1) The energy balance at the TOA is: 4 (1 α) = σt (1 0.3) = ( ) 4. (1 α) 4σ = ( S 0 = 255 T 1

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1 EAS488/B8800 Climate & Climate Change Homework 2: Atmospheric Radiation and Climate, surface energy balance, and atmospheric general circulation Posted: 3/12/18; due: 3/26/18 Answer keys 1. (10 points) Use the two-layer model in Fig of the text (p62). Place the two layers at 2.5 km and 5.0 km. Assume a fixed lapse rate of 6.5 K km -1. Since the lapse rate is given, T1, T2 and Ts can be easily determined (hint: T1 is simply the emission temperature and T2 and Ts can be worked out from its vertical distance from layer 1). In this model, convective fluxes (in W m -2 ) are needed to balance the energy equation. Assume two convective fluxes are present: conv2 representing the flux from the surface to the lower layer, and conv1 representing the flux from the lower layer to the upper layer. Solve for 1) T1, T2, and Ts,and 2) conv1 and conv2. (Hint: write down four energy balance equations at, respectively, TOA, layer 1, layer 2 and surface. In addition to radiative energy fluxes, there are also convective fluxes.) 1) The energy balance at the TOA is: S 0 4 (1 α) = σt 4 1 T 1 = ( S 0 (1 α) 4σ (1 0.3) = ( ) 4 = ) T 1 = 255 [K] (2 pts) The fixed lapse rate is 6.5 [K/km], which means the temperature drops 6.5 [K] for every 1 [km] increase in height. Thus T 2 and T S can determine from its vertical distance from layer 1.

2 T 2 = ( ) + T 1 = 271 T S = ( ) + T = 287 T 2 = 271 [K] and T S = 287 [K] (4 pts) 2) In this model, convective fluxes are needed to balance the energy equation. Layer 1: +σt 2 4 2σT 1 4 = 0! =σ(2t 1 4 T 2 4 ) = (2 point) Surf: σt 4 =0! =σ(t T 4 2 T 4 1 +σt 4 2 σt 4 S S ) = (2 point) conv1 = [W/m 2 ] and conv2 = [W/m 2 ] 2. (5 points) Fig of the text (p62) represents the pure radiative equilibrium. Problem 1, on the other hand, is the radiative-convective equilibrium. Which one has larger lapse rate (defined as dt/dz), if we assume the two layers in Fig are also at 2.5 km and 5 km? (Hint: read Section 3.10 of the text, p66-72) The pure Radiative Equilibrium (RE) gives a larger lapse rate than Radiative-Convective Equilibrium (RCE). Radiation heats the surface and cools the atmosphere, while convection transfers heat from the surface to the atmosphere. Therefore, if convection is not allowed for the Earth s atmosphere (RE), surface would be warmer and atmosphere would be colder, resulting in an increase of lapse rate.

3 3. (10 points) If the top 100 m of ocean warms by 5 0 C during a 3 month summer period, what is the average rate of net energy flux into the ocean during this period? Express the energy flux in the unit of W m (25 points) Before analyzing the following problem, we review a few concepts and their calculations: 1) Sensible heat cooling of the surface can be written as SH=cpρCDU(Ts-Ta), where Ts and Ta are, respectively surface and air temperatures, cp=1004 J kg -1 K -1, CD=0.002, and ρ=1.2 kg m -3. 2) Latent heat can be writtenas LE= cpρcdube -1 (Ts-Ta), where Be is Bowen ratio. This is basically Equation 4.33 of the text and can also be found on slide 16 of Lec3b_surface.pdf. 3) Blackbody radiation flux follows the Stefan-Boltzmann relation: σt 4, where σ = W m -2 K -4. A non-blackbody radiates at ε σt 4, where ε is emissivity.

4 Here is the scenario: air with a temperature of 27 0 C (or K) moves across a dry parking lot at a speed of 5 m s -1. The solar insolation of the surface is 600 W m -2 and the albedo of the asphalt surface is Downward longwave radiation is 300 W m -2. The longwave emissivity of the surface is Do the following calculation: 1) (10 points) Calculate the surface temperature in equilibrium or surface energy balance. (Hint: to analytically solve the problem, you will need to linearize the Stefan-Boltzman relationship for the surface upward longwave radiation as: σts 4 σta 4 +4 σta 3 (Ts-Ta), where Ts is surface temperature and Ta is air temperature, which is 27 0 C for this case; but note the downward longwave radiation is fixed as 300 W m -2 ). 2) (5 points) If the asphalt is replace with concrete with an albedo of 0.3 and the same longwave emissivity, what is the new surface temperature? 3) (10 points) If this concrete parking lot is now wet (e.g., after a heavy rain) and the air above it is maintained just at saturation, we will need to include latent cooling of the surface. Ignore any effects of surface water on albedo (still 0.3) and Ta (still 27 0 C). Re-calculate the equilibrium surface temperature. Assume the Bowen ratio Be= 0.18 for the temperature range. 1) Since it s a dry parking lot, LE =0.

5 2) Still dry parking lot; only albedo increases a little. 3) Now, the parking lot is wet, so LE is no longer zero:

6 5. (10 points) Calculate the zonal velocity of an air parcel at the equator, if it has conserved angular momentum while moving to the equator from 20 0 N, where it was initially at rest relative to the surface.