Atmospheric Dynamics 1
Atmospheric Dynamics 1 ESCI 342
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This 8 page Class Notes was uploaded by Cordie Miller on Thursday October 15, 2015. The Class Notes belongs to ESCI 342 at Millersville University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/223520/esci-342-millersville-university-of-pennsylvania in Earth Sciences at Millersville University of Pennsylvania.
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Date Created: 10/15/15
ESCI 342 Atmospheric Dynamics I Lesson 1 Vectors and Vector Calculus Reference Schaum s Outline Series Mathematical Handbook of Formulas and Tables VECTORS Vectors have both a magnitude and a direction Vectors are denoted by either writing them in boldface A or by placing an arrow over the top A The magnitude of a vector is denoted either by A or Vectors are added by placing them head to tail Vector addition is 0 commutative A B B A 0 associative A B C A B C Vectors can be multiplied by scalars Scalar multiplication is 0 associative mnA nmA mA 0 distributive m nA mA nA mABmAmB COMPONENT S A vector can be written in terms of components along the coordinate system axes A ajayjal1 i and I are the unit vectors magnitude 1 along the x y and z axes respectively In component form vector addition is accomplished by adding the components AH ax bxiay byal bl1 In component form multiplication by a scalar is mAmajmayjmal1 The magnitude of a vector is found from its Cartesian components axes are normal to one another using the Pythagorean formula 2 2 2 Aiqaxayal DOT PRODUCT The dot or scalar product of two vectors is defined as A 0 E AB cos 0 where 0is the angle between the two vectors The result of the dot product is a scalar not a vector In component form the dot product is A I E axbx ayby azbz If two vectors are normal their dot product is zero The dot product is o commutative A 0 E E 0 A o distributive 12l ECKEADC CROSS PRODUCT The cross product is defined as AX E AB sin 0 12 where 0is the angle between the two vectors and 12 is the unit vector perpendicular to both A and B in the direction consistent with the right hand rule 0 Note In European texts the cross product is often denoted using a A rather than a The result of the cross product is a vector If two vectors are parallel their cross product is zero In component form the cross product is found by finding the determinant of a matrix whose first row is the unit vectors along the axes and the second and third rows are the components of the vectors 2 3 A AX E ax a b b X y y k az abe byazi axbz bxaz 3 axby l7xay b The cross product is not commutative AX E B x A The cross product is distributive AX E C A E A39x DERIVATIVES OF VECTORS A vector function is a vector whose magnitude and direction depends upon other scalars for example time The derivative of a vector function is written in component form as M dd A dd A do A di d dl 17J7zkaxiayi Z7 dt dt dt dt dt dt dt 0 If the coordinate system is not rotating then the unit vectors i j k do not change direction and their derivatives are zero In this case we can write dA dd A dd A do A 7 l 7 j z dt dt dt dt The rules for differentiating dot and cross products are analogous to the product rule for scalar differentiation insg Af ig dt dt dt 1Ax sxd d x dt dt dt THE GRADIENT a A a A a A O The del operator is defined as V E ail 87 87k x y z 0 The del operator applied to a scalar yields a vector that points in the direction of steepest ascent ie a vector that is normal to the contours and pointing toward higher values 3a A 3a A Va E ii i j 3x 3y 91 dz Va is called the gradient of a Worth repeating The gradient is a vector that is normal to the contours and points toward higher values If the scalar is uniform in space ie has the same value everywhere then the gradient is zero DIVERGENCE 3 3a The divergence of a vector field is defined as V 0 A aux 7y aaz x az The divergence is a scalar When meteorologists speak of divergence they are referring to the divergence of the velocity vector 7 ui v w and so we usually see divergence written as Negative divergence is called convergence 0 If VV gt 0 there is divergence 0 If VV lt 0 there is convergence A physical meaning of divergence can be illustrated as follows If the vector field is pointing away from a point the divergence at that point is positive If the vector field is pointing into a point the divergence at that point is negative Direction alone cannot always be used to determine divergence or convergence The vectors may be pointing in the same direction and yet have divergence or convergence see illustrations below gt gt gt gt In many cases you cannot tell just by looking whether there is divergence or convergence For example the illustration below shows a case where you would have to perform the calculations to determine the divergence since it is not obvious by examining the field CURL The curl of a vector field is defined as Vgtlt The curl is a vector whose components are found by finding the cross product of the del operator with the vector Therefore in component form the cur is f j k VXAB B B Bazi if BaziBax 3 aifBax Bx By Bz By Bz Bx Bz Bx By a ay az The curl of the velocity vector VXV is called vorticity THE LAPLACIAN O The Laplacian operator is defined as V2 E V 0V O For a scalar the Laplacian is Vza E O For a vector the Laplacian is V23 E THE DEL OPERATOR IS LINEAR O VmnVmVn o vAB39VA39VB39 o VXK VXKVX iii 8x2 ayz azz 3x2 Byz azz 3x2 Byz Bzz EXERICISES Three vectors are given by 1 Show that each of the following is true a ZlB39x M3 b Ax x A 5 2 Show that A 0 27A A remember that A is the magnitude of A t t 3 A vector is a function of time given by 30 2t f 3t3j51nt Find t 4 For the following scalar fields find the magnitude of the gradient at the point indicated 21 mm 2x3y2 xz z1ny x y z 4 2 2 b axyz cosxsiny z x y z 0 TE 1 6 Way x2 y2 16 x y 2 2 5 For the following vector fields find the divergence at the point indicated a 3mm 3xny xy2z 3 4x21ny 12 x y z 4 2 2 b 3xyzgtcosxsinyf z3x y zgt0 1 1 c 3xygtx2y2fx2y23xy21gt 6 The geostrophic wind is avector eld given by Vg ugfvg where ug ifigl P y and V g fp 3x a Show that iffand p are constant then the divergence of the geostrophic wind V Ni is zero b In realityfis not constant but instead increases as you go north Show that in this case the divergence of the geostrophic wind is given by V Ni ivg BLf y c For the following isobar pattern will the geostrophic wind be convergent or divergent N L 1008 1004 7 For the following vector elds nd the curl at the point indicated co a Axyz 3xy2iAixyzz i74x2 lny 1 x y z 4 2 2 13 3x7yyz cosxsiny fez x y z 07t 1 c Altxyx1y1i x1y1ltxylt21 For the following scalar elds nd the Laplacian at the point indicated a axyz 2x3312 ixz zlny x y z 4 2 2 b axyyz cosxsinyiz X y z 0 ml c axy x1 yz 16 X y 2 392 9 Show that the following identities involving the del operator are true V39SAVS39ASV39A VXSAVSXASVXA VKx VXK KVX VxA39xB B39vA393vA A39VB39A39VB39 WitiiiiVA39AVB39B39xVxA39A39xVxB39 vVxA390 VxVxA39VVA39 V2A39
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