ESI 34 tmospherc Dnmcs I Lesson 1 Vectors nd Vector lculus Reference: Schum s Outlne Seres: Mthemtcl Hndbook of Formuls nd Tbles Suggested Redng: Mrtn Secton 1 OORDINTE SYSTEMS n orthonorml coordnte sstem s one n whch the bss vectors re mutull perpendculr nd re of unt length rtesn coordntes re n orthonorml coordnte sstem where the coordnte lnes re not curved We wll be usng orthonorml coordntes n ths clss (ether rtesn or sphercl coordntes The unt bss vectors for our coordnte sstem re î ĵ nd k whch pont towrd the Est North nd up respectvel VETORS Vectors hve both mgntude (length nd drecton Vectors re denoted b ether wrtng them n boldfce ( or b plcng n rrow over the top ( The mgntude of vector s denoted ether b or Vectors re dded b plcng them hed to tl Vector ddton s ο commuttve: ο ssoctve: ( ( Vectors cn be multpled b sclrs Sclr multplcton s ο ssoctve: m( n n( m mn ο dstrbutve: ( m n m n ; m( m m OMPONENTS vector cn be wrtten n terms of components long the coordnte sstem es
k nd k re the unt vectors (mgntude 1 long the nd es respectvel ο The components of vector m hve negtve vlues For emple veloct 1 1 vector gven b m s V 3m s hs component vectors 1 V m s 1 V 3ms ο We would thnk of V s hvng mgntude of m s 1 nd drecton opposte to î whle V hs mgntude of 3 m s 1 nd n the drecton of ĵ ο nother emple s grvt g whch n component form s g gk We s tht the mgntude of grvt s g nd ts drecton s opposte to k In component form vector ddton s ccomplshed b ddng the components b b b ( ( ( k In component form multplcton b sclr s m m m m k The mgntude of vector s found from ts rtesn components (es re norml to one nother usng the Pthgoren formul DOT PRODUT The dot (or sclr product of two vectors s defned s cosθ where θ s the ngle between the two vectors The result of the dot product s sclr not vector! In component form the dot product s b b b If two vectors re norml ther dot product s ero The dot product s ο commuttve: ο dstrbutve: ( ROSS PRODUT The cross product s defned s snθ u where θ s the ngle between the two vectors nd û s the unt vector perpendculr to both nd n the drecton consstent wth the rght-hnd rule ο Note: In Europen tets the cross product s often denoted usng ^ rther thn
The result of the cross product s vector! If two vectors re prllel ther cross product s ero In orthonorml coordntes the component form the cross product s found b fndng the determnnt of mtr whose frst row s the unt vectors long the es nd the second nd thrd rows re the components of the vectors k b b b b b b ( ( ( k b b b The cross product s not commuttve: The cross product s dstrbutve: ( DERIVTIVES OF VETORS vector functon s vector whose mgntude nd drecton depends upon other sclrs (for emple tme The dervtve of vector functon s wrtten n component form s d d d d d d d k k ds ds ds ds ds ds ds The rules for dfferenttng dot nd cross products re nlogous to the product rule for sclr dfferentton d d d ( ds ds ds d d d ( ds ds ds THE GRDIENT The del opertor n rtesn coordntes s defned s k The del opertor ppled to sclr elds vector tht ponts n the drecton of steepest scent (e vector tht s norml to the contours nd pontng towrd hgher vlues k ο When pplng the del opertor to sclr feld the order of the unt vectors nd the prtl dervtves doesn t mtter Ths s wh ou wll often see k s clled the grdent of Worth repetng: The grdent s vector tht s norml to the contours nd ponts towrd hgher vlues! If the sclr s unform n spce (e hs the sme vlue everwhere then the grdent s ero 3
The del opertor cn lso be ppled to vector (contrr to wht some tetbooks stte the result of whch s second-order tensor When pplng the del opertor to vector t s mportnt to wrte the unt coordnte vectors before ech term rther thn fter t k Ths s becuse the order n whch vectors re drectl multpled s not commuttve (e DIVERGENE The dvergence of vector feld s defned s ο In rtesn coordntes The dvergence s sclr! When meteorologsts spek of dvergence the re referrng to the dvergence of the veloct vector ( V u v w k nd so we usull see dvergence wrtten s u v w V Negtve dvergence s clled convergence ο If ο If V > 0 there s dvergence V < 0 there s convergence The phscl menng of dvergence cn be llustrted s follows If the vector feld s pontng w from pont the dvergence t tht pont s postve If the vector feld s pontng nto pont the dvergence t tht pont s negtve Drecton lone cnnot lws be used to determne dvergence or convergence The vectors m be pontng n the sme drecton nd et hve dvergence or convergence (see llustrtons below 4
5 In mn cses ou cnnot tell ust b lookng whether there s dvergence or convergence For emple the llustrton below shows cse where ou would hve to perform the clcultons to determne the dvergence snce t s not obvous b emnng the feld URL The curl of vector feld s defned s The curl s vector whose components re found b fndng the cross product of the del opertor wth the vector In rtesn coordntes the component form the curl s k k The curl of the veloct vector V s clled vortct THE LPLIN The Lplcn opertor s defned s ο In rtesn coordntes For sclr n rtesn coordntes the Lplcn s For vector n rtesn coordntes the Lplcn s
THE DEL OPERTOR IS LINER ( m n m n ( ( SPHERIL OORDINTES On the Erth t mkes sense to use sphercl coordntes rther thn rtesn coordntes In sphercl coordntes for the Erth the poston of pont s gven b ts dstnce from the center of the Erth r; the lttude φ ; nd the longtude λ ο The unt vectors n sphercl coordntes re the sme s n rtesn coordntes wth î ĵ nd k pontng towrd the Est North nd up respectvel The reltonshps between the coordntes ( nd (r φ λ re d r cosφdλ d rdφ d dr The del opertor n rtesn coordntes s k The del opertor epressed n sphercl coordntes s derved from tht for rtesn coordntes s follows: d λ k d φ k d r λ d φ d r d whch usng the reltonshps n (1 gves 1 1 r cosφ λ r φ k (3 r The choce of whch coordnte sstem to use (rtesn or sphercl s strctl up to us The both hve dvntges nd dsdvntges Sphercl coordntes re ttrctve becuse the Erth s sphercl However the del opertor s more cumbersome n sphercl coordntes nd lso gvng poston n (r φ λ s more cumbersome nd less ntutve thn usng ( Let s fce t rtesn coordntes re smpler nd eser to use nd thnk bout! Plus locll the Erth ppers flt to us Fortuntel we cn stll use sphercl coordntes but epress poston n terms of nd rther thn r φ λ Ths s gong to mke our equtons look ver smlr to rtesn coordntes lthough t wll ntroduce some ddtonl terms clled (1 ( 6
curvture terms nto some of our equtons We wll dscuss these terms s the rse n our studes ο When we use sphercl coordntes but use nd n plce of λ φ nd r we wll cll ths prmetered sphercl coordntes 7
8 EXERIISES Three vectors re gven b k c c c k b b b k 1 Show tht ech of the followng s true: ( ( b ( ( ( c ( ( ( Show tht dt d dt d (remember tht s the mgntude of 3 vector s functon of tme gven b k t t t t 5ln 3 ( 3 Fnd dt d 4 For the followng sclr felds fnd the mgntude of the grdent t the pont ndcted ln ( 3 ; ( (4 b sn cos ( ; ( (0 π 1 c 16 ( ; ( ( - 5 For the followng vector felds fnd the dvergence t the pont ndcted k ln 4 3 ( ; ( (4 b sn cos ( ; ( (0 π 1 c ( ( ( ; ( ( 1
6 The geostrophc wnd s vector feld gven b V u g g v 1 p g where u g fρ 1 p nd v g f ρ Show tht f f nd ρ re constnt then the dvergence of the geostrophc wnd ( V g s ero b In relt f s not constnt but nsted ncreses s ou go north Show tht n ths ( ln f cse the dvergence of the geostrophc wnd s gven b Vg vg c For the followng sobr pttern wll the geostrophc wnd be convergent or dvergent? 7 For the followng vector felds fnd the curl t the pont ndcted ( 3 4 ln k ; ( (4 b ( cos sn ; ( (0 π 1 c ( ( ( ; ( ( 1 8 For the followng sclr felds fnd the Lplcn t the pont ndcted 3 ( ln ; ( (4 b ( cos sn ; ( (0 π 1 c ( 16 ; ( ( - 9
9 Show tht the followng denttes nvolvng the del opertor re true Do ths b wrtng the opertors nd vectors n rtesn coordntes nd showng tht the reltonshps hold ecuse these denttes re wrtten n coordnte-free notton f ou prove them to be true n rtesn coordntes the re true n ll coordntes ( s s s( ( s ( ( ( ( ( ( ( ( ( ( ( ( ( s 0 ( 0 ( ( 10