A Model of Ocean Waves
Waves are defined as a transfer of energy through a medium. If the sea surface
is disturbed from a steady state by some input of energy, waves are the
mechanism by which this energy is transported across the ocean. When a rock is
dropped into a flat pond a small disturbance in the surface of the water is
formed. Gravity acts to restore the water to its original state, and the energy
from the rock flows away in waves, seen as ripples that get smaller the farther
they travel away. A bigger rock will generate bigger waves, so by looking at
the waves something can be known about the size of the rock.
On a larger scale, waves generated by an upwelling or down-welling event in an ocean will tell us something about the nature of that event. Underwater landslides and earthquakes can cause disturbances at sea level that are transferred by waves; weather patterns are transferred by waves as well. A model that can follow the evolution of an initial disturbance will be a useful tool for the visualization and understanding of different types of waves generated under different conditions, and the way waveforms change as they travel.
The evolution of the velocity and shape of the initial disturbance depends on several factors. The simplest case, non-dispersive waves, are independent of wavelength, frequency and direction; all frequency components move at the same speed and the shape of the initial disturbance does not change as the wave moves (Allen, 2003).
This is a non-dispersive
wave. This link will show you the code used to generate the animation below
which can be viewed in a separate window by clicking on the image.
Dispersive waves occur when the waveform changes with time and direction. A waveÕs dispersion relation defines the way it will move.
The rotation of the earth has an effect on the way a wave propagates if the waves are long enough; i.e.: the disturbance is big enough. One type of these waves are PoincarŽ waves: long gravity waves combined with internal oscillation, also called inertia-gravity waves. The degree of rotation exhibited by PoincarŽ waves depends on the comparison of the wavelength with the Rossby radius of deformation, such that short waves will not be affected by the rotation and fall into the non-dispersive category, and long waves on the order of the Rossby radius are classified as PoincarŽ waves (Gill, 1982).
A PoincarŽ wave has the dispersion relation
s2
= f2 + gDk2
where f=1*10-4s-1,
g=9.8ms-2, D=depth of water, k=wavenumber
This is a PoincarŽ
wave. This link will show you the code used to generate the animation below
which can be viewed in a separate window by clicking on the image.
Due to changes in potential vorticity on the surface of the earth, the strength of the Coriolis force changes with latitude. So when a wave is long enough to move across several degrees of latitude, the effect of the changing Coriolis is included in the dispersion relation by way of a constant, b. These low frequency waves, called Rossby waves, travel slowly westward on the b-plane and can be many kilometers long. Rossby waves depend on the Rossby Radius, R:
R
= (gD)1/2/f
When R is large compared to the width of the disturbance the wave generated may not be big enough to be classified as Rossby. A Rossby wave has the dispersion relation
s = - bk/(k2 + 1/R2)
This is a Rossby
wave. This link will show you the code used to generate the animation below
which can be viewed in a separate window by clicking on the image.
The model plots the evolution of an initial disturbance of a given height and radius, and at a given depth. There is a separate model for each of non-dispersive, f-plane, and b-plane waves. The latter two are similar, and involve Fourier theory; the first is much less complex and simply plots the movement of the waveform along a flat surface. In addition, parameters can be adjusted to follow the evolution of internal, or baroclinic, waves, by adjusting the water depth and the gravitational constant, g.
In all three models, the initial disturbance takes the shape of a Gaussian function,
h (x) = H exp (-x2 / r2)
with H=height, r=radius,
and x the domain over which the disturbance
will propagate. In ÒNo RotationÓ, the model for a non-dispersive wave, h (x) is split such that for every x value in the domain, half the disturbance will travel one way, and the
other half in the opposite direction:
h (x,t) = 0.5*H exp (-(x+c*t)2
/ r2)+ 0.5*
H exp (-(x-c*t)2 / r2)
A vector is generated for each time step, and these are plotted in succession such that the disturbance appears to split apart and propagate away.
When rotation is included in ÒF-planeÓ and Òb-planeÓ, the theory of the model becomes slightly more complex. A Fourier transform is applied to h (x) to solve for the amplitude of the disturbance at a given wavenumber, A(k):
A
(k) = ˜
h(x) * exp (ikx) * dx
This expression is then summed over the whole domain, x, for values of k and c_p, the phase speed. The values of k, the resolvable wavenumbers, are determined using discrete Fourier theory. A vector for the displacement of the surface, h, is calculated for each time step:
h(x,t) = 1/N * S {A(k) * exp(i*k*(x Ð c_p*t))}
where N is the length of the domain. Again, plotting these vectors in succession allows one to trace the evolution of the initial disturbance. The phase speed depends on the dispersion relation. Rossby waves do not depend on the waveÕs group speed, sqrt(g*H), their propagation depends only on the b constant and wavenumber; PoincarŽ waves are affected by f, c, and k.
The models produce good representation of wave propagation in a specific environment. It is easy to tell how the wave moves in different situations, and how the size of the initial disturbance can change the wave dynamics. For example, in the f-plane model, a wave with small enough radius will behave in the same manner as a wave in the no-rotation case, because a wave this small will not be affected by the Coriolis effect.
It is possible to consider baroclinic instead of barotropic waves by changing the value of g to gÕ, a reduced gravity constant based on the difference in density between layers. The effect of this is that the waves travel more slowly, so other parameters in the model must be altered, such as radius, domain size, and time step, to produce meaningful plots.
A more complete picture of reality would include more than one type of wave spawned from a disturbance. For instance, Rossby waves only happen in the b-plane, but PoincarŽ waves and non-dispersive waves also appear in a b-plane. However, this model shows only one type of wave per plane for a simplistic picture of how that type of wave moves and evolves.
Baroclinic- these waves occur in a stratified
system, between the two layers of different density; u and v vary with z
Barotropic- these
waves occur on the surface, and density is a function only of pressure, ie:
there is no variation of u and v with z
Wavenumber- the
number of wave cycles per unit distance; k=2p/l=s/c
Vorticity- the
curl of the velocity field
Allen, Susan. Advanced Geophysical Fluid Dynamics.Advanced Geophysical Fluid Dynamics. Earth & Ocean Sciences, UBC. 2003.
Baum, Steven K. ÒGlossary of Oceanography and the Related Geosciences with ReferencesÓ. January 20, 1997. http://stommel.tamu.edu/~baum/paleo/paleogloss/paleogloss.html Texas A&M University.
Cushman-Roisin, Benoit. Introduction to Geophysical Fluid Dynamics. New Jersey: Prentice Hall, 1994.
Georgia Systematic Teacher Education Program. ÒThe GSTEP Ripple ReportÓ. 2003.
Gill, Adrian E. Atmosphere-Ocean Dynamics. New York: Academic Press, Inc., 1982.
Hoffman, Forrest. ÒAn Introduction to Fourier TheoryÓ. January 13, 2004. http://utcsl.phys.utk.edu/~forrest/papers/fourier/index.html.
LeBlond, Paul, and Mysak, L.A. Waves in the Ocean. New York: Elsevier Scientific Publishing, 1978.
University Corporation for Atmospheric Research. ÒThe Balancing of Geostrophic AdjustmentÓ. http://meted.ucar.edu/nwp/pcu1/d_adjust.html 2002.
Kate Collins, February 11, 2004.