Tide and Currents

23

TIDE AND

CURRENTS

ALBERTO TOMASIN

Department of Applied Mathematics University of Venice,Italy

I.

Tide Generation

A.

The Tide Generating Forces B. The Development of Tidal Potential C.The Equilibrium Tide

D. The Development by Doodson E. Shallow Water Tides, Types ofTides 11.Hannonic and Non-hannonic Analysis

A.

Non-hannonic Terms

B. Principles of Hannonic Analysis C. Least Square Methods

D.Tide Tables E. Cotidal Charts

lIl. Tidal Currents

A.

Considerations on Fluid Flow B. Current Observations

C.Tidal Currents in Seas and Estuaries IV. Storm Surges and Extreme High Tides

A.

Atmospheric Effects, Wind and Pressure Actions B. Predictability of Storm Surges

C. Extreme High Tides

V. Models and Computing Methods A. Basic Equations

B. Friction and Stresses C. Finite Difference Methods D. Miscellaneous Methods

24

ALBERTO TOMASIN

I. Tide Generation

A.

THE TIDE GENERATING FORCES

The earth, the moon and the sun are the celestial bodies to

he

considered in studying the tides, since the other ones have a negligible influence. Indeed, the tide generating forces exerted on the earth by a celestial body are (obviously) proportional to its mass and (reasonably) to the cubic inverse of its distance from OUTplanet (the derivative of the Newtonian square inverse). This could also easily show why the moon is more important than the sun in this field.

The mechanism to be considered in detail starts from the question of how much a point-Iike body on the earth is affected by the moon (or the sun). Consider the place where the moon is seen on the zenith (the closest one), then the one seen on the nadir (the farthest), then any intermediate one.

Due to the different distance from the moon, .there will be an obvious increase of gravitational attraction when moving from the farthest to the intermediate and then the closest one. Now, one should consider the force exerted by the moon on the center of the earth and remember that it has the special meaning of having a reference value for the terrestrial bodies. Indeed, we would not care of any possible force acting precisely the same way on all of us and on the center of the earth: by the same token, if the cable of an elevator is cut and it falls a1most freely, the objects in it behave as if they were

.weightless, since they have the same acceleration as the cabin. Likewise, what is relevant for the dynamics of the bodies on the earth is the deviation of the forces with respect to the reference one. One sees, by properly drawing the force vectors, that the farthest and the closest point will experience upward forces (thus they feel lighter), while the intermediate one will be almost unaffected. In a first approximation, the two upward forces will be identical for the two extremes (which corresponds to a linear change of the attractive force in passing through the earth).

We can immediately deduce that in two opposite points on the earth everything, and in particular sea water, is slightly lighter, giving origin to two bulges in the ocean, at the expense of the intermediate seas.

But things move: the moon turns around the earth in about a month, and much more quickly the earth has a spin motion. An observer, fixed on the earth close to the sea, will see the swell rising twice a day, when he passes close to the two particular points (since we know very weil that

they

are not the poles of the rotation: this would dismay OUT

argurnents).

The day to be considered is clearly a lunar day (1.035 solar days), as long as we are considering the moon effects, but similar considerations hold for the sun.

TIDE AND CURRENTS 25

B.THE DEVELOPMENT OF THE TIDAL POTENTlAL

The above description is correct

,

but to go into practical evaluation one needs more

detail. So far, it was only stated that due to the moon the observer sees two high- and

two

low-tide

conditions every lunar

day,

and there are also two smaller bulges due to the

sun

.

This is due to earth rotation

,

but the picture is not statie, due to the slightly varying

distances of the two celestial bodies from the earth (elliptical, no circular orbits are

involved) and due to the varying trajectoryof the moon and the sun in the sky on a yearly

time scale (what familiarly is the sun's seasons).

We can consider (as such was indeed the developmentoftidal prediction) adding step by

step the various details of the ce1estialmotions in order to fit better and better the

theoretical description

to the observed world

.

This is closely connected to the

mathematical trick of a series expansion for the tidal forcing or for a potential function

from which the forces are derived.

Out of an infinite set, only a few terms turn out to be

relevant: better details will be given below

.

C.THE EQUILIBRIUM TIDE

Now

,

another important step is required. It was always assumed that an observer

passing by the "c1osest"point (with respect to the moon) sees aflood tide condition in his

area

:

this

-

is true only for an ideal sea which is able to respond instantly to the

gravitational change. If one prefers, the equipotential surfaces can accomplish it, but

whatever is real and material win suffer some delay. Under these conditions, one works

easily with an ideal "equilibrium tide"

.

The separate problem is then considered of the

real world ocean, where a huge wave runs east to west after this attracting potential.

There are many obstacles for this wave

,

i.e. islands, shallow areas, interposedcontinents,

so that the precise time and level of the high tide in a given place becomes something

frightening to be computed, from a theoretical basis

.

The help of observations will be

vital. It is vital also to rernark that the periodicity of the celestial motion is maintained

also in real life

:

for example, the seasonalvariation of the hight of the sun at noon win

certainly be found some way in the tidal records of any place in the world. This win

turn

out to be the main help availablefor practical analysis.

26 ALBERTO TOMASIN

D. THE DEVELOPMENT BY DOODSON

The above idea of a series expansion of the tidal potential, giving origin to terms of decreasing importance, can now be connected to the periodicities in the tides. The work done mainly by Lord Kelvin, Darwin and Doodson shows tbat one can describe things

by

considering the six periodicities of interest:

- 1 day is the period ofthe earth's rotation (relative to the Sun)

- 1 month is the period of the moon's orbital motion - 1 year is tbe period of the sun's orbital motion - about 8.85 years is the period of the lunar perigee

- about 18.61 years is the period ofregression of lunar nodes

-about 20,000 years is the period of solar perigee.

Consider now the frequencies fl ' f2 ,...,f6 related to the six periods above, sinee they are more useful in calculations; for example, one can subtly discover tbat other periodicities (Iike tbe lunar day) do not appear in the above list, as tbey may be deduced from tbe reported ones. This is tbe case of the lunar day, whose related fi:equency fmoon can be expressed as fl -f2

+

f3 (indeed, real expansions are performed in terms of lunar frequencies).

Since the whole range of time scales is covered, one can see that the tidal behaviour is

.fully described by a relatively simple expansion.

Take the six frequencies defined above in a vector, f, and think of a similar vector k built by six integer numbers (ki = 0,

±l,

±2, ... and i=I,2, ...,6). Tben we express the tide in a certain place by

z (t) = l:k Ck cos(21t k·f t - ek) or, if one pref ers,

In other words, we Iinearly mix the six fundarnental frequencies In all possible combinations, and from each of these we have a harmonie term witb tbe resulting frequency: its amplitude C and phase e (obviously dependent on the location coordinates, but not on time) can he theoretically determined to get the equilibrium tide. This way, a reasonable number of components can he selected, with an amplitude large enough to

he

reasonably observed.

TIDE AND CURRENTS

₂₇

It

has already been remarked that this correspondsto taking into account step by step all

the eelestial mot

i

ons

:

i

f we dis

c

o

v

er that a h

arm

on

i

e function

,

a sine

, h

as a slight and

slow perturbation on its amplitude

,

we multiply it by a long period sinusoid

.

But then a

simple transformat

i

on g

iv

e

s, i

n

s

tead of the product

,

the sum of two harmoni

e

terms

,

where the frequenciesare in linear relat

i

onship with the originalones.

This gives origin t

o

the weil known tidal terms like M2 .Kj

,

S2

etc

,

where each one

corresponds to a definite choice of kl ,k2

,

.

.

.,k6

.

These are the basic ideas of the present

handling of the tides

.

E

.

SHALLOWWATERTIDES

,

TYPES OF TIDES

We have seen now a good

t

ooi to investigate the tides, since we have split the

astronomical input into a series of harmonie components

.

Do we really expect

to

find

these components in the real world

,

in spite of the disturbing effects that were

mentioned?

I

ndeed, it is essentially so, each component is found also in the ocean

,

usually lagged with respect to the theoretical one (but one could argue that this statement

is

,

strictly speaking

,

nonsense

)

and with an amplitude sometime reduced, sometimes

enlarged. What will be considerd now is the generationof spurious constituents, a fact to

be expected from the dissipative factors in the tide propagationaround the world

.

The equations that describe this propagation (LTE

,

the Laplace tidal equations are the

starting point

)

will necessarily include frictional terms

,

essentially nonIinear, mainly in

shallow water and in all coastal areas

.

Again, one can say that a monochromatic wave (i.e

.

, a perfect sinusoid) generates other

harmonies whose frequencies are twiee

,

three times, etc

.,

the fundamental one (they are

called overtides in this case). If, as it usually happens

,

a couple of extra terms are

suffieient, then the situation is easily manageable. A similar development is done for

compoundtides, i.e

.

to nonlinear interactions between the importantharmonie terms

.

Many studies have been developed in this field without using the harmonie technique,

from a purely hydraulic, so to say

,

point of view

:

one should remember that the estuaries

,

where the most important harbours in the wo

r

ld are situated, are typical plaees for these

phenomena and for centuries people have

been

concerned with them

.

An

important

remark is

,

for example

,

a very simple one about the wave d

i

storsion

,

sinee, in shallow

water

,

the high tide will progress sliding over a thicker water layer than the low tide, so

that it goes faster and we can guess the kind of distorsion

.

In harmonie development

,

simple considerationscan be made on the phase of a shallow

water term with respe

c

t to the parent component or components: this can be of

28

ALBERTO TOMASIN

substantialhelp if the number of tenns to be considered

is cumbersome.

As a conclusion for tidal considerations,one

is

left with a definite idea of a large variety

of observedtides

in differentparts of the world.

The latitude is certainlyrelevant, but the

morphology of the oceans and the

continents

count more.

We expect a large range for

possible amplitudes,

the possible presence of shallow water terms and even the overall

distinction of places

with predominant diumal or semidiumal tide

.

There are formal

methodsto c1assifythe tide of a certain place according to the latest distinctions.

IL Harmonie and Non-harmonie Analysis

A

.

NON HARMONICTERMS

Beforethe present developmentof the theory of tides,

or more exactly of the techniques

to handle it,

other methods were developed,

now called non-harmonie.

They follow

essentially the same step-by-step refinement in adequating observations to celestial

mechanics.

Much more interesting than describing a duplication of the present methods,we should

only remember the glossary of tbe non-harmonie approach, connected to empirical

aspectsand still used to give a picture of local tides.

One can start by from HWI,

the time interval hetween tbe moon's transit and the next

high water; next will be MHWI, the mean value of all HWI during 29 days. In the same

field is HWF&C,

high water full and change,

also called Establishmentof the Port,

i.e.

the HWI on the days of full and new moon.

Aremark can he made on the spring tide,

i.e.

the particularly strong

tidal effect related,

but usually not coinciding,

with full and new moon,

due to the relative alignment of the

sun,

the earth and the moon.

The "phase age"

is tbe

interval between the time of full or

new moon and the time of spring tide, which turns out to he different for variousplaces.

Other symbols like MHWS

(mean bigh water in spring tide)

and similar ones are even

more obvious.

.

B.

PRINCIPLESOF HARMONICANALYSIS

The developmentof the tidal potential (or of

the equilibrium tide)

was

introducedabove

to give a useful tooI to investigate something theoretical: it would be quite Iimited if it

had no return

in the practical use.

lndeed, if one thinks of the amplitude and phase of

each component as something to

he

deduced from observation,

he concludes that from

TIDE AND CURRENTS

29

t

heory onl

y

fe

w

bu

t i

mp

o

rtant hints ha

v

e been

k

ep

t,

namely the frequencie

s

and the

crite

ri

a to

c

hoose areas

ona

bie numbe

r of co

mpo

n

ent

s

t

o

be in

v

est

ig

ated

.

The pra

c

t

ic

al

user will t

hen ha

v

e th

e

problem to anal

y

se a cert

ai

n amoun

t

of tida

l

o

bse

rv

at

io

n

s

(

u

sua

ll

y

h

ou

r

ly v

a

l

u

es of sea l

e

v

el

i

n a c

ertai

n p

i

a

c

e

,

taken b

y

a record

ing

tide

g

auge

fo

r a

r

e

a

s

o

nabie peri

o

d

)

i

n

or

d

er

to obta

i

n the harm

o

n

ie

constants

(

amp

li

tude

and phase

)

for a

few

c

o

mp

one

nt

s.

How m

an

y?

Say

,

tw

en

t

y i

n the del

i

cate c

as

e

s of

shall

o

w water

in

fluenee

,

but more usually only seven

;

e

v

en four would he better than

noth

i

ng. An add

i

ti

o

nal d

etai

l

:

onl

y t

he fir

s

t th

r

ee frequencies are actually used to d

e

fine

what is called a const

i

tuent

(

M2 for e

x

ample

,

or Kj

)

. In this structure

,

the three long

term period

i

ci

ti

e

s

d

i

sappear and are kept

i

nto account in another way. The previous

decompositionformula becomes,

i

n p

r

actice

:

z(t

)

=

j C cos(wt

-

9 + v

)

where the summat

i

on is extended to the frequencies that are really taken into account:

the new factor j and the phase correct

i

on v are slowly varying (as an effect of the

igno

r

ed long term period

i

cities

)

and are tabulated for practical purposes

.

There are many practical recipes to est

i

mate the harmonie constants

,

depending on the

length of the observat

i

onperiod and on the availability of computingtools.

Generally speaking, one thinks of the classical methods (e.g. the Admiralty scheme or

the Tidal Institute Method

)

as numerical filters that acting on the available time series

make evident, in the different steps, one or another component.

Frequently a smart

principle is used

, i.

e

.

the fact that neighboring frequenc

i

es cannot he distorted quite

differently in passing from grav

i

tationalpotential to real tide. The world ocean and its

obstacles will presumably have a different action, since they are a selective filter

,

on a

diurnal or a semidiumal oscillation

;

but if two components have very similar frequencies

and theoretical amplitudes, say, in the rat

i

o 1

:

2 one expects that the practical realizations

of the two sinusoids

i

n the ocean will have an almost identical relation. This can he a

powerful help in calculations.

C. LEAST SQUARE METHODS

An approach to estimate the harmonie constants of a given place from the recorded

values of the sea le

v

el can be the least squares method

:

the best fit is sought between an

ideal summation ofharmonic terms and the observed values

(

at an hourly rate)

.

Clearly this method should be preferred in the case of irregular data

,

where a frequent

lack of information makes it impossible to rely on predetermined schemes. Should the

recording he uninterrupted

,

then the least squares could he equally satisfactory

,

but many

30

ALBERTO TOMASIN

alternative methods could have particular advantages, for example for short records. lt is just the case to mention that an expression like

C cos(wt- 9 ) can be wri tten as

c

cos

e

cos wt

+

C sin 9 sin wt or

x

cos wt

+

Y sin wt.

This way X and Y are Iinear unknowns, rnuch better to deal with in the least squares method. At the end, C and 9 are easily deduced.

The next refinement will allow the long term variation of the harmonie constants to be taken into account (the

j

and v quantities of the preceding section). In tbe Jatter expression we make evident the unknown part (X and Y), whilst cos wt and sin wt are something known. The algorithm is only slightly heavier if one considers, instead of cos wt

,j

cos(wt-v ), with

j

and v known from tables (or put once forever in the computer). The result, thus, is a1ready corrected for the long term variations and hence directly comparabie with other harmonie constants and ready for forecasting purposes.

To a certain extent, the present large availability of good computing tools (which was implicit in the last considerations) reduces the importanee of many approximations that were developed in the past to run the least square algorithms more quickly, so tbat this well-known and multi-purpose method can be used rather straightforward.

D.TIDE TABLES

Tide tables, with the predicted values and the hours of the level extremes, have a1ways been precious for shipping, fishing and other coastal activities.

Usually the annual books give extensive tables for one or few important harbors, with some possibility to interpolate the values for intermediate places. The present availability of computing tools makes things easier, it becomes only a matter of good will to produce the tables for any place where the constants are known.

A severe limitation for the tidal predictions is to consider only the astronomical effects. One can never feel sure, in planning his trip at sea, that a change in weather might cause the sea level to deviate from the astronomical prediction: we will analyze how tbis can happen, but we know very weil from the beginning that a forecast given years in advance (as it is the case for something related to earth and moon movements) will never be given for weather conditions.

TIDE AND CURRENTS 31

One can conclude that tidal tables are valid only for normal atmospheric conditions,

otherwise they show at least the essential trend of the sea level. Surges and all meteorological efTects will develop around it as a perturbation.

E.COTIDAL CHARTS

Another use ofthe tidal constants (usually in the harmonie form, as we know) is related to tidal charts.

The

availability of tidal records is in most cases limited to coastal areas,

but from them (and with the help of more or less complex modeis) one can draw a map of that specitic sea, or ocean or part of it. A map with isolines related to the tide, so it is a tidal map: the main lines are the co-tidal ones, sbowing the points of equal phase, then the co-range ones are given.

Thi

s

introduces a scientific way to understand tbe real world of tides, that we confined before to an essentially empirie analysis. Seeing tbe real tides on a large scale gives us again the possibility of an insight that immediately surprises the newcomer: most maps show a circular dynamics around certain points in the sea tbat are called amphidromic (i.e. run around). All this is due to tbe eartb rotation and the consequent sideways deviation of motion (in terms of currents, tbis will

he

discusserl shortly). At an amphidromic point the tidal range for that particular component will be zero and it will increase as long as we go away from tbe point. Delicate experiments are sometimes required to solve ambiguities for tidal pattems in tbe

ocean.

m

Tidal eurrents

A.CONSIDERATIONS ON FLUID FLOW

Among general considerations concerning currents, one sbould never forget tbe deflecting force of tbe eartb's rotation. Indeed, we want to consider tbe motion relative to tbe earth, i.e. not with respect to an absolute coordinate system but relative to a rotating one.

The laws of mecbanics tben rernain valid only if one adds tbe accelerations ax ,ay and

az

to all moving masses, as given by (x pointing to tbe cast,

y

to tbc north) ax=-msin , Uy - 21lcos , Uz

ay=

2llsin cjl

u,

az=2ncos cjl

U

x

where

n

is tbe angular velocity of tbe earth, cjl is tbe geograpbical latitude (negative to the Soutb) and Ux,Uy and Uz are the components oftbe velocity. If, out of these. only the horizontaI ones are considered, one speaks of tbe deflecting force of tbe earth rotation. Ir always acts perpendicular to the motion, towards tbc right in the Northem Hemispbere when one looks in tbc direction of the current. To avoid saying "right" and

32 ALBERTO TOMASIN

"I

eft"

,

the suggestionwas acceptedby oceanographersto speak of "cum sole

"

and

"

contra

solem

"

respe

c

ti

v

ely

,

i

.

e.

in ac

c

o

r

dance

o

r aga

i

nst the apparent motion of the sun in the

sky.Th

i

s term

i

nol

ogy,

describingthe d

i

rect

i

onoftuming, corresponds to the expressions

ant

ic

yclonic and cyclonic

,

us

e

d in mete

o

rology.

By the way

,

there

i

s no more need to

specifythe hemispherewe refer to

.

Under the above influence

,

a water particIe moving with horizontal velocity U and not

subjectto any other force in the horizontal des

c

ribes a

ci

rcular trajectory

"

cum sole" with

radius U

/ (

2U sin

cp

).

Since in order to go along a stra

i

ghttrajectory a partiele should be

subjec

t

t

o

a fo

rce o

pp

o

s

i

n

g

and balan

cing

th

e

d

evi

ati

ng

fo

r

ce

, i

t i

s e

a

sy t

o d

e

duce that all

other forces

(

not exactly calibrated

)

will entrainthe partiele in cycloidal trajectory, a kind

a c

i

rcularmotion plus a drift

:

these are

i

ndeedthe currentsobserved in the large seale

.

B.

CURRENTOBSERVATIONS

Mapping the water motion in the sea is a common task for oceanographers and

engineers.

However, we are faced with serious problems since the motions are

complicated and any sampling turns out to be very expensive. In the past

,

there were

three groupsof methodsfor current measurements:

I) Indirect methods

,

when an estimate of the currents

i

s given following the

hydrodynamiclaws

.

The measured surface inclinat

i

on

,

the pressure d

i

stribution or the

map of other hydrographic properties (salinityand temperature)have been good tools to

deduce the currents

,

mainly in the ocean

.

The present use of numeri

c

modeis

,

also for

small areas, is in the same line

.

2) Drift measurements,tracking the trajectory of by a float.

3) Current measurements at a fixed point. Such measurements have the same

relationship with the previous ones following

,

that the Eulerian point of view has with

the Lagrange one

.

According to the latter

,

the problem is solved when we know the

trajectory of each particle, whilst for Euler one has to know the current direction and

velocitya

t

each point.

To this distinction of methods we should add today the remote sensing, that gives the

possibilityto know the complete(Eulerian) distributionof currents

in

a certain area from

observationstaken from a few points

,

taking advantage of acoustic waves.

Sometimes

one gets only integrated information

(

say, the diseharge in the English Channel) using

electromagneticproperties(like the GEK)

.

The current directionis always g

i

ven by the direction

i

n which

(

o

r

towards which) water

is moving

,

in contrast to meteorology wherethe wind direction is given by the one from

which the wind is blowing.

C

.

TIDAL CURRENTSIN SEAS AND ESTUARlES

TIDE AND CURRENTS

33

lt

is well known that the most important currents in the oceans are not related to tides

:

think of the Gulf Stream and many

si

milar phenomena

.

Tidal currents exist all over but

there are much less measurementsof them than of tidal levels

,

except perhaps for places

of interest for shipping

.

The first

t

h

i

ng one comrnents

i

n t

i

dal current observations is that

the direct

i

on of the current varies

i

n the horizontal plane

,

in addition, of course

,

to

variations in intensity. Thus an extra degree of freedom is added with respect to level

observations

:

It

is obvious that the vert

i

cal component of the current is negligible

:

of course, being a

wave, the tide should show

,

for the motion of a single water particle, an elliptic pattern in

the vertical plane

.

But the special wave we are considering, thousands of kilometers

long

,

g

i

ves origin to a very compressedellipse

.

If we add the variations in the horizontal plane that we mentionedbefore

,

we realize that

(except for channels

,

river mou

t

hs or similar configurations) the current figures

,

as

described by the tip of the current vector, deviate from a straight line

.

Like tides, tidal currents can be analyzed by harmonie components. The simplest way is

to take the two perpendicular projections of the current vector (say, north and east

components)and separatelyrun the harmonie analysis.

A simple analysis shows that the maximum current U in a place should be related to the

tidal amplitudeC (thinking of a single sinusoid) by the relation

U

= (gIh)1I2

C

where h is the depth and g the gravity acceleration

.

This holds in the open sea, where

one assumes the tidal wave to be a progressive one, i.e

.

, having the current maximum

correspondingto a maximum(or minimum) level.

IV

.

Storm Surges and Extreme High Tides

A. ATMOSPHERICEFFECTS

,

WIND AND PRESSURE ACTION

The wind blowing on the sea has an obvious effect on it

,

since it pushes it by shearing

on the surface

,

g

iv

ing o

ri

gin to a mot

i

on that propagates down to a certain depth

.

Without entering in deta

i

l

,

that are object of many stud

i

es due to the difficulty in

describing precise

l

y such a fuzzy phenomenon

,

we will better mention the general

problems conceming the wind action

.

34 ALBERTO TOMASIN

Let us consider firstof all the relevanee of the fetch, or the size of the area where the wind actson the sea. Every surface unit adds up for the resulting action. In conneetion with the wind wave studies (which differ from the present ones for the different wavelength of interest) one can think of a different dragging effectiveness of the wind over a smooth sea surface or over weil developed waves. This would force us to use.

dragging coefficients depending on the history, so to say,ofthe surface.

Another consideration is interesting, conceming a kind of steady state in an enclosed sea,a lake or perhaps just a coastal area. Suppose the wind is blowing towards the shore,

then we think of the surface water moving, giving origin to a certain slope. Will this slope, in principle, grow indefinitely as long as the wind keeps blowing? No, it is reasonable to expect that the bottom water will hydrostatically react to the surface slope by flowing in a bottom countercurrent: in this simplified scherne we have circulation maintaining a steady state. This, in fact, is not very far from what happens.

The barometer (or inverse-barometer) effect of the atmosphere on the ocean is weIl known. To a certain extent, an assumption of meteorological uniformity was implicit when we described the tendency of the ocean to match the gravitational surface.

Hydrostatic considerations suggest that corresponding to a low pressure area there will

,be a rise in the water level in order to maintain the same pressure over any ideal horizontal surface in the sea. Needless to say,a meteorological map with lows and highs cannot be stabIe and will evolve, but the favorable condition for our study is that at any time the barometer effect gives the instantaneous modification of the equilibrium surface at which all the sea is aiming.

B.PREDICT ABILITY OF STORM SURGES

The problem arises of how easily and - above all - how effectively can storm surges be predicted. In fact, people are not concerned with the level a certain storm surge will attain. Indeed, the actual level is of concern, the one caused by the astronomical dynamics PLUS the surge. The difference is subtIe. There are many cases reported where storm surges of relatively short duration (say, six hours) occurred with ebb tide,

and since the two facts were of the same order, nothing really happened for what concerns floods,i.e. the real threat. Another surprise: are there only"positive" surges or also "negative" ones? Literature reports that frequently disasterous waves happened to

TIDE AND CURRENTS ₃₅

hit a coastal area immediately after a sudden retreat of the sea, but this was just a phase trick.

A

really negative surge

can

certainly occur, with unpleasant situations for sailors and coastal power plants: both wind and inverted barometer effects can act oppositely to the way we considered (wind from the coast and sudden pressure increase) and give origin to negative surge.

To predict these phenomena requires a good atmospheric knowledge (and forecasting)

and ability to compute the air-sea interaction. The first point concerns the availability of weather data(from the sea:islands, ships, satellites and obviously coastal areas). Suppose now that

the

observational data are available, and suppose they give enough information about

the

atmospheric phenomena of interest as tbey are now. Then a suitable modelling

can

predict

the sea

response a few hours ahead, using the significant time it takes to tbe

sea

to respond to the present conditions.

For a longer term prediction, an eflicient weather forecasting is required, and we all know the international effort in this direction.

The use of observed

and

predicted meteorological data to estimate the effect on the sea

(and hence the possible surges) will be considered below:the models are the main tooi to do it.

C. EXTREME HIGH TIDES

Technically speaking, extreme high tide is equivalent to storm surge: tbe daily tide greatly affected by atmospheric factors. This compels us to consider possible interactions between tides and surges: indeed, it is frequently wrong to estimate the surge by subtracting the predicted astronomical tide from the observed heights. There are more sophisticated methods to perform this separation effectively.

It should be clear, from the above considerations, that this difficulty is due to nonlinear terms in the equations describing the motion: it sounds staggering to attribute a physical fact to a mathematical formulation, but this only derives from our necessity to describe tbings by the tools we have.

Statistical analysis turns out to be significant when applied to the extreme tides as tbey are, i.e., including the astronomical tides (which have no interesting statistical behaviour) and to the differences between the observed values and the table forecast. One

can

see that there isn't any essential differenee if the sample is appropriate.

Obviously what these methods give are statistical results, sinee they concern distributions and probability: they cannot give any prediction for tomorrow or, say, for the storm approaching now. But they give the probability for a certain level to

he

attained

36 ALBERTO TOMASIN

(under exceptional conditions) during a certain time (say, ten years). The methods by

Gumbel, best known for the run-off of rivers, have been adapted (or completely

reformulated) for the present purposes and appear to be very useful in planning dikes and other sea defenses.

v.

Models and Computing Methods

A.

BASIC EQUATIONS

The basic hydrodynamie equations will

be

shown here, without too many comments.

Tbey are adequate to study storm surges but without great difliculty one can use them for

tides, in particular when the propagation is considered in a gulf or in a bay. Tbe

equations can be written as:

dQx/dt =f Qy -

gh

Bz/lix

+

'tsx/p - 'tbx

lp -

gh lipJBx dQyldt = - fQx - gh Bz/liy

+

'tsylp - 'tby

lp -

gh lipJBy

Bz/ot

+ liOx/Bx + liOy/liy

=0

Here z is the water level, h is the local depth,

Q

is the transport, i.e., depth times

speed

(averaged top to bottom). The symbols 't are stresses, eitber acting on the surface (s) or on tbe bottom (b), i.e. due to wind or bottom friction; p is the atmospheric pressure. The symbols g,t,x,y have the standard meaning, whilst fis defined as the Coriolis parameter, originated by the earth rotation.

Who is not familiar witb tbese relations will appreciate the hint that the first two of them (the momentum equations) simply say that tbe acceleration equals tbe total applied force per unit mass. The tbird one is the continuity equation, expressing the conservation of water. Instead, who knows them will remark that the rainfall contribution is omitted, tbat many terms could

be

better developed and that by writing tbings this way we restrain from investigating tbe details in the water column, so tbat this modelling is barotropie and not baroclinic.

It was anticipated before tbat the

direct

tidal motion cao be studied by tbese equations (i.e., in addition to surges or simple propagation). Indeed, one should first of all evaIuate the distribution of gravitational potential (or tbe equilibrium tide): it requires a certain effort, but the available literature helps. Tben one should consider the surface slope (tbe second term in tbe right hand side of tbe momentum equations) and reaIize that it contributes to motion whenever the surface is not horizontaI. But the effect of the

TIDE AND CURRENTS

37

gravitational perturbation consists in changing the horizontal reference: this way (provided one knows the equipotential surface) the tide can be really used as a driving agent for water flow.

B. FRICTION AND STRESSES

Very important for a good calculation, friction and stresses appear to

be

the most elusive terms in the equations: so to say, the "dirty" factors.

Friction is always present whenever water moves, whilst wind is dominant for surge

generation. By the way, one should observe that surge has the conventional meaning of a level perturbation occurring offshore, but as soon as it approaches the coast it will undergo special interactions with the bottom (in particular, when the shore is gently declinating to the sea) and the"wave setup" phenomena.

Conceming stress, one promptly realizes that a variety of conditions could occur; in the open sea, one thinks of a column of water, moving for some reason: it will stress the bottom in the direction of the column motion. Think now of the kind of equilibrium that was figured about the wind: after a certain time, a wind blowing inshore will determine a bottom stress in the opposite direction.

Many cases could

be

explored, to conclude that wind stress is a really difficult

parameter. Yet people working on models find successful results in assuming a simp Ie form for the bottom stress, by adopting in general the above choice of the moving column of water, with the transport

Q,

affected by a bottom friction

~=kIQlQ

The similar doubts conceming the wind stress are even worse (as mentioned above), but again the most generaI way people try to gel rid of these difficulties is to adopt a quadratic relationship (

I Y I

wind speed)

Is =k' IYI Y

Many precise measurements have been done on the stress, mostly in microscale

experiments: it is not that obvious that their use in the modelling is fruitful, whicb suggests substantial differences at different scales.

C. FINITE DIFFERENCE METHODS

The basic equations we saw before can

be

solved eitber analytically or numerica1ly. In the fust cast the basin of interest (the North Sea, a small bay, an estuary ...) is assumed to

38 ALBERTO TOMASIN

have a regular shape, sometimes not very simple but allowing a geometrical description which isadequate to condition the equations, given a proper boundary condition. Also tbe meteorological input needs a "regular" description: in spite of these limitations, interesting conclusions have been obtained in many cases.

Tbe numeri cal approach is by far preferred to the analytical one, due to the case

i

n

adapting its schemes to all possible situations, both for morphology, extemal input and different hypothesis on delicate points like friction and stresses.

Tbere is a variety oftechniques proposed. We are considering here the finite difference method, widely used in the sea in tbe last forty years (and much longer for the atmosphere by meteorologists).

Finite differences are what one uses to replace the differential quantities in our equations. Tbis way the continuity is lost in space and in time and one has to decide roughly how the topography has to be reduced by this quantization. Typical finite difference schemes use a space grid, regular and orthogonal, in the horizontal plane. At each knot the depthis given and, in the calculation, current and level will be estimated at each time step. Tbe grid is intended to help in subsituting spacial derivatives with differences between adjacent points. It would take a long time to enumerate all the refinements tbat have

been

adopted.

They

are necessary to avoid the troubles that a straightforward use of tbe above ideas would induce.

Here we make some simple remarks: tbe grid usually lies in a horizontal plane, but earth is a sphere. It is not difficult to

he

more correct, since there are relatively simple formulae to keep curvature into account, which is important when we cover a rather large area.

The grid itself has usually to be more complicated, with separate points where to estimate different variables.

Tbere is an interesting point conceming the proposed schemes. Suppose we do exactly what was described above, namely: at a certain time we assume to know tbe relevant quantities (level and current) at all points of the grid. Since we presume to know all the space derivatives in a eertam point of the grid through its neighboring points, we are able to estimate the time derivatives, hence also the new quantities at the new time, i.e.after a time step. But at a time step L\t a sudden perturbation originating at a certain point cannot travel more than one grid step (or mesh) L\s. This defines a kind of numeri cal velocity that, in order to describe properly what happens, should never be less than tbe velocity of the real facts at sea. We can expect that if this requirement is not satisfied, OUT simulation will soon be unsuccessful. Since from our basic equations one could easily derive wave equations witb celerity (g h)l/2 , one gets tbe so-called "Courant-Friedrich-Lewy criterion" (or requirement)

(g h)l/2 < As/L\t

TIDE AND CURRENTS 39

(with some numeri cal factor depending on the details of calculation).

Meteorologists had trouble for many years before this criterion was adopted. What it means is that, given a certain grid, one has to reduce the time step At until the above criterion is satisfied: it might mean a much longer computing time than one could hope.

Tbe operating scheme proposed so far is perhaps the most simple because each step has a clear meaning: it is called the explicit scheme. We have just tackled tbe amount of dîfficultiesthat one faces as a toll for simplicity. If,instead of solving step by step, point after point, one writes huge systems of simultaneous equations, it tums out tbat many difficulties can be encompassed and usually much longer time steps can he adopted. This is the implicit metbod, another of the many fruitful approaches that appeared in tbe field.

Another inconvenience of ordinary finite differences is the use of an ortbogonal grid, where, in addition, Ax is the same all over, and so is Ay. Now, the important features of a given area could go exactly the other way: for any possible orientation of the grid, we frequently find significant streamlines in oblique directions and the array-like scheme has obvious difficulties in handlin them. Further, it is usual to have certain parts tbat clearly require a much finer description (and hence smaller meshes) than other very uniform areas. Solutions and improvements have

been

proposed, with finite differences, but we will instead consider substantially different methods.

D.MISCELLANEOUS METHOOS

Tbere are many approaches to our problem, as we know: a good neighbour to finite differences is the one using finite elements: usually a net of triangles, instead of rectangles, covers the map; each triangle has arbitrary size, provided tbat each side is shared only by two adjacent "elements" (the triangles). In each element the relevant variables vary Iinearly: this is the transposition of the corresponding discretization by finite differences, but with a much better ease to fit the topography with its different relevanee for flow. Usually a system of conditions is written (like in the implicit method) and special techniques are available to handle tbis type of equations.

Another essentially different approach will be mentioned now, i.e. the statistical (or

40 _{ALBERTO TOMASIN}

Another essentially different approach will be mentioned now, i.e. the statistical (or

empirical) method

.

A scientist will appreciate the above method since (beyond technical

differences) they enable us to iIIustrate the behaviour of a water body while it follows

certain laws believed to be true. Instead

,

in an executive office people will appreciate

clear, numerical and possibly simple formulae to be used every day even by

unexperiencedclerks.

lt

is frequentlypossible to run the above schemes in such a way

to obtain the "numerical formulae"required for service

,

but another idea is suggested to

the scientistby the present requirements. One could really forget the equations and find

from the past records interestingrelationshipsbetween the variables.

This is a statistical attitude

.

not very different from the way the harmonie tidal constants

have been determined all over

.

For the prediction of storm surges, for exarnple,more or

less simple schemes have been prepared, sometimes with nomograms and special charts

for practical use. Most ofthe work to prepare them has been empirical,with"black box"

methods (a name that clearly implies the refuse to understand things in detail) or

"empirical orthogonal functions" (EOF), very

clever from the mathematical point of

view but certainly based on past experience.

In each case, one has to decide which method is most suitable and adequate to

expectations.

REFERENCES

DEFANT,A.(1960)

.

"PhysicalOceanography

.

"Pergamon, New Vork.

DIETRICH,G.(I967)

.

"General

Oceanography

An

Introduction."

Wiley-Interscience,NewVork.

DOODSON,A.T

.

,

and

WARBURG,H

.

D.(l94l)

.

"Admiralty

Manual

of

Tides."H.M.StationeryOffice, London

.

DRONKERS,J.J.(l964)."Tidal Computation in Rivers and Coastal Waters

.

",North-Holland, Amsterdam

.

MUNK,W

.

H.,

and

CARTWRIGHT,D.E

.

(1966).Tidal

Spectroscopy

and

Prediction

.

PhiI.Trans.oftheRoy

.

Soc

.

of London A 259,533-581.

WELANDER,P.(l961).Numerical Prediction of

Storm Surges.In "Advances in

Geophysics,",VoI.8,319-379.Academic

P

.

,New Vork

.