Calculus I -- Chapter 1

Calculus I -- Chapter 2

Jetstreaks and Associated Circulations

            A jetstream is a narrow band of strong winds that encircles the Earth in the mid-latitudes, containing regions where locally strong pressure gradients produce exceptionally strong winds, called jetstreaks, which migrate through curved flow patterns. A jetstreak is located in a region of strong pressure gradient and is indicated by the large values of isotachs (lines of constant wind speeds) and the close spacing of the pressure or height contours. As air moves through the jetstreak, air parcels are displaced northward in the entrance region and southward in the exit region. Divergence occurs in the right entrance region (looking in the direction of the flow) while convergence in the left entrance region displaces air from the right (south) to the left (north) side of the jet. Divergence aloft will result in lower surface pressure, whereas convergence will result in higher surface pressure.

            Although the jetstream is not always a single ribbon of fast-moving air encircling the pole. In nature, a single jetstream can split into two branches and then merge again at a downstream location. In fact, the most extreme low pressures associated with cyclones in the middle latitudes usually occur when two (or even three) jetstreaks, each embedded in a different branch, interact with one another as their parent jetstreams merge.

Virtual Temperature

     A fictitious temperature that dry air would need to attain in order to have the same density as the moist air at the same pressure is considered virtual temperature. The fact that moist air is less dense than dry air was first clearly stated by Sir Isaac Newton in his “Opticks.” However, the basis for this relationship was not generally understood until the latter half of the 18^th century.

At any rate, due to that fact, and at the same temperature and pressure, the virtual temperature is always greater than the actual temperature. The use of virtual temp allows us to use the gas constant for dry air, R_d, saving us from constantly having to calculate gas constants for moist air—the value of which would vary with water vapor content.

            However, virtual temperature correction is usually neglected except in certain calculations relating to the boundary layer. Nonetheless, in moving from a given pressure surface to another pressure surface located above or below it, the geopotential height (used as the vertical coordinate in most atmospheric applications in which energy plays a role—i.e. large scale motions) is related geometrically to the thickness of the intervening layer which, in turn, is directly proportional to the mean virtual temperature of the layer. The mean virtual temperature is used for determining the thickness of a layer between two pressure surfaces (p₁ and p₂).

where e is vapor pressure, p is pressure, and ε is approximately equal to 0.622.

Wien's Displacement Law

Wien’s displacement law says that the wavelength of the maximum emitted radiation is inversely proportional to the absolute temperature (°K). In other words, hotter objects radiate more energy at shorter wavelengths than do cooler bodies at all wavelengths. This allows us to determine the temperature of other stars depending on its color. Something that glows blue hot is much warmer than one that glow red hot!

Stefan-Boltzmann Law

            Blackbodies are purely hypothetical bodies—they do not exist in nature—that emit the maximum possible radiation at every wavelength. The single factor that determines how much energy a blackbody radiates is its temperature. Although, the amount of radiation emitted by an object is not linearly proportional to its temperature which is where the Stefan-Boltzmann law comes into play. The blackbody version of the Stefan-Boltzmann law expresses that the intensity of energy radiated by a blackbody increases according to the fourth power of its absolute temperature.

I = σT⁴

where I denotes the intensity of radiation in watts per square meter, σ (Greek lowercase sigma) is the Stefan-Boltzmann constant (5.67 × 10−8 watts per square meter per K4), and T is the temperature of the body in kelvins.

            At any rate due to the fact that blackbodies do not exist in nature most liquids and solids can be treated as graybodies, meaning they emit some percent of the maximum amount of radiation possible at a given temperature. Which brings us to the graybody version of the Stefan-Boltzmann law that includes the emissivity factor, meaning that the electromagnetic energy emitted by any graybody will be some fraction of what would be emitted by a blackbody.

I = ƐσT⁴

That percent of energy radiated by a substance relative to that of a blackbody is considered emissivity (ε), ranging from just above zero to just below 100 percent. However, the atmosphere is an exception to this because emission depends on a number of factors (i.e. the amount of water vapor and other gases in the air). Still, we can say that the atmosphere is not a perfect emitter of radiation because it emits less radiation at any particular temperature than would a blackbody.

The Three-Cell Model

            According to the three-cell model, the circulation of each hemisphere is composed of three distinct cells: the Hadley cell, a Ferrel cell, and a polar cell. Thought more realistic than the single cell model, the three-cell model is so general that only fragments of it actually appear in the real world. Nonetheless the names for many of its wind and pressure belts have become well established in our modern terminology, and it is important that we undertint where these hypothesized belts are located.

            The Hadley cell is a thermally direct (hot air rises, cool air sinks) circulation along the equator where strong solar heating causes air to expand upward and diverge toward the poles, creating a zone of low pressure at the equator. This zone of low pressure is known as the equatorial low or the intercontinental convergence zone (ITCZ), it is the rainiest latitude in the entire world where winds can become light or nonexistent for extended periods of time (doldrums). Nonetheless, air in the upper troposphere moves poleward toward the subtropics at about 20° to 30° latitude. Upon reaching about 20° to 30° latitude, air in the cell sinks towards the surface to from the subtropical highs (large bands of high surface pressure). The pressure gradient force (PGF) directs surface air from the subtropical highs to the ITCZ where the weak Coriolis force deflects the air slight to the right (left in the southern hemisphere), forming the northwest trade winds (southeast trade winds in the southern hemisphere).

            Immediately flanking the Hadley cell in each hemisphere is the Ferrel cell, which circulates air between the subtropical highs and the subpolar lows. On the equatorial side of the cell air flows poleward, the subtropical high then undergoes a deflection to the right (left in the southern hemisphere) due to the Coriolis force, creating the westerlies (easterlies in the southern hemisphere) wind belt. The Ferrel cell is considered a thermally indirect circulation (cool air rises, hot air sinks) meaning that, unlike the Hadley cell, this cell does not arise from differential heating but, instead, is caused by the turning of the polar cell and the Hadley cell.

            Finally, the polar cell’s surface air moves from the polar highs toward the subpolar lows. At the subpolar location air is slightly warmer, resulting in low surface pressure and rising air. The very cold conditions create high surface pressure and low-level motion towards the equator. The Coriolis force, in both hemispheres, deflects the air to form a zone known as the polar easterlies in the lower atmosphere. Like the Hadley cell, this cell is also considered to be a thermally direct circulation (hot air rises, cool air sinks).

The Bottom Line:

─     The three-cell model is not realistic at all.

─     ITCZ is real enough to observed from space—many deserts exist in their predicted locations

─     Trade winds are the most persistent winds on Earth.

─     The Hadley circulation provides a good account of low-latitude motions.

─     The Ferrel and Polar cells are not quite as well represented in reality—though they do have some manifestation in the actual climate.

─     It is difficult to observe a persistent pattern of polar easterlies—they emerge in long-term averages, but are not a prevailing wind belt.

Rayleigh Scattering--Why is the Sky Blue?

            Scattering agents smaller than about 1/10^th the wavelength of incoming radiation disperse radiation (both forward and backward) in a manner known as Rayleigh scattering. Rayleigh scattering is performed by individual gas molecules in the atmosphere and primarily affects shorter wavelengths. It is particularly effective for visible light, especially for those colors with the shortest wavelengths (i.e. blue).

Combined with greater effectiveness in scattering shorter wavelengths than longer wavelengths, this characteristic leads to three interesting phenomena:

1.    The blue sky on a clear day

2.    The blue tint of our atmosphere

3.    The redness of sunsets and sunrises

Why is the sky blue?

            Gases and particles in the atmosphere scatter some incoming solar radiation in all directions. Air molecules scatter shorter wavelengths most effectively and blue light is among the shortest (and therefore most readily scattered) of visible wavelengths thus the scatter contains a higher proportion of blue light. And, since Rayleigh scattering occurs at every point in a clear atmosphere and diverts energy towards a viewer from all directions, no matter where you look on a cloudless day, the sky is blue.

Some more interesting facts…

Weather Station Model

Temperature, pressure, moisture, and wind measurements are reported hourly at the surface (most are usually made 2 meters above the ground).

77: Temperature.

68: Dewpoint.

998: Pressure, to the nearest tenth of a millibar. Add either a 10 or 9 in front based on which would bring the value closer to 1000. The pressure here is 999.8 millibars (mb).

-03: Pressure tendency the last 3 hours, to the nearest tenth of a millibar. The pressure here has fallen .3 mb the last 3 hours.

Middle Circle (filled in w/ mostly black): Cloud cover. It's mostly black showing that this station is mostly cloudy. Technically, this represents a broken sky with 7/8 of the sky covered with clouds.

Black line, extending from circle: Wind barb. It points to where the wind is coming from. The wind here is from the southwest, hence a southwest wind. The two lines extending represent 20 knot winds with each line representing 10 knots.

Symbol between 77 and 68: This is the present weather field and in this case shows that there is a thunderstorm occurring at the station.

Symbol next to -03: That line is the pressure tendency. The 1st hour the pressure was steady, then fell the last two hours.

Triangle (with a dot above it): Previous weather, or the weather one hour ago. In this case it was a light rain shower.

How does the upper air station model differ from the surface station model?

─     Temperature is given in Celsius

─     Dewpoint depression is given not dewpoint temperature

─     Altitude of the pressure surface is given instead of pressure

─     Cloud cover is not noted

─     Circles indicating station locations are often omitted

Newton's Three Laws of Motions

            Isaac Newton, the father of mechanics, is one of the most important scientists who ever lived, changing the standards by which scientists think today. His genius in mathematics and mechanics is exemplified by his creation of calculus to explain observations of the world around him. In addition, his laws of motion opened the door to progressive new thinking, enlightening the minds of thousands to the nature of things. His laws of motion make up the foundation for which dynamic meteorology exists.

1)    Law of Inertia—a body at rest or in motion will tend to stay that way until acted upon by a net external force.

2)    Law of Acceleration—a change in motion relates directly to a force trying to move it.

F = ma    or    F = ρg

3)    Action-Reaction Law—for every action there is an equal and opposite reaction.

Laws of Thermodynamics

0^th Law of Thermodynamics—if body A is in thermal equilibrium with body T, and so it body B, then A and B are in thermal equilibrium.

1^st Law of Thermodynamics—a measure of heat transferred into a system will result in an increase in temperature and in the system’s ability to do work. In other words, energy is conserved property that is neither created nor destroyed but, may change form and travel from place to place. Or, in terms of an internal combustion engine in an automobile, the first law describes the underlying principle of what occurs in the cylinder.

Other forms of the First Law…

2^nd Law of Thermodynamics—only in transferring heat from a warmer body to a cooler body can heat be converted into work, in a cyclical process. A cyclic process is a series of operations by which the state of the substance (working substance) changes but the substance is finally returned to its original state in all respects. In other words, heat is always transferred from regions of high temperature to regions of low temperature. Heat can be transferred by three processes…

1.    Conduction—the movement of heat through a substance without appreciable movement of the molecules.

2.    Convection—the transfer of heat by mixing of a fluid.

3.    Radiation—the transfer of energy by radiation that can occur through empty space.

Forms of the Second Law, considering the Carnot Cycle…

3^rd Law of Thermodynamics—there is no finite series of steps that can get you to absolute zero. In other words, since absolute zero cannot be reached an engine cannot be perfectly efficient.

Conclusion—heat can be converted into work, in a cyclic process, but can only be perfectly efficient at absolute zero, which is unattainable.