A&OS C110/C227: Review of thermodynamics and dynamics IIRobert FovellUCLA Atmospheric and Oceanic Sciencesrfovell@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 air4

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

• Represents balance (stalemate) between vertical pressure gradient force and gravity resulting (formally) in no vertical acceleration and (practically) in no vertical motion

6

g = 9.81 m s-2 at sea-level

Potential temperature• Potential temperature is a system property that is conserved

for a dry adiabatic process

• We can also define a virtual potential temperature analogously to virtual temperature

7

p in millibarsT in KelvinR and cp in J kg-1 K

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

8

p, T, Td, V, r, r, all changeq, r do not

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

10

p, T, Td, V, r, r, q, r all changeqe does not

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.

12

• 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

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 section1. Adiabatic expansion approach to saturation2. Dew point approach to saturation3. 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

19

• 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.

Stuve diagram

20

• A parcel achieves its potential temperature by moving dry adiabatically to p = 1000 mb

Stuve diagram

21

• 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

Stuve diagram

22

• 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

Stuve diagram

23

• 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

Stuve diagram

24

• 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

Stuve diagram

25

• 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, whichmeans it will reach one, and only one, moist adiabat qe. That qe characterizesthe parcel, whether it ever becomes saturated or not. So qe is also fixed.

Stuve diagram

26

• 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.

Stuve diagram

27

• 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

Stuve diagram

28

• 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.

Stuve diagram

29

• 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

Stuve diagram

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

34

• 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.

Skew-T diagram

35

• 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

Skew-T diagram

36

• 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

Skew-T diagram

37

• 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

Skew-T diagram

38

• 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

Skew-T diagram

39

• 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.

Skew-T diagram

40

• 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.

Skew-T diagram

41

• Now let’s compare our raised parcel to an environmental sounding

Skew-T diagram

42

• 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

Skew-T diagram

43

• 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.

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

47

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

48

• Environmental potential temperature vs. height shown

• Environmental values indicated by overbars

• How does environmental water vapor vary with height? Keep in mind r ≤ rs and rs (exp[T]/p)∝

tropopause

49

• Environmental vapor mixing ratio vs. height added

• How does environmental qe vary with height? Keep in mind it depends linearly on q and exponentially on r (= rs at LCL)

tropopause

50

• Environmental qe vs. height added

• Note q and qe differ most when vapor content is highest, and qe → q as vapor → 0

tropopause

[end]

51