Lecture 1. Introduction - univ-brest.frstockage.univ-brest.fr/.../ENSTA/CRE3/2019/ENSTA_CRE3_v2.pdfLecture 1: Introduction 2 Plan for today |Introduction. |Functioning of the course,

Lecture 1: Introduction 1

Lecture 1. Introduction

Lecture 1: Introduction 2

Plan for today

— Introduction.

— Functioning of the course, resources, subject matter.

— Introduction to oceanography.

About the course 3

Practical information

— I will present 3 courses of 3 hours each.

— You can interrupt me at any time, just raise your hand.

— You can ask me questions via email :

[email protected]

— You can find my slides on my website :

http://stockage.univ-brest.fr/~scott/ along with

other documents, including the notes in French from

previous instructor, hereafter “Arzel’s notes”.

— Other resources (much behond this course, but useful

references) :

Textbook on descriptive oceanography (Talley et al., 2011)

Very widely used textbook on atmospheric and oceanic

dynamics, (Vallis, 2006).

About the course 4

Goal

— Our aim is to understand enough physical oceanography to

understand the essential elements of Ocean thermal energy

conversion (OTEC).

— This technology exploits the temperature contrast between

warm surface waters and cold waters below the thermocline.

— The oceanic thermocline is the key structure that we wish to

understand. In particular the thermocline varies most

strongly with latitude, being much shallower and stronger in

the tropics than polar latitudes, making OTEC development

most favourable in the tropics. Our goal is to understand, in

the next 9hrs of course work, this feature of the oceans.

About the course 5

Our Perpetual Ocean !

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

perpetual-ocean.html

About the course 6

Subject matter of this course

For future reference :

This short course is an introduction to oceanic circulation. In

particular, we will cover :

— The equations of ocean circulation.

— 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.

About the course 7

Notation and conventions

— We will no longer 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)

About the course 8

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.

About the course 9

Ocean Circulation : The science of

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 : geostrophy, hydrostatic

balance, thermal wind, and Ekman transport.

About the course 10

— A key example we have seen already in Lecture 2 of RAN3 :

∂p

∂z= −ρg (2)

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. The vertical pressure gradient

force balances the force of gravity.

description of the worlds oceans 11

Ocean dimensions

— The oceans cover about 71% of the surface of the Earth.

— The average depth of the World Ocean is 3700 m.

— The mass of the ocean is about 260 times that of the

atmosphere.

— But the ocean represents about 0.023% of the mass of the

Earth.

— The upper 10 m of the ocean have the same weight as all the

air above. This means, according to the hydrostatic balance,

that the pressure at 10 m depth is twice the pressure at the

surface.

— 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 flow is, to a very good approximation, in

hydrostatic balance.


Ocean thermal capacity

— The specific heat capacity at constant pressure cp of a

substance is a measure of the energy required to raise the

temperature of a given mass of the substance by a given

temperature amout. In the SI system, the units are Joules

per kilogram per degree Kelvin or Celsius, J /(kg C).

— The value of the specific heat of sea water is

cpO = 3.99 kJ /(kg C) (note that one kilo Jule equals a

1000 joules, 1 kJ = 1000 J.)

— In comparison to the atmosphere, cpA = 1.01 kJ /(kg C) or

cpO ≈ 4cpA. As a result, the upper 10 m of the ocean have 4

times the heat capacity of the entire atmosphere, or the

upper 2.5 m of the ocean has about the same heat capacity

as the entire atmosphere.


— We say that oceans dampen climatic variations. They are

able to absorbe large quantities of heat with little change in

temperature.

— An apparant result of this as that seasonal cycle in

temperature of the upper ocean is much less than the

seasonal cycle of air temperature. We can cool off in the

summer by swimming in the ocean. The coatal regions

(especially those down wind of the prevailing winds, like

Brest) have more mild winters.

— Another result of this large contrast in heat capacity is

revealed daily by the sea breeze effect and seasonally by the

monsoons.

— The sea breeze. During the night the atmosphere, surface of

the Earth and the upper ocean cool as they exchange

radiation with outspace at 2.7 K (degree above absolute

zero). Over land the atmosphere, because of its small heat


capacity, cools quickly. The land too cools (because it is a

solid it doesn’t turn over and mix so only the upper surface

is exposed to the cooling air and extremely cold outer

space). The oceans cool much less. The upper ocean is caped

by the so-called mixed layer, a boundary layer tens of meters

deep that is well-mixed by the action of the wind, breaking

waves and thermal convection : as the surface water cools it

becomes more density and sinks, mixing with the waters

below. So to cool the surface of the ocean you must remove

heat from throughout the mixed layer, with heat capicity ten

times or more than the entire atmosphere. The result is that

by morning the air over the land is colder than over the sea.

During the day the land surface absorbs sunlight, heats

quickly causing convective currents that heat the lower

troposphere. But the surface of the ocean heats more slowly

and the air above it remains colder than over land. Thus a


temperature contrast builds between the air over land and

the sea, resulting in a convective cell forming, with cold

ocean surface air blowing in-land giving a refreshing cool

breeze near the shore on a hot summer day.


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 about 2,550


km (1,580 mi) long and averaging 69 km (43 mi) wide.

The maximum-known depth is Challenger Deep, at

10,994 metres.



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.

properties of sea water 23

Seawater properties : temperature, T ,

and potential temperature, θ, sea surface

temperature, SST

— Temperature is a measure of the molecular kinetic energy.

— Temperature is measured either in Kelvin, K, or degree

Celsius, C.

— 0 K = −273.16C.

— A change in of temperature of ∆T = 1 K = 1C.

— Thermistors are most often used in oceanography for in situ

temperature measurement and typically have an accuracy of

about 0.002C and a precision 0.0005 to 0.001C.

— Potential temperature is an indispensible concept in

oceanography. Imagine you measured a temperature


T = 7C for a parcel of seawater at 1000 m depth where the

pressure is about 102 bar (about 100 times atmospheric

pressure !). If you could take a sample of this seawater and

bring it to the surface it in a thermally isolated container so

that its pressure slowly decreases to atmospheric pressure,

the water sample would cool as it expanded adiabatically

(no heat loss or gain).

— Potential temperature, θ, is defined as the temperature that

a water parcel would have if moved adiabatically to a

reference pressure. The reference pressure must be stated for

the potential temperature to have meaning, but the surface

is the most common reference level.

— The temperature at the sea surface is ubiquitously

abbreviated to SST.

— Similarly, potential density is defined as the density that a

water parcel would have if moved adiabatically to a


reference pressure. Again, the reference pressure must be

stated for the potential density to have meaning, but the

surface is the most common reference level.


Seawater properties : absolute salinity

and practical salinity, S

— Seawater properties depend upon its temperature, pressure

and chemical composition.

— The chemical composition is determined by the salts that

are disolved in the water. While the salinity of seawater

varies throughout the ocean, it has been known since the

early 1800s that the proportion of the various salts in the

open ocean remains quite constant throughout the

world(Fofonoff , 1985), see Table 1. So for open ocean

applications it suffices to give one number, the salinity

defined as the concentration of dissolved salt, to characterize

the chemical content.


Table 1 – Proportion of salts in seawater

ion symbol percentage (by mass)

Chloride Cl 55%

Sodium Na 30%

Sulfate SO4 8 %

Magnesium Mg 4 %

Potassium K 1 %

Calcium Ca 1 %

Trace Br, C, etc. 1 %

— In coastal waters and enclosed seas, such as the Baltic Sea,

have important deviations from this general constancy of the

relative proportion of constituents.

— The most commonly used salinity units are the PSU =


practical salinity units, based upon the electrical

conductivity of the seawater sample which is a measure a of

the absolute salinity expressed in grams of salt per kg of

water. Note this is a measure of parts per thousand by mass.

You will find the general permil symbol h used to denote

parts per thousand used in some science books.

— We always use S to denote salinity in PSU. Most (about

90% ) of the seawater in the world ocean has a salinity in

the range 34 ≤ S ≤ 35 PSU.

a. There is a known bias of about half a percent in the calibration between

PSU and absolute salinity.


Seawater properties : equation of state,

ρ = ρ(T, S, p)

— The density (the mass per unit volume) is an important

dynamical property and is given by the equation of state,

ρ = ρ(T, S, p) (3)

where T is the temperature and p is the pressure.

— In the 1970s complicated algorithms were developed to

calculate the seawater density as a function of T, S and p

that are accurate to better than 9 g m−3. Since the density

is a little over 1000 kg m−3, this corresponds to an accuracy

better than 5 mg kg−1. This is much smaller than the error

that results from inferring salinity from measurements of

conductivity, which can be 50 g m−3. See


http://fermi.jhuapl.edu/denscalc.html for an online

calculator of the UNESCO International Equation of State

(IES 80) as described in (Fofonoff , 1985).

— For temperatures less than T < 4C and salinity

34 ≤ S ≤ 35 PSU the corresponding density at atmospheric

pressure p = 101.325kPa, the density of seawater is

T < 4C

34 ≤ S ≤ 35 PSU

p = 101.325kPa

=⇒ 1027 kg m−3 ≤ ρ ≤ 1028 kg m−3

(4)

— One can linearize the equation of state Eq(3) about a

reference state, T0, S0, p0 and approximate the density by

ρ = ρ0(1− α(T − T0) + β(S − S0) + γ(p− p0)), (5)

for which the density is ρ0 = ρ(T0, S0, p0). The three


constants are each in turn a property of seawater and are

evaluated at the reference state. They are defined as follows :

— α = 2× 10−4K−1 is the thermal expansion coefficient

— β = 8× 10−4PSU−1 is the haline contraction coefficient

— γ = (ρ0c2s)−1 is the coefficient of compressibility of

seawater defined in terms of the speed of sound

cs ≈ 1500m s−1.

— Even a large temperature variaiton, like ∆T = 30C, the

variation from the surface water temperature in the tropics

to that in the arctic, leads to a fractional variation in density

of only

∆ρ

ρ0= α∆T = 2× 10−4

1

K· 30C = 0.006 (6)

— Even a large salinity variaiton, like ∆S = 10 PSU, the

variation from the surface water temperature in the tropics

to that in the arctic, leads to a fractional variation in density


of only

∆ρ

ρ0= β∆S = 8× 10−4

1

PSU· 10 PSU = 0.008 (7)

— Perhaps surprisingly, these tiny density variations leads to

important effects in the ocean circulation.

— As we saw in RAN3, the Mach number, M = Ucs

, where 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 and velocity field is non-divergent

∇ · ~u =∂

∂xiui =

∂ux∂x

+∂uy∂y

+∂uz∂z

= 0. (8)

None-the-less, a fluid parcel at the bottom of the ocean, say

at 5000 m depth, is under enormous pressure, and as a result

has greater density ∆ρ than a parcel of water at the surface


with the same temperature and salinity

∆ρ = γ∆p =ρ0g(5000− 0)

ρ0c2s=

5000× 9.8

15002= 0.022 (9)

or about 22 g kg−1. We used the hydrostatic balance Eq(2)

to calculate ∆p, with g = 9.8 m s−2.

Large scale distribution of T and S 34

Large scale distribution of temperature

and salinity


Figure 1 – SST winter (JFM for NH and JAS for SH) (Talley et al.,

2011, Fig. 4.1).


Figure 2 – SST : Warmest month minus coldest month (Talley et al.,

2011, Fig. 4.9).


Figure 3 – Surface salinity in winter (JFM for NH and JAS for SH)

(Talley et al., 2011, Fig. 4.15).


Figure 4 – Vertical profile of potential temperature (a) 5N in the

Western Pacific, (b) 24N in the Eastern (light line) Western (dark

line) Pacific, (c) subpolar North Pacific, from (Talley et al., 2011,

Fig. 4.2).


Large scale distribution of temperature

and salinity : Key points

— The SST is over 30 degrees in the warmest parts of the

tropics and shows little seasonal variation.

— SST decreases with latitude to freezing temperature,

−1.9C, and shows larger seasonal variation at mid

latitudes, especially in the western boundary currant regions

of the subtropical gyres.

— For most of the Atlantic Ocean, the seasonal variation in

monthly mean temperatures is less than 8C. That’s less

than a typical diurnal air-temperature variation !

— The upper ocean is capped by a mixed layer, a region of

uniform temperature with depth varying with location from

a few tens of meters to a few hundred meters.


— Below the mixed layer is a strong temperature gradient, in

some cases this is the thermocline, the transition region

between the warm upper ocean and cold deep ocean below

θ = 10C. Below 1000 m depth, the potential temperature is

rarely above 12C and often below 5C .

— The thermocline has a strong latitudinal dependence. Near

the equator the thermocline is shallow, typically just a few

hundred meters depth, but in the subtropics and mid

latitudes it 800 to 1000 m depth.

— The deep ocean is characterized by homogeneous cold water.

For example, 47% of the North Atlantic water is between 2

and 4C. The global mean temperature of the ocean is 3.5C.

—

— While latitude is the dominant factor, there are variations,

especially related to winds and currents. The eastern

equatorial Pacific is very different from the western


equatorial Pacific because of the ENSO pheonomenon. The

SST is on average 8 to 10C colder in the east than in the

west.

Ocean dynamics 42

Lecture 2. The equations of motion

Ocean dynamics 43

Fundamental equations

— We have derived the continuity equation

∂ρ

∂t+

∂

∂xj(ρuj) = 0, (10)

which expresses the conservation of mass of the continuous

fluid.

— As mentioned above, the Mach number, M = Ucs

, where 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(10) in this case

simplifies to

∇ · ~u =∂

∂xiui = 0, (11)

Ocean dynamics 44

meaning that the velocity field is non-divergent.

— The momentum equation for a continuous fluid can be

written in the reference frame fixed to the solid Earth :

ai =DuiDt

+ Coriolis + Centripetal (12)

= −1

ρ

∂p

∂xi− gδi3 + Viscose forces. (13)

The RHS of Eq(13) is the same as the general momentum

equation we saw in RAN3, except here we have not written

the viscose terms out in terms of the deviatoric stress tensor.

The RHS of Eq(12) differs from what we saw in RAN3

because we are no longer restricting ourselves to 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

Ocean dynamics 45

more in detail next.

— In Eq(12) we have the acceleration ai of a fluid parcel

relative to an inertial reference frame. We have decomposed

this into 3 contributions :

1. The acceleration of the fluid parcel relative to the

reference frame of the Earth,

DuiDt

(14)

2. The Coriolis acceleration of a point moving with velocity

~u in the rotating reference frame of the Earth,

Coriolis = 2~Ω× ~u (15)

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

Ocean dynamics 46

acceleration is perpendicular to the current and the

Coriolis force per unit volume is

~Fcor = −ρ2~Ω× ~u. (16)

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, (17)

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

Ocean dynamics 47

the effective gravity

~g∗ = ~g − ~Ω× ~Ω× ~r, (18)

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 48

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 49

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 (19)

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 (20)

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

Ocean dynamics 50

x, y and z axes.

Ocean dynamics 51

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. (21)

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 52

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

(22)

And so

2Ω =2π radian

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

rad

s(23)

Ocean dynamics 53

— 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. (24)

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

of motion Eq(13) 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. (25)

Ocean dynamics 54

— 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, (26)

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

gravity is such an important term. Instead we multiply

Ocean dynamics 55

through by ρρ0

to obtain

ρ

ρ0

(Dw

Dt− f∗u

)= − 1

ρ0

∂p

∂z− ρ

ρ0g +

ρ

ρ0Fz. (27)

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. (28)

— In summary we can write the incompressible, Boussinesq

Ocean dynamics 56

equations of motion

Du

Dt+ f∗w − fv = − 1

ρ0

∂p

∂x+ Fx, (29)

Dv

Dt+ fu = − 1

ρ0

∂p

∂y+ Fy, (30)

Dw

Dt− f∗u = − 1

ρ0

∂p

∂z− ρ

ρ0g + Fz,

∇ · ~u = 0. (31)

Ocean dynamics 57

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. (32)

— First consider the incompressibility condition.

∂u

∂x+∂v

∂y= −∂w

∂z

O

(U

L

)= O

(W

H

), =⇒ W = αU. (33)

Ocean dynamics 58

So the ratio of the two Coriolis terms in the Eq(29)

f∗w

fv=

cos(lat)

sin(lat)α (34)

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, (35)

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(31). 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(31) are similar to or smaller than those on the LHS of

Ocean dynamics 59

the horizontal momentum equations :

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

(36)

accept near the Equator. Furthermore,

O

(Dw

Dt

)= αO

(Du

Dt

)= αO

(Dv

Dt

)(37)

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. The primary balance must be the

hydrostatic balace we found for a static fluid.

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

Ocean dynamics 60

have found the simpler set of equations :

Du

Dt− fv = − 1

ρ0

∂p

∂x+ Fx, (38)

Dv

Dt+ fu = − 1

ρ0

∂p

∂y+ Fy, (39)

1

ρ0

∂p

∂z= − ρ

ρ0g, (40)

the hydrostatic Boussinesq equations.

Ocean dynamics 61

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. (41)

This ratio is of fundamental importance in geophysical fluid

dynamics, and defined as the Rossby number.

Ocean dynamics 62

Table 2 – 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 63

in the equations of motion, giving

−fv = − 1

ρ0

∂p

∂x+ Fx, (42)

+fu = − 1

ρ0

∂p

∂y+ Fy, (43)

1

ρ0

∂p

∂z= − ρ

ρ0g, (44)

— 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(45)

where we have used a subscript g to emphasize that these

Ocean dynamics 64

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. (46)

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 where

f > 0 Eq(46) implies that, if you stand with your back to

the wind (or current) the higher pressure is on your right.

Ocean dynamics 65

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(46) 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.

— The oceanic pressure field determined from the η using the

Ocean dynamics 66

Eq(40)

−∫ η

z

∂p

∂zdz =

∫ η

z

ρgdz, (47)

p(x, y, z)− patm = gρ(η − z), (48)

where we have introduced the mean density,

ρ =1

(η − z)

∫ η

z

ρdz. (49)

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(48) for the

geostrophic current relation Eq(46) we find

~ug =gρ0f~k ×∇η, (50)

Ocean dynamics 67

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

— The equation Eq(50) 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

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

Ocean dynamics 68

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 69

Lecture 3. Thermal wind and Ekman

currents and Ocean General Circulation

Ocean dynamics 70

Thermal wind

— The geostrophic balance holds throughout the ocean for

large-scale flows, i.e.

Ro =U

fL 1, (51)

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.

— The geostrophic balance leads to another important relation

for the vertical derivative of the geostrophic currents. Taking

Ocean dynamics 71

the vertical derivative of Eq(46) we find

∂

∂z~ug =

∂

∂z

1

fρ0~k ×∇p,

=1

fρ0~k ×∇ ∂

∂zp,

(52)

Now we use the hydrostatic relation Eq(2)

∂

∂z~ug = − g

fρ0~k ×∇ρ. (53)

This vertical gradient in geostrophic current, Eq(53), 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

Ocean dynamics 72

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(53) is useful in many applications. Before the

development of satellite altimeter, the currents in the ocean

were either measured directly with current meters or

inferred from the Eq(53). In the latter case, we need a level

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

are known. Then Eq(53) 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′ (54)

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

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

Ocean dynamics 73

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(54) to make these current estimates.

— With the advent of the Argo programme (global distribution

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

Ocean dynamics 74

of upper ocean density from the T and S measurements

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

wind-driven circulation 75

The wind-driven circulation

— We noted earlier that 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 formula

~τs = ρairCDU10~U10 (55)

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(38,39), as a boundary condition for the frictional forces

per unit mass, denoted Fx and Fy in

Du

Dt− fv = − 1

ρ0

∂p

∂x+ Fx, (56)

Dv

Dt+ fu = − 1

ρ0

∂p

∂y+ Fy. (57)


Recall from the fluid mechanics class that these are written

for a general (not necessarily Newtonian) continuous fluid as

ρ0Fx =∂

∂xiσxi =

∂

∂xσxx +

∂

∂yσxy +

∂

∂zσxz,

ρ0Fy =∂

∂xiσyi =

∂

∂xσyx +

∂

∂yσyy +

∂

∂zσyz. (58)

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. To a very good approximation we retain only


the components with vertical derivatives :

ρ0Fx =∂

∂zσxz, ρ0Fy =

∂

∂zσyz, (59)

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(42) and (43) apply

−fv = − 1

ρ0

∂p

∂x+

1

ρ0Fx, (60)

+fu = − 1

ρ0

∂p

∂y+

1

ρ0Fy, (61)

(62)


— Replacing the frictional terms by Eq(59) we arrive at

−fv = − 1

ρ0

∂p

∂x+

1

ρ0

∂

∂zσxz, (63)

+fu = − 1

ρ0

∂p

∂y+

1

ρ0

∂

∂zσyz, (64)

— We now write the total current in Eqs(63) and (64) as the

sum of the geostrophic currents ~ug and the Ekman currents

~uE

~u = ~ug + ~uE . (65)

Substituting Eq(65) into Eqs(63) and (64) gives and using

Eq(46) we find

−fvE =1

ρ0

∂

∂zσxz, (66)

+fuE =1

ρ0

∂

∂zσyz, (67)


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, (68)

My =

∫ η

−Hρ0vE(z)dz = −

∫ η

−H

1

f

∂

∂zσxzdz,

= − 1

fτs,x, (69)

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(68) and Eq(69) can be

written in vector form

~M = − 1

f~k × ~τs. (70)

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 sights.


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. (71)

Time permitting, we will derive this relation in class using

the continuity equation and the definition of ~M in Eq(68)

and Eq(69).

— From Eq(71) and Eq(70) we find that the Ekman pumping

is given by the wind-stress curl

wE =1

ρ0~k ·(∇× ~τs

f

). (72)

— 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. 5.

— 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.

— More to our point, 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 5 – Annual mean surface wind stress (vector) and zonal

component (colour), (Talley et al., 2011, Fig. 5.6).

— 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. 6, based upon the global

wind patterns and Ekman theory.


Figure 6 – Zonally averaged annual mean potential density in

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

1000 kg m−3.


Time scale for water to pass through the

thermocline.

— The data in Fig. 6 is for the annual average and clearly

reveals that, averaged over the year, the mid latitude

subtropical regions are much less favourable for OTEC than

the Equatorial region.

— Naively one might ask if at least part of the year OTEC

might be more favourable.

— Unfortunately this turns out not to be the case because the

thermocline structure does not change much seasonally. We

can understand this by looking at the long time scales

involved.

— Let’s estimate the Ekman pumping rate wE in the sub

tropics from Eq(72). From Fig. 5 we see that the zonal


(East-West) wind dominates the meridional (North-South)

wind, |τs,x| |τs,y|, so that Eq(72) simplifies to

wE =1

ρ0~k ·(∇× ~τs

f

)=

1

ρ0

(∂τs,x/f

∂y− ∂τs,y/f

∂x

),

≈ 1

ρ0

∂τs,x/f

∂y(73)

— Note that f varies on length scales about Earth’s radius,

R = 6371km, while τs,x varies on length scales

L = O(2000)km. Therefore, for an order of magnitude

estimate it is sufficient to use f = f(30) = 7.3× 10−5s−1 .

— We find, with O(τs,x = 0.1)Pa,

O(wE) =1

1000kg m−3 × 7.3× 10−5s−10.1

2000× 1000 m

= 6.9× 10−7m s−1 × 32× 106s year−1 = 22 m year−1.

(74)


— But the thermocline is roughly 1000m deep in the subtropics

so it takes many years for the water to pass through the

thermocline. This time scale, being much longer than the

seasonal time scale, shows that the thermocline is not

strongly affected by seasonal variations. The thermocline

water is a combination of many years of water containing all

the seasons. In other words, it is never dominated by a single

season.

Terminology in climate science 91

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 92

— 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 93

Summary of this course

— Traditionally this course was about giving you the

fundamental tools to understand the general structure of

they oceans, focusing especially on the thin thermocline of

the Equatorial ocean.

— I’ve tried to broaden this a bit by giving you tools to

understand the data you will encounter when you do your

own research into the observations of a given region of

interest to your project.

— The wind is essential in understanding the large-scale ocean

circulation. The surface stress wind results in an orthogonal

Ekman transport the divergence/convergence of which is

proportional to the curl of the surface wind stress. The

convergence (respectively divergence) of the Ekman

Summary 94

transport results in a vertical transport downard

(respectively upward) of fluid called Ekman pumping.

— The Ekman pumping in the Equatorial (and to a less extent

in the subpolar) region is upward, called upwelling. The

sufaces of constant density, isopycnals, are pulled upward up

the upwelling. In contrast, in the subtropical and mid

latitudes, the Ekman pumping is downards, downwelling,

and the isopycnals are pushed downwards. In this way it is

the wind that is the premier mechanism for setting the

thermal vertical structure of the upper oceans.

— Thanks to this mechanism the tropical oceans are the most

favourable for expoitation of OTEC. The surfac temperature

is elevated, typically 25 to 30C.

— Isopycnal slopes, generated by Ekman pumping, can be 250

time as steep as the air-sea interface slope.

— The steric effect is the expansion/contraction of the water

Summary 95

column because of changes of density due to T and S

variations, with T being the dominant contributer. For

example, the subtropical gyre sealevel is greater (by about

2 m) than that of the subpolar gyre.

— The sloping isopycnals give rise to the thermal wind that

most often opposes the geostrophic current induced by the

sloping air-sea interface. The thermal wind is generally

strongest at the base of the thermocline (at few hundred

meters depth – shallower in the tropics, deeper at mid and

high latitudes) and essentially cancels the surface geostrophic

current. The result is that below the thermocline, the

geostrophic currents are generally very weak.

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 (see Table 1 for

proportions).

— T , denotes temperature.

— θ, denotes potential temperature.

Summary 97

— thermal wind

— 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 listed in Table 1.

Summary 99

References

Fofonoff, N. (1985), Physical properties of seawater : A new salinity

scale and equation of state for seawater, Journal of Geophysical

Research : Oceans, 90 (C2), 3332–3342,

doi :10.1029/JC090iC02p03332.

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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