ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY

Lecture 9

1.  Waves-concepts: should understand & apply 2.  Surface gravity waves: should understand the physical causes and character (e.g., energy dispersion, propagation & solution)

Learning objectives:

Autumn: Lake Superior , Michigan

Concepts: Wavelength, height, amplitude

1. Concepts L

C=L/T

L

Radian wave number:

Radian frequency:

Phase speed:

Group velocity:

Note that:

So,

Therefore:

Group velocity: Non-dispersive waves

Wave energy density:

Steepness: ratio of H/L is called steepness.

(NOTE: This is the sum of both kinetic energy and potential energy)

The energy density of a wave is the mean energy flux crossing a vertical plane parallel to a wave’s crest. The energy per wave period is the wave’s power density.

2. Surface gravity waves Equations of motion: [1] transient response; [2] steady circulation. Recall that – Large scale, interior ocean:

Which means that we can ignore nonlinear (inertial) terms and mixing. Excluding nonlinear terms: linear inviscid: Frictionless

is Coriolis parameter.

Equations of motion for x, y, z components:

The equations of motion in an homogeneous (constant density) ocean by ignoring nonlinear and mixing terms (Rossby and Ekman numbers are small), and ignoring Coriolis force (this is crude) are:

(Obtain perturbation equations based on given background state)

Assume background state:

Then we obtain: equation for background state:

Equations that govern perturbation:

Write a single equation in p alone:

Boundary conditions:

At the ocean bottom:

At the sea level,

Assume a wave form in x & y direction, but leave z-direction structure undetermined,

Substituting these wavelike solutions to the single equation in p alone, we obtain:

Where,

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sinh x =ex − e−x

2

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cosh x =ex + e−x

2

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tanh x =sinh xcosh x

By applying bottom boundary condition,

and thus

and surface boundary condition

We obtain,

Since

We have,

Now we have p, w, and solutions. ⌘

Let w satisfy boundary condition:

at z=0, This is the dispersion relation for surface gravity waves!

When [1]

!

Surface, long gravity waves:

Non-dispersive: group velocity is not a function of frequency and wave number.

Since

P is approximately independent of depth z.

( )

p ≈ gρ0η0 cos(kx + ly−ωt)

p ≈ gρ0η0 cos(kx + ly−ωt)

So, long surface gravity waves are non-dispersive, their structure is barotropic and feel the bottom of the ocean.

[2] Short surface gravity waves.

HYDROSTATIC.

Short surface gravity waves are dispersive because c and are functions of frequency and wave number.

Z<0, solutions exponentially decay. Surface trapped and do not feel the bottom.

Non-hydrostatic (board demo)

Particle motion associated with Surface waves & Stokes drift