Question

1.1

A flow field describing an atmospheric flow (i.e., a weather pattern) is described by the velocity field

V=-Ky2πx2+y2i+Kx2π(x2+y2)j

where K=105 m2/s $Whd6n8Xscs0AAAAASUVORK5CYII=$ and the x $w7Q+QXmadKbgaLgAAAABJRU5ErkJggg==$ and y coordinate directions are parallel to the latitude and longitude lines, respectively.

First, plot the y-component of the velocity when y=0 and -1 km<x<1 km $e+Mf6ktHU+AAAAAASUVORK5CYII=$ , but exclude the region near the origin (-100 m<x<100 m $DygJv4lXwAAAABJRU5ErkJggg==$ ). What is the x-component of the velocity in this same range?

Second, plot the x-component of the velocity when x=0 and -1 km<y<1 km $xK0yBwb+5EUAAAAASUVORK5CYII=$ , but exclude the region near the origin (-100 m<y<100 m). What is the y-component of the velocity in this same range?

Obtain an expression for the magnitude of the velocity vector as a function of position x $w7Q+QXmadKbgaLgAAAABJRU5ErkJggg==$ and y. Can you simplify this result any further by recognizing that r $GWkUkgB9r1Z7ZFk4jAAAAABJRU5ErkJggg==$ ,the distance from the origin, can be expressed as r2=x2+y2 $ALPdvL9k+QvRIAAAAASUVORK5CYII=$ ?

What kind of weather pattern do you think this velocity field represents?

1.2

For the atmospheric flow field above, determine an expression for the acceleration a and evaluate it at the following (x,y) coordinates: (-250 m, 0), (250 m ,0), (0,-250 m), (0, 250 m), and (0,0).

1.3

For the atmospheric flow field above, determine an expression for the dilation rate ∇⋅V and vorticity ∇×V.

Answer 1

V= - ky २तर Genugal fonm %3D whire e direction velocity component direetion velaci component relocity 10° m/s Plating g component veložity i.e, -Ikm<n <Ikm at -1000m < Vclo00m and encluding - 100m<n<100m and k = put 6= lo=constant whire 1o00 -00 100

n- Componant relocity .e, u Plotting -1000m < y < 1000m at -Ikm < y <1km = and n=0 eneluding - ky < 100m -00wm 스 put r=0 and k=/05" 1D cmatent where +4 Kticoo -loo -lo00 Magnitde y velocity vecton Ÿ k*y2 Vu²+y² k? (m²+yz) %3D 시대 Tw마님)- N472 put rininy k2 171 =

Weathor paturn:- Hypo bolic flow putern is neprestntiy the velocity field components. 1. 2 Accelination n- dinecdion in v du Ae = u au Эж =u 21-ky dyl 2x (x*+y2) tky x2x+v| -K(x²+ty²) +2K y 27 (v²+y?) = Ux -ky akyn ke -ag" + 2g² - (-k(a*ty*] = -k?sc = k?u dinection in v dv + vid - 2 kyn 2n (mty) 2a (a*tj")

27 - (n*rg})+2 x* – 2n²] ay 2. (10) (250) 16.22 m, at (-250m, 0) = 16.22 m/see (25om, 0) (10 (-250)4 = 16. 22./se? :(0,-250) (105) 16.22 (0, 250) not defined (o, 0)

-KY KY 27T (4 y2) Vy Expression for dilation rate KY ১৭ (आ (स --2 21T КXУ - kxy Voraticity Vu Vy 3(-34 -3(-2 me 32, 2(5)-2(- Y).

d () 4'v- uv! K( w4y2)-KN(2n) 2 (AY) . रो ग kyley)? 21T 2. ku2 21T K4-kn*+ Ku?. 2TT A7
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