Tropical Meteorology
Bretherton
Lecture 1. Review of moist thermodynamics
Moist thermodynamics for cumulus convection
A good understanding of moist thermodynamics and of commonly used moistthermodynamic quantities is essential for the study of cumulus convection and reading its
literature.
Take-home points
1. Moist air can hold 7% more water vapor per degree it is warmer.
2. Water vapor makes air less dense; condensate makes it more dense; these effects are
accounted for by using of virtual or density temperature to compute air parcel buoyancy.
3. Equivalent potential temperature θe and moist static energy h are approximately
conserved (except for latent heating by freezing/melting) and linearly mixing for typical
conditions of tropical convection. For most purposes, θe and moist static energy h/cp are
functional equivalents, as are θe and moist static energy h/cp.
4. The temperature excess of an adiabatically lifted boundary-layer air parcel (a measure
of conditional instability that neglects virtual effects) can be assessed above its LCL by
comparing its θe or h with the environmental profiles of saturated ! e* or h*. This type of
plot is commonly used in tropical meteorology.
We roughly follow the discussion in Sec. 2.1-2.2 of Cotton and Anthes, Storm and Cloud
Dynamics (Academic Press, 1989).
(a) Moist air as a mixture of components at thermodynamic equilibrium
A rigorous treatment of moist thermodynamics starts by considering moist air as a
thermodynamic system at a pressure p and temperature T composed of a density ρd of dry
air, ρv of vapor, ρl of liquid water and ρi of ice. In C&A the subscript a is often used in
place of the customary d for dry air. We define the water vapor mixing ratio qv = ρv/ρd
and similarly for the mixing ratios ql, qi of liquid and ice. All of these mixing ratios are
small (typical qv ~ 10-2 = 10 g kg-1, ql, qi~ 10-3 = 1 g kg-1).
The ideal gas law is assumed to hold for dry air and water vapor, with molecular weights
md = 29 daltons and mv = 18 daltons and hence gas constants
Rd = Ru/md = 287 J kg-1 (dry air)
Rv = Ru/mv = 461 J kg-1 (water vapor),
where Ru is the universal gas constant and are the molecular weights of dry air and water.
The gas constant and heat capacity of moist air (per unit of dry air) are
Rm = Rd + qvRv ,
cpm = cpd + qvcpv + qlcl + qici .
The isobaric heat capacities are:
cpd = 1005 J kg-1 K-1 (dry air),
cpv = 1870 J kg-1 K-1 (water vapor)
cl = 4190 J kg-1 K-1 (liquid)
ci = 2100 J kg-1 K-1 (ice)
For water, the latent heat of vaporization is L = L0 + (cpv- cl)(T – T0), where T0 = 273.15 K
is the freezing temperature and L0 = 2.5×106 J kg-1. Thus L decreases by ~0.1% for each 1
K increase of temperature. The latent heat of freezing Lf = Lf0 + (cl - ci)(T – T0), where
Lf0 = 0.33×106 J kg-1. Lf increases by ~0.7% for each 1 K increase of temperature.
The saturation vapor pressure e*(T) satisfies the Clausius-Clapeyron relation
d ln e*
L
=
! 0.07 K "1 (doubles every 10 K)
2
dT
RvT
(1)
Bolton (1980, JAS) provided the following useful approximate formula:
" 17.67(T ! 273.15) %
e* = 6.112 hPa exp $
'& .
#
T ! 29.65
(2)
The saturation mixing ratio is
q* =
Rd e*
Rd e*
.
"
Rv p ! e* Rv p
(3)
At 1000 hPa and 20 C, q* = 15 g kg-1; at 30 C, q* = 28 g kg-1; a typical 10 m q seen over
the warm tropical oceans is about 20 g kg-1, corresponding to a dew point of 24 C.
(b) Virtual and density temperature
Since small differences in the density between moist air parcels are what drive density
differences, it can be important to consider the contribution of water to the air density,
including ‘virtual’ effects due to vapor and ‘loading’ effects due to condensate:
p = pd + pv = ρdRdT + ρvRvT = ρdRdT(1 + qv md/mv),
ρ = ρd + ρv + ρl + ρi = ρd(1 + qv + ql + qi)
= (p/RdT )(1 + qv + ql + qi)/ (1 + qv md/mv).
Since md/mv = 1.61 is larger than 1, more water vapor decreases the density, while the
extra weight or ‘loading’ of condensate (liquid + ice) adds to the density.
This is often expressed in terms of a density temperature Tρ = p/ρRd that is defined to
satisfy the ideal gas law for dry air:
Tρ = T(1 + qv md/mv)/(1 + qv + ql + qi) ≈ T(1 + 0.61qv - ql - qi).
The density temperature is the temperature of dry air that has the same density as the
moist air at the given pressure. The virtual temperature Tv is like the density
temperature, but without the condensate loading terms (many authors are sloppy about
this distinction). Some scientists also use virtual potential temperature θv and potential
density temperature θρ, in which θ replaces T in (2.4).
(4)
A 1 g kg-1 water vapor mixing ratio increase raises the virtual or density temperature by
0.18 K at near-surface temperatures; the loading due to 1 g kg-1 of suspended liquid or ice
would lower the density temperature by 0.3 K.
(c) The heat equation
The first law of thermodynamics for moist air is (C&A eq 2.34):
c pm dT !
dp
+ L dqv ! L f di qi = Q
"
(5)
The symbol di indicates an internal thermodynamic phase change (e.g. freezing of water
droplets) but does not include external exchanges such as removal by precipitation. Q
denotes diabatic heating. This is exact so far.
If we divide (5) by T and use the ideal gas law, we get the equivalent form
c pm d lnT ! Rm d ln p +
Lf
L
Q
dqv !
di qi =
T
T
T
It is possible to go further without approximation, but it is cumbersome enough that only
the most dedicated thermodynamicists do so. Instead, we usually approximate Rm ≈ Rd
and cpm ≈ cpd. Introducing potential temperature θ:
(
)
Lf
L
Q
d lnT ! Rd c pd d ln p +
dqv !
di qi =
!###
#"####
$ c pd T
c pd T
c pd T
d ln "
(6)
(d) Equivalent potential temperature (EPT)
The equivalent potential temperature θe is the potential temperature achieved by
adiabatically lifting moist air up to zero pressure without addition of heat Q, condensing
water vapor above saturation. Traditionally, ice is not considered because in subfreezing
clouds, ice and water are often far from bulk thermodynamic equilibrium (e.g.
supercooled droplets can exist down to -40°C). Hence (6) implies
ln ! e = ln ! + "
pLCL
0
L
dq* ,
c pd T
(7)
A common further approximation is to take L ≈ L0 (constant) and approximate T in the
integrand by the saturation temperature T* at the lifted condensation level, which implies
ln ! e " ln ! +
Lqv
c pd T *
$ Lqv '
# ! e " ! exp &
*)
% c pd T (
(8)
Eq. (8) can underestimate by θe for warm moist air by up to 3 K. Bolton (1980) worked
out a better approximation that I use (Matlab function theta_es.m on class web page):
! pref $
θe = T #
" p &%
0.2854(1' 0.28qv )
(
! 3376
$+
exp * qv (1 + 0.81qv ) # * ' 2.54 & - ,
" T
%,
)
(9)
where pref = 1000 mb and the saturation temperature is estimated as follows:
!1
" 1
ln(qv q* ) %
T =$
!
+ 55 K.
2840 '&
# T ! 55
*
(10)
θe is conserved in equilibrium vapor-liquid phase changes, and it is also approximately
conserved during precipitation formation and evaporation. Thus it is an excellent tracer
in moist air motions. However, freezing or sublimation can cause θe changes up to a few
K; from (2) we see that
d ln ! e "
Lf
c pd T
di qi +
Q
c pd T
(11)
If mass m1 of an air parcel with EPT θe1 is mixed with mass m2 of an air parcel with EPT
θe2, the mixture EPT is to a very good approximation (but not exactly) ‘linearly mixing’,
i. e. θem = (m1θe1 + m2θe2)/ (m1 + m2).
The saturation equivalent potential temperature ! e* is the θe of air of a given T and p
that is saturated with respect to liquid water, so
! e* ( p,T ) = ! e ( p,T , qv = q* ( p,T ))
(12)
At a given pressure, this is an increasing function of T.
The 350 K ‘moist pseudoadiabat’ is the temperature profile corresponding to ! e* = 350 K
(a typical near-surface θe value in regions of deep tropical convection), and similarly for
other temperatures. These are the moist adiabats usually shown on skew-T and other
sounding plots. Because warmer air can hold more water, the temperature decreases less
rapidly with p on a warmer moist pseudoadiabat. Comparing two moist pseudoadiabats,
the temperature rise in the upper troposphere will be about twice as large as at 900 hPa
(Slide 1.7, left).
Some authors use the moist entropy s = cp log θe/θ0 instead of θe.
(e) Moist static energy
Moist static energy (MSE) is a close relative to θe. It is useful because even in the
presence of moist convection, the atmosphere is nearly in hydrostatic balance, such that
dp = -ρg dz.
(13)
Bernoulli’s formula suggests that nonhydrostatic pressure perturbations pnh will be
O(ρV2/2), where V is the convective velocity perturbation. Even in the strongest
thunderstorms, V < 30 ms-1 and ρV2/2 < 5 hPa, which is small compared to the pressure
changes felt by an air parcel ascending several km through a cumulus updraft. For most
tropical convective drafts, the nonhydrostatic pressure perturbations are well under 1 hPa
and the hydrostatic approximation is quite accurate. Applying (6) to the heat equation
(5), we obtain
c pm dT + gdz + Ldqv ! L f di qi = Q
(14)
Approximating L = L0 (constant) and cpm ≈ cpd we can define moist static energy (MSE)
h = cpdT + gz + Lqv
(15)
and integrate (14) to obtain:
dh ! L f di qi " Q
For adiabatic displacements (Q = 0) of a moist air parcel, with no internal ice production
or melting (diqi = 0), we deduce that MSE is approximately conserved, even when there is
liquid precipitation or evaporation. It is also linearly mixing. MSE can be interpreted as
the sum of a dry thermodynamic energy cpdT, gravitational potential energy gz, and latent
energy of vaporization Lqv per unit mass. It is expressed in units of kJ/kg (roughly
equivalent to K).
We can write
h = s + Lqv,
where s = cpdT + gz is the dry static energy or DSE.
Similarly, saturated MSE
h* = cpdT + gz + Lq*
(14)
is analogous to cp ! e* . Saturated MSE is approximately constant along a moist adiabat.
This provides a nice formula for the lapse rate along a moist adiabat:
dh*
0=
dz
g + Lq* / H
L *q* Lq*
q*
" dT %
! $
=(
, ) =
+
+
# dz '& ma
c p (1 + ) )
c p *T c pTe 6 g kg (1
(15)
Here H is the density scale height and Te = RvT2/L ≈ 15 K is the e-folding temperature for
saturation mixing ratio. The term Lq*/gH is roughly 0.15γ. γ ranges from 3 near
tropical cloud base down to 1 at the freezing level (5 km in the tropics) to near-zero in the
upper troposphere.
MSE is an energy, and θe is an entropy proxy. MSE is roughly equal and functionally
equivalent to cp θe. Similarly, dry static energy is functionally equivalent to cp θ. Both are
roughly conserved and linearly mixing, but each involves separate approximations to
exact thermodynamics. For most purposes, one just should use the more convenient
choice, MSE if working in height coordinates or θe if working in pressure coordinates.
The right panel of Slide 1.7 shows adiabats computed assuming constant saturated MSE.
To compare with the left panel, height has been hydrostatically converted to pressure for
each adiabat, assuming a surface pressure of 1000 hPa and computing density using the
virtual temperature of saturated air . Dry adiabats based on constant dry static energy are
plotted for comparison. The moist adiabats computed using saturated θe and MSE are
essentially identical, as are the dry adiabats computed using θ and DSE.
Vertical profiles of θe and ! e* or h and h*
Plots showing vertical profiles of θe or h and ! e* or h* are commonly used in tropical
meteorology, because these quantities nicely indicate conditional instability and don’t
vary too much across the tropical troposphere. The θe of adiabatically lifted near-surface
air tells us its ! e* (i.e. its temperature) above the LCL. If the θe of near-surface air exceeds
the ! e* in an extensive layer of the troposphere, that indicates the lifted parcel has a
positive temperature excess, suggesting potential for surface-based moist convection in
that layer. Similarly for h and h*.
Slide 1.8 shows 52-day mean soundings from five locations near Kwajalein Island (9N
167W) in the western Pacific ocean, plotted using h and h*. The vertical line shows an
adiabatically lifted boundary layer air parcel. It is well to the right of the profiles of h*
throughout the free troposphere, showing substantial deep conditional instability.