lecture - univ-brest.frmespages.univ-brest.fr/~scott/master_iuem/m2_physmar/... · 2018. 11....

Lecture 1: Introduction 1

Lecture


Plan for today

— Introduction : quick reminder of key concepts we will use.

You must interrupt me and tell me if you have not seen

some of this before.

— Sverdup theory (starts p.48 of these notes, material before

p.48 is background material I assumed you knew, and only

referred to as needed).

— Ekman currents and Ekman transport.

— The Wind-driven gyre (Stommel) theory.

— Observations will be referred to as needed, especially the

surface wind stress but also the zonally averaged potential

density to see the effects of the Ekman pumping on the

isopycnals.


Goal

— Our aim is to understand large-scale ocean circulation,

espeically the basin scale, midlatitudes circulation.

— The principle tools are the equations of ocean circulation and

the primary forcing of the circulation, especially the wind.

— We will focus on large-scale, equilibrium circulation. In

particular the balances known as geostrophy, hydrostatic

balance, thermal wind, and Ekman transport will be

introduced and used to understand the meridional

(north-south) variation in the thermocline.


Our Perpetual Ocean !

https://www.nasa.gov/topics/earth/features/

perpetual-ocean.html

background – stop me for more details ! 5

Notation and conventions

— We do not use inertial reference frames, but rather a

reference frame fixed to the rotating Earth. We will use

either spherical coordinates with origin at the centre of the

Earth or rectlinear (Cartesian) coordinates denoted (x, y, z)

or (x1, x2, x3) or simply xi with it understood that the index

i takes values 1, 2, 3 or equivalently values x, y, z. Generally z

(or r for spherical coordinates) will be the vertical direction.

— Vectors are all three dimensional Cartesian vectors (no

distinction between contravariant and covariant

components) and are indicated with an arrow, ~F .

— The velocity

~u =d

dt(x(t), y(t), z(t)) = (u, v, w) = (u1, u2, u3) (1)


Or we can simply refer to an arbitrary component of the

vector, ui, with the index i taking the 3 possible values,

i ∈ 1, 2, 3.


Physical oceanography in prespective

— The ocean (and most of the atmosphere) are governed by

the Navier-Stokes equations and the laws of classical

thermodynamics.

— But the full Navier-Stokes equations are too complex to

solve, and the resulting solutions too complex to describe

usefully.

— So we make gross simplifications, leading to key balances

that we give different names :

1. geostrophy

2. hydrostatic balance

3. thermal wind

4. Ekman transport

background : stops me for me info ! 8

Ocean bathymetry

— There are five principal ocean basins :

— Atlantic Ocean

— Pacific Ocean

— Indian Ocean

— Arctic Ocean

— Southern Ocean

— The ocean bathymetry is divided into several regions :

— Continental shelf of depth between 100 and 200 m.

— Abyssale plain, 3000 m to 6000 m.

— Continental slope, the region between the shelf and the

abyssal plain.

— Marine trenches, long thin valleys. For example the

Marianas Trench, trough in the Earth’s crust averaging


about 2,550 km (1,580 mi) long and 69 km (43 mi) wide.

The maximum-known depth is Challenger Deep, at

10,994 metres


Large-scale circulation

quasi-two-dimensional

— The largest horizontal length scales in the ocean are the

basin scale, roughly the radius of the Earth or about

6400km, which is more than 1000 times the depth of the

ocean.

— As a result the ocean is circulation is mostly horizontal or

quasi two dimensional. This quasi two dimensional is, to a

very good approximation, in hydrostatic balance.


Ocean forcing

Ocean currents are driven by 3 main mechanics

— Wind stress. The stress of the wind on the ocean surface

drives currents, called Ekman transport, that are the

principle mechanism setting the basin scale ocean thermal

structure. The resulting circulation is called the wind-driven

gyre circulation.

— Moon and Sun tidal force. The time varying

gravitational attraction of the Moon and Sun drives the

ocean tides on many time periods, the most prominant being

the principal lunar semi-diurnal tidal , called the M2 tide,

with period about 12 hrs and 25 minutes and principal solar

semi-dirurnal tide, called S2, with period exactly 12 hrs. The

direct motions resulting from this forcing are very large scale


barotropic (essentially depth-independent) waves and

barotropic coastal Kelvin waves (waves that decrease in

amplitude exponentially from the coast) that we observe at

the shore. But these directly forced tidal motions also drive

turbulence and internal waves as they interact with the

rough sea floor, which in turn mix the ocean thereby

strongly affecting its thermal structure and resulting global

scale circulation.

— Heat fluxes, evaporation and freezing. These three

important mechanims take place at the ocean surface and

affect the sea water density. Evaporation increases the

salinity and therefore the seawater density. Cooling at high

latitudes increases the density. Freezing, in the formation of

sea ice, increases the salinity of the surrounding waters and

thereby increases the density. These mechanisms set the

density boundary conditions that influence the global scale


ocean circulation. In the past some oceanographers believed

that these boundary conditions lead directly to a global

overturning circulation called the thermohaline circulation.

Now its is more widely appreciated that winds and tides are

essential in determining this global overturning circulation

and this term thermohaline circulation is less often used by

physical oceanographers.

— While all the above forcing mechanisms are important it is

sometimes difficult to separate a given current as due to the

given mechanism.

Ocean dynamics 15

Background material 2. The equations of

motion

Ocean dynamics 16

Fundamental equations

— The continuity equation

∂ρ

∂t+

∂

∂xj(ρuj) = 0, (2)

expresses the conservation of mass of the continuous fluid.

— The Mach number, M = Ucs

, where cs is the sound speed and

U is a typical ocean current velocity, is very small, which

implies that seawater is effectively incompressible ; the

volume of a fluid parcel undergoes negligible variations in

volume during its motion. The continuity equation Eq(2) in

this case simplifies to

∇ · ~u =∂

∂xiui = 0, (3)

meaning that the velocity field is non-divergent.

Ocean dynamics 17

— The momentum equation for a continuous fluid can be

written in the reference frame fixed to the solid Earth :

ai =DuiDt

+ Coriolis + Centripetal (4)

= −1

ρ

∂p

∂xi− gδi3 + Viscose forces. (5)

The RHS of Eq(5) is the net force per unit mass, the viscose

terms result from the divergence of the deviatoric stress

tensor. The RHS of Eq(4) includes inertial terms that do not

appear in an inertial reference frame. Because the reference

frame rotates with the Earth we have the additional

acceleration terms arising from the acceleration of the fixed

coordinates, the Coriolis acceleration and Centripetal

acceleration. We discuss these more in detail next.

— In Eq(4) we have the acceleration ai of a fluid parcel relative

to an inertial reference frame. We have decomposed this into

Ocean dynamics 18

3 contributions :

1. The acceleration of the fluid parcel relative to the

reference frame of the Earth,

DuiDt

(6)

2. The Coriolis acceleration of a point moving with velocity

~u in the rotating reference frame of the Earth,

Coriolis = 2~Ω× ~u (7)

where ~Ω is the rotation rate of the Earth expressed as a

vector pointing in the direction of the axis of rotation

and magnitude equal to the rate of rotation. Note that

this term vanishes if the fluid is stationary ! The Coriolis

acceleration is perpendicular to the current and the

Ocean dynamics 19

Coriolis force per unit volume is

~Fcor = −ρ2~Ω× ~u. (8)

You can feel the effect of the Coriolis force if you try to

rotate a spinning object, like a hair dryer or power drill.

3. The centripetal acceleration of a fixed point of the

rotating reference frame (accelerating because the Earth

is spinning on its axis).

Centripetal = ~Ω× ~Ω× ~r, (9)

where ~r is the position vector of the fluid parcel relative

to the centre of the Earth. Because this last term

depends only upon position we can consider this as a

modification of the acceleration of gravidty ~g. We define

the effective gravity

~g∗ = ~g − ~Ω× ~Ω× ~r, (10)

Ocean dynamics 20

which is no longer directed toward the centre of the

Earth. In fact this term has deformed the solid Earth

such that the Earth is no longer precisely a sphere but

has radius is about 22 km greater at the equator than at

the poles. To simplify the notation, we will never include

the star, and always just write ~g it being understood that

this term includes the gravity and a small correction

from the centripetale acceleration.

Ocean dynamics 21

Oceanic reference frame

— We naturally use a reference frame fixed to the rotating

Earth. Because of the Earth’s rotation, this frame is not

inertial, and the additional acceleration terms discuss above

arise.

— The most natural coordinate system in this reference frame

is a spherical coordinate system (r, θ, φ), with origin at the

centre of the Earth, with r measuring the distance from the

centre of the Earth, the polar angle θ measuring the angle

from the axis of rotation (so θ = 90 − lat, where lat is the

latitude in degrees, and the azimuthal angle φ measure the

angle from Greenwich meridian (where longitude is zero so

that φ is the longitude). We idealize the Earth as being

exactly spherical with radius R corresponding to the

Ocean dynamics 22

sealevel. Then relation then to the more familiar latitude

(lat) and longitude (lon) and altitude z are

r = R+ z, θ = 90 − lat φ = lon (11)

This is the best coordinate system to use for dealing with

large (basin size) calculation.

— In practice we often want to analyze local processes. Then

the more familiar Cartesian coordinate system is adequate

and simpler. We choose an origin at some convenient

location (latc, lonc) and apply a simple map projection to

local Cartesian coordinates

x = R cos(lat)(lon− lonc),

y = R(lat− latc),

z = z (12)

We will use the conventional unit vectors ~i,~j,~k along the

Ocean dynamics 23

x, y and z axes.

Ocean dynamics 24

Equations of motion in the oceanic

reference frame

— First we must find how the Coriolis acceleration can be

written for the three components of velocity (u, v, w). The

vector ~Ω points along the axis of the Earth, so geometry

gives us that at latitude lat this vector has components

2~Ω = 2Ω cos(lat)~j + 2Ω sin(lat)~k,

= f∗~j + f~k. (13)

What is Ω = ‖~Ω‖ ? It is the rate of rotation of the Earth in

an inertial reference frame. The stars provide an (extremely

good) approximation to an inertial reference frame so that

we can measure the rotation rate relative to the stars. The

Ocean dynamics 25

Earth takes by definition 24 hours to find the Sun in the

same position in the sky. But that means it turns slightly

more than 2π radians in a day, in fact it turns

2π +2π

365.2425rad day−1 =⇒ Tsidereal = 23 hr 56 min 4.09 sec

(14)

And so

2Ω =2π radian

(23 · 60 · 60 + 23 · 60 + 4.09)second= 7.292× 10−5

rad

s(15)

Ocean dynamics 26

— Recall our convention for velocity in Eq(1). We have

Coriolis = 2~Ω× ~u = det

~i ~j ~k

0 f∗ f

u v w

,

= (−fv + f∗w)~i+ fu~j − f∗u~k. (16)

— Denoting the viscose force as ~F we can write the equations

of motion Eq(5) as

Du

Dt+ f∗w − fv = −1

ρ

∂p

∂x+ Fx,

Dv

Dt+ fu = −1

ρ

∂p

∂y+ Fy,

Dw

Dt− f∗u = −1

ρ

∂p

∂z− g + Fz,

∇ · ~u = 0. (17)

Ocean dynamics 27

— The first approximation we make is called the Boussinesq

approximation. As noted earlier, the density in the ocean

does not changes much, at most a few percent, from the

value 1028 kg m−3. So we will make very little error in the

first two equations by setting ρ = ρ0, a constant reference

density in the first two equations.

— We can write the equations of motion

Du

Dt+ f∗w − fv = − 1

ρ0

∂p

∂x+ Fx,

Dv

Dt+ fu = − 1

ρ0

∂p

∂y+ Fy, (18)

— In the third equation we have to be careful because the

gravity is such an important term. Instead we multiply

Ocean dynamics 28

through by ρρ0

to obtain

ρ

ρ0

(Dw

Dt− f∗u

)= − 1

ρ0

∂p

∂z− ρ

ρ0g +

ρ

ρ0Fz. (19)

We have seen previously that the primary balance in this

equation is the hydrostatic balance, the underlined terms on

the RHS. The other terms provide at most a small

correction to this hydrostatic balance. So we argue thatρρ0≈ 1 and this is a good enough approximation for all the

small correction terms,

Dw

Dt− f∗u = − 1

ρ0

∂p

∂z− ρ

ρ0g + Fz. (20)

— In summary we can write the incompressible, Boussinesq

Ocean dynamics 29

equations of motion

Du

Dt+ f∗w − fv = − 1

ρ0

∂p

∂x+ Fx, (21)

Dv

Dt+ fu = − 1

ρ0

∂p

∂y+ Fy, (22)

Dw

Dt− f∗u = − 1

ρ0

∂p

∂z− ρ

ρ0g + Fz, (23)

∇ · ~u = 0. (24)

Ocean dynamics 30

The large scale motions

— We now scale the terms in the Boussinesq equations by

associating a typical value for horizontal velocity U , vertical

velocity W , horizontal length scale L, vertical length scale

H. As we are interested in the large scale motions, we

assume aspect ratio α = H/L is small :

α = H/L 1. (25)

— First consider the incompressibility condition.

∂u

∂x+∂v

∂y= −∂w

∂z

O

(U

L

)= O

(W

H

), =⇒ W = αU. (26)

Ocean dynamics 31

So the ratio of the two Coriolis terms in the Eq(21)

f∗w

fv=

cos(lat)

sin(lat)α (27)

which is small everywhere accept very near the Equator,

permitting us to simplify the first equation to

Du

Dt− fv = − 1

ρ0

∂p

∂x+ Fx, (28)

for large scale motion away from the Equator.

— An important observation in fluid mechanics is that the

pressure gradient term is almost always important ; no term

dominates the pressure gradient term.

— Now consider the vertical momentum equation Eq(23). The

pressure gradient in this equation is 1/α times stronger than

in the horizontal equations, and yet the terms on the LHS of

Eq(23) are similar to or smaller than those on the LHS of

Ocean dynamics 32

the horizontal momentum equations :

O(f∗u) = 2Ω cos(lat)U ≈ O(fu) = O(fv) = 2Ω sin(lat)U

(29)

accept near the Equator. Furthermore,

O

(Dw

Dt

)= αO

(Du

Dt

)= αO

(Dv

Dt

)(30)

The viscose terms involve the velocities and only in special

circumstantce could be much larger than the acceleration

terms. So outside the equator region, there is nothing to

balance this strong vertical pressure gradient force for the

large scale circulation except the gravity term. The primary

balance must be the hydrostatic balace we found for a static

fluid :∂p

∂z= −ρg (31)

Ocean dynamics 33

where p is the fluid pressure, z the vertical coordinate in our

inertial reference frame with Cartesian coordinates, ρ is the

fluid density, and g = 9.81 m s−2 the acceleration due to

gravity near the surface of the Earth. This is the very

important hydrostatic balance.

— In summary, for the large scale, non Equatorial ocean we

have found the simpler set of equations :

Du

Dt− fv = − 1

ρ0

∂p

∂x+ Fx, (32)

Dv

Dt+ fu = − 1

ρ0

∂p

∂y+ Fy, (33)

1

ρ0

∂p

∂z= − ρ

ρ0g, (34)

the hydrostatic Boussinesq equations.

Ocean dynamics 34

Geostrophic currents

— For time scales T long compared to the sidereal day, the

relative acceleration terms DuDt and Dv

Dt will be small relative

to the Coriolis terms, except possibily very near the

Equator. Notice we are considering the time scale following

the fluid parcel.

— We can anticipate when the time scales will be long using

the advection time scale T = L/U . Then we find the ratio of

the accerlation and Coriolis terms is

O(DuDt

)O (vf)

=U

L/U

1

Uf=

U

fL≡ Ro. (35)

This ratio is of fundamental importance in geophysical fluid

dynamics, and defined as the Rossby number.

Ocean dynamics 35

Table 1 – Rossby number of typical ocean (and atmospheric ) phe-

nomena

Feature L U Ro

Gulf Stream ring 100 km 1 m s−1 0.1

Gulf Stream 50 km 1 m s−1 0.2

Mid ocean eddy 50 km 0.1 m s−1 0.02

Rossby wave 1000 km 0.1 m s−1 0.001

Rossby wave (atmosphere) 1000 km 10 m s−1 0.1

Anticyclone (midlatitude “high”) 2000 km 10 m s−1 0.05

Cyclone (midlatitude “low”) 1000 km 20 m s−1 0.2

Category 3 hurricane 500 km 50 m s−1 1

Tornado 100 m 50 m s−1 5000

— So for small Ro we can ignore the relative acceleration term

Ocean dynamics 36

in the equations of motion, giving

−fv = − 1

ρ0

∂p

∂x+ Fx, (36)

+fu = − 1

ρ0

∂p

∂y+ Fy, (37)

1

ρ0

∂p

∂z= − ρ

ρ0g, (38)

— The viscose terms are generally of less secondary importance

accept in turbulent boundary layers.

— The conclusion is that the large scale, long time scale

motions outside turbulent boundary layers and not near the

equator are in geostrophic balance,

−fvg = − 1

ρ0

∂p

∂x, fug = − 1

ρ0

∂p

∂y(39)

where we have used a subscript g to emphasize that these

Ocean dynamics 37

currents are the geostrophic currents. One can take this as a

definition of the geostrophic currents, regardless of the

setting, and say that the currents are well approximated by

the geostrophic currents for the large scale, long time scale

motions outside turbulent boundary layers and not near the

equator.

In vector form

f~k × ~ug = − 1

ρ0∇p,

or ~ug =1

fρ0~k ×∇p. (40)

Notice the geostrophic currents are, surprisingly, orthogonal

to the pressure gradient. This relation works even in the

Southern Hemisphere where f is negative. In the NH

(northern hemisphere) where f > 0 Eq(40) implies that, if

you stand with your back to the wind (or current) the higher

Ocean dynamics 38

pressure is on your right. Furthemore, around a low pressure

system (or sea surface depression) the winds (or currents)

rotate in the same sens as the Earth’s rotation ; that is why

low pressure atmospheric systems are called cyclones and

high pressure systems are called anticyclones. In the NH, the

Earth appears to spin anticlockwise when you look down at

the North Pole.

— Note that Eq(40) implies that if we knew the pressure field

we could calculate the winds and currents. In the

atmosphere, there is a global network of ballons that

measure the temperature of the air to determine the

pressure, and thereby determine the large scale wins. For the

ocean, since the early 1990s, the sea surface height is

monitored using several satellites equiped with radar

altimeters that measure the sea surface height η to with a

cm or so precision.

Ocean dynamics 39

— The oceanic pressure field determined from the η using the

Eq(34)

−∫ η

z

∂p

∂zdz =

∫ η

z

ρgdz, (41)

p(x, y, z)− patm = gρ(η − z), (42)

where we have introduced the mean density,

ρ =1

(η − z)

∫ η

z

ρdz. (43)

For shallow depths ∇ρ is negligible. Furthermore typically

we ignore the pressure gradients from the atmosphere, which

are small because of the much larger length scales in the

atmosphere ; i.e. we generally assume that ∇patm gρ0∇η.

— Using these assumptions and p from Eq(42) for the

Ocean dynamics 40

geostrophic current relation Eq(40) we find

~ug =gρ0f~k ×∇η, (44)

because ∇z = 0 because z is a constant height level.

— The equation Eq(44) is valid near the surface (but below the

turbulent boundary layer where shear stresses are

important ; i.e. the so-called Ekman layer discussed later)

but at shallow enough depths that the horizontal variations

in density are not important. Observations reveal that the

horizontal mean density variations compensate the pressure

gradient induced by the sea surface height variations. In the

mid latitudes the geostrophic current becomes negligible

below about 1000 to 1500 m depth. In the tropics and

equatorial ocean, the geostrophic currents are even more

strongly surface trapped.

— Since 1992, accurate global sea surface height η observations

Ocean dynamics 41

have been available, permitting the global calculation of near

surface geostrophic currents. The orbital period of one

satellite is about 10 days, so the time resolution was initially

about 20 day (Nyquist period = shortest period

unambiguously resolvable at 10 sampling). When several

satellites observe the sea surface height simultaneously this

sampling is of course improved. I was just starting my career

at this time and I recall many researchers were initally

skeptical of the validity of this data. But it gradually because

trusted and has since revolutionized physical oceanography.

Ocean dynamics 42

Background Material 3. Thermal wind

Ocean dynamics 43

Thermal wind

— The geostrophic balance holds throughout the ocean for

large-scale flows, i.e.

Ro =U

fL 1, (45)

away from the equator and away from turbulent boundary

layers.

— We exploited this balance in the previous section for the

determination of near-surface geostrophic currents from sea

surface height observations, one of its important

applications.

Ocean dynamics 44

— The geostrophic balance

f~k × ~ug = − 1

ρ0∇p,

or ~ug =1

fρ0~k ×∇p. (46)

leads to another important relation for the vertical

derivative of the geostrophic currents. Taking the vertical

derivative of Eq(40) we find

∂

∂z~ug =

∂

∂z

1

fρ0~k ×∇p,

=1

fρ0~k ×∇ ∂

∂zp,

(47)

Ocean dynamics 45

Now we use the hydrostatic relation Eq(31)

∂

∂z~ug = − g

fρ0~k ×∇ρ. (48)

This vertical gradient in geostrophic current, Eq(48), is valid

in many conditions : for atmospheric winds as well as ocean

currents. For historical reasons, it is called the thermal wind,

even when applied to ocean currents.

— Returning to the discussion of the surface trapped current

geostrophic currents, we can say that it is the thermal wind

that tends to compensate the near-surface geostrophic

currents reducing the geostrophic current with depth so that

it eventually vanishes around by 1500 m depth in the mid

latitudes, shallower in the tropics.

— Eq(48) is useful in many applications. Before the

development of satellite altimeter, the currents in the ocean

were either measured directly with current meters or

Ocean dynamics 46

inferred from the Eq(48). In the latter case, we need a level

of reference zref for which the geostrophic currents ~ug(zref)

are known. Then Eq(48) can be integrated from this level∫ z

zref

∂

∂z′~ugdz

′ =

∫ z

zref

− g

fρ0~k ×∇ρdz′,

~ug(z) = ~ug(zref)−∫ z

zref

g

fρ0~k ×∇ρdz′ (49)

Historically it was assumed that there was a so-called level

of no motion around 2000 m depth, so this was choosen as

zref = −2000m. That is, the geostrophic currents were

considered so weak at this level that they could be

neglected, ~ug(zref) ≈ 0. Then temperature and salinity

measurements throughout depths z > zref where enought to

determine ρ, the horizontal gradient of which could be

integrated in Eq(49) to make these current estimates.

— With the advent of the Argo programme (global distribution

Ocean dynamics 47

of floats that divide up and down in the upper 1500m depth

measuring T and S and park at 1000 m depth for 10 days),

we can do better than the level of no motion assumption.

The Argo floats effectively measure the ocean current

velocity at 1000 m. (The Argo float parks at 1000 m depth

for 10 days before returning to the surface to communicate

its position and data to satellites. From these 10-day float

displacements one can infer the ocean current at the parking

depth. ) Furthermore, we have regular global measurements

of upper ocean density from the T and S measurements

taken by the Argo floats as they dive up and down.

Ocean dynamics 48

IUEM Lecture. Sverdup theory, Ekman

currents and Ocean General Circulation

Sverdrup Theory 49

Sverdrup Theory

Reference : See Chapter 7 of the course notes “Elements de

dynamique des fluides planetaires,” by Xavier Carton.

— Our immediate goal is to derive the Sverdrup relation

βvg = f∂w

∂z(50)

where vg is the merdional geostrophic velocity, w is the

vertical velocity, f is the local vertical component of the

Coriolis parameter f = 2Ω sin(θ), θ is the latitude and Ω is

the rate of rotation of the Earth with respect to the fixed

stars.

— This surprisingly simple relation

1. applies to a good approximation for the incompressible

Sverdrup Theory 50

geostrophic circulation,

2. is useful, when combined with Ekman theory to explain

the basin-scale circulation driven by the wind-stress curl.

Sverdrup Theory 51

Derivation of the Sverdrup relation

— Our starting point is the following two approximations.

— The geostrophic balance

~ug =1

fρ0~k ×∇p,

(51)

where p is the pressure, ρ0 is the fluid density, and ~k is a

unit vector in the local vertical direction. We will use these

relations on a basin scale so we use a local Cartesian

coordinate system (constructed from a map projection from

Sverdrup Theory 52

the spherical coordinates, see Eq(12) ) :

ug = − 1

fρ0

∂p

∂y,

vg =1

fρ0

∂p

∂x(52)

where x is zonal coordinate, y is the meridional coordinate,

z is the local vertical coordinate.

— The second starting assumption is the incompressibility

condition :

∇ · ~u = 0, (53)

which in the local Cartesian coordinate system becomes

∂u

∂x+∂v

∂y+∂w

∂z= 0 (54)

— Applying the incompressibility Eq(54) to the geostrophic

Sverdrup Theory 53

currents Eq(52) gives

−∂w∂z

=∂ug∂x

+∂vg∂y

,

=∂

∂x

(−1

ρ0f

∂p

∂y

)+

∂

∂y

(1

ρ0f

∂p

∂x

),

=

−1

ρ0f

(∂2p

∂x∂y

)+

1

ρ0f

(∂2p

∂x∂y

)+

(1

ρ0

∂p

∂x

)∂

∂y

(1

f

),

= −(

1

ρ0f2∂p

∂x

)∂f

∂y,

= −vgβ

f. (55)

βvg = f∂w

∂z(56)

— When does this apply ? When the circulation is geostrophic

Sverdrup Theory 54

and incompressible. These are very general conditions that

apply for the large scale, interior circulation (away from

turbulence boundary layers near the surface and bottom of

the ocean) away from the Equator.

— We can see this in more detail by considering which terms in

the complex momentum balance were ignored in the

geostrophic relation. These were the relative acceleration and

the turbulence stresses, the justification of which we now

recall.

— Recall geostrophy applies for the large scale L circulation

Ro =U

fL 1, (57)

with time scale T long compared to the “pendulum day”

(1/f),

1

T f. (58)

Sverdrup Theory 55

so that we can ignore the relative acceleration

∂ui∂t

+ uj∂ui∂xj

(59)

in the ith momentum equation, and away from the Equator

and the boundary layers where turbulence stresses∂σji

∂xj

become important (especially at the air-sea interface at the

ocean surface and the sea-solid-Earth interface at the ocean

bottom).

— Recall incompressibility applies when the current speed U is

much less than the speed of sound cs. The continuity

equation Eq(2) in this case simplifies to

∇ · ~u =∂

∂xiui = 0, (60)

meaning that the velocity field is non-divergent.

— To go beyond the Sverdrup relation we must consider these

Sverdrup Theory 56

turbulence boundary layers in more detail, especially the

surface layer and its relation to the surface wind stress.

wind-driven circulation 57

The wind-driven circulation

Reference : See Chapter 6 of the course notes “Elements de

dynamique des fluides planetaires,” by Xavier Carton.

— The wind is a principle source of the ocean circulation. The

mechanism is via the Ekman transport, which we now

describe.

— The theory of Ekman currents was first discovered by the

Swedish scientist Vagn Walfrid Ekman in the early 1900s in

attempting to explain the observations by Arctic explorer

Fridtjof Nansen that his ship drifted about 30 to the right

of the wind direction.

— The wind applies a stress ~τs at the surface of the ocean that

depends upon the near surface wind ~U10, air density

ρair ≈ 1.2 kg m−3, and sea surface roughness expressed as a


so-called drag coefficient CD which in turn is a function of

‖~U10‖ = U10 and density stratification. An order of

magnitude estimate of CD is O(10−3) except in extreme

conditions such as hurricanes. The wind stress is generally

estimated by the quadratic drag formula

~τs = ρairCDU10~U10 (61)

where ~U10 is the wind at the reference height of 10 m above

the sea surface (a height convenient for observation from a

large ship).

— The wind applies a stress ~τs enters the equations of motion,

the horizontal components of the hydrostatic Boussinesq

Eqs(32,33), as a boundary condition for the frictional forces


per unit mass, denoted Fx and Fy in

Du

Dt− fv = − 1

ρ0

∂p

∂x+ Fx, (62)

Dv

Dt+ fu = − 1

ρ0

∂p

∂y+ Fy. (63)

These are written for a general (not necessarily Newtonian)

continuous fluid as

ρ0Fx =∂

∂xiσxi =

∂

∂xσxx +

∂

∂yσxy +

∂

∂zσxz,

ρ0Fy =∂

∂xiσyi =

∂

∂xσyx +

∂

∂yσyy +

∂

∂zσyz. (64)

We seek equations valid in the upper boundary layer next to

the air-sea interface where the horizontal scales are set by

the length scales of the winds (generally hundreds to

thousands of kilometers for the most energetic scales) which

are much much greater than the vertical scales (observed to


be tens of meters). This boundary layer is now called the

Ekman layer, and generally lies within the upper part of the

mixed layer.


zx

-hEs

0

-H+hEb

-H

hEs

hEb

(H-hEb-hE

s)

surface Ekman layer

bottom Ekman layer

interior layer

Figure 1 – Three ocean layers : (i) surface Ekman layer, (ii) ocean

interior layer , (iii) bottom Ekman layer.


To a very good approximation we retain only the

components with vertical derivatives :

ρ0Fx =∂

∂zσxz, ρ0Fy =

∂

∂zσyz, (65)

with boundary conditions σxz = τs,x and σyz = τs,y.

— As argued above, when the time scale following the fluid

parcel is long compared to the f−1, so time scale longer than

a few days outside the Equatorial region, the relative

acceleration Dui/Dt is small and the Eqs(36) and (37) apply

−fv = − 1

ρ0

∂p

∂x+

1

ρ0Fx, (66)

+fu = − 1

ρ0

∂p

∂y+

1

ρ0Fy, (67)

(68)


— Replacing the frictional terms by Eq(65) we arrive at

−fv = − 1

ρ0

∂p

∂x+

1

ρ0

∂

∂zσxz, (69)

+fu = − 1

ρ0

∂p

∂y+

1

ρ0

∂

∂zσyz, (70)

— We pause to interpret this physically : The surface Ekman

layer also involves a 3-way momentum balance between

Coriolis, horizontal pressure gradient, and vertical derivative

of the horizontal turbulent stress.

— We now write the total current in Eqs(69) and (70) as the

sum of the geostrophic currents ~ug and the Ekman currents

~uE

~u = ~ug + ~uE . (71)

Substituting Eq(71) into Eqs(69) and (70) gives and using


Eq(40) we find

−fvE =1

ρ0

∂

∂zσxz, (72)

+fuE =1

ρ0

∂

∂zσyz, (73)

That is, the Ekman currents result from a balance between

the wind stress and the Coriolis force.

— A turbulent closure is required to be more quantitative. But

independent of the turbulent closure we can find the very


important Ekman transport

Mx =

∫ η

−Hρ0uE(z)dz =

∫ η

−H

1

f

∂

∂zσyzdz,

=1

fτs,y, (74)

My =

∫ η

−Hρ0vE(z)dz = −

∫ η

−H

1

f

∂

∂zσxzdz,

= − 1

fτs,x, (75)

where H is a depth sufficiently below the turbulent

boundary layer such that the stress has dropped to a

negligible amount. Mx and My and the eastward and

northward components of the depth-integrated mass

transport associated with the Ekman currents. Because they

depend only upon the stress boundary condition at the top

of the ocean, which is set by the winds, this incredible


theory lets us infer a complex, difficult to observe oceanic

mass transport with an atmospheric variable.

— The Ekman transport equations Eq(74) and Eq(75) can be

written in vector form

~M = − 1

f~k × ~τs. (76)

Check your understanding : This implies to the Ekman

transport in the Northern Hemisphere is directed (a) parallel

to the wind, (b) 90 to the right of the wind, or (c) 90 to

the left of the wind ?

— Historically is was not feasible to observe the wind stress

throughout the world ocean on a daily basis.

Oceanographers collected observations throughout the world

over many years and created approximate atlases of the

monthly means values taken (so averages of a given month

with observations from many different years). This common


averageing practice leads to climateological variables. Much

of the data came from ships of opportunity, these are

commercial ships that take climate data. As a result the

shipping lanes between North America and Europe and

Asian were well sampled, but much of the Southern Ocean

and Arctic were very poorly sampled.

— The 1980s saw a revolution in atmospheric observations with

the development of the technology to estimate surface wind

stress from satellite observations. The satellite-based radar

infer the sea surface roughness from echo return of radar

pulses. The surface wind stress is then calibrated to give

surface wind stress from detailed analysis of given sites.

Because winds decorrelated quickly in time, higher temporal

resolution is needed ; several daily wind stress products are

available, and some with multiple times per day.

— The divergence of the Ekman transport, ∇ · ~M , leads to


vertical velocity wE called the Ekman pumping upwelling,

wE =1

ρ0∇ · ~M. (77)

This relation can be derived using the continuity equation

and the definition of ~M in Eq(74) and Eq(75).

— From Eq(77) and Eq(76) we find that the Ekman pumping

is given by the wind-stress curl

wE =1

ρ0~k ·(∇× ~τs

f

). (78)

We will use this later !

— The most dominant wind patterns are the prevailing easterly

trade winds in the tropics and westerly winds between about

30 and 60 in both hemispheres, see Fig. 2.

— There is a positive wind-stress curl centred are 60N,

resulting in upwelling of the subpolar gyres in the North


Atlantic and Pacific oceans. This brings up nutrient rich

water to upper ocean that makes these waters very

productive marine life.

— We can understand the narrow thermocline in the equatorial

region from upwelling in this region due to the Ekman

pumping associated with the prevailing easterly trade winds.


Figure 2 – Annual mean surface wind stress (vector) and zonal

component (colour), (Talley et al., 2011, Fig. 5.6). Red τx > 0 ; blue

τx < 0.

— The convergence of the Ekman transport, −∇ · ~M , leads to a

negative vertical velocity wE called the Ekman pumping


downwelling. This is most prevalent in the subtropical gyre

regions of the mid latitudes, especially in the North

Hemisphere. The result is a much deeper thermocline. The

cold water necessary for the efficient OTEC is much deeper

in the subtropical gyre.

— In summary, we can understand the zonally averaged density

structure in the upper ocean, Fig. 3, based upon the global

wind patterns and Ekman theory.


Figure 3 – Zonally averaged annual mean potential density in

the global ocean. Units are kg m−3, expressed as a departure from

1000 kg m−3.

Sverdrup Transport 73

Derivation of the Sverdrup transport

Reference : See §7.1 of the course notes “Elements de dynamique

des fluides planetaires,” by Xavier Carton.

— We can integrate the geostrophic meridional velocity vg in

the vertical over its range of validity from the base z = −hsEof the surface turbulence boundary layer (Ekman layer) to

the top of the bottom turbulence boundary layer

z = −H + hbE .

— First note that

β =∂2Ω sin θ

∂y=

2Ω

a

∂ sin θ

∂θ,

=2Ω cos θ

a, (79)


where again a is the radius of the Earth. Thus

f

β= 2Ω sin θ

(2Ω cos θ

a

)−1= a tan θ. (80)

which in the Sverdrup relation Eq(56) we obtain

vg =f

β

∂w

∂z= a tan θ

∂w

∂z,∫ −hs

E

−H+hbE

vgdz = a tan θ

∫ −hsE

−H+hbE

∂w

∂zdz,

VS = a tan θ [w(z)]−hs

E

−H+hbE

(81)

— Now we can use Eq(78) for the vertical velocity at the base

of the surface Ekman layer,

w(−hsE) = wE =1

ρ0~k ·(∇× ~τs

f

). (82)

— At the base of the inertior layer, z = −H + hbE , we


traditionally set the vertical velocity to zero,

w(−H + hbE) = 0, commonly used assumption (83)

This assumption can be violated when the bottom

topography is strongly sloping, or when there are strong

boundary layers. The latter possibility will be discussed in

more detail later.

— The meridional volume transport, VS , then becomes

VS = a tan θwE = a tan θ1

ρ0~k ·(∇× ~τs

f

),

=f

β

1

ρ0~k ·(∇× ~τs

f

). (84)

This volume transport is called the Sverdrup transport.

— If we write the depth-integrated volume transport VS as the

product of a depth-averaged velocity v times the depth of

the ocean interior H − hsE − hbE ≈ H, since the Ekman layers


are 10s of meters thick while the ocean is thousands of

meters deep, then

VS = vH. (85)

In fact, we note that for a homogeneous interior,

ρ(z) = ρ0,−H + hbE < z < −hsE , we have constant interior

geostrophic velocity, vg = v.

— Substituting Eq(85) into Eq(84)

Hvβ =f

ρ0~k ·(∇× ~τs

f

). (86)

we can interpret it in terms of a vorticity balance. The large

scale wind stress curl is positive for example in the subpolar

gyres in the North Atlantic and North Pacific (see Fig. 2),

imparting positive vorticity (cyclonic, or in the same sense

as the rotation of the Earth) on the ocean. The relative

vorticity ζ f of the ocean interior being weaking relative


to the planetary voriticty f , the dominant term balancing

this voriticity input is the merdional advection of planetary

voriticity Hvβ ; i.e. v > 0 as the water drifts towards the

pole, gaining planetary vorticity. In the subtropical gyres in

the North Atlantic and North Pacific (see again Fig. 2), the

wind stress curl is negative and as a result v < 0 as the

water drifts towards the equator.

— Similarly we can interpret the Sverdrup relation Eq(56)

βvg = f∂w

∂z(87)

in terms of vorticity balances. When a water parcel, which is

spinning at f about the local vertical due to the Earths

rotation is subjected to vortex stretching (∂w∂z > 0), this

tends to increase the vorticity. This positive forcing term is

balance by the merdional advection of planetary voriticity

βvg > 0 as the water parcel moves poleward. This occurs in


the interior of the Northern Hemisphere subpolar gyres

where the Ekman pumping is upward ; the positive wind

stress curl drives upwelling. In the subtropical gyres the

negative wind stress curl drives downwelling, tending to

reduce vorticity which is achieved by drifting equatorward.

— Finally we note a limitation of the Sverdrup theory. Writing

the geostrophic velocity vg in terms of the pressure in the

Sverdrup relation Eq(94), we have

β1

ρ0f

∂p

∂x= f

∂w

∂z. (88)

On the other hand the RHS Eq(100) is determined by the

Ekman pumping and the local wind stress curl. Consider a

constant latitude, y =constant, where ∂w∂z > 0, for example

in the NA subpolar gyre. Integrating Eq(100) from the


western boundary, the pressure will monitonically increase

p(x, y) =ρ0f

2

β

∫ x

0

∂w

∂z(x′, y)dx′. (89)

On the other hand, we must meet the boundary condition

p(x, y) =constant along the boundary so that geostrophic

flux into the wall vanishes. Clearly this is impossible.

— This suggests that the friction term that we neglected above,

and which depends upon second derivatives of the pressure

field, becomes important in a boundary layer and physically

dissipates the vorticity provided by the wind and

mathematically allows two BCs to be meet on pressure on

either side of the basin.

bottom Ekman layer 80

Bottom Ekman Layer


Bottom Ekman Layer

— Like the surface Ekman layer, the bottom Ekman layer also

involves a 3-way momentum balance between horizontal

pressure gradient, Coriolis, and vertical derivative of the

horizontal turbulent stress.

— But in the bottom Ekman layer source of the turbulence

stresses is the interaction with the interior circulation and

the sea floor.

— A more complete description of this friction would involve

using a quadratic drag law as we used for the surface wind

stress, Eq(61). But the resulting equations become difficult

to treat analytically (see (Arbic and Scott , 2008) for a

numerical treatment).

— We follow the treatment of Stommel (Stommel , 1948) who


simplified the interaction by parameterizing the bottom

stress as linearly dependent upon the interior geostrophic

velocity above the bottom Ekman layer, ~ub

~τb = ρhbE |f |

2~ub. (90)

Comparing Eq(90) with Eq(61)

~τs = ρairCDU10~U10 (91)

we see that the velocity-dependent factor CDU10 has been

replaced by a constant hbE |f |/2.

— Using the linear bottom stress Eq(90) in the relation we


derived earlier between Ekman pumping velocity and stress

wE =1

ρ0~k ·(∇× ~τb

f

),

=1

ρ0~k ·(∇× ρhbE |f |~ub

2f

),

= sign(f)hbE2~k · (∇× ~ub) ,

= sign(f)hbE2ζb, (92)

where ζb is the vorticity of the interior geostrophic flow

above the Ekman layer. As anticipated by the argument

above, this term will turn out to be important only in a

narrow western boundary layer, where ζb is much larger than

in the deep-ocean interior.

Stommel Gyre 84

Stommel Gyre

Stommel Gyre 85

Stommel Gyre

— Stommel (1948) realised that the Sverdrup circulation could

be closed via a narrow boundary current in which the effects

of turbulent stresses become important and locally break the

Sverdrup meridional transport Eq(84).

— Correspondingly, we reconsider Sverdrup transport Eq(93),

replacing the bottom BC of vanishing vertical velocity,

Eq(83), with the Ekman pumping Eq(92) from the linear

drag, giving

VS = a tan θ [w(z)]−hs

E

−H+hbE

,

=f

β

(1

ρ0~k ·(∇× ~τs

f

)− sign(f)

hbE2ζb

)(93)

— A priori this boundary current could be either on the left or

Stommel Gyre 86

the right of the basin.

— But there is a simple argument that says that it should be

on the western boundary. Consider the NH subtropical gyres

where the vorticity input from the wind stress curl in the

central basin is negative, and the Sverdrup transport is

southward. The narrow boundary current must provide a

compensating, intense northward transport wherein the term

−hbE

2 ζb must be large and positive. This requires a negative

relative vorticity ζb < 0. An intense, north-flowing boundary

current on the eastern boundary would create a region of

positive vorticity, so −hbE

2 ζb < 0 would have the wrong sign.

On the other hand, a north-flowing boundary current on the

western boundary would create a region of negative

vorticity, so −hbE

2 ζb > 0 would have the correct sign.

— The argument still holds in the NH subpolar gyres with all

the signs reversed by the boundary current still on the

Stommel Gyre 87

western boundary.

— Consider the SH subtropical gyre. The easterly trade winds

to the north and the westerlies to the south (see Fig. 2)

create positive wind stress curl. Choosing our y-axis of the

local Cartesian coordinate system still pointing in the

northward direction even in the SH, we have f < 0 but

β > 0. There are two f in Eq(93) so their sign cancels, so

positive wind stress curl drives a northward (i.e.

equatorward) VS . The narrow boundary current must

provide a southward (poleward) flow in a region such that

f(−sign(f)hbE

2 ζb) < 0. This must be a narrow and positive

vorticity region, which we have seen must be adjacent the

western boundary.

Stommel Gyre 88

Stommel’s idealized subtropical gyre

— To obtain a simple idealized model to illustrate these ideas,

Stommel (1948) considered the case of a homogeneous

interior water column, ρ = ρ0 =constant. As we mentioned

briefly earlier, this implies vg = v, so we can replace

VS = Hvg = (H/fρ0)∂p/∂x in the Sverdrup transport,

VS =f

β

(1

ρ0~k ·(∇× ~τs

f

)− hbE

2ζb

),

H

fρ0

∂p

∂x=f

β

(1

ρ0~k ·(∇× ~τs

f

)− hbE

2

1

ρ0f∇2p

),

fhbE2H∇2p+ β

∂p

∂x=f2

H~k ·(∇× ~τs

f

)(94)

— This is a second-order linear inhomogeneous PDE for the

Stommel Gyre 89

pressure p. The wind stress curl represents a prescribed

forcing term.

— Stommel considered a rectangle basin on the β-plane,

f = f0 + βy. We consider the example of zonal extent

Lx = 5000km and meridional extent Ly = 3300km (as in

Carton §7.2).

— Following Stommel we prescribe a zonal wind stress

τx = −τ0 cos(π

Lyy), τy = 0,

~k · ∇ × ~τsf0

= τ0π

Lysin(

π

Lyy) (95)

noting that the β-plane approximation ignores the variation

in 1/f for simplicity.

— In the central basin (away from the western boundary) the

Stommel Gyre 90

Sverdrup transport applies and gives

∂p

∂x=

f0βH

τ0π

Lysin(

π

Lyy). (96)

This is easily integrated to give the interior solution,

pI(x, y) = p(x0, y) + (x− x0)f0βH

τ0π

Lysin(

π

Lyy). (97)

Imposing the boundary condition pI(Lx, y) = 0 (implying a

choice of p(x0, y) ) gives

pI(x, y) = (x− Lx)f0βH

τ0π

Lysin(

π

Lyy). (98)

— We now add on a narrow boundary layer solution pS(x, y),

with p(x, y) = pS(x, y) + pI(x, y). In the narrow boundary

layer,

∂

∂x ∂

∂y=⇒ ∇2 =

∂2

∂x2(99)

Stommel Gyre 91

and the dissipation is much larger than the forcing so the

full PDE Eq(94) reduces to

f0hbE2H∇2p+ β

∂p

∂x= 0,

f0hbE2H

∂2p

∂x2+ β

∂p

∂x= 0. (100)

This can be integrated to give

∂pS∂x

(x, y) = A(y)e− (x−x1)

LS , (101)

where LS is the Stommel boundary layer width,

LS =f0h

bE

2βH. (102)

and

A(y) =∂pS(x1, y)

∂x(103)

Stommel Gyre 92

and x1 = 0 is the western boundary.

— Another integration gives

pS(x, y) = pS(0, y)−A(y)LSe−x/LS , (104)

— This boundary solution pS must be matched to the interior

solution pI(x, y). This means when well outside the

boundary layer ( x/LS 1 ) but still near the western

boundary (x/Lx 1) (which is possible because LS Lx),

the boundary solution approaches the interior solution

pS → pS(0, y)→ −Lxf0βH

πτ0Ly

sin(πy

Ly). (105)

This gives our function pS(0, y) (it comes from the interior

solution), so that

p = pS + pI = −A(y)LSe−x/LS + (x− Lx)

f0βH

τ0π

Lysin(

π

Lyy)

(106)

Stommel Gyre 93

— We now impose p(0, y) = 0.

p(0, y) = pS + pI = −A(y)LS +−Lxf0βH

τ0π

Lysin(

π

Lyy),

=⇒ A(y) = −LxLS

f0βH

τ0π

Lysin(

π

Lyy). (107)

— Finally the total soluton is

p(x, y) =f0βH

τ0π

Lysin(

π

Lyy)[(x− Lx) + Lxe

−x/LS ]. (108)

Terminology in climate science 94

Terminology in climate science

— Monthly mean climatology : mean of each month, taken from

data over many years. For example, March SST is obtained

by averaging data only from the month of March but over

many years.

— Season climatology : like the monthly mean climatology, but

grouping months together to form seasons. Typically we

devide the year into 4 seasons of 3 months each. There are

different conventions, but a typical choice is January,

February and March (JFM) for winter, April, May, June

AMJ for spring, JAS for summer, OND for autumn. But

another commonly found convention is DJF for winter etc.

Again the JFM climatology will be an average over many

years of historical data.

Terminology in climate science 95

— Meridional average. Is an average in the meridional

direction. Meridians run North-South so this is an average

over latitude at fixed longitude.

— Zonal average. The zonal direction is East-West so this is an

average over longitude at fixed latitude.

Summary 96

Thesaurus

— Diapycnale, is a surface of constant density.

— downwelling (see also upwelling)

— Ekman transport

— Ekman pumping

— geostrophy

— Potential temperature, θ, is the temperature that a water

parcel would have if moved adiabatically to a reference

pressure. The surface is the most common reference level.

— S, denotes salinity in PSU (practical salinity units). The

salinity is the concentration of dissolved salt.

— T , denotes temperature.

— θ, denotes potential temperature.

— thermal wind

Summary 97

— thermocline

— upwelling (see also downwelling)

Summary 98

Practical resources for later use

— The MatLab routines that implement the EOS for seawater,

based upon the PSU for salinity, are available here http://

www.cmar.csiro.au/datacentre/ext_docs/seawater.htm

— Notice that they encourage the user to update to the latest

version of these routines, which are based upon absolute

salinity rather than PSU. The problem here is that you will

find most available historical data in PSU, as we discussed in

class. If you are working in a coastal area (as opposed to a

floating deep-sea platform) then the key quantity you need

to verify to know if your EOS calculations are accurate are

the proportions of the various salts.

Summary 99

References

Arbic, B. K., and R. B. Scott (2008), On quadratic bottom drag,

geostrophic turbulence, and oceanic mesoscale eddies, J. Phys.

Oceanogr., 38, 84–103.

Stommel, H. (1948), The westward intensification of wind-driven

ocean currents, Trans. Am. Geophs. Union, 29, 202–206.

Talley, L., et al. (2011), Descriptive Physical Oceanography : An

Introduction, Elsevier Science.

Vallis, G. K. (2006), Atmospheric and Oceanic Fluid Dynamics :

Fundamentals and Large-scale circulation, 744 pp., Cambridge

University Press.

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