f self = 1/T self (b) With revolution, rotaton period T rot in second and the frequency Ω rot are T yr T yr + T day T rot = T self > f self

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1 Problem : Units : Q-a Mathematically exress the relationshi between the different units of the hysical variables: i) Temerature: ) Fahrenheit and Celsius; 2) Fahrenheit and Kelvin ii) Length: ) foot and meter; 2) mile and meter iii) Seed: meter er second and mile er hour iv) Date: Eastern Standard Time and Universal Time v) Mass: ound and kilogram -a i) Tem: ) C = F 32) 9/5; 2) K = F 32) 5/ ii) Length: ) ft = m ; 2) mi = m iii) Seed: mi/hr = m/s iv) Date: GMT = EST + 5hr v) Mass: lb = kg Problem 2: Earth System : Frequency of earth rotation Q i) Mathematicl exression with a) self-rotation only; b) self-rotation and revolution. ii) Numerical values associated with a) and b), i) a) Self rotation only with T self in second: f self = /T self b) With revolution, rotaton eriod T rot in second and the frequency Ω rot are T rot = T self T yr T yr + T day f rot = /T rot = f self T yr + T day T yr > f self ii) Numerical values a) Hz; b) Hz Problem 3: : Vector Calculus: a) u) = 0 u = i + j + k ) iu + jv + kw) = w )i + w )j + )k u) = w ) + w ) + ) b) u )u = 2 u u) u u) Left hand side LHS): = w = 0 + u )u = u + v + w )ui + vj + wk) = u + v + w )i + u + v + w w + )j + u w + v w + w w )k

2 Right hand side RHS): u u) = u 2 + v + w w )i + u + v + w w )j + u + v + w w )k u u) = iu + jv + kw) w )i + )j + ) )k = v v w + w w ) i + w w w u ) + u j + u u w v w + v ) k + w )i + u + v + w )j u u) u u) = u 2 + v Hence c) fu) = f u + u f LHS +u w + v w + w w )k u )u = u u) u u) 2 fu) = i + j + k ) fui + fvj + fwk) = fu + fv + fw = f + + w ) + u f + v f + w f RHS f u = f + + w ) u f = u f + v f + w f and hence fu) = f u + u Problem 4: Horizontal flow : Q 2D Velocity field is given on a 3x3 grid with dx=20km and dy=20km as in Figure. a) t each grid oint, comute wind seed and sketch consistently scaled) velocity vector b) For i) Divergence D); ii) Vorticity ζ); iii) Deformation σ); iv) Deformation 2 γ). Write down the mathematical definition 2. Derive comutational formula at Grid 22 using dy, dy, and u ij and v ij i=,..3, and j=,,3). 3. Obtain the numerical value according to 2 2

3 c) For the four elements searately,. Give the mathematical formulat of the velocity otential or streamfunction using D, ζ, σ, γ 2. Give the mathematical formula of the corresonding velocity at Grid using D, ζ, σ, γ 3. Give the mathematical formula of the corresonding velocity at Grid using dy, dy, and u ij and v ij. 4. Sketch iso-curves of the velocity otential or streamfunction, along with the roerly scaled) velocity vector at each grid. d) Exlain the relationshi between the velocity comonents obtained for the four roerties and u, v) at Grid. a) Figure.a. b). Mathematically D ζ σ γ 2&3 at Grid 22, use unit is in Hz) c). Velocity otential and streamfunctions are φ D ψ ζ ψ σ ψ γ 2. Velocity at Grid x, y) = dx, dy) u D = D ) dx u 2 dy ζ = ζ ) dy 2 dx 3. Use D ζ σ γ u 23 u 2 u 32 u 2 v 23 v 2 v 32 v 2 D 4 x2 + y 2 ) ζ 4 x2 + y 2 ) σ 2 xy γ 4 x2 y 2 ) u σ = σ 2 dx dy ) u 23 u 2 + v 32 v 2 u 32 u 2 + v 23 v 2 u 23 u 2 v 32 v 2 u 32 u 2 + v 23 v 2 u γ = γ 2 dy dx 4. See Figure.b. d) By emlying the Taylor series exansion around Grid 22 and the straightforward substitution, we find mathematically. u u 22 = u D + u ζ + u σ + u γ. where on the RHS refers to grid. The numerical values are unit is in ms ) ) 7 u u 22 =, 3 ) ) ) ) u D =, u ζ =, u 5 σ =, u 0 γ =. 3 ) 3

4 Problem 5: Dynamic Persective : Q Imagine a stably stratified, steady-state flow in which temerature is conserved. What must be the relationshi between horizontal advection of T and vertical motion? Give a hysical exlanation of this relationshi. Using the relationshi for Lagrangian vs. Eulerian derivatives with horizontal and vertical directions searately If T is conserved dt dt = t + u H H T + w w = dt t u H H T. =0) and flow is steady t =0 ), w = u H H T This means that, horizontal advection of warm cold) air u H H T >0 <0) and rising sinking) vertical motion corresond in the trooshere where < 0 holds. Problem 6: dvection : In Figure 2, assume that black curves are streamfunctions and blue ones are iso-temerature curves with values in the figure). Q a) t the three locations,, B, and C, draw in the ma the vectors of i) Wind direction ii) Temerature gradient b) Identify and in the ma the areas of i) High temerature gradient indicate directions) ii) High wind seed indicate directions) and exlain the reason. c) Identify in the ma the areas of i) Warm advection ii) Cold advection and exlain the reason.. a) In the ma b) Using Lagrangian-Eulerian derivative with dt dt t 0 in 2D: = t + u H H T + w u H H T. Hence the warming t > 0 cooling t < 0) occurs where u H H T > 0 u H H T < 0). Problem 7: Newton s Law : Consider a case in which only force acting on the mass is gravity. Q a) Derive the time for a mass released at height H) to reach the earth surface, assuming that the standard gravity g) is constant during the dro. b) Comute numerical value of the time for H=50m, 2km, & 0km, using g=9.8ms 2. 4

5 a) Newton s law is Using deal gas low for each density with initial condition z0) = H and dz0) = 0, and hence for zt) = 0, we obtain b) 3.9, 20.20, 45,8 s. Problem 8: Virtual Temerature : d 2 zt) 2 = g zt) = H gt 2 /2 t = 2H/g) /2. Q The air has highly variable amount of water mixed into it. lication of the ideal gas law to the air with water vaor thus requires a variable gas constant. lternately, we can emloy the gas constant for dry air in conjunction with an adjusted temerature, called virtual temerature. Derive the virtual temerature as a function of moist air temerature T ), mixing rate w) of water vaor as the mass ratio m v /m d ), and ratio ɛ) of the molecular weight M v /M d ). Density ρ of air is sum of that of vaor ρ v and dry air ρ d, and ressure of air is the sum of that of vaor v and dry air d. Using deal gas low for each density Virtual temeratue is defined by ρ = ρ v + ρ d = v R v T + d R d T. ρ = R d T v = v + d R d T v. By equating the two density equations, we obtain T v = T v R d R v )) > T because v R d R v ) <. In terms of the mass ratio and ratio of the molecular weight, this is Problem 9: Potential Temerature : T v = T w + ɛ ɛw + ) > T Q The otential temerature θ) is the temerature of fluid at ressure ) and temerature T ) that would the fluid would be if it is brought adiabatically to a reference ressure 0 ). a) Derive mathematical formula for the otential temerature θ) as a function of ressure ) and temerature T ), and a reference ressure 0 ). b) Discuss whether the otential temerature θ) is higher than temerature T ) deending on the conditions. For adiabatic rocess dq = C dt αd = 0 5

6 Using the ideal gas law, C dt RT d = 0 d = C dt R T R C ln 0 0 d ln = θ T d ln T ) R/C ) θ = ln T θ T = 0 ) R/C { for 0 < for 0 < Figures : 3 x x 0 4 Figure.: a) Total velocity field; b) Corresonding velocity fields note γ < 0) 6