ESCI Cloud Physics and Precipitation Processes Lesson 4 - Convection Dr. DeCaria

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1 References: ESCI Cloud Physcs and Precptaton Processes Lesson 4 - Convecton Dr. DeCara Glossary of Meteorology, 2nd ed., Amercan Meteorologcal Socety A Short Course n Cloud Physcs, 3rd ed., Rogers and Yau, Ch. 4 Adabatc Mxng of Ar Parcels If two ar parcels are adabatcally mxed together, many thermodynamcs propertes of the mxture are a mass-weghted mean of ther propertes before mxng. A mass-weghted mean of some property s of two ar parcels of masses m 1 and m 2 s gven by the formula s m = m 1 m 2 s 1 + s 2. (1) m 1 + m 2 m 1 + m 2 Formula (1) apples exactly f s s specfc humdty q, and approxmately for mxng rato r and potental temperature θ. If the ar parcels beng mxed are also at the same pressure (sobarc mxng), then temperature and vapor pressure also mx as mass-weghted means, and (1) also apples. Adabatc mxng of two ntally unsaturated ar parcels may actually result n a saturated ar parcel. hs s why we can sometmes see our breath on cold days. he concept of mass-weghted mean can be appled to a contnuous layer of ar as follows: We magne the layer consstng of a seres of N very thn ar parcels, each havng a horzontal area A and thckness z. he mass-weghted mean s gven by the sum m s s m = m. (2) Each parcel has a mass gven by ρ A z, so that (2) becomes ρ A z s ρ z s s m = =. (3) ρ A z ρ z 1

2 In the lmt as the thcknesses of the ar parcels go to zero the summaton turns nto an ntegral, and the formula for the mass-weghted mean of a layer becomes s m = z 2 ρsdz z 1 z 2. (4) ρdz z 1 Formula (4) apples only to those parameters s that do not change as the ar parcel moves up or down. hus, t can be appled to specfc humdty, mxng rato, and potental temperature. Formula (4) cannot be appled to temperature. Formula (4) can be wrtten as a dervatve wth respect to pressure. he hydrostatc equaton allows us to wrte ρdz = gdp, so that (4) becomes 1 where p = p 1. s m = p 1 gsdp p 1 gdp p 1 = 1 sdp, (5) p Durng adabatc mxng the specfc humdty and potental temperature mx as massweghted means, and take on constant values n the mxed layer. he mxng rato also mxes to a close approxmaton as a mass-weghted mean From (5) these values are p 1 q m = 1 qdp (6) p p 1 r m = 1 rdp (7) p p 1 θ m = 1 θdp. (8) p On a dagram usng a logarthmc pressure axs the averages for (6), (7), and (8) can be approxmated graphcally, by usng the method of equal areas. he mass-weghted mean mxng rato, r m, s the sohume that splts the dewpont profle nto two equal areas, as shown n Fgure 1. 1 We can usually gnore, wthout sgnfcant error, the fact that g decreases wth alttude. hs allows t to be removed from the ntegrals. 2

3 he mass-weghted mean potental temperature, θ m, s the adabat that splts the temperature profle nto two equal areas, as shown n Fgure 1 Fgure 1: Equal-area method for determnng mass-weghted mean of mxng rato and potental temperature. Adabatc mxng s why surface temperatures on a wndy nght wll be warmer than on a calm nght. Convectve Mxng and the CCL Another mportant mxng process for the atmosphere s convectve mxng, whch occurs when a layer of ar s heated from below and the upward and downward overturnng of the layer mxes the ar parcels. Convectve mxng s not adabatc. In a convectvely mxed layer θ stll takes on a constant value, but t s not gven by (8). Instead, θ n a convectvely mxed layer wll be the value of the adabat through the surface temperature. Mxng rato n a convectvely mxed layer wll stll be gven by (7). A mornng soundng usually has a surface nverson. As the solar heatng warms the ground a convectvely mxed layer begns to form, eatng away at the surface nverson. A parcel at the surface can rse up the dry adabat untl t reaches the top of the convectvely-mxed layer. As the parcel rses t conserves t mxng rato. 3

4 If the parcel reaches an alttude where the saturaton mxng rato s equal to the parcels mxng rato, a cloud wll form at ths alttude. hs s known as the convectve condensaton level, or CCL. On a skew- dagram the CCL s found by followng the sohume through the surface dewpont up to where t crosses the temperature profle. hs assumes that the parcel does not mx wth t s envronment durng ascent. herefore, the CCL found from ths procedure s called the parcel method CCL. A more accurate assumpton s that the parcel wll mx and become dluted as t ascends. herefore, a better approxmaton to the CCL s found by averagng the mxng rato through a layer of ar near the surface, and followng ths average mxng rato lne up to the temperature soundng. he CCL found from averagng the mxng rato n a layer of ar s known as the mxng method CCL. Fgure 2 llustrates the locatons of the CCL for both the mxng and the parcel methods. Fgure 2: Example Skew- showng locaton of convectve condensaton level (CCL) usng both the parcel and the mxng methods. he convectve temperature, c, s the temperature that the surface must reach n order for an ar parcel to ascend dry adabatcally to reach the CCL. he convectve temperature s found by followng a dry adabat downward from the CCL to the surface, and readng the temperature at that pont. Fgure 3 llustrates the determnaton of the convectve temperature. Fgure 3: Example Skew- showng how to fnd the convectve temperature, c. 4

5 Lfted Ascent For ar parcels lfted mechancally rather than convectvely, the cloud bases wll be at the lftng condensaton level, or LCL. he LCL s located by followng the mxng rato contour through the surface dewpont and fndng where t crosses the adabat through the surface temperature (see Fg. 4). If the parcel s lfted beyond the LCL t wll follow a most adabat. If the most adabat eventually crosses the envronmental soundng, then the parcel wll be warmer than t s envronment and wll contnue to rse on t s own. he level at whch ths occurs s known as the level of free convecton, or LFC (see Fg. 4). Not all soundngs wll have an LFC. Fgure 4: Example Skew- dagram showng the lftng condensaton level (LCL), level of free convecton (LFC), level of neutral buoyancy (LNB), and areas of convectve avalable potental energy (CAPE) and convectve nhbton (CIN). Once a parcel s beyond the LFC t wll contnue to ascend along a most adabat untl t once agan ntersects the envronmental soundng. Beyond ths pont the parcel wll 5

6 be cooler than the surroundng ar, and wll have negatve buoyancy. he level at whch ths occurs s known as the level of neutral buoyancy, or LNB. 2 Convectve Avalable Potental Energy In the prevous lesson we establshed that the vertcal acceleraton on an ar parcel s gven by a z = g. (9) We also know that the acceleraton can be wrtten as a z = du/dt where U s the vertcal velocty of the ar parcel. 3 From the chan rule we can wrte a z = du dt = du dz dz dt = du dz U = 1 du 2 2 dz. (10) Combnng (9) and (10) and rearrangng results n ( ) du 2 = 2g dz. (11) We can use (11) to calculate the vertcal velocty of the parcel once t reaches the LNB. We do ths by ntegratng (11) from the LFC to the LNB we get U 2 LNB U 2 LF C = 2g z LNB z LFC ( ) dz, (12) and f we assume that the parcel s velocty at the LFC s zero, ths becomes U 2 LNB = 2g z LNB z LFC ( ) dz. (13) Equaton (12) has another useful nterpretaton. Recognzng that knetc energy per unt mass s E = U 2 2, then (12) becomes z LNB ( ) E LNB E LF C = g dz. (14) 2 he level of neutral buoyancy s also called the equlbrum level, or EL. 3 Normally meteorologsts use w to ndcate vertcal velocty, but n order to be consstent wth Rogers and Yau I wll use U for the vertcal velocty of the ar parcel. he prme on U s not needed, snce the only vertcal velocty we are dealng wth s that of the ar parcel. 6 z LFC

7 he dfference n knetc energy per mass between the LNB and LFC, E LNB E LF C, s called the convectve avalable potental energy, or CAPE, and so we have CAPE = g z LNB z LFC ( he unts for CAPE are energy per mass, J kg 1. ) dz. (15) he equaton for CAPE can be converted to pressure coordnates (see Exercses), and s CAPE = R d p LNB p LFC ( ) d ln p. (16) On a skew- dagram s represented by the temperature soundng and s represented by the most adabat through the LFC. he rght-hand sde of (16) s represented by the green shaded area shown n Fg. 4. CAPE s also called the postve area on the skew-. Convectve nhbton, or CIN, s the work requred to lft the parcel from the surface to the LFC. he equaton for CIN s smlar to that for CAPE, only wth dfferent lmts of ntegraton, CIN = R d where p 0 s the surface pressure. p LFC p 0 ( ) d ln p, (17) CIN s represented by the orange shaded area on the skew- dagram n Fg. 4. CIN s also called the negatve area on the skew-. o account for humdty s effects on buoyancy the vrtual temperature, v should really be used when calculatng CAPE and CIN. When vrtual temperature s used n the calculatons the CAPE s often denoted as CAPEV or CAPE v, and CIN s denoted as CINV or CIN v. he equatons for these quanttes are p LNB CAPE v = R d ( v v ) d ln p (18) p LFC p LFC CIN v = R d ( v v ) d ln p. (19) p 0 7

8 Forecastng CAPE CAPE s often computed from the mornng soundng. However, convecton doesn t usually fre up untl the afternoon. So, the mornng value of CAPE may not be partcularly relevant for the afternoon s thunderstorms. he Storm Predcton Center (SPC) computes a forecast CAPE value by makng assumptons about what the temperature and dewpont of the surface parcel wll be n the afternoon. her assumptons are: 4 he forecast surface temperature s the temperature that the parcel at 850 mb would have f t were brought dry-adabatcally to the surface, plus 2 C. he forecast surface mxng rato s the mean mxng rato of the lowest 100 mb of the soundng. he LCL and CAPE are then computed based on ths forecast surface parcel. Lmtatons of CAPE From (13) and (15) we see that the maxmum updraft speed s related to CAPE va U max = U LNB = 2 CAPE. (20) Equaton (20) wll overestmate the maxmum updraft speed because of several factors. Water burden: he expressons for CAPE assumed that any lqud water mmedately falls from the ar parcel. In realty there s a sgnfcant amount of lqud water that remans wth the parcel, whch adds to the mass of the parcel, and therefore slows t down. Compensatng subsdence: In areas of convecton the regons between updrafts are occuped by downward moton, or subsdence. hs warms the ar outsde of the updraft, and decreases the temperature dfference between the parcel and the envronment, reducng the buoyancy. Entranment: Along the edges of the convectve updraft there wll be mxng of drer and cooler ar nto the buoyant plume. hs cools the ar wthn the updraft through both evaporatve coolng and mxng, and reduces the buoyancy. Aerodynamc resstance: he rsng plume s slowed down by havng to push ar out of ts way

9 Exercses 1. Derve (16) from (15). Hnt: Use the hydrostatc equaton to substtute for dz. hen use the deal gas law to substtute for ρ. 9