Atmospheric Dynamics: lecture 3

Transcription

1 Atmospheric Dynamics: lecture 3 Moist convection Dew point temperature/lapse rate/lcl Equivalent potential temperature Conditional and potential instability Thermodynamic diagram CAPE Introduction to Python (Problem 1.11) problem 1.10 (sec. 1.18); problems in Boxes 1.6 & extra problems (a.j.vandelden@uu.nl) (

2 Cumulus clouds and convective precipitation Parcel of air rises spontaneously due to hydrostatic instability Figure 1.27: (Espy, 1841)

3 Section 1.15 & last week Vertical acceleration of an air parcel Governed by: d 2 δz dt 2 = g θ 0 dθ 0 dz δz N 2 δz N 2 g dθ 0 θ 0 dz Brunt Väisälä-frequency, N: N is about s -1 The solution: δz = exp ( ±int) If N 2 = g dθ 0 θ 0 dz < 0 Exponential growth instability If N 2 = g θ 0 dθ 0 dz > 0 oscillation stability

4 Radiosonde measurements De Bilt, 26 Sep. 2013, 12 UTC Launching a radiosonde at KNMI on 30 November 2012

5 Radiosonde measurements De Bilt, 26 Sep. 2013, 12 UTC θ Will an air parcel at the surface accelerate upwards spontaneously?

6 Lifted condensation level (LCL) Figure 1.27: (Espy, 1841) LCL The air parcel will cool as it ascends. At some level it will become saturated. This level is the Lifted Condensation Level (LCL)

7 Radiosonde measurements De Bilt, 26 Sep. 2013, 12 UTC θ Will it reach the LCL below this height, i.e. will clouds form?

8 Lifted condensation level (LCL) Figure 1.27: (Espy, 1841) LCL The LCL can be determined with information of the dewpoint and the temperature at the surface

9 Dew point temperature =temperature of air if it were cooled to saturation at constant pressure Teten s formula (based on Clausius Clapeyron equation): T[ C] e s = 611.2exp T[ C] Substitute e s =e and T=T d and invert : e ln T d [ C] = e ln (box 1.5)

10 Equations of state Dry air: p d = ρ d R d T Water vapour: e = ρ v R v T (ideal gases) Water vapour mixing ratio: r = ρ v ρ d r = ρ v ρ d = e R v R d p d = εe p d εe p with ε R d R v

11 Finding the lifted condensation level Dew point lapse rate Substitute e s =e and T=T d in the Clausius Clapeyron equation: de dt d = Le R v T d 2 de Lr v p dt 2 d R v εt d Previous slide: e r v p ε r v is constant (see problem 1.3)! Therefore: de r v dp dt d ε dt d 2 dt d dp = R v T d Lp With hydrostatic equation: dt d dz = ρgr v T 2 d Lp gr v T d 2 LR d T gt d Lε

12 Finding the lifted condensation level Dew point lapse rate dt d dz gt d Lε K m-1 Γ dew dt d dz Γ d g c p Figure 1, Box 1.5 Derive this expression for the dry-adiabatic lapse rate

13 Dew point lapse rate dt d dz Γ dew gt d Lε Figure 1, Box 1.5 Extra problem (see box 1.6) When is dt d dz Γ dew > Γ d? Investigate the consequences for conditional instability, cloud formation, precipitation and global water cycle if this is indeed the case.

14 Radiosonde measurements De Bilt, 26 Sep. 2013, 12 UTC θ Determine the lifted condensation level (LCL). Will clouds form?

15 Radiosonde measurements De Bilt, 26 Sep. 2013, 12 UTC θ The lifting condensation level is found by solving: T s + dt dz z LCL = T ds + dt d dz z LCL 16.0 g z LCL = c p 10 3 z LCL z LCL =1 km

16 Convective clouds organized into streets Thursday 26 Sep. 2013

17 What happens above the LCL? If the air parcel continues its ascent after reaching the LCL, condensation of water vapour will occur, which will be accompanied by release of latent heat..

18 Section 1.16 Latent heat release in updraught The rate of heating due to condensation is mj = L dm v? J L dr s dt dt r s is saturation mixing ratio L (= J kg -1 ) is the latent heat of condensation

19 Section 1.16 Latent heat release in updraught The rate of heating due to condensation is mj (m is mass of air parcel): mj = L dm v? J = L dr s dt dt r s is saturation mixing ratio L (= J kg -1 ) is the latent heat of condensation Change in r s following the motion is primarily due to ascent: dr s dt w dr s for w > 0; dz dr s 0 for w 0. dt

20 Section 1.16 Conditional instability Assume that θ=θ 0 (z)+θ, with θ <<θ 0. We have (eq 1.55), dθ dt dθ' dt + w dθ 0 dz = J Π. { dθ' dt = θ 0 g N 2 w if w 0; dθ' dt θ 0 g N m 2 w if w > 0, J=0 if w 0 and J=-Lwdr s /dz if w>0 Latent heat release only in the updraught!

21 Section 1.16 Conditional instability Assume that θ=θ 0 (z)+θ, with θ <<θ 0. We have (eq 1.55), dθ dt dθ' dt + w dθ 0 dz = J Π. dθ' { dt = θ 0 g N 2 w if w 0; dθ' dt θ 0 g N m 2 w if w > 0, J=0 if w 0 and J=-Lwdr s /dz if w>0 N 2 m N 2 + gl dr s θ 0 Π 0 dz, N m is the "moist" Brunt Väisälä frequency If N m <0 and w>0 then dθ' dt > 0: positive buoyancy and upward acceleration

22 Section 1.16 Conditional instability Assume that θ=θ 0 (z)+θ, with θ <<θ 0. We have (eq 1.55), dθ' dt = θ 0 dθ { g N 2 w if w 0; dt dθ' dt + w dθ 0 dz = J Π. dθ' dt θ 0 g N m 2 w if w > 0, J=0 if w 0 and J=-Lwdr s /dz if w>0 N 2 m N 2 + gl dr s θ 0 Π 0 dz, N m is the "moist" Brunt Väisälä frequency Frequently: N m 2 <0 and N 2 >0. In these circumstances the atmosphere is statically or buoyantly unstable only with respect to saturated upward motion. This is called conditional instability.

23 Section 1.16 Equivalent potential temperature Previous slide: N 2 m N 2 + gl dr s θ 0 Π 0 dz, Define a pseudo- or moist adiabatic process in which a equivalent potential temperature, θ e, is constant. That is, θ e is constant following saturated ascent. Simply define: then N m 2 = g d( θ e ) 0 ( ) 0 dz θ e θ e θ exp Lr s θπ Extra problem: Show that equivalent potential temperature is approximately conserved in saturated upward motion.

24 cold and dry air Warm conveyor belt

25 cold and dry air Warm conveyor belt

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27 Tropical cyclone Nadine : warm/moist core

28 cold and dry air Cyclone with cold/dry core

29 Section 1.16 Equivalent potential temperature θ e θ exp Lr s θπ Equivalent potential temperature unsaturated air parcel: actual mixing ratio θ e =θexp Lr θπ LCL Lr =θexp c p T LCL (LCL: lifting condensation level)

30 Potential Instability: θ e z < 0 De Bilt, 26 Sep. 2013, 12 UTC θ θ e θ z < 0 θ e z < 0 Potential Instability (PI) up to 650 hpa (3600 m)

31 Three-dimensional convection: open convection cells Special type of threedimensional flow! Narrow rings of clouds (upward motion) surrounding broad clear areas (downward motion) satellite image, 18 April 2001, 1245 UTC

32 Precipitation pattern associated with open convection cells radar image, , 1245 UTC

33 De Bilt, 12Z 18 Apr 01 Absolute instability: θ z < ! PRES HGHT TEMP DWPT RELH MIXR DRCT SKNT THTA THTE THTV! hpa m C C % g/kg deg knot K K K! ! ! ! ! ! ! ! ! ! ! ! ! Problem ! ! Compute the height of 50 the 0.71 LCL ! ! Is 3246 this level reached spontaneously? ! ! ! ! ! ! ! !

34 De Bilt, Absolute and Conditional Instability: 18 April UTC ! PRES HGHT TEMP DWPT RELH MIXR DRCT SKNT THTA THTE THTV! hpa m C C % g/kg deg knot K K K! ! ! ! ! θ ! ! z < ! ! Absolute Instability (AI) up to 800 hpa ! ! ! ! ! ! ! ! θ ! e ! z < ! ! Conditional (potential) Instability 0.41 up 322 to hpa ! ! ! !

35 South America Manaus Station latitude: Station longitude: Station elevation: 84.0!

36 82332 SBMN Manaus (Aeroporto) Observations at 00Z 08 Oct ! Manaus (Brazil) PRES HGHT TEMP DWPT RELH MIXR DRCT SKNT THTA THTE THTV hpa m C C % g/kg deg knot K K K! No 3688 absolute 7.2 instability; Only 5258 potential instability!! How far up will a parcel 55 at 0.49 the ground 219 have to rise to reach its lifting 42 condensation level?

37 Oct UTC Layered clouds tropical Cu convection Forced lifting in the ITCZ M

38 Tephigram : pref R /c p θ = T p Explanation constant θ constant pressure constant θe Lrs θ e θ exp θπ constant saturation mixing ratio R e rs = d s? Rv p isotherm T

39 Typical profiles in the tropics Dew point temperature temperature θ = T p ref p R/c p θ e θ exp Lr s θπ

40 Typical profiles in the midlatitude summer 18 April UTC Dew point temperature θ = T p ref p R/c p temperature θ e θ exp Lr s θπ tropopause

41 Tephigram Parcel of air is lifted from the ground. What temperature-profile does it follow? temperature LNB LNB=level of no buoyancy dew point temperature diluted parcel undiluted parcel LCL=lifted condensation level LCL

42 Tephigram 0 C Close-up Temperature (environment) Dew point temperature (environment) diluted parcel undiluted parcel 700 hpa LCL 800 hpa 900 hpa dt d /dz=1.8 K/km dt/dz=9.8 K/km 1000 hpa 10 C 20 C 30 C

43 Convective Available Potential Energy (CAPE) Force on air parcel: F = m d2 z θ' mg mgb 2 dt θ 0 B = buoyancy B θ' θ 0 Assuming a stationary state and horizontal homogeneity we can write: dw dt w dw dz = Bg. or wdw = Bgdz.

44 Convective Available Potential Energy (CAPE) wdw = Bgdz. A parcel starting its ascent at a level z 1 with vertical velocity w 1, will have a velocity w 2 at a height z 2 given by w 2 2 = w CAPE, z 2 CAPE g Bdz. z 1

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46 De Bilt, 12Z 18 Apr 01 Potential Instability: θ e z < ! PRES HGHT TEMP DWPT RELH MIXR DRCT SKNT THTA THTE THTV! hpa m C C % g/kg deg knot K K K! ! ! ! ! ! ! ! ! ! ! ! ! CAPE= J/kg ! ! If 50 CAPE= m 2 s ! ! w 2 56 = m s ! ! ! ! ! ! Potential Instability (PI) up to 568 hpa ! !

47 82332 SBMN Manaus (Aeroporto) Observations at 00Z 08 Oct ! Manaus (Brazil) PRES HGHT TEMP DWPT RELH MIXR DRCT SKNT THTA THTE THTV hpa m C C % g/kg deg knot K K K! CAPE= J/kg If 96 CAPE= m 2 s w70 2 =? 2.05 m s

48 Sounding Bordeaux Problem 1.10, p. 71 Spanish plume (warm and dry) Air parcels with high relative humidity must be forced to rise through the stable layer

49 Surface weather map of 20 July 1992, 15 UTC. The position of a surface station is indicated by a square. The number inside the square indicates the cloudiness (in octas). Also indicated are the temperature ( C) (upper left), the dew point temperature ( C) (lower left) and the pressure (hpa-1000) (upper right). Also shown are sea level isobars drawn every 1 hpa (thick line corresponds to 1012 hpa), according an objective analysis scheme. The letters 'Za" indicate the position of Zaragoza. Arrows indicate the movement and sources of moisture for the thunderstorm. The confluence line is indicated by a "hooked" solid line Infrared Meteosat satellite image, 20 July 1992,19:30 UTC. Illustration of the outbreak of summer thunderstorms due to forced lifting of potentially unstable air in advance of a cold front over southwest Europe. Height, temperature and temperature advection at 850 hpa on 20 July 1992, 12 UTC. The height, labeled in units of m, and the temperature, labeled in units of C, are shown in panel (a), while temperature advection, labeled in units of 10-4 K s-1, is shown in panel (b).

50 Homework: Problem 1.10 Plot the data shown in table 1.2 in a tephigram (figure 1.30). Determine from the tephigram the lifting condensation level (LCL) of an air parcel at the ground. Determine the height of the LCL from the theory described in Box 1.5. Determine the equivalent potential temperature of the air parcel. Will this air parcel reach the LCL spontaneously? Once it has reached the LCL, over how large a vertical distance will it rise? Estimate this vertical distance from the tephigram and also by using the theory of Box 1.6. Verify the value of θ e at the surface using eq Estimate the value of CAPE. Repeat this for the data in table 1.3.

51 Next week: large scale circulation geostrophic balance, vorticity and potential vorticity 11:00-12:45: Room David de Wied Gebouw (DDW) :30-16:00: Room 165 BBG

52 Next week sections 1.19 to 1.24 and Boxes 1.7 and 1.8. (geostrophic balance and thermal wind balance: (in)stability, cyclones) Problems 1.11, 1.12 and Box 1.7 First project: construct a numerical model of an inertial oscillation Solve this in the programming language Python

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54 Python programming language Python: E-book on the use of Python in atmospheric sciences: Package for scientific computing in python (Numpy): Package for plotting in python (Matplotlib): This site is useful in demonstrating how to make plots and how to customize your plots: The Canopy Python distribution might now be an easy way to get to know python: If you want to plot some figures that you want to use in latex the best way is to save your figure as a pdf using the plt.savefig command. That way the plot scales with the resolution so your figure will appear smooth in latex. You can use the pandas package ( to load text files, excel sheets and other databases easily into python and convert them to pandas dataframes. An easy way to load external data into python.

55 Example: Problem 0.1

56 Python script of solution to problem 0.1