A&OS C110/C227: Review of
thermodynamics and dynamics II
Robert Fovell
UCLA Atmospheric and Oceanic Sciences
rfovell@ucla.edu
1
Notes
• Everything in this presentation (except perhaps the Stuve
diagram) should be familiar
• Please feel free to ask questions, and remember to refer to
slide numbers if/when possible
• If you have Facebook, please look for the group
“UCLA_Synoptic”. You need my permission to join. (There are
two “Robert Fovell” pages on FB. One is NOT me, even
though my picture is being used.)
2
Water substance
• Water vapor (wv) mixing ratio
• mv, md vapor and dry air masses
• Gas constants for dry air and wv
• The ratio e
3
IGL and virtual temperature
• Ideal gas law
• Virtual temperature Tv
• At the same pressure, warm air is less dense than cold air
• At the same pressure and temperature, moist air is less dense
than dry air
4
More moisture variables
• Vapor pressure e and saturation vapor pressure es
• es = es(exp[T])
• Determined by the Clausius-Clapeyron equation
• Mixing ratio r and saturation mixing ratio rs
• rs = rs(exp[T], p)
• r = r(exp[Td], p)
• Td = dew point temperature
• Relative humidity RH = 100(r/rs)
5
Hydrostatic equation
g = 9.81 m s-2 at sea-level
• Represents balance (stalemate) between vertical pressure
gradient force and gravity resulting (formally) in no vertical
acceleration and (practically) in no vertical motion
6
Potential temperature
• Potential temperature is a system property that is conserved
for a dry adiabatic process
p in millibars
T in Kelvin
R and cp in J kg-1 K
• We can also define a virtual potential temperature
analogously to virtual temperature
7
Dry adiabatic process
•
•
•
•
•
Control mass: no mixing with environment
No heat source, external or internal
Conserves two system properties: q and r
Process is thermodynamically reversible
Should be termed subsaturated adiabatic
p, T, Td, V, r, r, all change
q, r do not
8
V = volume
Dry adiabatic lapse rate (DALR)
• Start with 1st law and use hydrostatic equation. Note dq = 0.
• Since cp ~ 1004 J/kg/K for dry air, the dry adiabatic lapse rate
Gd ~ 10˚C/km
9
Moist adiabatic process
•
•
•
•
Control mass: closed to environment
System is and remains at saturation
Internal heat source from water phase change
Conserves one system property: qe
• Equivalent potential temperature
• Process is thermodynamically reversible
• …if no condensate is lost from parcel
• Should be termed saturated adiabatic
p, T, Td, V, r, r, q, r all change
qe does not
10
Heat from vapor-liquid phase
change
• A saturated parcel is disturbed, forcing water phase change
(vapor to liquid or liquid to vapor) to regain saturation
• The parcel’s change in vapor mixing ratio is drs, the change in the
parcel’s saturation value. drs is negative if the parcel is
condensing vapor.
• The heat source or sink to the parcel is –Lvdrs, where Lv is the
latent heat of vaporization
• Lv is a function of temperature, but is Lv ~ 2.5E6 K kg-1 K-1 at 0˚C
11
Moist adiabatic lapse rate (MALR)
• Start with 1st law and use hydrostatic equation. Note dq = Lvdrs.
• The MALR is the DALR
modified by the drs/dz term
• Since rs = rs(exp[T], p), and
p varies with height, then
drs/dz varies with T and height
• Thus, MALR is (very) variable
12
Equivalent potential temperature
• The MALR was used to create a new potential temperature that is
conserved for moist adiabatic processes – the equivalent potential
temperature, qe
• The equation above can only be used when the parcel is saturated.
• This equation is an approximation, and more accurate versions exist.
In reality, specific heats of vapor and condensed water should also
be included. However, this form is sufficiently accurate for our
applications.
• Note that qe is also conserved for dry adiabatic processes. Can you
see why? (See slide #25)
13
Nomenclature
• A process that is dry adiabatic, conserving potential temperature q, is
called isentropic (constant entropy).
• Lines of constant potential temperature are called dry adiabats, subsaturated
adiabats and isentropes (all equivalent)
• After saturation, the situation is more complicated and depends on the
fate of condensed water and how it is handled
• The moist adiabatic or saturated adiabatic process presumes all condensate
remains within the parcel, and even receives some of the condensation
warming. This process is completely reversible
• The pseudoadiabatic process presumes all condensate is immediately
removed from the parcel (so it’s not a true CM). This process is
thermodynamically irreversible
• There is very little difference between moist adiabatic and pseudoadiabatic
ascent, so we will generally term lines of constant qe as moist adiabats when
plotted on thermodynamic diagrams (although technically they are
pseudoadiabats since condensed water is neglected)
• There is an enormous difference between moist adiabatic and
pseudoadiabatic descent, as we will see
14
Routes to saturation
• An air parcel is a sample of air, often but not always a closed
and isolated CM, that we follow and monitor. It is assumed
that its internal pressure equals environmental pressure
(mechanical equilibrium)
• There are three distinct ways of bringing an air parcel to
saturation. These will be illustrated on the thermodynamic
diagrams presented in the next section
1.
2.
3.
Adiabatic expansion approach to saturation
Dew point approach to saturation
Wet bulb approach to saturation
15
Thermodynamic diagrams
16
Diagrams
• The Skew-T and Stuve represent two commonly employed
thermodynamic diagrams
• The Skew-T is an “area-equivalent” diagram, in that the same area
placed in different parts of the diagram represents the same amount
of energy or work
• It also maximizes the contrast between dry adiabatic and isothermal
processes
• However, it is not Cartesian
• The Stuve diagram, while not area-equivalent, is Cartesian and has
these useful properties
• The horizontal axis is temperature, and the vertical axis is a function
(pR/cp) of pressure, so isotherms are precisely vertical and isobars are
precisely horizontal (although not equally spaced)
• Dry adiabats and mixing ratio lines are also straight
• The only curvilinear property on the diagram is the moist adiabat
• The vertical axis is not linear in height, but it’s fairly close
17
The Stuve diagram
18
Stuve diagram
• Isentropes labeled by where they cross p = 1000 mb (since T = q there)
• Isentropes are straight, with slope –g/cp
• Isentropes are not actually parallel, and actually converge as T, p → 0.
19
Stuve diagram
• A parcel achieves its potential temperature by moving dry adiabatically
to p = 1000 mb
20
Stuve diagram
• Mixing ratio lines are used for actual (r) and saturation (rs) mixing ratios
• Values increase swiftly with increasing T and slowly with decreasing p, since
rs ∝ exp(T)/p
21
Stuve diagram
• At a given p, T reveals rs and Td reveals r. For subsaturated air Td < T
• Suppose I cool the parcel isobarically, without change of vapor content
• During this process, T decreases but Td remains fixed
• When T = Td, saturation is achieved and dew has formed
• Dew point approach to saturation
22
Stuve diagram
• Lift subsaturated parcel instead. External and internal p drop, allowing expansion
• T decreases @ DALR, while Td drops slowly.
• Saturation is achieved at the lifting condensation level (LCL).
• Before saturation, q and r are conserved
• Adiabatic expansion approach to saturation
23
Stuve diagram
• After saturation, further ascent follows the moist adiabat, or line of constant qe
• Upon ascent, vapor is condensed, increasing potential temperature q
• Meanwhile, r = rs decreases, as vapor is lost to condensation
• When all vapor is exhausted, q = qe is achieved
• Each moist adiabat merges with an isentrope, and shares its label
24
Stuve diagram
• The preceding implies that the dry adiabatic process, which conserves q,
also conserves qe.
• Any subsaturated parcel with a given q and r shares a single, common LCL, which
means it will reach one, and only one, moist adiabat qe. That qe characterizes
the parcel, whether it ever becomes saturated or not. So qe is also fixed.
25
Stuve diagram
• Suppose we lift a parcel until no vapor remains uncondensed
• If the process was moist adiabatic, all condensate remained in the parcel
• Now cause the parcel to descend. Condensate is forced to return to vapor
• Parcel takes the same path down, back to LCL, even back to its origin. Reversible.
26
Stuve diagram
• Suppose we lift a parcel until no vapor remains uncondensed
• If the process was pseudoadiabatic, all condensate is irretrievably lost
• Parcel descent is dry adiabatic, as no condensate remains to oppose compression.
Irreversible, as original state cannot be regained (without further input)
• If parcel moved to 1000 mb, its temperature becomes qe
27
Stuve diagram
• The wet bulb temperature Tw of the original parcel can be approximated by
descending from the LCL along a moist adiabat to the original pressure level.
• Descending along the moist adiabat implies vapor is added to the parcel. This
is done by evaporating liquid into the parcel.
• Wet bulb approach to saturation. Note Td ≤ Tw ≤ T.
28
Stuve diagram
• By the way, an equivalent label for the moist adiabat is qw, the wet bulb potential
temperature, representing the T where the moist adiabat crosses p = 1000 mb
29
Stuve diagram
• Now let’s compare our raised parcel to an environmental sounding
• Temperature typically decreases with height between the surface and the
tropopause. Farther above, in the stratosphere, temperature remains constant or
increases with height.
30
Stuve diagram
• At first, a parcel rising from the surface may be colder than the environment.
• If it has a level of free convection (LFC), it becomes warmer than its surroundings
between that level and its equilibrium level (EQL), where the parcel and
environmental temperatures become the same again
• For deep convective storms, an EQL near the tropopause is common
31
Stuve diagram
• The parcel’s convective available potential energy (CAPE) is the positive buoyancy
between the LFC and EQL.
• To tap into the CAPE, the convective inhibition (CIN), or negative buoyancy
below the LFC, has to be overcome.
32
The Skew-T diagram
33
Skew-T diagram
• Isobars are horizontal on the Skew-T/Log-p, and values decrease upward.
• Isotherms are inclined upwards from left to right, and values increase downward
and to the right.
34
Skew-T diagram
• Isentrope values increase to the right and upward. A parcel achieves its
potential temperature via dry adiabatic displacement to p = 1000 mb
• Isentropes and isotherms meet at right angles
35
Skew-T diagram
• Mixing ratio lines are used for actual (r) and saturation (rs) mixing ratios
• Values increase swiftly with increasing T and slowly with decreasing p, since
rs ∝ exp(T)/p
36
Skew-T diagram
• At a given p, T reveals rs and Td reveals r
• Lift a subsaturated parcel. T decreases at the DALR, while Td drops slowly.
• Saturation is achieved at the lifting condensation level (LCL).
• Before saturation, q and r are conserved
37
Skew-T diagram
• After saturation, further ascent follows the moist adiabat, or line of constant qe
• Upon ascent, vapor is condensed, increasing potential temperature q
• Meanwhile, r = rs decreases, as vapor is lost to condensation
• When all vapor is exhausted, q = qe is achieved
• Each moist adiabat merges with an isentrope, and shares its label
38
Skew-T diagram
• A parcel achieves its qe by first ascending along the moist adiabat until all
vapor has condensed and fallen out, and then descending dry adiabatically to
p = 1000 mb. This is the irreversible pseudoadiabatic process.
• Remember, the moist adiabatic process is reversible.
39
Skew-T diagram
• The wet bulb temperature Tw of the original parcel can be approximated by
descending from the LCL along a moist adiabat to the original pressure level.
40
Skew-T diagram
• Now let’s compare our raised parcel to an environmental sounding
41
Skew-T diagram
• At first, a parcel rising from the surface may be colder than the environment.
• If it has a level of free convection (LFC), it becomes warmer than its surroundings
between that level and its equilibrium level (EQL), where the parcel and
environmental temperatures become the same again
• For deep convective storms, an EQL near the tropopause is common
42
Skew-T diagram
• The parcel’s convective available potential energy (CAPE) is the positive buoyancy
between the LFC and EQL.
• To tap into the CAPE, the convective inhibition (CIN), or negative buoyancy
below the LFC, has to be overcome.
43
CAPE and CIN
44
CAPE and CIN
• CAPE and CIN are defined using the virtual temperature difference
between the parcel (Tv) and its surrounding environment (Tv). Prove
to yourself their units are m2/s2 or J/kg.
• These expressions can also be written in terms of parcel and
environmental virtual potential temperatures
45
How do q and qe vary with height in the
environment?
46
Stuve diagram
• A typical tropospheric lapse rate is 6.5˚C/km
• The DALR is 10˚C/km.
• As a consequence, potential temperature in the environment tends to increase
with height, slowly in troposphere, more quickly in stratosphere
47
• Environmental potential
temperature vs. height shown
tropopause
• Environmental values indicated
by overbars
• How does environmental water
vapor vary with height? Keep in
mind r ≤ rs and rs ∝ (exp[T]/p)
48
• Environmental vapor mixing
ratio vs. height added
tropopause
• How does environmental qe vary
with height? Keep in mind it
depends linearly on q and
exponentially on r (= rs at LCL)
49
• Environmental qe vs. height
added
tropopause
• Note q and qe differ most when
vapor content is highest, and
qe → q as vapor → 0
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[end]
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