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http://www.cpc.noaa.gov/products/ctb/funded_projects.shtml
http://www.esrl.noaa.gov/psd/data/gridded/tools.html#excel
(due 12/1/2010 before class)
(due 12/1/2010 before class)
Effect of viscosity
1. Solve Prob 1 of Chap. 6.
compare the magnitude of Earths eastward coriolis force with the northward or southward coriolis force, so that you know the frictional force is negligible.
http://en.wikipedia.org/wiki/Earth%27s_rotationhttp://en.wikipedia.org/wiki/Earth%27s_rotation
Earth's rotation is the rotation of the solid Earth around its own axis. The Earth rotates towards the east. As viewed from the North Star Polaris, the Earth turns counter-clockwise
Evidence of Earth's rotation
In the Earth's rotating frame of reference, a freely moving body follows an apparent path that deviates from the one it would follow in a fixed frame of reference. Because of this Coriolis effect, falling bodies veer eastward from the vertical plumb line below their point of release, and projectiles veer right in the northern hemisphere (and left in the southern) from the direction in which they are shot. The Coriolis effect has many other manifestations, especially in meteorology, where it is responsible for the differing rotation direction of cyclones in the northern and southern hemispheres. Hooke, following a 1679 suggestion from Newton, tried unsuccessfully to verify the predicted half millimeter eastward deviation of a body dropped from a height of 8.2 meters, but definitive results were only obtained later, in the late 18th and early 19th century, by Giovanni Battista Guglielmini in Bologna, Johann Friedrich Benzenberg in Hamburg and Ferdinand Reich in Freiberg, using taller towers and carefully released weights.[n 5]
The most celebrated test of Earth's rotation is the Foucault pendulum first built by physicist Léon Foucault in 1851, which consisted of an iron sphere suspended 67 m from the top of the Panthéon in Paris. Because of the Earth's rotation under the swinging pendulum the pendulum's plane of oscillation appears to rotate at a rate depending on latitude. At the latitude of Paris the predicted and observed shift was about 11 degreesclockwise per hour. Foucault pendulums now swing in museums around the world.
Comapring the eastward force and northward coriolis force here:
rotation rate of earth going to the east=7.27*10^-5)s^-1
f east for altitude of 10km =2*omega*sin(angle)=2(7.27*10^-5)s^-1)*(sin(0.1)degrees) = 2.5*10^-5s^-1
f north for altitude of 3300km =2*omega*sin(angle)=2(7.27*10^-5)s^-1)*(sin(30degrees)) = 0.73*10^-4
therefore we can be convinced that the frictional force is negligible.
Earths rotation rate:
http://www.analysis-guru.com/rotation.htmlhttp://www.analysis-guru.com/rotation.html
2. Effect of friction on geostophic balance
for large scale flow over a region in the southern hemisphere.
Effect of friction on geostrophic balance
2. The following are two hypothetical maps of 500 mb height (top) and surface pressure
(bottom, assuming that the surface is flat), each for a large-scale flow over a region in the
Southern Hemisphere. Try to draw the anticipated horizontal velocity vectors, in the fashion
of Figs. 7.4 and 7.25, given the pressure/height pattern. For the surface map, you should
consider the effect of friction. Provide a brief explanation of your drawing (e.g., by sketching
the balance of forces for a selected wind vector). (2 points)
Energy conversion in baroclinic instability
4. Solve Prob 5 of Chap. 8. (4 points)
http://en.wikipedia.org/wiki/Baroclinity
Baroclinic instability
Baroclinic instability
Baroclinic instability is a fluid dynamical instability of fundamental importance in the atmosphere and in the oceans. In the atmosphere it is the dominant mechanism shaping the cyclones and anticyclones that dominate weather in mid-latitudes. In the ocean it generates a field of mesoscale (100 km or smaller) eddies that play various roles in oceanic dynamics and the transport of tracers. Baroclinic instability is a concept relevant to rapidly rotating, strongly stratified fluids.
Whether one is rapidly rotating or not is determined in this context by the Rossby number, which is a measure of how close the flow is to solid body rotation. More precisely, a flow in solid body rotation has vorticity that is proportional to its angular velocity. The Rossby number is a measure of the departure of the vorticity from that of solid body rotation. The Rossby number must be small for the concept of baroclinic instability to be relevant. When the Rossby number is large, other kinds of instabilities, often referred to as inertial, become more relevant.
The simplest example of a stably stratified flow is an incompressible flow with density decreasing with height. In a compressible gas such as the atmosphere, the relevant measure is the vertical gradient of the entropy, which must increase with height for the flow to be stably stratified. One measures the strength of the stratification by asking how large the vertical shear of the horizontal winds has to be in order to destabilize the flow and produce the classic Kelvin-Helmholtz instability. This measure is the Richardson number. When the Richardson number is large, the stratification is strong enough to prevent this shear instability.
Before the classic work of Jule Charney and Eric Eady on baroclinic instability in the late 1940s, most theories trying to explain the structure of mid-latitude eddies took as their starting points the high Rossby number or small Richardson number instabilities familiar to fluid dynamicists at that time. The most important feature of baroclinic instability is that it exists even in the situation of rapid rotation (small Rossby number) and strong stable stratification (large Richardson's number) typically observed in the atmosphere.
The energy source for baroclinic instability is the potential energy in the environmental flow. As the instability grows, the center of mass of the fluid is lowered. In growing waves in the atmosphere, cold air moving downwards and equatorwards displaces the warmer air moving polewards and upwards.
Baroclinic instability can be investigated in the laboratory using a rotating, fluid filled annulus. The annulus is heated at the outer wall and cooled at the inner wall, and the resulting fluid flows give rise to baroclinically unstable waves.
The term "baroclinic" refers to the mechanism by which vorticity is generated. Vorticity is the curl of the velocity field. in general, the evolution of vorticity can be broken into contributions from advection (as vortex tubes move with the flow), stretching and twisting (as vortex tubes are pulled or twisted by the flow) and baroclinic vorticity generation, which occurs whenever there is a density gradient along surfaces of constant pressure. Baroclinic flows can be contrasted with barotropic flows in which density and pressure surfaces coincide and there is no baroclinic generation of vorticity.
The study of the evolution of these baroclinic instabilities as they grow and then decay is synonymous in dynamical meteorology with the problem of developing theories for the fundamental characteristics of midlatitude weather.
Beginning with the equation of motion for a fluid (say, the Euler equations or the Navier-Stokes equations) and taking the curl, one arrives at the equation of motion for the curl of the fluid velocity, that is to say, the vorticity.
In a fluid that is not all of the same density, a source term appears in the vorticity equation whenever surfaces of constant density (isopycnic surfaces) and surfaces of constant pressure (isobaric surfaces) are not aligned. The material derivative of the local vorticity is given by
where is the velocity and is the vorticity, p is pressure, and ρ is density). The baroclinic contribution is the vector
This vector is of interest both in compressible fluids and in incompressible (but inhomogenous) fluids. Internal gravity waves as well as unstable Rayleigh-Taylor modes can be analyzed from the perspective of the baroclinic vector. It is also of interest in the creation of vorticity by the passage of shocks through inhomogenous media, such as in the Richtmeyer-Meshkov instability.
Divers may be familiar with the very slow waves that can be excited at a thermocline or a halocline; these are internal waves. Similar waves can be generated between a layer of water and a layer of oil. When the interface between these two surfaces is not horizontal and the system is close to hydrostatic equilibrium, the gradient of the pressure is vertical but the gradient of the density is not. Therefore the baroclinic vector is nonzero, and the sense of the baroclinic vector is to create vorticity to make the interface level out. In the process, the interface overshoots, and the result is an oscillation which is an internal gravity wave. Unlike surface gravity waves, internal gravity waves do not require a sharp interface. For example, in bodies of water, a gradual gradient in temperature or salinity is sufficient to support internal gravity waves driven by the baroclinic vector.
Rotating tank experiment
6. (a) The Rossby number, Ro, is defined as the ratio of the "inertial term" ( −v⋅∇v or
∂v /∂ t ) to the Coriolis force in momentum equation. For example, from the analysis in Sec
7.1 (p. 110), large-scale circulation in the atmosphere typically has Ro ~ 0.1, indicating the
dominance of Coriolis effect (also see Fig. 7.5). Try to estimate the typical value of Ro for the
flow fields in our rotating tank experiments (the 2nd and 3rd experiments with rotation turned
on). (b) Compare the magnitudes of the centrifugal and Coriolis forces in our rotating tank
experiments. Suppose that you have a very delicate rotating tank that allows a precise
control of the rotation rate and flow velocity, could you envision a situation when the Rossby
number of the flow is small (ideally Ro ~ 0.1) and, at the same time, Coriolis force is
significantly greater than centrifugal force? (Try to see what combinations of U, W, and R
might lead to such a flow regime. Here, U is the velocity scale of the flow, W the rotation rate
of the tank, and R the radius of the tank.) (2 points)
Note: (1) Since we have included centrifugal and Coriolis forces in our equation, we are
considering a fluid system in the rotating frame; The "velocity scale" U is for the velocity as
measured by an observer rotating with the tank. It should not be confused with the "absolute
velocity" of the fluid which would be on the order of WR. In our experiments, we did not have a
direct measurement of U but a reasonable number to use would be a few cm/s, based on
tracking of colored dye in the water. (2) The length scale of the flow, L, needs not be the same
as the radius of the tank, R. For the last part of the problem, however, we want L to be large
in order to make Ro small. Then, a natural choice would be L ~ R.
2 of
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