3. (a) (b) 4. (a) (b) A sample of moist air has a temperature of 5 C, a pressure of 80 kPa, and a relative humidity of 65%. Solve for the following properties, using approximations only when necessary. (i) Saturation vapour pressure (ii) Dew point temperature (iii) Mixing ratio Assume that the Martian atmosphere is completely transparent to shortwave radiation, totally opaque to longwave radiation (i.e., absorbs all long-wave radiation), and can be represented by a single layer of uniform properties, including uniform temperature. (i) Show that the ground temperature on Mars, 3i"g , is related to its atmospheric temperature, Ta , by: T}, = 21/47}, . (A sketch of incoming and outgoing radiation would be heipd). (ii) If the ground temperature of Mars is determined to be 238 K. What is: a. The total energy ux emitted by the Martian atmosphere? b. The estimated solar constant for Mars if its albedo is 0.29? The equation governing motion in the atmosphere in all three dimensions can be written as: dii_ 1!? "+13; 211 _' dt_ ,0 p+g m xv Indicate what force is represented by each term on the right hand side of the equation. When the above equation is decomposed into the x, and y dimensions we get the following equations. (The extra terms on the left are called curvature terms.). d\"""Wtanlll+\"1'1"=16p+22vsinaft-me'costMAir'3r ----- (1) dt a a p at m 2 i 4' F' dv u tan+1n = 1 6p 29nsin+\" ""'(2) dt a a pay m (i) Derive the rst term on the right hand side in the ydimension. (Start by considering an air parcel at the center of a box of sides 5x, 6y, 5:, and pressure pa.) (ii) What is meant by geostrophy? (iii) Using scale analysis show that for midlatitudes the geostrophic approximation is an appropriate one and write the equations for the x and y components of the geostrophic wind. You can ignore friction and assume: U ~ 10 m 5'1 horizontal velocity scale W - 1 cm 5'1 vertical velocity scale L w 106 m length scale a ~ It}4 km radius of earth scale Ap/p ~ 103 Pa horizontal pressure uctuation scale .0. ~ 104rad s '1 angular velocity of earth scale (iv) At a particular midlatitidue location (0 = 60) the pressure increases by 0.5 kPa eastward across a distance of 400 km, and by l kPa northward across a 500 km distance. Determine the magnitude and direction of the geostrophic wind. (Use pair = 1.25 kg m4). (v) Why is geostrophy not a valid approximation in the tropics? MMW be H