101: Wind II

For the 50th post on this blog, I will finish up the "101: Wind I" post with the descriptions of various types of wind. Although there are countless types of local wind phenomena, and even more names for them, I'll be grouping the winds in this post into four distinct groups: geostrophic wind, gradient wind, valley wind, and costal breezes. Unlike previous posts, I be using a fair amount of equations. Understanding what all the parts of the equations mean or how to solve them is not important, I just want to use them to show how changes in the equations affect the behavior of the wind. I would highly suggest referring back to "101: Wind I" for guidance.
 
 
 
Geostrophic Wind
(Much of the material in this section was also covered in the post "101: Geostrophic Balance")
Geostrophic wind is the result of a balance between the Coriolis force and the pressure gradient force (PGF). This wind always flows parallel to isobars (or height contours at upper levels) with low pressure to the left in the northern hemisphere (to the right in the southern hemisphere). In Cartesian coordinates the u and v components of the geostrophic wind are given by these equations:




ρ is the density of the air, p is the air pressure, and f is the "Coriolis parameter" which is proportional to sin(latitude)

However, in the reality, wind is never perfectly geostrophic because other forces, such as the centrifugal force and friction cause the wind to deviate from pure geostrophic motion. This deviation is called the ageostrophic wind. Therefore the true wind is obtained by adding the geostrophic and ageostrophic winds. The maps below are from 2003 as Hurricane Isabel was approaching North Carolina, a low was centered over central Canada with a sharp trough extended south across the northern plains, and a high was centered over New England. The maps are on layers of constant pressure with the height contours in white, the heights used were 950mb (close to the ground), 500mb (mid-way up, above any fronts or terrain), and 200mb (very high, at the level of the jet stream). At numerous points are wind barbs depicting the true wind in black, the geostrophic wind in blue, and the ageostrophic wind in red. 

950mb: Notice how strong the ageostrophic wind is on the barbs near the center of the low (Isabel), they show that the true wind is much less than the geostrophic wind, and is pointed slightly toward the low center. This is due to both friction with the surface and the centrifugal force associated with the Hurricane Isabel.

500mb: While friction plays very little role at this level, centrifugal force is still keeping the true wind from reaching the speed of the geostrophic wind.

200mb: The impact of the hurricane on height contours is practically gone, however, if you estimate the storm's location using the previous maps, you might notice that the true wind north of the center is faster than the geostrophic wind while the true wind south of the center is slower than the geostrophic wind. I believe this is due to the radial outflow from the top of the storm

950mb: Around the low center (on the left) the wind barbs are behaving similarly to the 950mb barbs around the hurricane. Further away however, and near the high (on the right) the ageostrophic barbs indicate that the true wind speed is not too much different from the geostrophic wind. The direction is still being affected, primarily due to friction, causing the true wind to point slightly toward the low center and away from the high center.

500mb: The largest deviation from pure geostrophic flow at this level occurs where there is a sharp turn. Otherwise, the flow is relatively geostrophic.

200mb: This is similar to the 500mb map with only a few wind barbs showing significant amount ageostrophic wind.

950mb: The high terrain near the west coast of North America often completely dominates the wind flow. Notice that near the coast, several true wind barbs are at nearly a 90 degree angle from the geostrophic barbs.

500mb: Above terrain and with relatively straight height contours, this flow is highly geostrophic.

200mb: Similar to the situation at 500mb, but even more geostrophic.

 
 
 
Gradient Wind
The equations for the gradient wind can be rather complicated, but their implications are highly important, so I'll boil it down to just the important points. Gradient wind is usually solved using natural coordinates and is highly dependent on the radius of curvature R and the sign of the PGF.
The raw version of the gradient wind equation is:


All this may seem complicated, but what is really important is that the first term on the left represents centrifugal force, the next term represents the Coriolis force, and the term on the right represents the PGF.

Two solutions can be obtained from this with just a little thought experiment regarding the importance of the two terms on the left.

Geostrophic Flow
Imagine the radius of curvature was very large, almost infinitely large. In that case, the left most term would essentially equal zero, leaving just one term on each side. If you divide that by f, the result is simply the geostrophic wind in natural coordinates. Here is this description in equation form:

Cyclostrophic Flow
Now consider a somewhat opposite case: small scales (or low latitudes). In this case f, the Coriolis parameter is very small, so the second term on the left pretty much equals zero, leaving only the first term on the left and the term on the right. solving for v yields the cyclostrophic wind equation.

Gradient Flow
This is where the gradient wind equation becomes much more complex when it is solved for the gradient wind: V.

There are a total of eight solutions to this equation each representing the flow around low or high pressure centers. The tricky thing is, since taking the square root of a negative number produces an unphysical solution and V must be a non-negative, only four of the solutions can actually exist and two of those are rarely seen.
 
 
 
 
Valley Winds
Terrain can have a large affect on wind. In the case of valleys, buoyancy often plays a role in creating a pair of local winds. As mentioned in several previous posts, warm air rises and cool air sinks due to relative density. As a valley heats up during the day the air near the surface also heats up and begins to ascend. In the confines of a mountain valley, this rising air will travel up the valley and up the valley walls producing upslope wind known as anabatic wind. When the flow reaches the top of the slope  it will converge with the anabatic wind that was traveling up the other side the mountain. If conditions are right, small cumulus clouds will form as a result of this convergence above, or near, the crest of mountain.
At night as the surface cools, downslope wind, known as katabatic wind, occurs as the air flows down the mountains into the valley. This cool air will collect along the valley floor in what is called a cold pool. Under the right conditions, this cooling will cause fog to form in the valley.
 
Below is a decent case of these winds from the Pocatello Valley (just north of the Great Salt Lake) on November 1, 2013.

The Pocatello Valley looking roughly north. The terrain is exaggerated to make it easier to see the valley borders. The head of the valley is the mountain near the center of the image that has a small white patch on it.

 


Here is one day's worth of observational data from inside the valley. The graph begins at 00Z which happens to be near sunset, and sunrise is around 12Z. The red line is the temperature and along the bottom are wind barbs showing wind speed and direction. Notice that as the valley cools the wind has a northerly component, this is the katabatic wind descending out of the valley. The opposite is true when the air heats up and anabatic wind ascends the valley.

 

This is a Terra/MODIS image of the valley a little after 18Z. By this time anabatic wind had become well established. Notice that small cumulus clouds have formed near the crests of the valley walls.

 

This is the same image as above, but the basic outline of the valley has been outlined in yellow.

 
 
  
Coastal Breezes
The key to understanding these winds is the fact that water has a high specific heat. This means it takes a lot of energy to raise its temperature. This is the property that is behind the phrase "a watched pot never boils". On the other hand, land can change temperature much more quickly. To compound all this is the fact that energy entering the surface of a body of water is spread out through a depth of many meters. Land on the other hand only has to be heated to a depth of a few millimeters.
 
 


The sea breeze circulation

Sea Breeze
In the tropics and subtropics, the temperature of the land exceeds that of the adjacent ocean after a few hours on sunlight, while the ocean's temperature essentially will remain constant throughout the day. The hot land will heat the air above it which will begin to rise. With all this air rising away from the surface, the surface pressure will drop slightly. Because the atmosphere always seeks to reach equilibrium, air from just above the ocean will move inland to take the place of the rising air. Much of the ascended air will make its way over the ocean and sink to replace the air that had moved onshore. This circulation is likely to persist as long as the land remains warmer than the ocean.
 
 
 


The land breeze circulation

Land Breeze
This type of flow is very nearly the opposite of sea breezes. At night, or whenever the land is cooler than the ocean, air will rise from over the ocean to be replaced by air from over the land moving offshore. The cycle is complete when the air that was over the ocean sinks over land. Since land breezes are not powered directly by solar energy, they tend to be weaker.