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Chapter 8 Coordinate Systems

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**(1) Conventional Coordinate Systems**

Cartesian: Three axes perpendicular to each other.

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**Plane polar coordinates are a natural way to describe circular motion.**

Instead of describing the position of an object using x and y coordinates, this system uses two coordinates that locate the object's position using its distance from the origin (called "r" for radius) and its angle (theta) from the positive x-axis . Angles measured counterclockwise from the x-axis are defined to be positive angles, and angles measured clockwise from the x-axis are negative angles. The distance from the origin is ALWAYS positive.

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**Spherical Coordinate System.**

The angle (ϕ) defines the elevation above (or below) the plane of (Θ). Latitude and longitude are examples of a spherical coordinate system. r is the distance from the center of the Earth. Θ is the longitude with the Greenwich Meridian as the starting point. ϕ is the latitude with the equator as the starting point. Up, where the elevation angle/latitude = 90o is the north pole. Looking down from the up position, longitude increases in a counterclockwise direction.

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**Use of spherical coordinates with radar**

The elevation angle, , is zero along a line parallel to the horizontal. The azimuth angle, , begins with zero at north, and moves clockwise just as the azimuth angle for wind. r is the radial distance from the radar location. Up refers to moving outward directly from the center of the Earth. Looking down from the up direction, moves in a clockwise direction just as wind.

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Because the Earth rotates, an object moving in a rotating coordinate system with the Earth experiences an apparent force outward from the axis of rotation of the Earth, the centrifugal force, FCN = Ω2R. That object also experiences gravity trying to move it toward the center of the Earth. The resultant of these two forces can be expressed as a single force, FEG, effective gravity. Since the centrifugal force is small compared to gravity, the direction of the effective gravity force is close to, but not exactly in the same direction as gravity. We, air molecules, etc. experience this and call it the force of gravity, except it is really the result of true gravity and the Earth’s motion.

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An object moving around and experiencing the same “effective” gravitational force on it is said to be moving on a surface of constant geopotential. A surface of constant geopotential is a surface along which a parcel of air could move without undergoing any changes in its potential energy. Also known as equigeopotential surface or a level surface.

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**Because the Earth rotates, it bulges at the equator.**

For large scale motions, the spherical coordinate system works well for identifying the location of air parcels. For small scale motions, the Cartesian coordinate system works well, even though the axes of the coordinate system must really change direction as parcels move about in the atmosphere.

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Pressure coordinate. In the atmosphere in the vertical, pressure is easier to measure than height. Often, the location of a parcel of air, or other characteristics of the air, is expressed in terms of the pressure where it is located rather than the height above some surface (as sea level). Isentropic coordinate. An isentrope is a surface along which the potential temperature is constant. Parcels remain on isentropic surfaces as long as conserved quantities (like mixing ratio) don’t change; i.e., the parcel doesn’t become saturated and start converting water vapor to liquid or ice.

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**(3) Hydrostatic equation**

The hydrostatic equation represents a balance of the vertical pressure gradient force (acting upward) and the gravitational force (effective gravitational force) acting down. When these two forces are not in balance, there is an acceleration of a parcel of air either upward or downward. This is expressed by the equation below from chapter 6.

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When they are in balance, the right side of the equation is zero and we have the hydrostatic equation. Using the Ideal Gas Law Equation to get an expression for density (as done in chapter 5) and substitute that in for density gives.

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**Divide both sides by gives.**

This can be written as:

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**(4) Hypsometric equation**

Integrate from some low level to some high level. Multiply both sides by -1, reverse the limits on the left (gets rid of the - sign on the left) and perform the integration on the right. Gives:

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**Graphically, integration can be expressed as “computing the area under a curve.”**

For the integration of the term on the right, the graph would look like the following. The “integrand” is constant and equal to 1 along the distance (z2-z1).

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For the left side of the equation, there is the temperature term, and temperature changes as we move vertically in the atmosphere. However, we can simplify the equation by using the average value of temperature between the levels represented by ln(P1) and ln(P2).

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**Then, we can write the equation as:**

The quantity is just a constant by which we are multiplying the integrand between the limits ln(P1) and ln(P2). Integrating the left side between those limits gives:

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**Graphically, it would look like:**

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Geopotential height. Remember, a surface of constant geopotential is a surface along which a parcel of air could move without undergoing any changes in its potential energy The geopotential height is the height at which an object (such as a volume of air) would have the same geopotential (assuming constant value for gravity) as it would in the real atmosphere (where gravity changes). It is expressed as:

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Virtual Temperature The real atmosphere has water vapor in it, which varies. Therefore, Rd is not a good value to use for the gas constant to express how the volume is going to behave. An R value should be used that expresses how each particular volume would behave based on the particular molecules (including water vapor of varying amounts) it has in it. However, that means calculating a new R value for every different volume of air, or coming up with an equation that expresses how R changes with different types of air.

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**We can get around that problem with water vapor and still use Rd by using virtual temperature.**

Virtual temperature is defined as the temperature a dry parcel of air would have if it had the same density and pressure as a moist parcel of air. Virtual temperature is always greater than actual temperature of the parcel because having moisture in the parcel makes it less dense. Thus, virtual temperature accounts for the effect of having water vapor in a volume of air.

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**Virtual temperature is given by:**

Where “ω” is the mixing ratio of the air. Sometimes it is expresses as g/kg. Thus, by using virtual temperature, we can use Rd and still account for the effects of water vapor being in the volume, as long as that water vapor amount doesn’t change.

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So, now we can use gravity as a constant as long as we realize we are dealing with geopotential heights (Z), not true height(z) - as measured with a ruler. And, we can use Rd as long as we use Tv, virtual temperature, to account for moisture.

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Hypsometric equation This integrated form of the hydrostatic equation shows that the vertical distance between two pressure surfaces in the atmosphere is proportional to the average temperature between those two surfaces.

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By knowing pressure and temperature, the height between pressure surfaces can be calculated (with only a small error if water vapor is ignored). By also knowing the water vapor amount (which can be obtained from knowing the dew point) a more accurate determination of the thickness between pressure surface can be made.

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By knowing the pressure at the ground, and temperatures up through the atmosphere, one can work their way up through the atmosphere determining the thickness of each layer between pressure surfaces and thus the height of pressure surfaces. It’s easy to see that where temperatures are cold, pressure surfaces are going to be at a lower height in the atmosphere. And, where temperatures are warm, pressure surfaces will be at a higher height in the atmosphere.

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(8) Critical thickness The thickness of a layer is used for forecasting (e.g., snow) because it gives a good indicator of the average temperature of the layer. The smaller the thickness between the layers, the colder the air. Some use mb thickness. Some use mb thickness. Some use both. The critical thickness varies with the elevation of the station and the stability of the air mass.

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Large changes in the thickness across a short distance (thickness gradient) can also give an indication of the strength of a front. So, on a thickness chart, thickness contours close together indicate a rapid change in the average temperature of the air perpendicular to those thickness contours.

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Homework Do: 1, 3, 4, 5, 6, 7

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