Dynamic M athematica Platform for the Coriolis Effects on the Global Atmospheric Circulations of the 3-Cell Model

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Dynamic M athematica Platform for the Coriolis Effects on the Global Atmospheric Circulations of the 3-Cell Model

Bogyeong Kim · Yu Yi

Department of Astronomy and Space Science, Chungnam National University, Daejeon 305-764, Korea

Hee-Joong Yun

^∗

Korean Institute of Science and Technology Information, Daejeon 305-806, Korea (Received 17 January 2014 : revised 12 May 2014 : accepted 16 May 2014)

The Coriolis force is an important factor which requires calculation of fictitious-force effects on atmospheric flow viewed from the rotating Earth. To analyze the fictitious-force effects on the frame, we solve the Lagrange equation in a non-inertial frame of reference numerically with M athematica and make a platform to visualize and compare the vectorial characteristics of the Coriolis effects with the effects of other effective forces on the atmosphere. Our M athematica platform presents a wind deflection due to the effective forces, which is confirmed in the global atmospheric 3-Cell model in 2D or 3D graphics.

PACS numbers: 01.30.Mn, 07.05.Jp, 92.60.Gn

Keywords: Fictitious forces, M athematica, Coriolis force, Atmospheric flow, Non-inertial frame of reference

U

6 M 6 Å U Øò & ÿ 3-Cell { ¢¨ | ù p § X ì Ä Ó Þ X ¢ W

Ä Ò ÞX ì Äß Ã Å ÂP ù v ÚP ÇÊ Ý 8 0 8 cX þ u § » Û

» ¹# Ü · T ¤

Ø

æ z @ / < Æ § ;ë H Ä ºÅ Òõ < Æõ , @ / 305-764

*

× <r )^ ï B

^∗

ô

Ç² D G õ < Æl Õ ü t& ñ Ð ½ ¨" é ¶, @ / 305-806

(2014¸ 1 Z 4 17{ 9 ~ Ã Î6 £ §, 2014¸ 5 Z 4 12{ 9 Ã º& ñ : r ~ Ã Î6 £ §, 2014¸ 5 Z 4 16{ 9 > F S X & ñ )

ïo ` ¦ o (Coriolis)j Ë µ É r r H t ½ ¨\ " f @ /l @ /í H¨ 8 ` ¦ { 9 Ü ¼v H X < © × æ כ ¹ô Ç כ ¹ è Ð 6 x

H Ðl j Ë µs . Ä ºo H @ /l @ /í H¨ 8 1 l x% i < Æ B j m 7 £ §` ¦ 7 ' K $ 3 l 0 AK " f q ' a$ í ý a³ ð> \

"

f Õ ª| ½ ÓÅ Ò ~ ½ Ó& ñ d ` ¦ B Û ¼B jw (Mathematica)r Û ¼% 7 \ " f Ã ºu K $ 3 & h Ü ¼ Ð Û ¦ ¦ ïo ` ¦ o j Ë µ` ¦

í < Êô Ç ¸ H 6 x§ 4 Ü ¼ Ð í H¨ 8 ÷ & H @ /l @ /í H¨ 8 õ & ñ ` ¦ Ð# Å Ò H B Û ¼B jw e ¦Ï ? @; §` ¦ ] j % i .

s

e ¦Ï ? @; § É r @ /l í H¨ 8 3-Cell ¸4 S q\ ´ ú 2 X e _ _ t ½ ¨ ³ ð 0 Au \ " f @ /l s 1 l x õ & ñ ` ¦ 2D Õ ªA i õ

3D Õ ªA i Ü ¼ Ð ½ ¨ & ³ < ÊÜ ¼ Ð+ t ½ ¨ @ /l í H¨ 8 õ & ñ ` ¦ % i 1 l x& h Ü ¼ Ð r Ó ý t Y Us ô Ç .

PACS numbers: 01.30.Mn, 07.05.Jp, 92.60.Gn

Keywords: Ðl jËµ, BÛ¼Bjw, ïo`¦o jËµ, @/l íH¨8,q'a$íýa³ð>

∗E-mail: heejy@reseat.re.kr

610

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

(2)

I. " e Â ] Ø

t

½ ¨ H I ª ` ¦ × æd Ü ¼ Ð / B N " f H ½ ¨ ^

Ð" f t ½ ¨\ " f ' a8 £ ¤ ÷ & H Ó ü t o & ³ © É r q ' a$ í ý a³ ð> _ î r 1

l

x Ü ¼ Ð l Õ ü t ) a [1–4]. t ½ ¨ ç H{ 9 ô Ç U ·s _ @ / ª Ü ¼ Ð W =

) e ¦ I ª Ü ¼ Ð Â Ò' \ -t Ä »{ 9 s { 9 & ñ F G t

~

½ Óõ & h ¸t ~ ½ Ó_ \ P ¨ î + þ A` ¦ Ä »t l 0 A # F G t ~ ½ Ó\

"

f & h ¸ t ~ ½ ÓÜ ¼ Ð @ /l s 1 l x ÷ & ¦ & h ¸\ " f © 5 p x l À Ó

H @ /l Ý ¶ © 8 £ x Â Ò\ " f r F G t ~ ½ ÓÜ ¼ Ð Ã º§ 4 ÷ & " f x 9

¸ 7 £ x # y © " f @ /í H¨ 8 ÷ & H Hadley Cell [5]` ¦ s

À Ò> | ¨ c כ s . Õ ª Q t ½ ¨ H @ / ª õ ¹ ¢ ¤ t _ 4 ¤ ¸ ú ô Ç t

+ þ A½ ¨ ¸ Ð ½ ¨$ í ÷ &# Q e Ü ¼ 9 @ /l 8 £ x _ ¿ ºa ¸ ô Ç& ñ ÷ &# Q e

Ü ¼ " f t % i & h > ] X & h כ ¹ [ þ t s 4 ¤ ½ + Ë& h Ü ¼ Ð 6 x l M

:ë H \ t ½ ¨& h @ /l s 1 l x É r B Ä º Ô ¦ ½ ©g Ë :ô Ç + þ AI Ð { 9

#

Qè ß . Õ ª Q © l ç ß ' a8 £ ¤ ) a l © ' a8 £ ¤ õ H ¨ î ç H& h t

½ ¨ @ /l í H¨ 8 õ & ñ s 3-Cell ¸4 S q\ ¸ ú [ O " î ÷ & ¦ e

. t ½ ¨ ³ ð \ f ¨ Ã º÷ & H \ -t Ô ¦ç H+ þ A M :ë H \ & h ¸t

~

½ Ó\ " f H & h ¸Ã º§ 4 @ / + þ A$ í ÷ & ¦, 0 A ¸ 30

^◦

Â Ò H \ " f

\ P

@ / ¦l · ú Ý ¶, 0 A ¸ 60

^◦

Â Ò H \ " f H ô Ç@ /$ l · ú @ /, F

G t ~ ½ Ó\ " f H F G ¦· ú @ / y ì ø Í½ ¨\ + þ A$ í H d Ü ¼ Ð t ½ ¨

³

ð @ /l H ² D G 7 > h_ @ /l Ý ¶ Ü ¼ Ð ½ ¨ì r ) a . s \ t

½ ¨@ /l H & h ¸t ~ ½ Ó\ " f H Hadley Cell, × æ 0 A ¸ t ~ ½ Ó\

"

f H Ferrel Cell, F G t ~ ½ Ó\ " f H Polar Cell \ í H¨ 8

÷

& H 3-Cell @ /l í H¨ 8 ¸4 S q Ð @ /l @ /í H¨ 8 õ & ñ s [ O " î

÷

& ¦ [5], © l ç ß ' a8 £ ¤ ) a 0 A$ í ' a8 £ ¤ õ ¸ 3-Cell ¸4 S qõ

¸ ú

Â Ò½ + Ë ) a [6]. 3-Cell ¸4 S q\ @ /l _ s 1 l x É r l · ú

â

¸§ 4 õ ¹ 1 Ï§ 4 , Õ ªo ¦ t ½ ¨ r M :ë H \ Ò q tl H Ð l

j Ë µs 4 ¤ ½ + Ë& h Ü ¼ Ð 6 x ÷ & H q ' a$ í ý a³ ð> \ " f Õ ª| ½ Ó Å

Ò î r1 l x ~ ½ Ó& ñ d Ü ¼ Ð l Õ ü t ) a . Õ ª Ql M :ë H \ ïo ` ¦ o

´

òõ \ ¦ [ O " î l 0 Aô Ç ª ô Ç §¹ ¢ ¤& h ¸½ ¨[ þ t õ á Ô ÐÕ ª Ï þ

[ þ t s Ö ¸6 x ÷ & ¦ e [7–9]. s [ þ t × æ Merry-Go-Round _

s ^ ¦ [7], Bath-Tub Vortex [10,11]1 p x É r ïo ` ¦ o ´ òõ

\

¦ [ O " î H Ä »6 x ô Ç ¸½ ¨ Ð Ö ¸6 x ÷ & ¦ e . Õ ª Q s [ þ t s

Õ ª ¼ # ¾ Ós # Q* ô Ç 6 x§ 4 \ _ ô Ç כ t \ ¦ ì rZ > K Å Ò t

H 3 l w ô Ç . s H Ó ü t ^ \ 6 x H 6 x§ 4 _ $ í ì r` ¦ r

o t H 3 l w l M :ë H s . þ j H \ H q ' a$ í ý a³ ð> î

r1 l x` ¦ 7 ' > í ß # r oô Ç r Ó ý t Y Us á Ô ÐÕ ªÏ þ [ þ t s

Ð ¦÷ & ¦ e [12–14]. : £ ¤ y l ñ> í ß õ Õ ªA i ½ ¨ & ³ s

Ã Ì Z 4ô Ç B Û ¼B jw (Mathematica) r Û ¼% 7 [15]\ " f

H 7 ' p ì r ~ ½ Ó& ñ d ` ¦ Ã ºu K $ 3 & h Ü ¼ Ð Û ¦ ¦ Õ ª > í ß õ

\

¦ ( { 9 õ & ñ \ O s ç ß é ß ô Ç ï` ç Ü ¼ Ð Õ ªA i ½ ¨ & ³s 0

p

x l M :ë H \ Ó ü t o < Æ ¸[ þ t \ > Ä »6 x ô Ç ¸½ ¨ Ð s 6 x ÷ & ¦ e

[15–18]. Ä ºo H B Û ¼B jw s ! Q \ " f 3-Cell

¸4 S q` ¦ & h 6 x ô Ç @ /l í H¨ 8 B j& m 7 £ §` ¦ 7 ' > í ß # @ /l í

H¨ 8 ½ ¨ ¸\ ¦ { 9 QÛ ¼à ÔY Us Ü ¼ Ð ] jr ô Ç e % 3 [17].

Fig. 1. (Color online) Real and fictitious forces on the rotating Earth’surface at the latitude of λ = π/2-θ in a topocentric frame of XYZ axes system. The angular velocity of the Earth is ~ ω

0

and θ is the polar angle in the geocentric reference frame [18].

Õ

ª Q e _ _ t & h \ " f @ /l í H¨ 8 õ & ñ ` ¦ f ] X r Ó ý t Y U s

H e ¦Ï ? @; §` ¦ ] jr t H 3 l w % i . s \ Ä ºo H t

½ ¨ © _ # QÖ ¼ t & h \ " f ¸ @ /l í H¨ 8 õ & ñ ` ¦ r Ó ý t Y Us

½

+ É Ã º e H B Û ¼B jw á Ô ÐÕ ªÏ þ Ü ¼ Ð @ /l í H¨ 8 B j& m 7

£

§` ¦ 2D < Ê É r 3D Õ ªA i Ü ¼ Ð K $ 3 ½ + É Ã º e H D h Ðî r B Û

¼B jw e ¦Ï ? @; §` ¦ ] j % i .

II. R å ¾ ËV R Ë Ò ÷ »4 ; c" e § S ê sU ¹ Å Æ

U Ø Ò ÞU ê sX N ËÅ k Ä

'

a$ í ý a³ ð> \ " f H Ó ü t o & ³ © ` ¦ l Õ ü t H 7 ' ~ ½ Ó& ñ d s

ý a³ ð> _ " é ¶& h õ ~ ½ Ó ¾ Ó\ @ / # 1 l qw n & h s # Q ô Ç .

7 £

¤ 1 p x ~ ½ Ó$ í õ ç H| 9 $ í _ r / B N ç ß { 9 9 כ ¹ e . Õ ª Q t

½

¨ H ' a$ í ý a³ ð> ( ¦& ñ ) a ½ Ó$ í )\ @ / # # î î r1 l x õ r

î r1 l x` ¦ H > e Ü ¼ Ð q ' a$ í ý a³ ð> \ " f l Õ ü t ) a . ' a

$ í

ý a³ ð> ( ½ Ó$ í ) K

0

\ @ / # # î î r1 l x ~ V (t)` ¦ H ý a

³

ð> \ ¦ K

⁰

¦ ¢ ¸ s ý a³ ð> K

⁰

\ @ / # { 9 & ñ ô Ç y

5 Å q ¸ ~ω Ð r H ý a³ ð> \ ¦ K ½ + ÉM : K ý a³ ð> \

"

f î r1 l x` ¦ l Õ ü t l 0 AK " f ý a³ ð ¨ 8 s 9 כ ¹ . [ þ v

÷

& H ¨ 8 K

₀

−→ K

^~^{V (t)} ⁰

−→ K

^~^ω

s Ã º' ÷ & " f ý a³ ð ¨ 8 ¸

¿

º 9 כ ¹ . ¦& ñ ý a³ ð> ( ' a$ í > ) K

0

ý a³ ð> \ " f Õ

ª| ½ Ót L

₀

É r [3]

L

₀

= 1

2 mv

₀²

− U (1)

(3)

s

. # l " f ~ v

0

H K

0

' a$ í ý a³ ð> \ " f ' a8 £ ¤ ÷ & H Ó ü t ^ _

5 Å q ¸s ¦ U H 0 Au 7 ' < ÊÃ º ( J $ [ > s . ¨ 8 K

₀ ^{V (t)}

−→ K

^~ ⁰

_ ý a³ ð ¨ 8 É r ~ v

₀

= ~ v

⁰

+ ~ V (t) s ÷ & ¦, ¨ 8 K

⁰

−→ K

^~^ω

_ ý a³ ð ¨ 8 É r ~ v

⁰

= ~ v + ~ ω s . s ] j K> \ " f

Õ ª| ½ Ót L É r

L = 1

2 mv

²

+ m~ v · ~ ω × ~ r + 1

2 m(~ ω × ~ r

²

) − m ~ W · ~ r − U (2) s

) a .

^{d ~}dt^V

= ~ W s . s ] j K ý a³ ð> \ " f þ j è 6 x" é ¶ o

\ ¦ ë ß 7 á ¤ H Õ ª| ½ ÓÅ Ò ~ ½ Ó& ñ d ,

_∂t^∂ ^∂L_∂~_v

=

^∂L_∂~_r

É r

m d~ v

dt = −∇U −m ~ W +m~ r×~˙ ω+2m~ v×~ ω+m~ ω×(~ r×~ ω) (3) s

) a . r y 5 Å q ¸ { 9 & ñ ¦ # î 5 Å q ¸\ ¦ Á ºr

m~r × ~˙ ω ü < ~ W \ ¦ Ò q t| Ä Ì½ + É Ã º e 6 £ § Ü ¼ Ð d (3) É r ² D G

m d~ v

dt = − ~ ∇U + 2m~v × ~ ω + m~ ω × (~ r × ~ ω) (4) s

) a . s \ ¦ m d~ v

dt = ~ f

ef f ective

(5)

m d~ v

dt = ~ f

real

+ ~ f

f ictitious

(6) ü

< ° ú s ¾ » : r ~ ½ Ó& ñ d Ü ¼ Ð j þ t Ã º e . @ /l H ² D G f ~

ef f ective

_ Ä »´ ò 6 x§ 4 Ü ¼ Ð s 1 l x ) a . z ´] j 6 x§ 4 f ~

_real

\ H ¹ 1 Ï§ 4 , l · ú â ¸§ 4 (−

^m_ρ

∇P ) ~ s s \ 5 Å q ¦ f ~

real

É r × æ§ 4 −~ ∇U = −m~g ë ß 2 [/ å L ô Ç . ~ f

f ictitious

H ï o

` ¦ o j Ë µ 2m~v × ~ ω ü < " é ¶ d § 4 m~ ω × (~ r × ~ ω) Ü ¼ Ð" f ¼ #

¾ Ó` ¦ { 9 Ü ¼v H Ðl j Ë µs . ±Y ~ ½ Ó ¾ Ó_ ~v × ~ω 7 ' \ ¦ Fig. 1 \ ³ ðr % i . 7 á § 8 © [ jô Ç q ' a$ í ý a³ ð> î r6 x

É

r §F \ ¸ ú l Õ ü t ÷ &# Q e [1–3].

III. 6 M 6 Å U Øò & ÿ 3-Cell { ¢¨ | ù p § ü X ¢ 8

0 8 cX ³ z º§ q

1. ¥ ÇX c l Ó ÞI í Ä8 ý s ð ' [V R Ë Ä Z Ø

t

© _ e _ _ t & h \ " f 3-Cell ¸4 S q\ É r @ /l _

¼

# ¾ Ó´ òõ \ ¦ K $ 3 l 0 AK " f d (4)\ ¦ B Û ¼B jw r Û ¼

% 7

\ " f / å L Ã ºK Ð 0 Au < ÊÃ º point[t, θ] ` ¦ ½ ¨ % i [13, 14]. 0 A ¸(π/2 − θ) & ñ K t 3-Cell ¸4 S q\ _ # @ / l

_ s 1 l x É r < ÊÃ º v0[θ]\ _ K & ñ ÷ & ¦ @ /l _ ¼ # ¾ Ó

É

r Õ ª t & h \ " f × æ§ 4 õ ïo ` ¦ o j Ë µõ " é ¶ d § 4 \ _

#

^point[t,θ] \ ¦ > í ß ô Ç ( 6 £ § M athematica coding Ã Ð

¸).

Fig. 2. (Color online) The table of the deflections at eight cities on the Earth’s surface. The cities are selected in the different atmospheric region of the world.

In[1]=

Clear[Global‘*"];

In[2]=

IF[θ > 1/2π, ω[t ]:= ω S[t], ω[t ]:=ω N[t]];

{ω N[t ]:={ - ω0 Sin[θ],0,ω0 Cos[θ]};

(N-Hemisphere)

{ω S[t ]:= { - ω0 Sin[θ],0, -ω0 Cos[θ] };

(S-Hemisphere)

In[3]=

fInertial = {0,0,- mg } ;

In[4]=

fCoriolis = 2 m Cross[r’[t],ω[t]];

In[5]=

fCentrifugal =

m Cross[ω[t],Cross[r[t],ω[t]]]

//Simplify]

In[51]=

v0[θ ]:= Which[ θ == 0, 10, 1/8 π ≤ θ < π/6 , 10, 1/6 π ≤ θ < π/3, -10, 1/3 π ≤ θ < π/2, 10, 1/2 π ≤ θ < π2/3, -10, 2/3 π ≤ θ < π5/6, 10, 5/6 π ≤ θ < π17/18, -10, θ == π , -10, ];

In[6]=

eq1 = - r’’[t]- fInertial -fCoriolis- fCentrifugal

In[7]=

eq2 =(Series[eq1/m, {t,0,nOrder }]==0) /.initialRule//Normal//Thread //Simplify

In[8]=

vars = Table[D[{x[t],y[t],z[t]},{t,i}]

,{i,2,nOrder+2}]/.t->0 //Flatten

In[9]=

sol =Solve[eq2,vars]//First

In[10]=

inertialRule = Thread/@r[0]->0, r’[0]->{1,0,0}//Flatten

In[11]=

point[t ,θ ]=Series[r[t],{t,0,nOrder+2}]

/.sol/.initialRule/.v0[θ]->v0x//Normal]

Out[11]=

{t v0x,−t

²

v0xω0 Cos[θ],-

^g₂

t

²

}

d

(4)\ ¦ Û ¦ l 0 Aô Ç B Û ¼B w ï` ç É r

In[1]- In[10]

s 9 s _

Ø ¦§ 4 É r

Out[11]=

s . 7 £ ¤, point[t,θ] = {tv0x,−t

²

v0x ω0Cos[θ],-

^g₂

t

²

} s .

In[22]=

plot[θ ]:=

ParametricPlot[point[t,θ][[θ][[{1,2}]]

/.{ω0 ->1, g ->9.8}//Evaluate,{t,0,510}

,PlotStyle-> {Thickness[0.0336],Hue[0.01]}

,AxesLabel-> {"x,S","Y,E " }

,PlotRange-> {{-50,50},{30,-30},{0,-50}}

,PlotLabel-> {"theta", θ } ,Ticks> False, ImageSize->85, ,DisplayFunction ->Identity ];

Table[plot[θ],{θ,(1/8)π,(8/8)π,(1/8)π}]

Out[22]= : Fig. 2.

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Table 1. The accelerations of the effective force calculated with those vector components at the eight cities. The calculation parameters: ω

0

= 7.292 × 10

⁻⁵

sec

⁻¹

, g = 9.8 m/sec

²

, and the unit of the acceleration is m/sec

²

. Minus sign of the ~ v

_0x

stands for the north direction of wind in unit of m/sec [18].

city (latitude) v ~

_0x

Coriolis force (×10

⁻⁴

) Centrifugal force (×10

⁻²

) Gravity Effective force (×10

⁻³

)

X Y

^a

Z X

^b

Y Z X Y Z X Y Z

^c

Murmansk (68

^◦

58’N) 10 0 -13.612 0 1.1346 0 0.4370 0 0 -9.8 11.346 -1.3612 -9.7956 New York (40

^◦

42’N) -10 0 9.5151 0 1.6744 0 1.9483 0 0 -9.8 16.774 0.9515 -9.7802 Honolulu (21

^◦

18’N) 10 0 -7.1424 0 1.4485 0 2.5788 0 0 -9.8 14.485 -0.7142 -9.7742

Equator (0

^◦

0’N) ±10 0 0 0 0 0 3.3924 0 0 -9.8 0 0 -9.7661

Lima (12

^◦

03’S) -10 0 -3.0445 0 -0.6259 0 3.2446 0 0 -9.8 -6.9259 -0.3044 -9.7675 Santiago (33

^◦

27’S) 10 0 8.0314 0 -1.5594 0 2.3636 0 0 -9.8 -15.594 0.8031 -9.7764 Queen Mary (66

^◦

45’S) -10 0 -13.399 0 -1.2304 0 0.5286 0 0 -9.8 -12.304 -1.3399 -9.7947 Daejeon (36

^◦

19’N) -10 0 8.6379 0 1.6189 0 2.2023 0 0 -9.8 16.189 0.8637 -9.7779

adeflection to East(+) or West(−)

bSouth(+) or North(−)

cno factor

{ 9

é ß ^point[t,θ] & ñ ÷ & r ç ß ¸B j 7 ' # Q Y

Us Ð @ /l _ s 1 l x â Ð\ ¦ ParametricPlot Ð Õ ªw n = Ã º e

. Fig. 2 H point[t,θ] \ ¦ s 6 x # t ½ ¨ © 8> h _

@ /l Ý ¶% i \ " f Ä »´ ò 6 x§ 4 \ _ ô Ç | Ã Ð_ ¼ # ¾ Ó` ¦ Table[plot[θ]] Ð Õ ª 2 ; כ s (

In[22]-Out[22])

. Fig. 2 \ " f Ð

H ü < ° ú s | Ã Ð_ ~ ½ Ó ¾ Ó\ ' a > \ O s , · ¡ ¤ ì ø Í½ ¨\ " f H ¸

É

rA á ¤ Ü ¼ Ð ¼ # ¾ Ó÷ & ¦ z ì ø ÍÂ Ò\ " f H ¢ , a¼ # Ü ¼ Ð ¼ # ¾ Ó H d` ¦ · ú Ã

º e . & h ¸(θ = π/2)\ " f H ¼ # ¾ Ó÷ &t · ú § H . & h ¸\

"

f H ïo ` ¦ o j Ë µ ~ v

₀

[θ] × ~ ω 7 ' 0s ÷ &l M :ë H s (Fig. 1 Ã Ð ¸). Õ ª Q Fig. 2 H ¼ # ¾ Ós # Q " Ðl j Ë µ

\

_ ô Ç כ t ì rZ > K Å Òt H · ú § H . s H ¼ # ¾ Ós õ

& h Ü ¼ Ð Ä »´ ò 6 x§ 4 ~ f

ef f ective

\ _ K s À Ò# Qt l M :ë H s

. s \ ¦ ì rZ > l 0 AK " f H ' a ¹ 1 Ï 9 H 0 Au \ " f 6

x H Ä »´ ò 6 x§ 4 _ כ ¹ èZ > $ í ì r 7 ' \ ¦ S X ½ + É 9 כ ¹

e

.

In[31]=

gomega={-ω0Sin[θ],0,ω0Cos[θ]};

rveector={0,0,rho};

velocity={v0[θ],0,0};

gravity={0,0,-9.8};

corioacc[θ ]=

2 Coross[velocity,gomega]//Thread centriacc[θ ]=

Cross[gomega,Cross[rvector,gomega]]

//Thread effectacc[θ ]=

{corioacc[θ]+centriacc[θ]+gravity}//Thread

Out[31]=

{0,-2ω0 Cos[θ]v0[θ],0}

Out[32]=

{ρ ω0

²

Cos[θ]Sin[θ],0,ρω0

²

Sin[θ]

²

}

Out[33]=

{ρω0

²

Cos[θ] Sin[θ],

-2ω0Cos[θ]v0[θ],-9.8+ρω0

²

Sin[θ]

²

} Ä

ºo H y y _ 0 Au \ " f Ä »´ ò 6 x§ 4 _ Ä »´ ò5 Å q ¸

~a

ef f ecive

= ~ f

ef f ective

/m\ ¦ כ ¹ èZ > Ð y y 7 ' > í ß

%

i . Ý ¶% i Z > Ð t ½ ¨ © 8> h ¸r _ Ä »´ ò5 Å q ¸\ ¦ > í ß

# Table 1\ & ñ o % i . Table 1\ " f Ä »´ ò 5 Å q ¸_

$ í

ì r s 0 6 x§ 4 \ _ K " f H Õ ª $ í ì r ~ ½ Ó ¾ ÓÜ ¼ Ð H ¼ # ¾ Ó s

{ 9 # Q t · ú § H . õ & h ¼ # ¾ Ó É r X, Y, Z ~ ½ Ó ¾ ÓÜ ¼ Ð { 9

# Q t ë ß ïo ` ¦ o ¼ # ¾ Ó É r ±Y ~ ½ Ó ¾ ÓÜ ¼ Ðë ß { 9 # Q ¦,

"

é

¶ d § 4 \ _ ô Ç ¼ # ¾ Ó É r ±X, Z ~ ½ Ó ¾ ÓÜ ¼ Ð, × æ§ 4 É r −Z ~ ½ Ó

¾ ÓÜ ¼ Ðë ß { 9 # Q H כ ` ¦ · ú Ã º e . ¢ ¸ô Ç ¼ # ¾ Ó§ 4 _ ß ¼ l

H ïo ` ¦ o j Ë µ:" é ¶ d § 4 :× æ§ 4 = 10

⁻⁴

: 10

⁻²

: 1 s ÷ &# Q

j Ë µ 2m~v × ~ ω

₀

s @ /l _ s 1 l x (~ v) \ Ã ºf Ü ¼ Ð 6 x #

¼

# ¾ Ó` ¦ { 9 Ü ¼v H Ä »{ 9 ô Ç Ðl j Ë µe ` ¦ S X K ï r . ë ß { 9

§

|

. s Û æ É r t % i \ I Û æ (Typhoon), s 9 þ t : r (Cyclone), ) o H (Hurricane)1 p x É r s 2 £ § Ü ¼ Ð Ô ¦ o 0

>| 9 ÷ r ° ú s @ /l s 1 l x ~ ½ Ó ¾ Ó\ Ã ºf Ü ¼ Ð 6 x H ï o

2. 3-Cell { ¢¨ | ù p § X ì Ä Ó Þ X ¢ 8 0 8 cX þ u § » Û

t

½ ¨ @ /l _ í H¨ 8 õ & ñ É r 4 ¤ ¸ ú ô Ç 6 x§ 4 õ l Ê ê o כ

¹ M :ë H \ { 9 F c& h Ü ¼ Ð K $ 3 ½ + É Ã º \ O Ü ¼ 9 3-Cell ¸ 4

S q\ @ /l @ /í H¨ 8 õ & ñ ` ¦ f ' a& h Ü ¼ Ð s K ¦ K

$ 3 l H ~ 1 t · ú § . Ä ºo H B Û ¼B jw r Û ¼% 7 \ " f Manipulate < ÊÃ º\ ¦ 6 x # 0 A ¸ © e _ _ 0 Au \ " f 3- Cell ¸4 S q\ @ /l _ s 1 l x õ & ñ s % i 1 l x& h Ü ¼ Ð r Ó ý t Y

Us ÷ & H e ¦Ï ? @; §` ¦ ] j % i . á Ô ÐÕ ªÏ þ ` ¦ z ´' r v

e ¦ Ï @; § _ 2D Graphics Û ¼è s Fig. 3(a) è ß .

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Fig. 3. (Color online) Snapshots of the M athematica platform of the general atmospheric circulation to the 3-Cell model. (a) is the 2D Graphics presentation and (b) is that of 3D Graphics. Simulations are running on click I in the popup when you click the ⊕ right of t panel. Anytime you may change the Graphics dimension and Graphics menu by select the panel menu. And you may change the Graphics mode even if the simulation is pause, if so the platform will show the changed simulation.

Fig. 3(b) H s _ 3D Graphics ¸× ¼ . á Ô ÐÕ ªÏ þ s Ã º'

×

æ s ¸ | Ã Ð_ 5 Å q§ 4 (W ind Speed), t ½ ¨_ r y 5 Å q

¸ (Earth

⁰

/ B I Ð ¨ 8 ) a r Ó ý t Y Us s Ã º' ) a . t ½ ¨@ /l _ í

H¨ 8 õ & ñ É r T heta(θ) ° ú כ` ¦ 1/8π ç ß Ü ¼ Ð ½ ¨ì r # t ½ ¨

³

ð ^ \ ¦ 8 > h_ @ /l Ý ¶% i \ " f @ /l _ s 1 l x õ & ñ ` ¦ r Ó

ý

t Y Us ½ + É Ã º e > [ O & ñ % i . Õ ª Q 3-Cell ¸4 S q\

É r @ /l s 1 l x õ & ñ ` ¦ ¸¿ º ' a ¹ 1 Ï½ + É Ã º e . T heta ¼ 1 Ñ

\ O

B j¾ »\ " f θ = 1/8π\ ¦ × þ 1 l x& h Ü ¼ Ð @ /l _ s

1 l x õ & ñ s r Ó ý t Y Us ÷ & " f · ¡ ¤F G¼ # 1 l xÛ æ(North Polar Easterlies, NPE)` ¦ Ð# ï r . J V , \ 3- Cell ¸4 S q\

É

r ¼ # ¾ ÓÛ æ _ s 2 £ § ¸ < Êa ³ ðr ) a . ¢ ¸ θ = 2/8π\ ¦ × þ

z " f¼ # " fÛ æ (South-West Westerlies, SWW)` ¦, θ = 3/8π\ ¦ × þ · ¡ ¤1 l x Á º% i Û æ (North-East Trade Wins, NEP) s r Ó ý t Y Us ) a . ð ø Ít Ð θ = 5/8π\ " f H z

1 l x Á º% i Û æ (South-East Trade Winds, SET), θ = 6/8π \

"

f H · ¡ ¤ " f¼ # " fÛ æ (North-West Westerlies, NWW), θ = 7/8π \ " f H z F G ¼ # 1 l xÛ æ (South Polar Easterlies, NPE)

`

¦ ' a ¹ 1 Ï½ + É Ã º e [17, 18]. 3-Cell ¸4 S q\ É r @ /l

@

/í H¨ 8 õ & ñ ` ¦ ô Ç e ¦Ï ? @; § \ " f / B I Ð r Ó ý t Y Us ½ + É Ã

º e . & h ¸\ " f_ ¼ # ¾ Ó` ¦ 2D Graphics \ " f H

t · ú §t ë ß , 3D Graphics\ " f H | Ã Ð_ 5 Å q ¸ (X~ ½ Ó ¾ Ó)

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o Ð Z~ ½ Ó ¾ Ó_ ¼ # ¾ Ó` ¦ ' a ¹ 1 Ï½ + É Ã º e [18]. ¢ ¸ & h ¸\

"

f H ~ ω

0

° ú כ\ ' a > \ O s Y¼ # ¾ Ós 0e ` ¦ S X ½ + ÉÃ º e .

ïo ` ¦ o j Ë µ 7 £ ¤, 2m~ v × ~ ω

₀

= 0 H d` ¦ Õ ªA i Ü ¼ Ð S X K

ï r . r Ó ý t Y Us s Ã º' × æ s ¸ ¼ 1 Ñ\ O B j¾ »_ B

j' \ ¦ Ë ¨ Ë ¨# Q B j' _ r Ó ý t Y Us s / B I

Ð Ã º' ) a . á Ô ÐÕ ªÏ þ _ Ã º' ¸× ¼\ ¦ 2D Graphics

<

Ê É r 3D Graphics Ð Ð ¨ 8 ½ + É Ã º e . e ¦Ï ? @; §` ¦ 2D Graphics ü < 3D Graphics` ¦ 1 l x r \ î r6 x < ÊÜ ¼ Ð+ @ / l

_ 2 " é ¶ ¼ # ¾ Ó÷ r ë ß m Table 1õ ° ú É r 3 " é ¶ ¼ # ¾ Ó

´

òõ \ ¦ S X ½ + É Ã º e .

s

e ¦Ï ? @; § É r | Ã Ð_ 5 Å q ¸ (~v), t ½ ¨_ r y 5 Å q ¸ ( ~ ω

₀

), 0

Au (~r)_ 7 ' oü < s [ þ t 7 ' _ 7 ' ½ + Ë\ É r

%

i < Æ½ ¨ ¸\ _ ô Ç 0 Au < ÊÃ º ^point[t,θ] \ ¦ 2D < Ê É r 3D Graphics Ð ³ ðr K º ¡ § Ü ¼ Ð" f @ /l _ s 1 l x õ & ñ ` ¦ o \ O

>

r Ó ý t Y Us ô Ç . & ñ _ ô Ç < ÊÃ º\ ¦ Animate < ÊÃ º Ð s 1 l x r

v H ~ ½ Ód õ [12] H l : r& h Ü ¼ Ð ½ ¨ & ³~ ½ Ód s Ø Ô . Ä º o

m 6 x§ 4 ` ¦ ~ Ã Î ¦ ¹ ¡ §f s H @ /l _ 7 ' & h 1 l x` ¦ r

Ó ý t Y Us ô Ç . B Û ¼B jw r Û ¼% 7 \ " f H s á Ô ÐÕ ªÏ þ _

º' ½ + É Ã º e > cdf { 9 [20] ¸ < Êa ] j ÷ &% 3 . s á Ô Ð Õ

ªÏ þ _ ¢ - a ô Ç { 9 ` ¦ s à Ô [21]\ " f î r ~ Ã Î Ã º' ½ + É Ã

º e .

IV. + s Ç Â ] Ø

t

# î

î r1 l x õ r î r1 l x` ¦ " f 5 Å q( r )÷ & ¦ e 6 £ § Ü ¼

Ð q ' a$ í ý a³ ð> î r1 l x s . q ' a$ í ý a³ ð> \ " f Ó ü t ^ _ î

r1 l x` ¦ l Õ ü t l 0 AK " f H ý a³ ð ¨ 8 ` ¦ K ¦ ¨ 8 ) a ý

a³ ð> _ Õ ª| ½ Ót Ü ¼ Ð Â Ò' þ j è 6 x" é ¶ o \ ¦ ë ß 7 á ¤

H Õ ª| ½ ÓÅ Ò ~ ½ Ó& ñ d (d 5) m

^d~_dt^v

= − ~ ∇U +2m~v×~ ω+m~ ω×

(~ r × ~ ω) ` ¦ & h 6 x ô Ç . # l " f ïo ` ¦ o j Ë µ(2m~v × ~ ω) õ " é ¶ d

§ 4 (m~ ω × (~ r × ~ ω) ) É r t ½ ¨ r y 5 Å q ¸ü < @ /l _ s 1 l x 5

Å

q ¸M :ë H \ µ 1 ÏÒ q t÷ & ¦ | Ã Ðs \ O Ü ¼ Ò q tl t · ú § H Ð l

j Ë µs . s [ þ t É r t ½ ¨@ /l í H¨ 8 õ & ñ \ " f B Ä º × æ כ ¹ô Ç כ

¹ è Ð 6 x ô Ç . ¢ ¸ô Ç ïo ` ¦ o j Ë µ É r \ P @ /; ¤Û æ, [ tÛ æ õ )

o H ` ¦ µ 1 ÏÒ q tr v H 6 x כ ¹ è [5]. ¸Z þ t ± ú , r

H t ½ ¨ ¨ 8 â \ " f q ' a$ í ³ ð> î r6 x _ Ó ü t o < Æ& h s K ü <

Õ

$ 3 K H ¸ H ½ ¨ [ þ t \ > B Ä º × æ כ ¹ô Ç õ ] j Ð F

{ 9 õ / B N 0 A$ í , ½ Ó/ B N l î r ½ Ó, Õ ªo ¦ MEMS s Ð Û

¼ ïá Ô î r6 x` ¦ 0 AK & ñ S X ô Ç ïo ` ¦ o ´ òõ > í ß s 9 כ ¹

[9,19]. Ä ºo H B Û ¼B jw r Û ¼% 7 \ " f ^Manipulate < Ê Ã

÷ & H e ¦Ï ? @; §` ¦ ] j % i . e ¦Ï ? @; § É r d (4)_ 7 ' p

ì r ~ ½ Ó& ñ d ` ¦ Û ¦ # Q ½ ¨ô Ç 0 Au < ÊÃ º ^point[t,θ] _ r ç ß ¸ B

j 7 ' # QY Us \ ¦ ParametricPlot Ð Õ ª 9Å ÒÙ ¼ Ð" f @ / l

_ s 1 l x â Ð\ ¦ o \ O > F & ³K ï r . á Ô ÐÕ ªÏ þ É r

| Ã

Ð_ 5 Å q ¸, t ½ ¨_ r y 5 Å q ¸, 0 Au 7 ' _ oü < s [ þ t _

7 ' ½ + Ë\ É r @ /l s 1 l x _ % i < Æ& h ½ ¨ ¸\ ¦ & ñ S X

>

s K ½ + É Ã º e > 7 ' > í ß # Õ ª õ \ ¦ e ¦Ï ? @; § \ " f r

Ó ý t Y Us ô Ç . s e ¦Ï ? @; § s q ' a$ í ý a³ ð> _ % i < Æ& h ½ ¨

¸\ ¦ Ó ü t o < Æ& h Ü ¼ Ð K $ 3 ¦ @ /l @ /í H¨ 8 ½ ¨ ¸\ ¦ s K

¦ & h 6 x 9 H ¸ H õ < Æ ¸ü < ½ ¨ [ þ t \ > Ä »e ô

Ç ¸½ ¨ Ð Ö ¸6 x| ¨ c כ ` ¦ l @ /ô Ç .

P

c p 8 ý ò k >

s

7 Hë H É r p A ½ Ó ¸õ < ÆÂ Ò_ §¹ ¢ ¤ õ < Æl Õ ü t < É ª l F K õ

4 ¤ Ý ¶ l F KØ ¦ \ O ô Ç² D G õ < Æl Õ ü t& ñ Ð ½ ¨" é ¶ s Ã º '

H ReSEAT á Ô ÐÕ ªÏ þ t " é ¶ Ü ¼ Ð Ã º' ÷ &% 3 _ þ v m .

REFERENCES

[1] K. R. Symon, Mechanics (Addison-Wesley, 1971), Chap. 7.

[2] J. B. Marion, Classical dynamics, (Academic Press, 1970), Chap. 11.

[3] L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed.

(Pergamon Press,1976), Sec 39.

[4] D. Hestens, New Foundations for Classical Mechan- ics, 2nd ed. (Dordrecht: Kluber Academic Pub- lisher,1999), Chap. 5.

[5] C. D. Ahrens, Essentials of Meteorology, 3rd ed.

(Brooks/Cole, 2001), Chap. 7.

[6] AMNH-Weather and Climate Events, http://www.

amnh.org/sciencebulletins/climate (accessed Mar 21, 2014).

[7] Merry-Go-Round, http://www.nasa.gov/audience/

forstudents/brainbites/nonflash/bb home coriolis

effect.html(accessed Mar 21, 2014).

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[8] D. H. Mclntyre, Am. J. Phys. 68, 1097 (2000).

[9] C. M. Graney, Phys. Today 64, 8 (2011).

[10] A. H. Shapiro, Nature 196, 1080 (1962).

[11] L. M. Treefethen, R. W. Bilger, P. T. Fink, R. E.

Luxton and R. I. Tannery, Nature 207, 1084 (1965).

[12] Wofram Demonstrations Project: Coriolis Force, http://demonstrations.wolfram.com/MotionOnSur faceOfTheEarthCoriolisForce/ (accessed Mar 21, 2014).

[13] R. L. Zimmerman and F. I. Olness, M athematica for Physics (Addison-Wesley,1995), Chap. 2.

[14] P. T. Tamm, A Physicist’s Guide to M athematica (AcademicPress, 1997), Chap. 4.

[15] http://www.wolfram.com/solutions/highered/ (ac- cessed Mar 21, 2014).

[16] H. J. Yun, Sae Mulli 50, 134 (2005).

[17] H. J. Yun, Sae Mulli 52, 87 (2006).

[18] B. Kim and H.-J. Yun, J. Astron. Space Sci. 31, 99 (2014).

[19] K. Bikonis and J. Demkovicz, TransNav 7, 401 (2013).

[20] Wolfram CDF Player for Interactive Com- putable Document Format(free download), http://

www.wolfram.com/cdf-player/(accessed Mar 21, 2014).

[21] hcoriolis.nb, http://home.mokwon.ac.kr/

^∼

Dynamic M athematica Platform for the Coriolis Effects on the Global Atmospheric Circulations of the 3-Cell Model