friction, frontogenesis, and the stratification of the surface mixed...

Friction, Frontogenesis, and the Stratification of the Surface Mixed Layer

LEIF THOMAS*

Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

RAFFAELE FERRARI

Massachusetts Institute of Technology, Boston, Massachusetts

(Manuscript received 11 March 2007, in final form 8 May 2008)

ABSTRACT

The generation and destruction of stratification in the surface mixed layer of the ocean is understood toresult from vertical turbulent transport of buoyancy and momentum driven by air–sea fluxes and stresses.In this paper, it is shown that the magnitude and penetration of vertical fluxes are strongly modified byhorizontal gradients in buoyancy and momentum. A classic example is the strong restratification resultingfrom frontogenesis in regions of confluent flow. Frictional forces acting on a baroclinic current eitherimposed externally by a wind stress or caused by the spindown of the current itself also modify thestratification by driving Ekman flows that differentially advect density. Ekman flow induced during spin-down always tends to restratify the fluid, while wind-driven Ekman currents will restratify or destratify themixed layer if the wind stress has a component up or down front (i.e., directed against or with thegeostrophic shear), respectively. Scalings are constructed for the relative importance of friction versusfrontogenesis in the restratification of the mixed layer and are tested using numerical experiments of mixedlayer fronts forced by both winds and a strain field. The scalings suggest and the numerical experimentsconfirm that for wind stress magnitudes, mixed layer depths, and cross-front density gradients typical of theocean, wind-induced friction often dominates frontogenesis in the modification of the stratification of theupper ocean. The experiments reveal that wind-induced destratification is weaker in magnitude than re-stratification because the stratification generated by up-front winds confines the turbulent stress to a depthshallower than the Ekman layer, which enhances the frictional force, Ekman flow, and differential advectionof density. Frictional destratification is further reduced over restratification because the stress associatedwith the geostrophic shear at the surface tends to compensate a down-front wind stress.

1. Introduction

The surface mixed layer of the ocean is a weaklystratified layer often encountered below the air–sea in-terface, where turbulent mixing is strong in response toatmospheric forcing. The processes that set the stratifi-cation and ventilation of the mixed layer are an essen-tial part of the coupled climate system, because thislayer regulates the exchange of heat, freshwater, and all

other climatically relevant tracers between the atmo-sphere and the ocean. Traditional models assume thatthe vertical structure of the mixed layer is set by thevertical mixing of buoyancy and momentum driven byatmospheric surface fluxes and stresses. Lapeyre et al.(2006) and Fox-Kemper et al. (2008) have recentlypointed out that during times of weak air–sea fluxes,lateral dynamics become leading order and tend to re-stratify the mixed layer through ageostrophic slumpingof lateral buoyancy gradients. The goal of this paper isto extend recent work of Thomas (2005) and show thatfrictional forces acting on buoyancy fronts can alsomodify the stratification of the mixed layer. These ef-fects are currently ignored in models and theory of theupper ocean and are likely to introduce biases in ourunderstanding of ocean–atmosphere interactions.

Most studies of the ocean mixed layer are cast interms of the impact of air–sea fluxes on the momentum

* Current affiliation: Department of Environmental Earth Sys-tem Science, Stanford University, Stanford, California.

Corresponding author address: Raffaele Ferrari, Department ofEarth Atmospheric and Planetary Sciences, 54–1420, Massachu-setts Institute of Technology, 77 Mass. Ave., Cambridge, MA02139.E-mail: [email protected]

NOVEMBER 2008 T H O M A S A N D F E R R A R I 2501

DOI: 10.1175/2008JPO3797.1

© 2008 American Meteorological Society

JPO3797

and buoyancy budgets. Less attention has been paid tothe potential vorticity (PV) budget. However, potentialvorticity is an extremely useful tracer to study the dy-namics of rotating stratified fluids. First, potential vor-ticity can only be changed by diabatic heating/coolingor friction. The amount of potential vorticity possessedby a water mass in the ocean is therefore a convenientway of tagging it, unaffected by changes of depth, lati-tude, or shear. Second, potential vorticity distributionsstrongly constrain the large-scale circulation throughthe “invertibility principle.” The invertibility principlestates that if the total mass under each isentropic sur-face is specified, then knowledge of the global distribu-tion of potential vorticity on each isentropic surfacetogether with boundary conditions is sufficient to de-duce all other dynamical fields, such as currents, strati-fication, geopotential heights, and so forth. Third, po-tential vorticity is generated and destroyed through dia-batic and frictional processes mediated by velocityshears and buoyancy gradients. Hence, the potentialvorticity budget provides a natural framework to studythe interaction of air–sea fluxes and lateral fronts insetting the stratification of a fluid.

Heating and cooling generate and destroy potentialvorticity at the ocean surface and modify the stratifica-tion. Haine and Marshall (1998) show that the destruc-tion of potential vorticity during cooling events can beassociated with residual vertical stratification in thepresence of lateral density gradients. Hence, potentialvorticity is a more natural variable than stratification todescribe the dynamics of the mixed layer. Lapeyre et al.(2006) show that restratification at an outcroppingocean front can also be conveniently described in termsof potential vorticity. They consider an idealized frontwith no-flux and free-slip surface boundary conditionsto minimize nonconservative processes. Under suchconditions, the potential vorticity of the fluid is con-served and restratification occurs through a rearrange-ment of potential vorticity through surface frontogenesis:a frontogenetic strain field drives a thermally directsecondary circulation that increases the stratificationeven in a fluid with a spatially homogeneous PV field(Hoskins and Bretherton 1972).

In this paper, it is shown that in the real ocean, fric-tion can also modify the potential vorticity of the mixedlayer and hence modify the stratification of the upperocean. The frictional generation–destruction of poten-tial vorticity has two contributions: PV change inducedby frictional forces with curl, which induce Ekman ver-tical stretching of isopycnals, and baroclinic generationdue to horizontal Ekman flows advecting light fluidover dense, or vice versa, at lateral buoyancy gradients.

The baroclinic generation has not been given much at-tention in the oceanographic literature, but it is knownto play an important role at atmospheric fronts (Cooperet al. 1992; Davis et al. 1993; Adamson et al. 2006).Using simple scaling arguments and numerical experi-ments, it is shown that the modification of the stratifi-cation by these potential vorticity sources–sinks is asstrong or exceeds that associated with frontogenesis.Furthermore, it is emphasized that restratification–destratification by frontogenesis and friction is most ef-ficient in the surface mixed layer with a weaker effect inthe main thermocline.

2. Governing equations for changes in the mixedlayer stratification

The average stratification over a rectangular volumeof widths Lx, Ly in the x and y directions and boundedby the vertical levels zt and zb is

N2 �1H �

zb

zt

�z�b� dz ��b�|z�zt

� �b�|z�zb

H, �1�

where b��g�/�o is the buoyancy, � � � (LxLy)�1Lx/2�Lx/2

Ly/2�Ly/2 dx dy denotes the lateral average, and H � zt �

zb. The buoyancy equation

�tb uh � �b w�zb � D �2�

(where term D encompasses all diabatic processes andsubscript h signifies a horizontal velocity vector) can belaterally averaged to yield an equation governing therate of change of the average stratification:

�tN2 � �

1H �

uh � �b��z�zb

z�zt �1H �

w�zb�|z�zb

z�zt�

1H �

D�|z�zb

z�zt . �3�

This equation states that the average stratification of awater column can be modified through diabatic irre-versible processes, differential horizontal advection ofbuoyancy, or vertical advection of buoyancy.

As described in Lapeyre et al. (2006), an alternativeequation for the rate of change of N2 can be derivedfrom the potential vorticity equation

�tq � �� J, �4�

where q � � • (�ab) is the Ertel PV, �a � fz � � uis the absolute vorticity, and J is the PV flux. The PVflux

J � uq �b � F � �aD �5�

has an advective component uq and a nonadvectivecomponent that arises from diabatic processes and fromfrictional and nonconservative body forces F (Marshall

2502 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 38

and Nurser 1992). Calculating the volume average of(4) over the same rectangular volume used in (3), uti-lizing Gauss’s theorem, and assuming that the horizon-

tal flow at the lateral boundaries of the volume goes tozero or is periodic yields the alternative equation forthe rate of change of N2

�tN2 � �

1fH ��t��b�|z�zb

z�zt �wq�|z�zb

z�zt �JzF�|z�zb

z�zt �JzD�|z�zb

z�zt �, �6�FRONT ADV FRIC DIA

where

JzF � ��hb � Fh� � z �7�

and

JzD � � � f ��D �8�

are the vertical components of the frictional and dia-batic PV fluxes, respectively, and � �x� � �yu is thevertical vorticity.

Equation (6) highlights the various phenomena thatcan result in restratification (or destratification) of theocean: frontogenesis or frontolysis (FRONT), advec-tion of PV (ADV), friction (FRIC), and diabatic pro-cesses (DIA). If the vertical levels zt and zb do notintersect surface and benthic boundary layers, then ad-vection of PV through the bounding surfaces dominatesover the other three terms: diabatic and frictional ef-fects are weak away from boundaries, while FRONTscales like ADV times a Rossby number. In the oceaninterior mean and eddy motions are close to geo-strophic balance, Rossby numbers are small andFRONT K ADV. The relative strengths of FRIC andDIA were explored in Thomas (2005), who showed thatat wind-forced ocean fronts, frictional PV change canbe at least as strong as diabatic creation–destruction ofPV–stratification for typical air–sea buoyancy and mo-mentum fluxes and cross-front density gradients.

The effect of frontogenesis/frontolysis on the modi-fication of the stratification can be isolated by consid-ering an adiabatic, frictionless flow, where w � 0 at z �zt, zb. Comparing (3) and (6) in this limit yields

FRONT � f�uh � �b�|z�zb

z�zt . �9�

This result highlights the importance of differentialhorizontal advection in re–de-stratification by frontogen-esis–frontolysis. Indeed, in the classic work of Hoskinsand Bretherton (1972), where the connection betweenfrontogenesis and restratification was established, itwas demonstrated that frontogenetic confluent (fron-tolytic diffluent) flow will always drive a thermally di-rect (indirect) ageostrophic secondary circulationwhose horizontal velocity tends to flatten (steepen)isopycnals.

At the ocean top and bottom boundaries all fourterms in the right-hand side of Eq. (6) participate inrestratification (or destratification). The relation be-tween surface frontogenesis and restratification is in-vestigated in Lapeyre et al. (2006), who use numericalsimulations to diagnose how frontogenetic convergentflows generate correlations between buoyancy and ver-tical vorticity, which through the term FRONT in-creases the stratification. It is fairly obvious to see howadvection of PV and diabatic processes can result inchanges of PV (e.g., upwelling of high PV from thestratified pycnocline and surface heating will both re-sult in restratification of the mixed layer), but it is notimmediately transparent why friction should change theaverage stratification. Indeed, when comparing Eqs. (3)and (6), the apparent lack of influence of friction on thestratification in (3) led Lapeyre et al. (2006) to concludethat while friction might change the PV, its effect is felton the vertical vorticity but does not directly contributeto restratification. It will be demonstrated in the nextsection that this conclusion is not generally true andthat surface friction can contribute significantly to bothrestratification and destratification of the mixed layer.

3. Modification of stratification by friction

A frictional force in the surface boundary layer willinduce an Ekman flow that can modify the stratifica-tion. While frictional forces appear in the formulationbased on the PV budget in (6), they do not appearexplicitly in the more traditional budget in (3). To il-lustrate how friction enters in (3), the horizontal veloc-ity can be decomposed into two parts: uh � unf ue,that is, one associated with all nonfrictional processessuch as frontogenesis (unf) and the other with Ekmanflow. The Ekman flow is related to the frictional forceas ue � Fh � z/f. Substituting this expression into (3), itis found that the contribution to the horizontal differ-ential advection by the Ekman flow is proportional tothe frictional PV flux

�1H�

ue � �b�|z�zb

z�zt � �1

fH�Fh � z � �b�|z�zb

z�zt

� �1

fH�Jz

F�|z�zb

z�zt . �10�


Owing to Ekman advection of buoyancy, friction onceagain explicitly appears in the equation for the stratifi-cation in terms of the frictional PV flux (7).

The manner by which differential advection of buoy-ancy by Ekman flow, and equivalently vertical fric-tional PV fluxes, modify the stratification of the surfacemixed layer is illustrated in Fig. 1. Friction can eitherinput or extract PV from the fluid, depending on theorientation of the frictional force and the lateral buoy-ancy gradient. For wind-forced flows, the frictionalforce at the sea surface is dominantly in the direction ofthe wind stress. Down front winds (i.e., oriented in thedirection of the baroclinic shear) drive vertical fric-tional PV fluxes that extract PV from the ocean. Forthese winds, Ekman flow advects denser water overlight, and convection ensues that mixes the stratifica-tion and reduces the PV (Thomas 2005). Friction injectsPV into the fluid when a baroclinic current is forced byup front winds because the Ekman flow advects lighterwater over dense and restratifies the fluid.

Even without wind forcing, frictional spindown willmodify the stratification of baroclinic geostrophic cur-rents. In this case, the turbulent stress at the surfacemust be zero, that is, (�x, �y) � 0 at z � 0; however, thegeostrophic shear and the stress associated with thatshear, �o� (�zug, �z�g) (� is an eddy viscosity and ug isthe geostrophic flow), is nonzero. Because the geo-strophic flow does not satisfy the no-stress boundarycondition, an Ekman flow must be induced to cancelthe geostrophic shear at the surface:

�v��zue|z�0, �z�e|z�0� � ��v��zug|z�0, �z�g|z�0�

� ��xg,�y

g� �11�

(Garrett and Loder 1981; Thompson 2000). As de-scribed in Thomas and Rhines (2002), this Ekman flowthat acts to spin down the geostrophic current can bethought as being driven by an effective “geostrophicstress” (�g

x, �gy) that is directed opposite to the geo-

strophic shear. The Ekman transport is therefore al-ways directed down the buoyancy gradient; hence, ad-vection of buoyancy by the Ekman flow tends to stratifythe fluid (Fig. 1). As shown in (10), the rate at whichthis restratification occurs is proportional to the fric-tional PV flux. An expression for the frictional PV fluxinduced during spindown can be derived using the Ek-man spiral solution, which satisfies boundary condition(11), and is given by �JF

z �|z�0 � �1/2��e|�hb| 2�|z�0,where �e��2�/f is the depth of the Ekman layer. Thefrictional PV flux during spindown is a negative definitequantity that results in the increase of both the PV andstratification of the fluid. Notice that if there is no lat-eral shear in the geostrophic flow, the geostrophicstress and frictional force are horizontally uniform andhence do not modify the vertical vorticity (the same istrue if such a geostrophic flow were forced by a spa-tially uniform wind stress). This demonstrates that fric-tion can change the PV of the fluid by modifying thestratification rather than the vertical vorticity.

4. Modification of stratification by frontogenesis

Whether frictional restratification and destratifica-tion of the surface Ekman layer is of leading-order im-portance in upper ocean dynamics depends on howlarge the contribution of friction is compared to surfacefrontogenesis, diabatic processes, and vertical advec-

FIG. 1. Schematic illustrating frictional re–de-stratification at a baroclinic current in theupper ocean. (left) Up-front winds blowing against a baroclinic geostrophic flow ug will drivean Ekman flow �e that flattens isopycnals (gray) and hence restratifies the fluid via differentialhorizontal advection of buoyancy. (middle) When the wind is oriented down front, Ekmanadvection of buoyancy destabilizes the water column, driving convective mixing and a reduc-tion of the stratification. (right) With no wind forcing, the frictional spindown of a barocliniccurrent arising from the mismatch of the geostrophic shear and the zero stress boundarycondition induces a restratifying Ekman flow.


tion in (6). In this section, classic results on frontoge-netic restratification are reviewed. A comparison of thevarious processes on the upper ocean stratification isthe focus of the next section.

Equation (9) shows that the rate of change of strati-fication due to frontogenesis–frontolysis is controlledby the differential horizontal advection at the top andbottom of the front. A scaling for this ageostrophicsecondary circulation (ASC) is obtained from the two-dimensional quasigeostrophic approximate version ofthe Eliassen–Sawyer equation for the ageostrophicstreamfunction �:

f 2�zz� N2�yy� � �2Q2, �12�

where (�ag, w) � (�z�, � �y�) and

Q2 � ��ug

�y� �hb �13�

is a component of the “Q vector” introduced byHoskins et al. (1978), which drives a thermally directASC (�ag, w) in frontogenetic conditions. Taking thevolume integral of (12) over the rectangular controlvolume described in section 2 with zt � 0 and zb � �H,and using Gauss’s theorem, yields

LxLy��z��|zb��Hzt�0 �

�Lx 2

Lx 2 ��H

0

�y*�|y��Ly 2

y�Ly 2 � �LxLyH2

f 2 Q2 , where y* � � fN�y. �14�

Let us assume that the region of enhanced frontalgradients in the fluid is finite, so that the geostrophicforcing Q2 is confined to a region in y of width Lf. Ifthe meridional edges of the control volume are faraway from the frontal region (i.e., Ly k Lf), then�y*�|y�Ly/2

y��Ly/2 � 0 and the second term on the left-handside of (14) can be neglected. If we also choose thebottom surface of the control volume to coincide with adepth H representative of the vertical length scale ofthe frontogenesis-driven ASC, then (14) can be used tocalculate the difference in mean horizontal velocity be-tween zt � 0 and zb � �H,

�front � ��z��|zb��Hzt�0 � �H

2

f 2 Q2

� H2f��yug�z�g � �y�g�zug�. �15�

The change in the geostrophic flow over the depth Hscales as H |�zug| � �ug, so an appropriate scaling forthe drop in velocity (15) is

�front � Roug �16�

where the lateral shear �yug and confluence ��y�g ofthe geostrophic flow averaged over the control volumehave been assumed to scale as f times a bulk Rossbynumber, Ro. Notice that Ro characterizes the strainflow that drives frontogenesis and not the large Rossbynumber motions that develop as a result of frontogen-esis.

5. Scalings for the relative contributions of frictionand frontogenesis to modifications ofstratification

Expressions (9) and (10) show that the relative con-tributions of friction and frontogenesis to changes in

stratification, governed by (6), are determined by therelative drop in horizontal velocities over the front ver-tical scale associated with frictional processes and front-ogenesis, respectively, that is,

FRICFRONT

� � ��fric

�front. �17�

In this section, scalings for the horizontal flows associ-ated with friction ��fric will be derived and used toestimate this ratio.

Frictionally driven flows are largest at the surfaceand decay to zero over the Ekman layer depth. A scal-ing for the surface horizontal flow associated with spin-down ��sd and wind forcing ��wind is given by Ekmansolutions:

�fric ��g

�of�e

�w

�of�e, �18�

�sd �wind

where �g and �w are representative values for the geo-strophic stress and the component of the wind stressparallel or antiparallel to the frontal jet. A scaling forthe drop in velocity associated with frontogenesis isgiven by (16), that is, ��front � Ro�ug.

The scalings (18) and (16) can now be used to esti-mate the ratio in (17). Considering the frictional flowsdriven by spindown and wind stress separately, then theratio for spindown is

�sd �vsd

vfront�

�g

�of�eRoug. �19�

In spindown problems, the stress at the surface is pro-portional to the vertical shear through an eddy viscosityas in (11), and the Ekman layer thickness is given by


�e��2�/f. The Ekman flow driven by the geostrophicshear therefore scales as �g/�of�e � (�e /H)�ug; hence

�sd � Ek12Ro�1, �20�

where Ek� �2e /H2 is the Ekman number. When friction

is very weak, like in the simulations of Lapeyre et al.(2006) (they use � � 10�5 m2 s�1), �e K H and front-ogenesis dominates the restratification process.

When a stress is applied at the ocean surface, theEkman layer thickness is given by �e � 0.4u*/f, whereu* � ��w/�o is the friction velocity and � is the stressat the surface (Wimbush and Munk 1970). These rela-tionships can be used to calculate the relative contribu-tion to mixed layer restratification by frontogenesis andfrictionally induced flows,

�wind ��wind

�front�

u*wug

Ro�1, �21�

where u*w ��w/�o. The friction velocity for a typicalwind stress of 0.1 N m�2 is u*w � 0.01 m s�1, so thateven in a relatively strong baroclinic current with �ug�0.1 m s�1 and Ro � 0.1, �wind � 1. This scaling argu-ment suggests that winds, in typical oceanic conditions,are as important as frontogenesis for modifying thestratification in the mixed layer.

In the next section, numerical simulations designedto pit restratification by frontogenesis against frictionaldestruction and creation of PV, which were used to testthe above scaling arguments, are described.

6. Numerical experiments

a. Configuration of experiments

The configuration of the experiments is designed toisolate the effects of frontogenesis and friction and de-emphasize the role of vertical advection in the modifi-cation of mixed layer stratification. To accomplish this,the model domain is confined to a layer of thicknessD � 100 m, and the vertical velocity and advective fluxof PV, ADV in (6), are both zero at the bottom. Theinitial density field used in the simulations is character-ized by two isolated fronts with a spatially uniformbackground stratification (Fig. 2). An analytical formfor the initial condition of the density that incorporatesthese features is

��y, z� � �yY�y� ��o

gNo

2z, �22�

where

FIG. 2. Initial condition for the numerical simulations. Density is contoured in gray and thevelocity vectors of the flow at the surface are shown in black.


Y�y� � �0.5�1 � tanh� y

Lf� tanh�y � Ly 2

Lf�� 0 y Ly 2,

0.5�tanh�y � Ly 2Lf

� � tanh�y � Ly

Lf� � 1� Ly 2 y Ly,

�23�

and N2o � 5 � 10�6 s�2 is the background stratification.

The frontal width, Lf � 8 km, and the north–south andeast–west extent of the domain, Ly � 200 km and Lx �100 km, are the same for all the experiments. Periodicboundary conditions are imposed on all variables onthe northern, southern, eastern, and western bound-aries of the domain. All experiments, excluding one,have a horizontal resolution, �x � �y, of 1 km. Oneexperiment was performed with an enhanced horizon-tal resolution of �x � �y � 500 m. The vertical reso-lution of the model, �z, is uniform and equal to 5.3 m.The fluid is on an f plane with the Coriolis parameterset to f � 1 � 10�4 s�1; consequently, the density fieldis associated with a geostrophically balanced zonalflow. Given the Coriolis parameter, background strati-fication, and depth of the mixed layer used in the simu-lations, the mixed layer Rossby radius of deformation,NoD/f � 2.2 km, is resolved in the numerical experi-ments.

The experiments are forced by a combination ofstrain associated with a barotropic velocity field in-serted into the flow as an initial condition and by aspatially uniform wind stress, �w

x , oriented in the posi-tive x direction. The barotropic velocity field is given bythe streamfunction

� � �LxLy

8�2 cos�2�

Lxx� sin�4�

Ly�y �

Ly

2 ��, �24�

where the velocity is equal to �� z, and � is themaximum value of the confluence. At t � 0, the baro-tropic flow is superposed on the baroclinic geostrophicflow of the mixed layer fronts, resulting in the velocityfield shown in Fig. 2. At the fronts, confluence (difflu-ence) is centered at x � Lx/4 (x � 3Lx/4). The windstress is applied at the start of each simulation and keptat a constant value through the length of the experi-ment. The bottom stress follows a quadratic drag lawwith a drag coefficient of 1� 10�8, which was chosen tobe extremely small to minimize the Ekman flow in thebottom boundary layer.

Vertical mixing of momentum and tracers is attainedthrough the K-profile parameterization (KPP) mixingscheme of Large et al. (1994). To avert frontal collapse,biharmonic friction and diffusion are used, with mixingcoefficients equal to 6 � 107 m4 s�1 and 3.8 � 106

m4 s�1 for the lower- and higher-resolution runs, re-

spectively. Advection of tracers is accomplished usingthe recursive flux-corrected MPDATA advectionscheme of Smolarkiewicz and Margolin (1998) to sup-press under–overshooting of density that tends to occurin frontogenetic situations. A third-order upstream-biased advection scheme is implemented for momen-tum.

A total of 18 experiments were performed, each last-ing 10 inertial periods, a duration long enough for thedevelopment of frontogenesis. The key parameters ofthe problem (��y, �, and �w

x ) that were varied in eachexperiment are listed in Table 1. The confluence � usedin the experiments spans 0–0.25f and was chosen torepresent horizontal strain typical of mixed layer ed-dies. The range of cross-front density contrasts em-ployed in the runs corresponds to lateral buoyancy gra-dients of 0.66–3.0 � 10�7 s�2 at t � 0, which fall on thehigh end of frontal gradients in the ocean. The experi-ments are forced by weak to moderately strong windstresses of magnitude 0–0.25 N m�2. Given these pa-rameters, the scaling �wind ranges from 0.21 to 6.7 (ex-cluding the wind-only and frontogenesis-only runs).Therefore, these experiments cover both the frontoge-netically and frictionally dominant regimes of mixedlayer re/destratification.

b. Basic properties of the numerical solutions

To illustrate the basic properties of the numericalexperiments, the evolution of the density field in run 4is plotted in Figs. 3–5. Run 4 is chosen as a control casebecause the frictional and frontogenetic forcings usedin the experiment are predicted to have a comparableeffect on the mixed layer stratification, because usingthe scaling in (21) �wind � 0.67 � 1.

The evolution of the surface density field reveals thatfrontal intensification occurs in the experiments (Fig.3). Frontogenesis driven by the confluent barotropicflow dominates over frontolysis associated with the dif-fluent flow, resulting in a net strengthening of the zon-ally averaged cross-front density gradient. Wind forcingleads to a southward translation of both fronts, with thefront to the north experiencing a larger displacement.

To ascertain the effect that wind forcing has on themixed layer stratification, the zonally averaged densityfields for runs 3 (which was forced by strain only, notwinds) and 4 (which was forced by strain and a mod-


erate wind stress of 0.1 N m�2) are compared in Figs. 4,5. For the frontogenesis-only run, the evolution of thezonally averaged density at both fronts is similar and ischaracterized by an intensification of the horizontaldensity gradient and a flattening of isopycnals underthe action of the thermally direct ageostrophic circula-tion. This frontogenetically induced restratification isdramatically altered in the presence of wind forcing. Atthe front forced by down-front winds, in the center ofthe domain, frontogenetic restratification is over-whelmed by Ekman advection of denser water overlight, causing the isopycnals to be nearly vertical in theEkman layer. In contrast, at the front forced by up-front winds, Ekman advection augments frontogeneticrestratification, resulting in a significant enhancementof the stratification at the base of the Ekman layer.These figures provide a qualitative picture of the rela-tive effects of friction and frontogenesis in modifyingthe stratification. To quantify these effects, diagnosticsfor the evolution of the volume-averaged stratificationand PV were performed and are described in the nextsection.

c. Diagnostics for the change in mean stratification

As was described in section 2, changes in the volume-averaged stratification

N2 � N2 � N2|t�0, �25�

and PV (normalized by f )

q

f�

1fV �

V�q � q|t�0� dV, �26�

are caused by friction, frontogenesis, diabatic pro-cesses, and advection. To quantify the contribution ofeach of these processes in the modification of the meanstratification, the following quantities were calculated:the frictional change in PV due to vertical mixing ofmomentum,

FRIC� �1

f V �0

t �V

� � F� � �b dV dt, �27�

the frictional change in PV due to lateral mixing ofmomentum,1

FRICl �1

f V�0

t�V

� � Fl � �b dV dt, �28�

the change in stratification due to frontogenesis,

FRONTi�1

f V��A

��b� dA|z�0,t� �A

��b� dA|z�0,t�0��

A

��b� dA|z��D,t��A

��b� dA|z��D,t�0��,

�29�

(primes denote a deviation from the areal average) thechange in PV due to diabatic processes,2

DIAi �1

f V �0

t �V

�a � �D dV dt, �30�

and the change in PV due to advection,

ADVi � �1

f V �0

t �V

u � �q dV dt. �31�

1 The lateral mixing of momentum, Fl, is associated with bihar-monic friction, while vertical mixing of momentum comprises theremainder of the frictional force, i.e., F� � F � Fl and was diag-nosed from the KPP mixing scheme.

2 The term encompassing diabatic processes, D, includes bothlateral mixing of buoyancy by biharmonic diffusion and verticalmixing as diagnosed from the KPP mixing scheme. It does notinclude implicit mixing associated with the advection scheme,which is accounted for in (31).

TABLE 1. Experimental parameters for numerical simulations,i.e., the cross-front density contrast, the zonal wind stress, and thestrength of the maximum confluence normalized by the Coriolisparameter. In the scaling of �wind, Ro � �/f. For all of the runs, ahorizontal resolution of �x� �y� 1 km is used, except for run 20in which the resolution is �x � �y � 0.5 km.

Run �y� (kg m�3) �xw(N m�2) �/f �wind � u*w�Ro�ug

3 0.25 0 0.1 04 0.25 0.1 0.1 0.675 0.25 0.1 0.01 6.76 0.25 0.1 0.02 3.37 0.50 0.1 0.05 0.678 0.25 0.1 0 �9 0.25 0.01 0.1 0.21

10 0.25 0.05 0.1 0.4711 0.25 0.028 0.1 0.3512 0.476 0.1 0.1 0.3513 0.476 0.037 0.1 0.2114 0.25 0.025 0.02 1.6515 0.25 0.1 0.19 0.3516 0.166 0.1 0.1 1.0017 0.25 0.0 0.02 018 0.25 0.25 0.1 1.0519 0.25 0.1 0.03 220 0.25 0.1 0.1 0.67


FIG. 3. Evolution of density at z � 0 for run 4. Contours of density with a contour intervalof 0.08 kg m�3 are shown for t� (a) 0.1, (b) 2.6, (c) 5.1, and (d) 7.6 inertial periods. The regionshaded in gray denotes domain 1 used in the PV budget. Domain 2 encompasses the rest ofthe model domain.


For convenience (26), (27), (28), (30), and (31) havebeen normalized by f so as to be expressed in units ofstratification, so that

N2� FRONTi

q

f

� FRONTi FRIC� FRICl DIAi ADVi.

�32�

Note that in calculating the terms in the PV budget(27), (28), (30), and (31), Gauss’s theorem has not beenapplied, but instead a volume integral of the PV equa-tion

�q

�t� �u � �q �a � �D � � F � �b �33�

has been used because this is easier to evaluate numeri-cally. In the diagnostic calculations, two control vol-umes were used, one encompassing the front in thecenter of the domain (referred to as domain 1) and theother occupying the rest of the domain (domain 2).Both volumes extend to the bottom, remain fixed intime, and are indicated in Fig. 3.

Time series of changes in the mean stratification andPV and terms (27)–(31) are plotted in Fig. 6 for run 4.In domain 1, the mean PV initially decreases, as is ex-

pected for a front forced by down-front winds. How-ever, because of frontogenesis, the decrease in meanstratification is slower than that of PV (Fig. 6a), that is,�N

2� �q/f � FRONTi � 0. The bulk of the decrease

in the volume-averaged PV is due to vertical friction,FRIC�, with diabatic processes, DIAi, resulting in PVreduction as well (Fig. 6c). In contrast, lateral frictionand advection of PV contribute to increasing the meanPV in domain 1.

After several inertial periods, both FRIC� andFRONTi asymptote to nearly a constant value, indicat-ing that not only is frontogenesis arrested but that someprocess limits erosion of PV by winds. The mechanismby which PV removal by down-front winds is sup-pressed will be described below, but first the strengthsof frontogenetic restratification versus frictional de-stratification will be compared. Averaged over the lasttwo inertial periods of the experiment, FRIC� contrib-utes to a reduction in �q/f by an amount 2.3� 10�6 s�2,more than double the 9.9 � 10�7 s�2 increase in strati-fication by FRONTi, suggesting that frictional PVchange is not only comparable in magnitude to fronto-genetic restratification but actually the dominant of thetwo processes.

Frictional reduction of PV is eventually suppressedbecause of a partial cancellation of the imposed windstress by the geostrophic stress [e.g., (11)], which is en-

FIG. 4. Zonally averaged density field near the front in the center of the domain for run 3(thick gray) and run 4 (black) at times (a) 0.1, (b) 2.6, (c) 5.1, and (d) 7.6 inertial periods. Thecontour interval is 0.05 kg m�3.


FIG. 5. Same as Fig. 4, but for the front on the edges of the domain.

FIG. 6. Temporal change in the mean PV and stratification; and the processes that give riseto this change as diagnosed from run 4. (a), (b) The change in mean stratification, �N2 (thinblack) and PV, �q/f (dashed) are plotted along with the frontogenetic contribution to re-stratification, FRONTi (thick gray), for domains 1 and 2, respectively. (c), (d) The change inPV due to vertical friction, FRIC� (dashed), lateral friction, FRICl (thick gray), diabaticprocesses, DIAi (dotted), and advection, ADVi (thin black), are plotted for domains 1 and 2,respectively.


hanced through frontogenesis (Fig. 7). Comparing timeseries of the maximum geostrophic stress in runs 4 and8 (which was forced by winds only), reveals that front-ogenesis, through intensifying the geostrophic shear,strengthens �g

x to such a degree that it becomes compa-rable in magnitude to the wind stress (Fig. 7b). At thefront in domain 1, �g

x opposes the wind stress, whilenear the front in domain 2 it reinforces the wind forc-ing. This behavior reveals a coupling between fronto-genesis and frictional processes that leads to an aug-mentation of frictional injection of PV in domain 2 anda quelling of PV removal by down front winds.

It might be argued that because frontogenesis is ar-rested by horizontal mixing in the simulations, fronto-genetic restratification is repressed, and hence its effectis underestimated (although our simulations are notunique in this respect because all numerical simulationsof ocean fronts have some form of explicit or implicitlateral mixing that arrests frontogenesis to some de-gree). To address this issue, FRIC� and FRONTi fromrun 4 are compared to the amount of restratificationthat would occur if frontogenesis could continue to thepoint of the formation of a discontinuity in density. Thederivation for this upper bound on frontogenetic re-stratification is given in appendix A, Eq. (A1). Giventhe cross front buoyancy contrast �b � 2.4 � 10�3

m s�2, depth of the layer H � 100 m, width of domain1 Ly � 60 km, and estimate for the displacement ofthe front from its initial location, Y � 3 km (based onsolutions from run 3; see Fig. 4), the maximum restrati-fication by frontogenesis for run 4 in domain 1 is�N 2

front � 2.4 � 10�6 s�2. This restratification is be-tween 2 and 3 times greater than FRONTi and approxi-mately equal to the destratification by FRIC� at the endof experiment 4. Hence, even if frontogenesis couldcontinue to frontal collapse, the resulting stratificationwould be at most comparable to that removed by fric-tion, emphasizing the importance of PV destruction bydown-front winds in the modification of the mixed layerstratification.

In domain 2, where the winds are up front, frictionalPV creation completely dominates over frontogeneticrestratification (Figs. 6b,d). Both mean stratificationand PV increase throughout the duration of the experi-ment and are almost indistinguishable from one an-other, evidencing the negligible effect of frontogenesison the mean stratification. In this case, nearly all of theincrease in mean PV and stratification is associatedwith vertical friction, while diabatic processes and ad-vection lead to a slight reduction in PV.

The difference between FRIC� in the two domainsattests to an asymmetry between the efficiency of PV

FIG. 7. Surface integral of the vertical frictional PV flux for run 8 (blue) and run 4 (red), fordomains (a) 1 and (c) 2. (b) Time series of the maximum absolute value of the zonal com-ponent of the geostrophic stress � g

x for run 8 (blue) and run 4 (red). (d) The density field(magenta contours of interval 0.1 kg m�3) and � g

x (shades) for run 4 at t � 6.5 inertial period.


Fig 7 live 4/C

injection and removal by up- and down-front winds.3

As shown in Fig. 8, most of this asymmetry is due to thefact that as up front winds restratify the fluid, the tur-bulent stress is confined to a depth shallower than theEkman layer. In contrast, stress induced by down frontwinds is felt through the entire weakly stratified Ekmandepth �e � 0.4u*w /f. Reducing the depth over which thestress is distributed enhances the frictional force, Ek-man flow, differential horizontal advection of buoy-ancy, and hence frictional change in stratification. Inaddition, the suppression (augmentation) of frictionalremoval (injection) of PV by the geostrophic stress de-scribed above contributes to the asymmetric responseto up and down front wind forcing.

d. Sensitivity to forcing and flow parameters

In the previous section it has been shown that forfronts forced by the relatively strong confluence (� �0.1f ) and moderate winds (�w

x � 0.1 N m�2) of run 4,friction played a greater role than frontogenesis in thechange of mean stratification in the mixed layer. Toassess the sensitivity of this result to the forcing andflow parameters, a suite of experiments was performedin which �, �w

w, and �� were varied (e.g., Table 1). Themetric used to determine the relative contributions offriction and frontogenesis to the re–de-stratification ofthe mixed layer is the same as has been used in theprevious section, that is, the ratio FRIC� to FRONTi

averaged over the last two inertial periods of each ex-periment. The absolute value of this quantity is plottedin Fig. 9 against the scaling �wind.4 As in run 4, for all of

3 Notice that this asymmetry is even more pronounced thanwhat can be inferred from Figs. 6c,d. A fair comparison betweenthe amount of PV injected and removed in domains 2 and 1 entailscalculating the relative contributions of FRIC� in each domain tothe volume-integrated PV of the whole model domain, which in-volves multiplying FRIC� by f times the volume of the respectivedomains. Consequently, the relative importance of frictional in-jection to removal of PV is greater than that seen in the figure bya factor of 2.3, i.e., the ratio of the volumes of domain 2 to 1.

4 Note that �wind and FRIC�/FRONTi averaged over the lasttwo inertial periods are not measures of the exact same quantity:one scales the relative rates of re/destratification while the otheris a measure of the relative total change in stratification. However,because the time dependence of FRIC� and FRONTi seen in theexperiments is not too different from being linear, FRIC�/FRONTi is a representative metric of the relative rates of strati-fication increase/decrease, and hence it can be compared to �wind.

FIG. 8. Comparison of the stratification, zonal friction force Fx and zonal stress �oz�D Fxdz

in domains 1 and 2 averaged between t � 2 � 3 inertial periods for run 4. (left) Square of thebuoyancy frequency, (middle) zonal frictional force, and (right) zonal stress averaged overfrontal regions where the magnitude of the zonally averaged N–S buoyancy gradient at thesurface exceeded 3.5 � 10�8 s�2, for domains 1 (solid) and 2 (dotted). The scaling for thedepth of the Ekman layer 0.4u*w/f (dashed line) is indicated in the right panel.


the experiments, frictional injection of PV by the up-front winds in domain 2 was stronger than frictionalremoval of PV in domain 1. Restratification by frictionranged from being 2 to �500 times greater than fron-togenic restratification in domain 2, while destratifica-tion by down-front winds exceeded FRONTi for themajority of the experiments, except for runs where�wind� 1 and the density contrast was large (i.e., runs 7,12, and 13). In both domains, the ratio of FRIC� toFRONTi generally increased with �wind, indicating thatthe scaling arguments presented in section 5 have someskill in predicting the dependence of the basic featuresof re–de-stratification at wind- and strain-forced frontson the key parameters involved. However, the scalingdid not predict the asymmetric response to up anddown front wind forcing. This is to be expected becausein deriving �wind, it was assumed that the wind stresswas distributed over the turbulent Ekman depth forboth down and up front winds, whereas, as illustrated inFig. 8, the stress is confined to a shallower depth for upfront winds as a consequence of the stratification.

The arrest of frontogenesis by horizontal mixing doesnot affect the general result that friction is at least as

important as frontogenesis in the modification of thestratification. In run 20, where the horizontal resolutionwas enhanced and the biharmonic diffusivity–viscositywas reduced, |FRIC� /FRONTi|� 1, as in the equivalentlower-resolution run. In addition, the ratio FRIC� /FRONTi| calculated using the upper bound on fronto-genetic restratification, (A1), and FRIC� from domain1 is never much smaller than one for any of the experi-ments, which suggests that the importance of friction inthe dynamics of the stratification of the upper ocean isa robust result.

7. Discussion

In the previous sections, the focus has been on quan-tifying changes in the vertically averaged stratificationof the surface mixed layer generated by tilting a baro-clinic buoyancy front via frictionally and frontogeneti-cally driven ageostrophic flows. In this section, it isshown that the changes in PV and stratification inducedby friction and frontogenesis penetrate to differentdepths into the upper ocean.

Frictional flows driven by spindown and up-front

FIG. 9. The absolute value ( y axis) of the ratio of term FRIC� to FRONTi averaged over thelast two inertial periods of the experiment vs �wind � u*w/Ro�ug for all of the runs forced byboth winds and confluence. Asterisks and circles correspond to values taken from domains 2and 1, respectively. Results from run 20, where the resolution was doubled, are given by theupward (domain 2) and downward (domain 1) triangles. The ratio of |FRIC�| in domain 1(averaged over the last two inertial periods) to the scaling for the upper bound on therestratification by frontogenesis �N2

front is denoted by the xs.


winds restratify the water column within the Ekmanboundary layer depth. A layer of enhanced stratifica-tion confined to the surface Ekman layer is well visiblein all panels of Fig. 5. Down-front winds, instead, de-stratify the water column, and their destabilizing effectis felt all the way down to the mixed layer base asshown in Fig. 4 and can penetrate into the pycnoclinethrough the formation of intrathermocline eddies(Thomas 2008).

Restratification by frontogenesis occurs over thedepth H of the frontogenesis-driven ASC. In appendixB, it is shown that if the Q vector is confined to a regionof width L , chosen to represent the frontal width, andthe background vertical stratification No is constant,then H� L f /No is a good measure of the vertical extentof the restratifying, frontogenesis-driven ASC. Whenthe stratification is not constant but has a mixed layer ofthickness hml with stratification Nml above a thermo-cline with stratification Ntc, the penetration depth andstrength of frontogenesis will be affected. As shown byBoccaletti et al. (2007), the presence of a mixed layer ina baroclinic current allows for the growth of submeso-scale, shallow baroclinic instabilities whose horizontalflow can be frontogenetic. The geostrophic forcing as-sociated with these mixed layer instabilities (MLI) islimited to the depths of the weakly stratified surfacelayer (i.e., h � hml, which for the Burger number Bu �1 implies that the horizontal length scale of the Q vec-tor scales with the mixed layer Rossby radius: L ml �Nmlhml /f. For such situations, solutions to the Eliassen–Sawyer equation in (12) are nearly completely confinedto the mixed layer because there is very little penetra-tion of the ASC into the thermocline because of thelarge difference in stratification. In this case, a goodmeasure for H is the mixed layer depth H � hml. Forthe case of deep fronts that penetrate into the ther-mocline and that are forced by mesoscale eddies andmeanders, the vertical scale of the frontogenesis-drivenASC scales with the thermocline depth given by H �Ldf/Ntc, where Ld is the deformation radius associatedwith the mesoscale straining flow.

The strength of the frontogenesis-driven restratifyingageostrophic flow is also affected by the presence of amixed layer for two reasons: 1) the magnitude and hori-zontal scale of the Q vector differ for MLI and meso-scale eddies and 2) the stratification affects the magni-tude of the overturning streamfunction. A scaling forthe relative strengths of the ageostrophic flows drivenby frontogenesis for MLI and mesoscale eddies can beconstructed from (B1), that is,

�zMLI

�zmeso �

QoMLI

Qomeso

NtcLml

NmlLd. �34�

The Rossby numbers characteristic of submesoscaleMLI are larger than those associated with mesoscaleeddies, suggesting that submesoscale straining domi-nates over mesoscale confluence. Cross-front densitygradients typically intensify moving from the ther-mocline to the surface. Both of these flow characteris-tics will conspire to make QMLI

o /Qmesoo k 1. However,

the deformation radius of mesoscale eddies is muchlarger than the mixed layer Rossby radius of deforma-tion so that for typical conditions, (NtcLml)/(NmlLd) �hml/htc K 1. This scaling argument suggests that foroceanic flows where both MLI and deep mesoscalestrain is present, frontogenesis-driven ageostrophicflows associated with both mechanisms will be of simi-lar strength. Having said this, frontogenesis by MLI willbe faster than that driven by mesoscale eddies becausethe time scale of frontogenesis goes as the inverse of thestrain rate, and hence as Ro�1f�1. However it is likelythat mesoscale frontogenesis remains important onlong time scales for restratification below the mixedlayer base. Numerical simulations that resolve both me-soscale and submesoscale instabilities are needed tosettle the issue.

Finally, the ageostrophic circulations associated withMLIs are very fast and redistribute material propertieswithin the whole mixed layer on the order of a day. Inparticular, Boccaletti et al. (2007) find that the en-hanced stratification resulting from surface friction inthe Ekman layer is rapidly communicated to the rest ofthe mixed layer. Hence, in the real ocean any change ofPV and stratification in the Ekman layer is likely to befelt throughout the whole mixed layer as a result ofthree-dimensional instabilities not considered in thispaper.

8. Conclusions

Traditional models of turbulent boundary layers atthe ocean surface assume that the buoyancy and mo-mentum budgets are essentially one-dimensional; thatis, stratification and shears are set through turbulentdownward transport of fluxes and stresses applied atthe air–sea interface. This description is appropriate aslong as buoyancy at the surface ocean is horizontallyhomogeneous. In practice, however, the ocean bound-ary layers are populated with lateral fronts that pro-foundly affect the vertical mixing of tracers and mo-mentum. Lapeyre et al. (2006) show that frontogeneticslumping of density gradients effectively restratifies theocean surface even without any air–sea flux. In thispaper, it was shown that frictional forces acting onbaroclinic currents are also effective at increasing–re-ducing the stratification beyond what a one-dimension-


al budget would predict. Friction modifies the stratifica-tion of a rotating fluid through two different processes:stretching and squeezing of the water column inducedby horizontal variations of the frictional forces and Ek-man advection of the lateral stratification, the latter ofwhich is illustrated in Fig. 1. Simple scaling argumentsand numerical simulations suggest that for typicalocean conditions, Ekman advection is as important ormore important as frontogenesis at modifying thestratification of the mixed layer.

Numerical simulations support the simple scalinganalysis but further reveal that there is asymmetry inthe changes of stratification when the wind forcing isoriented up or down front (i.e., directed against or withthe frontal shear). Up front wind restratification is typi-cally larger than down wind front destratification forequal wind stress. Two effects contribute to the asym-metry. First, stratification generated by up front windsleads to a reduction in vertical penetration depth of theturbulent stress and hence an enhancement of the fric-tional force and stratification change relative to thatassociated with the down front wind forcing. Second,the frictional destratification is reduced over restratifi-cation because the inherent spindown of a baroclinicgeostrophic flow always tends to increase the PV of thefluid, and hence counteracts the destratifying tendencyof down front winds. This effect is accentuated by front-ogenesis because the surface frictional stress ascribableto geostrophic shear that drives spindown is amplifiedduring frontal intensification.

A question arises as to whether the effect of frictionis overemphasized by ignoring the compensation be-tween the restratification driven by up front winds andthe destratification resulting from down front winds.Satellite observations show that the orientation offronts at the ocean surface is nearly isotropic at smallscales, suggesting that up wind and down wind condi-tions (as well as fronts with intermediate orientation)might occur roughly with the same frequency (e.g.,Castelao et al. 2006). The compensation between thesurface PV fluxes over many fronts, however, does notimply that the oceanic response to these fluxes averagesout as well. For example, the ocean response to thediurnal cycle of heating and cooling is very asymmetric,even though the daytime generation of PV is largelycompensated by PV destruction at night: daily heatingrestratifies mostly in a thin surface boundary layer,while nighttime cooling penetrates down to the mixedlayer base. Similarly, the generation of high PV by upfront winds concentrates stratification within the sur-face Ekman layer, while destruction of PV by downfront winds triggers convection and reduces stratifica-

tion all the way to the mixed layer base. Thomas (2007)shows that the low PV anomalies generated throughconvection are eventually subducted into the ocean in-terior as intrathermocline eddies, while the high PVanomalies remain at the surface. Hence, winds generatepositive PV anomalies close to the ocean surface andnegative PV anomalies in the thermocline and the ef-fects of up front and down front winds do not cancel outdespite the compensation in surface PV fluxes. This PVredistribution is likely to occur in the real ocean, be-cause the scaling arguments and numerical simulationsdescribed here suggest that wind-driven PV forcing is aleading-order process in setting the PV of the upperocean. A detailed study of the oceanic response to avariable and compensating PV flux is left for futurework.

The importance of frictional forces in setting thestratification of the global ocean should not come as asurprise. Frictional forces and diabatic processes areultimately the only processes that can generate and de-stroy stratification. Frontogenesis and advection canonly redistribute the stratification. Hence, the steady-state stratification of the upper ocean is maintained bya balance of frictional and diabatic fluxes. Frontogen-esis and advection regulate how the stratification gen-erated/destroyed at the surface is redistributed in thevertical. Only during transient events can any one pro-cess dominate over the others. The surprise is that lat-eral fronts can substantially modify the surface fluxesthat enter in the ocean. This modulation occurs at fron-tal scales between tens of kilometers and hundreds ofmeters, which are typically subgrid in ocean modelsused for climate studies. It is therefore an open ques-tion: What is the role of frontogenetic and frictionalprocesses in maintaining the observed ocean meanstate?

The scaling arguments described in this paper focusprimarily on two-dimensional flows. In three dimen-sions, submesoscale instabilities can develop alongmixed layer fronts that act to slump outcropping fronts(Boccaletti et al. 2007). The numerical simulations de-scribed in this paper were deliberately configured tosuppress frontal instabilities by imposing large steadystrain fields (Bishop 1993; Spall 1997) and strong windsaligned with the fronts (Thomas 2005). Frontal insta-bilities, however, developed if the strain field or thewinds were substantially reduced (not shown). Boccal-etti et al. (2007) and Capet et al. (2008) show that sub-mesoscale instabilities develop when strain and windsare not steady. These instabilities might play an impor-tant role in setting the mixed layer stratification, be-cause they are associated with overturning circulations


that restratify the surface mixed layer and subduct sur-face waters into the ocean interior. The details of theexchange of waters between the mixed layer and theinterior are not well understood. For example, Lapeyreet al. (2006) argue that frontogenetic ASC associatedwith the mesoscale straining field can also be importantand compete with ASC generated by instabilities withinthe mixed layer. We plan to investigate the dynamics ofthese three-dimensional circulations and their effect onthe mean stratification of the ocean in future work.

A final comment pertains to the implication of fric-tional re–de-stratification for numerical models. Thescaling arguments presented above suggest that to ac-curately simulate the stratification in the upper oceanusing numerical models, wind-driven frictional pro-cesses should be accounted for. If a numerical modeldoes not resolve the Ekman layer, frictional re/destrati-fication will be underestimated. The strength of the Ek-man velocity is proportional to the frictional force. Thefrictional force at the top grid point of a numericalmodel, using a simple centered difference approxima-tion, is

F �1�o

��

�z��w� � � |z��z

�oz. �35�

The turbulent stress in a fluid decays with depth, ap-proximately going to zero for depths beneath the Ekmanlayer. If the top grid of a numerical model is thickerthan the Ekman layer, this suggests that � |z��z � 0,and (35) is F � (�w)/�z, which compared to the appro-priate scaling for the frictional force F � �w/�o�e, issmaller by a factor of �e /�z. This argument implies thata numerical model that does not resolve the Ekmanlayer, ratio (21) will be reduced by a factor of �e /�z andthe impact of friction on modifying the stratificationwill be artificially reduced. To avoid this pitfall, numeri-cal models should use mesh grids with a vertical reso-lution smaller than the Ekman layer depth at theboundaries.

Acknowledgments. This research was supported byNSF Grants OCE-0549699 and OCE-0612058 (L.T.)and OCE-0612143 (R.F). We thank Baylor Fox-Kemper and Patrice Klein for useful discussions.

APPENDIX A

Bounds on the Change in Stratification byFrontogenesis

An upper bound on the stratification increase due tofrontogenesis can be calculated. Frontogenesis will pro-ceed until frontal collapse—that is, until the buoyancy

forms a discontinuity. After this point, frontogenesiswill not drive further restratification. At the time offrontal collapse, the location of the maximum buoyancygradient will be displaced relative to its initial locationby a distance Y; for example, in the solution of Hoskinsand Bretherton (1972) the front is shifted to the cy-clonic side of the front. If, for example, the initial buoy-ancy field had a negative buoyancy gradient in the ydirection and was centered at y � 0 and stays approxi-mately two-dimensional, then at frontal collapse the sur-face and bottom buoyancy will take the form b|z�0 ��b[H(y � Y) � 1] and b|z��H � �b[H(y Y) � 1],where H is the Heaviside step function and �b � 0 isthe buoyancy contrast crossing the front. Substitutingthese expressions into (1) to calculate the maximummean stratification increase that can be attainedthrough frontogenesis, one finds

N front2 � 2

bY

HLy. �A1�

It is difficult to determine analytically how the displace-ment Y of the maximum frontal gradient depends onthe key parameters of the problem because it is a quan-tity that is set during the nonlinear evolution of thefront. Having said this, it can be stated with certaintythat Y will be less than the initial width of the front, Lf ,because confluence and convergence during frontogen-esis will always push the isopycnals comprising the fron-tal interface toward the center of confluence. With thisupper bound on Y, it can be deduced that

N front2 � 2

b

H

Lf

Ly.

APPENDIX B

Vertical Extent of Frontogenetic AgeostrophicSecondary Circulations

The Eliassen–Sawyer equation in (12) can be used tocompute the vertical penetration H of the frontogene-sis-driven ASC. As an illustrative example, the problemis solved using an idealized form of the stratificationand Q vector. Consider a Q vector that has a periodiclateral structure: Q2 � ℜ[Z(z)ei2�y/L], with a wave-length L. With this form, the solution for the stream-function is � � ℜ[ (z)ei2�y/L] and (12) becomes zz �l2N2/f 2 � �2Z/f 2. Assuming that (1) the Q vector isconfined to the surface over a depth h, (2) its verticalstructure is approximated by a Heaviside step function,that is, Z � Qo � a constant for z � �h and zeroeverywhere else, and (3) the stratification is constant,N2�N2

o, then the solution for is


� �2Qo

f 2�2� �1 � cos�z� �1 � cos�h � sin�h�

�cos�h sin�h�sin�z z � � h

�cos�h � 1��cos�h sin�h�

e��zh� z � � h,

�B1�

where ! � 2�No/fL . The key parameter for the verticalstructure of the ASC is the Burger number Bu � N2

oh2/f 2L 2 � !2h2/(2�)2. Because the Q vector is generatedby the geostrophic flow, both the Q-vector and geo-strophic flow have similar Burger numbers. Geo-strophic flows have Burger numbers that are typicallyof order one. For Bu � O(1), (B1) predicts that theASC penetrates a finite distance beneath the region offorcing, decaying to 10% of its maximum value atz��1.25 h, indicating that the vertical scale h� L f/No

is a good measure of the vertical extent H of the re-stratifying, frontogenesis-driven ASC.

REFERENCES

Adamson, D. S., S. E. Belcher, B. J. Hoskins, and R. S. Plant,2006: Boundary layer friction in midlatitude cyclones. Quart.J. Roy. Meteor. Soc., 132, 101–124.

Bishop, C. H., 1993: On the behaviour of baroclinic waves under-going horizontal deformation. II: Error-bound amplificationand rossby wave diagnostic. Quart. J. Roy. Meteor. Soc., 119,241–267.

Boccaletti, G., R. Ferrari, and B. Fox-Kemper, 2007: Mixed layerinstabilities and restratification. J. Phys. Oceanogr., 37, 2228–2250.

Capet, X., J. McWilliams, M. Molemaker, and A. Shchepetkin,2008: Mesoscale to submesoscale transition in the CaliforniaCurrent System. Part II: Frontal processes. J. Phys. Ocean-ogr., 38, 44–64.

Castelao, R. M., L. P. Mavor, J. A. Barth, and L. C. Breaker, 2006:Sea surface temperature fronts in the California Current Sys-tem from geostationary satellite observations. Geophys. Res.Lett., 111, C09026, doi:10.1029/2006JC003541.

Cooper, I. M., A. J. Thorpe, and C. H. Bishop, 1992: The role ofdiffusive effects on potential vorticity in fronts. Quart. J. Roy.Meteor. Soc., 118, 629–647.

Davis, C. A., M. T. Stoelinga, and Y. H. Kuo, 1993: The integratedeffect of condensation in numerical simulations of extratrop-ical cyclogenesis. Mon. Wea. Rev., 121, 2309–2330.

Fox-Kemper, B., R. Ferrari, and R. Hallberg, 2008: Parameter-

ization of mixed layer eddies. I: Theory and diagnosis. J.Phys. Oceanogr., 38, 1145–1165.

Garrett, C. J. R., and J. W. Loder, 1981: Dynamical aspects ofshallow sea fronts. Philos. Trans. Roy. Soc. London, A302,563–581.

Haine, T., and J. C. Marshall, 1998: Gravitational, symmetric, andbaroclinic instability of the ocean mixed layer. J. Phys.Oceanogr., 28, 634–658.

Hoskins, B. J., and F. P. Bretherton, 1972: Atmospheric fronto-genesis models: Mathematical formulation and solution. J.Atmos. Sci., 29, 11–37.

——, I. Draghici, and H. C. Davies, 1978: A new look at the"-equation. Quart. J. Roy. Meteor. Soc., 104, 31–38.

Lapeyre, G., P. Klein, and B. L. Hua, 2006: Oceanic restratifica-tion forced by surface frontogenesis. J. Phys. Oceanogr., 36,1577–1590.

Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanicvertical mixing: A review and a model with a nonlocal bound-ary layer parameterization. Rev. Geophys., 32, 363–403.

Marshall, J. C., and A. J. G. Nurser, 1992: Fluid dynamics of oce-anic thermocline ventilation. J. Phys. Oceanogr., 22, 583–595.

Smolarkiewicz, P. K., and L. G. Margolin, 1998: Mpdata: A finite-difference solver for geophysical flows. J. Comput. Phys., 140,459–480.

Spall, M., 1997: Baroclinic jets in confluent flow. J. Phys. Ocean-ogr., 27, 381–402.

Thomas, L. N., 2005: Destruction of potential vorticity by winds. J.Phys. Oceanogr., 35, 2457–2466.

——, 2007: Dynamical constraints on the extreme low values ofthe potential vorticity in the ocean. Proc. 15th ‘Aha Huliko ‘aHawaiian Winter Workshop, Honolulu, HI, University of Ha-waii at Manoa, 117–124.

——, 2008: Formation of intrathermocline eddies at ocean frontsby wind-driven destruction of potential vorticity. Dyn. At-mos. Oceans, 45, 252–273, doi:10.1016/j.dynatmoce.2008.02.002.

——, and P. B. Rhines, 2002: Nonlinear stratified spin-up. J. FluidMech., 473, 211–244.

Thompson, L., 2000: Ekman layers and two-dimensional fronto-genesis in the upper ocean. J. Geophys. Res., 105, 6437–6451.

Wimbush, M., and W. Munk, 1970: The benthic boundary layer.The Sea, A. E. Maxwell, Ed., Vol. 4, Wiley Interscience, 731–758.


friction, frontogenesis, and the stratification of the surface mixed...

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