University:
Boston University
Course:
CAS EE 144 | Introduction to Oceanography
Academic year:

2023
Views:

288

Pages:

21
Author:

cengladaqwt9

Dynamical Theory of Tides • • Equilibrium theory ignored the inertia of the water Laplace formulated the basic form of the governing shallow water equations in 1775 • Tides considered hydrostatically as waves that respond to periodic forcing and thus have the same frequencies as the forcing • Included depth of the ocean, configuration of the ocean basins, frictional forces, and Coriolis (rotation) effects • Result are the conservation of momentum and vertically integrated continuity equation (mass conservation) equations Momentum Φi g ⎡csc γ ⎤ ∂ ⎛ ∂U η ∂ − ∓ 2ω U cos γ = − ⎢ ⎜ i ∂t a ⎣ 1 ⎥⎦ ∂μ ⎝ g ⎞ ⎟ + dissipation ⎠ Continuity ∂ηi csc γ ⎡ ∂ ⎡ Hv sin γ ⎤ ⎤ + ⎢ ⋅⎢ ⎥=0 ⎥ a ⎣ ∂μ ⎣ Hu ⎦ ⎦ ∂t Tidal Potential Tidal Potential • Tides are a measure of changes in gravity, caused by the attraction of the moon and sun. • Tidal potential is the gravitational potential that varies with the position of the Moon and Sun relative to the Earth • Components of the tidal potential – Deformation of the solid earth due to gravitational potential (solid earth tide) – Movement of ocean water due to changing potential (ocean tides) – Deformation of the solid earth due to the changing load of ocean tides (ocean tidal loading) Tides as Harmonic Constituents • Tidal variations can be described as a sum of harmonic terms – any periodic motion can be resolved into a sum of a series of single harmonic motions Harmonic of the tidal potential Ω = A cos(ωt + V ) Dynamic theory predicts the component for constituent, i: η (t ) = ∑ Ai cos (ωi t − φi + Vi ) Location (latitude) dependence i Frequency Amplitude Phase angle of equilibrium tide (phase lag relative to equilibrium tide) Local phase of constituent Tides as Harmonic Constituents Since harmonic decomposition of tides is done on records that are not sufficiently long to separate the 8.85 and 18.6 year tides, add corrections: η (t ) = ηo + ∑ fi Ai cos ⎡⎣ωi t − φi + (Vi + ui ) ⎤⎦ i Mean sea level Nodal factor Nodal angle Harmonic Analysis • Orbital paths are very nearly circular, so sinusoidal variations are suitable for tides. • The tidal patterns are decomposed into many sinusoids having many fundamental frequencies, corresponding to many different combinations of the motions of the Earth, the moon, and the angles that define the shape and location of their orbits. • For tides, harmonic analysis is not limited to harmonics of a single frequency. Harmonics are multiples of many fundamental frequencies. Their representation as a Fourier series having only one fundamental frequency and its (integer) multiples would require many terms, and would be severely limited in the time-range for which it would be valid. The study of tide height by harmonic analysis was begun by Laplace, William Thomson (Lord Kelvin), and George Darwin. A.T. Doodson extended their work, introducing the Doodson Number notation to organize the hundreds of resulting terms. Doodson’s Frequencies Frequency (°/hour) Period Source 1 lunar day Local mean lunar time 0.54901653 1 month Moon's mean longitude f3 0.04106864 1 year Sun's mean longitude f4 >0.00464184 f1 14.49205211 f2 8.847 years Longitude of Moon's perigee f5 -0.00220641 18.613 years Longitude of Moon's ascending node f6 0.00000196 20,940 years Longitude of sun's perigee Doodson (1922) - Fourier Series Expansion using 6 frequencies f = n1 f1 + n2 f 2 + n3 f 3 + n4 f 4 + n5 f 5 + n6 f 6 The Tidal Constituents Tidal Species Name Semidiurnal n1 = 2 n1 n2 n3 n4 n5 Equilibrium Amplitude* (m) Period (hr) Principal lunar M2 2 0 0 0 0 0.242334 12.4206 Principal solar S2 2 2 -2 0 0 0.112841 12.0000 Lunar elliptic N2 2 -1 0 1 0 0.046398 12.6584 Lunisolar K2 2 2 0 0 0 0.030704 11.9673 Diurnal n1 =1 Lunisolar K1 1 1 0 0 0 0.141565 23.9344 Principal lunar O1 1 -1 0 0 0 0.100514 25.8194 Principal solar P1 1 1 -2 0 0 0.046843 24.0659 >Q1 1 -2 0 1 0 0.019256 26.8684 Elliptic lunar Long Period n1 = 0 Fortnightly Mf 0 2 0 0 0 0.041742 327.85 Monthly Mm 0 1 0 -1 0 0.022026 661.31 Semiannual Ssa 0 0 2 0 0 0.019446 4383.05 Nonlinear Tides If a forced frequency gives rise to motions at the forced frequency as well as non-forced frequencies, the tidal response is non-linear. • Nonlinear tides arise from the advective terms resulting in sums and differences between frequencies • Overtides – interactions of a tidal frequency with itself – M2 interacting with itself gives rise to M4, M6, M8, etc. • Compound Tides – interactions between tidal frequencies – M2 interacting with M2 gives rise to MS4, MSf, etc. These tides are know as shallow water tides and are generally smaller than their primary frequencies • Residual Tide – non-oscillatory component of the tide – important for transport, interacts with wind, density, other processes; hard to separate tidal contribution Tidal Current Ellipses • Horizontal currents are two dimensional • Harmonic analysis performed on component vectors • Results for each constituent are combined and reported using ellipse parameters North The size of the current ellipse is proportional to the current magnitude Ma Mi g θ Ma = major axis – max. current velocity Mi = minor axis θ = inclination – angle semi-major axis makes to the East g = phase lag – time of maximum velocity behind max tidal potential Direction CCW rotation CW rotation Coastal Tidal Currents • Reversing current • Flood current or tide – approaching high tide (crest moves into harbor) • Ebb current or tide – approaching low tide (trough moves into harbor) • Slack tide – period of little current at high or low tide • High velocity flow in restricted channels Coastal Tidal Currents • • Peak flood and ebb currents occur in the middle of the tidal cycle ~1/2 of the tidal flow moves within the middle two hours of the tidal cycle For semi-diurnal tide - “rule of 12ths” Hr 1 – 1/12 of volume Hr 2 – 2/12 of volume Hr 3 – 3/12 of volume Hr 4 – 3/12 of volume Hr 5 – 2/12 of volume Hr 6 – 1/12 of volume - slack tide - fastest current - fastest current - slack tide Tidal range is not a good predictor of tidal currents Currents are modified by: • Shape and volume of basin • Restriction of flow at basin mouth • Winds • Friction at the basin boundaries Considerations for Tidal Modeling and Prediction • Allow applied tidal forcing to be applied gradually using a ramp function for a period of 12-15 days to eliminate the excitation of numerical artifacts in a modeled solution • Account for both the tidal potential (body force) and tides entering at the open ocean boundary. • Maintain consistency between model forcing and harmonic analysis - same constituents, same nodal and phase corrections • Select constituents for the analysis that are appropriate to the applied forcing and subsequent nonlinear interactions • Ensure the time series length for harmonic analysis is sufficient to separate the desired constituents • Comparisons between modeled and observed tides should be done by constituent, amp vs. amp (magnitude), phase vs. phase (timing) or by using a complex-valued amplitude Primary Considerations for Accurate Tidal Modeling • • • • • Bathymetric variations Representation of the coastal boundary geometry Specification of tidal constituents at the open ocean boundary Frictional characteristics of the region Model resolution appropriate for scales associated with the nonlinear tidal interactions Project: Modeling Tides in the South Atlantic Bight Finite element model domain and mesh for South Atlantic Bight Follow Blanton et al. 2004, J. Geophs. Res. Provided: Fort.14 – grids with and without estuarine detail Fort.15 – input file Fort.51, 52, 53, 54 – harmonic output files Fort.61, 62 – station time series for elevation and velocity maxele.63, maxvel.63 – max elevation and velocities over the simulation Importance of Including Estuarine/Tidal Inlet Morphology Semi-diurnal Observations Model with estuarine detail Model without estuarine detail Tidal current ellipse model-data comparisons Dotted lines do not include estuary/inlet detail Diurnal Enhanced Tidal Dissipation Semi-diurnal Diurnal No estuary/inlet detail Enhanced tidal dissipation in upper reaches of estuary/inlet complex Semi-diurnal Diurnal Tidal dissipation effects turbulent energy available for mixing and dispersion Project: Modeling Tides in the South Atlantic Bight Suggested activities: 1) 2) 3) 4) 5) 6) 7) 8) 9) Run the ADCIRC model for SAB Plot max tidal water level and velocity variations Plot tidal time series at station locations Apply harmonic analysis to time series data Compare derived and model computed harmonic results Convert tidal velocity amplitudes and phases into tidal ellipse parameters Plot tidal ellipses on bathymetry contours Compute energy flux and tidal dissipation Compare solutions from both grids using quantitative errors Project: Modeling Tides in the Bight of Abaco Finite element model domain and mesh for Bight of Abaco Project: Modeling Tides in the Bight of Abaco Suggested activities: 1) Run the ADCIRC model for the Bight of Abaco for two grid resolutions 2) Evaluate the effect of grid resolution – examine global time series, station time series, harmonically analyzed constituents 3) Examine the interaction between tidal constituents (Cases O-1 to O-6) 4) Examine effects of various nonlinear terms in the model (Cases O-7 to O-11, O-4)

The Importance of Tides 3

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