UNITS ESCI 41 Meteorolog Lesson 0 Mth nd Phsics Review A numer is meningless unless it is ccompnied unit telling wht the numer represents. The stndrd unit sstem used interntionll scientists is known s the SI unit sstem. The sic units needed for sstem of units re length, mss, nd time. In the SI sstem, these re the meter (m), kilogrm (kg), nd second (s). Nerl ever other unit cn e derived from these three sic units. The SI unit sstem is sometimes referred to s the m-k-s unit sstem (s opposed to the c-g-s sstem, which uses centimeters, grms, nd seconds s the sic units). Importnt units to rememer re: Phenomenon Unit nme Bsic units Alternte units (non-si) Force Newton (N) kg-m-s dne; pound Energ Joule (J) N-m erg*; foot-l; clorie Power Wtt (W) J-s 1 Horsepower Pressure Pscl (P) N-m l-in ; r; torr; tmosphere; in-hg Temperture Kelvin (K) none Celcius; Fhrenheit Prefies for units: Multiplier Nme A. 10 9 gig G 10 6 meg M 10 3 kilo k 10-3 milli m 10-6 micro µ 10-9 nno n Though interntionll meteorologists dhere to SI units, in the U.S. we continue to use some trditionl units tht differ from SI units. Some of these re ο Pressure: millir (m) = 10 P = hect-pscl (hp) 1
tmosphere (tm) = 10135 P = 1013.5 m inches of mercur (in-hg) 9.9 in-hg = 1013.5 m = 1 tm ο Temperture: Celcius ( C) = K 73.15 Fhrenheit ( F) = (9/5) C 3 ο Length: sttute mile (mi) = 1.61 km = 1760 ds nuticl mile (M) = 1.15 mi = 000 ds ο Speed: 1 Knot (kt) = 1 nuticl mile per hour = 1.15 mph 0.5 m-s 1 ο Energ: Clorie (cl) = 4.184 J CONVERTING UNITS When multipling or dividing numers, ou must keep trck of units. ο Tret units just like the were numers or vriles. ο Units cn cncel if the pper to the sme power on oth the top nd the ottom of frction. ο You cn multipl or divide oth sides of n eqution unit to tr to cncel them. ο Both sides of n eqution must hve the sme units, or something is wrong! ο Units must mke sense phsicll Emples of Unit Conversion 1m Convert 5 cm to meters: 5 cm = 0. 5 m 10 cm Convert 5 cm to µm: 6 1m 10 um 5 cm =.5 10 5 µ m 10 cm 1m Convert 14.66 l-in to m: l 4.46N 39.37 in 5 N 14.66 = 1.013 10 = 1.013 10 in 1l m m 3 1m 3 1.013 10 hp = 1.013 10 m = 1013m 1hP 5 10 hp P = 1.013 10 P 3 hp
CAUTION WHEN CONVERTING TEMPERATURE CHANGES Students often get confused when converting temperture chnges, rther thn tempertures themselves. This is illustrted the following prolem: ο If the temperture is flling 3 C per hour, wht is the rte of temperture chnge in Kelvin per hour? ο Mn students will mke the mistke of converting 3 C to 76.15 K using the formul K = C 73.15, nd then giving the nswer of 76.15 K/hr. DON T DO THIS!!! When deling with chnges in temperture, rther thn temperture itself, ou don t convert temperture using the trditionl formuls of ο K = C 73.15 ο Fhrenheit ( F) = (9/5) C 3 Insted, ou hve to relie tht chnge of 1 C is the ect sme s chnge of 1K,, so chnge of 3 C/hr = 3 K/hr. If ou re converting temperture chnge from Celsius to Fhrenheit, to need to recognie tht chnge of 1 C is the sme s chnge of 1.8 F. This often comes up when deling with specific hets, lpse rtes, or n plce where it is the chnge in temperture tht is importnt. You re now forewrned! RATIOS OF TEMPERATURE VS RATIOS OF OTHER UNITS When we hve rtios of pressures such s p1 p it doesn t mtter wht units for pressure we use s long s the re the sme on the top nd ottom. 500 m 14.76 in Hg 1 = = 1000 m 9.53 in Hg But when we hve rtios of temperture such s T1 T, we must use Kelvin! 373.15K = 1.37 73.15K 100 C = 0 C 3
The difference etween the two is tht the different pressure scles ll hve the sme ero vlue, while the different temperture scles do not shre the sme ero vlues! COORDINATES AND VELOCITY In meteorolog we use the following coordinte sstem: ο The -coordinte increses estwrd ο The -coordinte increses northwrd ο The -coordinte increses upwrd The velocit components long ech coordinte direction re defined s ο u d/ ; u is the speed in the estwrd direction (onl velocit) ο v d/ ; v is the speed in the northwrd direction (meridionl velocit) ο w d/ ; w is the speed in the upwrd direction (verticl velocit) DERIVATIVES A derivtive tells us how one vrile chnges due to chnges in nother vrile. ο For emple, dt/d refers to how temperture (T) chnges s we move estwrd (towrd incresing ). If temperture increses towrd the Est then dt/d is positive. If temperture decreses towrd the Est then dt/d is negtive. 4
The derivtive of function is the slope of tngent to the function. A positive slope mens positive derivtive; negtive slope mens negtive derivtive; ero slope mens ero for the derivtive (see picture elow). We will often use prtil derivtive nottion in this clss, which mens sometimes derivtives will e written s T/ insted of dt/d. ο For those who hven t een eposed to prtil derivtives, don t pnic we will use prtil derivtives prett much in the sme w s norml derivtives. Just think of it s how T chnges with. Velocit nd ccelertion re relted to the time derivtives of the coordintes. Coordinte Velocit Accelertion d u d v d w d = d = d = du dv dw INTEGRALS An integrl is n ntiderivtive. df d = f ( ) const. d 5
6 An integrl of function represents the re under the function. VECTORS A vector consists of direction nd mgnitude. Two vectors re dded plcing them hed to til A vector cn e written in terms of its components long ech coordinte direction. k j i B k j i A = = When two vectors re dded or sutrcted, their components re dded or sutrcted. k j i B A k j i B A ) ( ) ( ) ( ) ( ) ( ) ( = = The mgnitude of vector cn e found from its components. A GRADIENT The grdient (or del) opertor is defined s k j i. The grdient of sclr field (such s pressure) is vector pointing in the direction of mimum increse in the field. It is defined s k p j p i p p.
If contour of the sclr field is plotted (such s isotherms or isors) the grdient t given point is vector tht is oriented t 90 to the contours nd pointed towrd higher vlues. ο The emple elow shows the direction of the pressure grdient t severl points. ο Since the grdient is vector, it hs components in the - nd -directions. The tle elow shows the sign of the components of the pressure grdient t the five points from the emple ove. Point p/ p/ A 0 B C D E 0 0 ο If it isn t pprent to ou wh t point A is ero, imgine tht ou re wlking from west to est cross point A nd re crring rometer (or rogrph). The pressure trce would look something like the following figure. 7
The slope of the pressure trce is ero t point A, nd therefore p/ = 0 t point A. The mgnitude of the grdient t point on contour mp cn e found dividing the contour intervl the shortest distnce etween contours cross the point. In the emple elow the mgnitude nd direction of the pressure grdient is shown. The mgnitude ws found 101m 1008m p = 0.04m / km 100km ο The mgnitude of the grdient increses if the contours ecome closer together, nd decreses s the get frther prt. ο When the contours re pcked closel together it is often referred to s tight grdient. 8
ο When the contours re fr prt it is often referred to s loose grdient. READING EQUATIONS Meteorolog is rell rnch of pplied phsics, nd the lnguge of phsics is mthemtics. Our understnding of the tmosphere could not hve occurred without mthemtics s founion. Mthemtics nd equtions re rell just nother lnguge. If ou ttempted to write down the dvnced concepts of meteorolog without using mthemtics or equtions, ou would quickl run out of pper nd time. Equtions re shorthnd w of epressing phsicl concepts. The ke to using mth nd equtions properl in the stud of meteorolog is to lern how to red equtions. ο The w ou lern to red them is to prctice (just like if ou were lerning nother lnguge). ο Don t immeditel tr to plug numers into n eqution! Insted, tr to figure out wht the eqution is sing out the phsicl world. Often we will use nd write down equtions without ever using them to find n ctul numericl vlue. 9
EXERCISES dp 1. For the digrm elow, tell whether is positive, negtive, or ero t ech d point A, B, C, D, nd E.. For the vector pir (A nd B) shown elow, drw dditionl vectors tht represent A B nd A B. A B 10
3.. The digrm elow represents tempertures in F. At ech point A, B, C, D, nd E drw n rrow indicting the direction of the grdient of temperture, T. (Note tht E lies t the lowest temperture).. Fill in the tle elow telling whether T nd T re positive, negtive, or ero t ech point. Point A B C D E T T c. Which point hs the strongest temperture grdient? d. Which point hs the wekest temperture grdient? 11
4. For the following isotherm pttern, find the mgnitude nd direction of the temperture grdient t the two points A nd B. 1