Lecture 1+: Introduction, Dry Convection,
Meteorological Basics and Moist Thermodynamics
Adam Sobel
Columbia University
ahs129@columbia.edu

December 3, 2013

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Overview1

Introduction: Why do we care about moist convection?

2

Rayleigh-Benard Convection

3

Basics of the compressible atmosphere4

A little basic geophysical fluid dynamics

5

Moist thermodynamics

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Introduction, 1: Why do we care?

Image from Jean Lynch-Stieglitz, Georgia Tech
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Introduction, 1: Why do we care?

Net infrared radiative flux at earth’s surface is relatively small; there isa rough balance between insolation and latent + sensible heat flux;
Latent  sensible globally (and over oceans especially);
Infrared emission to space (balancing all non-reflected sunlight at topof atmosphere) comes mostly from high altitude;
Moist convection is essential for moving energy from the surface tohigh altitude where it can be emitted to space.
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Introduction, 2

Also, moist convection controls the morphology of weather on earth.

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Pictures	
  of	
  clouds	
   	
  	
  
Adam	
  Sobel	  FDEPS	
  Kyoto	  stratocumulus	
  Rob	
  Wood,	
  U.	
  Washington	
  stratocumulus	
  Rob	
  Wood,	
  U.	
  Washington	
  shallow	
  cumulus	
  (non-­‐precipitaDng)	
  shallow	
  cumulus	
  (non-­‐precipitaDng)	
  Isolated	
  precipitaDng	
  cumulus	  Isolated	
  precipitaDng	
  cumulus	
  (more	
  developed,	
  note	  anvil)	
  Mesoscale	
  convecDve	
  system	  in	
  visible	  satellite	
  image	
  hIp://cimss.ssec.wisc.edu/goes/blog/archives/494	  Super	  Typhoon	  Haiyan	  Image:	
  NASA	  John	  Allen,	  Columbia	  Rayleigh-Benard convection, 0This treatment of dry convection between parallel plates(Rayleigh-Benard) is taken from Kundu, Fluid Mechanics, but also draws
from Emanuel, Atmospheric Convection (1994).

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Boussinesq fluid with Cartesian velocity component ũi , pressure p̃,temperature T̃ . Two parallel plates, infinite in horizontal extent, separated
by distance d; one is at z = −d/2, other at z = +d/2. Lower plate is attemperature T̃ = T0 , upper at T̃ = T0 − ∆T . Linear equation of state,
thermal expansion coefficient α, thermal diffusivity κ, viscosity ν.

ρ = ρ0 [1 − α(T − T0 )]

(1)

1
d ũ
= − ∇p̃ − g [1 − α(T̃ − T0 )]k̂ + ν∇2 ũi
dt
ρ0
d T̃
= κ∇2 T̃dt∇·u=0Adam Sobel (Columbia)

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(2)
(3)
(4)

December 3, 20137 / 59Rayleigh-Benard convection, 2
Now define perturbations about a basic state of no motion as variables
without tildes:ũi = ui (x, t)(5)T̃ = T (z) + T (x, t)(6)p̃ = P(z) + p(x, t)(7)

The basic state must satisfy0=−1∇P − g [1 − α(T − T0 )]k̂ρ0with k̂ the vertical unit vector. Thus
1 ∂P
= g [1 − α(T − T0 )]k̂
ρ0 ∂z
∂2T
∆T
0 = κ 2 → T = T0 − Γ(z + d/2), Γ =.
∂zdIs this state stable to small perturbations?
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(9)

8 / 59Rayleigh-Benard convection, 3
Subtract the equations for the basic state from the total equations, andlinearize, assuming small perturbations, to obtain
∂ui
1 ∂p
=−
+ g αT k̂ + ν∇2 ũi∂tρ0 ∂xi∂T− Γw = κ∇2 T̃
∂t∇ · u = 0.

(10)
(11)
(12)

The boundary conditions are (using no-slip):u = v = w = T = 0, z = ±d/2.
Also, by continuity ∂w∂z = 0 at the walls. (Or, we can use stress-free,
∂2w
= 0.)
∂z 2
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Now in steady state the advection and diffusion terms must balance, so
scaling analysis requires that
Γw ∼

κT
,
d2

but T ∼ Γd, soκ.
dThere are two non-dimensional parameters. The ratio of the buoyancy toviscous forces is:
w∼

g αT
g αΓd
g αΓd 4∼∼≡ Ra,
νw /d 2
νw /d 2νκ
the Rayleigh number. Then the other is the ratio of the diffusivities, or
Prandtl numberνPr = .
κAdam Sobel (Columbia)

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Rayleigh-Benard convection, 5
Now we skip a lot of math, summarizing the steps in words. By
substitution we can reduce the equations to just two, for w and T . Use
the method of normal modes. Since the PDEs have constant coefficients,
we can assume exponential solutions:w = ŵ (z) exp(ikx + ily + σt),(13)

T = T̂ (z) exp(ikx + ily + σt),(14)where σ is complex,σ = σr + iσi . T̂ and ŵ are also complex. It is useful√
to define K = k 2 + l 2 . (Note there is rotational symmetry, x and y areinterchangeable.)
The first key result is that σ is pure real in this problem, σi = 0. There
are no oscillatory solutions. This means that the marginal stability curve(in K , Ra space) is that for which σ = 0 (rather than just σr = 0).
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Rayleigh-Benard convection, 6
We do not solve the full stability problem. Rather, we find the value of K
for which σ = 0 at a given Ra.
Problem reduces to a 6th order linear elliptic PDE with 6
homogeneous BCs.
We non-dimensionalize distance by d, T by ∆T , w by dκ .
Pr drops out of the problem (for marginal stability).
Because the BCs are symmetric, the eigenfunctions in z can be either
symmetric or antisymmetric about the midpoint z = 0.
The smallest Ra for which σ = 0 — the value above which we expect
to find motion — comes from the gravest even mode. This is verified
by experiment.
The problem is mathematically much simpler if we use stress-freeBCs; the results are quantitatively, but not qualitatively different fromthe no-slip case (in all ways other than the horizontal velocity at theboundary) so we focus on that case.
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Rayleigh-Benard convection, 7
The even solution for the stress-free eigenfunctions is
w = A sin nπz, n = 1, 2, 3...
The critical Rayleigh number isRa =(n2 π 2 + K 2 )3.
K2

The lowest value this can take on for any K is found by differentiatingwith respect to K ; the result is
Kcr =

π2
.
2

This is (almost) O(1); since we have non-dimensionalized by d, thehorizontal scale of the motion is the same order as the vertical scale.
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The Rayleigh number at which marginal stability of the system occurs is
Racr =

27π 4≈ 657,
4(somewhat larger for no-slip, but same order of magnitude). For
Ra < Racr , there will be no motion; heat will be transported diffusively
from bottom to top.

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Rayleigh-Benard convection, 8

The horizontal structure of the circulation cells at marginal stability is notpredicted by the theory, since only K , not k or l separately, appears. Any
regular polygons are possible. The detailed pattern is somewhat sensitive
to experimental details, but often begins as hexagons, and forms long rolls
later.
youtube video

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Rayleigh-Benard convection, 9Taken from Emanuel, 1994. (pp.84)

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Rayleigh-Benard convection, 10
In the atmosphere, the Rayleigh numbers are MUCH higher than critical— by many, many orders of magnitude. (Diffusivities are small, lengthscales are large, heat flux also large.) So it is of interest to say something
about large Ra.
The flow quickly becomes turbulent as one gets much past critical.
youtube video of turbulent 2D convection
The Nusselt number, or nondimensional heat flux,
Nu =

wT
κ∆T

increases with Ra, in a somewhat complex way (several regimes ofdifferent turbulent flow types).

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The details of all the turbulent transitions in Nu are not really relevant tothe atmosphere, because the heat flux in the atmosphere is more nearly
constrained externally than is the temperature.
For our purpose the main quantity of interest in the turbulent solutions is
the mean temperature, T (z). This is close to homogeneous in the interior,with all the gradient trapped in very thin layers near the boundaries.

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Stable stratification between parallel plates,1
Now consider same situation, but with the top boundary warmer than thebottom. Then it is completely stable. Just for sake of argument consider
the linear equations in 2D with both diffusivities set to zero:
1 ∂p
∂u
=−
,
∂t
ρ0 ∂x
∂w
1 ∂p
=−
+ αgT ,∂t
ρ0 ∂z∂T+ w Γ,
∂t
∂u ∂w
+
=0
∂x
∂z
where now Γ = (T (d/2) − T (−d/2))/d > 0.

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Stable stratification between parallel plates, 2The eigenfunctions are the same, and we can show that we get pureneutral oscillatory gravity wave solutions,w , T ∼ sin(nπz) exp(i(ωt + kx),
with frequency
ω2 =N 2k 2,k 2 + n2 π 2

where
N 2 = αg Γ
defines the buoyancy frequency, N.

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Stable stratification between parallel plates, 3

Since for the disturbances of interest in the real atmosphere havehorizontal scales much longer than vertical scales, the most relevant
analogs in the parallel-plate system are also those with k  n, so
ω2 =N 2k 2N 2k 2≈,k 2 + n2k 2 + n2 π 2
Nk
.
ω≈
nπ

(15)

Thus modes of higher vertical wave numbers (shorter vertical scales)
propagate more slowly. This approximation is equivalent to assuminghydrostatic balance.

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Stable stratification between parallel plates, 4

The phase speed of wave modes c = ω/k, so
c=

N
,
nπ

but remember that we have nondimensionalized distance by d.
Dimensionally,
Nd
.
c=
nπ
To fix numbers, for the atmosphere N ≈ 2 × 10−2 s −1 , and we may takethe vertical scale d to be the depth of the troposphere, 10 km. That is
broadly consistent with the value c ≈ 50m/s for waves with n = 1.

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A few points about internal gravity waves
1

If there were no boundaries, they would also propagate vertically.
(The atmosphere has no upper lid, though for some purposes canbehave as though it almost does.) The vertical wave numbers would
not be quantized in that case.

2

Putting the diffusivities back in would just make them weakly
damped.

3

Horizontal structure: w and T are in quadrature — as with convective
solutions, but opposite phasing (upward motion cooling vs. warming).

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Basics of the compressible atmosphere, 1
Ideal gas law:p = ρRT ,with p pressure, ρ density, T temperature, R gas constant. On large scales
it is a very good approximation to assume hydrostatic balance in thevertical:pg∂p= −ρg = −.
∂z
RT
T is measured in Kelvin and doesn’t vary that much compared to its
absolute value, so pressure drops approximately exponentially with height:p ∼ p0 e z/Hwhere H = RT0 /g is the scale height, T0 a reference temperature.
We can also show that the potential temperature,p0
θ = T ( )κ ,pwith κ = R/cp = 2/7, is conserved in the absence of friction or heating of
the dry fluid. It is related to specific entropy by s = cp ln θ + constant.
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Basics of the compressible atmosphere, 2
Because of hydrostatic balance on large horizontal scales, we can usepressure as a coordinate. In this coordinate the vertical velocity is
dp
ω=
,
dt
and the mass conservation equation is isomorphic to the incompressible
one (although we do not assume incompressibility)
∂u ∂v
∂ω
+
+
= ∇ · u = 0.
∂x∂y∂p
In the horizontal momentum equation, instead of the pressure gradient on
a constant height surface, we have to write the gradient of geopotential
height, φ = gz, on a constant pressure (isobaric) surface. The
geopotential height is found by integrating hydrostatic balance using theactual temperature
Z ps
Z
Z ps
RT
dp =
RTd ln p.
φ = gdz =pppAdam Sobel (Columbia)

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Basics of the compressible atmosphere, 3
Including Coriolis parameter f = (2Ω sin λ)ẑ, (λ = latitude) the primitive
equations in p coordinates are:dv
1
+ f × v = − ∇p φ + F
dt
ρ
1
∂φ
=α=−
∂p
ρ
dθ
=Q
dt
∂u ∂v
∂ω
+
+
=0
∂x
∂y
∂p
dq
= e − c.
dt
where v = (u, v ), d/dt = ∂/∂t + u∂/∂x + v ∂/∂y + ω∂/∂p, and∂φ∇p φ = ∂φ∂x x̂ + ∂y ŷ , and all horizontal derivatives are understood to betaken along isobaric surfaces. Heating is Q, friction is F, q specifichumidity (defined soon), evaporation and condensation of water are e, c.
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Basics of the compressible atmosphere, 4We are working in a local Cartesian planar approximation, so there are no
metric terms, and horizontal coordinates are x, y instead of latitude andlongitude.
We will represent the Coriolis parameter by the β - plane approximation(first two terms in Taylor expansion of the sine function),f = f0 + βy ,df
= (2Ω cos λ0 )/a, a is earth radius, λ0 awhere f = 2Ω sin λ0 and β = dyreference latitude.

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Basics of the compressible atmosphere, 4

If we wish, instead of using potential temperature, we can use dry static
energy,s = cp T + gz
(not to be confused with entropy although the letter s is also used). s isconserved for dry adiabatic transformations in which hydrostatic balance ismaintained, and thus for large-scale motions we can write
ds
=Q
dt

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Basics of the compressible atmosphere, 5
The primitive equations describe the motions on large scales. On
sufficiently small horizontal scales (L ≤ 10 km) hydrostatic balance may
break down. We may consider the vertical momentum equation forconvective motions to be (in height coordinates)dw∂p=−− ρg ,dt∂z
or defining a hydrostatically balanced component∂p− ρg ,
∂z
then
dw
∂p 0
=−
− ρ0 g ,dt∂zwhere p = p + p 0 , ρ = ρ + ρ0 .
In parcel theory, we will neglect the pressure term, and consider justdw
= −ρ0 g .
dt
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Basics of the compressible atmosphere, 6
While temperature decreases with height in the atmosphere, potential
temperature increases with height. This means the atmosphere is stableto small vertical displacements — at least as long as condensation ofwater vapor does not occur.
The buoyancy frequency in the compressible atmosphere, N, is
N2 =

g ∂θ
,
θ ∂z

N2 = −

ρg 2 ∂θ.
θ ∂por in pressure coordinates,

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Annual mean temperature vs. latitude and height (source: ECMWF,
ERA40 Rreanalysis).

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Annual mean potential temperature vs. latitude and height (source:
ECMWF, ERA40 Rreanalysis).

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A little basic GFD, 1

If we neglect friction, time rate of change, and inertia, the horizontal
momentum equation becomes1
f × v = − ∇p φ,
ρ
or
∂φ
∂y
∂φ
−fv = −∂x
fu = −

which is geostrophic balance — flow parallel to isobars.

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A little basic GFD, 2
500 hPa geopotential height, wind and vertical vorticity (∂v /∂x − ∂u/∂y )over USA one day last week

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A little basic GFD, 3
Combining geostrophic and hydrostatic balance with the ideal gas law
leads to the thermal wind relation
∂u
R ∂T
=
,
∂p
fp ∂y
∂v
R ∂T
=−
,
∂p
fp ∂x
which says that the vertical shear of the horizontal wind is
proportional (and perpendicular) to the horizontal temperature
gradient.
Note the factor ∼ 1/f on the RHS, which implies that to the extent
thermal wind balance holds, large horizontal temperature gradientscannot exist in the tropics, unless accompanied by very large shears.
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A little basic GFD, 4

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A little basic GFD, 5
Surface pressure map (cint=5hPa) from a random day, May 15, 2005.

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A little basic GFD, 6
Surface temperature map (cint=5C) from a random day, May 15, 2005.

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A little basic GFD, 7
Effective black body temperature (C) seen from space, from a random day,May 15, 2005.

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A little basic GFD, 8
The natural horizontal scale for motions which are close to geostrophic
balance is the deformation radius, or Rossby radius, Ld ,
Ld =

c
f

where c is the gravity wave speed. (Thus motions with different verticalscales have different Ld . This scale emerges, for example, from thegeostrophic adjustment problem.
For tropical dynamics we may take the reference latitude in our β-plane
approximation to be λ0 = 0, thus f = βy is the equatorial β-plane
approximation. In this case the natural scale that emerges is theequatorial deformation radius,pLβ = c/β.
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Moist Thermodynamics, 0This treatment of moist thermodynamics draws on Emanuel’s Atmospheric
Convection as well as Bohren and Albrecht’s Atmospheric
Thermodynamics.

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basic moisture variables, 1
Basic variables to describe the water vapor content of the atmosphereincludemixing ratio r , which is the mass of water vapor divided by massof dry air;specific humidity q, which is the mass of water vapor divided by
the total mass of air (dry air plus water vapor);vapor pressure e, which is the partial pressure of water vapor;vapor density ρv , which is the mass of water vapor per unit volume.
Then, r = ρv /ρd , where ρd is dry air density;q = ρv /(ρd + ρv ) = r /(1 + r ).
In the earth’s atmosphere, since it is always the case that r  1(maximum values are a few percent), q ≈ r .

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basic moisture variables, 2Analogously to r , we can also define the atmospheric content ofcondensate — liquid and ice. These are assumed to be in the form ofsmall particles falling at their terminal velocities. If the terminal velocities
are small compared to the vertical air velocity, we call them cloud wateror cloud ice; if the fall velocities are comparable to or greater than the air
velocity we call them precipitation (rain, snow, graupel, hail).
liquid water mixing ratio rl ,ice mixing ratio ri ,total water, rT = r + rl + ri .
We can define also specific humidities of liquid and ice analogously tovapor.

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basic moisture variables, 3Some more relationships between variables will be useful. Using the ideal
gas law separately for both dry air and vapor, with gas constants Rv andRd we can writer=e/Rv T
Rd e
e
e
=
=
≈
pd /Rd T
Rv p − ep−ep(since we can assume e  p). Here p is total pressure, pd dry air pressure,andRd≈ 0.622.
=Rv
We also define the relative humidity H:
H=

e
e∗

Where e ∗ is the saturation vapor pressure. The saturation vapor
pressure is the maximum vapor pressure which can exist in equilibriumover a liquid water surface (or a solid ice surface).
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basic moisture variables, 4
The saturation vapor pressure e ∗ is a function of temperature, defined bythe (nearly exact) relation:de ∗Lv e ∗=,dT
Rv T 2(16)

where Lv is the latent heat of vaporization. (16) is theClausius-Clapeyron relation. Lv is itself a function of temperature, but aweak one; in many cases we can assume it constant. An approximate
integration yields
17.67T),e ∗ = 6.112 exp(
T + 243.5
where T is in Celsius (not Kelvin) and e ∗ is in hPa. This is valid overliquid; a similar relation holds over ice, but with different numbers.

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basic moisture variables, 5

It is also useful to define the dew point, or temperature to which a given
parcel would have to be cooled at constant pressure in order for saturation
to occur. Dew point is a function of specific humidity and pressure.
The difference between the temperature and dew point, T − Td , issometimes called the dew point depression. It is related to saturation
deficit, which may be defined e ∗ − e, q ∗ − q, or r ∗ − r .

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phase changes of water in the atmosphere,1When e > e ∗ , condensation is possible. (Any excess e − e ∗ > 0 is calledsupersaturation.) Water vapor can condense into liquid or solid particles
by homogeneous nucleation (spontaneous clumping of water molecules),or heterogeneous nucleation (condensation onto a pre-existing liquid or
solid particle). Pre-existing particles on which condensation can occur are
called cloud condensation nuclei (CCN). Heterogeneous nucleation will
occur first if CCN are present.
In the atmosphere, there are always some CCN present for condensation
into liquid. Homogeneous nucleation of liquid droplets is believed not to
occur, nor is significant supersaturation over liquid. One can assumee ≤ e ∗.

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phase changes of water in the atmosphere, 2

On the other hand, ice nuclei, onto which either liquid drops or water
vapor can freeze, are not so universally present. Significant supercooling(existence of liquid water at T < 0◦ C ) can occur, as can significant
supersaturation of vapor with respect to ice.
At temperatures T < −40◦ C , homogeneous freezing of liquid takes place
and all condensed water can be assumed to be ice. However in the range−40◦ C < T < 0◦ C , the phase of condensate may not be known a priori,and depends on the presence or absence of ice nuclei.
At T > 0◦ C , melting of ice takes place quickly. (This can be visualized inradar images by the thin bright band, we will see later.)

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phase changes of water in the atmosphere, 3
Although there are always enough CCN for homogeneous nucleation,
variations in the distribution of CCN can be important to the properties ofany clouds that form.
Where there are few aerosols (atmospheric liquid or solid particles), clouddrops will be fewer and larger for the same amount of liquid. Where thenumber density of aerosols is higher, cloud drops will be greater in number
and smaller in size.
In aerosol-rich environments, clouds may be optically thicker — morereflective of sunlight — and less likely to form precipitation (large drops).
These effects are most apparent in shallow, non- or weakly-precipitating
clouds.

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phase changes of water in the atmosphere, 4Adam Sobel (Columbia)

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50 / 59Effect of water on air density, 1
Water vapor has a different heat capacity than does dry air, and thus the
mixture of the two also has a different heat capacity than dry air would(slightly, since e  p, ρv  ρd etc.). The effective gas constant for the
mixture is
1 + r /> Rd
R 0 ≡ Rd
1+r
The margin by which R 0 exceeds Rd depends on the humidity. The
mixture obeys the ideal gas law
p = ρR 0 T
R 0T
α=
,
p
where α = 1/ρ is the specific volume.

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51 / 59Effect of water on air density, 2
The direct effect of water vapor on air density is usually measured by theeffective temperature which dry air would need to have in order to havethe same density as moist air with a particular temperature and humidity.
This is called the virtual temperature, Tv , defined by
Rd Tv = R 0 T ,thus

1 + r /≈ T (1 + 0.61r ),1+rwith T now in Kelvin. The difference term 0.61rT is sometimes known as
the ”virtual temperature effect”. It is small, generally, since r is.
Note that the virtual temperature effect is only the direct effect of vapor
on density. It has nothing to do with phase change.
Tv = T

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The moist enthalpy (really specific moist enthalpy)k = (cp + rT cl )T + Lv r ≈ cp T + Lv q
is conserved under phase changes between liquid and vapor at constantpressure. (cl is liquid water heat capacity).
Thus the temperature increase for condensing a quantity ∆q of vapor is∆T = Lv ∆q/cp .
We can now show that compared to the virtual temperature effect, phasechange is by far the more important process by which water vapor
influences the atmospheric circulation.

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The virtual temperature effect is 0.61rT ; on earth r ≤ a few percent. Letr = 0.03, then for T = 300K , the virtual temperature effect is ∼ 5K . This
seems large, but water vapor gradients tend not to be large, and in any
case the dynamics will simply adjust; the pressure field simply
hydrostatically adjusts to Tv instead of T .
By contrast condensation tends to be localized, and much larger. The
effect on temperature of condensing all the water in our parcel withr = 0.03 is
Lv r /cp ≈ 75K !
It will not all condense at once, but the enthalpy is there.

Adam Sobel (Columbia)

Lecture 1

December 3, 201354 / 59Effect of water on air density, 5

Once there is condensate in the air, we assume it to be in suspension,
falling at terminal velocity. The mass of the condensate then simply canbe considered to contribute directly to the density of the air. This effect iscalled condensate loading.
Condensate loading, particularly by large precipitation particles, can beimportant in the generation of convective downdrafts. However in mostother circumstances we will ignore it.

Adam Sobel (Columbia)

Lecture 1

December 3, 201355 / 59

Conserved variables under phase change, 1Condensation and evaporation of water, if done adiabatically — slowly, inequilibrium, with the vapor always at saturation — is a thermodynamicallyreversible process. Thus we can define a moist entropy which is conserved.
That entropy can be related to a potential temperature - like quantity,
called equivalent potential temperature, θe :(cpd + rT cl ) ln θe = s + Rd ln p0 ,where cpd , cl are dry air and liquid water heat capacities and p0 is aconstant reference pressure; s is the moist entropy (an expression for which
we will not explicitly write here). This is the reversible θe , conserved if all
water remains in the parcel. It is changed by any irreversible process —heating, friction, non-equilibrium water vapor transport (diffusion), phasechanges which don’t occur at equilibrium. If the air is purely dry, rT = 0,
it reduces to the dry potential temperature, θ.
Adam Sobel (Columbia)

Lecture 1

December 3, 201356 / 59

Conserved variables under phase change, 2

The reversible θe is not conserved if precipitation occurs, since then watermass is irreversibly removed from the air. We define pseudo-adiabatic
process as one in which all condensate is removed from the air as soon as
it is formed. Then the pseudoadiabatic potential temperature, θep , is thetemperature reached by pseudoadiabatic ascent to p = 0 (so all water isremoved) and then dry adiabatic descent to the reference pressure.
If atmospheric processes were fully reversible, there would be no rain (orsnow). If they were fully pseudoadiabatic, there would be no clouds.
Clearly, the real atmosphere lies between these two extremes.
θep can be computed by an approximate formula which is sufficiently
complicated as to not be worth writing down. (E.g., see Emanuel, p. 132)

Adam Sobel (Columbia)

Lecture 1

December 3, 201357 / 59

Conserved variables under phase change, 3
The moist static energy is defined as
h = (cpd + rT cl )T + Lv r + (1 + rT )gz,
where g is gravitational acceleration and z is altitude. h is conserved for
adiabatic transformations in which mass is conserved, and in which thepressure remains hydrostatic. Since rT can be taken small, we can (andwill) usually make the approximationh ≈ cpd T + Lv q + gz.
We have also defined the dry static energy, s = cpd T + gz (not to be
confused with moist entropy), such that h = s + Lv q.
In many situations one has a choice whether to work with h or θe (or θep ).
This is to some extent a matter of taste. h is an energy, θe is (basically)an entropy. In practice they are conserved (or not conserved) tocomparable extents. h is algebraically simpler to use.
Adam Sobel (Columbia)

Lecture 1

December 3, 201358 / 59

Mixing, etc.

Because of the nonlinearity of the Clausius-Clapeyron relationship, things
can happen in a moist fluid that are impossible for a dry fluid.
1

A mixture of two unsaturated parcels can become saturated. (This is
how you can see your breath on a cold day.)2

A mixture of a positively buoyant (but saturated) parcel and a neutral(unsaturated) parcel can become negatively buoyant. (This is one way
convective downdrafts can form, though not the most important.)
Condensate can fall through the air (precipitation). Precipitation can
generate downdrafts by

3

1
2

Direct momentum transfer (condensate loading);
Evaporative cooling in unsaturated air.

Adam Sobel (Columbia)

Lecture 1

December 3, 201359 / 59

References• Bohren, C. F., Albrecht, B. A.: Atmospheric Thermodynamics. Oxford
University Press, New York.
• Emanuel, K., 1994: Atmospheric Convection. Oxford University Press, NewYork.
• Kundu, P. K.,2000: Fluid Mechanics. Academic Press, New York.