Synoptic Meteorology I: Finite Differences September Partial Derivatives (or, Why Do We Care About Finite Differences?

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1 Synoptic Meteorology I: Finite Differences September 2014 Prtil Derivtives (or, Why Do We Cre About Finite Differences?) With the exception of the idel gs lw, the equtions tht govern the evolution of fundmentl tmospheric properties such s wind, pressure, nd temperture (known s the primitive equtions) re fundmentlly relint upon prtil derivtives. Indeed, mny thermodynmic nd kinemtic properties of the tmosphere re typiclly expressed in terms of prtil derivtives. We will explore mny specific exmples of such equtions throughout both this nd next semester. Mthemticlly speking, the prtil derivtive of some generic field f with respect to some generic vrible x cn be expressed s: f = lim ( 0 In other words, x is equl to the vlue of f Thus, for smll (or finite) vlues of, we cn pproximte f how do we compute from vilble tmospheric dt? s pproches (but does not equl) zero. by f. Tht begs the question: To do so, we use wht re known s finite differences to pproximte the vlue of f over some finite. Applied to isoplethed nlyses of meteorologicl fields, finite differences enble us to evlute the sign nd/or mgnitude of given quntity tht depends upon one or more prtil derivtives. Applied to gridded dt, such s is used nd produced by numericl wether prediction models, finite differences re one mens by which the primitive equtions cn be solved so s to obtin numericl wether forecst. In the following, we wish to describe how finite difference pproximtions re obtined, the degree to which ech is n pproximtion, nd begin to describe how they cn be pplied to the tmosphere. Developing Finite Difference Approximtions First, let us consider generic continuous function f(x), grphicl exmple of which is depicted below in Figure 1. This function doesn t necessrily represent meteorologicl field, but it doesn t not necessrily represent one either; it is simply generic function. Along the curve given by f(x), there re three points of interest: x, x +1, nd x -1. The function f(x) hs the vlues f(x ), f(x +1 ), nd f(x -1 ) t these three points, respectively. The distnce between x nd x -1 is equl to the distnce between x nd x +1, nd we cn denote this distnce s. Finite Difference Approximtions, Pge 1

2 The Tylor series expnsion of f(x) bout x = b, where b is some generic point, is given by: f ''( b) 2 f '''( b) 3 f ( x) = f ( b) + f '( b)( x b) + ( x b) + ( x b) +... (2) In other words, f(x) is equl to the vlue of f(x) t x = b plus series of higher-order terms, ech of which hs different derivtive (primes), exponent on x - b, nd fctoril (!) order. Figure 1. Grphicl depiction of generic function f(x) evluted t three points. Plese see the text for further detils. Let us consider the cse where x = x +1 nd b = x. The distnce x b, or x +1 x, is equl to. Conversely, let us consider the cse where x = x -1 nd b = x. The distnce x b, or x +1 x, is equl to -. Mking use of this informtion, we cn expnd (2) for ech of these two cses: f ''( x ) 2 f '''( x ) 3 f ( x + 1 ) = f ( x ) + f '( x ) + ( ) + ( ) +... (3) f ''( x ) 2 f '''( x ) 3 f ( x ) = f ( x ) f '( x ) + ( ) ( ) +... (4) Note the similr ppernce of (3) nd (4) prt from the leding negtive signs on the first nd third order terms in (4). These rise becuse x b = - here, s noted bove. f From (3) nd (4), we re interested in the vlue of f (x ). This is equivlent to. We cn use (3) nd (4) to obtin n expression for this term; we simply need to subtrct (4) from (3). Doing so, we obtin the following: Finite Difference Approximtions, Pge 2

3 2 f '''( x ) 3 f ( x + 1 ) f ( x = 2 f '( x ) + ( ) +...(odd order terms) (5) Note how the zeroth nd second order terms in (3) nd (4) cncel out in this opertion. If we rerrnge (5) nd solve for f (x ), we obtin: f ( x+ f ( x f '''( x ) 2 f '( x ) = ( ) +... (6) 2 At this point, we wish to neglect ll terms higher thn the first order term from (6). Doing so, we re left with: f '( x ) = f ( x + 1 ) f ( x 2 ) (7) Eqution (7) is wht is known s centered finite difference. It provides mens of clculting t x = x by tking the vlue of f t x = x +1, subtrcting from it the vlue of f t x = x -1, nd dividing the result by the distnce between the two points (2). Note tht x here nd in lter exmples cn be ny vrible; it does not hve to represent the x-xis or the est-west direction. Eqution (7) is equivlent if x is replced by y, z, p, or ny number of other vribles. There exist other wys for us to use (3) nd (4) to get expressions for f (x ). For instnce, we cn solve (3) for this term. If we do so, we obtin: f ( x+ f ( x ) f ''( x ) f '''( x ) 2 f '( x ) = ( ) ( )... (8) Neglecting ll terms higher thn the first order term in (8), we obtin: f '( x ) = f ( x+ f ( x ) (9) Eqution (9) is wht is known s forwrd finite difference. It provides mens of clculting t x = x by tking the vlue of f t x = x +1, subtrcting from it the vlue of f t x = x, nd dividing the result by the distnce between the two points (). Alterntively, we cn solve (4) for f (x ). If we do so, we obtin: f ( x ) f ( x f ''( x ) f '''( x ) 2 f '( x ) = + ( ) ( ) +... (10) Finite Difference Approximtions, Pge 3

4 Neglecting ll terms higher thn the first order term in (10), we obtin: f '( x ) = f ( x ) f ( x (1 Eqution (1 is wht is known s bckwrd finite difference. It provides mens of clculting t x = x by tking the vlue of f t x = x, subtrcting from it the vlue of f t x = x -1, nd dividing the result by the distnce between the two points (). Finite Differences s Approximtions Note tht we do not necessrily need to neglect the higher-order terms in obtining ny of the bove expressions for f (x ); we hve done so here primrily for simplicity. If we were to retin the higher order terms, we would end up with more ccurte pproximtions for f (x ). This highlights key point: ll finite differences re pproximtions. All finite differences re ssocited with wht is known s trunction error, which is determined by the power of on the first term tht is neglected in obtining the finite difference pproximtion. For instnce, consider our centered finite difference given by Eqution (7). In obtining (7), the first term tht we neglected in (6) included () 2 term. As result, we sy this finite difference is second-order ccurte. By contrst, consider our forwrd nd bckwrd finite differences, given by Equtions (9) nd (1, respectively. In obtining ech eqution, the first terms tht we neglected in (8) nd (10) included () term. As result, we sy tht these finite differences re first-order ccurte. The higher the order of ccurcy, the more ccurte the finite difference. In synoptic meteorology, where exct vlues for prtil derivtives re often not necessry, we typiclly utilize the centered finite difference. Forwrd nd bckwrd finite differences re rrely utilized except long the edges of the dt, where the -1 nd +1 points my not exist. Higherorder finite differences, typiclly fourth- or sixth-order ccurte, re necessry for numericl wether prediction models given chos theory, which sttes tht very smll differences in dt cn led to very lrge forecst differences. A Finite Difference Approximtion for Second Derivtives While the first prtil derivtive of some field provides mesure of its slope, sometimes we re interested in evluting the second prtil derivtive of some field. Recll from clculus tht the second prtil derivtive of field provides mesure of its concvity; positive second prtil derivtives infer tht field is concve up (or convex), while negtive second prtil derivtives infer tht field is concve down. We cn obtin finite difference pproximtion for the second prtil derivtive by dding (3) nd (4). Doing so, we obtin: Finite Difference Approximtions, Pge 4

5 f ''( x ) 2 f ( x + 1 ) + f ( x = 2 f ( x ) + 2 ( ) +... (12) If we solve (12) for f (x ), we obtin: f ''( x ) = f ( x + 1 ) + f ( x 2 f ( x ) ( ) 2 (13) 2 f Eqution (13) provides fourth-order ccurte mens of evluting, or f ''( x ) 2, by dding the vlue of f t x +1 to the vlue of f t x -1, subtrcting two times the vlue of f t x, nd dividing the result by the squre of the distnce between points () 2. Just s for the finite difference pproximtion for the first prtil derivtive, x here nd in lter exmples cn be ny vrible; it does not hve to represent the x-xis or the est-west direction. Eqution (13) is equivlent if x is replced by y, z, p, or ny number of other vribles. Likewise, just s for the finite difference pproximte for the first prtil derivtive, higher-order ccurte finite difference pproximtions for the second prtil derivtive re possible if dditionl terms re not truncted. Applying Finite Differences: An Exmple One of the most importnt ttributes of the wind is its bility to trnsport. The trnsport of some quntity by the wind is known s dvection. We re most often interested in its horizontl trnsport, or horizontl dvection, where the horizontl surfce cn be tken to be Erth s surfce, constnt height surfce, n isobric surfce, or even n isentropic surfce. For convenience, we sometimes refer to horizontl dvection simply s dvection. In synoptic meteorology, we re prticulrly interested in temperture dvection, referring to the horizontl trnsport of energy (recll tht temperture is simply mesure of the verge kinetic energy of the ir) by the wind. Ptterns of cold ir dvection nd wrm ir dvection reflect the (horizontl) motion of ir msses nd, s we will see next semester, ply crucil role in forcing verticl motions, cn bring bout chnges in the mplitude of troughs nd ridges, nd cn influence cyclone nd nticyclone development. Mthemticlly, temperture dvection is expressed s the product of the pproprite component of the wind whether est-west (u) or north-south (v) nd the locl chnge of temperture in some direction est-west (x) or north-south (y) where: dvection = u v (14) y Finite Difference Approximtions, Pge 5

6 In vector nottion, (14) cn be written s: dvection = v T (15) The units of temperture dvection re the units of wind m s -1 multiplied by the units of temperture either C or K divided by distnce units m. As result, temperture dvection hs units of C s -1 or K s -1 ; in other words, how temperture is chnging loclly over some finite mount of time t. We cn evlute (14) from chrts of wether dt using our centered finite difference pproximtion developed bove. Consider the hypotheticl nlysis presented in Figure 2. We re interested in computing the horizontl temperture dvection t the point mrked by the closed circle nd wind observtion. We hve lredy completed n isotherm nlysis using temperture dt from this point s well s the other loctions tht surround it. We thus hve everything we need to compute horizontl temperture dvection. Figure 2. Hypotheticl surfce temperture observtions ( F, red numbers), isotherm nlysis (every 5 F, blck lines), nd single wind observtion (10 kt = 5.15 m s -1 out of the northwest). Depicted for reference re horizontl scles nd the north nd est crdinl directions. Dt re plotted on mp constructed using the Merctor mp projection. To compute horizontl temperture dvection, we must first set up our x- nd y-xes. Fortuntely, since we re told tht the dt re plotted on Merctor mp projection, the positive x-xis points to the right, or due est, while the positive y-xis points up, or due north. Since our centered finite difference pproximtion is only vlid over finite distnces here, nd y we must set up smll grid centered on the loction of our wind observtion. This is done so tht we cn estimte the temperture t points x+1, x-1, y+1, nd y-1 in other words, the terms tht Finite Difference Approximtions, Pge 6

7 enter into the numertor of our centered finite differences. The result of doing so is given in Figure 3. Figure 3. As in Figure 2, except with finite grid drwn in centered on our wind observtion. Both in this exmple nd in prctice, the distnce is tken to be equl to the distnce y. In this cse, using the distnce references on the edges of the mp, both nd y re 50 km (or 50,000 m). Next, we use our isotherm nlysis to estimte the vlue of temperture t points x+1, x-1, y+1, nd y-1. We must do so becuse we do not hve n exct temperture observtion t ny of these loctions. Visully doing so, we stte tht the temperture t x+1 is 72 F, t x-1 is 67 F, t y+1 is 67 F, nd t y-1 is 73 F. This enbles us to compute the finite difference pproximtions to our prtil derivtives, where: dvection = u v y T = u T 2 T v T 2 y 72 F 67 F 67 F 73 F = u v m m x+ 1 x y+ 1 y (16) Note tht, per Figure 3 s cption, we know tht = y = 50,000 m, such tht 2 = 2 y = 100,000 m. Now, we need to know the vlues of u nd v, the zonl (est-west) nd meridionl (north-south) wind components, respectively. To obtin these vlues, we need to use bit of trigonometry. Recll tht in meteorologicl convention, from the north = 0 /360, from the est = 90, from the south = 180, nd from the west = 270. If the wind direction (in degrees) is known, then the u nd v components of the wind cn be obtined using the following equtions: Finite Difference Approximtions, Pge 7

8 u = v sin π * wdir (17) 180 π v = v cos * wdir (18) 180 In both (17) nd (18), v is the mgnitude of the wind vector v. In pplied terms, v is simply equl to the wind speed. The π/180 fctor in both the sin nd cos sttements converts the wind direction from degrees to rdins. Returning to our exmple given by Figure 2, we know tht the wind speed is equl to 10 kt = 5.15 m s -1. We lso know tht our wind is out of (or from) the northwest. Expressed in degrees, from the northwest = 315 (e.g., hlfwy between 270 /west nd 360 /north). If we substitute these vlues into (17) nd (18), we obtin: v u π sin *315 = = ms ms 5.15 π cos *315 = = ms ms 5.15 (19) (20) A bit of snity check is in order before proceeding. The positive x-xis is to the est, while the positive y-xis is to the north. Our wind is blowing from the north nd west nd, thus, to the south nd est. Our wind thus blows in the positive x but negtive y directions. Since u is long the x-xis (est-west) nd v is long the y-xis (north-south), we would expect tht u should be positive nd v should be negtive for northwest wind nd, indeed, we find tht this is true. If we plug (19) nd (20) into (16) nd run through the clcultions, we obtin: F 67 F m 67 F 73 F ( 3.64ms ) = dvection = ms Fs ( m In other words, due solely to horizontl dvection, the temperture t the loction of our wind observtion is cooling by F every second. If we multiply this by 3,600 (the number of seconds in one hour) or 84,600 (the number of seconds in one dy), we cn convert this to F h -1 or F dy -1, respectively. Doing so, we obtin vlues of F h -1 nd F dy -1. In other words, due solely to horizontl dvection, the temperture t the loction of our wind observtion is cooling by 1.44 F every hour nd F every dy. Before we proceed further, it is gin time for nother snity check. In Figure 2, we see tht the wind is blowing towrd the sttion from where it is colder. As result, we would expect the wind Finite Difference Approximtions, Pge 8

9 to be dvecting (or trnsporting) colder ir towrd the observtion sttion. Our clcultion suggests tht this is true due to dvection, the temperture t the observtion sttion is cooling. The bove clcultion process represents firly complex mens of evluting horizontl temperture dvection. By contrst, our snity check hints t nother, fr less complex mens of doing so. Insted of using Crtesin (x,y) coordintes, s we did before, we my use nturl coordinte system to ssess horizontl temperture dvection. Recll tht in the nturl coordinte system, the pproprite coordintes become s, or long (stremwise) the wind, nd n, or norml to the wind. For the exmple given in Figure 2, the positive s-xis points to the southest, in the direction tht the wind is blowing, nd the positive n-xis points to the northest, or 90 to the left of the positive s-xis. Figure 4 below provides grphicl depiction of the nturl coordinte system pplied to the exmple from Figures 2 nd 3. Figure 4. As in Figure 3, except with the finite grid drwn in the nturl (rther thn Crtesin) coordinte system. Both in this exmple nd in prctice, the distnce s is tken to be equl to the distnce n. In this cse, using the distnce references on the finite grid, both s nd n re 50 km (or 50,000 m). In the nturl coordinte system, dvection is expressed mthemticlly s: Ts+ 1 Ts 1 dvection = v = V = V (22) s s 2 s In the bove, V is the wind speed, equivlent to the mgnitude of the velocity vector ( v ). Here, we hve used second-order ccurte centered finite difference pproximtion to obtin the reltionship t the fr end of Eqution (22). Note tht we no longer need to brek down our wind Finite Difference Approximtions, Pge 9

10 into its u nd v components, nor del with chnge in temperture in both the x nd y directions. We only need to know the wind speed 10 kt or 5.15 m s -1 in this exmple nd the chnge in temperture long the s-xis. Becuse the wind is ligned with the s-xis, the wind component perpendiculr to the xis is zero, nd thus we do not need to know the chnge in temperture long the n-xis. Evluting from Figure 4, we estimte tht T s+1, or the temperture t the grid point long the positive s-xis, is 73.5 F nd tht T s-1, or the temperture t the grid point long the negtive s- xis, is 66 F. Plugging these vlues into (22), we obtin: ( 5.15ms ) 73.5 F 66 F 2 dvection = = Fs (23) ( 50000m) Note tht this result is very nerly identicl to tht in Eqution (2, s we expect using the sme dt. Tht this is true provides snity check upon our result. The two re not exctly equl to ech other becuse of the inherent pproximte nture to ech of our two nlyses, nmely in obtining the vlues of T t ech of our grid points. With the bove in mind, we cn stte severl generl rules relted to temperture dvection: Where wind blows from cold ir towrd wrm ir, cold ir dvection is occurring. Where wind blows from wrm ir towrd cold ir, wrm ir dvection is occurring. When the chnge in temperture over fixed distnce is lrge, the mgnitude of the dvection will be lrge. When the chnge in temperture over fixed distnce is smll, the mgnitude of the dvection will be smll. When the wind blows prllel to the isotherms, no horizontl temperture dvection occurs. When the wind blows perpendiculr to the isotherms, horizontl temperture dvection is mximized. Horizontl temperture dvection is lrger when the wind component tht blows perpendiculr to the isotherms is lrger. Horizontl temperture dvection is smller when the wind component tht blows perpendiculr to the isotherms is smller. Horizontl temperture dvection is one of mny processes tht cn chnge temperture! For Further Reding Any college-level Clculus textbook will contin extensive informtion regrding the mthemticl definition of limits, prtil derivtives, nd Tylor functions nd series. Sections nd of Mid-Ltitude Atmospheric Dynmics by J. Mrtin provides similr informtion from the perspective of their ppliction to the tmospheric sciences. Finite Difference Approximtions, Pge 10