The Form factor F = [ K1 + O1 ] / [ M2 + S2 ] is customarily used to characterize the tide. When 0.25 < F < 1.25 the tide is mixed - mainly semidiurnal When 1.25 < F < 3.00 the tide is mixed - mainly diurnal F > 3 the tide is diurnal F < 0.25 the tide is semidiurnal

F > 3 the tide is diurnal F < 0.25 the tide is semidiurnal When 1.25 < F < 3.00 the tide is mixed - mainly diurnal When 0.25 < F < 1.25 the tide is mixed - mainly semidiurnal Superposition of constituents generates modulation - e.g. fortnightly, monthly This applies for both sea level and velocity

Subtidal modulation by two tidal constituents

In Ponce de León Inlet: M2 = 0. 41 m; N2 = 0. 09 m; O1: 0. 06 m; S2: 0 In Ponce de León Inlet: M2 = 0.41 m; N2 = 0.09 m; O1: 0.06 m; S2: 0.06 m; K1= 0.08 m F = [K1 + 01] / [S2 + M2 ] = 0.30 GNV

Panama City In Panama City, FL: M2 = 0.085 m; N2 = 0.017 m; O1: 0.442 m; S2: 0.035 m; K1= 0.461 m F = [K1 + 01] / [S2 + M2 ] = 7.52

The momentum balance is then: Progressive wave Assume linear, frictionless motion in the x direction only, under homogeneous conditions. The momentum balance is then: linear, partial differential equation (hyperbolic) And the continuity equation is: The solution is d’Alembert’s solution, which can be studied with the sinusoidal wave form:: a time

This indicates that the flow is in phase with the elevation time This indicates that the flow is in phase with the elevation a C/H time

Standing wave The momentum balance is also: And the continuity equation is: The solution is:

This indicates that the flow is out of time This indicates that the flow is out of phase with the elevation by 90 degrees a C/H time

Resonance L At the mouth x = L, Substituting into at x = L For resonance to exist, the denominator should tend to zero, i.e., and The natural period of oscillation is then:

H (m) L (km) C (m/s) TN (h) Long Island Sound 20 180 14 Chesapeake Bay 10 250 28 Bay of Fundy 70 26 10.7 Mode 1 (n =1) Merion’s Formula

, u , u , u , , , u u u (kLw)-1 r/ Perillo (2012)

Tidal Waves With Friction Momentum balance for a progressive wave: Integrating vertically: The bottom stress becomes: Expanding this in a Fourier cosine series, to lowest order: With continuity: These are the governing equations for progressive tidal motion with friction.

If we let and We obtain the solution: where maximum U precedes maximum eta

Effects of Friction on a Standing Tidal Wave in an Estuary For resonance, we have again U =0 at the head of the estuary, i.e.,

Winant (2007, JPO)

Effects of Rotation on a Progressive Tidal Wave in a Semi-enclosed basin Solution: R = C / f KELVIN WAVE

Effects of Rotation on a Standing Tidal Wave in an Estuary Two Kelvin waves of equal amplitude progressing in opposite directions. Instead of having lines of no motion, we are now reduced to a central region -- amphidromic region-- of no motion at the origin. The interference of two geostrophically controlled simple harmonic waves produces a change from a linear standing wave to a rotary wave.

Effects of Bottom Friction on an amphydromic system Parker (1990)

Virtual Amphidromes Parker (1990)

Virtual amphidromes in Chesapeake Bay Fisher (1986)

A EFFECTS OF CHANNELS ON TIDAL FLOWS B H (y) Euler’s formula x y Huijts, et al.(2006, J. Geophys. Res., 111, C12016, doi:10.1029/2006JC003615)

Example of bathymetry effects on tidal flows in the James River

(Looking into the estuary) Strongest tidal flows in channels relative to shoals (Looking into the estuary)

(Looking into the estuary) Tidal flows in channels lag behind tidal flows over shoals Surface flows lag behind bottom flows (Looking into the estuary)

Stronger tidal flows in channels than over shoals Tidal currents lead over shoals relative to channels