Coupled Ocean-Atmosphere Interaction
ATM S 560
24 APR 01

1   A five minute diversion on the ``hockey stick" plot.

A canonical plot of the change of the global mean temperature (of lower troposphere) that was constructed from tree-ring data shows a slow decrease in temperature from a thousand years ago until the last century (when we introduced greenhouse gasses)...there the plot shoots upward like the blade of a hockey stick (thus the hockey stick plot). When compared with similar plots of temperature proxies from tree ring data, however, this trend is not seen. In fact, the more carefully interpreted tree-ring data shows much more decadal variability (such as the little ice age (1300-1600) and warming in the middle ages) that the hockey-stick plot fails to capture. These potentially ``more correct" plots of mean global temperature = variability also fail to show a large increase within the last century due to the production of greenhouse gasses. The validity of these ``more correct" temperature proxy plots is reinforced by the fact that the same trends are seen in tree rings from New Zealand.

2   Continuation of the discussion of the family of equatorial wave solutions.

2.1   Background

= We started with the equatorial b plane (f=3Db y), linearized equations of motion with= forcing initially included in the form of wind stress. Remember that we're talking about the ocean here and not the atmosphere. These equations are valid for three cases
1. barotropic fluid, where ce2=3D(gH)...H being the water depth.
2. 1 1/2 layer system where the interface wave speed, ce2=3D(g'H), g' being reduced gravity.
3. a continuously stratified system.
A side note on the speed of the barotropic and the 1 1/2 layer waves: For the ocean, the barotropic mode will ALWAYS have a greater wave speed than the baroclinic mode (often an order of magnitude difference). HOWEVER, there is a surface displacement signature h that is associated with the baroclinic mode and travels at the same speed as the baroclinic wave. Gill (6.2) notes that this surface signature is 1/400 of the interface displacement, but is still sufficient for baroclinic motions to be detected by sea-surface measurements. To be further enlightened on this issue, see Gill chapter 6.
Our family of waves includes:
1. Kelvin waves (v=3D0). (exist only at the equator and boundaries!) Balance is geostrophic meridionally and is a gravity wave zonally.= These waves travel at the long wave speed and are non-dispersive. Also, their group velocity (which is their phase velocity since they are non-dispersive) travels TO THE EAST! This makes sense if we view the equator as a boundary along which the waves are propagating. These waves decay exponentially away from the equator with an e-folding scale of (2ce/b)<= FONT SIZE=3D2>1/2.
2. The rest (v=B90). These waves can be viewed as sinusoidal modes that exponentially decay away from the equator at a critical latitude. The larger the meridional mode number, the greater the critical latitude. The more baroclinic the disturbance (where the wave speed decreases) the shorter the trapping distance. These waves come in three forms:
1. Gravity and Inertio Gravity waves (Poincare).
2. Mixed Rossby Gravity (Yanai).
3. Long and Short Rossby waves.
The dispersion relation for these waves is:
Ce
b
=E6
=E7
=E7
=E8		 -
k
w
b-k2+
w2
Ce 2
=F6
=F7
=F7
=F8		 =3D2m+1



This dispersion relation is derived from the initial equations with forcing removed...these are the free-wave solutions. All three types of waves can be seen in this dispersion relation:
1. When k is small (wavelength is long) and w is also sufficiently small, the first term in the parentheses, -k<= /I>/wb, is a leading order term with the 2m+1 on the right side of the equation. This gives = long Rossby waves.
2. When k is large (wavelength is short) the first and second terms in the parentheses, -k/wb-k2, dominate with the 2m+1 on the right side of the equation. These are short Rossby waves.
3. Inertia-gravity waves are given by a balance of the second and third terms in the parentheses, -k2+w2/Ce2, with 2m+1. At high values of w, the balance is between the second and thi= rd terms in the parentheses, -k2 and w2/Ce2, which gives gravity waves.
The propagation characteristics of these various wave solutions can be represented in a dispersion diagram--on which the wavenumber, k, is plotted vs. the frequency, w
On these diagrams, the phase speed, w/k, is a line drawn from a point on the curve to the origin. Thus we can see that the phase speed for Rossby waves (long and short) is always to the WEST and the phase speed for Kelvin waves is always to the EAST. The slope of the curves, =B6w/=B6 k, is the group velocity or the velocity of the energy. From the dispersion diagram we can see that the phase velocity and group velocity for Kelvin waves are the same...thus they are non-dispersive. The dispersion diagram shows us that Rossby waves with long wavelengths (small k's) will have group velocities that are to the WEST and Rossby waves with short wavelengths (large k's) will have group velocities that are to the EAST. The diagram also shows us that long Rossby waves are only weakly dispersive.
We noted that there was an interesting frequency range where the energy should only be going eastward...this is a fairly high frequency range with periods in the ocean of about 5 days and periods in the atmosphere of 1 day.
NOTE: If we're interested in low-frequency variability then we can focus primarily on the long Kelvin waves and both the long and short Rossby waves. We will get the solutions for just these waves if we make the long wave approximation in the initial equations of motion.

2.2   The Forced Equations

We discussed the effects of leaving the wind forcing terms in the initial equations of motion. For low frequency oscillations, |w/f|=AB1, with the observed scales of the wind stress forcing, it turns out that it is primarily the zonal wind stress that drives the currents and the thermocline. To see this result we scaled three forcing terms which represented the zonal wind stress, -ftx, meridional wind stress, tt<= /SUB>y, and curl of the wind stress, Ce= 2/r1u1(=D1=D7t)x. In the midlatitudes, it is the wind stress curl term that wins out when the forcing terms are scaled, and the balance is Sverdrupian: b V=3D1/r1(=D1=D7t), where V is the vertical integral of the meridional veloc= ity.
Because the lengthscale of the zonal wind forcing at the equator is much larger than the equatorial ocean trapping scale, Ycrit, at small w/f the dominant wind forcing is zonal. Thus, the linearized, equatorial b-plane EOM become:
ut-b yv=3D-Px+tx/r= 1u1
 -b yu=3D-Py
Pt+Ce2(ux+vy)=3D0
By leaving out the vt term in the second equation we are making the long wave approximation...which filters out all waves but the long Rossby and Kelvin waves (the ones we're concerned about for low-frequency oscillations).

2.3   Propagation Characteristics

We looked at the propagation of the long Kelvin and Rossby waves that would originate from a gaussian bump in the thermocline (or surface) at mid-basin. The Kelvin wave, travelling to the east at the shallow water wave speed, is a bump symmetric at the equator that is characterized by eastward currents decaying in magnitude away from the equator. The long Rossby wave travels to the west at a speed that is 1/3 of the speed of the Kelvin wave. Because higher modes of the Rossby wave travel more slowly to the west but also have a greater Ycrit, the Rossby wave signature moves faster toward the equator, and thus begins to slope poleward from west to east as the wave propagates. The model results that we looked at showed a Rossby wave that was composed of the first, third and fifth modes. The Rossby wave propagates as two bumps symmetric about the equator with geostrophic flow around each bump: anticyclonic north of the equator and cyclonic south of the equator. This results in a flow to the west at the equator.

2.4   Boundary Processes

Using the dispersion diagram, we analyzed the possible results of waves r= eflecting off eastern and western boundaries.

2.4.1   Eastern Boundary

First starting with eastward propagating waves...which will consist primarily of the long Kelvin waves since the short Rossby waves and gravity wa= ves will be damped out. So, the Kelvin waves propagate to the east and when they hit the boundary (South America if we're in the Pacific) several things can happen:
1. Kelvin waves can propagate along the coast moving energy poleward.
2. Low-frequency energy will reflect off the boundary as equatorially symmetric Long Rossby waves propagating back to the west.
3. High-frequency energy will reflect back as inerto-gravity and gravity waves (but will be damped quickly).
As previously mentioned, the long Rossby waves carrying the energy back to the west will develop a west-to-east poleward slant that is a result of higher modes travelling slower and having a greater Ycrit. Because we have a boundary condition that reflected frequencies will be preserved, then there will be a frequency gap over which there will be no reflected waves. This is the gap between westward travelling inerto-gravity waves and the long Rossby waves.
The percentage of the incoming energy that is reflected is dependent upon the frequency of the incoming energy. In the limit as w=AE0, all the energy is returned as a Rossby wave--half of which comes through the first mode and happens to remain very concentrated about the equator (10% goes poleward). At frequencies above those of long Rossby waves (and above frequency gap), the energy is returned primarily by mode = 1 I-G waves. The energy that is not reflected or dissipated propagates poleward along the boundary as Kelvin waves.

2.4.2   Western Boundary

The possible incident energy at the western boundary consist of the long Rossby waves and the westward propagating I-G waves. We have the same boundary condition where the incident frequency is preserved. Now there is a frequency gap of wave energy that won't be incident on the western boundary...which with the b.c. limits the frequencies of the reflected waves. The energy incident on the western boundary can reflect as:
1. Equatorial long Kelvin waves (low frequency) if an incident Rossby wave is symmetric (modes 1,3,5,..)
2. Short Rossby waves (very slowly propagating)
3. Eastward propagating I-G waves that are quickly damped out.
The most energy that will be reflected off the boundary is in the limit as w=AE0 with a symmetric incident long Rossby wave, invicid conditions and a N-S boundary. At this limit half of the energy will be reflected back as equatorial Kelvin waves (most from the gravest long Rossby mode) and the other half will go into the generation of short Rossby waves. The short Rossby waves, however, propagate very slowly and actually transfer their energy into the western boundary currents (see Pedlosky 'cause I can't explain why).

Finally we ended with a quick look at a the assymetry of the geostrophic flow of a geopoential bump near the equator. Because f changes greatly near the equator, there will be an assymetry of zonal circulation caused by the bump--with more water being transported to the west than the east. This arises from the geostrophic relation u=3D-fy/f.
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