Reading #2: Relations useful for understanding thunderstorms, interpreting weather maps and resulting from combination of equations discussed in Reading #1

A.  EQUATIONS RELATED TO DISCUSSIONS ON BUOYANCY: 

Generally, involving combinations of (2), (3), and (4).

 

(i) 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 , Tp, with respect to the temperature of the surrounding air, Te, AT A GIVEN LEVEL.:

 

                           

 

       The difference [-(Tap – Te) ]is called the LIFTED INDEX, commonly evaluated at the 500 mb level (negative for unstable conditions).

 

(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.

 

(iii) The vertical velocity due to buoyancy at the top of the "positive area" on a sounding can be evaluated from the following:

 

w = [2 X CAPE]1/2

 

where w is the vertical velocity in the Cartesian Cooridinate system

 

B. HYPSOMETRIC RELATION: Heating the atmosphere causes it to expand (special application of the gas law, explained below) obtained by substitution of (3) into (1).  This relation provides the basis of explaining many, many things that synoptic meteorologists see on weather maps and charts.

Thickness of layer between two pressure surfaces is directly related to the mean virtual temperature of the layer.  (For the purposes of this class, we will use mean temperature rather than virtual temperature. To be discussed in detail in JPM's portion of Metr 201). )

Also, if we consider the thickness of a layer that is often of importance to synoptic meteorologists, the layer approximately between the ground (1000 mb) and about 6 km (500 mb), the Hypsometric Relation is

where k = R/g ln 2

Applications:

  • Tropopause is higher over the Equator than at the Poles (generally, the tropopause corresponds in winter to the 300 mb surface)
  • Since the surface pressure is nearly 1000 mb, deep cold air masses are associated with areas of low heights (troughs) in the middle and upper troposphere (and vice versa)
  • Since the surface pressure is nearly 1000 mb, the 1000-500 mb thickness pattern can be used as a first guess approximation of the 500 mb height pattern
  • Since fronts are defined as the surface expressions of boundaries between deep air masses with significant temperature differences, as a first guess they can be found on the warm air side of the packing of, for example, 500 mb height (crudest), 1000-500 mb thickness (crude) fields
  • The relation of surface winds to thickness contours allows one to assess temperature advection, and to make a better estimate of both the location and type of surface fronts (Note: In reality, finding surface fronts requires a careful analysis of actual temperature fields, instead of the layer mean temperatures inferred from thickness or height maps. Here is the NCEP analysis; note that the general position of and type of fronts for the eastern two thirds of the US was well "guessed", but the complications associated with the actual wind and temperatures in southwest TX made our "first guess" poor there)

C. PRESSURE TENDENCY EQUATION: With respect to the sea level weather map, pressure changes occur because of net accumulation  or net deficit of air in the air column above.

The PRESSURE TENDENCY EQUATION is obtained by manipulation of (5) and the substitution into (5) of (1) to eliminate the density. (To be discussed in detail in JPM's portion of Metr 200/201).



The pressure tendency equation allows one to assess how pressure changes at a given level develop--thus accounting for the evolution of the surface pressure field, for example.   It is the result of an Application of the Principle of Conservation of Mass.

Since upper tropospheric divergence tends to be larger than the compensating lower level convergence, generally speaking, areas of divergence in the upper tropsophere identify regions in which the air columns underneath will be experiencing a net export of mass (or weight). Thus, surface low pressure areas tend to develop under regions of upper tropospheric divergence and vice versa.  (Careful, this relates to PRESSURE TENDENCIES, not necessarily to the position of Highs and Lows on weather maps).The Delta p (Depth) refers to the height of the air column in pressure coordinates, or, the difference in the the pressure from the bottom to the top (say, 1000 mb at bottom to 200 mb at top would yield a difference of 800 mb).