Løsningsforslag: oppgavesett kap. 9 (2 av 3) GEF2200 s.m.blichner@geo.uio.no Oppgave 1 a) The turbulent vertical flux of sensible heat (Q H ) in the atmospheric boundary layer often takes place through turbulent eddies. Which two main mechanisms generate these eddies? (Repitition) Turbulence is generated either thermally or mechanically (see M and B in equation 9.7 in the book). Mechanically generated turbulence M is the turbulence we get when we have a wind shear. This wind shear is often generated when wind blows over rough surfaces and the wind close to the ground is slowed. Thermally generated turbulence is obtained when air becomes lighter than its surroundings and thus becomes buoyant and rises in so called thermals. This typically happens when the sun heats the surface which again heats the surface air which becomes buouyant. b) The turbulent vertical flux of sensible heat (positive upward) is expressed by Q H = ρc p w θ. Explain what w θ describes. The turbulent vertical flux of sensible heat (positive upwards) is expressed by Q H = ρc p w θ. Den tubulente vertikalfluksen av følbar varme (positiv oppover) uttrykkes ved Q H = ρc p w θ. Forklar hva w θ beskriver. w θ is the covariance between w and θ telling us to what degree these two vary together. We use this quantity to determine if the atmosphere is stable (w θ < 0) or unstable(w θ > 0), and if the heat is transported up(w θ > 0) or down (w θ < 0). c) Give a physical explenation of why we can expect w θ to be greater that zero during the day (day time). During day, the surface is heated by the sun and the bottom part of the boundary layer becomes unstable. From the answer to b) we know that this indicates positive values of w θ. See also Figure 9.8 in the book. 1
d) In the horizontal boundary layer we have that dq H dz > 0. What can you say about how temperature changes with time in this layer? From equation 9.10 in the book we know that temperature increases with time in an atmosphere where w θ > 0, z which is the same as Q H z > 0. The temperature would thus increase with time. Another way to understand this is that as the vertical heat flux decreases with height ( Q H < 0), this z means that the flux into a layer will always be larger than the flux out. Thus heat is accumulated in a layer and it warms. Oppgave 2 From observations of the vertical flow, temperature and spesific humidity, we have the timeseries given in table 1. For w and q the perturbation/deviation from the mean is given, while for T the actual temperature is given. Estimate the turbulent vertical fluxes of sensible and latent heat. You will need: ρ = 1, 2 kgm 3, c p = 1004 J/K kg and L v = 2, 5 10 6 Jkg 1 We must first calculate the temperature deviations T and then Raynolds average to find w T and w q. The mean of T is 288.2. We then get the values in 2. The average of T w is 0.0822 while the average of q w is 0.0127. Use the equation for the turbulent vertical heat flux (equation 9.9 in the book) and equivalent for latent heat. This gives: Q SH = ρc p w T = 1, 2 1004 0, 0822 = 99 Q LH = ρl v w q = 1, 2 2, 5 10 3 0, 0127 = 38 Thus we have Q SH = 99 Wm 2, and that Q LH = 38 Wm 2. (1) (2) 2
Table 1: Tidsserie av perturbert vertikalbevegelse, w, perturbert spesifikk fuktighet, q, og temperatur, T. w [m/s] -0,6 0,4 0,8-0,3 0,3-0,5 0,3 0,7-0,5 T [K] 288 288,4 288,6 287,5 287,3 288,6 289 288,4 288 q [g/kg] -0,05 0,04 0,14-0,18 0,15-0,06-0,5-0,09-0,08 Table 2: Tidsserie av T, T w, og q w. T -0,2 0,2 0,4-0,7-0,9 0,4 0,8 0,2-0,2 T w 0,12 0,08 0,32 0,21-0,27-0,2 0,24 0,14 0,1 T w = 0.0822 q w 0,03 0,016 0,112 0,054 0,045 0,03-0,15-0,063 0,04 q w = 0.0127 Oppgave 3 a) Explain what is illustrated by figure 9.9 in the textbook. The figure shows the radiative fluxes out of and into the ground. Positive values indicate a positive downwards flux. F shows the net radiative flux at the ground. b) What does F L show and why is it greatest in the afternoon? F L shows the flux of downwardwelling longwave radiation at the surface. This is longwave radiation wmitted from the atmosphere and thus depends on the temperature of the atmosphere. The air is gradually heated during the day and will thus emit the most radiation in the afternoon when it s been heated over a long time. This is in contrast to downwelling short wave radiation, which follows the sun perfectly. c) F is vary large in the daytime, but this still does not result in a very strong heating of the surface. Why? Because the energy is transported away from the ground by non-radiative processes, namely the latent and sensible heat fluxes (see F Hs and F Es in Figure 9.10. 3
Oppgave 4 Use figure 1 to answer the following questions: a) Which of the figures represent respectively daytime over dry desert, daytime over wet surface and nighttime over wet surface? Explain your answers. In the first figure, we see daytime over wet vegetation. We see that the flux of latent heat is large since the ground is wet and that the flux of sensible heat is also semi-large because of heating during daytime and high surface temperatures compared to the air above. The other figure shows daytime over desert. The flux of sensible heat iis very large because of very high surface temperatures, while the latent heat flux is small because water is not available for evaporation. The third figure shoes nighttime over wet vegetation. The net radiative flux is upwards because of the lack of radiation from the sun. Since the ground is cooling radiatively and it s cooling more efficiently than the atmosphere, the other heatfluxes are downwards: Water vapor condensates on the ground giving a downward latent heat flux and the atmosphere heats the ground leading to a downwards sensible heat flux. b) Imagine a the same figure over the ocean. What would be different? The latent heat flux would have been larger due to the availability of water. F Gs, would be much larger since the turbulence in the ocean mixing layer is an efficient transporter. c) How does the variation during 24-hours of the surface temperature differ from that of the sea surface temperature? Explain why. Again due to mixing in the upper ocean layer and the fact that the heat capacity of the ocean is much larger than the earth surface, the ocean surface maintains its temperature more or less throughout the day, while the surface temperature will change a lot due to incoming solar radiation. d) Under windy condition, warm air is transported over a cold, moist surface. Sketch a figure as in figure 1 showing the different fluxes for this case. What do we call the response of the latent heat flux to these conditions? See Figur 9.11d) in the book. The oasis effect. 4
Figure 1: Flukser av netto stråling F, følbar varme F Hs og latent varme F Es Oppgave 5 a) In which three ways can energy get transfered from one place to another? 1) Conduction: molecules with high temperatures transfer vibration or collide with molecules with lower temperature, thus transferring heat. Mass does not move, but there must be contact. 2) Convection: mass with high temperature moves to a place with lower temperature. Example: ocean currents, winds ect. 3) Radiation: Transport of energy through electromagnetic waves. No need for contact, can happen through vacuum. No mass transfer. b) The flux of sensible heat between the ground and the overlying air happen through a combination of two mechanims of energy transference. Which two and how? In the lowest millimeters over the ground, we have mainly molecular conduction because mixing is almost zero (the ground is solide!). Higher up, the turbulent mixing takes over and here molecular conduction is neglectible. Here convection dominates. c) Explain what the following equation describes and explain the different factors: Q H = ρc p C H V (T s T air ) 5
This is the bulk aerodynamic formulae. It is a parameterization of the transfer of sensible heat between the ground and the air, in units of W m 2 (see section 9.2.3). The sensible heat flux is equal to the the kinematic heat flux [Kms 1 ] multiplied with the density of air ρ [kg/m 3 ], and Dette er den kinematiske fluksen av følbar varme [Kms 1 ] multiplisert med tettheten til luft, ρ [kg/m 3 ], the spesific heat capacity of air at constant pressure c p [J/kg/K]. C H is a dimensionless heat transport coefficient, V is the wind speed at 10,eter above the ground [ms 1, while T s is the surface temperature [K] and T air is the is the temperature of the air 2 meters above the ground [K]. d) Explain how we arrive at the expression above. Why can we use this expression? Why isn t w part of the equation? In the bottom couple of millimeters over the surface, molecular conduction dominates the transport of heat and moisture. Here, turbulence is close to zero. From the top of this layer, turbulent convection takes over and distributes heat and moisture to the rest of the boundary layer. The flux through the molecular layer is luckiely determined by the gradient (of temperature or moisture) and thus it is more or less determined by how quickly heat/moisture is transported away from the top of the layer. This means that it is equal to the turbulent flux. We thus define the effective turbulent flux as the sum through the molecular layer and the layer where turbulence takes over. e) Why doesn t w or the vertical turbulent transport w T figure in the expression above? Because in this approximation w is parameterized by C H V. We thus assume that w T can be approximated by the horizontal wind multiplied by a coefficient which includes effects of stability and surface roughness. f) What is a typical size for C H in c), and how does it vary with V? Between 0,001 and 0,005. When there is little wind, C H becomes large. When the wind weakens, so does C H. g) Write the equivalent expression for the latent heat flux. Q E = ρl v C E V (q(t s ) q a ). Oppgave 6 Study figure 9.16 in the textbook. Where and when do we have an inversion? Why did the inversion occure? What are the consequences of the inversion? 6
We remember from earlier that an inversion is a temperature increase with height. We see that we have an inversion at the top of the boundary layer in daytime and over the stable boundary layer pluss ant the top of the boundary/residual layer in nighttime. During the day, radiation from the sun will heat the ground, warm air will rise and generate turbulence. This leads to a well mixed bottom part of the troposphere (air with high potential temperature from above is mixed with air with low potential temperature from below). As long as the mixing if efficient, we will have more or less constant potential temperature with height. We will thus have a temperature jump at the top of the layer. See figure 9.15 in the book. During night, the surface will be efficiently cooled by emitting radiation. This sets up an inversion layer right over the surface and traps the air here. The inversion over the residual layer is mearly a surviver from the day. 7