Exemplar Answers
a) Run analyze_ex2_a in matlab. Check out all four options. Make movies if you like (these can be saved and run without matlab using a web browser or other software). To turn in: Describe the time evolution of the behavior that you see. Consider growth rates, wavelength, wavenumber, differences between the hemisphere, etc. Use your understanding of meteorology as best you can, or describe what you see generically as an instability problem. I encourage you to print out a couple of your favorite figures using the print icon on the figure window and turn them in. You should write about 300 words.
The map of evolving 850 hPa temperature reveals a perturbation to the zonal mean that develops slowly at first and then quite explosively. This is a case where the system starts slightly perturbed from an unstable equilibrium - like a ball at the top of a mountain that receives a slight nudge. The wave like structure grows first in the NH and then follows about a week later in the SH. The NH wave has a wavelength of about 50 deg of longitude but the wavelength is much shorter in the SH. These results show that the initial perturbation has a long lasting imprint on the structure of the wave, in location, timing, and structure. The structure is shorter wavelength in the SH because the initial zonal structure only experiences perturbations when the wave like structure in the north is large enough to reach into the SH. By that time, the NH wave has spread over a wide range of longitudes. Hence the first significant perturbation in the SH occurs over a wide range of longitudes and is quite noisy, compared to the nice neat Gaussian perturbation that was added to the NH.
As the wave develops it moves heat to the north and south, so the pole to equator temperature gradient weakens. The map of geopotential height at 500hPa confirms that the wave-like structure develops in the height too, and hence pressure gradient forces (PGFs) develop. The PGFs give rise to a wave in the jet, so that much higher winds develop over some longtitude sectors. The sea level pressure maps helps us see the consequences of the weakening pole to equator temperature, as most of the perturbations of higher pressure appear equatorwad and lower pressure appear poleward of the latitude of the maximum wave amplitude in temperature and wind speed.
b) Run analyze_ex2_b in matlab. The figures illustrate the relative phase of the waves at the upper and lower levels for day 8. Figure 1 is just the height of pressure surfaces contoured like a topo map. The heights are higher to the south. Thus their is a pressure gradient force perpendicular to the contours (on average pointing to the north) and a coriolis force pointing roughly opposite but not exactly because this flow is not balanced. We know it is not balanced because it is unsteady. Figure 2 shows the departure of the height from the zonal mean. It helps us to see where the waves crests and troughs lie. First from Fig 1, note how the wave crests line up along a curving "axis" north to south. Now look at Fig 2 to see how the red contours correspond to the ridge axis. Likewise for troughs. Now notice how easy it is to see in Fig 2 that the upper level ridge and trough axes are shifted westward of the lower level axis.
Theory tells us that temperature gradients fuel growth of waves because winds transport heat to deepen upper level wave amplitudes. However, for wave growth there must be a phase shift between upper and lower waves so the heat transported near the surface around highs and lows can deepen the upper level structure. With no phase shift, the heat transported near the surface is exactly inbetween crests and troughs aloft, and therefore cannot deepen the troughs or raise the crests. Instead it causes the crests and troughs to shift. Pure growth would happen if 90 degrees out of phase. Our phase shift is definitely less that 90 degrees and therefore we see some growth and some shifting to the east. This is a somewhat advanced topic. Do your best to see and interpret these behaviors in the figures. It may be challenging. Ask Cecilia for help if you wish.
To turn in: Estimate and discuss the characterstics of the phase shift in the upper and lower waves that you see in the simulation. Where is the phase shift between waves at upper and lower levels a maximum and minimum (eye-ball this) and how does this correspond to relative wave growth? Print the figures and turn them in. Turn in about 200 words.
The eliptical shaped anomalies in Figure 2, highlighting wave crests and troughs, in the eddy height field are shifted in phase by a few degrees of longitude between the upper and lower level. The shift is largest for the left most wave crests and troughs. Consistent with the theory, the troughs and ridges are growing in magnitude on the left side of the wave packet, while those on right are about at their asymptotic limit. The wave packet still moves to the right because it is being advected with the zonal wind. Notice also that the ridges have larger magnitude than the ridges.
Even though this model is idealized, the phase shift is not as large as you might expect from the textbook drawing of a developing baroclinic wave (like the one shown in class). The real atmosphere is not quite like a text book, and neither is a model of the primitive equations.
c) Run analyze_ex2_c in matlab. The figures are fairly self explanatory. To turn in: Discuss the winds at both levels. Print the figures and turn them in too.
The zonal wind at the upper level is much stronger, so the perturbation is a smaller relative contribution. Hence the left panel wind vectors appear to have a smaller wave amplitude. The vertical wind magnitude is much larger in updrafts than downdrafts. As a result the area with upward winds is smaller (so an eaqual amount of mass goes up and down). The wind is upward where the horizontal winds tend to be perturbed slightly northward, where there is strong advection of heat from the south at the surface.
d) Propose an experiment you would like to try to probe this system further. Explain why and offer a hypothesis as best you can. I won't evaluate if your hypothesis is correct. Famous modeler Syukuro Manabe said roughly, "Use the model to tell you the answer". If possible we will try some of these ideas later! One suggestion is to mess with the initial conditions. But say how.
It would be neat to run the model longer, change the horizontal scale of the Gaussian perturbation, change the magnitude of the Gaussian perturbation, perturb with random noise instead of a Gaussian mountain, vary the rotation rate of the planet, vary the initial temperature condition, add moisture back into the model so there is condensation heating, etc.