Tides

.J.

P.

Th.Kalkwijk

Edition 1984

tides

HYDRAULIC AND ENVIRONMENTAL ENGINEERING

.J.

P. Th.Kalkwijk

Edition 1984

T I DES

LEe T URE NOT E S

Contents

I. Introduetion 2

2. Origin and astronomiealanalysisof the tide; tide predietion 4

2.I. Origin of the tide 4

2.2. Astronomiealanalysisof the tide 15

2.3. Tidal predietion 26

3. Tidal analysis 30

3.1. Introduetion 30

3.2. Method of the least squares 32

3.3. Sample interval 36

3.4. Duration of a tide measurement 38

3.5. Closure remark 40

4. Tidal eomputations 41

4.1.Introduetion 41

4.2. The equations 42

4.3. Some basie solutions 51

4.3.1. Simplifiedsystem 51

4.3.2. Basie solution 52

4.3.3. Refleetionand transmissionat a transition 57 of ehannels

4.4. Harmoniewaves

4.4.1. Single progressiveharmonie wave 4.4.2. Standingwave

4.4.3. Tidal wave on a r~ver 4.5. Lorentz method

4.5.1. Lorentz linearization 4.5.2. Harmonie solution 4.5.3. Example of computation

4.5.4. Channel closed-offat one end 4.5.5. Couplingof seetions 4.5.6. Run-off discharge 4.6. Closure femark 59 59 62 64 73 73 76 83 89 94 97 104

5. Tidal motion in seas and oceans 105

References 113

115 Notations

I. Introduction

As long as people live on the borders of the sea and sail thereon they are interested in the rise and the fall of the water level as this is caused by the motion of the celestial bodies as moon, sun and earth. The qualitative explanation of the phenomenon, which is called tide, has been given in the

17th century. Due to the increasing navigation the quantitative aspects regarding analysis and prediction of the tid~ became more and more important and in the 19th much progress has been made r.nthis respect. This basis was extended in this century,

so that nowadays tidal analysis and prediction are common procedures. In addition the need to predict hydraulic consequences of new

civil engineering works become manifest in particular in this century. In this connection the execution of tidal studies is often necessary. In the first half of the century computations were based on the use of semi-analytical methods, which could be carried out by hand. For complicated situations or in cases of complex situations the support of a hydraulic or analog model was indispensable. With the development of the digital electronic computer this has changed completely. Nowadays almost all tidal calculations, both for one-dimensional and two-dimensional cases, are made on the computer. Physical models to study the tidal motion exclusively are hardly made nowadays. Analytical methods are only used in cases of a first impression or for control purposes.

This course deals with a brief introduction into tidal analysis and prediction (chapters 2 and 3). In chapter 3 it will be explained

that the tide arises from the influence of maan and sun on the earth and some attention will be paid to the astronomical analysis of the tide generating force. The treatment of this subject.is short: it only aims at making plausible that many components with particular frequencies occur in the tide generating force, which return in the real tide on earth. Chapter 3 deals with the method of the least squares and its application to a tidal registration in order

to determine the so-called tidal constants. The chapter is closed off with the method how to predict the tide if the tidal contents are known.

Chapter 4 deals with the hydrodynamic behaviour of tides 1n one dimensional situations, such as this can be described by the relevant equations. The treatment begins with the derivation of some basic cases of wave propagation as the single progressive wave and its reflection and transmission at a transition in a ehannel. Next harmonic waves are

considered, with examples as the standing wave and the propagation of a tidal wave on a river. The ehapter finishes with the

so-called Lorentz-method for tidal computations. It is an approximative method based on harmonie behaviour of the

tide.

Chapter 5 dealing with tides in seas and oceans is purely descriptive. Most important is the explanation of so-called amphidromic systems.

2. Origin and astronomical analysis of the tide; tide prediction

This part of the lectures aims to show, that moon and earth give r~se to the tide as it exists on earth. The understanding of tide generating force will be defined and derived and it will be made plausible, that this force can be decomposed into many sinusoidal components, each having a characteristic frequency. These frequencies are related to

the periodicity of the various modes of motion carried out by earth, moon and sun, respectively. Af ter having explained, that in addition

other frequencies can also occur in tidal signals, this chapter will be closed off with a consideration how to predict the tide

if the tidal constants are known.

2.1. Origin of the tide

The forces, which generate the tide, are the mutual attraction forces between earth, moon and sun. The influence of other celestial bodies can be disregarded.

The following, simple, considerations apply only in case of the system earth-moon; the considerations for the system earth-sun are identical.

The attraction force between two bodies is determined by Newton's law of gravity:

where a = universal gravitational constant

x distance between the centers of gravity of the respective bodies

masses of the respective bodies

There exists a relationship between the universal gravitational constant and the acceleration due to gravity, g. To derive this, consider the attraction force of a body on the surface of the earth with mass mI. This body is attracted by the earth with a force:

F mIme mI g

=

a --2- + a r 2 .E_ m e (2.1 )

where m

e

r

mass of the earth = 5.98 x 1024

radius of the earth

=

6.38 x 106 -2

9.81 ms

kg m g

Taking the mass of the moon equal to Mm (M

=

0.0123) and the

e

distance between the centers of gravity of moon and earth, respectively, equal to Kr (K

=

60.3), then the attraction force

between earth and moon is, using the relationship for a (see eq. 21):

F (2.2)

This force 1S 'responsible' for the rotation of the system

earth-moon around a common center of gravity with a position inside Mkr

the earth itself at a distance --- < r from the center of the earth.

I+M Kr moon earth common centre of gravity

If the angular speed of the system is rïpresented by w rad/s , the acceleration of the earth is equal to wI~r , which is caused by the mutual attraction force of earth and moon. Consequently:

gMm e

7

2 w MKr I+M me 2 w

with the known values for g, M, K and r the result for w

=

2.66x10-6 rad/se This means that the whole system makes one revolution in 27.32 days.

common centre of gravity the earth moves such, that its motion can be conceived of as a combination of a translation and a rotation. The latter rotation can be added to the natural own rotation of the earth, s~nce both rotations have an identical influence on the tide. The other mode of motion, the translation of the earth, will be treated in more detail; the rotation will be considered later on.

moon

orbit of cen tr e of gravity of earth

-:

Fa Fa

com mon een t re of gravity

In fig. 1 it is shown, that resulting from this translation each particle of the earth has the same acceleration (same magnitude, same direction). This is drawn again in the simplified figure above. As the line OP exclusively carries out translations the point P has

the same acceleration as the point

o.

The acceleration of the point P has a direction parallel to the connection of the centers of gravity

respectively. lts magnitude (force per unit mass) of moon and earth,

is (see eq. 2.2) :

F gM

a K2

(2.3)

The acceleration F has to be 'provided' by the attraction force between a

the particle on earth and the moon. This force depends on the distance from the moon to the location on earth. lf this distance is Rr then

it can be stated without further proof that this force is (per unit mass):

F m

=

gM R2

~

....

. -'-E 0&- (J ",

..

(\I lil

-

..- (LIOl Q.I Ol C C. ~ >-U (\I L <» ", r::: 11\

...

",

--

CQ.I C E 0 0 _L- _oC 0 E u ..L- 0 al Ol U VI l- C -.. al -.. 0 '0

-s,

'0 Q_C U :J 0 '0 L-c: ", ", .c

...

-

....

0 I"IJ 0 ~ U _a.

:::::

i: ...

c

o

:ç

o

...

o L... I \

,

\ \ <,

'-

./ /

c

o

o

~

\

\

\

')<_

/"-/

\

\

-_0 ,... <,

,

\

-,

I

\

'"

I

\

/

I

I

\

\

\

... ..c:; 0 ... ... L...

.a

0 L... Q) 0 ... o Cl) L...

...

C Q) u ... L...

o c

...

.b

...o:..ë

L_ L...

o

C 'ö

o

Q_ \L. 0.0 U r::: o 0 Ul .J-",

...

o L-C ~ o L... o ... L... o 4-J jJ L... o QJ U

o

4-L... .:J lil o ~ 4-J

m

F and a residual force

F - F ,

which is called tide generating force

a m a

since this small residual force is the direct cause for the existence of tides on earth. The tide generating force makes an angle with the surface of the earth. For the actual motion of the water, however, only the component tangential to the surface of the

earth is important. This component will be called tractive force, F (after Doodson) and is:

s F s gM - Sl.n(<I> R2 + 0.) - gM sin <I> • K2

Ta simplify this expression the following approximations will be introduced:

R 'V _{K - coscjJ} cos 0. 'V 1 ,

sin 0. 'V Sl.n<I> K

Then F becomes, disregarding small terms (K is large!) s

F ~ gM 2(sin <I> cos 0. + cos <I> sin 0.) _ gM sin _<I> s

(K - cos _<1» K2

~

[(1

₊ 2 cos cp) _{(sin <I> +} Sl.ncP cos <1»

cp]

'V 2 _K - Sl.n

K K

~ 2

F_s 'V 3 (3 Sl.n<I> cos cp + 2 sin cp cos cp)

K K

F 3gM sin 2cp

s _2K3

(2

.

5)

The sketch below shows how the tractive force is distributed over the surface of the earth.

c

~,

G)

0 ; 0 \

E

\

i

I 1 I I \ I , .

\

·1 I I

\

I

I

\

I

I

\

\

I

I

L-

\

I

~

\

\

I

i

I

CV

I

\ LI i U X L

I

~ 0 4-C 0

I

4-J _I U

;'

0

\

L I I 4-J

I

-~ C _I

uF-

,

\ \ 0 / \ U.. / C ~~

\

o 0 -. 4-J \ i

.

~ u

_L

!

o CV L

>

I

CV --' CV

I

u u

I

(1 LL-+-' I ~ / +

/,/;,1

/ §. / I

TT

/

' /

!f.

/

I " I

>

U .

/..

,..

/

~2 ~

/

.

-

,

'._'. -, -4-' / C ~ !..J I I I I ./ ç:: o N

The magnitude of the tractive force is very small. lts max~mum

TI

occurs for ~ = + 4

these angles.

the table gives the numerical values for

moon sun M 0.0123 333,000 K 60.3 _23,500 3eM -6 2 -6 -2 2K3 0.82x1O ms O.38x10 ms

The ratio of the tractive forces as caused by moon and sun is

approximately 2:1, implying that the influence of the sun on the tide on earth is not negligible.

The respective tractive forces are so small, that the influence on fixed particles mostly can be ignored, although a wave motion in the earth (earth tides) can be detected. Contrary the influence on water

particles, which can move, is much greater. To compute the water motion the tractive force could be introduced ~n the equations, describing the motion of the water. Next they can be solved (numerically),

provided the necessary boundary conditions are known. Although great progress has been made in this field, the solutions for cases in which bottom and ocean configurations are realistically represented are not very satisfactory. The computations for academic cases as oceans with constant depth bounded by meridians do not yield good results as weil, since the tidal amplitude remains limited to centimeters in these cases.

This is peculiar when realizing that the actual vertical tide has the

order of magnitude of one meter. The explanation of this phenomenon probably is, that the response of the oceans to the exciting force

(the tractive force) very much depends on the natural frequencies of

oscillation of the ocean water bodies. If one of these natural frequencies is close to the frequency of the exciting force, resonance is possible.

This phenomenon occurs ~n var~ous cases.

In the continental seas the tide is mainly determined by the tide in the adjoining oceans. Furthermore the tidal amplitude is strongly influenced by bottom and land configuration.

The way of oscillation of the water masses in oceans and seas has a typical character due to the rotation of the earth. Some attention to this phenomenon will be paid in chapter 5. All these considerations, however, are fairly complicated and therefore some fundamental aspects of the behaviour of the tide will be shown by considering the earth sphere, fully covered with water, and to examine the shape of the water surface if inertia forces are dis-regarded. In that case the tractive force has to be balanced by the force arising from the gradient of the water level. The latter force can be derived by considering the net force acting on a water element

m

e

and assuming hydrostatic pressure distribution.

I

-

dx --___. ..x

Thus net force force left - force right

dh pgh dx dx

dh Consequently the resulting force per unit mass ~s equal to - g dx' which force has to be balanced by F . This results in a shape of

s

the water level as shown in the sketch below. In other words, an

increase of the water depth occurs at the side of the earth facing the moon and at the opposite side.

arising in this way is called equilibrium tide. The consideration was already developed by Newton.

So far the rotation of the earth was left out of account. In this

connection the axis of rotation of the earth is important. It namely makes a mean angle of about 23!0 with the connection of the centers of earth and moon (sun). To illustrate the consequences next two cases will be considered.

a) The axis of rotation ~s perpendicular to the connection of the

centers of earth and moon. If the location on earth is fixed and measuring the water depth~two times a day a high water and two times a day a low water can be observed: semi-diurnal tide.

b) The ax~s of rotation makes an angle unequal to 90 degrees with the connection of the centers of earth and moon. In this case locations on earth exist, where once a day a high water and a low water are

measured: diurnal tide.

As a wave phenomenon, tides will propagate and therefore above

considerations are very academie. It is a fact, however, that the reaI

tide contains both semi-diurnal and diurnal components. One of the

consequences is, for instanee, that the high waters on a certain day

have different heights. The same holds for the low waters. This phenomenon

is called daily inequality, see fig. 3, which gives some examples of mean

tidal curves along the Dutch coast.

In most places of the world the semi-diurnal tide is dominant, but there are some places where the diurnal tide is most important, e.g. Tanjong Priok.

In practice high water will not occur, when the moon crosses the meridian.

The real HW will occur later. This time lag, which mean value is fixed for each place on earth is called HWF

&

Ch (high water full and change) or port establishement.

I... ",I IC"> I..!. _"'I ~I c, 11 ~I 11 c « Ol ZI o, 3:1 lil1 0> -c _~I c c Z _{:::::1 11>}.... 11> _"0 ...JI .... "0 Cl A _:> 11> _J: I rtI >-E c >-:> . -::::;> C rtI- u-Il> ....- IJ Cl 0 _{E ~.!: 0)} c >-:>

'"

g'~ ~ A E

1'

,

,= ,

_A ..r ..lil> + I';: I/) l/) I"~ _{... ~'I'O} 1- l/) '" 13 .,..; Cl I";l/) "0 :> 1=_Cl 1"0 ₁₌ A I I~ 'win;, ₁₀ 1- ..r I: 0) A _{0 ~} 0

_'"

.win;, _I..: I,,:<d I~ 1 :> I ---1 ---T 0 cA ₀ 'win;, a._1 a._1 « « Z ZI a._1 0 «I l/)

'"

'0 ZI I _c lil Cl A _{3: c} A ...J 11> 0 -' 0> I C C I/) _Cl VI _> > l/) -'"

'"

111 N 0 J: 10 111 I/) A A .... '0_... ..: :>l/) "0 "";1 Cl ₀₎₁ 0> Ö ~I a> ..: :> A __ "0 ~ I

?i

l

ZI

sn

Moo~8 SPRINGTIDE

C)

SUN

C)

G

OMOON EARTH c!)!---OON

_---IC)

EARTH SUN ~~---NE-A-P--T-ID-E---() MOON HIGHEST OBSERVEDLEVEL+455 (1953) 400 HIGHEST OBSERVEDLEVEL+460 (1825)

2h

DAY AFTERNM OR FM 400 IS SPRINGTIDE

~

H

<t

..

400 300 f,%DAY AF~M OR~FM • 200

IS SPRINGTIDE NEAPTIDE

!'~!

111III

tlllIllllI!IUUIIIIIIII~

1111

Ull~

II~

:p

-100J1fiillmlinliii~mllllfflnllifflr"nlOo

-200 - LLWS-203 -200

LOWESTOBSERVEDLEVEL-333 (1956) -300 -300 _{LOWESToeSERVED LEVEL-348 (1915)} EXAMPLE OF VERTICALTIDE FOR FLUSHING

FROMNEW MOON TO FULL MOON

EXAMPLE OF VERTICAL TIDE FOR DELFZIJL FROM NEW MOON TO FULL MOON

Although the influence of the sun is hardly discussed in the foregoing, it is clear that its influence cannot be neglected.

Moreover the combined action of moon and sun can give rise to a high HW and a low LW, which is called springtide. This will occur when

the tractive forme has a maximum, so when the moon and sun cross a meridian at the same time or when they have positions at opposite sides of the earth (see fig. 4). Therefore springtide will occur approximately each two weeks. It is evident that a low HW, and consequently a high LW, which 1S called neaptide, will occur one week after springtide.

It should be noted that springtide and neaptide do not occur at the instant, that the tractive force attains a maX1mum or minimum value. Due to inertia effects they occur one to three days later. This time

lag 1S called the age of the tide. Fig. 4 also gives examples of tidal registrations for two Dutch ports, covering a period of two weeks (from new moon to new moon).

2.2. Astronomie analysis of the tide

The result for the horizontal component of the tide generating force, the tractive force F , is

s F s 3gM

3

s i.n 2<p 2K (2.5)

In this formula <p and K are no eonstants; they depend on the moti.ons carried out by the celestial bodies with respect to each other. All these motions have a periodic character, implying that each motion has a charateristic mean angular speed. Until now the attempts to compute the complex water motion in the oceans were not very successfull, but it can be assumed that the phenomena generated by the tractive force contain the same frequencies as this force itself. Therefore, when analysing a tidal signal, it would be advantageous if the important frequencies are known beforehand. This, namely, could simplify the analysis and the prediction.

Several investigators, such as Darwin and Doodson, have

succeeded to decompose the tractive into its sinusoidal components.

The results of this so-called astronomical analysis give an

impression of the relative importance of each component. This

aspect can be taken into account by neglecting unimportant components ~n the actual analysis.

In this series of lectures the methods of decomposition, which

make use of expansions with spherical functions of the force

potential induced by moon and sun on earth, will not be treated.

Only a small introduction will be given, ~n which it ~s made

plausible that the tractive force can be decomposed into sinusoidal

components.

In order to do so a conc~se description of the important mations

carried out by earth, moon and sun will be given. For that purpose

the understanding of the celestial sphere has to be introduced.

It can be defined as a non-rotating sphere, moving along with the

earth. The relative motions of the celestial bodies as they appear

to the celestial sphere are projected on this sphere. First the

-projection of the apparent path of the sun on the celestial sphere

will be considered. It is a circle, which is called the exliptic

(fig. 5). The angle between ecliptic and equator is constant with

a value of approximately 23!0. The ecliptic intersects the equator

on two places, the vernal and autumnal equinoxes. One of them, the

vernal equinox, ~s used as a point of reference for the description

for the motions of the celestial bodies. When the sun is in the

vernal equinox then spring beg ins on the northern hemisphere.

Now the vernal equinox is in the constellation of Aries. The position

of the equinoxes is not constant. They make one complete revolution

around the equator in about 26,000 year. This motion is so slow that

it can be ignored for the tide.

The earth rotates around the sun in 365.26 days, which means that

this is the period between two successive crossings of the sun through

the vernal equinox. The corresponding mean angular speed is equal

to

w

=

0.041069 °/hour ~ The real angular speed var~es

s

slightly around this value because of the ellipticity of the orbit

of the earth around the sun. It is namely greatest when the earth ~s

~n the perihelium (earth closest to the sun) and smallest in the

It 1S evident that the afore-mentioned angular speeds will also occur

1n the variation of the distance between the earth and the sun,

as represented by K.

The motion of the moon 1S more complicated. The lunar orbit on

the celestial sphere intersects the ecliptic at two points, the

so-called nodes (ascending and descending node). As a consequence

of the varying position of the lunaf orbit with respect to the

equator (variation approximately 5°) the nodes move along the

ecliptic with a period of 18.6 years (w

=

0.00221 o/hour). Also

n

the moon moves 1n an ellips, but the position of the ellips 1S not constant. lts perigeum and consequently also its apogeum rotate once in 8.85 years (w = 0.004642 o/hour) around the

p

earth. Furthermore the moon completes one revolution in about one month around the earth. With respect to the vernal equinox the

o

period is 27.32 days (w = 0~549016 /hour). m

Finally, the earth rotates once a day around its own axis. With respect to the vernal equinox this period is somewhat less, namely 0.997 day (w

=

15.041096 o/hour). The difference with respect to

e

the period of one day is one revolution a year precisely. It is here, where the correction of the rotation as mentioned in section

2.1. is taken into account.

The various relevant frequencies and periods are collected 1n the next table.

origin angular speed period

0

1n /hour

rotation earth

w

=

15.041069 0.997 day

e

moon around earth

w

=

0.549016 27.32 day m

earth around sun

w

=

0.041069 365.26 day s

perigee moon

w

₌

0.004642 8.85 year

p

nodes lunar orbit

w

=

0.002206 18.60 year n

,

\

\

\

\

\

\ I

\

'I

\

"

\\

\

\ I

\

i\

I1

----

.

--t

\

~-J---

I

\

,

I. ln \ 11

~1

_o

'

~\1

_I

1

_:

_\

₁

cl

I

I~

.!:

I

I \\

...,

\

\\

~ I

I

\

\

~

I

\, :

\ 1 \ \ \ \

-

-

.

_

-

-

~

.

_-

-

~

QJ -0

o

C o C -0 C QJ U U1 Q ç:: o -e (I) -1-1 () (I)

.

..., o ~ c, ç:: ::s Cl)

.

Lf'I

Decomposing systematically the expression for F we can

s

expect all these frequencies to occur in the contributing sinusoidal components. In general it can be put foreward, that F can be decomposed as:

s F s 3gM sin_2K3 2<p cos (wnt ~n which w n ~wems+ jw + kw + Iwp + m~n (i, j, k, 1, m ..., -2, -I, 0, 1,2, ...) A amplitude n

-K K - mean

<Pn

phase at t

o

To give an impression how this result can be derived, the celestial sphere will be again considered. The position of an observer on earth and the position of a celestialbody (moon or earth) at a certain instant will be projected on the celestial sphere (fig. 6).

The celestial body, projected at the point S, has a declination d.The corresponding angle at'the center M is also indicated in the figure.

The location on earth is projected in the point T; it has a latitude b , Thepoints S and T lie on meridians, which intersect at the pole P under an angle p.

The angle <p in the expression for the tractive force was the angle between the connection between the centers of gravity of earth and moon (SM) and the connection of the location of earth with the center of the earth (TM). Consequently the angle <p ~s g~ven by the angle SMT. The tractive force is tangential to the earth and lies in plane through S, Mand T. This plane intersects the celestial sphere according to a circle and so F ~s also

s

tangential to this circle. The angle F makes with the meridian s

--

--p

=

hour angle;

S:::

pro jee

t ion

0

f t

hem

0 0

non

th e e art h

5

ph

ere:

d ::: declinution

celes tial

body;

b:::

latitude

point

on earth;

S'Te

por t of circle, containing

5 and T, with

centre

0

The figure shows, that, if the angles d, band p are g~ven, the angels ~ and t can be derived. The relations between these various angles is given by formulae originating from spherical

trigoniometry. They will be given without deriviation:

cos ~ s~n ~ s~n t s~n ~ cos t

s~n b s~n d + cos b cos d cos p cos d sin p

s~n d cos b + sin b cos d cos p (2.7)

First F will be decomposed into components tangential to the

s

parallel and meridian of the point T.

Represented by F hand F ,respectively, they are:

s sv 3gM sin ~ cos ~ s~n t K3 'F sv 3gM -- s~n 2~cos t 2K3 3gM -- s~n ~ cos ~ cos t K3 (2.8)

Substitution of the formulae according to (2.7) into expressions

(2.8) yields:

3gM cos d s~n p (sin b s~n d + cos b cos d cos p) K3

F

sv 3gM (sin d cos b + s~n b cos d cos p)(sin b sin d +

K3 _{cos b cos d cos p)}

This yields after same elaboration:

3gM/2K3

2

s~n b sin 2d sin p + cos b cos d sin 2p

(2) (4)

F s v.

!(3 sin2d - 1) sin 2b + cos 2b s~n 2d cos p

(1) (3)

3gM/2K3

2

+

!

s~n 2b cos d cos 2p

Even after this simple decomposition some important features can be distinguished. For fixed latitude band fixed declination d, the following cases can be analysed:

a) the terms (4) and (5) contain S1n 2p and cos 2p. The angular speed (the rate of change) of 2p is equal to 2(w - w ) or 2(w - w ), so that apparently the s

emi-e mes .

diurnal components are involved. In this case the angular speeds are those of the principal tides, viz M2 with period 12 h 25 min and S2 with period 12 h.

b) the terms (2) and (3) contain a S1n pand a cos p. The angular speed of p 1S equal to w - w or w - w , so that diurnal

e mes

components are involved. These components would not.be present.

if the celestial bodies would move in the plane of the equator, because then sin 2d = O. Therefore these components are

called declination tides. In addition it should be noted, that semi-diurnal declination tides exist as weIl.

c) if the present declination d has a fixed value term (1) does not represent anything else as a permanent force directed to the equator. The declination, however, does vary, so that this

force also varies. Typical angular speeds for d are wand w ,

m s

yielding much longer periods than one day, viz about one month and one year, respectively. In other words in addition to diurnal and more-diurnal also tides which long periods, the long-period tides, can be expected.

This decomposition of the tractive force can be extended considerably, for instanee by eliminating the declination of the celestial body by expressing this quantity in the inclination of the lunar orbit (or ecliptic) and the longitude of the point S with respect to the vernal equinox. This further decomposition, in which also other steps are involved, will not be carried out here.

The final result is represented in table 1, which contains the most

important components. The most right column in the table gives the amplitude A, the so-called astronomical amplitude. As indicated before

they are made dimensionless and therefore they indicate the relative

importance of each component individually, but it does not mean that

the ratios of amplitudes of the real tidal constituents are the same.

As long as the constituents belong to the same group, e.g. the

semi-diurnal tides, the mutual ratio is more or less according to the figures

to the tabIe. It does not hold at all for constituents of different

groups. Then in genera1the semi-diurnal constituents are much more

important as one would expect from the tabIe.

The other columns indicate how the angular speed of each tidal constituent

is built up. The general expression for a component of the tractive

force is:

A cos (iw' + jw + kw + bw + mw )t + ~

ems p n

~n which w'

e we - wm

The numerical values of the angular speeds are given on page 17

The column under m only contains ciphers. This does not mean, that the

position of the nodes does not influence the components of the tractive

force. This influence, however, is taken into account in a different way.

Since this influence, the so-called 19-yearly variation, is small,

multiplication factors, the node-factors, close to unity represent it.

For instance, the amplitude of the principle lunar tide, MZ' is multiplied

with:

fMZ

=

1.001 - 0.039 cos wnt

The node factors are considered constant during a calender year.

On the steep coasts of the deep oceans or near oceanic islands the tide can

be reasonably described with the components having frequencies according to

table 1. Unfortunately this is an exception, more or less, as other and also

higher frequencies occur in the tide as given by the tabIe. This is caused

by non-linear phenomenema playing a prominent part in the propagation of

S_a 0 0 0 0 ].]60 Ss_a 0 0 2 0 0 0 7.299 MIn 0 -2 0 -] 0 0 8.254 Mf 0 2 0 0 0 0 ]5.642 2Q] -3 0 2 0 0 955 0] -3 2 0 0 0 ].153 QI -2 0 0 0 7.216 PI -2 2 -I 0 0 I.371 Ol -I 0 0 0 0 37.689 TI -I 2 0 0 0 491 M] 0 0 -] _{0 0 1.065} NO] 0 0 0 0 2.964

X]

0 2 -] _{0 0 566} 7T] -3 0 0 ].029 PI -2 0 0 0 17.548 S] -I 0 0 423 KI 0 0 0 0 53.050 1J!] 0 0 -I 423 <p] 2 0 0 0 756 el 2 -2 0 0 566 JI 2 0 -I 0 0 2.964 00] 3 0 0 0 0 ].623 E2 2 -3 _{2 0 0 671} 2N2 2 -2 0 2 0 0 2.301 112 2 -2 2 0 0 0 2.777 N2 2 -I 0 0 0 17.387 \)2 2 -I 2 -I 0 0 3.303 M2 2 0 0 0 0 0 90.812 À2 2 -2 _{0 0 670} L2 2 _{0 -I 0 0 2.567} T2 2 2 -3 0 0 2.469 S2 2 2 -2 0 0 0 42.358 R2 2 2 -I 0 0 -I 354 K2 2 2 0 0 0 0 11.506 1;2 2 3 -2 0 0 123 n2 2 3 0 -I 0 0 643 M3 3 0 0 0 0 0 I.]88

a) variable width of the waterlevel caused by shoalings and the like,

b) influence of bottom friction on the flow; normally the friction is proportional to the square of the velocity, c) non-linear influences for flows with relative high

velocities (convective acceleration term),

d) variable celerity of the tidal wave, caused by depth variation.

Some more attention will be paid to points b) and d).

In case b) the friction F is proportional with the square of the velocity u, so:

f u2

which generally for alternating flows is written as:

F ulul

Assuming u sin wt, F can be expanded using a Fourierseries:

F sin wt

I

si.n wtl

8 8

3~ s~n wt + 15 s~n 3wt + ••••

This expression implies, that the non-linear friction seems to genera te terms, components, with higher frequency than the basic frequency. If in this case the basic frequency would be

that one of M2, a so-called M6, six oscillations a day, would be generated. It is clear, that this M6 does not have an astronomical origin.

In case d) a rectangular horizontal channel will be considered, ~n which a wave propagates which initially had a sinusoidal shape. In principle the velocity of propagation of a fixed elevation of the water level is equal to Igh. This causes to change the shape of the wave after a certain time (see sketch). If again MZ would be the basic wave apparently a M4 (4 oscillations a day) is

superharmonics having little to do with the astronomy.

---- original wave - - - wave aller some time

direct ion of

___ ---11.p ropagat ion of wave

"

Now it may be clear, that non~linear interaction between the other tidal components can give rise to new components with frequencies deviating form the original ones. For instance the interaction between M2 and S2 (think on spring tide and neap tide) yields MS4, a quarterly diurnal tide with a frequency,

consisting of the combination of the frequencies of M2 and S2.

The interaction between M2 and N2 yields MN4, and so on. The nomenclature of these components indicates how its frequency

is composed, for instance w2MS2

=

2wM2 - wS2• All these tidal

components, created by non-linear effects in shallow water are called shallow water tides. Table 2 g~ves a survey of the most

important ones. Of course the table does not contain a column

with values for their astronomical amplitude; their origin is completely different. Therefore it is difficult to predict which shallow water tides are really important. Experience, however, has learned that at least M4' M6' MS' MS4, and MN4 have to be considered in an analysis of restraint size.

At last it should be mentioned that meteorological tides also exist.

For example monsoons, i.e. winds blowing in opposite directions during succeeding periods of half a year, can give rise to

considerable changes of the water level. The resulting changes ~n the water level can be interpreted as a meteorological tide with a

period of a year. Tides of this and similar nature should not be

overlooked in an analysis.

2.3. Tidal prediction

Name ~ _J k 1 m n same fre-quencie as NO} 0 0 0 0 SOl 3 -2 0 0 0 OQ2 2 -3 0 3 0 0 2MS2 2 -2 2 0 0 0 _~2 OP2 2 0 -} 0 0 MKS2 2 0 2 0 0 0 2MN2 2 0 -} 0 0 L2 MSN2 2 3 -2 -} 0 0 _1;2 28M2 2 4 -4 0 0 0 M03 3 -} 0 0 0 0 S03 3 -2 0 0 0 MK3 3 0 0 0 0 SK3 3 3 -2 0 0 0 MN4 4 -} ₀ 0 0 M4 4 0 0 0 0 0 SN4 4 -2 0 0 MS4 4 2 -2 0 0 0 MK4 4 2 0 0 0 0 S4 4 4 -4 0 0 0 SK4 4 4 -2 0 0 0 2MN6 6 -} _{0 0 0} M6 6 0 0 0 0 0 MSN6 6 -2 0 0 2MS6 6 2 -2 0 0 0 2MK6 6 2 0 0 0 0 2SM6 6 4 -4 0 0 0 MSK6 6 4 -2 0 0 0 3MNa

s

-] 0 0 0 Ma a 0 0 0 0 0 2MSNa a -2 0 0 3MSa a 2 -2 0 0 0 2(MS)a a 4 -4 0 0 0 2MSKa a 4 -2 0 0 0

forms the base of the prediction of the tide all over the world: n h h +

L

f H cos (w t + V + u ) o nn n n n ~n which h h 0 H n w n t V + u n n V n f n' un

tide (horizontal or vertical)

mean value (level or flow)

astronomical amplitude, depending on the tidal component angular speed

GMT (Greenwich Mean Time)

astronomical argument, ~n which

uniform changing part, since each day begins with t

°

correction for position of the nodes.

For alocation L degrees west of Greenwich the equilibrium

tide with t ~n GMT is: n

h h +

L

f H cos (w t + V + u - pL)

o nn n n n

~n which p

=

0, 1, 2, .•. (species number), depending on the

nature of the tidal component: p 0, for a long-period tide,

p

=

1 for a diurnal tide, and so on.

In genera1 this location uses another time as GMT.

Assuming the location to lie in a time zone, where it ~s S hours earlier than ~n Greenwich, the equilibrium tide becomes:

h

n

h +

L

f H cos (w t + V + u

o nn n n n pL +

w

nS)

After this derivation the step to the real tide will be made. Now

it will be assumed that hand H , the mean value and the amplitude

o n

respectively are quantities belonging to the real tide. The phase of the arbitrary tidal component, however, will deviate from that one of the equilibrium tide. So:

n

h h +

2

f H cos(w t + V + u - pL + w S - K )

o n n n n n n n

1n which K

=

kappa-number (USA:epoch) n

Mostly this 1S written as:

n

h h +

I

f H cos

(w

t + V + y - g )

o n n n n n n

in which g

n corrected kappa-number

=

pL - w Sn + Kn

Hand gare called tidal constants. They have to be determined

n n

from observations.

For completeness it should be noted that f and u are considered

n n

constant during a calendar year. They are derived from

astronomical analysis. This also holds for f , despite the fact n

that its precise effect on the tide in general is unknown. Experience has proved that this way of doing is acceptable.

3. Tidal analysis

3.1. Introduction

The general express~on for the horizontal or vertical tide as function of time ~s:

het)

=

n

h +

I

f H cos(w t + V + u - g )

o nn n n n n O.I.)

This formula indicates, that the tidal signal is composed of many sinusoidal functions, each with its own amplitude, angular speed and phase (at t

=

0). A measurement of het) will contain all these components and if the measurement provides enough data, in principle all sinusoidal functions can be computed from the measured signal. The real unknowns in expression (3.1.) are the tidal constants

Hand g . The other quantities are known from astronomical analysis;

n n

both the node factor f and the astronomical argument V and u are

n n n

derived from the equilibrium tide.

A measured signal has a beginning and an end. For the analysis of the signal a time base with a fixed time origin has to be defined. In

the computations each instant of measurement should be referred to this time base. Consequently this time base differs from the usual one, ~n that the usual daily phase shift of each component does not occur. If, for instance, the time origin is located at midnight of a certain day, the tidal signal can be written as:

het)

n

h +

I

h cos(w t - a )

o n n n

(3.2.)

~n which hand a are considered as the new unknowns. Dnce

n n

determined the tidal constants follow immediately from:

H n h n

f

n a + V + u n n n

(3.3.)

b >-u C <IJ ;::,

cr

<IJ ... o ._

~ f

~o ~

o

---:,3 d

o

o

o CD Ó

ó

0

apruudure

...

d

..

A result of an analysis for the amplitude H is given ~n fig. 7.

n

It gives the amplitude of the most important diurnal, semi-diurnal

and shallow water tides for Hook of Holland in the Netherlands. Apart from long period components which are not drawn in the figure one can distinguish,a coarse and a fine structure. The coarse structure is related to the grouping of components around frequencies of once a day, twice a day etc. In each group (species) a number of components can be distinguished. The differences in frequency within a group are small too very small in comparison with the differences ~n frequency between the various groups. The small differences ~n

frequencies within a group are equal to mult_iplesof basic frequencies,

such as wand w .

m s

If the measured signal would be undisturbed theoretically a small number of observations would be sufficient to determine amplitudes and phases of the tidal components. Then for n tidal components 2n measurements suffice. Nature, however, does not supply undisturbed measurements. In the case of the tide the observations are disturbed by meteorologic effects, oscillations ~n basins (both in small as in large basins) , and the like. In order to eliminate the disturbing effects much more observations have to be taken as follows from the number of unknowns. Nowadays computer methods based on the method of the least squares and/or Fourier analysis are applied to 'filter out' the tidal components. Only the method of the least squares will be treated in the context of this series of lectures.

3.2. Method of the least squares

In principle tidal analysis aims at the determination of constants, amplitudes and phases, such that they are as plausible as possible. Basically the behaviour of the functions, here sinusoidal, is known. In general this problem ~s classified under the theory of the most likely estimates and .in particular under the regression analysis. In other words it is a problem of the best fit. The method to determine the parameters of the function ~s the method of the least squares.

Suppose g(t) is the measured tidal signal during the time

interval (tl' t2) and het) 1S the basic function, but it has a

number of parameters a, b, c, .•., which have to be determined

from the observations. The measured signal g(t) is not equal to the function het); the difference (the error) is:

€(t) het) - g(t)

The method of the least squares requires that the integrated squared error

=1

't2 2

J

t2 [het) - g(t)] 2 dt

F(a, b, c,) E (t)dt (3.4.)

tI tI

1S minimal. F does not depend on t, but it does on a, b, c, ••••

A

necessary condition for F to reach this minimum is, that the

partial derivatives of F with respect to the unknown parameters a, b, c, ••• are equal to zero:

aF

aa

aF

0, ab 0,

aF

ac

0, ..•.•.• (3.5.)

This procedure will yield as many equations as unknowns, so that the latter can be solved.

The method will be demonstrated on a case in which two sinusoidal

functions are considered.

Of

course this is not sufficient for a real

tidal analysis, but the extension to more components is very easy.

The example is based on computer .treatment of the data and computational

procedure.

To make the procedure more elegantly, the following new unknowns are introduced:

so, het)

The parameters ~n this function het) are AI' A2, BI and B2; to determine them measurements are taken at the time instants:

t , t + t, t + 2t:;t, •••.., t + it:;t ,•.•.•, t + kt:;t.

o 0 0 0 0

The arbitrary instant of time t + it:;twill be denoted by t. ;

0 i.

the corresponding measured value by g(t.). The number of

~

measurements is k + I » 4 (= number of unknown parameters).

The integrated error F becomes according ta equation(3.4.), rep'lacing the integration by a summation:

or

k

F

I

[Alcosw1ti + B)sinw)ti + A2cos w2ti + B2sin w2ti - g(ti)] 2 t:;t. i=O

Differentiating F with respect to AI' BI' A2 and B2, respectively, and equating the derivatives to zero, yields:

k

[A) cos

I

w)ti + B)sin w)ti + A2cos w2ti + B2sin w2_{ti - g(ti)]} cos w1ti 0

i=O k

[

]

L

"

"

"

"

"

s~n w1ti 0 i=O k

[

I

"

"

"

"

"

]

cos w2ti 0 i=O k

L

[

"

"

"

"

"

_]

cos w2ti 0 i=O

Af ter some elaboration:

k k k

A}

L

cos w}ti cos w}ti + B}

L

sin w}ti cos w}ti + A2

L

cos w2ti cos w}ti +

i=O i=O i=O

k

+ B2

L

S1n w2ti cos w}ti i=O

k

L

g(t.)cos w}t.,

i=O 1 1

k k k

A

L

cosw}ti sin w + B}

L

S1n w}ti sin w}ti + A2

L

cos w2ti sin w}ti +

} i=O }ti i=O i=O

k + B2

L

S1n w}ti S1n w}ti = i=O k

L

g(t.)sin w}ti, i=O 1 k k k

A}

L

cos w}ti cos w2ti + B}

L

S1n w}ti cos w2ti + A2

L

cos w2t. cos w2t.

i=O i=O i=O 1 1 +

k

B2

L

sin w2ti cos w2ti i=O

k

L

g(t.) cos w2t.,

i=O 1 1

k k k

A}

L

cos w}ti sin w2ti + B}

L

S1n w}ti sin

"z

ti +~A2

L

cos w2ti S1n w2ti +

i=O i=O i=O

k k

B2

_L

S1n w2ti sin w2ti

_L

g(t.) sin w2ti•

i=O i=O 1

Remark, that all summations in the last equations can be computedx• This holds both for the left and the right members. In other words: a system of 4 linear equations with 4 unknown has arisen. In matrix form this can be represented as:

a}} a}2 a}3 al4 Al b}

a21 a22 a23 a24 _B} b2

=

a3} a32 a33 a34 A2 b3

a4} a42 a43 a44 B2 B4

x

To save computational work the summations in the left members of the equations can be replaced by simple products. Since this is not very essential in the context of this subject, no attention will be paid to this aspect.

The unknowns Al' BI' A2 and B2 can be solved by inverting the matrix,for example. If Al' BI' A2 and B2 have been found, hl'

al and h2, a2 can be determined as weIl, and so the tidal constants. The foregoing procedure can be easily extended to much more than two tidal constituents. In fact only the matrix to be inverted gets a bigger size and therefore computational time on the computer drastically increases.

Nowadays the method of the least squares is commonly used for tidal analysis. A great advantage of the method is, that gaps in a

registration (drop out of the instrument of ten occurs in practice)

are not disastrous. Of course the computational procedure has to be changed sliehtly, but basically the methad remains the same.

Once the constituents have been determined the residual, the difference between registration and theoretical signal, has to be examined whether it contains still some tidal contribution of components, which were not chosen beforehand. It is not very easy to find out then, which components have been overlooked and therefore same times aspectral analysis is made from the residual or from the original registration in order to determine the most important constituents. The theory of spectral analysis

(generalized Fourier analysis) is rather complicated and will be treated in another course. lts theory provides information on sampling interval (needed for computer computations) and duration of the measurement. Both aspects are of fundamental importance for the analysis of tidal signais. This is the reason that some additional attent ion will be paid to these problems, however, without treating the relevant theory.

3.3. Sample interval

Elaborating data on a digital computer or taking measurements while using a digital read-out implies that one cannot work with the entire signal.

For those purposes the signal has to be digitized , meaning that only discrete points, the samples, are used for further processing. The time interval between the points, usually being constant, is called sampling interval.

h(t)

i

sampling

i

nterval

..:

-t

•

=

values of tidal curve with wich computations are made.

digitization of a tidal curve

It is evident, that not each arbitrary choice will satisfy. ~his can be demonstrated, for instance, in the case of one tidal component,

being a sinusoidal function of time with period T. If the sample interval would be equal to the period T, the digitization will only yield samples all having the same value. In this way the computation of the mean level will be disturbed and it is impossible to compute the actual amplitude and phase of this sinusoidal function. It is plausible that a smaller Sa~pling interval has to be chosen and it can be proved, that a sinusoidal function should be sampled at least twice during one oscillation in order to obtain a proper result 1n an analysis. For instance, when a tidal component as M8, period about 3 hours, can be expected to occur the sampling interval should be less than

I!

hour. As the occurrence of a component as M8 is quite common

1n general a sampling interval of one (1) hour 1S taken 1n tidal analysis.

In this connection the following remark. Quite often tidal registrations are carried out in harbour bas ins and the like 1n which natura1

oscillations (seiches) can develop,having nothing to do with the actual tide. Seiches in general have periods varying between 1-30 minutes, (much) less than the periods of tidal components. Despite the fact that one may be not interested in the seiches the sampling interval should be adjusted to the period of the occuring natural oscillation. This can yield much more information as really necessary for tidal analysis. A common procedure

then is to 'smooth' the registration in order to filter out the higher

frequencies and to go on with hourly values of the smoothed tidal registration.

The preceeding remark also applies to measurements at locations where short waves (periods 1-20 s) play a role. A good instrument should damp out these kind of oscillations. Otherwise a proper analysis of the signal is hardly possible.

3.4. Duration of a tide measurement

As shown 1n chapter 2 tidal components exist, having angular speeds (periods) very close to each other. Consider,for instanc~ the angular speed of the principal semi-diurnal tides M2 and S2' having angular speeds of about 29 o/h and 30 o/h,respectively. Considering their joint behaviour during a day yields a sinusoidal function with

slightly varying amplitude. This can be demonstrated with the following case. Add two sinusoidal functions, same amplitude, but with slightly different angular speeds:

h(t)

with w1

=

~W and w₂

=

(n-l)~w with n large. This function is periodic

'h 'd 2w , h' h h ' h f '

W1t per10 ~w' 1n w 1C t e component W1t requency w1 carr1es out n oscillations and the other one n+l oscillations.

Composing the two sinusoidal function yields:

h(t ) 2 cos -2-~wt S1n w..;2'( ~w) t

1n which the S1ne represents the fast oscillating and the cosine the slowly oscillating part. The latter can be conceived of as the varying amplitude of the sine. It var1es so slowly that the amplitude

is more or less constant during a day. Taking account of natural disturbances it is clear that a single day measurement would not give sufficient information to separate the individual components.

A sharp criterion for the duration of a measurement does not exist. Generally, however, the so-called criterion of Rayleigh is accepted

as an indication for separation. This criterion says, that two components, with angular speeds w1 and w2' respectively, can beseparated, if the

observation period P satisfies the following condition:

P> 2w (with w 1n radians/s)

This condition implies, that the number of oscillations of the components should differ at least one in the measurement ?eriod, in order to be able to separate them.

Applying this criterion to a diurnal and a semi-diurnal

component the result is just one day. In other words the combined

diurnal effects can be easily separated from the semi-diurnal ones.

It is clear that components within a group will yield much longer

observation periods.

Next table givesa survey of the required observation periods to

separate diurnal tides. Such a period is sometimes called a

synodic periode PI Kl 01 Q1 PI Kl 01 x 14,8 13,7 9,6 9,1 27,6 period 1n days x 182,6 x

The corresponding survey for some semi-diurnal components follows.

S2 K2 112 U2 L2 2MS S2 x 182,6 14,8 9,6 31,7 7,4 K2 x 13,7 9,1 27,1 7.1 period M2 x 27,6 27,6 14,8 in days N2 x 13,8 9,6 L2 x 9,6

Remark, that a m1n1mum period of 15 days 1S required to separate the

very important tidal constituents M2 and S2. Many other components have

synodic periods approximately equal to 29/n (days) (n

=

1, 2, 3, 4) •

Some old hand methods (e.g. the Admiralty Method) used this property,

so that with a minimum effort a reasonable number of tidal components

could be detected in a very elegant way. Nowadays this period of

29 days is more or less accepted as a standard observation period

1n behalf of a minimal tidal analysis. There is no need, however, to use this period as an untouchable law.

It follows from the tables, that components as S2, K2 and KI, PI cannot be separated in a month analysis, since at

least half a year is required. There is a method, using results from astronomical analysis, to separate them, but it will not be treated here. Nevertheless a long observation is still required if accurate determination is necessary. In this respect a year analysis involves a period of 355 or 369 days. These periods are obtained in the same way as the preceeding period of 29 days.

3.5.

Closure remark

For quite a long time tidal analysis and to alesser extend tidal prediction has_ been considered as a highly specialized job, which therefore should be left to specialized institutes. The ma1n reason for this was the rather complex treatment of a larger number of data, giving rise to many laborious procedures in the pre-computer area Several very elegant hand methods were developed to reduce the

computational work, but the methods themselves required much experience. Nowadays the things have changed. It is much easier to handle large quantities of data and to elaborate them using a computer. There is much less need to support fully on a specialized institute. One point, however, should be clear. All procedures should be carried out with utmost care. This holds both for measurements as for calculations. So, the measuring instrument must have a stable position, it should nnt

rise or sink during long periods of measurements. Furthermore measurements shall be taken at constant time intervals; if the sample int~rval is one hour it should not vary between 50 and 70 minutes. The OCCurrence of other wave phenomena can completely spoil the measurements, and so on, and so on. In the framework of this short course it is not possible to mention all pitfalls. Doing this kind of work all steps required have to be considered very carefully.

4

.

Tidal computations

4.1. Introduction

This second part of the course a~ms at an introduction into long wave phenomena, in general, and into tidal wave propagation, in particular. It will provide some insight into basic wave behaviour and some basic principles of the behaviour of one-dimensional tidal waves. It is not intended to treat the subject such that

afterwards each arbitrary problem can be solved. So most considerations will be based on the use of analytical methods, which generally

illustrate the 'principles in a better way than numerical methods do. For actual cases, however, numerical methods, which are not treated

in this course, are adequate. Then, an analytical computation may serve for control purposes.

The reason why tidal computations are necessary when civil works are carried out in tidal regions is quite evident considering for il!~tanc'~, some typical oases, such as

a) construction of a dam in a river or estuary or ~n a network of river branches

dam

b) new channels, connecting a lagoon or r~ver with the sea

short-cut

c) reclamation of land ~n an estuary

More examples can be mentioned, but all of them illustrate that the tidal regime can be influenced by civil engineering works like this. Although the change of the water level may be of prime

interest, the change of the velocities is of ten much more important. Increase of the water level, namely will have consequences for the height of the dikes, but increase of the velocities can give rise

to big erosion, which asks for stabilizing measures. Contrary,

decreasing velocities, can give rise to shoalings, which necesitate continuous dredging. In many actual cases it is not possible to predict the consequences of the change in the tidal regime without making proper computations.

This course will comprise an introduction into basic one-dimensional wave behaviour as can be derived from the equations for nearly horizontal flow (ratio depth/wave length is smali). In particular attention will be paid to harmonic waves, such as the single tidal wave, the standing wave, the tidal wave propagation in upstream direction of a river. The most important part deals with the so-called Lorentz-method for tidal computations. In simple cases this semi-analytical method can be used. It will also be shown, how this method can be used when the cross-section of the channel changes in length direction, or when there is run-off discharge (fresh water discharge).

4.2. The equations

As it can be stated that the vertical velocities are very small in tidal flows, the equations holding for nearly horizontal flow (unsteady

flow) will satisfy. The considerations will be confined to one spatial dimension and therefore one spatial coordinate will be introduced. The equations can be derived formally from the complete Navier-Stokes equations, describing three-dimensional flow. This can be very instructive, but for simplicity a more

heuristic approach will be followed. In other words, the equations, namely the equation of continuity and the equation of motion,

will be derived directly, after introducing the assumptions:

i constant density (specific mass) of the water. As in general fresh water flows to the salt sea, the density may vary from

1000 to 1040 (kg/m3). This variation means, that the influence of the density on the behaviour of the water level is relatively small, so that a constant density can be taken in that kind.of computations. This also holds for stratified systèms, but in that case also internal wave phenomena can occur, which have to be described with more equations.

~~ the water is supposed to be incompressible. As a free surface flow is considered with disturbances with a celerity much less than the sound celerity as will be shown later on this

assumption is justified.

~~~ there is a hydrostatic pressure distribution. As the flow is supposed to be nearly horizontal and therefore has parallel streamlines the pressure is proportional to the depth.

iv width of the channel is relatively small, so that the water level in cross-direction can be assumed to be horizontal. To g~ve some idea the width should be less than approximately 10 km (order of magnitude). The reason of this limitation is the influence of the rotation of the earth, which gives rise to effects, which are not taken into account in the equations. The measure of 10 km should be conceived of as a rough value, as the effect of the rotation of the earth can be easily introduced for channels with widths of that order of magnitude. If the channel gets much wider a two-dimensional treatment is inevitable.

To derive the equations an x-axis ~s introduced being parallel to the bottom of the channel. So it has the same slope as the bottom. At first sight this may not seem very practicle since the slope of the bottom varies and thus also the x-axis. Later on it will be shown how this problem can be avoided. The water level ~s referred to the chosen position of the x-axis. Therefore it is not necessarily the water depth.

The cross-section may be irregular, but for conven~ence it will be assumed to be uniform, i.e. it does not depend on x. This may be true for small sections of tidal channels, but ~n general this simplification is not at all justified in practical situations. The resulting equations, however, also hold for diverging or converging channels.

The following property will be used in the derivations:

dA =bdn _(4.1)

~n which A b

wet cross-section

width of the channel at the water level (storage width)

n water level

It means, that a change of the water level dn will yield a change of the cross-section equal to bdn '.Since A and nare functions of x and t:

aA

ax b~ax

aA

at b~at

and

The equation of continuity will be derived considering a fixed control volume and the principle of conservation of mass will be applied .

Q ___ .._Q+ ÖQdx 8x

_X-Qxis

The control volume is formed by two vertical planes at a distance dx, the bed and bottom of the channel and the water level at a certain instant. As the water level changes the water particles leave the upper boundary of the control volume with a velocity an/at • A side-flow is not present. Considering the flow during a time increment dt: inflow - outflow = 0 aQ

ar,

Qdt - (Q + ax)dx - b

at

dxdt 0 ~+ b~ ax at

o

in which

Q

discharge (4.2)

The equatio~_<?__~Il_1<?tioncan be derived using the same control volume and applying the principle of conservation of momentum. In integral from this law 1S:

F amu

at

+

1n which m mass 1n the control volume

u velocity (in x-direction)

ul velocity perpendicular at the boundary

of the control volume

fi

surface integral

F forces act i.ngon the mass a.nthe control volume

The first term of the right member represents the local change

of momentum, the second one the momentum exchange. Applying above

formula the equation of motion can be derived in a straightforward way.

Another method is, the direct application of Newton's law:

force mass x acceleration

on the water particle.

Naturally both methods, being strongly related, yield the same result.

Here the latter method will be followed. First the acceleration of a

water particle will be considered. lts velocity is u(x1t) and its

acceleration is given by

Du Dt

dx

t'

"I

~n which uI and u2 are the veloeities of the partiele on

positions land 2 in the x,t-plance, respectively. The general change of the velocity is given by the totale derivative:

du aU dt + au dx

at ax

h . 1 h b f 11 d . l' dx

T e part~c e as to e 0 owe, ~mp y~ng dt = the acceleration is:

U, and consequently Du Dt = aU aû + u-at ax

Going on with the forces, they will be considered per unit mass. The following contributions can be distinguished:

~ from the slope of the bed: gsin I ~ gI (small slope)

~n which g acceleration due to gravity

x

I slope of the bed

ii from the slope of the water level: _g

an

_ax

This result can be derived by considering again an element dx per unit width and determining the residual force acting on the boundaries of the element

residual force force left - force ri8ht

I 2

:.1 pgn !pg(n+i!ldx)2 dX __{dn_ dn}

~ - pgn dX (= - g dX per unit mass)

This result does not depend on the actual depth, or on the width of the channel. Each water particle in the cross-section has therefore the same force from the slope of the water level.

LLL friction from the bed: - g ~ C2R

The proportion attributed to friction effects is always a difficult problem in tidal calculations,because it cannot be directly measured. The contribution above is derived from uniform flow, where Chezy's law is applicable:

u C

ViU

or

I

Ln which R hydraulic radius

In the case of uniform flow the friction balances the gravity influence, but this does not hold in tidal flow. Nevertheless it is assumed that a good estimate for the friction can be

made by taking the corresponding uniform flow friction. A justi-fication for this way of doing is the relatively slow oscillation due to the tide. In stead of u2ulul LS used in order to take account of the .ieLterna.ti.ng character of the flow and thus of friction. Otherwise the friction would always act in the same direction.

The usual procedure to determine the coefficient C is by making some test calculations. But sometimes C varies during the tidal cycle and then it is no more than a sort of mean estimate.

All forces per unit mass together determine the final acceleration. Thus the result for the equation of motion is:

au au ên

at + u dX + g ax (4.3)

The equations (4.2) and (4.3) also hold for channels with non-uniform cross-section. So the channel may diverge or converge,

the equations remain the same.

For all clarity, equations (4.2) and (4.3) hold for an x-axis parallel to the bottom. In principle this slope 1S a function of x, which is not very convenient in a computation. Therefore, normally a horizontal x-axis is used. The equations then slightly differ, because a transformation to the new spatial coordinate has to be carried out:

water level new X'-axis

original X-axis

If the new variables are denoted with a prime then the following relationship between new and old variables exists:

n

n

'

+ x!I

Differentation with respect to x yields:

l!:l_'

a

n

êx + I'" ax' + I

And substitution of this result into equation (4.3) yields

au au

anI

equation. This is not very surprising since in the original

equation a horizontal water level would yield a term gan/ax= gI,

which is balanced by gI in the right member. With the horizont al

x'-axis then gan'/ax'=

o

.

In addition, the equation of continuity does not change.

In the following the considerations will be based on the

system with horizontal x-axis. The primes will be omitted.

Executing tidal computations the consistent use of the discharge

Q

in stead of the velocity u appears to be more convenient. In

particular this becomes clear for a network of connected channels

where apart from the level condition a continuity condition for

the discharges must hold at the node points.

1 2 at node point n} n2 1 2 Q} + Q2 0 ~ at node point n} n2 n3 Q} + Q₂ + Q₃ 0

Transformation of equation (4.4) into an equation with

Q

as

dependent variable required some elaboration. So the various terms

in equation (4.4) become: au at

~g

arA

_!_~A at _!_ aQ A at

bQ

an A2

at

au ax

~g_

ax A _!_~ A êx Q aA ~2 ax _!_ aQ

A

ax

bQ

an A2

at

Substitution of these results into equation (4.4) yields:

Replacing ~~ again by ~~ using the continuity equations yields the desired result:

(4.5)

This equation of motion can be solved together with the equation of continuity (4.2.), provided sufficient boundary and initial conditions are given.

4.3. Some basic solutions 4.3.1. Simplified system

The system of equations (4.2) and (4.3) contains four non-linear terms, viz.:

which prevent analytical solution of the system. The second and the third term are not very important, but the first and the last cannot

be disregarded ~n tidal calculations. It is evident, that the third term can be left out if:

Above condition, small Froude-number, generally holds under practical

2

conditions (take, for example u

=

1 mis,

A/b

=

10 mand g

=

10 mis,

2

then bu IgA

=

0.01).

The smallness of the second term with respect to the.other one is not that evident. Later on it will be shown that in case of small Froude numbers it can be 'disregarded as weIl.