CAPE, LI, Helicity, Storm Motion, Vorticity, Thickness
CAPE
| CAPE stands for Convective Available Potential Energy. It is a measurement of the amount of engery available to a buoyant parcel of air during the process of convection. CAPE is measured in joules per kilogram (J/kg) The higher the amount the more productive the atmosphere to severe weather i.e. the higher the figure the more unstable the atmosphere is. |  Click on map to enlarge
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LIFTED INDEX or LI
LI is the difference in temperature between the air within a bouyant parcel of air and the observed temperature surrounding environment at a height of 500mb. The lifted index temperature is worked out by subtracting the parcels temperature from the environmental temperature. Negative values suggest an unstable atmosphere or positive buoyancy. Positive Lifted Index values suggest a stable atmosphere or negative buoyancy.
A lifted Index value that is positive represents stability of the troposphere with respect to boundary layer convection. A negative LI indicates instability. A LI of 0 is neutral. LI should only be used when forecasting in the warm season or in the warm sector of a mid-latitude cyclone. Parcels of air will NOT rise from the lower PBL on the colder side of a frontal boundary or within dense PBL polar air. The LI is best used when the troposphere has the potential to produce boundary layer based thunderstorms. The more negative the LI, the more potential acceleration an air parcel has if lifted to the Level of Free Convection (LFC). LI values from -1 to -3 are unstable, -4 to -6 are very unstable, -7 or less are extremely unstable. The LI value says nothing about if storms will occur. It gives a forecaster a general idea of convective forcing if thunderstorms do develop. Unstable LI values combined with a high RH indicates the troposphere is near saturation and has instability. A "trigger mechanism" such as a front will be able to produce thunderstorms and heavy rain in this high RH low LI environment. Again, LI is not of much use in the winter because the PBL tends to be dry (low dewpoints) and cold (stable). "Elevated convection", "dynamic forcing without thermodynamic forcing" and "isentropic lifting" do not mesh well with using the LI. The LI can be very stable but the troposphere produces precipitation when the three terms in parenthesis above occur.
HELICITY and STORM MOTION

HELICITY
Helicity is simply a measure of the amount of rotation found in a storm's updraft air. If there is significant rotation in a storm's updraft air, the storm will more than likely become a supercell and possibly spawn one or more tornadoes.
Helicity is a parameter that defines the amount of streamwise vorticity (i.e., directional shear) a steady storm updraft will ingest as a result of a given storm motion. For a straight hodograph, if the updraft moves off the hodograph to the right of the mean shear vector, it will tend to be correlated with cyclonic rotation; if the updraft moves to the left of the hodograph, the updraft rotation will tend to be anticyclonic. If the hodograph has clockwise (counterclockwise) curvature and the updraft moves within the curve of the hodograph, it will likewise be correlated with cyclonic (anticyclonic) rotation. This correlation between the mean shear vector and the updraft was derived analytically by Davies-Jones (1984) and the measure of this correlation has become known as the Storm-Relative Environmental Helicity (SREH), normally just referred to as "helicity." On a hodograph, the helicity value is proportional to twice the area swept out by the storm relative wind vectors from the surface to a specific height, usually 3 km.
STORM MOTION
STORM MOTION is the average wind speed in knots a storm will move and the direction the storm will move from. The storm moves slower than the ambient wind speed since a storm has a large mass of water that has to be pushed along. The turbulence within a storm also makes it more difficult to push along. Storms will move more quickly in cases where there is speed shear with height (wind speed increases with height).
The storm motion is given as the compass direction from which the storm will move from. The meteorological compass has 90 degrees being a wind from the east, 180 degrees being a wind from the south, 270 degrees being a wind from the west and 0 degrees / 360 degrees being a wind from the north..
Strong storms will veer (move to the right of the original path of motion) due to storm dynamics.
Storm motion gives insight into which direction supercells and tornadoes will move from on days in which supercell thunderstorms are favorable.
This information was taken in parts from the website of METEOROLOGIST JEFF HABY
Vorticity and thickness maps
Vorticity and thickness maps
| Relative vorticity is a measure of the rotation of fluids about a vertical axis relative to the earth's surface. Colors indicate the strength of relative vorticity, red for positive (counterclockwise rotation) and blue for negative (clockwise rotation) vorticity, respectively. | 
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Positive vorticity at the 500 hPa level are often associated with cyclones and troughs in the 500 hPa topography (see right-hand picture).
Positive Vorticity develops in a wind field with counterclockwise curvature and/or due to shear with higher velocities on the right, as seen in flow direction. |  positive Vorticity due to curvature
positive Vorticity due to shear
 positive Vorticity due to curvature
positive Vorticity due to shear
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Negative Vorticity develops in a wind field with clockwise curvature and/or due to shear with higher velocities on the left, as seen in flow direction.
Negative vorticity at the 500 hPa level is often associated with fair weather and ridges in the 500 hPa topography. |  negative Vorticity due to curvature
negative Vorticity due to shear
 negative Vorticity due to curvature
negative Vorticity due to shear
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Vorticity is an important measure and used to locate dynamically active zones and fronts. The omega-equation, an equation used to diagnose vertical motion (or the so called omega, in pressure units) links vorticity and vertical motion. It says that:
greater upward velocity occurs where there is greater advection of positive vorticity by the thermal wind |
The geostrophic vorticity at the 700 hPa level is often used as a representative value for the omega equation. Now the thermal wind is only a mathematical construct (vector difference between geostrophic winds at two different heights or pressures) and not an actual wind. To examine the thermal wind, thickness maps are needed: |  Thickness map (colors) with thermal wind (arrows) and surface pressure (black contours).
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A thickness map between two different pressures (e.g 1000 and 500 hPa) is a measure of the average virtual potential temperature within that layer, where blue is cold and red is warm. As can be seen the thermal wind is parallel to the thickness contours, with cold air to the left in the northern hemisphere. Closer packing of thickness colors indicates a stronger horizontal temperature gradient and thus a stronger thermal wind. By the thermal wind relationship, the horizontal temperature gradient causes the geostrophic wind to change with altitude (how much is shown by a thermal wind vector). | 
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Note that if thickness lines (layer temperature) cross pressure lines, there is a temperature advection (a transport of temperature by the wind. The wind is parallel to pressure lines and stronger if isobars [lines of constant pressure] are closer together). In the thickness map shown above, there is cold air advection over Great Britain.
Greater upward velocity favors clouds and heavier precipitation and that's another good reason to look for vorticity. It may be complicated to evaluate vertical motion from vorticity, but this has historical reasons. If you like it simple examine the plots of vertical velocity.