I. Convective Available Potential Energy (CAPE)
(i) The acceleration an air parcel experiences due to density differences at a given level 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 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 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).
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 equation tells you 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.
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.