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C H A P T E R

S7

Dynamical Processes for DescriptiveOcean Circulation

This is the complete chapter concerningdynamical processes; a truncated versionappears in the print text.Many additional figuresare included here, along with expanded descrip-tions and derivations. Tables for Chapter 7appear only on this Web site.

Water in the ocean at all time and space scalesis subject to the same small set of forces andaccelerations. What distinguishes one type ofmotion from another, for instance a surfacewave from the Gulf Stream, is the relativeimportance of the different accelerations andforces within this small set. In this chapter weintroduce a basic dynamical framework for themajor circulation and water mass structuresdescribed in ensuing chapters. We use as littlemathematics as possible, relying principally onword descriptions of the physical processes.Students are directed to dynamical oceanog-raphy textbooks for complete coverage of thesetopics, including scale analysis and derivations,such as Gill (1982), Pedlosky (1987), Cushman-Roisin (1994), Knauss (1997), Salmon (1998),Vallis (2006), and Huang (2010).

We proceed from the basic equations ofmotion (Sections 7.1 and 7.2) and densityevolution (Section 7.3) to mixing layers (Section7.4); direct wind response including Ekmanlayers (Section 7.5); geostrophic flow (Section

1

7.6); vorticity, potential vorticity, and Rossbywaves (Section 7.7); wind-driven circulationmodels of the gyre circulations (Section 7.8);equatorial and eastern boundary circulations(Section 7.9); and finally thermohaline forcing,abyssal circulation, and overturning circulation(Section 7.10).

7.1. INTRODUCTION:MECHANISMS

Ultimately, motion of water in the ocean isdriven by the sun, the moon, or tectonicprocesses. The sun’s energy is transferred tothe ocean through buoyancy fluxes (heat fluxesand water vapor fluxes) and through the winds.Tides create internal waves that break, creatingturbulence and mixing. Earthquakes andturbidity currents create random, irregularwaves including tsunamis. Geothermalprocesses heat the water very gradually withlittle effect on circulation.

Earth’s rotation profoundly affects almost allphenomena described in this text. Rotatingfluids behave differently from non-rotatingfluids in ways that might be counterintuitive.In a non-rotating fluid, a pressure differencebetween two points in the fluid drives the fluid

S7. DYNAMICAL PROCESSES FOR DESCRIPTIVE OCEAN CIRCULATION2

toward the low pressure. In a fluid dominatedby rotation, the flow can be geostrophic, perpen-dicular to the pressure gradient force, circlingaround centers of high or low pressure due tothe Coriolis effect.

Ocean circulation is often divided conceptu-ally into two parts, the wind-driven and the ther-mohaline (or buoyancy-dominated) components.Wind blowing on the ocean initially causessmall capillary waves and then a spectrum ofwaves and swell in the ocean (Chapter 8).Impulsive changes in wind lead to short time-scale inertial currents and Langmuir cells.Steady or much more slowly changing wind(in speed and direction) creates the ocean’snear-surface frictional Ekman layer, whichinvolves the Coriolis effect. As the windmomentum transfer persists, the geostrophic,wind-driven circulation results.

Thermohaline circulation is associated withheating and cooling (“thermo”), and evapora-tion, precipitation, runoff, and sea ice forma-tion d all of which change salinity (“haline”).Thermohaline-dominated circulation is mostlyweak and slow compared with wind-drivencirculation. Thermohaline forcing ranges fromvery local to very broad scale. An example oflocal forcing is the deep overturn driven by cool-ing and/or evaporation in which the horizontalscale of convection is at most a few kilometers.Broad-scale buoyancy forcing is associatedwith vertical diffusion that acts on the large-scaletemperature and salinity structure. Verticaldiffusion is very weak in the interior of theocean, but is essential for maintaining theocean’s vertical stratification. In discussing ther-mohaline effects, it is common to refer to themeridional overturning circulation (MOC)(Section 14.2.3). Overturning does not have tobemeridional to be of interest, and it is generallyuseful to simply refer to overturning circulation.The energy source for thermohaline circulationimportantly includes the wind and tides thatproduce the turbulence essential for the diffu-sive upwelling across isopycnals that closes the

thermohaline overturning. Both the wind-driven and thermohaline circulations are almostcompletely in geostrophic balance, with theforcing that drives them occurring at higherorder.

7.2. MOMENTUM BALANCE

Fluid flow in three dimensions is governed bythree equations expressing how velocity (ormomentum) changes, one for each of the threephysical dimensions. Each of the threemomentum equations includes an accelerationterm (how velocity changes with time), anadvection term (see Section 5.1.3), and forcingterms. These are the sameNewton’s Laws taughtin physics. Since a fluid is continuous, the massof a single object is replaced by the mass perunit volume (density); forces are also expressedper unit volume. In “word” equations:

Density� ðAcceleration þAdvectionÞ¼ Forces per unit volume (7.1)

Forces per unit volume

¼ Pressure gradient forceþGravity

þ Friction (7.2)

Expressions (7.1) and (7.2) are each three equa-tions, one for each of the three directions (e.g.,east, north, and up). The terms in Eqs. (7.1)and (7.2) are illustrated in Figure S7.1. Forocean dynamics, these equations are usuallywritten in Cartesian coordinates (x, y, z), wherex and y are westeeast and southenorth, and zis upward. Atmospheric dynamicists and someocean modelers use spherical coordinatesinstead (longitude, latitude, and the localvertical).

The inclusion of advection means thatEq. (7.1) is the expression of momentum changein a Eulerian framework, where the observersits at a fixed location relative to Earth.

Pressure gradient force

x1

Time t1

Time t2

Advection V T

T = 2° 3° 4° 5°

x

x2 x3 x4

2° 3° 4° 5°

Highpressure

xA

PA

Lowpressure

xB

PB

dpdx

PB – PAxB – xA

=~

pressure gradient

pressure gradientforce

Moving plate, speed u = uo

Fixed plate, speed u = 0

Acceleration associated with friction and viscosity



fluid velocity u(z)

time: just after top plate startsHigh flux divergence

High acceleration



time: laterLower flux divergence

Lower acceleration

x

z

Acceleration

t1 t2 t3x1 x2 x3

v2v1

a

Time

Position

Velocity

Acceleration

Gravitational force – g

– g

x-momentum flux = u/ z

time: -->No flux divergence

No acceleration

(a)

(c)

(e)

(d)

(b)

FIGURE S7.1 Forces and accelerations in a fluid: (a) acceleration, (b) advection, (c) pressure gradient force, (d) gravity,and (e) acceleration associated with viscosity y.

MOMENTUM BALANCE 3

Equation (7.1) can be written without theadvection term, in a Lagrangian framework,where the observer drifts along with the fluidflow. (See also Section S16.5 in the onlinesupplement.)

For a rotating geophysical flow, we, asobservers, sit within a rotating “frame ofreference” attached to the rotating Earth.For this reference frame, the accelerationterm on the left-hand side of Eq. (7.1) is


rewritten to separate local acceleration dueto an actual local force from the effects ofrotation. The effects that are separated outare the centrifugal and Coriolis accelerations(Section 7.2.3).

The “pressure gradient” force in Eq. (7.2)arises from external forcing. The frictional forcein Eq. (7.1) leads to dissipation of energy due tothe fluid’s viscosity.

7.2.1. Acceleration and Advection

Acceleration is the change in velocity withtime. If the vector velocity is expressed in Carte-sian coordinates as u¼ (u, v, w) where the boldu indicates a vector quantity, and u, v, and w arethe positive eastward (x-direction), northward(y-direction), and positive upward (z-direction)velocities, then

x-direction acceleration ¼ vu=vt (7.3a)

with similar expressions for the y- and z-direc-tions. In a rotating frame of reference, such ason the surface of Earth, the acceleration termincludes two additional terms: centrifugal andCoriolis acceleration (Section 7.2.3).

Advection is defined in Section 5.1.3. Advec-tion is how the flow moves properties(including scalars such as temperature orsalinity) and vectors (such as the velocity).Advection can change the flow property if thereis a gradient in the property through which thefluid moves. The advection term thus isa product of the velocity and the difference inthe property from one location to another. Thereare three advection terms in each momentumequation, since the flow bringing in a differentproperty can come from any of the three direc-tions. (The vertical advection term is sometimescalled convection.) In the x-momentum equa-tion, the advection term is

x-direction advection

¼ uvu=vxþ vvu=vyþwvu=vz (7.3b)

The substantial derivative is the sum of the
acceleration and advection terms:
Du=Dt ¼ vu=vtþ uvu=vx

þ vvu=vyþwvu=vz (7.4)

Eq. (7.4) represents the change in u at a fixedpoint (Eulerian framework). In a Lagrangianframework, following the particle of water,only the time derivative appears; the threeadvection terms do not appear since they arecontained in the movement of the particle.

7.2.2. Pressure Gradient Force andGravitational Force

Pressure is defined in Section 3.2. The flow offluid due to spatial variations in pressure (thepressure gradient force) is also described. In math-ematical form, the pressure gradient force is

x-direction pressure gradient force

¼ �vp=vx(7.5)

The pressure gradient force has a negative signbecause the force goes from high pressure tolow pressure.

The gravitational force between Earth andthe object or fluid parcel is directed towardthe center of mass of Earth. Gravitationalforce is mass of the object � gravitationalacceleration g, equal to 9.780318 m2/sec (atthe equator). The gravitational force per unitvolume is

z-direction gravitational force per unit volume

¼ �rg

(7.6)

7.2.3. Rotation: Centrifugal andCoriolis Forces

Earth rotates at a rate of U¼ 2 p/Twhere T isthe length of the (sidereal) day, which has 86,164

act

ual p

ath

actual path

0°

30°

30°

60°

60°

150°

180°

120°

90°

150°

Earthrotation

intendedtarget

actual finallocation

intendedpath

intendedpath

Coriolis"deflection"

centrifugalforce

centripetalforce

V

string

(a)

(b)

ball

FIGURE S7.2 (a) Centrifugal and centripetal forces and(b) Coriolis force.

MOMENTUM BALANCE 5

seconds; hence U¼ 0.729� 10�4 sec�1.1 We lookat motions and write our theories sitting ina “rotating reference frame,” that is, attached tothe rotating Earth. However, the reference framethat is correct for Newton’s Laws (Eq. 7.1) is an“inertial reference frame,” which is not rotating.To look at motions from within our rotatingreference frame, we must add two terms due tothe Earth’s rotation. The first is the “Coriolisforce” and the second is the “centrifugal force”(Figure S7.2). A derivation of these twopseudo-forces is given at the end of this section.

7.2.3.1. Centrifugal and Centripetal Force

Centrifugal force is the apparent outward forceon amasswhen it is rotated. Think of a ball on theend of a string that is being twirled around, orthe outward motion you feel when turninga curve in a car. In an inertial frame, there is nooutward acceleration since the system is notrotating. The ball or your body just moves inthe straight line that they were following origi-nally. But in the rotating reference frame of thestring or the car, they appear to be acceleratedaway. Since Earth rotates around a fixed axis,the direction of centrifugal force is alwaysoutward away from the axis. Thus it is oppositeto the direction of gravity at the equator; atEarth’s poles it is zero. (Centripetal force is thenecessary inward force that keeps the massfrom moving in a straight line; it is the samesize as centrifugal force, with the opposite sign.Centripetal force is real; centrifugal force is justan apparent force. For the rotating Earth, centrip-etal force is supplied by the gravitational forcetowards Earth’s center.)

If Earth was a perfect, rigid sphere, the oceanwould be 20 km deeper at the equator than atthe poles. But this is not observed, because the

1 The solar day, which is the time between consecutive highest p

sidereal day is the rotation period relative to the fixed stars, wh

shorter than the solar day, with 23 hours, 56 minutes, and 4.1

a sidereal day is the time it takes for Earth to rotate 360 degre

day¼ 5.7, 1.4, 1.2 sidereal days.

solid Earth is deformed by centrifugal force.That is, Earth is a spheroid rather than a sphere,with the radius at the equator approximately20 km greater than at the poles. Therefore the

oints of the sun in the sky, is 24 hours, or 86,400 seconds. The

ich is the inertial reference frame. The sidereal day is slightly

seconds. One pendulum day is one sidereal day/sin4, where

es and where 4¼ latitude. For 4¼ 10�, 45�, 60�, 1 pendulum


centrifugal force at the equator is balanced(canceled) by the extra gravitational force there(this is referred to as “effective gravity”).

The mathematical expression for centrifugalacceleration (force divided by density) is

centrifugal acceleration ¼ U2r (7.7)

where U is the rotation rate of Earth, equal to2p/T where T is the length of day, and r isEarth’s radius. Because the centrifugal accelera-tion is nearly constant in time and pointsoutward, away from Earth’s axis of rotation,we usually combine it formally with the gravita-tional force, which points toward Earth’s center.We replace g in Eq. (7.6) with the effectivegravity g, which has a weak dependence on lati-tude. Hereafter, we do not refer separately to thecentrifugal force. The surface perpendicular tothis combined force is called the geoid. If theocean were not moving relative to Earth, itssurface would align with the geoid.

7.2.3.2. Coriolis Force

The second term in a rotating frame of refer-ence included in the acceleration equation (7.1)is the Coriolis force. When a water parcel, airparcel, bullet, hockey puck, or any other bodythat has little friction moves, Earth spins outfrom under it. By Newton’s Law, the bodymoves in a straight line if there is no other forceacting on it. As observers attached to Earth, wesee the body appear to move relative to our loca-tion. In the Northern Hemisphere, the Coriolisforce causes a moving body to appear to moveto the right of its direction of motion (FigureS7.2b). In the Southern Hemisphere, it movesto the left.

The Coriolis force is non-zero only if the bodyis in motion, and is important only if the bodytravels for a significant period of time. Coriolisforce is larger for larger velocities as well. Forthe flight of a bullet there is no need to considerthe Coriolis force because the travel time isextremely short. For missiles that fly long paths

at high speeds, Coriolis force causes significantdeflections. For winds in the atmosphere’s JetStream, the timescale of motion is several daysto several weeks, so Earth’s rotation is veryimportant and the winds do not blow fromhigh to low pressure. The same holds true inthe ocean, where currents last for weeks or yearsand are strongly influenced by the Coriolisforce.

For large-scale ocean currents, and to someextent winds, the vertical velocity is muchweaker than the horizontal velocity. Certainlythe distance that a water parcel can move inthe vertical is much more limited than in thehorizontal, because of both the difference indepth and width of the ocean, and because ofthe ocean’s stratification. Therefore, Corioliseffects act mostly on the horizontal velocitiesand not on the vertical ones. As noted previ-ously, the Coriolis force apparently sendsobjects to the right in the Northern Hemisphereand to the left in the Southern Hemisphere. Atthe equator, the Coriolis effect acting on hori-zontal velocities is zero. Its magnitude is largestat the poles.

Mathematically, the Coriolis force is

x-momentum equation:

�2U sin 4 vh� f v (7.8a)

y-momentum equation:

�2U sin 4 uh f u (7.8b)

Coriolis parameter:

f ¼ 2U sin 4 (7.8c)

where “h” denotes a definition,U is the rotationrate, 4 is latitude, u is velocity in the x-direction,v is velocity in the y-direction, and where thesigns are appropriate for including these termson the left-hand side of Eq. (7.1). The Coriolisparameter, f, is a function of latitude and changessign at the equator, and it has units of sec�1. (Thenon-dimensional parameter called the Rossbynumber introduced in Section 1.2 is Ro¼ 1/fT

MOMENTUM BALANCE 7

or Ro¼U/fL, where U, L, and T are character-istic velocity, length, and timescales for the flow.)

7.2.3.3. Derivation of Centrifugal andCoriolis Terms

The Coriolis and centrifugal terms arederived by transforming Newton’s law ofmotion (Eq. 7.1) from its true inertial system,relative to the fixed stars, to the rotating Earth-centric system. This derivation is available inadvanced textbooks on classical mechanics,and is included here for completeness. Equa-tions are numbered separately to maintainconsistent numbering because they are notincluded in the print text. We write the three-dimensional vector version of Eq. (7.1) as

v v!s

vt¼ F

!=r (S7.1)

where the subscript “s” means that the velocityof the particle is measured in the inertial frameof reference relative to the stars. Rewrite thisvelocity as the sum of the particle’s velocity rela-tive to Earth’s surface and the velocity of Earth’ssurface due to rotation:

v!s ¼ v!e þ U/

� r! (S7.2)

where v!e is the particle velocity relative to localcoordinates on Earth’s surface, U

/

is Earth’srotation vector, pointing northward along theaxis of rotation with magnitude equal to therotation rate, and r! is the vector position of theparticle. Substituting this back into Eq. (S7.1)yields�

v v!s

vt

�s

¼�v v!e

vt

�s

þ v

vtðU/

� r!Þs

¼�v v!e

vt

�s

þ vU/

vt� r!þ U

/��v r!vt

�s

(S7.3)

Since Earth’s rotation is essentially constantcompared with the timescales of atmosphericand oceanic circulation, and using Eq. (S7.2),we find that

�v v!s

vt

�s

¼�v v!e

vt

�s

þ U/

��v r!vt

�s

¼�v v!e

vt

�e

þ U/

� v!e þ U/

��v r!vt

�e

þ U/

��U/

� r!�

(S7.4)

Since the derivative of any vector in the fixedframe is related to the derivative in the rotatingframe as�

v q!vt

�s

¼�v q!vt

�e

þ U/

� q! (S7.5)

we find finally that

�v v!s

vt

�s

¼�v v!e

vt

�e

þ 2U/

� v!e

þ U/

� �U/

� r!�(S7.6)

The first term on the right-hand side is the accel-eration relative to the rotating (Earth) frameof reference, the second term is the Coriolisterm, and the third term is the centrifugalacceleration.

7.2.4. Viscous Force or Dissipation

Fluids have viscous molecular processes thatsmooth out variations in velocity and slowdown the overall flow. These molecularprocesses are very weak, so fluids can often betreated, theoretically, as “inviscid” rather thanviscous. However, it is observed that turbulentfluids like the ocean and atmosphere actuallyact as if the effective viscosity were much largerthan the molecular viscosity. Eddy viscosity isintroduced to account for this more efficientmixing (Section 7.2.4.2).

7.2.4.1. Molecular Viscosity

We can think ofmolecular viscosity by consid-ering two very different types of coexisting

L‘ U‘

FLOW U

FIGURE S7.3 Illustration of molecular processes thatcreate viscosity. The mean flow velocity is indicated in gray(U). L0 is the distance between molecules. U0 is the speed ofthe molecules. Random molecule motions carry informationabout large-scale flow to other regions, thus creating(viscous) stresses. Viscous stress depends on the meanmolecular speed jU0j and mean molecular free path jL0j.


motion: the flow field of the fluid, and, due totheir thermal energy, the randommotion ofmole-cules within the flow field. The random molec-ular motion carries (or advects) the larger scalevelocity from one location to another, and thencollisions with other molecules transfer theirmomentum to each other; this smoothes out thelarger scale velocity structure (Figure S7.3).

The viscous stress within a Newtonian fluidis proportional to the velocity shear. The propor-tionality constant is the dynamic viscosity, whichhas meter-kilogram-second (mks) units ofkg/m-sec. The dynamic viscosity is the productof fluid density times a quantity called the kine-matic viscosity, which has mks units of m2/sec.For water, the kinematic viscosity is1.8� 10�6 m2/sec at 0�C and 1.0� 10�6 m2/secat 20�C (Table S7.1).

Flow is accelerated or decelerated if there isa variation in viscous stress from one location

TABLE S7.1 Molecular and Eddy Viscosities and Diffusiv

Molecular, at salinity[ 35

Viscosity 1.83� 10�6 m2/sec at 0 �C 1.05� 10�6

Thermal diffusivity 1.37� 10�7 m2/sec at 0 �C 1.46� 10�7

Haline diffusivity 1.3� 10�9 m2/sec

to another. This is illustrated in Figure S7.1e,where a viscous stress is produced by themotion of a plate at the top of the fluid, witha stationary plate at the bottom. The fluidmust stay with each plate, so the fluid velocityat each boundary equals that plate velocity.

1. At very small times (leftmost panel), just afterthe top plate starts to move, there is a largevariation in velocity in the fluid close to thetop plate, which means there is a large stressthere. The stress is associated with flux ofx-momentum down into the fluid from theplate. Since there is much smaller stressfarther down in the fluid, there is a netdeposit of x-momentum in the fluid, whichaccelerates it to the right.

2. At a later time (center panel), this accelerationhas produced velocity throughout the fluidand the change in viscous stress from top tobottom is reduced.

3. At a very large time (rightmost panel), theviscous stress is the same at all locationsand there is no longer any acceleration; atthis time the velocity varies linearly fromtop to bottom. (This is known as “Couetteflow.”) There is a stress on the fluid asa whole, which is balanced by the frictionalstress of the fluid back on the plates; thereis dissipation of energy throughout thefluid even though there is no localacceleration.

Formally, for a Newtonian fluid, which isdefined to be a fluid in which stress is propor-tional to strain (velocity shear), and if viscosity

ities (m2/sec)

Eddy: horizontal

(along-isopycnal)

Eddy: vertical

(diapycnal)

m2/sec at 20 �C 102 to 104 m2/sec 10�4 m2/sec

m2/sec at 20 �C 102 to 104 m2/sec 10�5 m2/sec

102 to 104 m2/sec 10�5 m2/sec

MOMENTUM BALANCE 9

has no spatial dependence, viscous stress entersthe momentum equations as

x-momentum dissipation

¼ y�v2u=vx2 þ v2u=vy2 þ v2u=vz2

�(7.9)

where y is the molecular (kinematic) viscosity.(The dynamic viscosity is ry.) This expressioncomes from the divergence of the viscous stressin the x-direction. For the example shown inFigure S7.1.e, this stress is yvu/vz, and there isan acceleration of the fluid only if this stressvaries with z.

Molecular viscosity changes flow veryslowly. Its effectiveness can be gauged bya non-dimensional parameter, the Reynoldsnumber, which is the ratio of the dissipationtimescale to the advective timescale: Re¼UL/y.When the Reynolds number is large, the flowis nearly inviscid andmost likely very turbulent;this is the case for flows governed by molecularviscosity. However, frommatching observationsand theory we know that the ocean currentsdissipate energy much more quickly than wecan predict using molecular viscosity. Howthis happens is described next.

7.2.4.2. Eddy Viscosity

Mixing at spatial scales larger than thosequickly affected by molecular viscosity is gener-ally a result of turbulence in the fluid. Turbulentmotions stir the fluid, deforming and pulling itinto elongated, narrow filaments. A stirred fluidmixes much faster than one that is calm andsubjected only to molecular motion. While stir-ring is technically reversible, mixing is not. Itis easier to think about this for a property,such as milk in a coffee cup, or salinity in theocean, than for velocity, but the same principlesapply to both. The filaments are deformed byturbulence on a smaller spatial scale. Eventuallymolecular viscosity takes over, when the spatialscales become very small. We refer to the effectof this turbulent stirring/mixing on the fluid aseddy viscosity.

There is no obvious way to derive the sizeof eddy viscosity from molecular properties.Instead, it is determined empirically, eitherdirectly from observations, or indirectly frommodels that work relatively well and includeeddy viscosity. For large-scale ocean circula-tion, the “turbulent” motions are mesoscaleeddies, vertical fine structure, and so on,with spatial scales smaller than the largerscales of interest. Like molecular viscosity,eddy viscosity should be proportional to theproduct of turbulent speed and path length.Therefore, horizontal eddy viscosity is gener-ally much larger than vertical eddy viscosity(Table S7.1). More specifically, although weoften refer to “horizontal” and “vertical”eddy viscosity, the relevant directions arealong isopycnals (adiabatic surfaces) andacross isopycnals (diapycnal mixing), since theseare the natural coordinates for uninhibitedquasi-lateral motion and the most inhibitedquasi-vertical motion (Redi, 1982; Gent &McWilliams, 1990).

To mathematically include eddy viscosity, theviscous terms in Eqs. (7.1) and (7.9) are replacedby the eddy viscosity terms:


¼ AH

�v2u=vx2 þ v2u=vy2

�þAV

�v2u=vz2

�(7.10a)

whereAH is the horizontal eddyviscosity andAV

is the vertical eddy viscosity. (The use of thesymbol A is from the early German definitionof an “Austausch” or exchange coefficient torepresent eddy viscosity.) AH and AV have unitsof kinematic viscosity, m2/sec in mks units.(Although we often use these Cartesian coordi-nates, the most relevant stirring/mixing direc-tions are along isopycnals (adiabatic surfaces)and across isopycnals (diapycnal mixing), so thecoordinate system used in Eq. 7.10a is bettermodeled by rotating it to have the “vertical”direction perpendicular to isopycnal surfaces,and replace AH and AV with eddy viscosities

S7. DYNAMICAL PROCESSES FOR DE10

that are along and perpendicular to thosesurfaces.)

In many applications and observations, it isuseful to include spatial dependence in theeddy viscosity coefficients because turbulenceis unevenly distributed. Equation (7.10a) isthen written in its original form, which includesspatial variation of stress:


¼ v=vxðAHvu=vxÞ þ v=vyðAHvu=vyÞþ v=vzðAVvu=vzÞ (7.10b)

Eddy viscosity coefficients (Table S7.1 andSection 7.3), also called eddy momentumdiffusion coefficients, are inferred from obser-vations of microstructure (very small scalevariations in velocity) and from eddy diffusiv-ities acting on temperature and salinity that arealso derived from observations, given that bothare due to similar structures that mix theocean. (Formally, in fluid mechanics, the non-dimensional ratio of viscous diffusivity tothermal diffusivity is called the Prandtl number;if we assume that eddy viscosity and eddydiffusivity were equal, we are assuminga turbulent Prandtl number of 1.) Numericalmodels typically use higher eddy viscositiesthan eddy diffusivities (e.g., Smith, Maltrud,Bryan, & Hecht, 2000; Treguier, 2006).

7.2.5. Mathematical Expression ofMomentum Balance

The full momentum balance with spatiallyvarying eddy viscosity and rotation is

Du=Dt� fv ¼ vu=vtþ u vu=vxþ v vu=vy

þw vu=vz� fv

¼ �ð1=rÞvp=vxþ v=vxðAHvu=vxÞþ v=vyðAHvu=vyÞþ v=vzðAVvu=vzÞ

(7.11a)

Dv=Dtþ fu ¼ vv=vtþ u vv=vxþ v vv=vz

þw vv=vzþ fu

¼ �ð1=rÞvp=vyþ v=vxðAHvv=vxÞþ v=vyðAHvv=vyÞþ v=vzðAVvv=vzÞ

(7.11b)

Dw=Dt ¼ vw=vtþ u vw=vxþ v vw=vy

þw vw=vz

¼ �ð1=rÞvp=vz� gþ v=vxðAHvw=vxÞþ v=vyðAHvw=vyÞþ v=vzðAVvw=vzÞ

(7.11c)

Here the standard notation “D/Dt” is thesubstantial derivative defined in Eq. (7.4).

The full set of equations describing the phys-ical state of the ocean must also include the massconservation equation (Section 5.1):

Dr=Dtþ rðvu=vxþ vv=vyþ vw=vzÞ ¼ 0

(7.11d)

If density changes are small, Eq. 7.11d is approx-imated as

vu=vxþ vv=vyþ vw=vz ¼ 0 (7.11e)

which is known as the continuity equation.The set is completed by the equations govern-ing changes in temperature, salinity, anddensity, which are presented in the followingsection.

SCRIPTIVE OCEAN CIRCULATION

7.3. TEMPERATURE, SALINITY,AND DENSITY EVOLUTION

Evolution equations for temperature andsalinity d the equation of state that relatesdensity to salinity, temperature, and pressure,and thus an evolution equation for density dcomplete the set of equations (7.11aed) that

TEMPERATURE, SALINITY, AND DENSITY EVOLUTION 11

describe fluid flow in the ocean. The boundaryand initial conditions required for solving thesystems of equations are beyond our scope.

7.3.1. Temperature, Salinity, andDensity Equations

Temperature is changed by heating, cooling,and diffusion. Therefore the most basic equationwould be that for heat (or enthalpy), but mostdynamical treatments and models use anexplicit temperature equation. Salinity ischanged by addition or removal of freshwater,which alters the dilution of the salts. Mostmodeling uses an explicit salinity equationrather than a freshwater equation. Density isthen computed from temperature and salinityusing the equation of state of seawater. The“word” equations for temperature, salinity,and density forcing include:

temperature change

þ temperature advection=convection

¼ heating=cooling termþ diffusion

(7.12a)

salinity change

þ salinity advection=convection

¼ evaporation=precipitation=runoff

=brine rejectionþ diffusion

(7.12b)

equation of state ðdependence of densityon salinity; temperature; and pressureÞ

(7.12c)

density changeþ density advection=convection

¼ density sourcesþ diffusion

(7.12d)

Written in full, these are

DT=Dt ¼ vT=vtþ u vT=vxþ v vT=vy

þw vT=vz

¼ QH=rcp þ v=vx�kHvT=vx

�þ v=vy

�kHvT=vy

�þ v=vz�kVvT=vz

�(7.13a)

DS=Dt ¼ vS=vtþ u vS=vxþ v vS=vyþw vS=vz

¼ Qs þ v=vxðkHvS=vxÞþ v=vyðkHvS=vyÞ þ v=vzðkVvS=vzÞ

(7.13b)

r ¼ rðS;T;pÞ (7.13c)

Dr=Dt ¼ vr=vtþ u vr=vxþ v vr=vyþw vr=vz

¼ ðvr=vSÞDS=Dtþ ðvr=vTÞDT=Dt

þ ðvr=vpÞDp=Dt

(7.13d)

where QH is the heat source (positive for heat-ing, negative for cooling, applied mainly nearthe sea surface), cp is the specific heat ofseawater, and QS is the salinity “source” (posi-tive for evaporation and brine rejection, nega-tive for precipitation and runoff, applied ator near the sea surface). (See also Chapter 5 fordiscussion of heat and salinity.) kH and kV arethe horizontal and vertical eddy diffusivities,analogous to the horizontal and vertical eddyviscosities in the momentum equations(7.11aed; Table S7.1. The full equation of stateappears in Eq. (7.13c), from which the evolutionof density in terms of temperature and salinitychange can be computed (Eq. 7.13d). The coeffi-cients for the three terms in Eq. (7.13d) are thehaline contraction coefficient, the thermalexpansion coefficient, and the adiabaticcompressibility, which is proportional to theinverse of sound speed (Chapter 3).


7.3.2. Molecular and Eddy Diffusivity

The molecular diffusivity k for eachsubstance depends on the substance and thefluid. The molecular diffusivity of salt inseawater is much smaller than that for heat(Table S7.1). This difference results in a processcalled “double diffusion” (Section 7.4.3).

Eddy diffusivity is the equivalent of eddyviscosity for properties like heat and salt. Eddydiffusivity and eddy viscosity typically havesimilar orders of magnitude (Table S7.1) sincethe same turbulent processes create both. Forlack of observations and for simplicity, diapyc-nal, quasi-vertical eddy diffusivity was onceconsidered to be globally uniform (e.g.,Stommel & Arons, 1960a, b; Section 7.10.3). Aglobally averaged vertical eddy diffusivity ofkv¼ 1� 10�4 m2/sec accounts for the observedaverage vertical density structure (Section7.10.2; Munk, 1966). However, the directlyobserved vertical (or diapycnal) eddy diffusivityin most of the ocean is a factor of 10 lower: kv ~1� 10�5 m2/sec, based on direct measurementsof mixing rates using dye release and spread(Ledwell, Watson, & Law, 1993, 1998), measure-ments of very small scale vertical structure(Osborn & Cox, 1972; Gregg, 1987), and large-scale property distributions within the pycno-cline (e.g., Olbers, Wenzel, & Willebrand, 1985).This implies regions of much higher diffusivityto reach the global average.

Measurements show huge enhancements ofdiapycnal eddy diffusivity in bottom boundaryregions, especially where topography is rough(Figure S7.4; Polzin, Toole, Ledwell, & Schmitt,1997; Kunze et al., 2006), and on continentalshelves where tidal energy is focused (Lien &Gregg, 2001). In these regions, the tides movewater back and forth over hundreds of metershorizontally (Egbert & Ray, 2001). If the bottomis rough, as it is over most mid-ocean ridgesystems (Figures 2.5 and 2.6), the internal tidecan break, causing enhanced turbulence anddiffusivity. Internal tides have been directly

observed and related to turbulence along theHawaiian Ridge (Rudnick et al., 2003). If theinteraction is strong, then the enhanced diffu-sivity can reach high into the water column,even reaching the pycnocline, as is seen overthe topographic ridges in Figure S7.4.

Diapycnal eddy diffusivity also depends onlatitude (Figure S7.4b). It is small at low lati-tudes (order of 10�6 m2/sec), increasing toa peak at 20e30� latitude, and then decliningsomewhat toward higher latitudes (0.4 to0.5� 10�4 m2/sec). The relation of this diffu-sivity distribution to the actual efficiency of mix-ing, which also depends on the currents, has notyet been mapped.

Within the water column, away from the topand bottom boundaries, internal waves aregenerally relatively quiescent, without muchbreaking, but nonlinear interactions betweeninternal waves and encounters with mesoscaleeddies could also produce higher velocityshears that result in a low level of breakingand turbulence.

In the surface layer, eddy diffusivities andeddy viscosities are also much greater than theMunk value (e.g., Large, McWilliams, & Doney,1994). In Section 7.5.3 on Ekman layers, wedescribe eddy viscosities in the surface layeron the order of 100 to 1000� 10�4 m2/sec. Largelateral variations in diapycnal diffusivity resultfrom the processes that create the turbulence,such as strongly sheared currents (such as theGulf Stream) and wind-forced near-inertialmotions near the base of the mixed layer.

Horizontal eddy diffusivities kH are estimatedto be between 103 and 104 m2/sec, with largespatial variability (e.g., Figure 14.17). kH ismuch larger than kV. The larger size is relatedto the larger horizontal length and velocity scalesthan in the vertical; turbulent motions and mix-ing are enhanced in the horizontal. Observa-tional estimates of horizontal diffusivity havebeen based on dye release (Ledwell et al.,1998) and on dispersion of floats and surfacedrifters (Section 14.5.1). Estimates of horizontal

FIGURE S7.4 (a) Observed diapycnal diffusivity (m2/s2) along 32�S in the Indian Ocean, which is representative of otherocean transects of diffusivity. (b) Average diapycnal diffusivity as a function of latitude range (color codes). Source: FromKunze et al. (2006).

MIXING LAYERS 13

diffusivity are also made from choices requiredto match observed and theoretical phenomenasuch as boundary current widths. It is empha-sized that, unlike molecular diffusivities, eddydiffusivities are not a property of the flow ingeneral, but depend on which space and time-scales are “resolved” and “unresolved.”

7.4. MIXING LAYERS

Mixing occurs throughout the ocean. Mixingof momentum is the frictional process whilemixing of properties is the diffusion process.While it is weak, it is essential for maintainingthe observed stratification and can regulate the


strength of some parts of the circulation. In thissection we look at mixing in the surface layerwhere there is direct atmospheric forcing; inbottom layers where mixing can be caused byinteraction between ocean currents, tides, andwaves and the bottom topography; and withinthe water column, away from top and bottomboundaries.

7.4.1. Surface Mixed Layer

The surface layer (Section 4.2.2) is forceddirectly by the atmosphere through surfacewind stress and buoyancy (heat and freshwater)exchange. The surface “mixed layer” is seldomcompletely mixed, so it is sometimes difficultto define. We consider it to be the upperboundary layer of the ocean, forced directly bythe atmosphere through: (1) surface stress ofthe wind and (2) buoyancy (heat and fresh-water) exchange through the surface. Windstress generates motion, which is strongest atthe surface and decreases with depth, that is,with vertical shear in the velocity. Thesemotions include waves that add turbulentenergy to increase mixing, particularly if theybreak. Wind-driven Langmuir circulations(Section 7.5.2) can promote mixing, possiblythrough the full depth of the mixed layer.

For a surface layer that is initially stablystratified (Figure S7.5a), sufficiently largewind stress will create turbulence that mixesand creates a substantially uniform density ormixed layer (Figure S7.5b). This typicallyresults in a discontinuity in properties at themixed layer base.

The upper layer can also be mixed by buoy-ancy loss through the sea surface, increasingthe density of the top of the surface layer andcausing it to overturn (convect) to a greater depth(Figure S7.5cee). This type of mixed layer typi-cally has no discontinuity in density at its base.Heat or freshwater gain decreases the density ofthe top of the surface layer, resulting in a morestably stratified profile. If the wind then mixes

it, the final mixed layer is shallower than theinitial mixed layer (Figure S7.5feh).

Mixed layer observations typically showmuch more vertical structure than might beexpected from these simple ideas. This isbecause the mixed layer is subject to greatlyvarying forcing, including diurnal heating thatrestratifies the mixed layer, cooling that convec-tively mixes the layer, wind-generated turbu-lence that mechanically stirs the layer, andsmall-scale instabilities of the many localizedfronts within the mixed layer that can changeits stratification (e.g., Boccaletti, Ferrari, & Fox-Kemper, 2007).

The thickest mixed layers occur at the end ofwinter (Figure 4.5), after an accumulation ofmonths of cooling that deepens the mixed layerand increases its density. For large-scale oceano-graphic studies, these end-of-winter mixedlayers set the properties that are subductedinto the ocean interior (Section 7.8.5). Maps oflate winter mixed layer depth and alsomaximum mixed layer depth are shown inFigure 4.5.

Several different parameterizations of surfacelayer mixing due to winds and buoyancy fluxeshave been used. The first parameterization used(“K-T”) was developed by Kraus and Turner(1967). The Price, Weller, Pinkel (1986; PWP)model largely replaced the K-T model and isstill used widely. Large et al. (1994) proposedthe most commonly used modern approach,called “K-Profile Parameterization” (KPP),where “K” is shorthand for diffusivity k. TheKPP model extends the response to surfaceforcing to below the completely mixed layer,since turbulence set up at the base of the well-mixed layer penetrates downward; for instance,through near-inertial motions (Sections 7.5.1and 14.5.3).

7.4.2. Bottom Mixed Layers

Near the ocean bottom, turbulence, and hencemixing, can be generated by currents or current

MIXING LAYERS 15

shear caused by the interaction with the bottom.In shallow (e.g., coastal) waters, complete mixingof the water column occurs if the depth (H) isshallow enough and the tidal currents (U) arefast enough (see reviews in Simpson, 1998 &Brink, 2005). Complete mixing of the watercolumn occurs if the depth (H) is shallow enoughand the tidal currents (U) are fast enough. Fromenergy dissipation arguments, a useful criticalparameter based on depth and velocity isH/U3. When H/U3 < a, where a is proportionalto the empirically determined mixing efficiencyand the buoyancy flux, there can be completemixing that destroys the stratification. Consider-able observational efforts have been made andare ongoing to quantify and understand theturbulence that creates the mixing (Doron,Bertuccioli, Katz & Osborn, 2001; Polzin, Toole,Ledwell, & Schmitt, 1997; Kunze et al., 2006;Lien & Gregg, 2001 and many others).

At longer timescales on the shelf, a bottomEkman layer can develop in which frictionaland Coriolis forces balance (Ekman, 1905 andSection 7.5.3), with the bottom slope alsoaffecting the layer. The bottom slope on theshelf, and its intersection with the water col-umn’s density structure, is important forbottom Ekman layers, which can have bothupslope and downslope flow. Eddy viscosityalso has important variations in space andtime, which affects the Ekman layer structure(Lentz, 1995).

Enhanced turbulence in a bottom boundarylayer can be created by movement of wateracross rough topography and by breaking ofinternal waves that reflect off the topographyand result in higher eddy diffusivity values(Figure S7.5). The higher turbulence createslocally mixed bottom boundary layers that canthen be advected away from the bottom topog-raphy, creating “steppy” vertical profiles nearthe bottom some distance from the mixing site(Figure S7.6a).

Bottom currents due to density differencescan also cause mixing. One example is

a turbidity current down an underlying bottomslope. (See Section 2.6. Another example is theoverflow of dense water across a sill, as seen atthe Strait of Gibraltar (Chapter 9.) The densewater flows down the continental slope asa plume, mixing vigorously with the lighterwater around it (Figure S7.6b). This turbulentprocess is called entrainment.

As it entrains, the outflow reaches an equilib-rium density with the ambient water andspreads thereafter along that isopycnal surface.The entrainment rate and the final density ofthe plume depend on the density of the straitoutflow and the density profile in the ambientwaters outside the strait.

Density differences due to the injection oflighter water into the ocean also cause mixingand entrainment. An example is hot hydro-thermal water injected at mid-ocean ridgesand hotspots that entrain ambient waters asthe plumes rise. A man-made example is waterfrom a sewer outfall where the discharged fluidis less dense than the seawater. In both cases,mixing (entrainment) takes place as the plumesrise due to their buoyancy.

7.4.3. Internal Mixing Layers

In the interior of the ocean (i.e., away fromboundaries), continuous profiling instrumentshave shown that vertical profiles of water prop-erties d temperature and salinity, and hencedensity d are often not smooth (Figure S7.5i)but “stepped” (Figure S7.5j). The vertical scaleof the steps can be decimeters to many meters.Turbulence (Section 7.4.3.1) and/or doublediffusion (Section 7.4.3.2) mix the water columninternally and can create such steps.

7.4.3.1. Turbulent Mixing

Breaking internal waves can create internalmixing (Section 8.4; Rudnick et al., 2003).Vertical shear from other sources can also resultin turbulence. On the other hand, vertical strat-ification stabilizes the mixing. One way to

FIGURE S7.5 Mixed layer development. (a, b) An initially stratified layer mixed by turbulence created by wind stress;(c, d, e) an initial mixed layer subjected to heat loss at the surface, which deepens the mixed layer; (f, g, h) an initial mixedlayer subjected to heat gain and then to turbulent mixing presumably by the wind, resulting in a thinner mixed layer; (i, j) aninitially stratified profile subjected to internal mixing, which creates a stepped profile. Notation: s is wind stress and Q isheat (buoyancy).


(a)

(b)

Descent and Entrainment

Marginal seadense water production

Ocean

Ocean-SeaExchange

ρ = constant

Equilibration

Air-SeaExchange

(buoyancy loss)

FIGURE S7.6 (a) Bottomboundary layers and their advec-tion away from the bottom Source:From Armi (1978). (b) Mixing ofa plume of dense water as it flowsout over a strait into less denseambient waters. After Price andBaringer (1994).

MIXING LAYERS 17

express this trade-off is through a non-dimen-sional quantity called the Richardson number:

Ri ¼ N2=ðvu=vzÞ2 (7.14)

where

N2 ¼ �g�vr=vz

��r0 (7.15)

N is the Brunt-Vaisala frequency (Section3.5.6) and the vertical shear of the horizontalspeed is (vu/vz). If the Richardson numberis small, the stratification is weak and theshear is large, so we expect mixing to bevigorous. From theory and observations,vigorous mixing starts when the Richardsonnumber falls below 1/4.

The initial steps of mixing between two hor-izontally adjacent waters with strong tempera-ture/salinity differences are visible at thefront between the waters. Stirring at the frontdraws layers of the adjacent waters into eachother along isopycnals, resulting in interleavingor fine structure, with layering of one to tensof meters on both sides of the front. The inter-leaving facilitates local vertical (diapycnal)mixing between the two water masses, whichis the next step to actual mixing betweenthem. The actual mixing can take place throughturbulent processes or the double diffusiveprocesses described in the next subsection. Inboth cases, much smaller scale vertical struc-ture, on the order of centimeters (microstruc-ture), is an indication of the actual mixing at


the interfaces between the interleaving layers.Such interleaving has been observed in thewestern equatorial Pacific, in the AntarcticCircumpolar Current (ACC), in the Kuroshioand Gulf Stream, and so forth; that is, in everyregion where there are strong water massfronts.

7.4.3.2. Double Diffusion

Heat diffuses about 100 times faster than salt(Table S7.1). Double diffusion is due to thesediffering molecular diffusivities, acting at scalesof centimeters to meters, and can also createwell-mixed internal layers. When warm, salty

WarmSaltyLower density

ColdFreshHigher density

Salt fingering layer(s)

Salt fingering

Diffusive layering

ColdFreshLower density

WarmSaltyHigher density

Diffusive layer(s)

(a)

(b)

FIGURE S7.7 Double diffusion:(a) salt fingering interface (cold,fresh water warms and rises;warm, salty water cools and sinks).(b) Diffusive interface. (c) NorthAtlantic Mediterranean eddysalinity profile with steps due tosalt fingering (25� 23’N, 26�W). (d)Arctic temperature profile withdiffusive layering. Source: From

Kelley et al. (2003).

water lies above cold, fresh water, and the inter-face between the two is disturbed so that smallcolumns of warm, salty water are next to cold,fresh ones, the fast heat exchange betweenthem will cool the saltier water and warm thefresher water while the salinity will mix muchless. The saltier water becomes denser and tendsto sink into the lower layer and vice versa(Figure S7.7a). The alternating columns arecalled salt fingers. In the laboratory, salt fingerscan be produced easily and can grow to a fewmillimeters across and up to 25 cm long. Lateraldiffusion occurs between the “fingers” andproduces a uniform layer. Then the process

Salty

Fresh

Warm

Cold

N. Atlantic “Meddy” salt fingering

5 10 15Potential temperature (°C)

500

1000

Pre

ssur

e (d

bar)

35 36Salinity (psu)

Saltfingering

layers

Arctic diffusive layering (Kelley et al., 2003)

-2150

200

250

-1 0 1Temperature (°C)

Dep

th (

m)

(c)

(d)

RESPONSE TO WIND FORCING 19

may start again at the two interfaces that arenow present, and eventually a number of indi-vidually homogeneous layers develop withsharply defined temperature and salinity inter-faces (as in Figure S7.5j). In the ocean the layersmay be meters to tens of meters thick, separatedby thinner interface zones of sharp gradients oftemperature and salinity. External horizontalvelocities can disturb the growing fingers, soprediction of salt finger growth under alloceanic conditions is complex.

When cold, fresh water lies above warm, saltywater (Figure S7.7b), the lower layer, losing heatthrough the interface but not much salt, willbecome denser and water will tend to sink, againwithin its own layer. This is called the diffusiveform of double diffusion. The original stratifica-tion is strengthened by this double diffusiveprocess. An important difference from saltfingering is that fluid does not cross the interface.

Salt fingering effects are observed in the oceanwhere there are strong contrasts in salinity, forinstance, where salty Mediterranean Waterenters the Atlantic through the Strait of Gibraltar(Figure S7.7c). The saline water intrudes atmid-depth (about 1000e2000 m) into the cooler,less saline Atlantic water (Section 4.3 andFigure 4.10). Step structures in temperature/depth and salinity/depth traces due to doublediffusion are clear in CTD profiles below theMediterranean water (Figures S7.7c and S9.33)..Diffusive interfaces are observed in high latituderegions where there is a fresh, cold layer at thesurface with an underlying saltier temperaturemaximum layer (a dichothermal layer; Sections4.2 and 4.3.2 and Figure S7.7d).

7.5. RESPONSE TO WIND FORCING

The wind blows over the sea surface exertingstress and causing the water to move within thetop 50 m. Initially thewind excites small capillarywaves that propagate in the direction of thewind.Continued wind-driven momentum exchange

excites a range of surface waves (Chapter 8). Thenet effect of this input of atmosphericmomentumis a stress on the ocean (wind stress) (Section 5.8).For timescales of about a day and longer, Earth’srotation becomes important and the Corioliseffect enters in, as described in the followingsubsections.

7.5.1. Inertial Currents

The ocean responds initially to a wind stressimpulse with transient motions known as “iner-tial currents.” These are a balance of the Coriolisforce and the time derivatives of the initial hori-zontal velocities caused by the wind stress. Inthe Northern Hemisphere, Coriolis force actsto the right of the velocity. So if the current isinitially to the north, then Coriolis will move itto the east, and then to the south, and so forth.Thus, the water particles trace out clockwisecircles (Figure S7.8a). In the Southern Hemi-sphere, Coriolis force acts to the left and inertialcurrents are counterclockwise.

(Mathematically, inertial currents are thesolution of

vu=vt ¼ �fv (7.16a)

vv=vt ¼ �fu (7.16b)

which is taken from Eq. 7.11a and b assumingthat advection, pressure gradient forces, anddissipation are very small and can be neglected.)

Since the Coriolis force is involved, inertialcurrents vary with latitude. They have shortertime and length scales for higher latitudes. Thefrequency of an inertial current (time for a fullcircle) is the Coriolis parameter f, so the time ittakes for the circle (the period) is 2p/f. Since therotation is to the right of the initial stress (windimpulse), the average flow over a full circle ofthe inertial current is perpendicular to the windstress and to the right in the Northern Hemi-sphere and left in the Southern Hemisphere.

Inertial currents are often observed in sur-face drifter trajectories and surface velocity

Northern hemisphere

T = 2π/f

Southern hemisphere

T = 2π/fNorth

East

(a)(b)

(c)

49.0

48.5

48.0

47.5

47.0

46.5

46.0-143 -142 -141 -140 -139 -138 -137

Longitude

Latitude

FIGURE S7.8 (a) Schematics of inertial currents in the Northern and Southern Hemispheres. (b) Hodograph of inertialcurrents at 45�N for a wind blowing in the y-direction; the numbers are in pendulum hours. Source: From Ekman (1905).(c) Observations of near-inertial currents. Surface drifter tracks during and after a storm. Source: From d’Asaro et al. (1995).


moorings in the wake of a storm (Figure S7.8c).Inertial periods are often very close to tidalperiods, so separating tidal and inertial effectsin time series is sometimes difficult.

After the wind starts to blow impulsively, thecurrent will initially oscillate around and then,after several days, settle frictionally to a steadyflow at an angle to the wind (Figure S7.8bfrom Ekman, 1905). This becomes the surfaceEkman velocity (Section 7.5.3).

7.5.2. Langmuir Circulation

“Langmuir circulation” is another transientresponse to impulsive wind forcing, in whichhelical vortices form near the sea surface.

Langmuir cells (LCs) were first discussed byLangmuir (1938) who carried out a number ofexperiments to identify their character. LCsare visually evident as numerous long parallellines or streaks of flotsam (“windrows”) thatare mostly aligned with the wind, althoughthey can deviate by 20 degrees (Figure S7.9).

The streaks are formed by the convergencecaused by the vortices (Figure S7.10). Alternatecells rotate in opposite directions so that conver-gence and downwelling occurs at the surface (toform streaks of flotsam) between pairs of adja-cent cells, while divergence and upwellingoccurs between alternate pairs. (LCs onlybecome apparent to the eyewhen there is flotsamon the surface to be brought together by the

FIGURE S7.9 “Windrows” of foam, associatedwith the Langmuir circulation in LochNess. The surfacewave field suggeststhe wind direction, which is parallel to the narrow bands of foam. Source: From Thorpe (2004).

Thermocline

Spiral

5° to 15°

Win

w' ~ u'

v' asymmetric?

Wav

es

v' ~ u'?

Plankton in "zones of retention"

bubbles

Con

verg

ence

Con

verg

ence

Ekman

bubbles

(Form of bottom part not well known)

u' ~ 1% Wind

d

FIGURE S7.10 Langmuir circu-lation, first described by Langmuir(1938). Source: From Smith (2001).


convergences.) The water in the cells progressesdownwind as well, so that its motion is helical.

LCs have typical depth and horizontalspacing of 4e6 m and 10e50 m, but they canrange up to several hundred meters horizontal

separation and up to two to three times themixed layer depth. The cells can be manykilometers long. Multiple scales have beenobserved simultaneously in strong wind condi-tions (Assaf, Gerard, and Gordon, 1971). The


downwelling zones are concentrated in jetsoccupying one-third or less of the cell widthunder the streaks while upwelling is morewidely distributed at smaller speeds. Velocitieswithin an LC are only a fraction of the windvelocities that create them. Thus, the horizontalflow speed at the surface in the streaks can add10 cm/sec to the non-Langmuir currents else-where in the surface layer. The vertical downw-elling at the convergences is about one-third ofthe surface water speed as driven by the wind.Downwelling velocities are several centimetersper second and up to 20 cm/sec.

Langmuir circulations only appear whenthere are wind waves on the water surface, asin Figure S7.9. Surface films that dampen smallwaves tend to inhibit the formation of the cells.LCs generally occur only for wind speedsgreater than 3 m/sec and appear within a fewtens of minutes of wind onset. The mechanismfor producing Langmuir circulation is beyondthe scope of this text. See Smith (2001) andThorpe (2004) for further discussions.

Langmuir circulations provide a mechanismfor converting wave energy to turbulent energyand mixing and causing the upper layer todeepen. Mixed layer observations suggest thatLangmuir downwelling can penetrate to at leastthe middle of the mixed layer, therefore, it isexpected that the downwelling plumes can pene-trate to the bottom of the actively mixing layer(Weller et al., 1985; Smith, 2001). Langmuir cellscan generate internal waves in the stratified layerbelow the mixed layer that contribute to movingmomentum from themixed layer into the interior(Polton, Smith, Mackinnon, & Tejada-Martinez,2008). Thus LCs are one of several processesthat may contribute to surface mixing.

Note that Ekman’s theory of the wind drift(Section 7.5.3) yields an upper layer motionthat is about 45 degrees to the right of the

2 Collected as part of Fridtjof Nansen’s Fram expedition, the s

explain as his Ph.D. thesis, which focused on the response of w

analysis of these data showed that the sea ice drifted 20 to 4

wind, whereas LCs are more closely alignedto the wind. This is because the timescales ofthe two mechanisms are quite different. LCsare generated within minutes of the windonset and die out soon after the strong windpulse, whereas the Ekman circulation takesmany hours to develop.

7.5.3. Ekman Layers

Wind stress is communicated to the oceansurface layer through viscous (frictional)processes that extend several tens of metersinto the ocean. For timescales longer thana day, the response is strongly affected by Cori-olis acceleration. This wind-driven frictionallayer is called the Ekman layer after WalfridEkman (1905), who based his theory on shipdrift observations of the Fram in the Arctic.2

The classical surface Ekman layer is thesteady frictional response to a steady windstress on the ocean surface (Figure S7.11). Thephysical processes in an Ekman layer includeonly friction (eddy viscosity) and Coriolis accel-eration. Velocity in the Ekman layer is strongestat the sea surface and decays exponentiallydownward, disappearing at a depth of about50 m. It coexists with, but is not the same as,the mixed layer depth or euphotic zone depth.

The two most unusual characteristics of anEkman layer (compared with a frictional flowthat is not rotating) are (1) the horizontalvelocity vector spirals with increasing depth(Figure S7.11) and (2) the net transport inte-grated through the Ekman layer is exactly tothe right of the wind in the Northern Hemi-sphere (left in the Southern Hemisphere).

The surface water in an Ekman layer movesat an angle to the wind because of Coriolis accel-eration. If eddy viscosity is independent ofdepth, the angle is 45 degrees to the right of

hip drift and wind measurements were given to Ekman to

ater movement in the upper ocean to the wind stress. Later

0 degrees to the right of the wind (Nansen, 1922).

“Ekman flow” - Water velocities decreasingand rotating with increasing depth:

Resultant volume transportat right angles to wind

Ekman spiral

Wind

Surface

HorizontalPlane

DE

Vo

Vo

V1

V1

V2

V2

V3

V3

V4

V4

V5

V5

V6

V6

V7

V7

V9

V9

45°

W

FIGURE S7.11 Ekman layer velocities (Northern Hemi-sphere). Water velocity as a function of depth (upperprojection) and Ekman spiral (lower projection). The largeopen arrow shows the direction of the total Ekman trans-port, which is perpendicular to the wind.


the wind in the Northern Hemisphere (and tothe left of the wind in the Southern Hemi-sphere). If viscosity is not constant with depth,for instance, if the turbulence that createsthe eddy viscosity changes with depth, thenthe angle between the surface velocity and thewind will differ from 45 degrees.

As the surface parcel moves, a frictional stressdevelops between it and the next layer below.This accelerates the layer below, which movesoff to the right (Northern Hemisphere) of thesurface parcel. This second layer applies stressto the third layer, and so on. The total stressdecays with depth, at a rate that depends onthe eddy viscosity coefficient AV. Since eachsuccessively deeper layer is accelerated to the

right of the layer above it (NorthernHemisphere)and has a weaker velocity than the layer above it,the complete structure is a decaying “spiral.” Ifthe velocity arrows are projected onto a hori-zontal plane, their tips form the Ekman spiral(Figure S7.11). The whole spiral is referred to asthe “Ekman layer.”

The Ekman layer depth is the e-folding depthof the decaying velocity:

DE ¼ ð2Av=fÞ1=2 (7.17)

Using a constant eddy viscosity of 0.05 m2/secfrom within the observed range (Section 7.5.5),the Ekman layer depths at latitudes 10, 45, and80 degrees are 63, 31, and 26 m, respectively.The vertically integrated horizontal velocity inthe Ekman layer is called the Ekman transport:

UE ¼Z

uEðzÞ dz (7.18a)

VE ¼Z

vEðzÞ dz (7.18b)

where uE and vE are the eastward and north-ward velocities in the Ekman layer, and UE

and VE are the associated Ekman transports.(Ekman “transport” has units of depth timesvelocity, hence m2/sec, rather than area timesvelocity.) Ekman transport in terms of thewind stress is derived from Eq. (7.11):

UE ¼ sðyÞ=ðrfÞ (7.19a)

VE ¼ �sðxÞ=ðrfÞ (7.19b)

where s(x) and s(y) are the wind stresses positivein the east and north directions, assuming notime acceleration, advection, or pressuregradient force, and setting the eddy frictionstress at the sea surface equal to the wind stress.The Ekman transport is exactly perpendicularand to the right (left) of the wind in theNorthern (Southern) Hemisphere (large arrowin Figure 7.5).

TABLE S7.2 Angle of Surface Flow, a to the Right ofthe Wind Direction (Northern Hemi-sphere), with Overlapping Surface andBottom Ekman Layers

h/DE a Net flow direction in the water column

>1 45� At 90� to right of wind

0.5 45� About 60� to right of wind




For applications of Ekman layers to generalcirculation (Sections 7.8 and 7.9), only theEkman transport matters. Thus, the actualeddy viscosity and Ekman layer thickness areunimportant.

Ekman layers also form in the ocean’s surfacelayer below sea ice. When the ice is blown by thewind, friction between the sea ice and waterdrives the water. If the timescale is longer thana day, the Coriolis effect is important, and anEkman layer develops.

Ekman layers also occur at the ocean bottom(Section 7.4.2). Because of friction, the flow at thebottom must be zero. When the timescale of thedeep flow is longer than a day, Coriolis acceler-ation is important, and an Ekman layerdevelops, also 50 to 100 m thick above thebottom like the surface Ekman layer. If there isa current (e.g., a geostrophic current), flowingin the lower part of the water column over thesea bottom, which we will assume for simplicityto be flat, then there is a bottom frictional stresson the water. The frictional stress at the bottomacts in the opposite direction to the current.The result of the stress is a frictional transportto the right of the stress (Northern Hemisphere).Therefore the frictional transport (bottomEkman layer transport) is to the left of thecurrent. The total current (interior plus Ekman)must be zero at the bottom. The net result is anEkman current spiral in the bottom layer withthe total current rotating to the left as the bottomis approached.

In shallow water, the top and bottom Ekmanlayers can overlap, so that the right-turningtendency in the top layer (Northern Hemi-sphere) will overlap the left-turning tendencyin the bottom layer. The opposing right- andleft-turning effects will tend to cancel moreand more as the water depth decreases. If thereis a wind stress at the top surface that wouldproduce an Ekman layer of depth DE in deepwater, then in water of depth h, the approxi-mate angle a between the wind and the surfaceflow is as listed in Table S7.2. That is, as water

depth decreases, the net flow is more in thedirection of the wind.

Tides or internal waves (Chapter 8) rubbingagainst the bottom can also generate bottomEkman-like layers, but with time-dependentspiraling currents in the frictional layer.

7.5.4. Ekman Transport Convergenceand Wind Stress Curl

When the wind stress varies with position sothat Ekman transport varies with position, therecan be a convergence or divergence of waterwithin the Ekman layer. Convergence resultsin downwelling of water out of the Ekman layer.Divergence results in upwelling into the Ekmanlayer. This is the mechanism that connects thefrictional forcing by wind of the surface layerto the interior, geostrophic ocean circulation(Section 7.8).

Divergence and convergence occur if thetransport varies in the same direction as thetransport. In Figure S7.12, with varying zonal(west to east) wind, the Ekman transport is tothe right of the wind, and is convergentbecause the zonal wind varies with latitude.Note that it is not necessary for the Ekmantransports to be in opposite directions to havedivergence or convergence, just that the trans-ports change.

The vertical velocity wE at the base of theEkman layer is obtained from the divergenceof the Ekman transport, by vertically integrating

North

East

Up

Wind

Ekman transport

Ekman convergence/pumping(downwelling)

Ekman divergence/suction(upwelling)

FIGURE S7.12 Ekman transport convergence anddivergence in the Northern Hemisphere due to variations ina zonal (eastward) wind. Ekman transport is southward, tothe right of the wind. Divergent transport causes downw-elling, denoted by circles with a cross. Convergent transportcauses upwelling, denoted by circles with a dot.


the continuity equation Eq. (7.11e) over thedepth of the Ekman layer:

ðvUE=vxþ vVE=vyÞ ¼ V,UE

¼ �ðwsurface �wEÞ ¼ wE

(7.20)

where UE is the horizontal vector Ekman trans-port and it is assumed that the vertical velocityat the sea surface, wsurface, is 0. WhenEq. (7.20) is negative, the transport is convergentand there must be downwelling below the seasurface (increasingly negative wE). The relationof Ekman transport divergence to the windstress from Eq. (7.19a, b) is

V,UE ¼ v=vxðsðyÞ=ðrfÞÞ � v=vyðsðxÞ=ðrfÞÞ¼ k,V� ðs=rfÞ

(7.21)

where s is the vector wind stress and k is theunit vector in the vertical direction. Therefore,in the Northern Hemisphere (f> 0), upwellinginto the Ekman layer results from positivewind stress curl, and downwelling results

from negative wind stress curl. Downwellingis referred to as Ekman pumping. Upwelling issometimes referred to as Ekman suction.

A global map of wind stress curl was shownin Figure 5.16d, and is referred to frequently insubsequent chapters because of its importancefor Ekman pumping/suction, although themapped quantity should include the Coriolisparameter, f, to be related directly to upwellingand downwelling.

Equatorial upwelling due to Ekman trans-port results from the westward wind stress(trade winds). These cause northward Ekmantransport north of the equator and southwardEkman transport south of the equator. Thisresults in upwelling along the equator, eventhough the wind stress curl is small becauseof the Coriolis parameter dependence inEq. (7.21).

At the equator, where the Coriolis parameterchanges sign, zonal (east-west) winds can causeEkman convergence or divergence even withoutany variation in the wind (Figure S7.13a,b).Right on the equator, there is no Ekman layersince the Coriolis force that would create it iszero (f¼ 0). However, it has been shown fromobservations (Eriksen, 1982) that the Coriolisforce is important quite close to the equator inthe ocean, starting at about 1/4� latitude. If theequatorial wind is westward (a trade wind),then the Ekman transport just north of theequator is northward, and the Ekman transportjust south of the equator is southward, and theremust be upwelling into the surface layer on theequator. This is roughly included in Eq. (7.21)because of the variation in f, although the equa-tion is not accurate right on the equator where fvanishes.

The coastline is the other place where Ekmantransport divergence or convergence can occur,and it is not included in Eq. (7.21), because thisdivergence is due to the boundary condition atthe coast and not wind stress curl. If the windblows along the coast, then Ekman transportis perpendicular to the coast, so there must be

EquatorTrade Winds

Ekman transport (northward)

Ekman transport (southward)

NorthernHemisphere

SouthernHemisphere

NorthernHemisphere

SouthernHemisphere Equator

Upwelling

Ekman transport

Ekman transport

Thermocline

Sea surface

TradeWinds

coldwarmwarm

North

EastEkman transport

Isopycnal

Isopycnal

Onshore transport

Upwelling region

NorthernHemisphere

Alongshore windEastern boundary curr.

Poleward undercurrent

(a)

(b)

(c)

FIGURE S7.13 Ekman transport divergence near the equator driven by easterly trade winds. (a) Ekman transports. (b)Meridional cross-section showing effect on the thermocline and surface temperature. (c) Coastal upwelling system due to analongshore wind with offshore Ekman transport (Northern Hemisphere). The accompanying isopycnal deformations andequatorward eastern boundary current and poleward undercurrent are also shown (see Section 7.9).


either downwelling or upwelling at the coast tofeed the Ekman layer (Figure S7.13c). This isone mechanism for creation of coastalupwelling and subtropical eastern boundarycurrent systems. The other mechanism iswind stress curl in the near-coastal region thatdrives upwelling (Section 7.9). One such

example is the California-Oregon coast, wherethe mean wind during most of the yearincludes a component that blows southwardalong the coast. This causes westward(offshore) Ekman transport to the right of thewind. This means there must be upwelling atthe coast.

(a)

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

East Velocity (cm/s)

No

rth

Vel

oci

ty (

cm/s

)

Mean wind

(m/s)

8

1216

OBSERVATIONS

(slab extrapolation)

EKMAN THEORY

0

4

8

12

16

De = 25 m

De = 48 m

A = 274.2 cm /s

A = 1011 cm /s

Average currents and wind (6 Jun − 4 Oct 1993)

2

2

(b)

FIGURE S7.14 Observations of an Ekman-like responsein the California Current region. (a) Progressive vectordiagrams (Section 6.5.2) at 8, 16, 24, 32, and 40 m depth.Because of the way the ADCP measures, the currents areshown relative to a deeper depth, rather than as absolutecurrents. The wind direction and speed for each day isshown by the small arrows on the 8 m progressive vectorcurve. (b) Observed mean velocities (left) and two theoret-ical Ekman spirals (offset) using different eddy diffusivities(274 and 1011 cm2/S). The numbers on the arrows aredepths. The large arrow is the mean wind. Source: FromChereskin (1995).


7.5.5. Observations of Ekman Responseand Wind Forcing

The Ekman theory has major consequencesfor wind-driven ocean circulation. Thus it hasbeen important to confirm and refine Ekman’stheory with ocean observations, beyond theoriginal ice, wind, and ship drift observationsused by Ekman (1905) and Nansen (1922).

For instance, one assumption, that the eddyviscosity in the water column is constant withdepth, is not accurate. (Recall that the Ekmantransport is independent of viscosity, so thevariability of eddy viscosity does not matterfor large-scale circulation.) Eddy viscosity ishighest near the sea surface because of turbu-lence resulting from wind waves and inertialcurrents generated at the surface. Also, Ekmanassumed a steady wind. The speed with whichthe Ekman circulation develops depends onlatitude, because the Coriolis force dependson latitude. Observations of Ekman spiralsand Ekman response are very difficult becauseof the time dependence of the wind. It takesabout one pendulum day for inertial and Lang-muir responses (Sections 7.5.1 and 7.5.2) to dieout and an essentially Ekman circulation todevelop.

Ekman layer observations are also difficultbecause the spiral is thin compared with theusual vertical resolution of current measure-ments. Davis, deSzoeke, and Niiler (1981)measured currents in the mixed layer in thenortheast Pacific. By filtering the data and look-ing at responses at short and long timescales,they found that the currents at timescales oflonger than about one day looked like Ekman’stheory. Chereskin (1995) measured currents inthe mixed layer in the California Current usingan Acoustic Doppler Current Profiler (SectionS16.5.5.1 of Chapter S16 in the online supple-ment). Because the wind direction there wasrelatively steady, the Ekman-like responsewas clear (Figure S7.14) even without filteringthe data. The high eddy viscosity values that


Chereskin reported (Section 7.5.3) were obtainedby fitting the observed spiral to an Ekman layerwith depth-dependent viscosity.

An Ekman response to the wind for a largepart of the Pacific Ocean is apparent in theaverage 15 m velocity from surface driftersdeployed in the 1980s and 1990s. The surfacedrifters were drogued at 15 m depth, within theEkman layer. The drifter velocities from manyyears of observations were averaged and theaverage geostrophic velocity was subtracted.The resulting “ageostrophic” velocities, whichare likely the Ekman response, are to the rightof the wind stress in the Northern Hemisphereand to the left in the Southern Hemisphere(Figure S7.15).

The Ekman volume transports (horizontaland vertical) for each ocean and for the WorldOcean are shown in Figure S7.16. The easterlytrade winds (blowing westward) cause pole-ward horizontal Ekman flows in the tropicalAtlantic and Pacific. The westerlies (blowingeastward) cause equatorward flows at higherlatitudes. The Pacific Ekman transports arelarger than the Atlantic transports mainly

FIGURE S7.15 Ekman response. Average wind vectors (reThe current is calculated from 7 years of surface drifters daverage density data from Levitus, Boyer, and Antonov (1994aequator because the Coriolis force is small there.) This figure cNiiler (1999).

because the Pacific is so much wider, notbecause the wind stress differs. The near-equa-torial Indian transports are of the opposite signcompared with the Pacific, Atlantic, and totaltransports because of the large annual monsooncycle; the westerly winds dominate the annualmean in the equatorial Indian Ocean.

Associated with the convergences and diver-gences of the horizontal Ekman flows arevertical flows due to Ekman pumping (FigureS7.16b). Between approximately 40�S and40�N, downwelling prevails and the windscause convergent Ekman transport. Polewardof about 40 degrees, there is upwelling causedby divergent Ekman transport. The narrowregion of Ekman upwelling at about 5 to 10�Nis associated with the Intertropical ConvergenceZone in the winds. Not shown is the majorupwelling along the equator that must resultfrom the divergent Ekman transports theredue to the change of sign in the Coriolis param-eter. Again the Pacific and Atlantic have similardistributions and the Indian Ocean differsbecause of its strong annual (monsoonal) varia-tion north of the equator.

d) and average ageostrophic current at 15 m depth (blue).rogued at 15 m, with the geostrophic current based on) removed. (No arrows were plotted within 5 degrees of thean also be found in the color insert. Source: From Ralph and

FIGURE S7.16 (a) Zonally integrated meridional Ekman fluxes (Sv) for the three oceans by latitude and month. (Positiveis northward, negative is southward.) (b) Zonally integrated vertical Ekman volume flux (Sv) at the base of the Ekman layerper 10 degrees latitude belt by latitude and month. (Positive is up, negative is down.) Source: From Levitus (1988).


High pressure

Low pressure

v (velocity)

PGF CF

(a)

(b)

x

y

High pressure

Low pressure

Low pressure

PGF

CF

.v (in) v (out)

PGF

CF

xz

FIGURE S7.17 Geostrophic balance: horizontal forcesand velocity. (a) Horizontal forces and velocity ingeostrophic balance. PGF¼ pressure gradient force.CF¼Coriolis force. (b) Side view showing elevated pres-sure (sea surface) in center, low pressure on sides, balance ofPGF and CF, and direction of velocity v (into and out ofpage).


7.6. GEOSTROPHIC BALANCE

7.6.1. Pressure Gradient Force andCoriolis Force Balance

Throughout most of the ocean at timescaleslonger than several days and at spatial scaleslonger than several kilometers, the balance offorces in the horizontal is between the pressuregradient and the Coriolis force. This is called“geostrophic balance” or geostrophy.3

In a “word” equation, geostrophic balance is

horizontal Coriolis acceleration

¼ horizontal pressure gradient force (7.22)

This is illustrated in Figure S7.17. The pressuregradient force vector points from high pressureto low pressure. In a non-rotating flow, thewater would then move from high to low pres-sure. However, with rotation, the Coriolis forceexactly opposes the pressure gradient force, sothat the net force is zero. Thus, the water parceldoes not accelerate (relative to Earth). The parcelmoves exactly perpendicular to both the pres-sure gradient force and the Coriolis force.

A heuristic way to remember the direction ofgeostrophic flow is to think of the pressuregradient force pushing the water parcel fromhigh to low pressure, but Coriolis force movesthe parcel off to the right (Northern Hemi-sphere) or the left (Southern Hemisphere). Inthe resulting steady geostrophic state, the waterparcel moves exactly perpendicular to the pres-sure gradient force.

The vertical force balance that goes withgeostrophy is hydrostatic balance (Section 3.2).The vertical pressure gradient force, whichpoints upward from high pressure to low pres-sure, is balanced by gravity, which pointsdownward. Thus vertical acceleration, advec-tion, and diffusion are assumed to be very

3 The other terms in the force balance d the actual acceleration

no flow is exactly geostrophic.

small, just as in the horizontal momentumequations. (We note that in full treatments ofrotating fluid dynamics, the student will learnthat hydrostatic balance holds for a very largerange of fluid flows, not just those that aregeostrophic.)

The mathematical expression of geostrophyand hydrostatic balance, from Eq. (7.11a, b, c), is

, the advection, and diffusion d never completely vanish, so

GEOSTROPHIC BALANCE 31

�fv ¼ �ð1=rÞvp=vx (7.23a)

fu ¼ �ð1=rÞvp=vy (7.23b)

0 ¼ �vp=vz� rg (7.23c)

An alternate form for Eq. (7.23c), used fordynamic height calculations (Section 7.6.3), is

0 ¼ �a vp=vz� g (7.23d)

wherea is specific volume.Note howmanyof theterms in Eq. (7.11) have been assumed to be verysmall and therefore are left out in Eq. 7.23a,b).4

From Eq. (7.23a,b), if the Coriolis parameteris approximately constant (f¼ fo) and if densityin Eq. (7.23a,b) is also very nearly constant(r¼ ro; the “Boussinesq approximation”), thegeostrophic velocities are approximately non-divergent:

vu=vxþ vv=vy ¼ 0 (7.23e)

Formally in fluid dynamics, such a non-diver-gent velocity field can be written in terms ofa streamfunction j:

u ¼ �vj=vy and v ¼ vj=vx (7.23f)

From Eqs. (7.23a, b) the streamfunction forgeostrophic flow is j¼ p/(foro). Therefore,maps of pressure distribution (or its proxieslike dynamic height, steric height, or geopoten-tial anomaly; Section 7.6.2) are maps of thegeostrophic streamfunction, and flow approxi-mately follows the mapped contours.

4 Rigorous justification of geostrophic balance is based on smal

defined in Section 7.2.3, and the Ekman number is the non-dim

term to the size of the Coriolis term. For the vertical direction

parameter and H is a characteristic vertical length scale; note

in Eq. (7.17). Hydrostatic balance (Eq. 7.23c,d) is valid when

vertical scale of motion (H) to the horizontal scale of motion

more strongly justified when the Rossby number is small; tha

scales as the square of the aspect ratio times the Rossby num

assumptions are rigorously applied to the full set of moment

beyond the scope of this text.

Geostrophic balance is intuitively familiar tothose with a general interest in weather reports.Weather maps show high and low pressureregions around which the winds blow (FigureS7.18). Low pressure regions in the atmosphereare called cyclones. Hurricanes, dramatic winterstorms, and tornados are all cyclones. Flowaround low-pressure regions is thus calledcyclonic (counterclockwise in the NorthernHemisphere and clockwise in the SouthernHemisphere). Flow around high-pressureregions is called anticyclonic.

In the ocean, higher pressure can be causedby a higher mass of water lying above the obser-vation depth. At the “sea surface,” pressuredifferences are due to an actual mounding ofwater relative to Earth’s geoid. Over thecomplete width of the Atlantic or Pacific Oceananticyclonic gyres, the total contrast in sea-surface height is about 1 m.

The geostrophic velocities at the sea surfacecould be calculated if the appropriately time-averaged sea-surface height were known (asyet not possible for the time mean, but definitelypossible from satellite altimetry for variationsfrom the mean). The geostrophic velocity atthe sea surface in terms of sea-surface height habove a level surface is derived from Eqs.(7.23a,b):

�fv ¼ �gvh=vx (7.24a)

fu ¼ �gvh=vy (7.24b)

l Rossby and Ekman numbers, where the Rossby number is

ensional parameter that is the ratio of the size of the viscous

, the Ekman number is EV¼ 2AV/fH2, where f is the Coriolis

the resemblance of this parameter to the Ekman layer depth

the non-dimensional aspect ratio, which is the ratio of the

(L); that is, d¼H/L, is small. Hydrostatic balance is even

t is, the substantial derivative in the z-momentum equation

ber. Performing a complete “scale analysis” in which these

um equations, thus deriving the balances in Eq. (7.23), is far

FIGURE S7.18 Example of a daily weather map for North America, showing high- and low-pressure regions. Winds aregenerally not from high to low, but rather clockwise around the highs and counterclockwise around the lows. Source: FromNOAA National Weather Service (2005).


To calculate the horizontal pressure differencebelow the sea surface, we have to considerboth the total height of the pile of water aboveour observation depth and also its density,since the total mass determines the actual pres-sure at our observation depth (Figure S7.19).(This is where the vertical hydrostatic balancein Eq. 7.23c enters.) Therefore, if a mound ofless dense water lies above us in one locationand a shorter column of denser water inanother location, the total mass in the two pla-ces could be the same. Close to the sea surface,there would be a pressure difference betweenthe two places since the sea surface is higherin one location than in the other, but at depththe pressure difference would vanish becausethe difference in densities cancels the difference

5 The thermal wind balance should not be confused with the

directly involving buoyancy fluxes (Section 7.10).

in heights. Therefore there would bea geostrophic flow at the sea surface, whichwould decrease with depth until it vanishes atour observation depth (h3 in Figure S7.19a),where the total mass of the two columns ofwater is the same.

The variation in geostrophic flow with depth(the geostrophic velocity shear) is therefore propor-tional to the difference in density of the twowater columns on either side of our observationlocation. The relation between the geostrophicvelocity shear and the horizontal change(gradient) in density is called the thermal windrelation, since it was originally developed bymeteorologists measuring temperature andwind, rather than by oceanographers measuringdensity and currents.5

thermohaline circulation, which refers to ocean overturning

ρΑ ρΒ

ΗΑ

ΗΒ

h1

h3

h2

PGF

v

(a) (b)

ρ1

ρ2

ρ3

p1

p2

p3

A B

xz

PGF

v

v

v

FIGURE S7.19 Geostrophic flow and thermal windbalance. (a) Schematic of change in pressure gradientforce (PGF) with depth, assuming that the left column (A)is shorter and denser than the right column (B), that is,rA> rB and HA<HB. The horizontal geostrophic velocityV is into the page for this direction of PGF and is stron-gest at the top, weakening with depth, as indicated by thecircle sizes. (If the densities of the two columns were thesame, then the PGF and velocity V are the same at alldepths.) (b) Same, but for density (red) increasing withdepth, and isopycnals tilted, and assuming that the seasurface at B is higher than at A so that the PGF at the seasurface (h1) is to the left. The PGF decreases withincreasing depth, as indicated by the flattening of theisobars p2 and p3.


The thermal wind relation is illustrated inFigure S7.19b. The sea surface is sloped, withsurface pressure higher to the right. This createsa pressure gradient force to the left, whichdrives a surface geostrophic current into thepage (Northern Hemisphere). The density rincreases with depth, and the isopycnals aretilted. Therefore the geostrophic velocitychanges with depth because the pressuregradient force changes with depth due to thetilted isopycnals. Because the isopycnals aresloped in the opposite direction to the sea-surface height, the into-the-page geostrophicvelocity is reduced with depth. That is, whenthere is light water under a high sea surfaceand dense water under a low sea surface, thehorizontal pressure gradients become smallerwith depth, since the mass of the two columnsbecomes more equalized with depth.

A useful rule of thumb for geostrophic flowsthat are surface-intensified is that, when facingdownstream in the Northern Hemisphere, the

“light/warm” water is to your right. (In theSouthern Hemisphere, the light water is tothe left when facing downstream.) This can besafely recalled by memorizing the examplefor the Gulf Stream recalling that the currentflows eastward with warm water to the south.

Geostrophic flow with vertical shear, whichrequires sloping isopycnals, is often called bar-oclinic. Geostrophic flow without any verticalshear is often called barotropic. Barotropicflow is driven only by horizontal variationsin sea-surface height. Most oceanic geostrophicflows have both barotropic and barocliniccomponents.

Mathematically, the thermal wind relationsare derived from the geostrophic and hydro-static balance Eq. (7.23):

�fvv=vz ¼ ðg=r0Þvr=vx (7.25a)

fvu=vz ¼ ðg=roÞvr=vy (7.25b)

(Here we have again used the Boussinesqapproximation, where r is replaced by theconstant ro in the x- and y-momentum equa-tions, whereas the fully variable density r mustbe used in the hydrostatic balance equation.)

To calculate geostrophic velocity, we mustknow the absolute horizontal pressure differ-ence between two locations. If we have onlythe density distribution, we can calculate onlythe current at one level relative to that atanother, that is, the geostrophic vertical shear.To convert these relative currents into absolutecurrents, we must determine or estimate theabsolute current or pressure gradient at somelevel (reference level).

The selection of a reference level velocity isone of the key problems in using the geostrophicmethod to compute currents. A common, butusually inaccurate, referencing approach hasbeen to assume (without measuring) that theabsolute current is zero at some depth (level ofno motion). In the case of western boundarycurrents or in the ACC where the currents


extend to great depth, this is not a good assump-tion. Nevertheless the relative geostrophicsurface current calculation can be revealingsince the surface currents are usually muchstronger than the deep ones, and small relativeerror in the surface currents might be toleratedwhile the same amount of error in the deepcurrents is untenable.

In the next subsection, we introduce the“dynamic” method widely used to calculategeostrophic velocities (shear), and continue thediscussion of reference velocity choices.

7.6.2. Geopotential and DynamicHeight Anomalies and Reference LevelVelocities

Historically and continuing to the present, ithas been too difficult and too expensive toinstrument the ocean to directly observevelocity everywhere. Density profiles, whichare much more widely and cheaply collected,are an excellent data set for estimatinggeostrophic velocities using the thermal windrelations and estimates of the reference levelvelocity. This approach to mapping oceancurrents is called the “dynamic method”; itwas developed in the School of Geophysics inBergen, Norway, more than a century ago.This is the same school where both Nansenand Ekman worked, contributing some verysignificant ideas to physical oceanography. Thedynamic method has origins in both oceanog-raphy andmeteorology. In the dynamic method,the distribution of mass in the ocean is used tocompute an important component of the currentfield. In this text the emphasis will be on howthis method is commonly used in descriptivestudies of ocean circulation rather than deriva-tion of the method.

In the ocean the distribution of mass is repre-sented by the distribution of density over boththe horizontal and vertical dimensions. Oncethe density profiles at two locations (“stations”)

are calculated from observed temperature andsalinity, the distribution of mass at the twostations can be used to calculate the verticalshear of geostrophic velocity at the midpointbetween the two stations at all depths that arecommon to the two stations (Section 7.6.1).Then, if the velocity is known at one depth (oris assumed to have a certain value, e.g., zero),the vertical shear can be used to give the velocityat all other depths. The assumed or measuredvelocity at one depth is called the referencevelocity, and its depth is called the “referencedepth” or reference level. Thus in this method,the horizontal change in the distribution ofmass creates the horizontal pressure gradient,which drives the geostrophic flow.

Oceanographers have created two closelyrelated functions, geopotential anomaly anddynamic height, whose horizontal gradientsrepresent the horizontal pressure gradient force.Another closely related concept, steric height, isused to study variations in sea level. All arecalculated from the density profiles computedfrom the measured temperature and salinityprofiles. Sverdrup, Johnson, and Fleming(1942), Gill and Niiler (1973), Gill (1982), Pondand Pickard (1983), and Stewart (2008) area few of the many useful references for thesepractical quantities.

The gradient of the geopotential, F, is in thedirection of the local force due to gravity (modi-fied to include centrifugal force). The geopoten-tial gradient is defined from hydrostatic balance(Eq. 7.23c) as

dF ¼ g dz ¼ �a dp (7.26a)

where a is specific volume. The units of geopo-tential are m2/sec2 or J/kg. For two isobaricsurfaces p2 (upper) and p1 (lower), the geopo-tential is

F ¼ g

Zdz ¼ gðz2 � z1Þ ¼ �

Zadp (7.26b)


Geopotential height is defined as

Z ¼ �9:8 m s�2

��1Z

g dz

¼ ��9:8 m s�2��1

Za dp (7.26c)

and is nearly equal to geometric height. Thisequation is in mks units; if centimeter-gram-second (cgs) units are used instead, the multipli-cative constant would change from 9.8 m s�2 to980 cms�2.Most practical calculations, includingcommon seawater computer subroutines, use thespecific volume anomaly

d ¼ aðS;T;pÞ � að35; 0;pÞ (7.26d)

to compute the geopotential anomaly

DF ¼ �Z

d dp: (7.26e)

The geopotential height anomaly is then definedas Z

Z0 ¼ �ð9:8 m s�2Þ�1 d dp: (7.26f)

Geopotential height anomaly is effectivelyidentical to steric height anomaly, which isdefined by Gill and Niiler (1973) as

h0 ¼ �ð1=roÞZ

r0 dz (7.27a)

in which the density anomaly r’¼ r� ro. Usinghydrostatic balance and defining ro asr(35,0,p), Eq. (7.27a) is equivalent to Tomczakand Godfrey’s (1994) steric height (anomaly)

h0 ¼Z

d ro dz (7.27b)

which can be further manipulated to yield

h0 ¼ ð1=gÞZ

d dp: (7.27c)

This is nearly identical to the geopotentialheight anomaly in Eq. (7.26f), differing only inthe appearance of a standard quantity for g. InSI units, steric height is in meters.

Dynamic height, D, is closely related to geopo-tential, F, differing only in sign and units ofreporting. Many modern publications andcommon computer subroutines do not distin-guish between dynamic height and geopotentialanomaly. The unit traditionally used fordynamic height is the dynamic meter:

1 dyn m ¼ 10 m2=sec2: (7.28a)

Therefore dynamic height reported in dynamicmeters is related to geopotential anomaly as

DD ¼ �DF=10 ¼Z

d dp=10: (7.28b)

Its relation to the geopotential height and stericheight anomalies is

10 DD ¼ �9:8 Z0 ¼ gh0: (7.28c)

The quantities DD and Z0 are often used inter-changeably, differing only by 2%. With use ofthe dynamic meter, maps of dynamic topog-raphy are close to the actual geometric heightof an isobaric surface relative to a level surface;for example, a horizontal variation of 1 dynmmeans that the isobaric surface has a horizontaldepth variation of about 1 m. Note that the geo-potential height anomaly more closely reflectsthe actual height variation, so a variation of1 dynm would be an actual height variationcloser to 1.02 m.

Geostrophic velocities at one depth relative tothose at another depth are calculated using Eq.(7.25) with geopotential anomalies, steric heightanomalies, or dynamic heights. In SI units, andusing dynamic meters for dynamic height, thedifference between the northward velocity vand eastward velocity u at the pressure surfacep2 relative to the pressure surface p1 is


f�v2 � v1

� ¼ 10 vDD=vx ¼ �vDF=vx

¼ gvh0=vx (7.29a)

f�u2 � u1

� ¼ �10 vDD=vy ¼ vDF=vy

¼ �gvh0=vy (7.29b)

where the dynamic height or geopotentialanomalies are integrated vertically from p1 top2. The surface p1 is the reference level. (Compar-ison of Eq. 7.29 with Eq. 7.23 shows that thedynamic height and geopotential anomaliesare streamfunctions for the difference betweengeostrophic flows from one depth to another.)

How is the velocity at the reference levelchosen? Since the strength of ocean currentsdecreases from the surface downward in many(but not all) regions, for practical reasons,a deep level of no motion has often beenpresumed. A much better alternative is to usea “level of knownmotion”. For example, currentmeter measurements, or the tracks of sub-surface floats, may be used to define the currentat some level, and then dynamic or stericheights can be used to compute currents at allother levels relative to the known referencelevel. Another modern practice is to requirethat the entire flow field, which is defined bymany density profiles, satisfy some overallconstraints. An obvious one is that there canbe no net transport into a region enclosed bya set of stations (otherwise there would be anincreasing mound or hole in that region).Another one is that the chemistry must makesense d there can no net production of oxygenwithin the ocean outside the surface layer forinstance. Another type of constraint is that theflows match measured velocities from currentmeters or floats, but allowing for some error inthe match. The constraints then help narrowthe choices of reference level velocities. Formalversions of these methods, first applied to thereference level problem by Carl Wunsch inthe 1970s, are called inverse methods because ofthe mathematics used to connect the constraints

to the choices of reference velocities (seeWunsch, 1996).

Another apparently attractive option is to usesatellite altimeters to measure the sea-surfaceheight, which would give the pressure distribu-tion and hence geostrophic currents at the seasurface. These can be used to reference thegeostrophic velocities calculated at all depthsbelow the surface using dynamic heightprofiles. However, while the sea surface eleva-tion is measured very precisely by satellitealtimeters, the height includes Earth’s geoid,which has large spatial variations that are notyet well measured; this leads to spurious surfacecurrents if one simply calculates the gradient inmeasured surface height. The geoid does notvary in time, so satellite altimetry does provideexcellent information on time changes of thesurface geostrophic currents. The GRavity andEarth Climate Experiment (GRACE) satellite,launched in 2002 to measure the shorter spatialscales of Earth’s gravity field, is helping toresolve this geoid problem. Satellite altimetersand GRACE are described in the online supple-mentary Chapter S16.

As an example of the geostrophic method,we calculate dynamic height and a geostrophicvelocity profile from two density profiles thatstraddle the Gulf Stream (Figure S7.20 andTable S7.3). The isopycnals sloping upwardtoward the north between 38 and 39�N markthe horizontal pressure gradient associatedwith the Gulf Stream (Figure S7.20a). Thegeostrophic velocity profile is calculatedbetween stations “A” and “B” relative to anarbitrary level of no motion at 3000 m. (If itwere known, the velocity at 3000m can beadded later to the full velocity profile.) StationA has lower specific volume (higher potentialdensity) than station B (Figure S7.20b). Thesurface dynamic height at A is therefore lowerthan at B (Figure S7.20c) and the surface pres-sure gradient force is toward the north, fromB to A. Therefore, the geostrophic velocity atthe midpoint between the stations (Figure

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000100 200 300

27.527.7

27.8

27.88

27.0

26.5

26.0

Dep

th (

met

ers)

Distance (km)

38°N 39°N 40°N

Stations

Potentialdensity(kg/m3)

(a) (d)(b) (c)

0 200 400 0 2 0 50 100

Specific volume anomaly(x 10–8 m3/kg)

Dynamic height(dyn m)

Geostrophic velocity(cm/sec)

B A

B B

A A

FIGURE S7.20 (a) Potential density section across the Gulf Stream (66�W in 1997). (b) Specific volume anomalyd (� 10�8 m3/kg) at stations A and B. (c) Dynamic height (dyn m) profiles at stations A and B, assuming reference level at3000 m. (d) Eastward geostrophic velocity (cm/sec), assuming zero velocity at 3000 m.


S7.20d) is eastward and is largest at the seasurface. This means that the sea surface musttilt downward from B to A. The vertical shearis largest in the upper 800 m where the differ-ence in dynamic heights is largest.

For practical applications in which themaximum depths of density profiles vary, it isoften most convenient to first calculate dynamicheight by integrating from the surface down-ward for every profile (Table S7.3), and thencalculate the associated geostrophic velocityrelative to 0 cm/sec at the sea surface (column4 in Table S7.3). This geostrophic velocity profile

is likely not close to the actual velocity profile,since velocities are usually small at depth andnot at the sea surface. Then the assumed or inde-pendently measured velocity at the chosen deepreference level is compared with the velocity atthe reference level calculated relative to 0 cm/sec at the sea surface, and the entire geostrophicvelocity profile is offset by the difference. Forinstance, if our reference velocity choice is0 cm/sec at 3000 m, then we look for the calcu-lated geostrophic velocity at 3000 m relative to0 at the sea surface and subtract this from thevelocities at all depths (column 5 in Table

TABLE S7.3 Computation of Dynamic Height and Geostrophic Current Between Stations A and B Relative to anAssumed Zero Velocity at 3000 m and at the Deepest Common Level (DCL)

Eastward speed (cm/sec)

Depth (m) DA (dynm) DB (dynm) Relative to sea surface Ref. 0 cm/sec at 3000 m Ref. 0 cm/sec at DCL*

0 0 0 0 94.83 95.13

50 �0.230 �0.211 �2.78 92.04 92.32

100 �0.416 �0.340 �10.70 84.12 84.40

150 �0.552 �0.425 �17.89 76.94 77.22

200 �0.657 �0.495 �22.85 71.98 72.26

300 �0.835 �0.611 �31.52 63.31 63.59

400 �1.005 �0.708 �41.80 53.03 53.31

500 �1.164 �0.785 �53.27 41.56 41.84

600 �1.300 �0.846 �63.79 31.04 31.32

800 �1.511 �0.947 �79.37 15.45 15.73

1000 �1.652 �1.039 �86.29 8.53 8.81

1500 �1.904 �1.267 �90.19 4.64 4.92

2000 �2.142 �1.489 �91.87 2.96 3.24

2500 �2.377 �1.712 �93.45 1.37 1.65

3000 �2.602 �1.928 �94.85 0.0 0.28

3500 �2.814 �2.136 �95.34 �0.52 0.24

4000 �3.024 �2.347 �95.26 �0.43 0.15

4500 �3.243 �2.566 �95.13 �0.31 0.03

4710 (DCL*) �3.343 �2.667 �95.10 �0.28 0.0

Note: Although D is called “height” and is quoted in units of “dynamic meters,” it has physical dimensions of energy per unit mass as it

represents work done against gravity.

Distance between the two stations¼ 78.0 km; latitude¼ 38.65�N.

The SI units for D are J/kg¼m2/s2.


S7.3). If our best estimate of a reference velocityis 0 cm/sec at the bottom (deepest commonlevel; DCL), then we offset the velocities by thevalue at the bottom (column 6 in Table S7.3). Ifwe have measured the bottom current to be5 cm/sec, then we add an offset to the completevelocity profile so as to yield 5 cm/sec at thebottom.

The DCL is the maximum depth at whichgeostrophic velocity can be calculated for thisparticular station pair, since the shallower ofthe two stations extends to 4710 dbar (thedeeper of the pair extends to 4810 dbar). Espe-cially for transport calculations in which thebottom current is of interest, the geostrophicvelocity below the DCL is needed, but there is


only one density profile available. There area variety of ways to assign velocity to this“bottom triangle,” including (1) no assignment,(2) assignment of velocity at the DCL, (3) extrap-olation of velocity profile from above the DCL,(4) extrapolation of the velocity horizontallyfrom the next station pair if there is one, or (5)objective mapping of the velocity field into thetriangle. The last is the best way, but an objectivemapping scheme might not be readily available.

7.6.3. Dynamic Topography andSea-Surface Height Maps

Dynamic height at one surface relative toanother is the streamfunction for thegeostrophic flow at that surface relative to theother, as an extension of Eq. (7.23f). Flows arealong the contours with the high “hills” to theright of the flow in the Northern Hemisphere(to the left in the Southern Hemisphere). Thespeed at any point is proportional to the steep-ness of the slope at that point; in other words,it is inversely proportional to the separation ofthe contours.

Dynamic topographymaps (equivalently, stericheight or sea-surface height) are shown inChapter 14 and throughout the ocean basinchapters (9e13) to depict the geostrophic flowfield. As an illustration of the common featuresfor all basins, we show here dynamic topog-raphy maps for the Pacific and Atlantic Oceans(Figures S7.21 and S7.22). These were the firstmodern basin-wide maps in common use andthus have some historical interest; both showdynamic height relative to a deep level of nomotion. For comparison, Figures 9.2a and 10.2aare the surface steric height maps from Reid(1994, 1997) that we use to illustrate circulationin the Pacific and Atlantic Ocean chapters. Thesteric height in these maps has been adjustedto represent the full flow, hence incorporatingestimates of deep geostrophic velocities at allstation pairs.

At the sea surface, all five ocean basins havehighest dynamic topography in the west in thesubtropics. The anticyclonic flows around thesehighs are called the subtropical gyres. TheNorthern Hemisphere oceans have lowdynamic topography around 50e60�N; thecyclonic flows around these lows are thesubpolar gyres. Tightly spaced contours alongthe western boundaries indicate the swiftwestern boundary currents for each of the gyres.Low values are found all the way around Ant-arctica; the band of tightly spaced contours toits north marks the eastward ACC. The contrastin dynamic height and sea-surface height fromhigh to low in a given gyre is about 0.5 to 1dynamic meters.

In the subtropical gyres in Figures S7.21 andS7.22, close contour spacings, hence largegeostrophic velocities, are found at the westernboundaries. These include the energeticsubtropical western boundary currents justeast of Japan (Kuroshio), east of North America(Gulf Stream), east of Australia (East AustralianCurrent), east of Brazil (Brazil Current), and eastof southern Africa (Agulhas Current).

The similarity between the two Pacificsurface maps (and the two Atlantic surfacemaps) indicates that indeed the flow at 1000dbar (700 dbar) is relatively weak. The addi-tional advantage of the Reid (1994, 1997) anal-yses is that he also produced maps of absolutedynamic topography at 1000 dbar, and at 500dbar intervals to the ocean bottom, whereasthe simple dynamic topography methodassuming a level of no motion clearly does notyield a reasonable flow field at these depths.

7.6.4. A Two-Layer Ocean

It is frequently convenient to think of theocean as composed of two layers in the vertical,with upper layer of density r1 and lower layer ofdensity r2 (Figure S7.23). The lower layer isassumed to be infinitely deep. The upper layer

FIGURE S7.21 Mean annual dynamic topography of the Pacific Ocean sea surface relative to 1000 dbar in dyn cm(DD¼ 0/1000 dbar). Source: From Wyrtki (1975).


thickness is h +H, where h is the varying heightof the layer above the sea level surface and H isthe varying depth of the bottom of the layer. Wesample the layers with stations at “A” and “B.”Using the hydrostatic equation (7.23c), wecompute the pressure at a depth Z at thestations:

pA ¼ r1g�hA þHA

�þ r2g�Z�HA

�(7.30a)

pB ¼ r1g�hB þHB

�þ r2g�Z�HB

�: (7.30b)

Here Z represents a common depth for bothstations, taken well below the interface. If we

FIGURE S7.22 Dynamic topography of 100 dbar surface relative to 700 dbar surface (DD¼ 100/700 dbar) in dyn cm in theAtlantic Ocean. Source: From Stommel, Niiler, and Anati (1978).

Sea surface

Ideal level surface

Pycnocline

(b)

HA

HB

hA

hB

Pycnocline

(a)

z

ρ1

ρ2

ρ

FIGURE S7.23 The two-layer ocean. (a) Vertical density profile with upper and lower layers of density r1 and r2. (b) Seasurface and pycnocline for two stations, A and B, where the thickness of the layer above the “ideal level surface” is hA andhB and the thickness of the layer below the level surface is HA and HB, respectively. Both h and H are part of the “upper”layer shown in (a).


Sea surface

Level surface

Pycnocline

Cyclonic eddy

Anticyclonic eddy

Sea surface

Level surface

Pycnocline

(a)

(b)

FIGURE S7.24 Two-layer oceandepiction of a (a) “cold,” cyclonicocean circulation showing the“ideal” sea surface and the subsur-face thermocline structure and a (b)“warm” anticyclonic circulation.


assume thatpA¼ pB,which amounts to assuminga “level of no motion” at Z, we can computea surface slope, which we cannot measure interms of the observed density interface slope:

hA � hBDx

¼ r2 � r1r1

HA �HB

Dx(7.31a)

We then use Eq. (7.30a) to estimate the surfacevelocity v:

fv ¼ ghA � hB

Dx¼ g

r2 � r1r1

HA �HB

Dx(7.31b)

This says thatwe can estimate the slope of the seasurface (hA � hB) from knowledge of the subsur-face slope of the density interface (HA�HB),which then allows us to estimate the surfaceflow velocity from the shape of the pycnocline.

This simple construct is also useful in depict-ing various forms of geostrophic circulation

features. For example, a cyclonic feature in eitherhemisphere is drawn in Figure S7.24a where thesubsurface pycnocline slope ismuch greater thanthe surface topographic change. These cyclonicfeatures are also known as cold features due tothe upwelling of the central isopycnals in thecenter of the feature. This is true even if thefeature is not a closed circulation. Likewisea warm feature looks like Figure S7.24b regard-less of hemisphere. What will change with thehemisphere is the direction of the flow wherea warm feature rotates anticyclonically (clock-wise in the Northern Hemisphere) and a coldfeature is cyclonic (counterclockwise). The two-layer depiction of the ocean is convenient forquickly evaluating new measurements in termsof the corresponding geostrophic currents. Notethat the two-layer assumption results ina mapping of only geostrophic currents and notthe entire current field.

VORTICITY, POTENTIAL VORTICITY, ROSSBY AND KELVIN WAVES, AND INSTABILITIES 43

7.7. VORTICITY, POTENTIALVORTICITY, ROSSBY AND KELVIN

WAVES, AND INSTABILITIES

Ocean currents are mostly geostrophic. Thismeans that the equation for velocity includesonly the pressure gradient force and Coriolisforce. This poses an apparent problem: Howdo we insert external forces such as the wind?In formal geophysical fluid dynamics, wewould show that these forces are in themomentum equations, but are so weak that wesafely consider the flows to be geostrophic (tolowest order). To reinsert the external forces,we have to consider the “vorticity” equation,which is formally derived from the momentumequations by combining the equations in a waythat eliminates the pressure gradient forceterms. (It is straightforward to do.) The resultingequation gives the time change of the vorticity,rather than the velocities. It also includes dissi-pation, variation in Coriolis parameter with lati-tude, and vertical velocities, which can be setexternally by Ekman pumping.

7.7.1. Vorticity

Vorticity in fluids is similar to angularmomentum in solids, and many of the intuitions

Current

North

East

Paddlewheel circulation

Right-hand rule, thumb up:positive vorticity

(a)

Up Up

FIGURE S7.25 Vorticity. (a) Positive and (b) negative vorticithe direction of the thumb (upward for positive, downward fo

developed about angular momentum froma standard physics course can be applied tounderstanding vorticity.

Vorticity is twice the angular velocity at a pointin a fluid. It is easiest to visualize by thinking ofa small paddle wheel immersed in the fluid(Figure S7.25). If the fluid flow turns the paddlewheel, then it has vorticity. Vorticity is a vector,and points out of the plane in which the fluidturns. The sign of the vorticity is given by the“right-hand” rule. If you curl the fingers onyour right hand in the direction of the turningpaddle wheel and your thumb points upward,then the vorticity is positive. If your thumbpoints downward, the vorticity is negative.

Vorticity is exactly related to the concept ofcurl in vector calculus. The vorticity vector u

is the curl of the velocity vector v, expressedhere d in Cartesian coordinates:

u ¼ V� v

¼ iðvv=vz� vw=vyÞ þ jðvw=vx� vu=vzÞþ kðvv=vx� vu=vyÞ

(7.32)

where (i, j, k) is the unit vector in Cartesian coor-dinates (x, y, z) with corresponding velocitycomponents (u, v, w). Vorticity, therefore, hasunits of inverse time, for instance, (sec)�1.

North

East

Paddlewheel circulation

Current

Right-hand rule, thumb down:negative vorticity

(b)

ty. The right-hand rule shows the direction of the vorticity byr negative).


Fluids (and all objects) have vorticity simplybecause of Earth’s rotation. This is called plane-tary vorticity. We do not normally appreciatethis component of vorticity since it is only impor-tant if a motion lasts for a significant portion ofa day, and most important if it lasts for manydays, months, or years. Since geostrophic motionis essentially steady compared with the rotationtime of Earth, planetary vorticity is very impor-tant for nearly geostrophic flows. The vectorplanetary vorticity points upward, parallel tothe rotation axis of Earth. Its size is twice theangular rotation rate U of Earth:

uplanetary ¼ 2U (7.33)

where U¼ 2p/day¼ 2p/86160 sec¼ 7.293�10�5 sec�1, so uplanetary ¼ 1.4586� 10�4 sec�1.

The vorticity of the fluid motion relative toEarth’s surface (Eq. 7.32) is called the relativevorticity. It is calculated from the water velocitiesrelative to Earth’s surface (which is rotating).The total vorticity of a piece of fluid is the sumof the relative vorticity and planetary vorticity.The total vorticity is sometimes called absolutevorticity, because it is the vorticity the fluid hasin the non-rotating reference frame of the stars.

For large-scale oceanography, only the localvertical component of the total vorticity isused because the fluid layers are thin comparedwith Earth’s radius, so flows are nearly hori-zontal. The local vertical component of the plan-etary vorticity is exactly equal to the Coriolisparameter f (Eq. 7.8c) and is therefore maximumand positive at the North Pole (4¼ 90�N),maximum and negative at the South Pole(4¼ 90�S), and 0 at the equator.

The local vertical component of the relativevorticity from Eq. (7.32) is

z ¼�vv

vx� vu

vy

�¼ curlzv (7.34)

The local vertical component of the absolutevorticity is therefore (z + f). The geostrophic

velocities calculated from Eq. (7.23) (Section7.6) are often used to calculate relative vorticity.

7.7.2. Potential Vorticity

Potential vorticity is a dynamically importantquantity related to relative and planetaryvorticity. Conservation of potential vorticity isone of the most important concepts in fluiddynamics, just as conservation of angularmomentum is a central concept in solid bodymechanics. Potential vorticity takes into accountthe height H of a water column as well as itslocal spin (vorticity). If a column is shortenedand flattened (preserving mass), then it mustspinmore slowly. On the other hand, if a columnis stretched and thinned (preserving mass), itshould spin more quickly similar to a spinningice skater or diver who spreads his or her armsout and spins more slowly (due to conservationof angular momentum). Potential vorticity,when considering only the local vertical compo-nents, is

Q ¼ ðzþ fÞ=H (7.35)

where H is the thickness, if the fluid is unstrati-fied. When the fluid is stratified, the equivalentversion of potential vorticity is

Q ¼ �ðzþ fÞð1=rÞðvr=vzÞ: (7.36)

When there are no forces (other than gravity) onthe fluid and no buoyancy sources that canchange density, potential vorticity Q isconserved:

DQ=Dt ¼ 0 (7.37)

where “D/Dt” is the substantial derivative (Eq.7.4). This means that a water parcel keeps thevalue of Q that it obtains wherever a force actson it. For instance, parcels of water leaving theocean surface layer, where they are subject towind forcing, which changes their potentialvorticity, keep the same value of potential


vorticity after they enter the ocean interior whereforces (primarily friction) are much weaker.

Considering the potential vorticity (Eq. 7.35),there are three quantities that can change: rela-tive vorticity z, the Coriolis parameter f, andthe thickness H (or equivalent thickness�(1/r)(vr/vz) in Eq. 7.36). The variation in fwith latitude has huge consequences for oceancurrents and stratification. Therefore, a specialsymbol b is introduced to denote the change inf with northward distance y, or in terms of lati-tude f and Earth’s radius Re:

b ¼ df=dy ¼ 2U cos F=Re (7.38)

We often refer to the “b-effect” when talkingabout how changes in latitude affect currents,or the very large-scale, mainly horizontalRossby waves for which the b-effect is therestoring force, described in Section 7.7.3.

All three components of potential vorticitycan change together, but we learn more aboutwhat happens if we consider just two at a time.

First we consider changes in relative vorticityz and Coriolis parameter f, holding thickness Hconstant (Figure S7.26). When a water parcel ismoved northward, it experiences an increase inf. Its relative vorticity z must then decrease to

Relativevorticity

ζ = 0

H

Q = f(θ1)/H

move northwardsLatitude θ1 L

Q = (f(θ2)

Conservation of potential vorticity Q in thestretching (northern hemispher

balance of planetary vorticity and relati

keep the numerator of Eq. (7.35) constant. If z iszero to start with, z will become negative andthe water parcel will rotate clockwise. If theparcel is moved southward, f decreases and itsrelative vorticity will have to become more posi-tive; the parcel will rotate counterclockwise.

Secondly, we consider changes in relativevorticity z and thickness H, holding the latitudeand hence f constant (Figure S7.27). (This wouldbe appropriate for mesoscale eddies with highrelative vorticity that do not move far from theirinitial latitude, Arctic dynamics, or for flows inrotating laboratory tanks.) An increase in thick-ness H (“stretching”) must result then in anincrease in relative vorticity, and the waterparcel will rotate more in the counterclockwisedirection. A decrease in thickness (“squashing”)results in a decrease in relative vorticity, and thewater parcel will rotate more in the clockwisedirection.

Thirdly, if the thickness H is allowed to vary,and if the relative vorticity is very small (such asin the very weak currents in the mid-ocean),then a northward move that increases f mustresult in column stretching (Figure S7.28). Simi-larly, a southward move would cause H todecrease or squash. (Since neither thickness

Relativevorticity

ζ < 0

atitude θ2

+ ζ)/H = f(θ1)/H

absence of e):ve vorticity

FIGURE S7.26 Conservation ofpotential vorticity: changes in rela-tive vorticity and Coriolis param-eter f, if thickness is constant.

Relativevorticity

ζ = 0

Relativevorticity

ζ > 0

H1

Q = f(θ1)/H1

Latitude θ1 Latitude θ1

Q = (f(θ1) + ζ)/H2 = f(θ1)/H1

Conservation of potential vorticity Q in the absence of planetary vorticity change (northern hemisphere):

balance of relative vorticity and stretching

H2

(same latitude)

FIGURE S7.27 Conservation of potential vorticity:changes in thickness and relative vorticity, assumingconstant latitude (constant f).

Relativevorticity

ζ = 0

Relativevorticity

ζ = 0

H1

Q = f(θ1)/H

Latitude θ1 Latitude θ2 Q = f(θ2)/H2 = f(θ1)/H1

Conservation of potential vorticity Q in the absence of relative vorticity (northern hemisphere):

balance of planetary vorticity and stretching

1

H2

move northwards

FIGURE S7.28 Conservation of potential vorticity:changes in thickness and latitude (Coriolis parameter f),assuming negligible relative vorticity (NorthernHemisphere).


nor relative vorticity can change without limit,there is an inherent restoring force to northwardand southward movements in the ocean andatmosphere. This restoring force creates Rossbywaves.)

In the Southern Hemisphere, f is negative. Asouthward move of a water column makes feven more negative, and requires stretching(increase in H). A northward move makes fless negative, and requires squashing (decreasein H). Therefore, looking at both hemispheres,we can say that poleward motion, toward largermagnitude f, requires stretching. Equatorwardmotion requires squashing.

The equator is a special place in terms ofpotential vorticity, since f changes from negativeto positive crossing the equator and is zero onthe equator. Any water parcels moving intothe equatorial region must become more domi-nated by relative vorticity, as in Figure S7.28.We see this in the much stronger horizontalcurrent shears near the equator than at higherlatitudes. (Geostrophy also breaks down righton the equator; slightly off the equator, smallpressure gradients result in large geostrophiccurrents, so we also see high velocities in theequatorial region compared with otherlatitudes.)

7.7.3. Rossby Waves

The adjustment of any fluid to a change inforcing takes the form of waves that move outand leave behind a steady flow associatedwith the new forcing. We describe some generalproperties of waves in Chapter 8. The large-scale, almost geostrophic circulation adjusts tochanging winds and buoyancy forcing mainlythrough “planetary” or Rossby waves and Kelvinwaves (Section 7.7.6). Pure Rossby and Kelvinwaves are never found except in simplifiedmodels and lab experiments. However, muchof the ocean’s variability can be understood interms of Rossby wave properties, particularlythe tendency for westward propagation relativeto the mean flow. We describe these waveswithout derivations, which can be found in themany geophysical fluid dynamics textbooksreferenced at the start of this chapter.

A first important fact is that Rossby waveshave wavelengths of tens to thousands of kilo-meters. Since the ocean is only 5 to 10 km deepand is stratified, particle motions in Rossbywaves are almost completely transverse (hori-zontal, parallel to the surface of Earth), which


differs from intuition that we build from watch-ing surface gravity waves.

Second, the restoring force for Rossby wavesis the variation in Coriolis parameter f with lati-tude, so all dispersion information includesb (Eq. 7.38). As a water column is shoved offto a new latitude, its potential vorticity mustbe conserved (Eq. 7.35). As with all waves, thecolumn overshoots, and then has to be restoredagain, creating the wave. Therefore the watercolumn height or relative vorticity begin tochange. These cannot change indefinitelywithout external forcing, so the water columnis restored back toward its original latitude. Aswith all waves, the column overshoots, andthen has to be restored back again, creating thewave. For a short wavelength Rossby wave,the relative vorticity changes in response to thechange in Coriolis parameter f as in FigureS7.26 d for a parcel moving northward tohigher f, the relative vorticity becomes negative.This pushes columns to the east of the parceltoward the south and pulls columns to thewest of parcel toward the north. The net effect

Long Rossby wave: f/H conserveGeostrophic flow due to pressure ridges, moves

producing westward propagat

North

Column moved northStretched

Southward

geostrophic flow

Latitude θ1

Latitude θ2

H1

H2

westward phase propagation

FIGURE S7.29 Schematic of a long wavelength Rossby wa

is a westward propagation of the wave. Fora long wavelength Rossby wave (Figure S7.29)the column height changes in response to thechange in f, as in Figure S7.28. For northwardmotion of the parcel, height increases; the down-ward slope in height to the east causes south-ward geostrophic flow on that side while thedownward slope in height to the west of theperturbation causes northward geostrophicflow on the west. The net effect again is west-ward propagation of the disturbance.

Third, Rossby wave crests and troughs moveonly westward (relative to any mean flow,which could advect them to the east) in boththe Northern and Southern Hemispheres;that is, the phase velocity is westward (plusa northward or southward component). Onthe other hand, the group velocity of Rossbywaves can be either westward or eastward.The group velocity of Rossby waves is west-ward for long wavelengths (more than about50 km) and eastward for short wavelengths(even though the zonal phase velocity iswestward).

d (f1/H1 = .f2/H2)columns northward or southward,ion of the wave

East

Northward

geostrophic flow

ve.


Fourth, velocities in Rossby waves are almostgeostrophic. Therefore, they can be calculatedfrom variations in pressure; for instance, asmeasured by a satellite altimeter, whichobserves the sea-surface height. Behaviorsimilar to a Rossby wave (westward phasepropagation) can be seen at almost all latitudesin each of the subtropical oceans in the satellitealtimetry images in Figures 14.18 and 14.19.

Although pure Rossby waves do not occur,many variable flows such as eddies or meandersof currents like the Gulf Stream Extension orAAC can be interpreted in terms of Rossbywave properties, in the sense that Rossby wavesare the basic set of linear wave solutions forflows with a small Rossby number and smallaspect ratio. If the mean flows are removedfrom observed variability, the variability oftenappears to move westward. The atmospherehas the same Rossby-wave-like phenomena,such as those seen in daily weather mapsshowing large loops or meanders in the JetStream (Figure S7.18).

7.7.4. Rossby Deformation Radius andRossby Wave Dispersion Relation

Turning slightly more analytical, we intro-duce, again without derivation, the Rossbydeformation radius and the dispersion relationfor Rossby waves with simple stratification.

The length scale that separates long fromshort wavelength Rossby waves is called theRossby deformation radius. It is the intrinsic hori-zontal length scale for geostrophic or nearlygeostrophic flows, relative to which all lengthscales are compared. The Rossby radius charac-terizes the observed mesoscale (eddy) lengthscales and also the spatial decay scale ofboundary-trapped waves such as Kelvin waves(Section 7.7.6) and the latitudinal width of equa-torially trapped waves.

The Rossby deformation radius in an unstrat-ified ocean is

RE ¼ ðgHÞ1=2=f (7.39a)

whereH is the ocean depth scale. RE is called thebarotropic Rossby deformation radius or “external”deformation radius. Barotropic deformationradii are on the order of thousands of kilome-ters. In an unstratified ocean, the horizontalvelocities for geostrophic flows are the same(in magnitude and direction) from the top ofthe ocean to the bottom. In the more realisticstratified ocean, there is a similar “barotropicmode,” with velocities in the same direction atall depths, and with a barotropic Rossby defor-mation radius also given by Eq. (7.39a).

The Rossby deformation radius associatedwith the ocean’s stratification is

RI ¼ NHS=f (7.39b)

where N is the Brunt-Vaisala frequency (Eq.7.14), and Hs is an intrinsic scale height for theflow. RI is called the baroclinic deformation radius(or “internal” deformation radius). “Baroclinic”means that the velocity structure changes withinthe water column, associated with isopycnalslopes. The first baroclinic mode has a singlevelocity reversal within the water column. Thevertical length scale Hs associated with the firstbaroclinic mode is about 1000 m, which is thetypical pycnocline depth. (The second baroclinicmode has two velocity reversals and hencea shorter vertical length scale, and so on forthe higher modes.) The vertical length scale Hs

associated with the first baroclinic mode isabout 1000 m, which is the typical pycnoclinedepth. RI for the first baroclinic mode variesfrom more than 200 km in the tropics to around10 km at high latitudes (Figure S7.30a; Cheltonet al., 1998).

The dispersion relation (Section 8.2) for firstmode baroclinic Rossby waves is

u ¼ �bk

k2 þ l2 þ ð1=RIÞ2(7.40)

+120° +150° –180° –150° –120° – 90° – 60° –30° 0° +30° +60° +90°

–75°

–60°

–45°

0°

0°

+45°

+60°

+75°

50

50

50

100

100

200 200

200

500

500

1000

1000

(b)

FIGURE S7.30 (a) Rossby deformation radius (km) for the first baroclinic mode. Source: From Chelton et al. (1998).(b) Shortest period (in days) for the first baroclinic mode, based on the deformation radius in (a). Note that the annual cycle,at 365 days, occurs around latitudes 40 to 45 degrees; poleward of this, all such waves are slower. Source: From Wunsch (2009).



where u is the wave frequency, k and l are thewavenumbers in the east-west (x) and north-south (y) directions, b is as in Eq. (7.38), and RI

is as given in Eq. (7.39b). Highest frequency(shortest period) occurs at the wavelength asso-ciated with the Rossby deformation radius(Figure S7.31). The shortest periods vary fromless than 50 days in the tropics to more than 2to 3 years at high latitudes (Figure S7.30b fromWunsch, 2009). Poleward of about 40 to 45degrees latitude there is no first baroclinicmode at the annual cycle, so seasonal atmo-spheric forcing cannot force the first baroclinicmode at these higher latitudes. This results ina fundamentally different response to atmo-spheric variability at higher latitudes than inthe tropics and at mid-latitudes.

In much of the ocean away from the equator,the barotropic and first baroclinic modes domi-nate the variability, and hence the space andtimescales of the eddy field. At the equator,a much larger set of baroclinic modes is typicallyobserved, resulting in much more complexvertical velocity structure than at higher lati-tudes. Equatorial Rossby waves are slightlydifferent from non-equatorial Rossby wavessince geostrophy does not hold at the equator,but the vertical structures and behavior aresimilar, with the restoring force for the equatorialRossby waves the same as at mid-latitude d thechange in Coriolis parameter with latitude.

7.7.5. Instability of Geostrophic OceanCurrents

Almost all water flows are unsteady. Whengyre-scale flows break up, they do so into largeeddies, on the order of tens to hundreds of kilo-meters in diameter or larger (see Section 14.5).The size of the eddies is often approximatelythe Rossby deformation radius. The eddiesusually move westward, like Rossby waves.

Instabilities of flows are often studied by con-sidering a mean flow and then finding the smallperturbations that can grow exponentially. This

approach is called “linear stability theory”; it islinear because the perturbation is alwaysassumed to be small relative to the mean flow,which hardly changes at all. When perturbationsare allowed to grow to maturity, when theymight be interacting with each other andaffecting the mean flow, the study has becomenonlinear.

We define three states: stable, neutrally stable,and unstable. A stable flow returns to its originalstate after it is perturbed. A neutrally stable flowremains as is. In an unstable flow, the perturba-tion grows.

The two sources of energy for instabilities arethe kinetic energy and the potential energy of themean flow. Recall from basic physics that kineticenergy is ½ mv2 where m is mass and v is speed;for a fluid we replace the mass with density r, orjust look at the quantity ½ v2. Also recall frombasic physics that potential energy comes fromraising an object to a height; the work done inraising the object gives it its potential energy.In a stratified fluid like the ocean, there is noavailable potential energy if isopycnals are flat,which means that nothing can be released. Forthere to be usable or available potential energy,isopycnals must be tilted.

Barotropic instabilities feed on the kineticenergy in the horizontal shear of the flow. Forinstance, the Gulf Stream and similar strongcurrents are jet-like, with large horizontal shear.Their speeds exceed 100 cm/sec in the center ofthe jet and decay to 0 cm/sec over about 50 kmon either side of the jet. Such currents also havelarge kinetic energy because of their highspeeds. The kinetic energy can be released ifspecial conditions on the potential vorticitystructure of the current are met. These condi-tions are that the horizontal shear be “largeenough” compared with a restoring b-effect(Eq. 7.38), which creates Rossby waves in theabsence of sheared flow (previous subsection).Barotropic instabilities can be thought of as the(unstable) waves that occur in the presence ofa horizontally sheared current and possibly

Wavenumber (km-1)

Fre

quen

cy (

day-

1 )

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

-0.1 −0.09 −0.08 −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0

Rossby wave dispersion relationLatitude 20°NDeformation radius RI = 50 km

100

km

20

km

200

km

RI

70 days

100 days

200 days

300 days

500 days400 days

(a) Period = 2π/frequency

Wavelength (km)

Per

iod

(day

)

3000

50

100

150

200

250

300

350

400

450

500

2500 2000 1500 1000 500 00

Rossby wave dispersion relationLatitude 20°NDeformation radius RI = 50 km

100

km

20

km 1

0 km

200

kmlength scales =

wavelength/2π

RI

(b)

FIGURE S7.31 Dispersion relation for first mode baroclinic Rossby waves (Eq. 7.40), assuming a deformation radius RI of50 km, latitude 20 degrees (north or south) and y-wavenumber l¼ 0. (a) Frequency u versus x-wavenumber k and (b) periodversus wavelength. The Rossby radius is shown with the dashed line. The highest frequency and shortest period are at theRossby radius length scale.



also the b-effect; see Pedlosky (1987). The neteffect of the barotropic instability is to reducethe size of the horizontal shear. For the GulfStream, for instance, this results in decreasingthe maximum speed at the core of the jet, andinducing flows in the opposite direction on theoutskirts of the jet. These flows look like “recir-culations” (Section 9.3.2).

Baroclinic instabilities draw on the availablepotential energy of the flow. The relativelyrecent study of sub-mesoscale eddies and insta-bilities generated in the ocean’s mixed layer isessentially that of baroclinic instability oper-ating on density fronts within the mixed layer(Boccaletti et al., 2007). The fronts are stronglytilted isopycnals, which are then subject to thiskind of potential energy release.

Baroclinic instability is peculiar to geostrophicflows, because Earth’s rotation makes it possibleto have amean geostrophic flowwithmean tiltedisopycnals. On the other hand, barotropic insta-bility is similar to instabilities of all sheared flowsincluding those without Earth’s rotation.

7.7.6. Kelvin Waves

Coastlines and the equator can supporta special type of hybrid wave called a “Kelvinwave,” which includes both gravity wave andCoriolis effects. Kelvin waves are “trapped” tothe coastlines and trapped at the equator, whichmeans that their amplitude is highest at thecoast (or equator) and decays exponentiallywith offshore (or poleward) distance. Kelvinwaves are of particular importance on easternboundaries since they transfer information pole-ward from the equator. They are also central tohow the equatorial ocean adjusts to changes inwind forcing, such as during an El Nino(Chapter 10).

Kelvin waves propagate with the coast to theright in the Northern Hemisphere and to the leftin the Southern Hemisphere. At the equator,which acts like a boundary, Kelvin waves prop-agate only eastward. In their alongshore

direction of propagation, Kelvin waves behavejust like surface gravity waves and obey thegravity wave dispersion relation (Section 8.3).However, unlike surface gravity waves, Kelvinwaves can propagate in only one direction.Kelvin wave wavelengths are also very long,on the order of tens to thousands of kilometers,compared with the usual surface gravity wavesat the beach. Although the wave propagationspeed is high, it can take days to weeks to seethe transition from a Kelvin wave crest toa Kelvin wave trough at a given observationpoint.

In the across-shore direction, Kelvin wavesdiffer entirely from surface gravity waves. Theiramplitude is largest at the coast. The offshoredecay scale is the Rossby deformation radius(Section 7.7.4).

Lastly, Kelvin wave water velocities in thedirection perpendicular to the coast are exactlyzero. The water velocities are therefore exactlyparallel to the coast. Moreover, the alongshorevelocities are geostrophic, so they are associatedwith pressure differences (pressure gradientforce) in the across-shore direction.

7.8. WIND-DRIVENCIRCULATION: SVERDRUPBALANCE AND WESTERNBOUNDARY CURRENTS

The large-scale circulation in the ocean basinsis asymmetric, with swift, narrow currents alongthe western boundaries, and much gentler flowwithin the vast interior, away from the sideboundaries. This asymmetry is known as west-ward intensification of the circulation; it occurs inboth the Northern and Southern Hemispheresand in the subtropical and subpolar gyres.

The Gulf Stream is the prototype of thesewestern boundary currents, as the first thatwas extensively studied, and as the examplefor which theories of westward intensificationwere developed. In a book that summarizes

WIND-DRIVEN CIRCULATION: SVERDRUP BALANCE AND WESTERN BOUNDARY CURRENTS 53

these theories, Stommel (1965) reviewed earlyknowledge of the Gulf Stream, dating back tothe first explorations of the North Atlantic, andsummarized theoretical attempts to understandit, dating back to the nineteenth century. Whenthe subtropical gyre and Gulf Stream werefinally modeled theoretically in the mid-twen-tieth century, the resulting theory was breath-takingly simple. The long delay in arriving atthis theory was due to the similarity betweenthe wind patterns above the subtropical NorthAtlantic and the circulation d both are highpressure systems d with anticyclonic flow(clockwise in the Northern Hemisphere). Butthe winds are clearly not westward intensifiedrelative to the ocean boundaries.

The primary originators of the theories thatprovide our present understanding wereHarald Sverdrup, Henry Stommel, WalterMunk, and Nicholas Fofonoff. Sverdrup (1947)first explained the mid-ocean vorticity balance,created by variations in Ekman transport, thatcreates what we now call the “Sverdrup inte-rior” solution (Section 7.8.1). Just a few yearsearlier, Sverdrup et al. (1942) were still suggest-ing that the Ekman transport variations wouldsimply pile water up in the central gyre, witha resulting geostrophic flow around the pile.Stommel (1948) and Munk (1950) provided thefirst (frictional) explanations for the westernboundary currents (Section 7.8.2), and Fofonoff(1954) showed how very different the circula-tion would be without friction.

Most of the physical effects described in thissection occur because the Coriolis parametervaries with latitude, that is, because of theb-effect (Eq. 7.38).

7.8.1. Sverdrup Balance

The gentle interior flow of the (non-equato-rial) oceans can be described in terms of theirmeridional (north-south) direction. In thesubtropical gyres, the interior flow is towardthe equator in both the Northern and Southern

Hemispheres. In the subpolar gyres, the interiorflow is poleward in both hemispheres. Theseinterior flow directions can be understoodthrough a potential vorticity argument intro-duced by Sverdrup (1947), so we call the appli-cable physics the “Sverdrup balance.”

Consider a schematic of the subtropicalNorth Pacific (Figure S7.32). The winds at thesea surface are not spatially uniform (Figure5.16 and Figure S10.2 in the online supplement).South of about 30�N, the Pacific is dominated byeasterly trade winds. North of this, it is domi-nated by the westerlies. This causes northwardEkman transport under the trade winds, andsouthward Ekman transport under the west-erlies. As a result, there is Ekman convergencethroughout the subtropical North Pacific(Figures 5.16d and S10.2).

The convergent surface layer water in thesubtropics must go somewhere so there isdownward vertical velocity at the base of the(50 m thick) Ekman layer. At some levelbetween the surface and ocean bottom, thereis likely no vertical velocity. Therefore thereis net “squashing” of the water columns inthe subtropical region (also called Ekmanpumping; Section 7.5.4).

This squashing requires a decrease in eitherplanetary or relative vorticity (Eq. 7.35). In theocean interior, relative vorticity is small, so plan-etary vorticity must decrease, which results inthe equatorward flow that characterizes thesubtropical gyre (Figure S7.28).

The subpolar North Pacific lies north of thewesterly wind maximum at about 40�N. Ekmantransport is therefore southward, witha maximum at about 40�N and weaker at higherlatitudes. Therefore there must be upwelling(Ekman suction) throughout the wide latitudeband of the subpolar gyre. This upwellingstretches the water columns (Eq. 7.35), whichthen move poleward, creating the polewardflow of the subpolar gyre.

The Sverdrup transport is the net meridionaltransport diagnosed in both the subtropical

Ekman transportEkman

downwelling

Thermocline

Sverdrup transport

Subtropical gyre

East

North

Ekman transport

Ekman upwelling

Ekman upwelling

Northern Hemisphere

Subpolar gyre

Tropical gyre

Westerlies

Trades

FIGURE S7.32 Sverdrupbalance circulation (NorthernHemisphere). Westerly and tradewinds force Ekman transportcreating Ekman pumping andsuction and hence Sverdruptransport.


and subpolar gyres, resulting from planetaryvorticity changes that balance Ekman pumpingor Ekman suction.

All of the meridional flow is returned inwestern boundary currents, for reasonsdescribedin the following sections. Therefore, subtropicalgyres must be anticyclonic and subpolar gyresmust be cyclonic.

Mathematically, the Sverdrup balance isderived from the geostrophic equations ofmotion with variable Coriolis parameter f(Eq. 7.23a,b). The x- and y-momentum equa-tions are combined to form the vorticity equa-tion, recalling that b = df/dy:

fðvu=vxþ vv=vyÞ þ bv ¼ 0 (7.41)

Using the continuity equation

vu=vxþ vv=vyþ vw=vz ¼ 0 (7.42)

Eq. (7.41) becomes the potential vorticitybalance

bv ¼ f vw=vz: (7.43)

This important equation states that watercolumn stretching in the presence of rotation isbalanced by a change in latitude (Figure S7.28).

In Eq. (7.43), the vertical velocity w is due toEkman pumping. From Eqs. (7.20) and (7.21):

w ¼ v=vx�sðyÞ=rf

�� v=vy

�sðxÞ=rf

�

¼ }curl s} (7.44)

where s is the vector wind stress, s(x) is the zonalwind stress, and s(y) is the meridional windstress. Assuming that the vertical velocity w iszero at great depth, Eq. (7.43) can be verticallyintegrated to obtain the Sverdrup balance:


b�MðyÞ �

�sðxÞ=f

��¼ v=vx

�sðyÞ

�� v=vy

�sðxÞ

�

¼ }curl s}

(7.45)

where the meridional (south-north) mass trans-port M(y) is the vertical integral of the meridi-onal velocity v times density r. The secondterm on the left side is the meridional Ekmantransport. Thus, the meridional transport in theSverdrup interior is proportional to the windstress curl corrected for the Ekman transport.

Themeridional transport M(y) is the Sverdruptransport. A global map of the Sverdrup trans-port integrated from the eastern to the westernboundary is shown in Figure 5.17. The size ofthe integral at the western boundary gives thewestern boundary current transport sinceSverdrup’s model must be closed with a narrowboundary current that has at least one additionalphysical mechanism beyond those in theSverdrup balance (a shift in latitude because ofwater column stretching driven by Ekman trans-port convergence). Physics of the boundarycurrents are discussed in the following sections.

7.8.2. Stommel’s Solution: WestwardIntensification and Western BoundaryCurrents

In the late 1940s, Henry Stommel (1948)added simple linear friction to Sverdrup’s

+150

+50

+100

0-10

-10

+10

1000 km 0

FIGURE S7.33 Stommel’s wind-driven circulation solutioncentral latitudes of Figure S7.32: (a) surface height on a uniformb-effect. After Stommel (1965).

model of the gentle interior flow in a basinwith eastern and western boundaries (Section7.8.1). Mathematically this is an addition ofdissipation of potential vorticity Q on theright-hand side of Eq. (7.37). The remarkableresult was that the returning flow can only bein a narrow jet along the western boundary(Figure S7.33). The potential vorticity balancein this jet is change in planetary vorticitybalanced by bottom friction.

Figure S7.33a shows the ocean circulation ifthere were no latitudinal variation in Coriolisparameter (no b-effect; Stommel, 1965). Thisis the solution if Earth were a rotating, flatdisk with westerlies in the north and tradesin the south. In this solution, the potentialvorticity input from the wind cannot bebalanced by a change in latitude, so the flowbuilds up relative vorticity (negative sign)that is balanced throughout the basin bybottom friction; the Sverdrup balance (Eq.7.40) cannot apply. In Figure S7.33b, for therealistic spherical Earth with a b-effect, theflow is southward throughout the interior(Sverdrup balance), and returns northward ina swift jet on the western boundary. Thisidealized circulation resembles the GulfStream and Kuroshio subtropical gyres inwhich the Gulf Stream and Kuroshio are thenarrow western boundary currents returningall southward Sverdrup interior flow back tothe north.

1000 km

40

30

20

10

for a subtropical gyre with trades and westerlies like thely rotating Earth and (b) westward intensification with the


Stommel’s frictional solution is somewhatunrealistic since the friction is between theboundary current and the ocean bottom. Thismeans that the wind-driven flow must reachto the ocean bottom. However, with stratifica-tion, it is not at all obvious that the circulationreaches so deep (although in fact one character-istic of strong western boundary currents suchas the Gulf Stream system is that the narrowcurrent does reach to the bottom even if theSverdrup interior flow does not). A subsequentstudy by Walter Munk avoids this restrictionand still yields westward intensification, asseen next.

EAST WESTZONAL WINDS

WESTERLY WINDS(ROARING FORTIES)

WEAK EASTERLIES PO

WE

SU

SU

SUBTROPICAL

ANTICYCLONES

CYCLONES

TRADE WINDS

TRADE WINDS

DOLDRUMS

EQ

EQ

EQ

Western boundaryvortices

Wes

tern

cur

rent

FIGURE S7.34 Munk’s wind-driven circulation solution: zocenter. After Munk (1950).

7.8.3. Munk’s Solution: WesternBoundary Currents

A few years after Stommel’s work, WalterMunk considered the effect of more realisticfriction on the ocean gyre circulations betweenthe currents and the side walls rather thanbetween the currents and the ocean bottom.Munk’s (1950) result was very similar toStommel’s result, predicting westward intensifi-cation of the circulation. A narrow, swift jetalong the western boundary returns theSverdrup interior flow to its original latitude(Figure S7.34).

MERIDIONAL WINDS

LAR CURRENT

ST WIND DRIFT

BTROPICAL GYRE

BPOLAR GYRE

UATORIAL CURRENT

UATORIAL CURRENT

UATORIAL COUNTER CURRENT

N

S

Wind-spun vortex

Eas

tern

cur

rent

nal wind profiles on left and circulation streamlines in the


How does the potential vorticity balancework in Munk’s model (which is combinedwith Sverdrup’s model)? Why do we find theboundary current on the western side ratherthan the eastern side, or even within the middleof the basin (if considering Stommel’s bottomfriction)? In the Sverdrup interior of a subtrop-ical gyre, when the wind causes Ekman pump-ing, the water columns are squashed, theymove equatorward to lower planetary vorticity.

Wes

tern

bou

ndar

y (c

oast

line)

North

Western boundary current

Frictional boundarylayer

Inputpositive relative vorticity

Frictional westelayer (Munk, 19positive relativenorthward boun(increasing pla

West

Southward interior (Sverdrup) flow Impermissible eastern boundary

What happens if the boundary current is on the eastern boundary? Input of negative relative vorticity cannot allow northward boundary current. This solution is not permissible as a balance for southward Sverdrup interior flow.

(a)

(b)

To return to a higher latitude, there must beforcing that puts the higher vorticity back intothe fluid. This cannot be in the form of planetaryvorticity or very, very narrow wind forcing,since the first is already contained in theSverdrup balance, and the second is unphysicalexcept in one or two extremely special locations(e.g., Arabian coast, Chapter 11). Therefore, theinput of vorticity must affect the relativevorticity.

East

Southward interior (Sverdrup) flow

rn boundary50): input of vorticity allowsdary current

netary vorticity)

Eas

tern

bou

ndar

y (c

oast

line)

North

current

Frictional boundarylayer

Inputnegative relative vorticity

FIGURE S7.35 (a) Vorticitybalance at a western boundary,with side wall friction (Munk’smodel). (b) Hypothetical easternboundary vorticity balance,showing that only western bound-aries can input the positive relativevorticity required for the flow tomove northward.


Consider a western boundary current fora Northern Hemisphere subtropical gyre(Figure S7.35), with friction between the currentand the side wall (Munk’s model). The effect ofthe side wall is to reduce the boundary currentvelocity to zero at the wall. Therefore, theboundary current has positive relative vorticity.This vorticity is injected into the fluid by the fric-tion at the wall, and allows the current to movenorthward to higher Coriolis parameter f. (Notethat there is negative relative vorticity in theboundary current offshore of its maximumspeed, but the current changes much moreslowly and the negative relative vorticity thereis much lower than the positive relative vorticityat the boundary.) On the other hand, if thenarrow jet returning flow to the north were onthe eastern boundary, the side wall frictionwould inject negative relative vorticity, whichwould make it even more difficult for theboundary current fluid to join the interior flowsmoothly. Therefore, vorticity argumentsrequire that frictional boundary currents be onthe western boundary. The reader can gothrough this exercise for subpolar gyres aswell as for both types of gyres in the SouthernHemisphere and will find that a westernboundary current is required in all cases.

7.8.4. Fofonoff’s Solution: Large-ScaleInertial Currents

In one further important simplified approachto large-scale ocean circulation, Nicholas Fofon-off, in 1954, showed that circulation can arise asa free, unforced mode. The idea is that a verysmall amount of wind, with very little frictionanywhere in the system, could set up sucha circulation. Indeed, aspects of the Fofonoffsolution are found in highly energetic regions,such as in the neighborhood of the Gulf Stream(which in actuality is not highly frictional, andwhich is stronger than predicted from theSverdrup interior balance). This type of circula-tion is called an “inertial circulation.” It is

easiest to describe using Fofonoff’s own figure(Figure S7.36).

In the Fofonoff circulation, there is noSverdrup interior with flow moving northwardor southward. The interior flow is exactly zonal(east-west). This is because there is nowind inputof vorticity, so flow cannot change latitude sinceit would then have to change its planetaryvorticity. This exact zonality therefore resultsfrom the b-effect. However, there are strongboundary currents on both the western andeastern boundaries, and there can be strong,exactly zonal jets crossing the ocean in its interior.

How do these strong currents with so muchrelative vorticity connect to each other?Consider westward flow across the middle ofthe ocean, as illustrated in Figure S7.36. This rea-ches the western boundary and must somehowget back to the eastern boundary to feed backinto the westward flow. It can do this by movingalong the western boundary in a very narrowcurrent that has a large amount of relativevorticity. This current can be to either the northor the south. Suppose it is to the north. Then therelative vorticity of this frictionless current ispositive, allowing it to move to higher latitude.It then jets straight across the middle of theocean, reaches the eastern boundary, and movessouthward, feeding into the westward flow inthe interior. There is no net input of vorticityanywhere in this model (no wind, no friction).

Following the Sverdrup, Stommel, Munk,and Fofonoff models, a number of theoreticalpapers explored various combinations of thedifferent types of friction, inertia, and boundarygeometries on the mean ocean flow, but theirresults can all be understood in terms of thesebasic models. Some of the earliest ocean circula-tion models (Veronis, 1966; Bryan, 1963) illus-trated the dynamical processes for variousstrengths of friction and inertia. Further realbreakthroughs in theoretical understanding ofwind-driven ocean circulation occurred thirtyto forty years later, with treatment of the effectof stratification, as discussed next.

1000 km

1000 km

N

S

W E

FIGURE S7.36 Inertial circulation, in the absence of friction and wind, but in the presence of the b-effect. Source: FromFofonoff (1954).


7.8.5. Wind-Driven Circulation ina Stratified Ocean

What happens to the wind-driven circulationtheories in a stratified ocean? Water moves

down into the ocean,mostly along very graduallysloping isopycnals.Where streamlines of flowareconnected to the sea surface, we say the ocean isdirectly ventilated (Figure S7.37). Where there isEkman pumping (negative wind stress curl), the


Sverdrup interior flow is equatorward (Section7.8.1). Water columns at the local mixed layerdensity move equatorward and encounter lessdense water at the surface. They slide down intothe subsurface along isopycnals, still movingequatorward. This process is called subduction(Luyten, Pedlosky, & Stommel, 1983), usinga term borrowed from plate tectonics. The sub-ducted waters then flow around the gyre andenter the western boundary current if they donot first enter the tropical circulation. Thedetails of this process are beyond the scope ofthis text.

40

30

–60

Latit

ude

(deg

ree)

(b)

20

Western Pool

(a)

WESTERNUNVENTILATEPOOL

N

W WESTERN BOU

FIGURE S7.37 (a) Subductionschematic (Northern Hemisphere).(b) Streamlines for idealizedsubduction on an isopycnalsurface. The light gray regions arethe western pool and easternshadow zone, where streamlinesdo not connect to the sea surface.The heavy dashed contour iswhere the isopycnal meets the seasurface (surface outcrop); in thedark gray area there is no water ofthis density. After Williams (1991).

In each subducted layer, there can be threeregions (Figure S7.37): (1) a ventilated region con-nected from the sea surface as just described, (2)a western unventilated pool with streamlinesthat enter and exit from the western boundarycurrent without entering the surface layer, and(3) an eastern quiet (shadow) zone between theeasternmost subducting streamline and theeastern boundary. A continuous range ofsurface densities is found in the subtropicalgyre; the water column is directly ventilatedover this full range, with waters at each densitycoming from a different sea-surface location

–50 –40 –30 –20Longitude (degree)

Ventilatedregion

Surface outcrop

Eastern shadow

zone

WINDWIND

SURFACE

OUTCROP

D

EASTERN

SHADOW

ZONE

ABYSSALOCEAN

SUBDUCTED

REGION

NDARY

EASTERN BOUNDARY

WIND-DRIVEN CIRCULATION: EASTERN BOUNDARY CURRENTS AND EQUATORIAL CIRCULATION 61

depending on the configuration of streamlineson that isopycnal. This is called the “ventilatedthermocline”; in water mass terms, this processcreates the Central Water. The maximumdensity of the ventilated thermocline is set bythe maximum winter surface density in thesubtropical gyre (Stommel, 1979). This usuallyoccurs at the most poleward edge of the gyre,around 40 to 50 degrees latitude. Themaximum depth of the ventilated thermoclineis the depth of this densest isopycnal, and isbetween 500 and 1000 m depending on theocean (see Chapters 9e11).

Subducting waters can leave the surface layerin two distinct ways: they can be pushed down-ward along isopycnals by Ekman pumping, andthey can also be included in the subsurface layerthrough seasonal warming and cooling of thesurface layer while they flow southward. Inwinter the surface layer is of uniform density.Entering spring and summer, this is glazedover by a surface layer of much lower density.All the while the geostrophic flow is southward.When the next winter arrives, the water columnis farther south and winter cooling does notpenetrate down to it. Therefore it has effectivelyentered the subsurface flow and does notre-enter the surface layer until it emerges fromthe western boundary, possibly many yearslater. Therefore the properties of the subsurfaceflows are set by the late winter conditions. Theother seasons have no impact other than toprovide seasonal isolation of the winter layeruntil it has subducted. Stommel (1979) calledthis phenomenon the “Ekman demon,” analo-gous to Maxwell’s demon of thermodynamics,which is a thought experiment about separatinghigher and lower energy molecules.

The opposite of subduction is obduction, bor-rowed again from plate tectonics by Qiu andHuang (1995). In obducting regions, watersfrom subsurface isopycnals come up and intothe surface layer. These are generally upwellingregions such as the cyclonic subpolar gyres andthe region south of the ACC.

Wind-driven circulation occurs in unventi-lated stratified regions as well. It is mostvigorous in regions connected to the westernboundary currents where water can enter andexit the western boundary. In these regions,the western boundary currents and their sepa-rated extensions usually reach to the oceanbottom. In a region that is closer and closer tothe western boundary with increasing depth,there can be a closed circulation region thatconnects in and out of the western boundarywithout connection to the sea surface; suchregions are characterized by constant potentialvorticity (stretching and planetary portionsonly, or f/H). These dynamics are beyond thescope of this text.

7.9. WIND-DRIVENCIRCULATION: EASTERN

BOUNDARY CURRENTS ANDEQUATORIAL CIRCULATION

7.9.1. Coastal Upwelling and EasternBoundary Currents

The eastern boundary regions of the subtrop-ical gyres have strong but shallow flow that isdynamically independent of the open oceangyre regimes. Upper ocean eastern boundarycirculation is driven by alongshore wind stressthat creates onshore (or offshore) Ekman trans-port that creates upwelling (or downwelling;Section 7.5.4). Beneath or inshore of the equator-ward eastern boundary currents there is a pole-ward undercurrent or countercurrent. Coastalupwelling systems are not restricted to easternboundaries; the southern coast of the Arabianpeninsula has the same kind of system. Thesecirculations are fundamentally different fromwestern boundary currents, which are tied topotential vorticity dynamics (Section 7.8).

The classical explanation of eastern boundarycurrents is that equatorwardwinds force Ekmanflow offshore, which drives a shallow upwelling


(on the order of 200 m deep) in a very narrowregion adjacent to the coast (on the order of10 km; Figure S7.13c). The upwelling speed isabout 5e10 m/day. Because of stratification,the source of upwelled water is restricted tolayers close to the sea surface, usually between50 and 300 m.

The zone of coastal upwelling can beextended to more than 100 km offshore by anincrease in longshore wind strength withdistance offshore; this is observed in eacheastern boundary upwelling system due totopographic steering of the winds by theoceaneland boundary. The offshore Ekmantransport therefore increases with distanceoffshore, which requires upwelling throughthe whole band (Bakun & Nelson, 1991). Thezone is identified by positive wind stress curl,notably in the California Current and Peru-Chile Current regions and the Arabianupwelling zone (Figure 5.16d).

Upwelled water is cooler than the originalsurface water. It originates from just below theeuphotic zone and therefore is also rich in nutri-ents, which results in enhanced biologicalproductivity characterized by high chlorophyllcontent (Section 4.6). Cool surface temperaturesand enhanced biological productivity are clearin satellite images that record sea-surfacetemperature and ocean color (Figure 4.28).

Upwelling is strongly seasonal, due to sea-sonality in the winds. Onset of upwelling canbe within days of arrival of upwelling-favorablewinds. In one example, off the coast of Oregon,the surface temperature dropped by 6�C in twodays after a longshore wind started.

Coastal upwelling is accompanied by a rise inupper ocean isopycnals toward the coast (Figure7.6). This creates an equatorward geostrophicsurface flow, the eastern boundary current. Thesecurrents are narrow (<100 km width and nearthe coast), shallow (upper 100 m), strong(40e80 cm/sec), and strongly seasonal. Theactual flow in an eastern boundary currentsystem includes strong, meandering eddies

and offshore jets/filaments of surface water,often associated with coastline features such ascapes (Figure 10.6). Actual eastern boundarycurrents are some distance offshore at the axisof the upwelling front created by the offshoreEkman transport.

Poleward undercurrents are observed at about200 m depth beneath the equatorward surfacecurrents in each eastern boundary upwellingsystem. When upwelling-favorable windsweaken or disappear, the equatorward flowalso disappears and the poleward undercurrentextends up to the surface (there is no longer anundercurrent). Poleward undercurrents arecreated mainly by the alongshore pressuregradient that drives the onshore subsurfacegeostrophic flow that feeds the upwelling. Theremay also be a contribution from positive windstress curl throughout the eastern boundaryregion that leads to poleward Sverdrup trans-port (Section 7.8.1; Hurlburt & Thompson,1973).

The only ocean without an equatorwardeastern boundary current is the Indian Ocean.The Leeuwin Current along the west coast ofAustralia flows poleward, even though thewinds are upwelling favorable and would drivea normal eastern boundary current there in theabsence of other forces. However, there isa much larger poleward pressure gradient forcealong this boundary than along the others, dueto the flow of water westward through the Indo-nesian archipelago from the Pacific to the IndianOcean.

7.9.2. Near-Surface EquatorialCurrents and Bjerknes Feedback

Circulation within about 2 degrees latitude ofthe equator is very different from non-equato-rial circulation because the Coriolis parameterf vanishes at the equator. The narrowness ofthis equatorial influence, that is, the equatorialbaroclinic deformation radius, is set by the vari-ation in Coriolis parameter with latitude and the

BUOYANCY (THERMOHALINE) FORCING AND ABYSSAL CIRCULATION 63

stratification of the ocean. Equatorial circulationis driven by easterly trade winds in the Pacificand Atlantic and by the seasonally reversingmonsoonal winds in the Indian Ocean. Wedescribe here only the equatorial circulationthat results from trade winds.

Since the Coriolis parameter vanishes andthere is no frictional Ekman layer, the easterlytrade winds drive equatorial surface flow duewestward in a frictional surface layer (FigureS7.38a). The westward surface current isshallow (50 to 100 m) and of medium strength(10 to 20 cm/sec). In each of the three oceans,this westward surface flow is a part of theSouth Equatorial Current. The water piles upgently in the west (to about 0.5 m height) andleaves a depression in the east. This createsan eastward pressure gradient force (fromhigh pressure in the west to low pressure inthe east). The pressure gradient force drivesan eastward flow called the Equatorial Under-current (EUC). The EUC is centered at 100 to200 m depth, just below the frictional surfacelayer. The EUC is only about 150 m thick. It isamong the strongest ocean currents(>100 cm/sec). (See illustrations of the PacificEUC in Section 10.7.3 and of the Atlantic EUCin Section 9.4.)

The pileup of waters in the western equato-rial region results in a deepened pycnoclinecalled the warm pool and a shallow pycnoclinein the eastern equatorial region. Coriolis effectsbecome important a small distance from theequator; the resulting off-equatorial Ekmantransport enhances upwelling in the equatorialband. This creates upwelling along the equatorand shoaling of the pycnocline toward theequator that drives a westward, nearlygeostrophic flow at the sea surface. Thisbroadens the frictional westward flow foundright on the equator.

Upwelling in the eastern equatorial regiondraws cool water to the surface because of theshallow thermocline there. This creates a coldsurface feature along the equator called the cold

tongue (see the sea-surface temperature map inFigure 4.1). Because of the thickness of thewarm pool in the west, even intense upwellingcannot cause cold surface temperatures. Thewarm pool’s high surface temperature, in excessof 28 �C, is maintained through radiative equilib-rium with the atmosphere (Jin, 1996).

The east-west contrast in temperature alongthe equator maintains the atmosphere’s Walkercirculation, which has ascending air over thewarm pool and descending over the easterncolder area. The Walker circulation is an impor-tant part of the trade winds that creates thewarm pool and cold tongue, so there can bea feedback between the ocean and atmosphere;this is called the Bjerknes feedback (Bjerknes,1969; Figure S7.38b). If something weakens thetrade winds, as at the beginning of an El Ninoevent (Chapter 10), the westward flow at theequator weakens and upwelling weakens orstops. Surface waters in the eastern regionstherefore warm. Water in the deep warm poolin the west sloshes eastward along the equator,thinning the pool. The change in sea-surfacetemperature weakens the Walker circulation/trade winds even more, which further exacer-bates the ocean changes. This is an example ofa positive feedback.

In the Indian Ocean, the prevailing equatorialwinds aremonsoonal, meaning that trade windsare only present for part of the year. This createsseasonally reversing equatorial currents andinhibits the formation of the warm pool/coldtongue structure. The Indian Ocean sea-surfacetemperature is high at all longitudes.

7.10. BUOYANCY(THERMOHALINE) FORCING AND

ABYSSAL CIRCULATION

Heating and cooling change the ocean’stemperature distribution, while evaporation,precipitation, runoff, and ice formation changethe ocean’s salinity distribution (Chapters 4

120ϒE

Equator

80ϒW

Normal Conditions

Convective Circulation

Thermocline

Bjerknes tropical feedback

Trade wind strength

Equatorial upwelling.High sea level and deepthermocline in west.Low sea level and shallow thermocline in east.

Zonal tropical SST difference

(+ positive feedback)

(b)

(a)

+ +

FIGURE S7.38 (a) Schematic of upper ocean equatorial circulation (large white arrows), surface temperature (red iswarm, blue is cold), and thermocline depth and upwelling, driven by the Walker circulation (“convective loop”). Source:From NOAA PMEL (2009b). (b) “Bjerknes feedback” between the trade wind strength and zonal (east-west) difference intropical surface temperature. (Arrows mean that increase in one parameter results in an increase in the second parameter.) Inthis positive feedback loop, increased trade winds cause a larger sea-surface temperature difference, which in turn increasesthe trade wind strength.


and 5). Collectively, these are referred to asbuoyancy, or thermohaline, forcing. Buoyancyprocesses are responsible for developing theocean’s stratification, including its abyssalproperties, pycnocline, thermocline, halocline,and upper layer structure (other than in wind-stirred mixed layers). Advection by currentsalso changes temperature and salinity locally,but it cannot change the overall inventory ofeither.

Abyssal circulation refers to the general cate-gory of currents in the deep ocean. The overturn-ing circulation, also called the thermohalinecirculation, is the part of the circulation associ-ated with buoyancy changes, and overlapsspatially with the wind-driven upper oceancirculation; it also includes shallow elements

that are independent of the abyssal circulation.In the overturning circulation, cooling and/orsalinification at the sea surface causes water tosink. This water must rise back to the warmsurface, which requires diffusion of heat (buoy-ancy) downward from the sea surface. Thesource of eddy diffusion is primarily wind andtidal energy. Thus aspects of the thermohalinecirculation depend on the magnitude of non-buoyancy processes through the eddy diffu-sivity (Wunsch & Ferrari, 2004).

Studies of the overturning circulationoriginated in the 1800s and early 1900s withGerman, British, and Norwegian oceanogra-phers. J. Sandstrom (1908) presented experi-ments and ideas about the simplest overturningcells driven by high-latitude cooling and a deep

Eddies (~10 km)

Chimney (50-100 km)

Plumes (< 1 km)

Mixed water

Stratified water

FIGURE S7.39 Processes in a deep convection region.After Marshall and Schott (1999).


tropical warm source that we now identify withdownward heat diffusion (see Figure S7.40).H. Stommel, in the 1960s, produced a series ofelegant papers on abyssal flow driven by isolatedsources of deep water and broad scale upwellingthat returns the water back to the upper ocean(Section 7.10.2). At the same time, Stommel pre-sented simple theories of the complementaryidea of ocean flows driven by very large-scaledensity contrasts (warm, saline tropics andcold, fresh poles: Section 7.10.3).

7.10.1. Buoyancy Loss Processes(Diapycnal Downwelling)

Water becomes denser through net cooling,net evaporation, and brine rejection duringsea ice formation. We have already describedbrine rejection (Section 3.9.2); it is responsiblefor creating the densest bottom waters in theglobal ocean (Antarctic Bottom Water and partsof the Circumpolar Deep Water) and also in theregional basins where it is operative (ArcticOcean, Japan Sea, etc.). Here we focus onconvection created by net buoyancy loss in theopen ocean, when surface water becomesdenser than water below, and advects andmixes downward. Convection creates a mixedlayer, just like wind stirring (Section 7.3).However, a convective mixed layer can behundreds of meters thick by the end of winter,whereas a wind-stirred mixed layer is limitedto about 150 m by the depth of wind-driventurbulence.

Convection happens on different timescales.Diurnal (daily) convection occurs at night inareas where the surface layer restratifiesstrongly during the day. During the annualcycle, cooling usually starts around theautumnal equinox and continues almost untilthe spring equinox. The resulting convectioneats down into the surface layer, reachingmaximum depth and density at the end ofwinter when the cumulative cooling reaches itsmaximum (FebruaryeMarch in the Northern

Hemisphere and AugusteSeptember in theSouthern Hemisphere).

Ocean convection is usually driven bysurface cooling. Excess evaporation can alsocreate convection, but the latent heat loss asso-ciated with evaporation is usually stronger.“Deep” convection is a loose term that usuallyrefers to creation of a surface mixed layer thatis thicker than about 1000 m. Deep convectionhas three phases: (1) preconditioning (reduc-tion in stratification), (2) convection (violentmixing), and (3) sinking and spreading. Pre-conditioning for deep convection includesreduced stratification through the watercolumn and some sort of dynamical featurethat allows stratification to become evenmore. The convection phase occurs when thereis large heat loss, usually due to high windspeeds along with very cold, dry air usuallyblowing from the land. Adjustment or restrati-fication, followed by spreading, occurs as theconvective features collapse (Killworth, 1983;Marshall & Schott, 1999.)

Convective regions have a typical structure(Figure S7.39). These include: (1) a chimney,which is a patch of tens to more than hundredsof kilometers across within which precondi-tioning can allow convection and (2) convec-tive plumes that are the actual sites ofconvection and are about 1 km or less across(Killworth, 1979; Marshall & Schott, 1999).


The plumes are about the same size across asthey are deep. It is not yet clearly knownwhat the vertical velocity structure is withinthe convective plumes. Observations in theLabrador Sea have suggested that there ismore downward motion than upward motion,and that the required upward motion mightoccur more slowly over a broader area withinthe chimney.

Deep convection occurs only in a few speciallocations around the world: Greenland Sea, Lab-rador Sea, Mediterranean Sea, Weddell Sea,Ross Sea, and Japan (or East) Sea. These sites,with the exception of the isolated Japan Sea,ventilate most of the deep waters of the globalocean. (The denser bottom waters, particularlyin the Southern Hemisphere, result from thebrine rejection process around Antarctica.)

7.10.2. Diapycnal Upwelling(Buoyancy Gain)

The structure of the basin and global scaleoverturning circulations depends on both theamount of density increase in the convectivesource regions and the existence of a buoyancy(heat) source at lower latitudes that is at leastas deep as the extent of the cooling (Sandstrom,1908; Figure S7.40). Since there are no significantlocal deep heat sources in the world ocean,waters that fill the deep ocean can only returnto the sea surface as a result of diapycnal eddydiffusion of buoyancy (heat and freshwater)

Cooling

Poleward

Up

Equatorward

Heating

Warming throughdiffusion

FIGURE S7.40 The role of vertical (diapycnal) diffusionin the MOC, replacing Sandstrom’s (1908) deep tropicalwarm source with diapycnal diffusion that reaches belowthe effect of high latitude cooling.

downward from the sea surface (Sections 5.1.3and 7.3.2).

Munk’s (1966) diapycnal eddy diffusivityestimate of kv¼ 1�10�4 m2/sec (Section 7.3.2)was based on the idea of isolated sources ofdeep water and widespread diffusive upwellingof this deep water back to the surface. From allof the terms in the temperature and salt equa-tions (7.12 7.13), Munk assumed that most ofthe ocean is dominated by the balance

vertical advection ¼ vertical diffusion (7.46a)

w vT=vz ¼ v=vzðkVvT=vzÞ (7.46b)

Munk obtained his diffusivity estimate from anaverage temperature profile and an estimate ofabout 1 cm/day for the upwelling velocity w,which can be based on deep-water formationrates and an assumption of upwelling over thewhole ocean. The observed diapycnal eddydiffusivity in the open ocean away from bound-aries is an order of magnitude smaller thanMunk’s estimate, which must be valid for theglobally averaged ocean structure. This meansthat there must be much larger diffusivity insome regions of the ocean d now thought to beat the boundariesd at large seamount and islandchains, and possibly the equator (Section 7.3).

7.10.3. Stommel and Arons’ Solution:Abyssal Circulation and Deep WesternBoundary Currents

Deep ocean circulation has been explainedusing potential vorticity concepts that are veryfamiliar from Sverdrup balance (Section 7.8.1).Stommel (1958), Stommel, Arons, and Faller(1958), and Stommel & Arons (1960a,b) consid-ered an ocean with just two layers, and solvedonly for the circulation in the bottom layer.They assumed a source of deep water at thenorthernmost latitude, and then assumed thatthis water upwells uniformly (at the same rate)everywhere (Figure S7.41). This upwellingstretches the deep ocean water columns.

Equator

φ1 φ2

S0(a)

(b)

FIGURE S7.41 (a) Abyssal circulation model. After Stommel and Arons (1960a). (b) Laboratory experiment results lookingdown from the top on a tank rotating counterclockwise around the apex (So) with a bottom that slopes towards the apex.There is a point source of water at So. The dye release in subsequent photos shows the Deep Western Boundary Current, andflow in the interior Si beginning to fill in and move towards So. Source: From Stommel, Arons, & Faller (1958).


Stretching requires a poleward shift of the watercolumns to conserve potential vorticity (Eq.7.35). The predicted interior flow is thereforecounterintuitive d it runs toward the deep-water source. (Actual abyssal flow is stronglymodified from this by the major topographythat modifies the b-effect by allowing stretchedcolumns to move toward shallower bottomsrather than toward higher latitude.)

Deep Western Boundary Currents (DWBCs)connect the isolated deep-water sources and

the interior poleward flows. Whereas unambig-uous poleward flow is not observed in the deepocean interior (possibly mostly because oftopography), DWBCs are found where theyare predicted to occur by the Stommel andArons abyssal circulation theory (Warren,1981). One such DWBC runs southward beneaththe Gulf Stream, carrying dense waters from theNordic Seas and Labrador Sea. Swallow andWorthington (1961) found this current afterbeing convinced by Stommel to go search for


it. Maps from the 1920s Meteor expedition(Wust, 1935) show the large-scale consequencesof this particular DWBC for deep salinity andoxygen distributions over the whole length ofthe Atlantic (see Chapter 9). Stommel’s (1958)map (Figure S7.42), revisited later by Kuo andVeronis (1973) using a numerical ocean model,shows the conceptual global pattern of DWBCsand abyssal circulation, including the deep-water source in the northern North Atlantic(the source of North Atlantic Deep Water,Chapter 9) and in the Antarctic (the source ofAntarctic Bottom Water, Chapter 13).

7.10.4. Thermohaline Oscillators:Stommel’s solution

An entirely different approach to the MOCfrom the Stommel-Arons abyssal circulationmodel considers changes in overturn associatedwith changing rates of dense water production.The prototype of these models is a very simplereduction of the ocean to just a few boxes, andwas also developed by Stommel (1961), whocan be appreciated at this point as a giant ofocean general circulation theory. Such boxmodels show how even the simplest model ofclimate change, for example, can lead to

FIGURE S7.42 Global abyssalcirculation model, assuming twodeep water sources (filled circlesnear Greenland and Antarctica).Source: From Stommel (1958).

complex results. In this case, multiple equilibriaresult, that is, the system can jump suddenlybetween quite different equilibrium states.

Stommel (1961) reduced the ocean to twoconnected boxes representing dense, cold,fresh high latitudes and light, warm, saltierlow latitudes (Figure S7.43). The boxes are con-nected, with the amount of flow between themdependent on the density difference betweenthe boxes. (This is a simplification of sinkingof dense water to the bottom and flowingtoward a region of lower bottom density, tobe fed in turn by upwelling in the lowerdensity box, and return flow at the sea surface.)In each box, the temperature and salinity areset by (1) flux of water between the boxes (ther-mohaline circulation) that depends on thedensity difference between the boxes and (2)restoring temperature and salinity to a basicstate over some set time period. Then theeffects on the flow between the boxes of slowheating and cooling, or of freshwater fluxes(evaporation and precipitation for instance),are studied.

Stommel (1961) found that several differentthermohaline circulation strengths exist fora given set of choices of model parameters(externally imposed temperature and salinity,

Low latitudes

WarmingEvaporation

TH, SHTL, SL

High latitudes

CoolingFreshening

surface flow

bottom flow

(a)

(b)

High latitude Freshwater input

Nor

th A

tlant

ic S

ST

12

High latitude Freshwater input

Nor

th A

tlant

ic S

ST

1

23

FIGURE S7.43 (a) Schematic of the Stommel (1961) two-box model of the meridional overturning circulation. Thedirection of the arrows assumes that the higher latitude box (blue) has higher density water. Each box is well mixed. (b)Schematic of the hysteresis in North Atlantic sea-surface temperature resulting from hysteresis in MOC strength. Thestarting point in freshwater is denoted by 1; starting at lower freshwater, hence higher salinity, in the left panel. Freshening isdenoted by the blue arrow, with the same total amount in both panels. Salinification is denoted by red arrow, and should beexactly opposite to the freshwater arrow. In the left panel, starting at higher salinity, the freshening allows the system toremain on the top branch, and so subsequent evaporation returns the system to original state. In the right panel, witha fresher starting point, the same freshening causes transition to lower curve (2), and subsequent evaporation returns systemto a different state, denoted by 3. After Stocker and Marchal (2000).


restoration timescales for temperature andsalinity, and factor relating the flow rate to thedensity difference between the boxes). As thebasic state was slowly changed, perhaps byreduction of the basic high-latitude salinity

(which reduces its density), the flow rate slowlychanged and then suddenly jumped toa different equilibrium rate. When the basicstate salinity was then slowly increased, thesystem jumped back to a higher flow rate but


at a very different basic salinity than during itsdecreasing phase. Thus this system exhibitshysteresis: it has different equilibrium statesdepending on whether the state is approachedfrom a much higher salinity or a much lowersalinity.

The coupled atmosphere-sea-ice-land-physics-biology-chemistry climate system isfar more complex than the two simple boxes inthis very simple Stommel oscillator model. Yetits multiple equilibria and hysteresis behaviorhave been useful in demonstrating the potentialfor abrupt and relatively large changes inclimate and, more specifically, for interpretationof numerical models of the changes in overturn-ing circulation that could result from changes inexternal forcing.

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