L10b - Tides

Tides: A Quick Introduction

Introduction to Tides

• Tides are primarily caused by the gradient of the Moon's gravitational field across the Earth's diameter.

• The fundamental equation governing gravitational force is: $F = \frac{GMm}{d^2}$, where:

  • $F$ is the gravitational force.

  • $G$ is the gravitational constant.

  • $M$ and $m$ are the masses of the two bodies (e.g., Earth and Moon).

  • $d$ is the distance between the centers of the two bodies.

Tidal Forces and Their Effects

• The tidal force on a parcel is about $10^{-7}$ times gravity.

  • Example: A 70kg man would effectively be about 8mg lighter when the moon is directly overhead.

• The tractive force, which is the component of the tidal force parallel to the Earth’s surface, is what causes water parcels to move.

• Tidal forces lead to both diurnal and semidiurnal tides.

• Semidiurnal tides occur because each point on the Earth's surface rotates beneath two ocean bulges each day.

• Declination causes the two tides in a day to have different amplitudes.

Lunar and Solar Influences

• The moon passes overhead at a given longitude every 24 hours and 50 minutes. Therefore, high waters are approximately 12 hours and 25 minutes apart.

• The Sun's influence on tides is considerable, despite its distance.

  • The sun is 27 million times heavier than the moon but is also 389 times further away.

• The solar tide is approximately 0.46 times the lunar tide.

Spring and Neap Tides

• Solar and lunar tides combine during full and new moon phases, creating spring tides.

• During half moon phases, solar and lunar tides oppose each other, resulting in neap tides.

• The spring-neap cycle occurs approximately every 15 days.

Tidal Range and Geographical Variability

• The equilibrium tide on a land-free Earth would be about 0.4m.

• Land masses interfere with tidal patterns, causing different bodies of water to respond differently.

• Examples of extreme tidal ranges:

  • Leaf Basin, Ungava Bay

  • Cobequid Bay, Bay of Fundy

Tidal Prediction and Harmonic Analysis

• Tidal motions can be modeled as a combination of sine waves with various periods (e.g., 12 hours, 24 hours, a fortnight, a month).

• Harmonic analysis of observations at a specific location can deduce the amplitude and phase of each tidal constituent.

• These constituents are used to forecast future tides.

• Weather conditions (atmospheric pressure, wind) can also affect tide height.

Lunar Recession and Energy Dissipation

• Lunar recession: The moon is moving away from the Earth at a rate of 38mm/year.

• Energy loss due to tides: 3,200 GW (2,500 GW from the M2 tide).

• Increase in the length of day: 1 second in 40,000 years.

• Reference: Dickie et al. (1994)

Tidal Power

• Tidal barrages (e.g., La Rance, average 68MW).

• Tidal stream turbines (potential application in places like Portland).

• Reference: Blunden and Bahaj (2006)

Deep Ocean Tides and Ocean Mixing

• Deep ocean tides, through internal tide generation, can help mix the ocean.

• This mixing affects ocean circulation and climate.

• Example: Hawaii Ocean Mixing Experiment