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PACS numbers: 01.30.Mn, 01.30.Pb, 02.60.Cb

Keywords: 7˜'>íߖ, q›'a$íýa³ð>, ïo`¦o j˵, BÛ¼Bw, @/líH¨8Š

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In[2]:= ω[t ]={−ω0 Sin[θ], 0, ω0 Cos[θ]};

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In[4]= fCoriolis = 2 m Cross[r’[t],ω[t]];

In[5]= fCentrifugal =

m Cross[ω[t],Cross[r[t],ω[t]]]

//Simplify]

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//Normal]

Out[11]={t,-t²ω0 Cos[θ], −^gt₂² }

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Fig. 3. Coriolis force deflects the winds flow to the right in Northern hemisphere and to the left in the Southern hemisphere. No deflect in the equator.

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Table 1. Calculations of the apparent forces over the earth surface in the vector calculation The calculation parameters:

ω0 = 7.292 × 10⁻⁵sec⁻¹,v0x = ±10.0m/s,g = 9.8m/sec², t=5 sec.

Coriolis acceleration(m/sec²) Centrif ugal acceleration(m/sec²) Gravitational(m/sec²) θ^a

X^b Y Z X Y Z X Y Z

π/8 ↓ −9.0772 × 10⁻⁷ −1.3473 × 10⁻³ −3.7599 × 10⁻⁷ 4.5386 × 10⁻⁷ 0.0 1.8799 × 10⁻⁷ 0.0 0.0 -9.8 2π/8 ↑ 5.3173 × 10⁻⁷ 1.0312 × 10⁻³ 5.3173 × 10⁻⁷ −2.6586 × 10⁻⁷ 0.0 −2.6586 × 10⁻⁷ 0.0 0.0 -9.8 3π/8 ↓ −1.5574 × 10⁻⁷ −5.5810 × 10⁻⁴ −3.7599 × 10⁻⁷ 7.7870 × 10⁻⁸ 0.0 1.8799 × 10⁻⁷ 0.0 0.0 -9.8 4π/8 ↑ 3.9870 × 10⁻³⁹ 8.9288 × 10⁻²⁰ 6.5116 × 10⁻²³ −1.9936 × 10⁻³⁹ 0.0 −3.2558 × 10⁻²³ 0.0 0.0 -9.8 5π/8 ↑ 1.5574 × 10⁻⁷ −5.5810 × 10⁻⁴ −3.7599 × 10⁻⁷ −7.7870 × 10⁻⁸ 0.0 1.8799 × 10⁻⁷ 0.0 0.0 -9.8 6π/8 ↓ 5.3173 × 10⁻⁷ 1.0312 × 10⁻³ 5.3173 × 10⁻⁷ 2.6586 × 10⁻⁷ 0.0 −2.6586 × 10⁻⁷ 0.0 0.0 -9.8 7π/8 ↑ 9.0772 × 10⁻⁷ −1.3473 × 10⁻³ −3.7599 × 10⁻⁷ −4.5386 × 10⁻⁷ 0.0 1.8799 × 10⁻⁷ 0.0 0.0 -9.8

a↑wind direction to north, ↓ to south

b± deflection to east(+) or west(-) direction respectively

Fig. 4. Calculations of the Coriolis deflections on the six latitudes on the earth. The winds (a)toward the the South and (b) toward to North. The Coriolis force deflects to the right in the Northern hemisphere and to the left in the Southern hemisphere when viewed along the line of motion.



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Table 2. Calculations of the apparent forces varies from city to city.The calculation parameters: ω0 = 7.292 × 10⁻⁵sec⁻¹, g = 9.8m/sec², t=5 sec. Minus sign stand for the north direction of the wind in a the unit of m/sec at the city.

Coriolis acceleration(m/sec2 ) Centrif ugal acceleration(m/sec2 ) Gravitational(m/sec2 ) City(± v0x)

X a Y Z X Y Z X Y Z

Nome(0.902) −7.8311 × 10−8 −1.1890 × 10−4 −3.7218 × 10−8 3.9155 × 10−8 0.0 1.8605 × 10−8 0.0 0.0 -9.8 London(-0.783) 5.0997 × 10−8 8.9365 × 10−5 4.0568 × 10−8 −2.5498 × 10−8 0.0 −2.0284 × 10−8 0.0 0.0 -9.8 New York(-0.652) 2.9439 × 10−8 6.1958 × 10−5 3.4272 × 10−8 −1.4719 × 10−8 0.0 −1.7136 × 10−8 0.0 0.0 -9.8 Atlanta(-0.555) 1.8234 × 10−8 4.4989 × 10−5 2.7271 × 10−8 −9.1174 × 10−9 0.0 −1.3635 × 10−8 0.0 0.0 -9.8 Panama( 0.157) −4.0859 × 10−10 −3.5818 × 10−6 −2.5797 × 10−9 −2.0429 × 10−10 0.0 1.2898 × 10−9 0.0 0.0 -9.8 Sydney(0.559) −1.8556 × 10−8 4.5547 × 10−5 2.7546 × 10−8 9.2782 × 10−9 0.0 −1.3773 × 10−8 0.0 0.0 -9.8 Daejon(-1.50) 5.6163 × 10−8 1.2980 × 10−4 7.6189 × 10−8 −2.8081 × 10−8 0.0 −3.8094 × 10−8 0.0 0.0 -9.8

a± deflection to east(+) or west(-) direction respectively

Fig. 5. ParametricPlot3D of the deflections by the Apparent forces: (a)None Apparent force (b) with the Apparent gravity (-mg), (c) with the apparent gravity and the centrifugal forces, (d) with the gravitational,centrifugal and Coriolis force. All the cases, the angular velocity ω0 varies according to the list of {0,0.002,0.004,0.006,0.008}. All the cases of flight line are degenerated for the ω0 except of (d)case.

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Fig. 6. ParametricPlot3D of the deflections due to the Coriolis force at the latitudes of θ = ^π₄ and θ = ^3π₄. The values of the angular velocity increasing with the thickness of the flight line of the winds. Both the cases, the angular velocity ω0 varies according to the list of {0,0.002,0.004,0.006,0.008}.

Fig. 7. Circulation of the atmosphere due to the Coriolis force and centrifugal and gravitational force to the earth.

In left column, the calculated wind directions in the specific region of the earth match to the winds in the region over the earth surface. Last character of the abbreviations in the figure shows the wind names: W for westerlies, T trade winds, PE polar easterlies respectively.

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Mathematica Ssimulations for the Analysis to the Coriolis Force in a Non-Inertial Frame of Reference

Hee-Joong Yun^∗

Department of Optical and Electronic Physics, Mokwon University, Daejon 302-729 (Received 29 September 2005, in final form 13 January 2006)

The coriolis force is an important factor which allows calculation of the apparent effects on atmospheric flow when viewed from the rotating earth and is elemental in solving the Lagrange’s equation in a non-inertial frame of reference. To analyze the effects of apparent forces on the frame, we solved Lagrange’s equation numerically with Mathematica and visualized the vector calculations of the effects of the Coriolis force on the flow near the earth’s surface in order to improve the teaching and learning of physics in the senior year and to enhance enhancing the ingenuity of students. This forms a fundamental basis for analyzing the 3 Cell model of air circulation near the earth’s surface.

PACS numbers: 01.30.Mn, 01.30.Pb, 02.60.Cb

Keywords: Apparent forces, M athematica, Coriolis force, Atmosphere flow.

∗E-mail: [email protected]