Class Notes April 19

Development of Ocean Models of Intermediate Complexity

During this class we looked at linear reduced-gravity wave theory on the
equatorial beta plane in the ocean.  This is important to how the equatorial
ocean adjusts to changes in forcing, which at the equator is primarily
zonal windstress.

First we looked at the equations of motion on an equatorial beta plane,
which consist of zonal and meridional momentum conservation equations and a
continuity equation.  This set of equations assumes the ocean is basically
hydrostatic.  The equations are adiabatic, so there can be no density
changes in the fluid; all forcing in this model is from mechanical stresses.

These Equations are valid for a variety of physical systems, with a suitable
choice of the parameter c:

1. A barotropic fluid, if c = g H1 and  | eta / H1 | <<  1.

2. A reduced gravity system (aka a 1 M-= layer model) if:
        c = g* H1, where g* = g ( rho2 - rho1 ) / rho2

        This model assumes that there is an upper fluid layer that lies atop
a much deeper fluid layer that is motionless.  Wave disturbances propagate
in the upper layer in such a way that the pressure gradient caused by the
surface slope is entirely canceled by the sloping of the interface between
the upper and lower layer.  This is model is supposed to represent a shallow
thermocline layer resting atop a deeper abyssal layer.  It tends to be
accurate only on long timescales.  For the real ocean, appropriate choices
of layer thicknesses would be: H1 = 100 - 200m, H2 = 4 -5km This is the
interpretation of the equations that we will be using in class.

3. A continuously stratified system with a rigid lid, if P, u and v are
assumed to be the component of pressure and velocity associated with a
particular vertical standing mode.  A continuously stratified system has an
infinite number of such vertical modes, with decreasing c values for higher
mode numbers.  The value of the first mode for a representative equatorial
density structure would be c 1 = 2 - 3 m/s.

4.The horizontal propagation component of a vertically propagating mode, if
c = N / m.

These equations can only represent horizonally propagating waves.  Real
annual period rossby waves generated at the eastern boundary tend to radiate
energy downwards, so that they dissipate by the time they get to the
dateline.

The simplest free solution of this set of equations is the Kelvin mode,
found by setting the meridional velocity = 0.  This represents a wave that
is in geostrophic balance meridionally, but moves like a gravity wave
zonally across the equator, propagating from west to east.  The Kelvin mode
tends to be trapped within 3-5 degrees of the equator.

The general free solution of this equation is found by solving the equations
for v and substituting a sinusoidal wave solution.  These solutions have
modes in the meridional direction, with a structure that decays towards the
poles for y > c (2m + 1) / beta and is sinusoidal closer to the equator.
The critical latitude where the solutions start decaying is proportional to
the wave speed and the mode number, so large-scale north-south disturbances
tend to be trapped near the equator, as are higher-mode baroclinic
disturbances, which tend to have smaller wave speeds.  These functions form
a linearly independent set for the decomposition of thermocline wave
motions.

We then derived the dispersion relationship, which shows the relationship
between omega and k for these wave modes.  The relationship is quantized in
m, so each meridional wave mode has a slightly different equation relating
omega and k.  The solutions we derive are divided into several separate
families, based upon their location in omega-k space.  Interio-gravity waves
(aka Poincare waves) exist for omega > f and all k and m.  Mixed Rossby
Waves exist for m=0, and represent a wave mode trapped near the equator that
propagates westward like a pure Rossby wave and eastward like a gravity
wave.  Finally, for omega < f, there are Rossby wave solutions, which have
a group velocity that is westward for long Rossby waves and eastward for
short Rossby waves.