(i) The acceleration an air parcel experiences due to density differences at a given level (buoyancy accelerations) can be related to the difference in temperature of the air parcel , Tap, with respect to the temperature of the surrounding air, Te, AT A GIVEN LEVEL.:
(ii) A true measure of the potential buoyancy is a measure of the "positive" area on a Skew-T Ln P diagram. This represents the portion of the parcel ascent curve in which the parcel is warmer and, thus, less dense than the air surrounding it.
The parameter is known as Convective Available Potential Energy (CAPE) or Positive Buoyancy (B+).
Note that this equation really states that CAPE is directly proportion to the total acceleration a parcel would experience due to buoyancy from the LFC to the EL.
The positive area represents a potential source of energy for parcels at the ground that are lifted to the elevation (LFC) above which they become warmer than their surroundings. To obtain this, one needs to algebraically add (that's what the summation sign means) this parameter at every level of the parcel's ascent until it reaches the point at which it becomes the same temperature as its surroundings again (Equilibrium Level).
Equation (2) states that CAPE is a function of the difference in temperature between the rising (or sinking) air parcel (Tap) relative to the air around it at the same elevation (Te). Note that the temperatures should be measured in Kelvin. CAPE is a measure of the area on a Skew T/Ln P diagram bounded by the curve of the ascending air parcel on the right and the environmental lapse rate (ELR) (sounding) on the left from the Level of Free Convection (LFC) (or, in the case of a forecast sounding for the afternoon based upon heating, the Convective Condensation Level c CCL) to the Equilibrium Level (EL). (Note: a similar expression is defined for Convective Inhibition Energy (CIN or CINH) in which the parcel curve lies to the left of the ELR.).
The summation sign in the equation means that the value of the expression in brackets needs to be evaluated at every level (and, of course, there are an infinite number of them) between the LFC (or CCL) and the EL. When you take calculus you will learn how to approximate the summation (called, finite differences) without making an extravagant number of calculations. But for the purposes of Homework 1, you will simply divide the atmosphere into 50 mb layers, from the LFC to the EL and calculate a "mean" CAPE for each of the layers and then sum those up to get the total or net CAPE.
The procedure is relatively simple and is best done on a spreadsheet program (for example, Excel). Of course, if you are unfamiliar with spreadsheet programs, then using an hand calculator or doing it by hand will work (but is very tedious).
We will start with a schematic sounding (Fig. 1)--Remember. this is only an example, and does not correspont to a real case. Only the portion of the sounding from the LFC to the EL is shown. Remember, the temperatures should be in Kelvin. So add 273o to convert from Centigrade.
Figure 1: Portion of schematic sounding from the LFC at 800 mb to the EL above the tropopause at 200 mb. Elevations of isobaric levels are shown. Procedure to estimate CAPE is summarized at bottom, as discussed in text.
The base of the portion of the sounding considered is indicated by the line at the 800 mb level. The thick solid line represents the ELR and the thin solid curve represents the wet adiabatic ascent curve of the rising air parcel. Note that the area of the polygon bounded by the two curves would be the CAPE.
To estimate the CAPE for a given layer, say the 650 mb to 500 mb layer, one would need to estimate the average value obtained from Equation 1 for the whole layer. A simple approximation of that would be to evaluate Equation 1 at the 500 mb level and at the 650 mb level. Add the result and average by dividing by 2. Multiplying the result by g (9.8 m s-2) and the thickness of the layer (in this case, 1500 m) would yield an estimate of the mean CAPE for the layer.
This procedure would need to be followed for each layer considered. In this example, there would be four values of layer CAPE estimates that you would need to add to estimate the total CAPE for the case.
In the case of this exercise, you will use layers 1500 m (1.5 km) thick. You'll be making the (bad) assumption that the conditions at each level represents conditions for a layer 1500 m thick centered at each level. Clearly, that is not estimating conditions correctly, but it is a start.
Procedure Detail for Homework 2 Assignment Using a Different Sounding (KOUN, 12 UTC 3 May 1999)
Step 1: After you have examined the sounding and determined the LFC (or CCL) and EL, you should tabulate the environmental temperature and air parcel temperature (remember, in Kelvin) for each pressure level from the LFC (or CCL) to the EL. Fill in the Table below.
|Pressure||"Layer" Number||Temp Parcel||Temp Env||Temp Diff||Temp Quotient*|
Table 1: Tally Table for CAPE Calculations
*Temperature Quotient = (Temp Diff/TempEnv)
Step 2: Using the information from Table 1, you will need to compute the values needed to
solve Equation 1 at each level. I suggest having a second table like the one given as Table 2.
|Layer||Delta z||gravity||Temp Quotient*||Layer CAPE**|
|9.8 m s-2|
|2||1500 m||9.8 m s-2|
|3||1500 m||9.8 m s-2|
|4||1500 m||9.8 m s-2|
|5||1500 m||9.8 m s-2|
|6||1500 m||9.8 m s-2|
Table 2: Computational Results (from Equation 1 and Table 1)
*Should really be Layer Average Quotient but what we will simply use is the values from Table 1
** Layer Cape = Layer Average Quotient X gravity X 1500 m
***Total CAPE = Add all the values above
Once you have complete Table 2, you simply sum the values in the fifth column to obtain your estimate of CAPE for the sounding. It's going to be an underestimate because we used values for levels instead of layer averages. You can see how a computer could do this better, by computing layer averages for layers that were very small.
II. Lifted Index
The difference [-(Tap Te) ]is called the LIFTED INDEX, commonly evaluated at the 500 mb level (negative for unstable conditions). You can calculate that from Table 1 for this particular case. Note that the algebraic negative sign in from of the expression means that the Lifted Index will return a negative value if the rising air parcel is warmer than its environment and vice versa.
The strength of the updraft will be greatest at the Equilibriium Level (EL). Thus, the greatest vertical velocities in cumulonimbus clouds are found at this level. Since, for each layer the air parcel traverses, it is given an additional acceleration (another bump up), the parcel will continue accelerating until the Equilibrium Level.
Above the Equilibrium Level the air parcel will have a negative (downward-directed) acceleration that will oppose the updraft. (In other words, the air parcel continues to rise because of its own upwards momentum, until the negative buoyancy due to the parcel finding itself colder than its surroundings produces a downward contribution of equal magnitude to the upward velocity.)
At that point, normally found about 1/4 to 1/3 the depth of the positive area on the sounding above the EL, the parcel will sink. The formula for the maximum upward motion (at the EL) due to buoyancy forces can be derived from the equations above. It is
w = [2 X CAPE]1/2
where w is the vertical velocity in the Cartesian Cooridinate system (x, y, z). For the homework problem, all you need to do is to multiply the CAPE value you got in Table 2 by 2 and then take the square root. Before you do that, make sure that the units work out.