg = CgH2by

xzµ(z)

|f |√

Ri, (37)

have substantially more scatter than Fig. 14a con-firming the scaling proposed in (20).5 Fig. 14f shows

5The similar scalings of Canuto and Dubovikov (2005) andVisbeck et al. (1997) are equivalent to Stone and Green andhave more scatter, too.

that this scatter is associated with erroneous depen-dence on the time-evolving Ri, rather than otherfactors. (Using the initial value of Ri instead of atime-evolving value produces an order of magnitudemore scatter for these scalings, not shown). Fig. 14eshows there is no systematic trend with Ri in thedepartures of Ψd from (20), nor is there a system-atic trend with the initial value of Ri (not shown).Dependence on Ro through the frontal width Lf ,as in (29) and as assumed by Haine and Marshall(1998) is irrelevant as soon as ∆y > Lf , which oc-curs soon after finite amplitude is attained. A figurelike Fig. 14e, but with Ro as ordinate shows no de-pendence on Ro (not shown).

Additional potentially relevant nondimensionalquantities appear in the problem, such as (H/Lf ),Smagorinsky coefficient (Sm), grid resolution tofront width (∆x/Lf ), Ekman number (Ek ≡νH−2f−1), diurnal cycle timescale to inertialtimescale f/Ω, and interior stratification to MLstratification (Nml/Nint). Nonlinear optimizationwas used to test sets of nondimensional parametersPi to find exponents b(i) and the efficiency factorCe that reduced the difference between Ψd and theproduct of parameters, CeH

2M2|f |−1ΠiPb(i)i . By

this method, an Ekman number factor of approx-imately Ek−0.2 was found to improve the results.No robust dependence on any other nondimensionalparameter was found (i.e.,the exponents were lessthan 0.1 in magnitude). Haine and Marshall (1998)note that the parameter space needed to distinguishpotential scalings is often unexplored. Even the241 simulations here neglect some part of parame-ter space. Neglected regions include nonhydrostaticeffects (H/Lf = O(1)), barotropic instabilities ofthe front (RiRo2 1), and viscosity sufficient tostabilize the ML instabilities. However, the scalingpresented here spans the regime relevant for MLEs.

5. Summary and Conclusion

Numerical simulations and theory reveal that theML is host to shallow frontal instabilities that actto restratify the ML. This paper presents a parame-terization of the restratification by these instabilitiescast as a streamfunction to represent the overturn-ing of the front. The parameterization depends onthe horizontal buoyancy gradients and provides afirst attempt at incorporating the interaction of lat-eral gradients and vertical mixing in the ML. Thisparameterization will provide GCMs with a novelclimate sensitivity, so far ignored by other ML pa-rameterizations. In three dimensions, the parame-


terization takes the form,

Ψ = CeH2∇b

z×z|f | µ(z), (38)

with Ce between 0.06 and 0.08. The vertical struc-ture µ(z) is well approximated by

µ(z) =[1−

(2zH + 1

)2] [

1 + 521

(2zH + 1

)2]. (39)

This parameterization produces fluxes and an eddy-induced velocity

u′b′ ≡ Ψ×∇b, u∗ = ∇×Ψ. (40)

Two companion papers (Fox-Kemper and Ferrari,2007; Fox-Kemper et al., 2007) give further insightinto the skill, implementation, and importance forclimate of the parameterization.

Previous attempts to include eddy driven restrat-ification by horizontal buoyancy gradients in MLmodels relied on ad hoc modification of the GMmesoscale eddy parameterization through taperingfunctions. This approach fails as the mesoscale hor-izontal fluxes–were they to flux along the shallowML slopes–imply excessive vertical fluxes and re-stratify the ML immediately. Indeed, the GM ta-pering schemes are introduced precisely to avoid in-stantaneous ML restratification. In contrast, MLEsprovide the correct amount of eddy restratificationfor the ML.

The approach in developing this parameterizationis novel in that scaling arguments are derived di-rectly for the overturning streamfunction instead ofrelying on diffusive closures for the horizontal eddyfluxes. The scaling simply constrains the stream-function to release PE at the rate expected for baro-clinic spindown. Working in terms of diffusivities of-fers less obvious constraints. Furthermore, the pa-rameterization avoids parameters that are difficultin modeling practice: Ri, deformation radius, in-stability length scale, or the width of a ’barocliniczone’. Only the readily available ML depth and hor-izontal buoyancy gradient are needed. The issue ofestimating the relevant horizontal buoyancy gradi-ent in a coarse model is discussed in Fox-Kemperet al. (2007). In principle, the approach here couldbe extended to a mesoscale parameterization for usein the ocean interior, but the nontrivial complica-tions of variable background stratification are leftfor a future investigation.

A few observational studies prove the existenceand ubiquity of MLEs. Flament et al. (1985) observethe development of small-scale eddies along a MLfront that compare favorably with the phenomena

here. Munk et al. (2000) have noted MLEs in photostaken by Astronaut Scully-Power. Recent observa-tions also suggest the tendency for MLEs to releasePE from fronts (D’Asaro, pers. comm.). Houghtonet al. (2006) detect submesoscale along-isopycnal fil-aments of tracer possibly indicating frontal instabil-ities, although somewhat below the surface ML. Re-peated MLE slumping of horizontal density fronts(formed from salinity and temperature variations)interspersed with strong vertical mixing events effec-tively eliminates the horizontal density fronts, butleaves behind compensated salinity and temperaturegradients (Young, 1994; Ferrari and Young, 1997;Ferrari and Paparella, 2003). ML density compen-sation is observed at the submesoscale (Rudnick andFerrari, 1999; Ferrari and Rudnick, 2000; Rudnickand Martin, 2002). Hosegood et al. (2006) demon-strate that density variability extends to the ML de-formation radius and not beyond, in agreement withour analysis of MLEs. Rudnick and Martin (2002)show that density compensation is stronger for deepMLs. All of these observations are consistent withrestratification by MLEs.

Now that a foundation has been laid, the effectsof MLEs may be studied in combination with effectsof wind (Thomas, 2005) and mesoscale frontogenesis(Spall, 1997; Oschlies, 2002; Lapeyre et al., 2006).Including the additional physics may improve thefundamental parameterization here. However, theresults here and in FF show that for the case ofnonlinear spindown of a mixed layer front this pa-rameterization has significant skill.

Acknowledgement This work was supported byNSF grant OCE-0612143. Discussions with A. Ma-hadevan, X. Capet, G. Flierl, W. Large, L. Thomas,W. Young, and J. Marshall contributed greatly tothis work. The CPT-EMILIE team served to ham-mer out many of the difficulties involved.

A. Mesoscale-Submesoscale ResolvingModel Configuration

The coupled mesoscale-submesoscale simulation is a200km×600km×800m channel on an f -plane withtemperatures restored near the walls to force a geo-strophic flow. A sloping bottom keeps the eddiesout of the temperature restoring region. The ver-tical resolution is 10m over the surface 100m, andthen enlarges by 20% for each deeper gridpoint. Themodel is spun up with ∆x = 8km for 900 days, in-terpolated and continued for 100 days at 2km reso-lution. Fig. 1 shows day 925.


Initially, H = 75m, and a 50m ML is preserved bya diurnal cycle of 200W/m2 nighttime cooling andjust enough daytime penetrating shortwave radia-tion (maximum heat flux −717W/m2) to give zerodiurnal average. KPP (Large et al., 1994) is usedto simulate ML turbulent processes. The heat fluxq is:

q = q0 + qdmax[cos(2πt), cos(πth)]− cos(πth).

The constants are q0 = 200W/m2, qd ≈−1834W/m2, th = 1/3day, and t is model time inunits of days. The temperatures in the upper layersare restored only on the warm side of the front.

Below H, the initial stratification is:

ρρ0

= 1− α[T0 + e(z+H)/δ(∆Tv + ∆Th

2 tanh y−y0L )]

∆Tv ≡ TL+TR−2T02

∆Th ≡ TR − TL

This stratification also is restored along the walls.Constants are ML depth (H), left temperature (TL),right temperature (TR), bottom temperature (T0),thermal expansion coefficient (α), center of channel(y0), thermocline depth scale δ and active channelwidth (L) and depth (D).

QG linear instability solutions are used to tunethe parameters so that the most unstable modesfit in the domain. The choices used (∆Tv = 5C,∆Th = 8C, δ = 100m and D = 800m) provide50 − 150km unstable modes. The fastest-growingmode is near 80km with an efolding time near 6days. These values are smaller than those expectedin the real ocean, but a sacrifice must be made forcost. Horizontal temperature gradients are rapidlymixed by the mesoscale to the boundary regions overthe sloping bottom and the efolding time decreasesto O(1 month). Thus, a temporal submesoscale tomesoscale scale separation is present. A robust andapproximately statistically-steady mesoscale eddyfield persists throughout.

B. Rossby Adjustment Model Configura-tion

The Rossby adjustment simulations begin with atemperature front above a stratified interior. Theinitial stratification is

b = N2(z + H) +LfM2

f

2tanh

[2(y − yo)

Lf

]+ bo,

N2 =

N2ml ∀ : z > −Ho

N2int ∀ : z <= −Ho

The channel is 300m deep. The initial vertical strat-ification has a ML, with parameters H,Mf , Nml, Lf ,which rests on a more strongly stratified interiorwith Nint. Rotation rate, and viscosities are alsovaried (f,Sm, ν). Unbalanced or balanced ini-tial conditions and a diurnal cycle (with 200W/m2

nighttime cooling as in (41)) were also used in manyof the simulations. Convective adjustment was usedin all simulations shown here, but test simulations innonhydrostatic mode and with KPP (Large et al.,1994) mixing parameterization were run and gavegenerally similar results (see BFF). A third-orderflux-limiting advection scheme was used for temper-ature that does not require explicit diffusion, so nonewas used. The selection of parameters for all 241simulations used are given in Table 2.

C. Computation of Diagnostics

Verification of (20) begins with an along-channelmean of the fluxes and buoyancy at every time snap-shot. While (20) was derived with a constant M2 inmind, in the simulations by varies in cross-channeldirection to isolate the front from the effects of thehorizontal boundaries of the channel. Thus, care isneeded in cross-channel averaging. One might aver-age over the initial location of the center of the front,use averages weighted by by, average only where by

is over a critical value, or use the average over thewhole channel (given that w′b′ and by are likely tobe nonzero over roughly the same region). All ofthese methods agree when MLEs dominate, and dif-fer only when the signal is contaminated (e.g.,bygravity wave w′b′, by the front sliding out of theaveraging window, or by boundary effects). Usingthe basin-average is closest to (12), but averagingonly over the center of the front reveals the rele-vant Ri. Experimentation determined that averag-ing over the center of the front (i.e.,where |by| wasmore than 10% of it’s median value) agrees withthe basin average to within 15%, so this was they-averaging used.

Another issue is quantifying the vertical structureof the diagnosed overturning streamfunction. Thisis readily accomplished by evaluating the best fit to(21) (via the singular value decomposition pseudo-inverse of the discrete form of µ(z) based on a di-agnosed ML depth), or more simply by taking themaximum absolute value of the streamfunction in zover the ML. Estimates agree to within a few per-cent, so the fit to (21) is used.

A suitable definition of H, the ML depth, is given


by the integral constraint,

N2(H) =Cm

H

∫ 0

−H

N2(z′)dz′, (41)

The base of the ML is the depth at which the localbuoyancy frequency is Cm times the buoyancy fre-quency averaged from the surface. The results arerelatively insensitive for 1.5 < Cm < 3, Cm = 2 wasused. To find H and Hs, one begins at the levelof the minimum of N2 and separates these boundsuntil (42) is satisfied.

N2(H) =Cm

H −Hs

∫ −Hs

−H

N2(z′)dz′. (42)

While this more complex method is used diagnos-tically to aid in determining the streamfunctionfrom the MLE-resolving simulations, it is probablymore complicated than needed in a parameterizationwhere (41) will suffice.

The parameterization focuses on the period ofstrong restratification by finite-amplitude ML ed-dies. Thus, for each simulation, a time window isdiagnosed. It begins when the rms v′ was more than10% of the initial maximum mean shear velocity athalf of the ML gridpoints, i.e., when finite amplitudeis acheived. It ends if the total buoyancy differenceacross the channel changes by 10% for half of theML gridpoints to avoid effects from the sidewalls.Finally, the window is restricted to times when thedifferent y-averaging methods agree to within twostandard deviations to eliminate the occasional mo-ment when M2 vanishes in a particular average. Forruns with a diurnal cycle, the averaging window isfurther restricted to afternoon times. This time win-dow generally agrees with the window one woulddesignate ’by eye’ as equilibrated, and the scaling re-lationships shown in all figures are supported withthe ’by eye’ window as well. This window simplyreduces the scatter over the ’by eye’ version.

The relevant diagnosed quantity is thus

1T

∫〈w′b′〉〈by〉

µ(z)−1dt. (43)

Where µ(z)−1 indicates the pseudo-inverse of (21),and the time-averaging occurs only over the timewindow specified above.

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a)10−3 10−2 10−1 100 10110−3

10−2

10−1

100

101

Ce H2 M2 |f|−1

Ψd

b)10−3 10−2 10−1 100 10110−3

10−2

10−1

100

101

Ce H2 M2 |f|−1

Ψd

c)10−3 10−2 10−1 100 10110−3

10−2

10−1

100

101

Cs H2 M2 |f|−1 (1+Ri)−1/2

Ψd

d)10−3 10−2 10−1 100 10110−3

10−2

10−1

100

101

Cg H2 M2 |f|−1 Ri1/2

Ψd

e)10

!110

010

110

210

!2

10!1

100

101

102

Ri1/2

!d/(

Ce H

2 M

2 |f|!

1)

f)100 101 102

10−1

100

101

Ri1/2

Ψd/Ψ

s,Ψd/Ψ

g

Figure 14: Magnitude of Ψd versus theories for magnitude of Ψ for simulations with diurnal cycle (blue) andwithout (red) starting from balanced (circles) or unbalanced (squares) initial conditions. Plus signs and crossesindicate balanced simulations where Ri0 > 1 or Ri0 < 1 initially. a) Shows Ψd in the balanced, no diurnal cycle

simulations versus Cebyxz

H2|f |−1yt

,Ce = 0.06, and b) includes unbalanced and diurnal cycle simulations, Ce = 0.08.

c) Shows Stone’s theory, (36), Cs = 0.53. d) Shows Green’s theory, (37), Cg = 0.0085. e) Ψd/Cebyxz

H2|f |−1yt

versusRi1/2. Also shown are lines parallel to Ri1/2 and (1 + Ri)−1/2. f) Ψd/Ψs (black dots) and Ψd/Ψg (green crosses)versus Ri1/2. Also shown are lines parallel to Ri±1/2. Ψd, Ψs, and Ψg are defined in (35), (36), and (37).


Symbol Name Typical ValueH ML depth 100mb buoyancy (b = −gρ/ρ0, ρ0 = 1035kg/m3, g = 9.81m/s2) ±0.04m/s2

u, v, w velocity components ±0.05m/suH horizontal velocity ±0.05m/sA along-channel mean of A and perturbation from A

Axy

along and cross-channel meanA

xyzhoriz. mean and vert. mean over ML

V,W typical eddy velocity scales 0.05m/sU mean shear velocity scale (M2H/f) 0.05m/s

M2 front-averaged horiz. buoy. gradient −(2f)2

N2 front-averaged vert. buoy. gradient (buoy. freq.2)M2

f initial maximum horiz. buoy. gradient −(2f)2

N2ml, N

2int initial ML and interior vert. buoy gradient (4f, 64f)2

Ω earth angular frequency (2π/day) 7.29× 10−5s−1

f Coriolis parameter Ωτs Stone growth timescale 1 day from (3)

Ls, ks Stone fastest-growing lengthscale/wavenumber 1km from (2)

E(κ) kinetic energy power density spectrum (|u′Hb′|2xyz

=∫

E(κ)dκ)Es(κ) kinetic energy power density spectrum prediction from (1)

KE,EKE kinetic energy, eddy kinetic energyPE,EPE potential energy, eddy potential energy

Ψtr traditional streamfunction −v′b′/N2

Ψhs Held & Schneider streamfunction w′b′/M2

Ψd diagnosed streamfunctionΨ 3d streamfunctionu∗ 3d eddy-induced velocity

∆x,∆z horizontal and vertical grid spacing Ls/10 from (2)Lf , Lb front width, basin width 40∆x, 150∆xx, y, z along-channel, cross-channel, and vert. coordinate 0 → Lb,−300 → 0m

C flux slope to isopycnal slope ratio, −M2v′b′/(w′b′N2) 2Ce efficiency factor 0.06− 0.08Cs efficiency factor (Stone parameterization) 0.1− 0.9Cg efficiency factor (Green parameterization) 0.001− 0.009k, l along- and cross-channel wavenumbers 1/Lb → 1/∆x

κ isotropic wavenumber (√

k2 + l2) 1/Lb → 1/∆x

ζ vertical excursion scale√

b′2/N2 0.2HRi0 initial cond. balanced Richardson number Ri0 = N2

0 f2/M40 0 → 256

Ri balanced Richardson number Ri = N2f2/M4 0 → 4500Sm Smagorinsky coefficient (horiz. visc.=

[Sm∆x

π

]2 √(ux − vy)2 + (uy + vx)2) 1

ν vertical viscosity 0.0001m2/sκv, κH vert. and horiz. MLE effective diffusivity

Table 1: Symbols used in this paper.


Symbol Name Value RangeH ML depth 50, 200mM2

f horiz. buoy. gradient − (1, 2, 4f)2

Nml |ML vert. buoy. gradient|1/2 (Buoy Freq.) 0, 4, 16, 32fNint |interior vert. buoy. gradient|1/2 (Buoy Freq.) 16, 64, 128f

f Coriolis parameter 2Ω,Ω,Ω/2∆x standard horiz. grid Ls/10 from (2)∆x tripled resolution test grid Ls/30 from (2)∆z standard vert. grid 5m∆z vertical test grid 1mq0 nightime cooling 0, 200W/m2

Lf front width 20, 40, 80∆xSm Smagorinsky coefficient 1, 2, 4, 8ν vertical viscosity 0.0001, 0.001, 0.01m2/s

Table 2: Parameters varied across simulations. Test grids were confirmed to agree with standard grids.

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