`This essay is entirely my own thoughts, so the standard
disclaimers apply. I am writing a Web essay rather than a formal
publication because I doubt that this is material worthy of even a
"Note" in some journal. Instead, I am hoping to correct some possible
misuse of the virtual temperature correction.`

In 1994, Erik Rasmussen and I published a short paper on the use of the virtual temperature correction for calculating convective available potential energy [CAPE].

Doswell, C.A. III, and E.N. Rasmussen, 1994: The effect of neglecting the virtual temperature correction on CAPE calculations.

Wea. Forecasting,9, 619-623.

It recently has come to my attention that there may be some continuing confusion about precisely how to apply the virtual correction to the problem of calculating CAPE and other parameters (such as convective inhibition [CIN], the lifting condensation level height [LCL], the level of free convection [LFC] height, or the equilibrium level [EL] height).

Therefore, I want to begin this discussion with some tutorial material, before I go on to the specific issues. An excellent treatment of the topic can be found in Hess (1959; §4.3):

Hess, S.L., 1959:

Introduction to Theoretical Meteorology.Holt, Rhinehart, and Winston, 362 pp.

Just what is the virtual correction? What purpose does it serve? The Equation of State should be familiar to most of my readers; it is:

where *p* is the pressure, *r* is the density, *R* is the so-called
Gas Constant (see below), and *T* is the temperature. I'm not
going to go into the issue of units here, although they can be
tricky, because most meteorologists think of pressure in terms of
millibars, which makes things a little difficult. The issue in
applying this version of the Equation of State is that the Gas
Constant is not exactly constant, when the gas involves a mixture of
air (which is, in turn, a mixture comprising mostly nitrogen, oxygen,
argon, and carbon dioxide) with some highly variable amount of water
vapor.

In the troposphere, the relative percentages of the dry components
in air remain pretty much constant (ignoring the anthropogenic
increase in CO_{2}) and so dry air has, in fact, a pretty
much constant value for its Gas Constant, *R _{d}, *where
the subscript denotes the fact that it applies only to

The mixing ratio, *q*, is defined as the mass of water vapor
per unit mass of dry air:

where *p* is the total air pressure, *e* is the partial
pressure of the water vapor constituent, *M _{v}* is the
mass of water vapor,

where the summations are over the *N* dry constituent gas
components (oxygen, nitrogen, etc.). The ratio
*m _{v}*/

The dry form of the Gas Constant, *R _{d}*, in the
Equation of State is derived from the so-called Universal Gas
Constant,

A similar process is required to incorporate an adjustment for the variable amount of water vapor. Basically, we have to find a mean molecular weight, , for the mixture including water vapor, where

In order to adjust for the variable amount of water vapor, then, it can be shown that the Equation of state would become

In deriving this result, use was made of an approximation; since the partial pressure of water vapor is typically about two orders of magnitude less then that of the dry air constituents, then

Rather than incorporating this correction term into the Gas
"Constant", thereby creating the rather awkward situation of a
*variable* "Constant", we incorporate it into the temperature,
which we call the *virtual temperature*:

so the Equation of State is simply

When *q* is expressed in terms of g g^{-1}, rather
than g kg^{-1}, it is obvious that *q* is a number much
smaller than unity. Typical atmospheric values of *q* range from
a low of near zero up to values as high as, say, 30 g
kg^{-1}, or 0.03 g g^{-1}.Therefore it is common to
make the simplifying approximation:

so that

Since *q* is a small number, we make the additional
simplification of neglecting the term involving *q*^{2}.
Note also that

Therefore, a good approximation to the virtual correction is what Erik and I used in our paper:

but we used the notation that = 0.608 rather than the more standard notation = 0.622.

Therefore, the virtual correction allows use of the Equation of State, without adjusting the dry air Gas Constant. The virtual temperature can be thought of as that which would be used to find the density of a parcel of air at a constant pressure level. Using the uncorrected temperature would give an inaccurate value for density, unless a correction was made to the Gas Constant for the variable contribution to density associated with water vapor. Since water molecules are lighter than those of dry air (which, recall, is a mixture of gases), the presence of water vapor means that the virtual temperature is always slightly higher than the actual temperature, corresponding to a slight decrease in density owing to the presence of water vapor.

This is important when calculating anything associated with density, notably such things as CAPE, CIN, etc. from soundings, as Erik and I discussed in our paper.

From recent experience, I believe that a number of people are applying the virtual correction to the sounding prior to doing any calculations of such things as CAPE and CIN. It's quite possible to do this incorrectly. What follows will be my attempt to clarify what is going on.

Imagine a pseudo-real world, in which pure parcel theory applies
with perfect accuracy. Within this pseudo-real world, any
"observations" would match perfectly with what is depicted on a
standard thermodynamic diagram, such as a Skew-*T*, Log *p*
diagram. In order to explore the issue of what is the proper
calculation of CAPE, CIN, etc., we need to begin by selecting a
parcel to lift. There are many ways to do this, as mentioned in
Doswell and Rasmussen (1994), and the choice may have a large impact
on the resulting calculations. Of all the possible ways to do this,
most involve choosing either a parcel rising from a particular level
in the sounding (e.g., the surface parcel), or a parcel that
represents a "well-mixed" parcel in some arbitrarily chosen layer
(e.g., the lowest 100 mb). In either case, the result is that there
are constant values of the mixing ratio and potential temperature
(implying a particular dry adiabat on the thermodynamic diagram)
associated with that parcel that we are going to lift.

At the start of its ascent, in general, the parcel is not
saturated; that is, its dewpoint is less than that of its
temperature. The mixing ratio associated with the parcel is
determined by the value of the *saturation* mixing ratio at the
dewpoint temperature. As the parcel ascends along this dry adiabat,
the parcel's mixing ratio and potential temperature are assumed to
remain constant. Hence, the dewpoint temperature of the rising parcel
during its unsaturated ascent is determined by the temperatures along
that line of constant mixing ratio. On a thermodynamic diagram, a
mixing ratio line is simply the dewpoint temperature lapse rate for a
parcel with a fixed value of the mixing ratio. The process of ascent
means that the parcel's air temperature decreases at the dry
adiabatic rate. Since the saturation vapor pressure of a parcel is a
function of its temperature alone (or very nearly so), the path of
the dry adiabat crosses mixing ratio lines, reflecting a decrease of
its *saturation* mixing ratio. When the parcel is lifted far
enough, the dry adiabat along which it is ascending eventually
intersects the parcel's dewpoint temperature (along the mixing ratio
line); the parcel's *saturation* mixing ratio has become equal
to its *actual* mixing ratio. That is, the parcel has reached
its *saturation point*. It should be noted that the saturation
point is determined uniquely by the parcel's original pressure,
temperature, and dewpoint (see Betts 1982). The saturation point is
also the *lifting condensation level*, or LCL.

Betts, A.K., 1982: Saturation point analysis of moist convective overturning.

J. Atmos. Sci.,39, 1484-1505.

Further ascent, above the saturation point, proceeds along a
*moist* adiabat. Note that the particular moist adiabat
(associated with a unique value of equivalent potential temperature,
*q _{e}*, or wet-bulb
potential temperature,

Although the moist adiabatic lapse rate is less than that along a
dry adiabat, it still involves a decrease in temperature, so moist
adiabats still must cross mixing ratio lines. That is, even though
the parcel ascending a moist adiabat remains saturated, its mixing
ratio is constantly decreasing. For a *pseudoadiabatic* process,
it is assumed that any water vapor that condenses out during the
parcel's ascent immediately falls out of the parcel. For a
*reversible adiabatic* process, it is assumed that all of the
condensed water remains within the parcel. Real moist adiabatic
processes typically lie somewhere in between these two extremes.
Although on most thermodynamic diagrams, the moist adiabats are
derived from the pseudoadiabatic assumption, the difference between
that and a reversible moist adiabat is generally minor.

As ascent proceeds along the moist adiabat, the parcel remains saturated, so its mixing ratio is determined everywhere along the ascent curve by the temperature (which equals the dewpoint temperature, for a saturated parcel) along that moist adiabat. If we were to do measurements with a perfect thermometer in my pseudo-real world, the temperatures we would actually find within ascending air parcels should correspond exactly to those shown on the uncorrected parcel ascent curves on a standard thermodynamic diagram.

The use of the virtual correction is required any time that parcel
density is involved in the calculation. It is *not* required if
we simply want to know what the process looks like in the psuedo-real
world. In particular, it is not necessary to determine a parcel's
saturation point (or LCL).

On the other hand, if we are interested in relative
buoyancy,^{*} we want to examine the relative density between
the parcel and its environment. For determination of CAPE, for
instance, we require

where is
associated with the parcel ascent curve and *T _{v}* is
associated with the environmental sounding curve. Hopefully, it's
clear that in order to determine the height of either the LFC or the
EL, it will be necessary to apply the virtual correction to the whole
of

`Figure 1. Schematic sounding, showing the processes with and
without the virtual correction (see the key). The parcel ascent curve
is for the surface parcel.
`

Since the relative buoyancy (CAPE) requires a comparison of the
relative *virtual* temperatures, both curves need to have the
virtual correction done *before* we can determine where the
parcel ascent curve crosses the sounding curve. Since the parcel
usually becomes saturated somewhere below the LFC, while the sounding
generally is not saturated, the virtual correction to the
*parcel* ascent curve is typically larger than that applied to
the sounding. Hence, we would expect to find the LFC in the corrected
curves at a *lower* level than in the original, uncorrected
traces. The virtual correction pushes the parcel ascent curve to the
right (on the diagram) farther than the virtual correction shifts the
*sounding*. This results in a lowering of the LFC, as shown in
Fig. 1.

There are several things to note about this schematic ... the
mixing ratio generally decreases as one gets up into the mid- and
upper troposphere, and the schematic example sounding shown is quite
dry above the moist surface boundary layer, so the virtual correction
to the sounding curve pretty much becomes negligible above the moist
layer. However, for the *parcel* ascent curve above the
saturation point (LCL), the parcel is assumed to be saturated, so
there is a non-negligible virtual correction through a much greater
depth than there is for the sounding. The EL for the corrected ascent
curve is virtually the same as for the uncorrected curve because the
EL generally is in the upper troposphere, where the mixing ratio even
in a saturated situation is negligible. However, for this example,
the LFC is notably lower whereas the LCL remains the same, for
reasons already discussed. The result is that the corrected CAPE is
notably higher, whereas the corrected CIN is not all that much
different. Not all soundings will have precisely these
characteristics, as discussed in Doswell and Rasmussen (1994) but
many will in situations where deep convection is possible.

Also, observe that the apparent saturation point of the corrected
*parcel *ascent curve is shifted to the right on the Skew-T, Log
p diagram as a result of the virtual correction. Because movement to
the right on the diagram is toward higher mixing ratios, this might
suggest to the unwary that the implied mixing ratio associated with
the saturation point has increased. This appearance is somewhat
deceptive; *the true saturation point remains unchanged*, but if
we want to compute the density at the saturation point, we must use
the *virtual* temperature, not the actual temperature. All we
are doing is applying the virtual correction to the parcel ascent
temperature curve. Since the correction is always non-negative, this
results in a rightward shift of the trace (each point moves to the
right at the *same pressure* where it started), but this does
*not* imply an increase in the actual dewpoint. __The virtual
correction is ____never____ applied to the sounding
dewpoint trace!__ Keep in mind that the virtual correction is only
used to calculate density, not to infer the thermodynamics of the
saturation point during the psuedo-real parcel ascent.

The process of making the virtual correction to the parcel ascent
trace occurs only *after* having computed the *uncorrected*
parcel ascent curve. __Never use the ____corrected____
sounding profile to compute the parcel ascent curve!__ The virtual
correction to the parcel ascent curve uses the dewpoint of the
ascending parcel (which is along the mixing ratio line below the
saturation point, and is equal to the temperature along the moist
adiabat which the parcel ascends at and above the saturation point).
The LCL is the same as that found using the *uncorrected* parcel
ascent process, whereas the CAPE, CIN, LFC, and EL should be found
from the *corrected* sounding and parcel ascent traces.

Hopefully, this clears up any misunderstandings. If any remain, you can contact me at cdoswell@hoth.gcn.ou.edu.

___________________________

^{*} I am describing the calculation of CAPE as the
determination of *relative* buoyancy. That is, CAPE involves the
difference in temperature between the parcel and some base state
sounding (in this case, the "environmental" sounding). There is no
simple way to define the notion of an "environment" for convective
situations and, therefore, the notion of the base state sounding is
actually rather nebulous and is actually unphysical. An ascending
parcel does *not* need to know its temperature *relative*
to anything outside of the vertical column within which it is found,
in order to be buoyant. In reality, buoyancy is an **unbalanced
vertical pressure gradient force**, and so an ascending parcel's
buoyancy must be *independent* of any "environment" or "base
state"! The traditional derivation of buoyancy as the difference in
temperature between the parcel and the "environment" is an incomplete
and misleading representation of buoyancy. Paul Markowski and I have
submitted a
paper
on this subject for publication.