AIR- SEA INTERACTION IN THE INDIAN-PACIFIC TROPICAL REGION

AIR- SEA INTERACTION IN THE INDIAN-PACIFIC TROPICAL REGIONAIR- SEA INTERACTION IN THE INDIAN-PACIFIC TROPICAL REGIONAIR- SEA INTERACTION IN THE INDIAN-PACIFIC TROPICAL REGIONAIR- SEA INTERACTION IN THE INDIAN-PACIFIC TROPICAL REGIONAIR- SEA INTERACTION IN THE INDIAN-PACIFIC TROPICAL REGIONAIR- SEA INTERACTION IN THE INDIAN-PACIFIC TROPICAL REGION

Đại học Quốc Gia Hà Nội

Đinh Văn Ưu

Tương tác biển-khí quyển

Khu vực nhiệt đới ấn độ -

Thái bình dương

Nhà xuất bản Đại học Quốc Gia Hà Nội - 2003

VIET NAM NATIONAL UNIVERSITY, HANOI

Dinh Van Uu

AIR- SEA INTERACTION

IN THE INDIAN-PACIFIC TROPICAL REGION

VIET NAM NATIONAL UNIVERSITY PRESS - 2003

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VIET NAM NATIONAL UNIVERSITY, HANOI Dinh Van Uu AIR- SEA INTERACTION IN THE INDIAN-PACIFIC TROPICAL REGION VIET NAM NATIONAL UNIVERSITY PRESS - 2003 1 Table of contents INTRODUCTION 3

Chapter I SMALL SCALE AIR-SEA INTERACTION 10

1.1 Introduction 10

1.2 Surface Processes 10

1.2.1 Equation of motion with Viscosity 12

1.2.2 Turbulence in boundary layer 13

1.2.3 Turbulent Stresses: The Reynolds Stress 14

1.2.4 Thickness of the lower atmospheric boundary layer 15

1.2.5 Vertical profile of wind speed within the surface layer 16

1.2.6 Characteristics of the lower atmospheric boundary layer on the sea -18

1.3 Influence of the stratification and wind wave on the turbulent characteristics of the lower atmospheric boundary layer 21

1.3.1 Stratification and its influence on the turbulent characteristics of the lower atmospheric boundary layer 21

1.3.2 Dependence of the sea surface roughness on wind wave characteristics 23

1.4 Calculation of turbulent fluxes on sea surface 25

1.4.1 Direct Calculation of Fluxes 25

1.4.2 Gradient method 26

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1.4.3 Indirect Calculation of Fluxes: Bulk Formulas 28

1.4.4 Field and model estimates of the sea surface drag 30

1.4.5 Climatic estimation of the turbulent fluxes 32

1.4.6 Calculation fluxes in the storm condition 35

2 Chapter II OCEAN’S RESPONSE TO THE ATMOSPHERE 36

2.1 Dynamic interaction and Ekman layer 37

2.1.1 Response of the upper ocean to winds and inertial motion 37

2.1.2 Wind circulation and Ekman Layer at the Sea Surface 38

2.1.3 Langmuir Circulation 41

2.1.4 Influence of stability in the Ekman Layer 42

2.1.5 Ekman Mass Transports 42

2.1.6 Application of Ekman Theory 43

2.2 Hydrodynamic interaction model of air-sea boundary layers 45

2.2.1 Boundary layers model with constant turbulent viscosity 45

2.2.2 Boundary layers model with variable turbulent viscosity 47

2.3 Wind Waves as the ocean’s response to atmosphere 48

2.3.1 Introduction 48

2.3.2 Linear Theory of Ocean Surface W aves 49

2.3.3 Waves and the Concept of a Wave Spectrum 51

2.3.4 Generation of Waves by W ind 52

2.3.5 Influence of atmospheric stratification on wave development 52

2.3.6 Wave Forecasting 53

2.4 The Oceanic Heat Budget 54

2.4.1 Introduction 54

2.4.2 Factors influencing Heat-Budget Terms 54

2.4.3 Geographic Distribution of Terms in the Heat Budget 56

2.4.4 Meridional Heat Transport 57

2.5 Thermodynamic interaction of air-sea boundary layers and the upper ocean active layer 59

2.5.1 Introduction 59

2.5.2 The Oceanic Mixed Layer and Thermocline 59

2.5.3 Geographical Distribution of Surface Temperature and Salinity 63

2.5.4 Thermal transformation of the lower atmospheric boundary layer on the sea -67

3 Chapter III AIR SEA INTERACION IN THE INDIAN-PACIFIC TROPICAL ZONE 69

3.1 Introduction 69

3.2 Thermal Land-Ocean-Atmosphere interaction and general atmospheric and ocean circulation 70

3.2.1 Importance of the Ocean in Earth’s Heat Budget 71

3.2.2 The development of a thermal circulation 71

3.2.3 The Coriolis effect (deflection caused by Earth’s rotation) 74

3.2.4 Sea level pressure and wind 75

3.2.5 Intertropical Convergence Zone (ITCZ) 76

3.2.6 Atmospheric circulation in tropical zone 78

3.2.7 General ocean circulation 81

3.3 Scales of variability in the ocean-atmosphere system 91

3.3.1 Diurnal and synoptic scales 92

3.3.2 Seasonal to interannual time scales 95

3.3.3 Interannual cycle 96

3.4 Variability of the monsoon system 98

3.4.1 Monsoon activity in the tropical ocean-atmosphere system 98

3.4.2 Variability of the Asian-Australian monsoon system 101

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3.5 Interannual climate variability in the Indian-Pacific tropical zone

1051 -100

3.5.1 ENSO activity and variability 105

3.5.2 The Quasi-Biennial zonal wind Oscillation (QBO) 112

3.5.3 Decadal variability in the Pacific and Indian Oceans 116

3.5.4 The variability in the Bien Dong (South China) Sea 121

REFERENCES 128

Air and water are the two fluids we know most about They are both essential to the maintenance of our lives, providing a hospitable environment for all living things We have a direct, lifelong experience in observing how they behave and how we utilize them In the natural environment we can see, feel or hear examples of almost all the kinds of fluid flows that we will study in this book

The atmosphere is a layer of gas held to the surface of the earth by gravitational attraction Most of the mass of the atmosphere is confined to the first 15 kilometers above sea level, yet the small amount above this level is responsible for filtering out the deadly high energy radiation from the sun which would otherwise destroy life The interaction of the atmosphere with sunlight helps to maintain the earth's surface temperature above that of an airless planet, like the moon (This increase in temperature, called the greenhouse effect, threatens to grow in the future because of anthropogenic emissions of heat absorbing gases.)

The motion of the atmosphere that we observe around us is driven by the diurnal pattern of heating by the sun and cooling

by radiation to outer space Over the year, these heating/cooling patterns shift to different latitudes, giving rise to annual

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climate variations An intimate part of this process is the

evaporation of water from the earth's surface, the formation of

clouds from this water at high elevations and its precipitation

back to the surface This distillation of ocean water, moved to

land by winds, provides the sweet water that maintains

terrestrial life

Local weather provides a variety of wind motions

Sometimes the wind speed is quite small, especially at night

when radiative cooling stabilizes the atmosphere But storms

driven by precipitation of water vapor, such as thunderstorms

and hurricanes, can have very high wind speeds Cold air is

more dense than warm air, so that a cold air mass tends to flow

under a warm air mass, forming a cold front Large scale

weather patterns drift past our locality, bringing changes that

are not greatly affected by local conditions As we can see from

watching the daily television weathercast, the main features of

the weather pattern extend over many thousands of kilometers,

a distance hundreds of times larger the atmospheric height Yet

the changes of pressure, temperature and humidity are much

greater in the vertical direction than in the horizontal direction,

despite the much greater horizontal size of a weather pattern

The pull of gravity is so strong over large distances that it

forces the atmosphere to flow mostly in the horizontal direction

Because we are so small compared to the vertical and

horizontal dimensions of the atmosphere, we can observe how

the wind blows in a tiny portion of the atmosphere that is

nearby us It is noticeable that the wind speed and direction are

somewhat variable, especially over time intervals of less than a

minute These changes are much more rapid than the changes

accompanying a weather pattern, which may take a day to pass

us by Fluid flows that exhibit variability over time and length scales which are small compared to that of the overall flow are called turbulent flows The atmospheric wind is a turbulent flow

The atmospheric motion is responsible for diluting air pollutants, such as those emitted by power plants and automobiles When these pollutant streams are marked by smoke, we can readily observe how the smoke intensity decreases as the wind turbulence mixes the pollutant stream with clean air, diluting the strength of the pollutant within the plume (or, if you prefer, dirtying more and more of the atmosphere) Most of these pollutants mix no higher than a few kilometers in the atmosphere , and are eventually carried far downwind and deposited back to the earth's surface Some, however, do not soon return to the earth and instead mix gradually throughout the entire atmosphere, including the stratosphere Some of these gases lead to the destruction of stratospheric ozone and to increased average surface temperature The mixing properties of the atmosphere are extremely important in determining the degree of atmospheric pollution in urban areas

The water in the ocean, lakes and rivers, as well as that in underground aquifers, is called the hydrosphere The volume of fresh water on the continents is small compared to the oceanic volume, but it is the part of the hydrosphere that is most important to the maintenance of terrestrial life The management and use of this water for agricultural and other purposes form an important branch of engineering But the ocean is important too It provides the source of precipitation over the land and, in its surface layer, an environment for the

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growth of microscopic plants and animals that form the base of

the oceanic food chain The ocean tends to make the climate

more uniform in the latitudinal direction by moving warm

tropical waters towards the poles and displacing cold polar

water towards the equator

We are all familiar with the downhill motion of streams

and rivers, flowing toward the sea The energy in this flow can

be tapped by building dams that force the river flow through

turbines to generate electric power Sometimes this energy is

dramatically dissipated as the river plunges over a precipice to

form a waterfall The violently turbulent motion at the base of

the falls converts the river's directed energy into heat When

the river reaches the sea, its fresh water, being lighter than the

sea water, floats on top of the sea, sometimes many miles

beyond the river's mouth before it is diluted with sea water

Most of the fresh water on the continents is out of sight,

below ground It exists in the pores between mineral deposits

and is fed by precipitation that percolates through the ground

under the influence of gravity The fluid velocity in the

underground aquifers is much lower than it is in rivers, the

water being impeded by the frictional force of the porous

medium through which it flows Underground water is often the

source of potable water Locating underground water and

pumping it from the ground for human use is limited by the

characteristics of the underground aquifer Preserving the

purity of this water from the contamination by toxic fluids

buried or dumped on the surface and subsequently traveling

down into the aquifer is a major problem worldwide

At the edge of the ocean we see the ocean waves crashing

on the shore The waves carry to the shore energy generated by

the wind blowing over the ocean surface Of course, the ocean surface doesn't move (on the average) in the direction of the waves, but it oscillates as the wave passes by Ocean waves are called gravity waves because the pull of gravity at the air-sea interface is responsible for the propagation of these waves, which do not penetrate far below the ocean surface

The other oceanic motion we notice at the sea shore is the tidal rise and fall of the sea surface This twice-a-day cycle is caused by the difference in gravitational pull of the moon (and

to a lesser extent, the sun) on opposite sides of the earth The differential gravity force gives rise to a bulging of the ocean surface in the direction of the moon, which passes a given location twice in the lunar day of 25 hours The tidal motion, consisting of both a vertical and horizontal oscillation, may be amplified greatly along the continental coastline, sometimes by

a factor of ten above the general oceanic amplitudes

Oceans may contain localized currents, like the Gulf stream, that are mighty rivers flowing across a nearly stationary ocean Earthquakes can generate tsunami waves that travel many thousands of kilometers before crashing ashore, sometimes wreaking devastation on lowlying coastlines Even hurricanes can generate storm surge waves that cause coastal flooding

The rise and fall of the tide can be utilized to produce mechanical power, but at the present time this is seldom economical compared to river power Mechanisms have been devised to extract power from ocean waves, but this has also proved to be uneconomical But the forces exerted on ships and wave barriers by the ocean waves can be very substantial, and protecting them against such forces very expensive Knowledge

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of the dynamics of the ocean is important to many of mankind's

pursuits

Air-sea interaction provides a comprehensive account of how the

atmosphere and the ocean interact to control the global climate, what

physical laws govern this interaction, and its prominent mechanisms In

recent years air-sea interaction has emerged as a subject in its own right,

encompassing small-scale and large-scale processes in both air and sea

By developing its subject from basic physical (thermodynamic)

principles, the book is broadly accessible to a wide audience It is mainly

directed towards graduate students and research scientists in meteorology,

oceanography, and environmental engineering The book will be of value

on entry level courses in meteorology and oceanography, and also to the

broader physics community interested in the treatment of transfer laws,

and thermodynamics of the atmosphere and ocean

Current research in air-sea interaction is attempting to

apply our understanding of molecular and turbulence scale

processes to global scale phenomena There are many

unknowns in the fundamental processes that control the global

climate For example, what processes control the amount and

distribution of water in the atmosphere; what is the effect of

cloud variability on the sea surface temperature? How do

changes in the ocean circulation affect the atmospheric

circulation and, hence, the distribution of wind stress,

temperature and precipitation, and how does this feed back to

the ocean? These, and many other related questions, encurage

some of the studies of the interaction of the atmosphere and the

ocean

Major uncertainties remain in our understanding of the

fundamental processes of air-sea interaction, particularly, in

heterogeneous and nonequilibrium conditions; for example, we

do not know enough about the relationships between the directional wave spectra, surface fluxes and the properties of the oceanic and atmospheric boundary layers to develop satisfactory predictive models The ocean and atmosphere are interdependent, or coupled, because of the dependence of the atmosphere on heat and moisture at the sea surface and the dependence of the ocean circulation on the wind Studies rarely combine investigations of both environments to determine the extent of the fedbacks between the ocean and the atmosphere Understanding the coupled ocean-atmosphere system depends largely on the scales of interaction between the two fluids and the processes that provide the strongest feedbacks Advances are most likely through multidisciplinary process studies that connect the upper ocean and lower atmosphere

For global studies more comprehensive parameterizations

of the surface processes are required as well as improvements

in satellite retrievals and assimilation in numerical models The high spatial and temporal variability of surface processes needs to be properly considered In situ measurements are revealing very complex horizontally and vertically heterogeneous fields that cannot be resolved by current remote sensing techniques High resolution models, which include the physics of the processes that contribute to this variability, combined with satellite data seems to be the best tool for global analysis and prediction

The large-scale dynamics of the ocean and the atmosphere are closely related Energy is transferred from the atmosphere

to the ocean surface mixed layer driving the circulation of the upper ocean In turn, energy from the ocean is fed back to the atmosphere affecting the atmospheric circulation, the weather

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and the climate The concept is deceptively simple, but as we

explore the coupled earth system, we are frequently limited by

our lack of understanding of the interchanges between the

atmosphere and ocean Kraus and Businger (1994) highlighted

several areas that continue to require more attention, for

example, the interaction of the wind and surface waves, the

parameterization of subgrid scale processes in large-scale

circulation models, and the transfer of gases across the air-sea

interface

It has been convenient to divide air-sea interaction studies

into two categories: small- and large- scale ocean-atmosphere

interactions However, this often belies the fundamental

precept that the basis of the interaction of the atmosphere and

the ocean is the exchange of matter and energy across a

material interface - the sea surface An exchange that, for the

most part, occurs on molecular scales, involving both turbulent

and laminar processes modified by wave breaking, surface

tension, the structure of the planetary boundary layer and the

ocean mixed layer and other effects A satisfactory

understanding of these processes remains elusive, but is

essential if we are to address adequately the larger scale

ocean-atmosphere problems

In the past many proponents of large-scale studies, such as

global climate and ocean circulation relied heavily on the

veracity of the parameterization of small-scale air-sea exchange

processes, often overlooking the uncertainties in the basic

measurements and their interpretation More recently, we have

recognized the importance of connecting small-scale process

studies, investigating the exchange of heat, moisture,

momentum and trace constituents across the air-sea interface,

with the large-scale problems of global climate change and ocean circulation that rely heavily on numerical models and highly averaged fields Studies, such as the Tropical Ocean Global Atmosphere Coupled Ocean Atmosphere Response Experiment (TOGA COARE) and the World Ocean Circulation Experiment (WOCE), are leading the way by highlighting the importance of process studies for a satisfactory understanding

of global climate and ocean general circulation problems The small-scale exchange processes are generally related to the global-scale problems via parameterizations of the fluxes that use mean quantities obtained by measuring on various platforms such as buoys, ships and satellites These parameterizations are also used extensively in operational meteorological models as well as many research general circulation models of the coupled ocean-atmosphere system Large uncertainties exist in the derivations of the bulk parameterizations due to the difficulty of measuring surface fluxes directly, and the difficulty of applying these measurements to scales greater than a few hundred kilometers and several hours The problems are particularly acute in the tropics where low wind speeds and very high sea surface temperatures result in primarily buoyancy-driven fluxes that are not well parameterized by most prevailing methods, and in coastal regions where fetch, topography and water depth vary considerably

While there is resistance to the establishment of canonical parameterizations of the fluxes, the TOGA COARE research community recognized that such an approach has the advantage of focusing the attention of a large group of researchers to exchange and compare data and rapidly transfer

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this information to a broader community whose interests

require surface fluxes Despite these efforts, it is unlikely that

in this decade of large scale climate and ocean circulation

studies we will know the surface fluxes as well as enough we

would like A continued effort will be required to improve our

knowledge of heat, moisture, momentum and trace constituent

fluxes, to increase our understanding of the uncertainties in the

fluxes we measure, and to improve our knowledge of the

relationship between boundary layer processes on each side of

the interface to the surface exchange mechanisms

The Indian- Pacific tropical zone is most important heat storage of

the World Ocean with the highest value of the mean sea surface

temperature (SST); there is the Western Pacific/ Eastern Indian or

Asian-Australian warm pool The formation of the warm pool and its variation

is the favourable condition for typhoon formation and development in the

western part of the Pacific Ocean The variability of the global-scale

ocean and atmosphere circulation including Trade’s wind, the Walker

circulation, the Asian-Australian monsoon system, the El Niño/Southern

Oscillation (ENSO) and sea surface temperature (SST) is related to the

oscillations of the ocean-atmosphere system

This course will acquentance physical oceanography and

meteorology students with the one-dimensional theories of turbulent

boundary layers, to observations of the planetary boundary layer of the

atmosphere, and to observations of the surface mixed layer of the ocean

Coupled one-dimensional models of air-sea interaction will be studied

with a view towards understanding the importance of interactions of the

turbulent boundary layers with each other, and with the interior of their

respective fluids Finally, we will review current progress in our

understanding of the surface processes, and the application of this

research to the global and Indian-Pacific regional issues of ocean climate and its variations

atmosphere-19 20

Chaptre 1

SMALL SCALE AIR-SEA INTERACTION

2.1 Introduction

This part of the course will introduce physical

oceanography and meteorology students to the one-dimensional

theories of turbulent boundary layers, to observations of the

planetary boundary layer of the atmosphere, and to

observations of the surface mixed layer of the ocean Necessary

concepts of turbulence theory will be studied, along with the

probability and statistical tools that are appropriate

Coupled one-dimensional models of air-sea interaction will

be studied with a view towards understanding the importance

of interactions of the turbulent boundary layers with each

other, and with the interior of their respective fluids Finally,

the subject of the influence of horizontal variability will be

opened, with the objective of appreciating the role that ocean

and atmosphere dynamics play in modulating the

thermodynamics of the turbulent boundary layers, and as an

introduction to the associated part of the course on Large-Scale

Ocean-Atmosphere Interactions: Air-sea interaction and

tropical meteorology and oceanography

2.2 Surface Processes

Interactions between the ocean and atmosphere occur at the air-sea interface The ocean surface is a material interface that is a barrier to the exchange of heat, moisture, momentum and trace constituents Away from the surface both fluids are usually in turbulent motion, but near the interface turbulence

is suppressed and the transport is controlled primarily by molecular processes To quantify the exchanges at the interface

it is necessary to understand how the turbulent layers of the ocean and the atmosphere are connected via the molecular sublayers in either side of the sea surface In turn, we need to understand how the turbulent layers transport the properties of the interface into the interior of these fluids, the extent to which changes in the interior feed back to the interface, and how processes at the surface affect the structure of the deep ocean and free atmosphere The fundamental processes that connect the atmosphere and the ocean are the energy input to the ocean by the wind, the net freshwater flux, expressed primarily as precipitation minus evaporation, and the net surface heat flux

As Charnock (1951) pointed out the energy transmitted by the wind to the ocean is a tiny fraction of the radiation received

at the surface, yet wind-driven currents largely determine the regions where the ocean energy is fed back into the atmosphere that sets the pattern of cloudiness, which in turn determines the radiation input The ocean-atmosphere system is intrinsically coupled, although feedbacks across the air-sea interface are often masked by temporal and spatial differences

As in the example above, we may understand the processes that connect the ocean and the atmosphere, but we do not understand well enough the distribution of energy within the

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system Many of the problems that vexed early attempts to

measure the structure of the upper ocean and lower atmosphere

remain paramount; for example, flow distortion around

measurement platforms, the difficulty of obtaining

measurements in the open ocean, and the relationship between

point measurements and large-scale fields The latter is of

particular importance when trying to apply the results of

process studies to large scale monitoring of the earth by

satellites

To the list of basic physical processes mentioned above we

must also add gas exchange across the air-sea interface The

most significant are carbon dioxide (CO2), a greenhouse gas,

dimethyl sulfide (DMS), which may be the main source of

non-sea-salt sulfate aerosols, and nitrous oxide (N2O), which is both

a greenhouse gas and may play an important role in the

destruction of ozone

From the 1950s most studies of wind stresses were

conducted over small bodies of water, through the late 1950s,

the open ocean measurements of the fluxes became more

prominent, to the more contemporary large scale field

programs: Barbados Oceanographic and Meteorology

Experiment (BOMEX), Atlantic Trade Winds Experiment

(ATEX), Global Atmospheric Research Program Atmospheric

Tropical Experiment (GATE), Joint Air-Sea Interaction

Experiment (JASIN), Marine Remote Sensing Experiment

(MARSEN), Storm Transfer and Response Experiment

(STREX), Humidity Exchange Over the Sea (HEXOS),

Marginal Ice Zone Experiment (MIZEX), Frontal Air-Sea

Interaction Experiment (FASINEX) More recent field

programs include TOGA COARE that was completed at the end

of February 1993, the Surface of the Ocean, Fluxes and

Interactions and Atlantic Stratocumulus Transition SOFIA/ASTEX experiment devoted to air-sea interactions and cloud development in the Azores region of the Atlantic Ocean that took place in June 1992, and the ongoing WOCE

Understanding of the transfer of heat, moisture, momentum and mass across the air-sea interface is fraught with difficulties One of the most complex problems is understanding the effect of wind waves on the momentum flux Also understanding what happens at the air-sea interface requires knowledge of how energy is transferred across the stable layers connecting the interiors of the atmosphere and ocean with their respective boundary layers These processes are frequently intermittent andinextricably linked to the larger scale circulations of these fluids

Normally the air-sea interaction processes are presented through energy and mass fluxes exchanged between atmosphere and ocean The atmosphere within 100 m above the sea surface is influenced by the turbulent drag of the wind on the sea and the fluxes of heat through the

surface This is the atmospheric boundary layer Its thickness Z i varies

from a few tens of meters for weak winds blowing over water colder than the air to around a kilometre for stronger winds blowing over water warmer than the air The structure of the layer is influenced by the exchange of momentum and heat between the surface and the atmosphere

In this section we will consider the exchange of mechanic energy, heat and water between the atmosphere and the ocean Theoretical aspects and calculation methods are based on the system of thermodynamic equations applied to boundary layer The details of one of these equations – equation of motion – are presented in the following section

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2.2.1 Equation of motion with Viscosity

Newton’s second law relates the change of the momentum

of a fluid mass due to an applied force The change is:

is force, m is mass, and vr

is velocity Here we

have emphasized the need to use the total derivative because

we are calculating the force on a particle We can assume that

the mass is constant, and (1.1) can be written:

m

f m

is force per unit mass

Four forces are important: pressure gradients, Coriolis

force, gravity and friction Without deriving the form of these

forces (the derivations are given in the next section), we can

write (1.1) in the following form:

r

F g v p

is acceleration of gravity,  is fluide density, F r is

friction, and the magnitude  of r

is the Rotation Rate of Earth,

2π radians per sidereal day or

Expanding the derivative in (1.3) and writing the

components in a Cartesian coordinate system give:

x

F v

x

P z

u w y

y

P z

v w y

z

P z

w w y

w v x

w u t

where F i are the components of any frictional force per unit

mass, and φ is latitude In addition, we have assumed that w <<

v, so the 2w cos φ has been dropped from equations (1.5)

Equations (1.5) appear under various names Leonhard Euler (1707–1783) first wrote out the general form for fluid flow with external forces, and the equation is sometimes called the

Euler equation or the acceleration equation Louis Marie Henri

Navier (1785–1836) added the frictional terms, and so the

equation is sometimes called the Navier-Stokes equation

The term 2u cos φ in (1.5c) is small compared with g, and

it can be ignored in geophysical fluid dynamics

For boundary layer we can consider the form of the term Fi

if it is due to viscosity Molecules in a fluid close to a solid boundary can strike the boundary and transfer momentum to the boundary Molecules further from the boundary collide with molecules that have struck the boundary, further transferring the change in momentum into the interior of the fluid This

transfer of momentum is molecular viscosity Molecules,

however, travel only micrometers between collisions, and the process is very inefficient for transferring momentum even a few centimetres Molecular viscosity is important only within a few millimetres of a boundary

Molecular viscosity is the ratio of the dominant component stress x tangential to the boundary of a fluid and

x-the shear of x-the fluid at x-the boundary So x-the stress has x-the form:

25 26

for flow in the (x, z) plane within a few millimetres of the

surface, where  is the kinematic molecular viscosity Typically

 = 10 -6 m2/s for water at 20°C

Generalizing (1.6) to three dimensions leads to a stress

tensor giving the nine components of stress at a point in the

fluid, including pressure, which is a normal stress, and shear

stresses A derivation of the stress tensor is beyond the scope of

this book, but you can find the details in Lamb (1945) or Kundu

(1990) For an incompressible fluid, the frictional force per unit

mass in (1.5) takes the form:

u z y

u y x

2.2.2 Turbulence in boundary layer

If molecular viscosity is important only over distances of a

few millimetres, and if it is not important for most oceanic

flows, unless of course you are a zooplankter trying to swim in

the ocean, how then is the influence of a boundary transferred

into the interior of the flow? The answer is: through turbulence

The turbulent viscous termes or turbulent shear stresses in

the (1.7) have the form similar as (1.6):

u u









, , etc The importance of these

terms is given by a non-dimensional number, the Reynolds

Number, which is the ratio of the non-linear terms to the

viscous terms:

2 2

2Terms

Viscous

Termslinear

NonNumber

Reynolds

L U L

U U

x

u v x

u u

typical cross-stream distance, or an along-stream distance

Typical values in the open ocean are U = 0.1 m/s and L = 1 mega meter, so Re = 1011 Because non-linear terms are

important if Re > 10 – 1000, they are certainly important in the

ocean The ocean is turbulent

The Reynolds number is named after Osborne Reynolds (1842–1912) who conducted experiments in the late 19th century to understand turbulence In one famous experiment (Reynolds, 1883), he injected dye into water flowing at various speeds through a tube If the speed was small, the flow was

smooth This is called laminar flow At higher speeds, the flow

became irregular and turbulent

The transition occurred at Re = VD/ ~ 2000, where V is

the average speed in the pipe, and D is the diameter of the pipe

As Reynolds number increases above some critical value, the flow becomes more and more turbulent Note that flow pattern is a function of Reynolds’s number All flows with the same geometry and the same Reynolds number have the same flow pattern Furthermore, the boundary layer is conned to a very thin layer close to the cylinder

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2.2.3 Turbulent Stresses: The Reynolds Stress

Those who studied fluid mechanics in the early 20th

century hypothesized that parcels of fluid in a turbulent flow

played the same role in transferring momentum within the flow

that molecules played in laminar flow The work led to the idea

of turbulent stresses

To see how these stresses might arise, consider the

momentum equation for a flow with mean and turbulent

T

u

U

X T

11

) ( )

U U x

u U

u

U

x

u u x

U u x

u U x

U U x

u U

'

' ' '

' '

v x

W x

V x U

' '

U v y u u x

U v

Thus the additional force per unit mass due to the turbulence is:

' ' '

' '

z v u y u u x

The terms (u’u’),  (u’v’), and  (u’w’) transfer eastward

momentum (u’) in the x, y, and z directions For example, the

term (u’w’) gives the downward transport of eastward

momentum across a horizontal plane Because they transfer momentum, and because they were first derived by Osborne

Reynolds, they are called Reynolds Stresses

The Reynolds stresses such as u ' w'

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2.2.4 Thickness of the lower atmospheric boundary

layer

The lowest part of the atmospheric boundary layer is the

surface layer Within this layer vertical fluxes of heat and

momentum are nearly constant

Normally, the limits of the flux variation are from 5% to

10% within viscos layer, the gradient of momentum, heat and

moisture fluxes (

z z

of motion, heat transfer and diffusion

z

p z

Apply the same operation for total heat flux (H+F), with

the sensible heat flux:

T w

T w

radiation heat flux F

In the condition without phase transformation, the heat balance equation has the following form:

z

F Η ρC

dt

-dθ θ

t

-Δθ θ

T

p Δ

1Δ



The layer thickness h' u, with the scale of left side term about 3/h and total heat variation is from 10% to 20%, will be determined as:

10

F H z

h

) ( )

( '

where: [H, F] = cal/cm2.s, [h’ u]= cm

In the condition unstable stratification (during summer) the amplitude of H is about 0.005 cal/cm2.s, the thickness h’u is

31 32

about 50 meters For H < 0 (night stable stratification) the

turbulent heat flux is only 0.0005 cal/cm2.s, the thickness h’u is

about 5 meters

For moist flux E =

z

q q

For the characteristic scale value of left side is about 0.5

g/kg.h the constant moist layer thickness h’’u ( E: g/cm2.s; h’’u:

cm) will be estimed as:

Normally the scale amplitude of E is varied from 10-6 to 10-5

g/cm2.s thus layer thickness h'' u is corresponded from 10 to 100

meters

2.2.5 Vertical profile of wind speed within the surface

layer

Let’s return to the simple example shown in figure 1.1,

which shows a boundary layer above a flat plate in the x, y

plane Now let’s assume that the flow above the plate is

turbulent This is a very common type of boundary layer flow,

and is a type of flow that we will describe various times in later

chapters It can be wind flow above the sea surface or flow at

the bottom boundary layer in the ocean or flow in the mixed

layer at the sea surface

Molecules carry horizontal momentum perpendicular to wall through perpendicular velocity and collisions with other molecules

Figure 1.1 Molecules colliding with the wall and with each other transfer momentum

from the fluid to the wall, slowing the fluid velocity (from Stewart, 2002)

For flow above a boundary, we assume that flow is constant

in the x, y direction, that the statistical properties of the flow vary only in the z direction, and that the mean flow is steady

Therefore

y x

We now assume, in analogy with (1.6)

z

U A w

viscosity  in (1.6)

With this assumption,

2 2

z

U A z

U A z

varies more slowly in the z direction than

33 34

The x and y momentum equations for a homogeneous,

steady-state, turbulent boundary layer above or below a

horizontal surface are:

where f = 2 sinφ is the Coriolis parameter, and we have

dropped the molecular viscosity term because it is much

smaller than the turbulent eddy viscosity

Note (1.26) follows from a similar derivation from the

y-component of the momentum equation Equations (1.26) will be

used later as the starting point for describing oceanic flow near

the sea surface

The assumption that an eddy viscosity A z can be used to

relate the Reynolds stress to the mean flow works well for

describing the flow near a horizontal surface where U is a

function of distance z from the surface, and W, the mean

velocity perpendicular to the surface is zero This is the

approach first described in 1925 by Prandtl, who introduced the

concept of a boundary layer, and by others Please notice that a

value for A z cannot be obtained from theory Instead, it must be

calculated from data collected in wind tunnels or measured in

the surface boundary layer at sea See Hinze (1975) and

Goldstein (1965) for more on the theory of turbulence flow near

a flat plate

Prandtl’s theory based on assumption (1.24) works well

only where friction is much larger than the Coriolis force This

is true for air flow within tens of meters of the sea surface and

for water flow within a few meters of the surface

The application of the technique to other flows in the ocean

is less clear For example, the flow in the mixed layer at depths below about ten meters is less well described by the classical turbulent theory Tennekes and Lumley (1970) write: Mixing-length and eddy viscosity models should be used only to generate analytical expressions for the Reynolds stress and mean-velocity profile if those are desired for curve fitting purposes in turbulent flows characterized by a single length scale and a single velocity scale The use of mixing-length theory in turbulent flows whose scaling laws are not known beforehand should be avoided

The theory for the mean velocity distribution in a turbulent boundary layer over a flat plate was worked out from 1915 to

1935 independently by G.I Taylor (1886–1975), L Prandtl (1875–1953), and T von Karman (1881–1963) Their empirical

theory, sometimes called the mixing-length theory predicts well

the mean velocity profile close to the boundary Of interest to

us, it predicts the mean flow of air above the sea Here’s a simplified version of the theory applied to a smooth surface

We begin by assuming that the mean fow in the boundary

layer is steady and that it varies only in the z direction Within

a few millimeters of the boundary, friction is important and (1.6) has the solution



z u u

u *

*

 (1.28)

is the friction velocity

Further from the boundary, the flow is turbulent, and

molecular friction is not important In this regime, we can use

effective in mixing momentum than small eddies, and therefore

A z ought to vary with distance from the wall Karman assumed

that it had the particular functional form k u = κzu., where κ is a

dimensionless constant With this assumption, the equation for

the mean velocity profile becomes

Because u is a function only of z, we can write du = u./(κz)

dz, which has the solution

2

z

z u

u * ln

 (1.33)

where z0 is distance from the boundary, at which velocity

goes to zero, called the sea surface roughness

For air flow over the sea, κ = 0.4 and z o is given by

Charnock’s (1955) relation

z0 = 0.0156 u.2/g (1.33) The mean velocity in the atmospheric boundary layer just above the sea surface fits well the logarithmic profile as does the mean velocity in the upper few meters of the sea just below the sea surface

Furthermore, using (1.29) in the definition of the friction velocity, then using (1.33) gives Charnock’s form of the drag coeficient as a function of wind speed

2.2.6 Characteristics of the lower atmospheric boundary layer on the sea

The wind generates two boundary layer flows in opposite direction with respect to the reference position, with are in general considered separately

After inspection of all research data in wind-wave tunnels the quite general conclusion may justified that both the air- and water-side boundary layers behave quite similarly to solid wall boundary layers with respect to the time average turbulent parameters outside the immediate near-interface layer, although these parameters show some quantitative alterations From this conclusion it may be deduced that the same basic processes and features as described for the wall boundary layer should be found also in both the air- and water-side boundary layers

The obvious reason for the earlier-mentioned quantitative alterations with respect to wall boundary layers is the wind induced waviness of the air-water interface Wave dynamics is indeed an important part of ocean dynamics Here, however, quite a specific aspect of wavy surfaces is of interest – their transfer characteristics governing diffusive momentum, heat

37 38

and mass exchange between the two media If the analogy to

solid wall transfer characteristics postulated earlier is justified,

the attention should be focused on length scales which also

govern the transfer processes in wall boundary layer

In the wall boundary layer theory, there are two processes

characterized by quite a violent ejection of fluid from sub layer,

followed by a return of more quiescent flows towards the wall

These processes are now called in literature burst and sweep

Futher investigations led to the more detailed model with the

counter-rotating streamwise vortices

The representation of the turbulence production and of the

frequency of burst events as function of the wall distance

provides strong evidence of association of these processes

Typical scales and parameters related to the

three-dimensional structure of the viscous layer are:

(the value for zero pressure gradient is about 1000)

wall distance scale, l3:

burst frequency, f:

010003

of wave, these numbers indicate that the smallest, youngest waves should be the relevant one of the processes investigated here

For neutral stratification, to compute momentum, heat and mass fluxes through the boundary layer it is assumed that the velocity, temperature and humidity distribution in the viscous layer (z < hn = 5a/v *) can be represented by linear law and in the outer flow (z > 30hn): logarithmic law Wind Reynolds stress

on the sea surface included two components: tangential t and normal p:

h - vector normal of wave surface s

For wind flow on the wavy interface with celerity c0, (x,t) =

a cos(kx- c 0 t), the wind profile and streameline are schematised

in figure 1.2

Figure 1.2 Streamline on the moving wave surface, after Miles, 1957

39 40

The measurements reveal a behavior characterized by a

strong decrease of the wave length with increasing friction

velocity

A specific feature of flow over wavy surfaces is periodic

variation of shear stress and pressure in flow direction For

potential flow conditions these variations are always symmetric

with respect to the wave crest For flow with friction they may

become asymmetric due to flow separation Below a steepness

of about 0.08 (which is defined as wave amplitude divided by

wave length), no flow separation occurs For increasing

steepness flow separation occurs and the pressure profiles

become more and more asymmetric, with a phase lag reducing

towards 120° At the same time, the shear stress variation

steepness more and more reaching quite extreme values near

the crest and decreasing towards zero along most of the

descending part of the wave

Generally the wind wave field is depicted schematically as

composed of old, further growing long wave of length , and

just generated short waves of length  riding on the long waves,

the generation of which depends on a Phillips kind mechanism

With the young short waves, air and water side elementary

boundary layers are characterized by “microscopic” shear stress

and pressure variations The essential feature of the old long

waves is “macroscopic” shear stress and pressure variations

Firstly it is assumed that, independent of the age of the

wave field, the length of new waves is inversely proportional to

the local shear stress Due to the “macroscopic” shear stress,

the length of the wavelets shortens towards the crest of the

“mother” wave

From the wave trough the air flow starts to accelerate driven by external shear and by the negative pressure gradient, which was built up during the burst-sweep period Correspondingly an accelerating laminar boundary layer developed along the ascending wave surface, characterized by increasing interface shear stress Near the wave crest the atmospheric boundary layer lifts up and bursts in the vicinity of the crest At the same time the shear stress at the interface drops abruptly towards zero The burst is followed by a sweep Due to the increasing shear stress, the interface velocity ui is accelerated from a low value in the wave trough – corresponding to about the bulk water velocity below – and reaches about phase velocity at the wave crest By the abrupt drop of the shear stress, the interface velocity drops again towards the bulk water velocity along the descending wave surface

This nearly stepwise change of the interface velocity ui near the crest acts on the water below the crest in a way similar to a sudden acceleration of the water bulk with respect to the interface A decelerating boundary layer starts to develop agianst wind direction towards the wave trough This kind of boundary layer is characterized be decreasing interface shear, which is in correspondence with the interface shear variation of the accelerating atmospheric boundary layer, as requires the continuity condition Near the wave trough the water boundary layer lifts away, bursts, and new bulk water sweeps in

From this qualitative description it follows that along the ascending wave surface (in wind direction), momentum, heat and mass is exchanged between the air and water, and that along the descending wave surface – where no exchange

41 42

between air and water occurs – momentum, heat and mass is

exchanged within the fluids themselves, providing, by burst

and sweep, transport between the near interface boundary

layer and the respective outer flow region

This qualitative picture composed of long old waves and

young, elementary boundary layer like short waves, which are

concentrated on the ascending wave part, implies potentially a

duality of the driving forces for the further growth of the old

waves The generally accepted mechanism for growth is that

proposed by Miles (1957) which presumes the existence of the

earlier discussed phase shifted pressure fluctuations

We can consider the assumption that the momentum

exchange between sea and atmosphere is governed by the

boundary layer processes related to the wavelets, which cause

the frictional roughness effect and which should have no form

resistance due to pressure variations Formally it would have to

be supplemented by additive term accounting for the roughness

effect due to form resistance

Within the lower part of the turbulent boundary layer on

the sea interface there are two kind of velocity fluctuations:

purely turbulent and wave (u',v',w' and u'w, v'w , w'w) The

investigations show that it is no correlation between these

different kind of fluctuations: u'w u'0,. v'w v'0,. u'w w'0,

but within them there are some correlations: u'w w'w0,

in think layer h aw about 0, 1 (- wave length), in outer layer

the velocity distribution could be represented as for solid wall boundary layer

To calculate as wind stress on the wave sea surface  = t +

w as vertical wind profile it is introduced  = t ( 1 + ) where 

= f(v * /c 0 ) is function of wave age

2.3 Influence of the stratification and wind wave

on the turbulent characteristics of the lower atmospheric boundary layer

2.3.1 Stratification and its influence on the turbulent characteristics of the lower atmospheric boundary layer

In the lower atmospheric boundary layer, the fluctuation of

humidity q' is considerably influenced on the turbulent

fluctuation of density ' - directly related to buoyancy force

g'/ , it is necessary to take into account the atmospheric stability

The state equation for air-water mixture (wet air) is then:

p = RT v = RT [1+q((R h /R) -1)] ~ RT [1 + 0.61 q] (1.40)

(R=287 J/kg.K, R h = 461J/kg.K – universal constant for

dry and wet air corresponding) q = h / - specific humidity, T -

temperature, T v – virtual temperature Normally the difference

between T and T v is only about 2-3 % but its fluctuations – depending of q – are considerable It needs take account its influences on ' and Archimedes bouyancy force in the equation for turbulent kinetic energy:

43 44

01

6102

gc z

u

u

p p

' u u

u

Q i j - the mean energy flux due to the

fluctuation of velocity in the vertical direction, - dissipation of

energy

The flux Richardson number – parameter for atmospheric

stability – the ratio between the terms of the convection and the

mechanic energy in the previous equation:

R f* = R f (1 + I)

where R f =

-z

u v

z

u w u

w g

' '

I = 0,61 c p T E/ H = 0,61 T q * /T * (1.43)

q * and T * the specific values similar as u *characterized for

intensity of turbulent moisture and heat exchange

q * = -E / v * , T * = - H / c pu * (1.44)

The turbulence will be decreased while R f >1

Often I is expressed through the Bowen ratio Bo:

I = m/Bo

a

a a

a p E

e e q

q

c E L H

) (

where: m = 0.61 cp T / L E is dimensionless coefficient depending on temperature, there are some values of m listed in the following table

The Bowen ratio is also used to calculate total heat fluxes knowing only temperature data

Table 1.1 Values of m for calculating the humidity influence on the turbulent

exchange

Observation results show that the influence of stratification

will increase when I > 1 For summer, the mean values of m ~ 0.07 and Bo ~- 0.1, the value I is about – 0.1 For negative I, and the absolute value is bigger than 1 (I < -1), the

atmospheric stratification still unstable when the potential temperature increases with altitude, this is due to the moist stratification influence

The calculations of the turbulent fluxes in stratified atmosphere are carried out on the base of Monin-Obukhov similarity theory that introduced the Monin-Obukhov length scale for the stratification parameter:

L = u * 2 /2T * (1.46)

Taking account the humidity then the length scale results in:

L* = L/(1 + I) (1.47) Considering the vertical gradient of velocity is related to

mechanic turbulent intensity v *, height z, and Monin-Obukhov

45 46

length scale, the vertical profile of the velocity can be

established on the base of similarity theory:

) (

L

z z

The wind stress  =  u * 2 could be calculated after the

dimensionless function (z/L) or f(z/L)- solution of previous

equation:

u =(u * / ) f(z/L) (1.49)

The form of f(z/L) could be determined in different critical

cases As was shown in the section 1.5 the value of (0) = 1 then

(z/L) approximation in the first term of the Taylor’s

development is:

(z/L) = 1 +  (z/L) (1.50)

the coefficient  is constant for land condition:  ~ 0,6

The function f (z/L) will be in form logarithm + linear:

u 2 -u 1 = (u * / ) [ln (z/L) +  (z 2 - z 1 )/L] (1.51)

Using the roughness length z0 this equation becomes:

u 2 = (u * / ) [ln (z/z0) +  (z 2 - z 0 )/L] (1.52)

As the wind blows over the sea there's an exchange of

momentum between atmosphere and ocean, leading to wave

growth The magnitude of this vertical momentum transfer or

the surface stress depends on the size and shape of the surface

waves Hence a complex feedback exists with the overlying air

Combining this equation with the wind profile

approximation for the surface boundary layer winds gives a

unique relationship between the turbulent flux and roughness

length The two are equivalent measures of the sea surface

roughness and need to be parametrized using some measure of wave development

2.3.2 Dependence of the sea surface roughness on wind wave characteristics

The dependence of the sea surface roughness on wind speed and wave age has been investigated using both field data and a numerical model that calculates the sea surface drag Data collected include wind speed and direction, plus surface fluxes

of heat, moisture and momentum From this data, drag and Charnock coefficients have been calculated, giving estimates of the sea surface roughness

For the lower atmospheric boundary layer on the sea, the roughness length is not constant as on the land surface, it depends on wind-wave age:

z 0 = z m exp (-c 0 /u *) (1.53)

where z m is the height at which the wind velocity is equal the speed of the dominant/most energetic waves in the energy-wavenumber spectrum

Charnock (1955) argued that the roughness length, z0 could

47 48

wave age, c0/ u* describes how developed the waves are

relative to the surface winds Here c is the speed of the 0

dominant/most energetic waves in the energy-wavenumber

spectrum Many claim that the surface roughness is greater

when the wave age is small since the surface will then be

dominated by short, steep, slow-moving waves Hence ch is

often written as some power of the inverse wave age, u /* c0

The parameterization of Bortkovskii et al (1993) is in

following form:

g

u c

z

z ~ ( ~ * ) *

2 0

00 0

at

6040

6060

6072

4at

724at

027

0

z z z

c c z z

z

c c

a

c c

z

n n

~ ) (

~ ) (

~

~ ,

~

.

~ ,

~

p

Parameterization by equation of dimensionless sea surface

roughness length, ~z 0 z0g/u2* dependence on wind wave age

parameter, c~ *c0/u* is carried as generalization of Charnock

approximation using the data of laboratory and field

measurements The value ~z00 0.033a g/u3* in equation is z~ 0

at regime of aerodynamically smooth flow According to

equation 1.56, at initial stage of wind wave development,

72

4.

~

*

c , dimensionless roughness length grows proportionally

to rising of wind wave age parameter, whereas at further

increasing of c the dimensionless roughness length is ~ *

decreasing function of ~c Thus the maximum value * z~ =0.127 is 0

achieved at ~c =4.72 In the region 4.72<* ~c  60, which is the *

most probable for real open ocean conditions, the decreasing of 0

z~ is represented by mean of 4th degree polynomial with coefficients: a0 = 0.204681, a1 = -1.9606 x 10-2, a2 = 7.4002 x 10-4,

a3 = -1.1917 x 10-5, a4 = 6.8427 x 10-8 The mean square deviation of experimental value z~ from the polynomial 0

approximation is 0.0149, and it is several less than the value 0

z~ =0.018, determined by polynomial at c = 25 i.e at fully ~ *

developed sea Eventually for region ~c > 60, field *

measurements are scarce and a linear decrease of z~ until 0

achieving the value z~ is taken 00

The similarity theory shows that the ratio between the

roughness z 0 and wave characterized scale h g : z 0 /h g could be

expressed as wave Reinolds number R eg :

where R eg = (h g u * )/ and h g = u * /g

It is possible to determine asymptotic expression for

function P 0 (R eg ) for each critical values of R eg

In the weak wind condition the value of Reg is very small, in other hand the gravitational wave is in stage developing therefore the role of gravitational acceleration g will be

disappeared, that means the function P0 has the form Reg-1:

z 0 = m 0 h g (R eg -1 ) = m 0 (u * /g)(g/u * ) = m 0/u * = m 0 h v (1.58)

h v is the characteristic length for molecule viscous layer – its value is about some millimetres, the dimensionless coefficient m0 for rigid surface is about 0.1

49 50

In strong wind condition, R eg has big value, the mean

velocity profile is not depended from molecule viscosity  and

function P 0 will take constant value The expression for

roughness z 0 becomes:

z 0 = m 1 (u * /g) =m1 h g (1.59)

The value of coefficient m1 for rigid surface is about 0.03

These above asymptotic formulas are the same as Charnock

empirical formulas, where the coefficient m1 is equal 0.035

Figure 1.3 Relations between roughness length and dimensionless wave height after

Kitaigorotxkii (1970)

After the analysis of results of observed data on the sea,

Kitaigorotxkii (1970) found the following formula for P0:

R for

R

R for

R

p

p503

20480

501

0

.

is the order of magnitude differences in values of the Charnock coefficient between the model runs since the only differences between them relate to the assumed forms for the wave spectrum and wind input to it, both of which up to now have been determined empirically from field observations

2.4 Calculation of turbulent fluxes on sea surface

2.4.1 Direct Calculation of Fluxes

Before we can describe the geographical distribution of fluxes into and out of the ocean, we need to know how they are measured or calculated

There is only one accurate method for calculating fluxes of

sensible and latent heat and momentum at the sea surface: from direct measurement of turbulent quantities in the atmospheric boundary layer made by gust probes on low-flying

51 52

aircraft or offshore platforms Very few such measurements

have been made They are expensive, and they cannot be used

to calculate heat fluxes averaged over many days or large areas

Table 1.3 Notation describing fluxes

Cp Specific heat capacity of air 1030 J.kg -1 °K -1

Cθ Latent heat transfer coefficient 1.2x10 -3

Cq Sensible heat transfer coefficient 1.0x10 -3

LE Latent heat of evaporation 2.5x10 6 J/kg

q Specific humidity of air kg (water vapour)/kg

u10 Wind speed at 10 m above the sea m/s

w’ Vertical component of fluctuation of

wind

m/s

The gust-probe measurements are used only to calibrate

other methods of calculating fluxes

 Measurements must be made in the surface layer of the

atmospheric boundary layer, usually within 30 m of the

sea surface, because fluxes are independent of height in this layer

 Measurements must be made by fast-response instruments (gust probes) able to make several observations per second on a tower, or every meter from a plane

Measurements include the horizontal and vertical components of the wind, the humidity and the air

temperature.Fluxes are calculated from the correlation of

vertical wind and horizontal wind, humidity, or temperature: Each type of flux is calculated from different measured

variables, u’, w’, T’, and q’:

' '

' ' '

'

' ' '

w q L LE

w T C w T H

u w u w u

where the brackets denote time or space averages, and the

notation is given in table 1.3 Note that specific humidity

mentioned in the table is the mass of water vapour per unit mass of air

2.4.2 Gradient method

Proceeding just as in the previous section we can now establish the formulas for vertical profiles of the velocity, potential temperature and humidity:

u z

u z

T z

q z



z f u z

z f z

z f q z

Η  p and moisture q ' w' turbulent fluxes can be

calculated by the gradient method using the given velocity,

potential temperature, humidity gradient and turbulent

viscosity K u , heat K h and moisture K D diffusion coefficients:

g

L u

K  * Generally we can introduce the inverse Prandtl (H) and Shmidt (D) number in form:

D

g

g K

D

) ( ) (

0

100

0

100

q D

In the condition of weak stratification they fit the logarithmic + linear profile:

z u z u z

1

2 1

z z

z

H

1 2 1

2 0 1

z q

z q z

1

2 0 1

2

1

ln )

( )

These formulas can be used to investigate the air-sea interaction processes in condition taking account the variability

of the surface characteristics as the mean value of roughness h0,

the height of turbulent viscous layer h = /u *, sea surface temperature s and humidity q s The relation between h0, and h

is named as Reynolds number R e0 = h 0 /h

u R

h

z u

cm e q

The mean value of molecular Prandtl and Shmidt numbers

(Pr m and Sc m ) for air is about 0.72, the variation of this value

for gas mixture is relatively large: Pr m = 0.34 – 1.96 In

turbulent flow the value of Pr and Sc is more stable: Pr = 0.7 –

0.75 and Sc = 0.83 – 0.77 In lower atmospheric boundary layer

on the sea we can take the same values for Pr and Sc

considering as function of atmospheric stratification In the

condition of neutral stratification they have value 1, for strong

instability their value is about 0.3 and in the condition strong

stability they become greater than 1

2.4.3 Indirect Calculation of Fluxes: Bulk Formulas

The use of gust-probes is very expensive, and radiometers

must be carefully maintained Neither can be used to obtain

long-term, global values of fluxes To calculate these fluxes from

practical measurements, we use observed correlations between

fluxes and variables that can be measured globally

For fluxes of sensible and latent heat and momentum, the

correlations are called bulk formulas They are:

LE E D  , 1.75)

Air temperature θ is measured using thermometers on

ships It cannot be estimated from space using satellite

instruments θ 0 is measured using thermometers on ships or

from space using infrared radiometers such as the AVHRR The specific humidity of air is the mass in kilograms of water vapor in a kilogram of air The specific humidity of air

(normally at 10 m) above the sea surface q is calculated from

measurements of relative humidity made from ships Gill (1982) describes equations relating water vapour pressure, vapour density, and specific heat capacity of wet air The

specific humidity at the sea surface q 0 is calculated from θ 0

assuming the air at the surface is saturated with water vapour The wind velocity is measured or calculated using the instruments or techniques described in previous section

The coefficients C u , C θ and C D are calculated by correlating

direct measurements of fluxes made by gust probes with the variables in the bulk formulas

Smith (1988) investigated the accuracy of published values

for the coefficients, and the values for C θ and C D in table 5.1 are

his suggested values Smith (1988) also gives fluxes as a function of observed variables in tabular form Liu, Katsaros, and Businger (1979) discuss alternate bulk formulas Note that wind stress is a vector with magnitude and direction It is parallel to the surface in the direction of the wind

In order to investigate the relation between C u , C and Cq,

it will do following transformation

From above mentioned coefficient’s definition:

H

z u c

z  p 

While the momentum and heat exchange processes are

similar, the Prandtl number Pr =K v /K h =1, then C= C u These

conditions are not always variables, both the heat exchange and

moist diffusion processes are full similar and Pr ~ Sc In this

case the values of coefficients C and C D are the same

The coefficients Cand C D are depended from stratification

as Cu: when instability increases then these coefficients

increase too, for strong stability these coefficients decreased

significantly

Returning to formula (1.13), applying the similarity

hypotheses we can write these formulas of wind, temperature

and humidity profiles in the following forms:

u =

0

z

z u

ln

*

0 0 0

z

z T

z

z q q q

where  and q are new parameters independed with z

and can be expressed in the forms:

Pr) , (

0 T P R e

  ;

) , (

0 q P q R e

q 

The universal functions P and P q are also determined

similarly as for C and C D According to previous discussions, between Pr and Sc as

between C and C D there are no difference then we can suppose

that P and P q have the same values

Using similar formula as for roughness length z 0 : z 0 = h g

P 0 (R eg ), we can write the formulas for 0 and q 0:

0T*P* Reg , Reg u*3/g (1.81)

P q

q0 * q*Reg , g  *2/

As was indicated in previous section, while R eg << 1, z 0 =

m 0 *h =m 0 * /u * , thus we can get the following formulas: 0 =

m*T 0 and q 0 = m q *q 0 For big R eg , the similarity between z 0 ,

0 and q 0 may be perturbed due to the occurrence of water bubbles in lower atmospheric boundary layer The mechanism

of heat, mass and momentum transfer will be different

59 60

Comparing the temperature and humidity value at levels z2

= 2 m , z0 and on sea surface, it is shown the existing

permanent relation between them:

2 - 0 = (2 - s ), q 2 - q 0 = q (q 2 - q s), 1.83)

with  andq have the value corresponding 0.99 and 0.81

Applying formula (1.16) is conduced to determine functions

of P* and P q *:

P*

0

211

The ratio 0 /(z -s )  0,01, q 0 /(q z -q s)  0.2 helps us to

determine the value P* = P q * = 0 and C = C D = C u for

weak stratification to calculate the heat and moisture fluxes

2.4.4 Field and model estimates of the sea surface drag

For different stratification conditions we can use vertical

velocity profiles to calculate the values of u z , z 0 , L (or L*) the

same as for momentum flux  = u *

2.

The surface stress, , can be represented by the bulk

aerodynamic formula (1.73) Hence, the height of a wind

measurement is important Usually, winds are reported as the

value of wind at a height 10 m above the sea u10

For neutral stratification, the drag coefficient C u0 is easy

get from velocity profile:

For unstable stratification (L < 0), the value of C u will be

bigger than C u 0, and inverse relation is applied for stable

stratification ( L > 0): C u < C u 0

In the condition of weak stratification, the coefficient C u 0

could be considered as function of wind speed u, if the velocity

at 10 meters above sea (u 10) used, it yields:

C u 0 = 10-3 (a + cu 10 ) (1.87)

where the values of a: 0.70 < a < 1.10; c: 0.04 < c < 0.12 in

SI units: [u] = (m/s) Because this condition is frequently

observed in the real oceanic condition, the formula 1.87 is often used

In this condition, wind frofile fit well logarithm law:

- above the rigid surface:

In the condition of free convection or strong instability

(z/L) < -0.05  -0.10, turbulent viscosity is exponential function

of 4/3 and vertical wind profile is fit to law -1/3 The drag coefficient becomes:

C u =2 [ln(z k /z 0 ) +  (z k /L) + C 1 (z 10 -1/3 - z k -1/3 )/L -1/3 ] -2 (1.90)

61 62

z k is the level of beginning of free convection: z k ~ -0.07 L

Due to buoyancy effect, the convection is conduced to

increase exchanged fluxes between atmosphere and ocean

In the condition of strong stability z/L > 0.04, turbulent

viscosity is not more depended on distance, the drag coefficient

Cu could be calculated by following formula:

C u =2 [ln(z i /z 0 ) +  (z i /L) + C 1 (z 10 - z i )/LR fcr] -2 (1.91)

where R fcr is critical value of Richardson flux number, z iis

lower limit of thermal inverse layer, normally z i ~ 0,4L, = 10

The stable condition is conduced to decrease exchanged fluxes

between atmosphere and ocean

Figure 1.4 Theoretical (1) and observed fit function (2) of drag

coefficient on the sea, after Bortkovxki P C et al (from Doronin, 1980)

Using the definition of the friction velocity, then using (1.79) gives Charnock’s form of the drag coeficient as a function

of wind speed in figure 1.5

This figure shows the 10m drag coefficient against 10m wind speed The individual data points are from FASTEX measurements, whilst the lines show the relationship between drag coefficient and wind speed for constant Charnock coefficients of 0.018 and 0.04 respectively In all cases there is a steady increase in the drag coefficient with wind speed, but the larger increase in the FASTEX data compared to with a constant Charnock coefficient suggests that the sea surface roughness may have an inverse wave age dependence Typical

of surface flux measurements over the ocean, the field data contains a great deal of scatter One reason is that the measurement technique assumes spatial and temporal homogeneity in the turbulent fluctuations above the surface In reality this assumption does not hold Also the presence of ocean swells propagating in various directions and temporal variability in mean wind speed and direction will affect the surface roughness

Assuming of a constant stress in the lowest few tens of metres of the atmospheric boundary layer, it calculates the surface stress given the 10m wind speed and wave age Drag or Charnock coefficients can then be calculated The model involves assuming a form for the energy wavenumber spectrum This spectrum is split into an energy containing part consisting of waves longer than a few metres including the spectral peak, and a high wavenumber part, which consists of all waves shorter than a few metres It is the latter that supports the bulk of the stress However its dependence on

u10(m/s)

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u10

63 64

wave age is uncertain and the model can be made to have either

a wave age dependent or independent high wavenumber

spectrum

Figure 1.5 Observations of the drag coefficient as a function of wind speed u10 ten

meters above the sea The solid line is 1000Cu = 0.44 + 0.063u 10 proposed by Smith

(1988) and the dashed line follows from Charnock (1955)

In conclusion, both field and numerical studies of the sea

surface roughness suggest that there is still a great deal of

uncertainty as to its dependence on sea state

2.4.5 Climatic estimation of the turbulent fluxes

The problem now becomes: How to calculate the fluxes

across the sea surface required for studies of ocean dynamics?

The fluxes include: 1) stress; 2) solar heating; 3) evaporation; 4)

net infrared radiation; 5) rain; 6) sensible heat; and 7) others

such as CO2 and particles (which produce marine aerosols)

Furthermore, the fluxes must be accurate We need an accuracy

of approximately ±15 W/m2 This is equivalent to the flux of

heat which would warm or cool a column of water 100 m deep

by roughly 1°C in one year

Normally, for turbulent fluxes as stress, evaporation and sensible heat, the gradient method could be used to calculate the small-scale variation Beside observed wind, humidity and temperature profiles, it needs knowledge of small-scale

variation law for the dynamical parameters as z 0 , u * and thermal ones as 0 , H, E, q 0 (see previous section)

In the spatial fields calculation we have not enough observed data on several layers, it needs teak account synoptic analysis data Within synoptic analysis data, only standard level (surface, 850 mb, 200 mb, etc.) values will be available There are different semi-empirical formulas deduced from similarity for applied to planetary atmospheric boundary layer According these approximations there are two principal thermo-dynamical combined parameters:

(i) Rossby number Ro = G/ fz 0 ,

(ii) Thermal stratification: S = /fG,

where G – wind velocity at the upper limit of atmospheric

boundary layer (h) or geostrophic wind, f - Coriolis frequency,

 = h - 0 temperature difference In taking account the

moisture stratification, the thermal stratification S becomes S *:

S * = S(1 + I), I = 0,61 c p TE/H = m/Bo (1.92)

Using the similarity approximation between heat and

moisture exchange processes, the ratio I will be expressed as

65 66

for geostrophic drag coefficient, Stenton and Dalton numbers in

the following form:

) ,

J G

J q

G

D    0



Using these coefficients we can estimate the fluxes  (u * ),

H, E based on available data of geostrophical wind G and

another thermal and moisture parameters of atmospheric layer

on the sea as  and q

Certainly, for determining the parameters Ro vµ S*, it

needs know the roughness length z0, the last could be calculated

as function of R eg and friction velocity u* determined by using

surface wind velocity u * = C u u a

2

Empirical estimations give us the following general feature

of these important parameters:

- While Ro decreases (weak geostrophic wind G or hight

roughness), the geostrophic drag coefficient  increases together

with the Stenton and Dalton numbers For characteristic values

of Ro varying within interval from 10+5 to 10+13 (Ro = 10+7 at G =

10 m/s, f = 10-4 s-1 and z 0 = 10-2 cm),  varies from 0.01 to 0.2

- While S* increases (weak geostrophic wind G, strong

stability), the geostrophic drag coefficient  increases together

with the Stenton and Dalton numbers For Ro = 10+7 at S* =

-10-2 (unstable)  = 0.06, at S* = 10+3 ( stable)  = 0.02

Normally, the value of  is varied from 0.02 to 0.03 and St = D

= 10-3

Now, let’s look at all above mentioned variables

Wind Speed and Stress

Stress is calculated from wind observations made from ships at sea and from scatterometers in space Beaufort observations give mean wind velocity and wind stress, and scatterometers measurements yield global maps of day to day variability of the winds used to produce maps of monthly-mean

wind stress

Latent Heat Flux

Latent heat flux is calculated from ship observations of relative

humidity, water temperature, and wind speed using bulk formulas and ship data

The fluxes are difficult to calculate from space because satellite instruments are not very sensitive to water vapour close to the sea Liu (1988) however showed that monthly averages of surface humidity are correlated with monthly averages of water vapour in the column of air extending from the surface to space This is easily measured from space; and Liu used monthly averages of microwave radiometer observations of wind speed, water vapour in the air column, and water temperature to

calculate latent heat fluxes with an accuracy of ±35 W/m2 Later, Schulz et al (1997) used AVHRR measurements of sea-surface temperature together with SSM/I measurements of water vapour

and wind, to calculate latent heat flux with an accuracy of ±30 W/m2 or

±15 W/m2 for monthly averages

Perhaps the best source of flux information is the values calculated from numerical weather models

Sensible Heat Flux

Sensible heat flux is calculated from observations of air sea temperature deference and wind speed made from ships, or

67 68

from the output of numerical weather models Sensible fluxes

are small almost everywhere except offshore of the east coasts

of continents in winter when cold, Arctic air masses extract

heat from warm, western, boundary currents In these areas,

numerical models give perhaps the best values of the fluxes

Historical ship report gives the long-term mean values of the

fluxes

Insolation

Insolation is calculated from cloud observations made from

ships and from visible-light radiometers on meteorological

satellites Satellite measurements are far more accurate than

the ship data because it’s very hard to estimate cloudiness from

below the clouds Satellite measurements processed by the

International Satellite Cloud Climatology Project are the basis

for maps of insolation and its variability from month to month

(Darnell et al 1988; Rossow and Schiffer 1991)

The basic idea behind the calculation of insolation is very

simple Sunlight at the top of the atmosphere is accurately

known from the solar constant, latitude, longitude, and time

Sunlight is either reflected back to space by clouds, or it

eventually reaches the sea surface Only a small and nearly

constant fraction is absorbed in the atmosphere Thus

insolation is calculated from:

where S = 1365 W/m2 is the solar constant, A is albedo, the

ratio of incident to reflected sunlight, and C is a constant which

includes absorption by ozone and other atmospheric gases and

by cloud droplets Insolation is calculated from cloud data

(which also includes reflection from aerosols) collected from

instruments such as the AVHRR on meteorological satellites

Ozone and gas absorption are calculated from known distributions of the gases in the atmosphere QSW is calculated

from satellite data with an accuracy of 5–7%

Recent work by Cess et al (1995) and Ramanathan et al (1995) suggest that the simple idea may be wrong, and that atmospheric absorption is a function of cloudiness Schmetz (1989) gives a good review of the technique, and Taylor (1990) describes some of the relationships between satellite observations and terms in the radiation budget

Water Flux (Rainfall)

Rain rate is another variable that is very difficult to measure from ships Rain collected from gauges at different locations on ships and from gauges on nearby docks all differ by more than a factor of two Rain at sea falls mostly horizontally because of wind, and the ship’s superstructure distorts the paths of raindrops Rain in many areas falls mostly as drizzle, and it is difficult to detect and measure

The most accurate measurements of rain rate in the tropics

(±35°) are calculated from microwave radiometer and radar

observations of rain at several frequencies using instruments

on the Tropical Rain Measuring Mission TRMM launched in

1997 Rain for other times and latitudes can be calculated accurately by combining microwave data with infrared observations of the height of cloud tops and with rain gauge data Rain is also calculated from the reanalyses of the output from numerical weather forecast models (Schubert, Rood, and Pfaendtner, 1993), from ship observations (Petty, 1995), and from combining ship and satellite observations with output from numerical weather prediction models (Xie and Arkin, 1997)

69 70

The largest source of error is due to conversion of rain rate

to cumulative rainfall, a sampling error Rain is very rare, it is

log-normally distributed, and most rain comes from a few

storms Satellites tend to miss storms, and data must be

averaged over areas up to 5 on a side to obtain useful values of

rainfall

Net Long-Wave Radiation

Net Long-wave radiation is not easily calculated because it

depends on the height and thickness of clouds, and the vertical

distribution of water vapour in the atmosphere It can be

calculated by numerical weather-prediction models or from

observations of the vertical structure of the atmosphere from

atmospheric sounders The net flux is:

F = <e> F d - S T4 (1.96)

where <e> is the average emissivity of the surface, F d is

downward flux calculated from satellite, microwave-radiometer

data or numerical models, T is sea-surface temperature, and S is

the Stefan-Boltzmann constant The first term is the downward

radiation from the atmosphere absorbed by the ocean

Frouin, Gautier, and Morcrette (1988) describe how F d can

be calculated The second term is the radiation emitted from

the ocean Both terms are large, and the net flux is the

difference between two large quantities

Another researchers estimated the accuracy of monthly

averaged values is ±5–15 W/m2 Improvements will come from

more data, which reduces sampling error, and from a better

understanding of daily cloud variability Note, however, that

the flux tends to be relatively constant over space and time; so

much improved accuracy may not be necessary

2.4.6 Calculation fluxes in the storm condition

In the condition of strong wind, especially during storm with wind speed higher than 15 m/s, the momentum, heat and mass exchange processes are considerably perturbed The perturbation is due to occurrence of water bubbles ejected from wave and water surface The direct influence of these bubbles

on momentum flux could be explained by following mechanisms:

(i) The mass of these bubbles moving together with air flow transfered momentum to water after returning to sea surface Occurrence of air bubbles in the sea surface layer is also contributed to increase the momentum flux into sea

(ii) In the condition strong wave, the humidity in the lower boundary layer increases thus stratification changed then the momentum flow changed too

The value of drag coefficient C u in storm condition is very difficult to determine by observation data, but all different investigations indicated that they are considerably big The figure 1.6 shows the values of drag coefficient in different wind

condition included storm wind It is recommended to use C u

from 2x10-3 to 4 x10-3 for climatic calculation

The influence of storm wind on another fluxes is also considerable; normally the heat and moisture fluxes in storm condition are about two or more times greater than in normal condition While the wind speed is over 25 m/s the growth rate may be reached from 5 to 6 times

1100 mm, Halong has cold winter, Phan Rang has hot summers

If we look closely at the characteristics of the atmosphere along the two coasts, we find more differences that may explain the climate For example, during north-east monsoon, when the wind blows onshore toward Ha Long, it brings a cold, continental dry air from Chinese main-land During south-west monsoon, the wind in Ninh Thuan blows offshore, the boundary layer a few hundred meters thick capped by much warmer, dry air In Ha Long the wind blows onshore, it brings a warm, moist, marine boundary layer that is much thicker Convection, which produces rain, is much easier on Quang Ninh coast than

on the south-east coast Why then is the atmospheric boundary layer over the water so different on the two coasts? The answer can be found partly by studying the ocean’s response to local winds, the subject of this chapter

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73 74

3.1 Dynamic interaction and Ekman layer

3.1.1 Response of the upper ocean to winds and inertial

motion

To begin our study of currents near the sea surface, let’s

consider first a very simple solution to the equations of motion,

the response of the ocean to an impulse that sets the water in

motion For example, the impulse can be a strong wind blowing

for a few hours The water then moves under the influence of

Coriolis force and gravity No other forces act on the water

Such motion is said to be inertial The mass of water

continues to move due to its inertia If the water were in space,

it would move in a straight line according to Newton’s second

law But on a rotating earth, the motion is much different

Equations of motion are two coupled first-order, linear,

differential equations which can be solved with standard

techniques If we solve the second equation for u, and insert it

into the first equation we obtain:

fv dt

v d

Rearranging the equation puts it into a standard form we

should recognize, the equation for the harmonic oscillator:

02

which has the solution (2.3) This current is called an

inertial current or inertial oscillation:

2 2

Notice that (2.3) are the parametric equations for a circle

with diameter D i = 2V/f and period T i = (2π)/f = T sd /(2 sin φ) where

T sd is a sidereal day

T i is the inertial period, and it is one half the times required

for the rotation of a local plane on Earth’s surface (Table 2.1)

The direction of rotation is anti-cyclonic: clockwise in the

northern hemisphere, counter clockwise in the southern Notice

that at latitudes near 30°, inertial oscillations have periods very

close to once-per-day tidal periods, and it is difficult to separate inertial oscillations from tidal currents at these latitudes

Table 2.1 Inertial Oscillations

75 76

Inertial currents are usually caused by wind, with rapid

changes of strong winds producing the largest oscillations The

forcing can be directly through the wind stress, or it can be

indirect through non-linear interactions among ocean waves at

the sea surface (Hasselmann, 1970) Although we have derived

the equations for the oscillation assuming frictionless flow,

friction cannot be completely neglected With time, the

oscillations decay into other surface currents

3.1.2 Wind circulation and Ekman Layer at the Sea

Surface

Steady winds blowing on the sea surface produce a thin,

horizontal boundary layer, the Ekman layer By thin, it means a

layer that is at most a few-hundred meters thick, which is thin

compared with the depth of the water in the deep ocean A

similar boundary layer exists at the bottom of the ocean, the

bottom Ekman layer, and at the bottom of the atmosphere just

above the sea surface, the planetary boundary layer or

frictional layer The Ekman layer is named after Professor

Walfrid Ekman, who worked out its dynamics for his doctoral

thesis

Ekman’s work was the first of a remarkable series of

studies conducted during the first half of the twentieth century

that led to an understanding of how winds drive the ocean’s

circulation (Table 2.1) In this chapter we consider Nansen and

Ekman’s work

Fridtjof Nansen, while drifting on the Fram, noticed that

wind tended to drive ice at an angle of 20°–40° to the right of

the wind in the Arctic, by which he meant that the track of the

iceberg was to the right of the wind looking downwind He

subsequently worked out the basic balance of forces that must

exist when wind tried to push icebergs downwind on a rotating earth

Nansen argued that three forces must be important:

1 Drag must be opposite the direction of the ice’s velocity;

2 Coriolis force must be perpendicular to the velocity;

3 The forces must balance for steady flow

Ekman assumed a steady, homogeneous, horizontal flow with friction on a rotating Earth Thus horizontal and temporal derivatives are zero:

u

A z yz w z

w xz

w is the density of sea water

With these assumptions the x and y components of the

momentum equation have the simple form:

77 78

00

2 2 2

where f is the Coriolis parameter

It is easy to verify that the equations (2.7) have solutions:

) cos(

) exp(

) sin(

) exp(

az az

V

v

az az

f a fA

V

2

and2





(2.9)

and V 0 is the velocity of the current at the sea surface

Now let’s look at the form of the solutions At the sea

surface z = 0, exp(0) = 1, and

) sin(

)

(

) cos(

)

(

40

40

The current has a speed of V0 to the northeast In general,

the surface current is 45° to the right of the wind when looking

downwind in the northern hemisphere

The current is 45° to the left of the wind in the southern

hemisphere Below the surface, the velocity decays

exponentially with depth (figure 2.1):

u2(z) v2(z) 2 V0exp(az)



Figure 2.1 Vertical distribution of current due to wind blowing on the sea surface

(From Dietrich, et al., 1980)

To proceed further, we need values for any two of the three

parameters: the velocity at the surface, V0; the coeficient of eddy

viscosity, Az; or the wind stress  The wind stress is well known, and Ekman used the bulk formula:

10 2

79 80

100127

With this information, he could then calculate the velocity as a

function of depth knowing the wind speed u10 and wind direction

Figure 2.2 Vertical structure of drift current in shallow sea – the projections of

current on the horizontal plan (after Ecman)

The thickness of the Ekman layer is the depth Df at which the current

velocity is opposite the velocity at the surface Because the Ekman

currents decrease exponentially with depth, this condition occurs at a

depth D f = π/a and the Ekman layer depth is:

in SI units: wind in meters per second gives depth in meters

The constant in (2.14) is based on w = 1027 kg/m3, air = 1.25

kg/m3, and Ekman’s value of C f = 2.6 × 10 -3 for the drag coeficient Using (2.14) with typical winds, the depth of the Ekman layer varies from about 45 to 300 meters (Table 2.2), and the velocity of the surface current varies from 2.5% to 1.1% of the wind speed depending on latitude

Table 2.2 Typical Ekman Depths