equilibrium beaches

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equilibrium beaches and logarithmic spirals
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Swash and backwash Longshore drift
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Swash and backwash Longshore drift
Longshore drift depends on prevailing wave angle
of incidence
E.g. S sin f cos f
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Swash and backwash Longshore drift
S sin f cos f
f0 or fp/2 No longshore drift. Static
equilibrium
fk Constant longshore drift. Dynamic
equilibrium
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Dynamic equilibrium beaches

• River of sand influxoutflux for each
  control volume
• Simplest case Straight beach
• Stable equilibrium?

Ninety Mile Beach, New Zealand
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Dynamic equilibrium beaches
Conjecture Under statistically
stationary-in-time conditions, all beaches will
reach equilibrium. Dynamic equilibrium is only
possible with constant-flux boundary conditions
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Logarithmic spirals
r r0ekf
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Logarithmic spirals
Fun property 1 Scale invariant (apart from a
rotation) Jakob Bernoulli, 1692
Ar0ekf r0ekfln A
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Logarithmic spirals
Fun property 2 Finite length (Torricelli,
1645, and Wallis, 1657)
Fun property 3 Circle in the limit
Fun property 4 A coiled-up straight
cone (Christopher Wren, 1669)
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Logarithmic spirals
Fun property 5 Equiangular (Descartes, 1638)
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Breakwaters/groins/headlands
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Breakwaters/groins/headlands
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Complications -(

• Real logarithmic beaches are often seemingly
• in static equilibrium, with waves parallel to
  shore
• Refraction
• Position of center of spiral
• Boundary conditions

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Questions

• What is the local mechanism for trajectory
• towards equilibrium?
• Are logarithmic beaches in static or dynamic
• equilibrium?
• Why the logaritmic spiral shape?

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