Tidal Prediction Tidal Prediction

November 3November 3

Equilibrium TheoryEquilibrium Theory Predicts periodicities, but not actual Predicts periodicities, but not actual

movement of tidesmovement of tides

Dynamic TheoryDynamic Theory Modifies equation theory to take into account: Modifies equation theory to take into account:

i) irregular shape and varying depth of oceansi) irregular shape and varying depth of oceans

ii) Coriolis modifies water motion (rotation of ii) Coriolis modifies water motion (rotation of the earth)the earth)

iii) inertia of water motionsiii) inertia of water motions Developed by Laplace – considers wave propagation of tides rather than “bulges”Developed by Laplace – considers wave propagation of tides rather than “bulges”

Tidal wave (not to be confused with Tidal wave (not to be confused with tsunamis or storm surges) travels at tsunamis or storm surges) travels at shallow water gravity wave speedshallow water gravity wave speed

Velocity of the wave is governed by the Velocity of the wave is governed by the depth hdepth h

ghC p

Tidal forces set up standing waves:Tidal forces set up standing waves: Consider a rectangular basin:Consider a rectangular basin:

W E

l

node

Tidal wave reflects from walls - Incoming waves interfere with reflected waves to produce standing wave

http://www.kettering.edu/~drussell/Demos/superposition/superposition.html

Period of oscillation:Period of oscillation:

Natural period: time for a wave to go across Natural period: time for a wave to go across basin and backbasin and back

Waves are not free waves – tidal forces Waves are not free waves – tidal forces continually act on fluid in complicated way, continually act on fluid in complicated way, always varying in direction and magnitudealways varying in direction and magnitude

Forced waves must respond at forcing Forced waves must respond at forcing frequencyfrequency

- example: pendulum analogy- example: pendulum analogy

gh

lTn

2

Give the pendulum a single push, the Give the pendulum a single push, the pendulum swings at a natural period pendulum swings at a natural period determined by the lengthdetermined by the length

If you keep pushing, you can make it If you keep pushing, you can make it swing at any period you wishswing at any period you wish

If you happen to push at the natural If you happen to push at the natural period in a way that supports natural period in a way that supports natural oscillation, the amplitude of the oscillation, the amplitude of the swing increases swing increases

Known as resonance conditionKnown as resonance condition

In the same way, if TIn the same way, if Tnn is close to the is close to the period of the tide generating force, period of the tide generating force, then you get resonance and the then you get resonance and the amplitude of the standing wave amplitude of the standing wave increasesincreases

→ → Basin geometry determines which Basin geometry determines which tide generating forces and tide generating forces and periodicities are most effective in periodicities are most effective in generating tidesgenerating tides

Bay of Fundy – extreme example of how bay shape Bay of Fundy – extreme example of how bay shape augments tideaugments tide

Amplification is due to combination of Amplification is due to combination of resonance and convergence resonance and convergence

- Narrowing bay “wedges” water together - Narrowing bay “wedges” water together → increases height of tide→ increases height of tide

If length of bay (in the direction of tide If length of bay (in the direction of tide advance) and depth are just right, can set advance) and depth are just right, can set up standing oscillation with tidal period – up standing oscillation with tidal period – ResonanceResonance

Natural Period of bay very close to Semi-Natural Period of bay very close to Semi-diurnal tidal perioddiurnal tidal period

At ocean basin scale, Earth’s rotation – Coriolis force – deflects tidal currents to the right in the N. hemisphere

Affects standing wave pattern: - water moving to the west veers to the north, piling water up in the north side of the basin - water moving east veers to the south and piles water up there

Wave moves in a counter-clockwise direction around nodal point, instead of sloshing about nodal line – Kelvin Wave

Nodal point is called Amphidromic PointNodal point is called Amphidromic Point

- from Greek- from Greek

Amphi = aroundAmphi = around dromas = runningdromas = running Wave moves as Kelvin WaveWave moves as Kelvin Wave

N

E

S

WH at t1

H at t2H at t3

This is for a flat-bottom, square Ocean. The real ocean is much more complicated.

Co-Tide lines (red): High or low tide occurs at same timeCo-Range lines (blue): Tidal range the same at all points

Amphidromic points: Intersection of Co-Tide lines, zero tide range

In mid-ocean, tide range is small In mid-ocean, tide range is small ~50 cm~50 cm

In shallow water, amplitude In shallow water, amplitude increases, particularly in gulfs and increases, particularly in gulfs and embayments along coastembayments along coast

Tide classification by spring tidal Tide classification by spring tidal range – range – - Microtidal – less than 2 m- Microtidal – less than 2 m

- Mesotidal – 2-4 m- Mesotidal – 2-4 m

- Macrotidal - > 4 m- Macrotidal - > 4 m

Remember, tide travels at:Remember, tide travels at:

In some very long, narrow estuaries and In some very long, narrow estuaries and rivers, the velocity of water in tidal rivers, the velocity of water in tidal currents becomes larger than Ccurrents becomes larger than Cpp

- causes tidal wave to steepen and break, - causes tidal wave to steepen and break, just like gravity waves on the beach, just like gravity waves on the beach, creating a tidal bore traveling up the rivercreating a tidal bore traveling up the river

- largest bores in China (7.5 m) and on the - largest bores in China (7.5 m) and on the Amazon (5 m)Amazon (5 m)

ghC p

Measuring Tides

180 Primary Water Level Gauges Nationally

(http://tidesandcurrents.noaa.gov/publications/tidal_datums_and_their_applications.pdf)

Tidal PredictionTidal Prediction If tides in the ocean were in equilibrium with the tidal potential, If tides in the ocean were in equilibrium with the tidal potential,

tidal prediction would be much easier tidal prediction would be much easier

Tidal Prediction for Ports and Shallow WaterTidal Prediction for Ports and Shallow Water - Two methods are used to predict future tides at a tide-gauge - Two methods are used to predict future tides at a tide-gauge station using past observations of sea level measured at the gauge:station using past observations of sea level measured at the gauge:(1) The Harmonic Method:(1) The Harmonic Method:

- traditional method, and it is still widely used. - traditional method, and it is still widely used. - uses decades of tidal observations from a coastal tide - uses decades of tidal observations from a coastal tide

gauge from gauge from which the amplitude and phase of each tidal which the amplitude and phase of each tidal constituent (the tidal constituent (the tidal harmonics) in the tide-gage record are harmonics) in the tide-gage record are calculated calculated

(2) The Response Method:(2) The Response Method:- method developed by Munk and Cartwright (1966), - method developed by Munk and Cartwright (1966),

calculates the calculates the relationship between the observed tide at relationship between the observed tide at some point and the tidal some point and the tidal potential. potential.

- The relationship is the spectral admittance between the - The relationship is the spectral admittance between the major tidal major tidal constituents and the tidal potential at each station. constituents and the tidal potential at each station. The admittance The admittance is assumed to be a slowly varying function of is assumed to be a slowly varying function of frequency so that the frequency so that the admittance of the major constituents admittance of the major constituents can be used for determining the can be used for determining the response at nearby frequencies. response at nearby frequencies. Future tides are calculated by Future tides are calculated by multiplying the tidal potential multiplying the tidal potential by the admittance function.by the admittance function.

Tidal ConstituentsTidal ConstituentsConstituents Amplitude (m) Phase (°) Period (hr)

M2 0.175 197 12.4206

S2 0.057 211.7 12.0000

N2 0.03 191.3 12.6583

K1 0.167 49.9 23.9345

O1 0.155 37.7 25.8193

Q1 0.029 26.2 26.8684

P1 0.049 57.6 24.0659

K2 0.025 215 11.9672

Constituent – One of the harmonic elements in a mathematical expression for the tide-producingForce in corresponding formulas for the tide or tidal current. Each constituent represents a Periodic change or variation in the relative positions of the Earth, Moon, and SunAmplitude – One-half the range of a constituent tide, may be applied also to the maximum speed of a constituent currentPhase – phase lag, may be expressed in angular measure as 360°Period – Time between two consecutive like phases of the tide or tidal current

Constituents and definitions are from www.tidesandcurrents.noaa.gov

Eight main tidal constituents for Tampa Bay

Harmonic Constituents Name Definitions (first 37 most important)

M2- Principal lunar semidiurnal constituent S2- Principal solar semidiurnal constituent N2- Larger lunar elliptic semidiurnal constituent K1- Lunar diurnal constituent M4- Shallow water overtides of principal lunar constituent O1- Lunar diurnal constituent M6- Shallow water overtides of principal lunar constituent MK3- Shallow water terdiurnal S4- Shallow water overtides of principal solar constituent MN4- Shallow water quarter diurnal constituent NU2- Larger lunar evectional constituent S6- Shallow water overtides of principal solar constituent MU2- Variational constituent 2N2- Lunar elliptical semidiurnal second-order constituent OO1- Lunar diurnalLAM2- Smaller lunar evectional constituent S1- Solar diurnal constituent M1- Smaller lunar elliptic diurnal constituent J1- Smaller lunar elliptic diurnal constituent MM- Lunar monthly constituent

Harmonic Constituents Name Definitions – Continued

SSA- Solar semiannual constituent SA- Solar annual constituent MSF- Lunisolar synodic fortnightly constituent MF- Lunisolar fortnightly constituent RHO- Larger lunar evectional diurnal constituent Q1- Larger lunar elliptic diurnal constituent T2- Larger solar elliptic constituent R2- Smaller solar elliptic constituent 2Q1- Larger elliptic diurnalP1- Solar diurnal constituent 2SM2- Shallow water semidiurnal constituent M3- Lunar terdiurnal constituent L2- Smaller lunar elliptic semidiurnal constituent 2MK3- Shallow water terdiurnal constituent K2- Lunisolar semidiurnal constituent M8- Shallow water eighth diurnal constituent MS4- Shallow water quarter diurnal constituent

Plot showing predicted water level, observed water level, and observed – predicted at the St. Petersburg station from September 26, 2006 – October 24, 2006Plot is from www.tidesandcurrents.noaa.gov

Tidal Prediction for Deep-Water - Prediction of deep-ocean tides is much more difficult than prediction of shallow-water tides because tide gauges were seldom deployed in deep water.

- All this changed with the launch of Topex/Poseidon. The satellite was placed into an orbit especially designed for observing ocean tides (Parke et al., 1987) , and the altimetric system was sufficiently accurate to measure many constituents of the tide. - Data from the satellite have now been used to determine deep-ocean tides with an accuracy of ± 2cm. For most practical purposes, the tides are now known accurately for most of the ocean

- Prediction Using Hydrodynamic Theory: Purely

theoretical calculations of tides are not very accurate, especially because the dissipation of tidal energy is not well known.

Tidal Datums – Reference levels for water level measurements

Computed from Water Level Observations over a 19-year Tidal Epoch

MHHW – Mean Higher High Water – Average of all Higher High Water observations

MHW – Mean High Water – Average of all HW observations

MSL - Mean Sea Level – Average of all hourly Water Level observations

MTL – Mean Tide Level – Average of all HW and LW observations or ½(MHW+MLW)

MLW – Mean Low Water – Average of all LW observations

MLLW – Mean Lower Low Water – Average of all Lower Low Water observations – Reference level for tide gauges and depth measurements – “Chart Datum”

Geodetic Datums – fixed reference system used by surveyors, topo maps, etc.

NGVD29 – National Geodetic Vertical Datum of 1929 - Also known as the Sea-level Datum of 1929

NAVD88 - North American Vertical Datum of 1988

Datums are transferred from location to location by leveling – Geodetic datums carried across continent in this way – NAVD88 from Rimuski, Canada

(http://tidesandcurrents.noaa.gov/publications/tidal_datums_and_their_applications.pdf)

(http://tidesandcurrents.noaa.gov/publications/tidal_datums_and_their_applications.pdf)

(http://tidesandcurrents.noaa.gov/publications/tidal_datums_and_their_applications.pdf)

Heights relative to MLLW

St. Johns River, FL8720218

St. Petersburg, FL8726520

Corpus Christy, TX8775870

MSLMSL 0.752 m 0.366 m 0.282 m

NAVD88NAVD88 0.934 m 0.443 m 0.136 m

NGVD29NGVD29 0.598 m 0.172 m -0.010 m

NAVD88-MSL

0.182 m 0.077 m -0.146 m

Relationship between Tidal and Geodetic Datums varies with location and with Time

(http://tidesandcurrents.noaa.gov/publications/tidal_datums_and_their_applications.pdf)

Tidal Datums determine position of Shoreline – determine private ownership, EEZ, etc.

Storm SurgeStorm Surge Occur when storm winds blowing over shallow, Occur when storm winds blowing over shallow,

continental shelves pile water against the coast, continental shelves pile water against the coast, increasing sea level. increasing sea level.

Several processes are important in storm surge:Several processes are important in storm surge:(1)(1) Ekman transport by winds parallel to the coast Ekman transport by winds parallel to the coast

transports water toward the coast causing a rise in transports water toward the coast causing a rise in sea level. sea level.

(2)(2) Winds blowing toward the coast push water directly Winds blowing toward the coast push water directly toward the coast.toward the coast.

(3)(3) Wave run-up and other wave interactions transport Wave run-up and other wave interactions transport water toward the coast adding to the first two water toward the coast adding to the first two processes. processes.

(4)(4) Edge waves generated by the wind travel along the Edge waves generated by the wind travel along the coast. coast.

(5)(5) The low pressure inside the storm raises sea level by The low pressure inside the storm raises sea level by one centimeter for each millibar decrease in pressure one centimeter for each millibar decrease in pressure through the inverted-barometer effect. through the inverted-barometer effect.

(6)(6) Finally, the storm surge adds to the tides, and high Finally, the storm surge adds to the tides, and high tides can change a relative weak surge into a much tides can change a relative weak surge into a much more dangerous one. more dangerous one.

Winds can overcome Astronomical Tide

Katrina Storm Surge

Waveland, MS

Biloxi, MS

Figure 17.9 in Stewart. Probability (per year) density distribution of vertical height of storm surges in the Netherlands. The distribution function is Rayleigh, and the probability of large surges can be estimated from extrapolating the observed probability of smaller, more common surges. From Wiegel (1964: 113).