Major Tidal Constituents

Tidal Constituents

Constituent

Doodson Number

Frequency (cpd)

Description

sa

056.554

0.00273791

Solar annual

ssa

057.555

0.00547582

Solar semiannual

mm

065.455

0.03629165

Lunar monthly

msf

073.555

0.06772638

Lunisolar synodic fortnightly

mf

075.555

0.07320220

Lunar declinational fortnightly

mt

085.455

0.10949385

Termensual

2q1

125.755

0.85695241

Smaller elliptical diurnal

σ1

127.555

0.86180932

Lunar variational diurnal

q1

135.655

0.89324406

Larger lunar elliptical diurnal

ρ1

137.455

0.89810097

Larger lunar evectional diurnal

o1

145.555

0.92953571

Lunar diurnal

τ1

147.555

0.93501153

m1

155.555

0.96613681

Smaller lunar elliptical diurnal

χ1

157.455

0.97130317

Smaller evectional diurnal

π1

162.556

0.99452418

Solar elliptical diurnal

p1

163.555

0.99726209

Principal solar diurnal

s1

164.555

1.00000000

Raditional solar diurnal

k1

165.555

1.00273791

Principal declinational diurnal

ᴪ1

166.554

1.00547582

Smaller solar elliptical diurnal

φ1

167.555

1.00821373

Second-order solar diurnal

θ1

173.655

1.03417265

Evectional diurnal

j1

175.455

1.03902956

Smaller lunar elliptical diurnal

oo1

185.555

1.07594011

Second-order lunar diurnal

ϵ2

227.655

1.82825558

2n2

235.755

1.85969032

Second-order lunar elliptical semidiurnal

μ2

237.555

1.86454723

Lunar variational

n2

245.655

1.89598197

Larger lunar elliptical semidiurnal

ν2

247.455

1.90083888

Larger lunar evectional semidiurnal

m2

255.555

1.93227362

Principal lunar semidiurnal

λ2

263.655

1.96370835

Smaller lunar evectional

l2

265.455

1.96856526

Smaller lunar elliptical semidiurnal

t2

272.556

1.99726209

Larger solar elliptical semidiurnal

s2

273.555

2.00000000

Principal solar semidiurnal

r2

274.554

2.00273791

Smaller solar elliptical semidiurnal

k2

275.555

2.00547582

Lunisolar declinational semidiurnal

η2

285.455

2.04176747

m3

355.555

2.89841042

Principal lunar terdiurnal

From Cartwright and Edden [11], Cartwright and Tayler [12], Doodson and Lamb [17]

Compound Constituents

Two or more constituents can interact harmonically in shallow-water to form overtides or compound constituents. The properties of these compound constituents can be derived from the properties of their parent constituents.

Constituent Notations

Every tidal constituent corresponds to a specific combination of astronomical cycles [see Arguments], and several notation systems exist for encoding that combination compactly.

pyTMD supports three (interchangeable) formalisms:

  • Cartwright numbers: stores the multipliers as signed integers in an ordered list [12].

  • Doodson numbers: compact decimal representation designed for human-readable identification of constituents [17].

  • Extended Doodson numbers (XDO): compact and human-readable representation used by the UK Hydrographic Office (UKHO)

Cartwright Numbers

Cartwright numbers are an ordered list of signed integers for the multipliers of the astronomical arguments [see Equation 1.2]:

\[\text{Cartwright numbers} = [d_1, d_2, d_3, d_4, d_5, d_6]\]

  • \(d_1\): multiples of the spherical harmonic dependence (\(\tau\))

  • \(d_2\): multiples of the mean longitude of the Moon (\(S\))

  • \(d_3\): multiples of the mean longitude of the Sun (\(H\))

  • \(d_4\): multiples of the mean longitude of the perigee of the Moon (\(P\))

  • \(d_5\): multiples of the mean longitude of the node of the Moon (\(N\))

  • \(d_6\): multiples of the mean longitude of the perigee of the Sun (\(Ps\))

Doodson Numbers

Doodson numbers are an unsigned notion where the second through sixth multipliers (\(d_{2-6}\)) are encoded by adding 5 (the first digit \(d_1\) is not offset). This offset maps the range of multipliers \([-5, +4]\) into \([0, 9]\). This encoding can be extended by mapping \(+5\) to \(\mathrm{X}\), \(+6\) to \(\mathrm{E}\), and \(+7\) to \(\mathrm{T}\) (which would map to \(10\), \(11\) and \(12\) in the standard notation).

\[\text{Doodson number} = \{d_1\}\{d_2+5\}\{d_3+5\}.\{d_4+5\}\{d_5+5\}\{d_6+5\}\]

Extended Doodson Numbers

The UKHO Extended Doodson Number (XDO) system was designed to address two limitations of the standard Doodson notation:

  1. Range: the +5 digit offset only covers multipliers in \([-5, +4]\). Tidal catalogs larger than the original from Doodson and Lamb [17] can have constituents with multipliers outside that range

  2. Disambiguation: the format carries a seventh character encoding the index \(k\) which resolves ambiguities when constituents share the same Doodson number

The XDO system maps \(0\) to \(\mathrm{Z}\), the range \([1,15]\) to \([\mathrm{A},\mathrm{P}]\), and the range \([-8,-1]\) to \([\mathrm{R},\mathrm{Y}]\).

Tidal Classifications

Doodson Number Classification

Classification

Description

Species

shared \(d_1\)

Group

shared \(d_1\) and \(d_2\)

Subgroup

shared \(d_1\), \(d_2\) and \(d_3\)