Arthur Thomas Doodson

Dr. Arthur Thomas Doodson (1890-1968) spent much of his life developing the analysis of tidal motions mainly in the oceans but also in lakes, and was the first to devise methods for shallow water as in estuaries. Tide height and current tables are of great importance to navigators, but the detailed motions are complex. The thorough analysis he excelled at became the international standard for the study of tides and the production of tables through the method of determination of Harmonic Elements by Least-Square fitting to data observed at each place of interest. That is, by proper association of the astronomical phases, observations made at one time can enable predictions decades away with different astronomical phases.

Doodson Numbers

The usual analysis of a periodic function is in terms of Fourier series, that is, over a period of observation covering a time interval $T$, the behaviour is analysed in terms of sinusoidal cycles having zero, one, two, three, etc. cycles in that period; in other words, a collection of frequencies all being a multiple of a particular fundamental frequency. If for example, measurements are made at $N$ equally-spaced times (thus at times $0$, $h$, $2h$, $3h$, $...$, $(N\; -\; 1)h\; =\; T$) then there are $N$ observations, and the standard analysis provides an amplitude and phase figure for $N/2$ different frequencies of $0$, $T$, $T/2$, $T/3$, etc.

In the case of tidal height (or similarly, tidal current) analysis the situation is more complex. The frequency (or period) and phase of the forcing cycle is known from astronomical observations, and, there is not just one such frequency. Relevant periods are the time of the earth's revolution, the completion of the moon's orbit around the earth, and the earth's orbit around the sun. Notoriously, none of these cycles are convenient multiples of each other. At this stage, a component of the tidal force has the form $Acos(omega\; t\; +\; phi)$ - that is, an amplitude $A$, an angular frequency $omega$, and a phase $phi$ related to the choice of a zero time and the orientation of the astronomical attribute at that zero time. However, because the orbits are not circular, the magnitude of the force varies, and this variation is also modeled as a sinusoidal factor (or cosnusoidal), so that the amplitude is given by $A\; [1\; +\; A\_acos(W\_at\; +\; P\_a)]$ where $A\_a$ represents the size of the variation around the average value of $A$, $W\_a$ the angular speed of this variation and $P\_a$ its phase with regard to the time $t\; =\; 0$.

Because $cos(a)cos(b)\; =\; [cos(a\; +\; b)\; +\; cos(a\; -\; b)]\; /2$, a product of cosine terms can be split into the more convenient addition of two simple cosine terms, but having frequencies that are the sum and difference of the frequencies of the two product terms. Thus, where there was one cosine term whose amplitude varied, there are now three terms, with frequencies $omega$, $omega\; +\; W\_a$, and $omega\; -\; W\_a$. Further, although a variation is well-represented by a cosine curve, it is not exactly represented by a cosine curve and so each spawns further terms that are multiples of its fundamental frequency just as in the simple Fourier analysis with one fundamental frequency where the variation being analysed is not exactly sinusoidal.

A determined analysis, such as Doodson excelled at, generates not just dozens of terms but hundreds (though many are tiny: tidal prediction might be performed with one or two dozen only) and the Doodson Number is a part of organising the collection. A particular component will be described with a name (M2, S2, etc.) and its angular frequency specified in terms of the Doodson Number, which specified what astronomical frequencies have been added and subtracted for that component. Thus, if $f\_1$, $f\_2$, $f\_3$, $f\_4$, $f\_5$, $f\_6$ are the basic astronomical frequencies and a particular component has the frequency $(f\_2\; +\; f\_3\; -\; 3.f\_5)$ then its Doodson Number would be given as 0110-3 meaning $0.f\_1\; +\; 1.f\_2\; +\; 1.f\_3\; +\; 0.f\_4\; -\; 3.f\_5\; +\; 0.f\_6$. To avoid the typographical inconvenience of negative signs, the digit string might be presented with five added to each component so that fanciful example would be presented as 566525, except that the first digit may not have five added.

Precise usage depends on the precise choice of the component frequency definitions, whether or not five is added (if not, the string might be called an Indicative Doodson Number), and also, as some forces vary only slowly with time, a calculation once a month (say) might suffice so certain components might not be separated into additive terms following that variation.

Example

This is adapted from a script for the MatLab system, and its main merit is that it actually does generate a suitable curve. In more general work, times and phases are usually referenced to GMT, and the prediction would be annotated with actual dates and times.

%Speed in degrees per hour for various Earth-Moon-Sun astronomical attributes, as given in Tides, Surges and Mean Sea-Level, D.T. Pugh. clear EMS; EMS.T = +360/24; %+15 w0: Nominal day, ignoring the variation followed via the Equation of Time. %T + h - s =+14.492054485 w1: is the advance of the Moon's longitude, referenced to the Earth's zero longitude. EMS.s = +360/(27.3217)/24; %+0.5490141536 w2: Moon around the earth in 27.3217 mean solar days. EMS.h = +360/(365.2422)/24; %+0.0410686388 w3: Earth orbits the sun in a tropical year of 365.24219879 days, not the 365.2425 in 365 + y/4 - y/100 + y/400. Nor with - y/4000. EMS.p = +360/(365.25* 8.85)/24; %+0.0046404 w4: Precession of the moon's perigee, once in 8.85 Julian years: apsides. EMS.N = -360/(365.25*18.61)/24; %-0.00220676 w5: Precession of the plane of the moon's orbit, once in 18.61 Julian years: negative, so recession. EMS.pp= +360/(365.25*20942)/24; %+0.000001961 w6: Precession of the perihelion, once in 20942 Julian years. %T + h = 15.041068639°/h is the rotation of the Earth with respect to the fixed stars, as both are in the same sense. % Reference Angular Speed Degrees/hour Period in Days. Astronomical Values. %Sidereal day Distant star ws = w0 + w3 = w1 + w2 15.041 0.9973 %Mean solar day Solar transit of meridian w0 = w1 + w2 - w3 15 1 %Mean lunar day Lunar transit of meridian w1 14.4921 1.0350 %Month Draconic Lunar ascending node w2 + w5 .5468 27.4320 %Month Sidereal Distant star w2 .5490 27.3217 27d07h43m11.6s 27.32166204 %Month Anomalistic Lunar Perigee (apsides) w2 - w4 .5444 27.5546 %Month Synodic Lunar phase w2 - w3 = w0 - w1 .5079 29.5307 27d12h44m02.8s 29.53058796 %Year Tropical Solar ascending node w3 .0410686 365.2422 365d05h48m45s 365.24218967 at 2000AD. 365.24219879 at 1900AD. %Year Sidereal Distant star .0410670 365.2564 365d06h09m09s 365.256363051 at 2000AD. %Year Anomalistic Solar perigee (apsides) w3 - w6 .0410667 365.2596 365d06h13m52s 365.259635864 at 2000AD. %Year nominal Calendar 365 or 366 %Year Julian 365.25 %Year Gregorian 365.2425 % Obtaining definite values is tricky: years of 365, 365.25, 365.2425 or what days? These parameters also change with time.

clear Tide; % w1 w2 w3 w4 w5 w6 Tide.Name{1} = 'M2'; Tide.Doodson{ 1} = [+2 0 0 0 0 0] ; Tide.Title{1} = 'Principal lunar, semidiurnal'; Tide.Name{2} = 'S2'; Tide.Doodson{ 2} = [+2 +2 -2 0 0 0] ; Tide.Title{2} = 'Principal solar, semidiurnal'; Tide.Name{3} = 'N2'; Tide.Doodson{ 3} = [+2 -1 0 +1 0 0] ; Tide.Title{3} = 'Principal lunar elliptic, semidiurnal'; Tide.Name{4} = 'L2'; Tide.Doodson{ 4} = [+2 +1 0 -1 0 0] ; Tide.Title{4} = 'Lunar semi-diurnal: with N2 for varying speed around the ellipse'; Tide.Name{5} = 'K2'; Tide.Doodson{ 5} = [+2 +2 -1 0 0 0] ; Tide.Title{5} = 'Sun-Moon angle, semidiurnal'; Tide.Name{6} = 'K1'; Tide.Doodson{ 6} = [+1 +1 0 0 0 0] ; Tide.Title{6} = 'Sun-Moon angle, diurnal'; Tide.Name{7} = 'O1'; Tide.Doodson{ 7} = [+1 -1 0 0 0 0] ; Tide.Title{7} = 'Principal lunar declinational'; Tide.Name{8} = 'Sa'; Tide.Doodson{ 8} = [ 0 0 +1 0 0 0] ; Tide.Title{8} = 'Solar, annual'; Tide.Name{9} = 'nu2'; Tide.Doodson{ 9} = [+2 -1 +2 -1 0 0] ; Tide.Title{9} = 'Lunar evectional constituent: pear-shapedness due to the sun'; Tide.Name{10} = 'Mm'; Tide.Doodson{10} = [ 0 +1 0 -1 0 0] ; Tide.Title{10} = 'Lunar evectional constituent: pear-shapedness due to the sun'; Tide.Name{11} = 'P1'; Tide.Doodson{11} = [+1 +1 -2 0 0 0] ; Tide.Title{11} = 'Principal solar declination'; Tide.Constituents = 11; %Because w0 + w3 = w1 + w2, the basis set {w0,...,w6} is not independent. Usage of w0 (or EMS.T) can be eliminated. %For further pleasure w2 - w6 correspond to other's usage of w1 - w5.

%Collect the basic angular speeds into an array as per A.T. Doodson's organisation. The classic Greek letter omega is represented as w. %w(0) = EMS.T; %This should be w(0), but MatLab doesn't allow this! w(1) = EMS.T + EMS.h - EMS.s; w(2) = EMS.s; w(3) = EMS.h; w(4) = EMS.p; w(5) = EMS.N; w(6) = EMS.pp;

%Prepare the basis frequencies, of sums and differences. Doodson's published coefficients typically have 5 added %so that no negative signs will disrupt the layout: the scheme here does not have the offset. disp('Name °/hour Hours Days'); for i = 1:Tide.Constituents Tide.Speed(i) = sum(Tide.Doodson{i}.*w); %Sum terms such as DoodsonNumber(j)*w(j) for j = 1:6. disp( [int2str(i),' ',Tide.Name{i},' ',num2str(Tide.Speed(i)),' ',num2str(360/Tide.Speed(i)),' ',num2str(15/Tide.Speed(i)),' ',Tide.Title{i}] ); end;

clear place; %The amplitude H and phase for each constituent are determined from the tidal record by least-squares %fitting to the observations of the amplitudes of the astronomical terms with expected frequencies and phases. %The number of constituents needed for accurate prediction varies from place to place. %In making up the tide tables for Long Island Sound, the National Oceanic and Atmospheric Administration %uses 23 constituents. The eleven whose amplitude is greater than .1 foot are: Place(1).Name = 'Bridgeport, Cn'; %Counting time in hours from midnight starting Sunday 1'st September 1991. % M2 S2 N2 L2 K2 K1 O1 Sa nu2 Mm P1... Place(1).A = [ 3.185 0.538 0.696 0.277 0.144 0.295 0.212 0.192 0.159 0.108 0.102] ; %Tidal heights (feet) Place(1).P = [-127.24 -343.66 263.60 -4.72 -2.55 142.02 505.93 301.5 45.70 86.82 340.11] ; %Phase (degrees). %The values for these coefficients are taken from http://www.math.sunysb.edu/~tony/tides/harmonic.html %which originally came from a table published by the US. National Oceanic and Atmospheric Administration.

%Calculate a tidal height curve, in terms of hours since the start time. PlaceCount = 1; Colour=cellstr(strvcat('g','r','b','c','m','y','k')); %A collection. clear y; step = 0.125; LastHour = 720; %8760 hours in a year. n = LastHour/step + 1; y(1:n,1:PlaceCount) = 0; t = (0:step:LastHour)/24; for it = 1:PlaceCount i = 0; for h = 0:step:LastHour i = i + 1; y(i,it) = sum(Place(it).A.*cosd(Tide.Speed*h + Place(it).P)); %Sum terms A(j)*cos(speed(j)*h + p(j)) for j = 1:Tide.Constituents. end; %Should use cos(ix) = 2*cos( [i - 1] *x)*cos(x) - cos( [i - 2] *x), but, for clarity... end;

figure(1); clf; hold on; title('Tidal Height'); xlabel('Days'); for it = 1:PlaceCount plot(t,y(1:n,it),Colour{it}); end; legend(Place(1:PlaceCount).Name,'Location','NorthWest');

Results

This shows the common pattern of two tidal peaks in a day, though remember that the repeat time is not exactly twelve hours but 12.4206 hours. The two peaks are not equal: the twin tidal bulges beneath the moon and on the far side of the earth are aligned with the moon. Bridgeport is north of the equator, so when the moon is north of the equator also and shining upon Bridgeport, Bridgeport is closer to its maximum effect than approximately twelve hours later when Bridgeport is on the far side of the earth from the moon and the high tide bulge at Bridgeport's longitude has its maximum south of the equator. Thus the two high tides a day alternate in maximum heights: lower high (just under three feet), higher high (just over three feet), and again. Likewise for the low tides.

This shows the spring tide/ neap tide cycle in the amplitudes of the tides as the moon orbits the earth from being in line (sun - earth - moon, or sun - moon - earth) when the two main influences combine to give the spring tides, to when the two forces are opposing each other as when the angle moon - earth - sun is close to ninety degrees producing the neap tides. Note also as the moon moves around its orbit it also changes from north of the equator to south of the equator. The alternation in the heights of the high tides becomes smaller, until they are the same (the moon is above the equator), then redevelops but with the other polarity, waxing to a maximum difference and then waning again.

This shows just over a year's worth of tidal height calculations. The sun also cycles between being north or south of the equator and as well the earth - sun and earth - moon distances change on their own cycles. None of the various cycle periods are commensurate.

"Remember always" that calculated tidal heights take no account of weather effects, nor include any changes to conditions since the coefficients were determined, such as movement of sandbanks or dredging, etc.