Global Atmospheric Circulations problems I hope to update and expand upon this set of problems in the future. As of this writing, I have not yet worked out how to scan in the diagrams referred to in some of the problems. The problems listed here were transfered using DOS TEXT output from original WordPerfect 5.1 files; hence, the equations, Greek letters, and other symbols may not reproduce. I shall work on the problem as time permits. Please be patient. For obvious reasons, I am not providing the solutions to these problems. The intention here is to provide instructors who use my textbook with some ideas about the types of problems that they might ask in a course of their own design. Thanks for accessing this file.\wpm\course\probs94.240 ATM 240 Winter 1994 Problem set #1 1. a. Estimate the total number of bytes of weather information collected each day. Use information given in Chapter 2. Assume that each weather datum value is stored as a 16 bit (2 byte) word. Express your answer in MB. Do not include satellite imagery, but do include satellite observations of other variables like T, u, and v. Explain your reasoning in estimating the amount of data for each observing system. b. Estimate the amount of data produced by satellite imagery. Use the following information: GOES: 3 satellites are reliably archived. They have different properties, but assume the following average capabilities. For IR assume 3 channels, having 4km (nadir) to 10km (edge) with average 6 km resolution for an area of the globe that is 50o latitude in radius. For visible: 2km (nadir) to 5 km (edge) with average of 3 km resolution for the same area as the IR, but only 1 channel. These resolutions and areas define the number of pixels in each full-disk image. Each pixel stores 256 shades of gray in one 8-bit byte. Assume that images are kept each hour: 24 IR images, 12 visible images per day. Polar: 2 satellites are active; one crosses 7 am and 7 pm, the other 2 am and 2 pm. Resolution varies, but assume that each scan has 2048 pixels across and that each scan is made every 2 km along the track. They orbit the pole every 90 minutes, so assume that there are 16 complete orbits for each satellite each day. The information in one pixel is stored as one 8-bit byte. c. If one compact disc stores 500 MB, how many discs are needed each day to store all the data in parts a and b? 2. Estimate the error as a percentage of the correct value for geopotential height (è); meridional kinetic energy of the geostrophic wind (þè/þx)2; and meridional geostrophic vorticity (þ2è/þx2). Use geopotential heights from the following scenario. You are trying to measure a 4000 km long wave in geopotential height which in reality has an amplitude of 300 m. However, your instrument is only able to measure to within 5 m accuracy. And, there are numerous gravity waves which have an average wavelength of 50 km and net amplitude of 15 m. The observing site is at Davis (approximately 38o N). We want a maximum error in each case, so assume that the crest of all your waves happen to line up at the location of your measuring device. Calculate the percent error this way: calculate the correct value using the actual amplitude; then the value when the errors are maximized. Take the absolute value of the difference between the former and latter values; divide that by the correct value and multiply by 100. Some Notes on Problem Set #1 Problem 1a: Note that the follow items need to be incorporated and can greatly alter the numeric storage depending upon whether they were noted or not. other variables than those mentioned in table 2.1 (e.g. sky cover, present weather, etc.) many sfc stations report more frequently than every 12 hours ignoring ASDAR forgetting that P must be stored if you keep non-standard levels header information to match station (or location) to the measurement. In my experience, here are some common errors that cannot be ignored: "V" is not realized as being more than one variable (2 horiz. components) commercial aircraft do not fly at 2 or more levels simultaneously too many temps from a satellite (e.g. 10 temps from 4 channels instead of just 4 data values) radiosondes & pibals with too few levels (e.g. they have more than 1) missing "major" observing systems like CTW ATM 240 Problem Set #2 Winter 1994 Figure 1 shows the radiation reaching a given latitude is measured by how much passes through a unit area tangent to the surface. Figure 1 shows the situation for solar radiation at, say, 25 N. Figure 2 shows how the location of the sun might appear to an observer on the surface of the earth as the observer turns to face the direction of the sun during the course of 24 hours. (Direction is indicated by the letters N for north, E for east, etc.) Also indicated are the times of sunrise (t=0) and sunset (t=ç). Assume that the sun's angle above the horizon is given by A, where A = èm sin(xt) where xç = ã and where the length of the daylight hours ç is a function of date and latitude. This formula is valid as long as the day includes a sunrise and sunset. In the calculations that follow we shall only consider that situation. Radiation reaching the top of the atmosphere when the sun is directly overhead (noon when èm = 90o) is equal to the solar constant. When the sun is just at the horizon, the radiation passing through the unit area equals the solar constant multiplied by a factor: f. Note that 1.0 ò f ò 0. From the information above and in Figure 2, a formula for f might be f = sin(A) = sin(èm sin(xt)) This formula is hard to integrate analytically (easily integrated numerically), so we shall use an average daily angle defined as where both integrals are over 24 hours. 1. Derive an equation for Aa as a function of ç and èm that does not have integrals. You may use units of hours for time. 2. Derive a formula for the area under f (Figure 2) over a 24 hour period. Let F designate the area under f. Note that the maximum area can be Fmax=24 units (using hours) and the minimum 0 units. The maximum possible area, Fmax corresponds to a unit area that is perpendicular to the solar radiation all day long. F corresponds to the fraction of the maximum possible solar insolation when you divide that area by the maximum possible area Fmax. Note that solar insolation I can be defined: Where Es is in W/m2 you must convert F from hours to seconds by multiplying by 3600. 3. Derive a formula for èm as a function of Julian day (J) of the year and latitude (L). You may use a formula that includes an "if" test such that when the formula specifies èm < 0 (sun below the horizon) then èm is set to zero. Assume that the subsolar point migrates between 23.5 S (negative value for L) to 23.5 N in a sinusoidal manner. Assume that one solstice happens 8 days before the end of the year and that there are 364 days in the year. Remember that the maximum value of èm = 90 degrees. 4. Estimate the amount of insolation, I that reaches the top of the atmosphere at a day in mid-July when the earth's rotation axis is tilted 20 degrees. Assume that Es = 1380 W/m2 and make your calculation for the 7 latitudes from 60 S to 60 N in multiples of 20 degrees. Express your answer in MJ/m2 each day. Compare your answer with the standard reference, Figure 3. 5. Estimate the cloud amount at each latitude for mid-July as follows. Use Figure 3.25b in your text at the 7 latitudes from 60 S to 60 N at intervals of 20 degrees. Use ISCCP data only. 6. Using information in part 5, estimate the albedo at those 7 latitudes using this approximation: for clouds use 95%, for surface use 25%. Hence a cloud amount of 40% gives an albedo à = 0.4x0.95 + 0.6x0.25 = 0.53. Assuming that mid-July values equate with June-August conditions, how do your numbers compare with Figure 3.7c? 7. Calculate the amount of absorbed radiation per second, on average, over the course of that day in mid-July at those 7 latitudes. Use your albedos from part 6. Note that this problem needs the insolation absorbed over one day divided by the number of seconds in a day at each of the 15 latitudes. How do your numbers compare to Figure 3.8a? \wpm\course\probs94.240 ATM 240 Winter 1994 Problem set #3 1. Figure 3.5 illustrates the first two steps in a simple model for terrestrial radiative balance. Draw a diagram (with numeric labels) that would show the third step. Essentially, make a "Figure 3.5c." 2. Use information attached, given in class, and on page 44 to estimate surface temperature T for the observed radiative balance. Assume the earth is a blackbody and round off your answer to the nearest whole degree. What would the temperature be if the earth had an emissivity that was 0.87 instead of 1.000? 3. Use the following mathematical functions to describe the winter Hadley circulation during December, January, and February. Based on Figure 3.16, assume that all the northward flow is confined between 300-100 mb at latitude 10 N; the southward flow is assumed confined between 800-1000mb at this same latitude. Assume the northward flow is given by this formula: [v] = [vn] sin(ã{300-P}/200) where P is pressure in mb and [vn] is 3.0 m/s. Assume a similar formula for the southward motion which has a maximum at the surface: [v] = [vs] cos(ã{1000-P}/400) The meridional and vertical motion are assumed zero on other pressure levels at 10 N. a. Using mass conservation, determine [vs]. b. Assume the rising motion from this Hadley cell occurs only between 5-0 N. Let the rising motion vary with latitude such that [þ] = [þu] cos(íã/10) at 500 mb, where í is in degrees latitude. From mass conservation, find [þu]. c. Assume that the upward motion is concentrated within thunderstorm updrafts which have two additional features. 1) The upward motion [þT] in the updraft is uniform over the area of the updraft. 2) The updrafts occupy 1/10 of 1 percent of the surface area in the latitude band between 5-0 N. Convert this pressure velocity [þT], into a vertical velocity [wT], by using the Hydrostatic equation. This should give a reasonable average updraft speed. \wpm\course\probs94.240 ATM 240 Winter 1994 Problem set #4 1. Calculate how much the earth's length of day (LOD) is changed in going from July to January. Make the following assumptions: a. Assume that the zonal wind changes by this amount during that time: þ[u] = [U] sin(Pã/Po) sin(2í) where [U] = 20 m/s (est. from fig 3.15) and Po=105 Pa is surface pressure (assumed constant). b. Assume that the density of the solid earth is given by the constant: þg = 2x103 kg/m3. c. Assume that only the mantle and crust of the earth respond to the changes in atmospheric angular momentum. Assume that the mantle starts at radius r=3185 km. Hints: you want to have conservation of angular momentum, so the changes to atmosphere and solid earth must be equal. The solid earth moves in unison, so you need to find the change to the angular velocity first; then approximate that change by a change to LOD. Note that ê = 2ã/LOD. 2. Eddy momentum fluxes. Assume that a streamfunction field þ is defined: þ = sin(x + c y) cos(y) where 0 ó x ó 2ã, and -ã/2 ó y ó ã/2. Also, c is a constant, u = - þþ/þy, and v = þþ/þx. a. Derive the general form of the zonal mean of the eddy momentum flux at y=0. Evaluate all integrals. Your result should only be a function of c. b. Which direction is the eddy momentum flux if c > 0? c. Sketch the þ field when c = ã/2 as follows. Let solid lines show the 0.5, -0.5, and zero contours. Mark the locations and values of maxima and minima. Mark the trough and ridge axes with dashed lines. For ease of comparison, make the size of your rectangular figure close to 6 cm in y and 12 cm in x. \wpm\course\probs90.240 AS 240 Weekly Question Alternate Prob. Set #4 3. Zonal mean variables Check the consistency of the variables with respect to the following "laws". Decide upon which areas satisfy (approximately) and which do not (if that applies). Note that you have to decide several aspects about how you will solve this problem, therefore, you should give some justification for the procedure you choose. For example, you must explicitly define the criteria you use to determine if the given balance is satisfied; the criteria should be quantitative, perhaps based upon a scale analysis. a) U, í and P with geostrophic balance b) T and P with hydrostatic balance (do for global horizontal average values; a reasonably accurate way to get T vs P values is to use the USSA line on a Skew-T chart. Alternatively, you could find the USSA book.) c) adiabatic balance (this is a trick question) d) Next calculate the zonal mean surface temperature as a function of latitude if the atmosphere was in local radiative equilibrium. Compare those values with a latitudinal profile of observed T. \wpm\course\probs94.240 ATM 240 Winter 1994 Problem set #5 This is the first of several problem sets to examine energy and storm growth. These problems examine î near two areas of frequent storm formation. 1. Estimate the efficiency factor î in January at two locations: (40 N, 100 E) and (40 N, 160 E). Follow the indicated steps; the result should be consistent with Fig. 4.18. First, we assume that [î] = 0 at the surface at 40 N (recall Fig. 4.17). Second, assume that the average sea level pressure at that latitude equals the reference pressure (Pr) of the average sea level potential temperature (given as 278 K). Third, use Fig 5.8 to estimate zonal mean sea level pressure along 40 N by using the values at integer multiples of 20 degrees longitude. Fourth, read off the surface T at 160 E and 100 E from Fig. 5.3. Fifth, at 100 E you need to calculate a "sea level temperature," since this locale is about 200mb above sea level; use the lapse rates given next. Sixth, assume that over land the temperature increases with pressure by 6.6 x 10-4 K/Pa, while over ocean assume that the rate is 2.2 x 10-4 K/Pa. Seventh, calculate the surface potential temperature at the two locations. Eighth, using the lapse rate information above, estimate the pressure of the 278 K potential temperature surface at the two locations by adding an increment to the measured surface pressure at that location. Note: the pressure you find can be greater than the actual surface pressure; also the increment (þP) can be negative. Ninth, use the answers to parts labeled "third" and "eighth" to estimate î at the two longitudes: 100 E, and 160 E. 2. Using an approximate technique that differs slightly from problem 1 above, calculate î at two locations south of South Africa: (20 E, 35 S) and (20 E, 45 S). Let the global mean surface pressure be 1013 mb; and global mean surface T be 285. First, from this information calculate global mean surface é. Second, calculate surface é at the two locations using figs. 5.3 and 5.8. Third, use the answers to the last step AND use the static stability estimate from item: "sixth" above over water to calculate the pressure of the global mean surface é at each of the two locations. Fourth, use the answers to the last step to calculate î at both locations. \wpm\course\probs94.240 ATM 240 Winter 1994 Problem set #6 1. Using the closed system model of Fig 4.21 as a guide, estimate the following energy- related quantities. The "box" is 800 mb deep and is a rectangle with sides: 2000 km in the east-west direction and 1000 km in the north-south direction. Assume these things about the eastern half of the domain: the surface low exactly covers only the eastern half of the domain. it is entirely located over water. the volume average virtual temperature is 264 K. the average surface P = 1000 mb on the eastern side; the average top pressure is 200 mb on the eastern side the volume average efficiency factor is î = 0.02 the horizontal average rain rate is 2.16 cm/day. the eddy wind components are given by v'= G(p,t)sin(þ) and u'=-G(p,t)cos(þ) where þ is a compass angle measured from true north. If p is in mb, then G(p,t)=U(t) ({p-600}/400). a. Estimate the total latent heat energy release rate (Q) in the eastern half of the domain. You may use latent heating and density of liquid water at 0oC. Q has units J/s. b. Estimate the volume average heating rate (q) in the eastern half of the domain. q has units K/s. The following formula is helpful: Q (from part 1a) = þþþ Cp q dM = Cp q þþþ dM c. Using q from part b, estimate the volume average T change over 24 hours. d. Assume that half the diabatic energy conversion from the rainfall is simultaneously converted into kinetic energy. Estimate the volume average vertical velocity (þ) above the surface low. Note that using (4.31), we are assuming that: îQ/2 = -îþ/g þþþ à dP dx dy (Your answer should be in Pa/s.) e. Using (4.34) as a guide, set the local derivative of eddy KE in the eastern half equal to the baroclinic conversion (=îQ/2, from part d). Assume that the storm is initially weak, with wind magnitude U(0)=3 m/s. What is the wind magnitude U(ç) after 24 hours? Express your answer in m/s. f. Assume that the rain continues even as the low reaches maturity (fig 4.22). To simplify matters, assume that the information is the same as above with these three exceptions: î =-0.02 (since the low now has a cold core); U(0) =23 m/s; and the vertical structure is barotropic: G(p,t)=U(t). What is the wind magnitude (U) after 24 hours in this case? \wpm\course\probs94.240 ATM 240 Winter 1994 Problem set #7 1. Consider cyclone formation South of Africa. a. From charts of surface sensible heat flux, the average sensible heat flux from ocean into the atmosphere in the area: 30-40 S and 0-30 E, is +10 W/m2. At higher latitudes, between 0-30 E, the flux is -10 W/m2 when averaged over an equal area. Calculate the southernmost latitude "Y" such that the area: 40 S -"Y" and 0-30 E, has the same area as the area: 30-40 S, 0-30 E. Y must be further south than 40 S. b. Calculate the heat transport by the atmosphere through a meridional wall at 40 S that is needed to balance the source and sink of heat given in part a. Hence, you assume that there is no heat transport across 30 S or across "Y"; and no transport in the zonal direction. c. Using (3.3) as a guide, estimate the zonal (sector) and pressure average meridional flux of heat: {vT} across 40 S. Assume that the pressure layer is from 1000 mb to 800 mb and that the zonal sector is still 0-30 E. 2. Imagine you are on the planet þ-Cardasia. The planet has such a strong meridional circulation that the meridional gradient of [é] is negligible and can be assumed to be zero. þ-Cardasia has no axis tilt and no clouds; the solar heating and IR cooling dominate the circulation forcing. Both occur at the planet's surface with turbulent transfer occuring in a bottom "boundary layer." Assume that there are no eddies in þ-Cardasia's atmosphere and that friction is negligible. After choosing appropriate scales of motion, you are able to derive the following non-dimensional form of the Kuo-Eliassem equation to describe the planet's zonal mean meridional cells: A þ2þ/þy2 + C þ2þ/þp2 = þH/þy (1) where A=20/ã2 and C=8.1/ã2. The diabatic heating H is input only into the lowest 0.18 units of the atmosphere. The domain in P is from P=1 to P=0.1 units; where P=1 is the bottom of the atmosphere. The domain in y is 0 to 2, where the equator is at y=0. Thus: H=(120./ã)cos(ãy/2.) for Pò0.82 H=0 for P<0.82 The streamfunction is zero along the boundaries of the domain. From the boundary conditions you note that solutions to (1) can be represented by a double Fourier sine series: a. Verify that the series satisfies the given boundary conditions. b. Draw a schematic diagram of the heating H as a function of y (choose P=1.). c. Find the analytic formula for the Fourier sine coefficients Fn,m. d. Calcuate the numerical value of F1,2. \wpm\course\probs94.240 ATM 240 Winter 1994 Problem set #8 1. Consider the area S. of Africa bounded by 0-30 E and 30 S - 52 S. (Hint: these are "negative" latitudes.) In the area 30-40 S the sensible surface heat flux is 10 W/m2 upwards; while it is -10 W/m2 downwards in the remaining area. The atmosphere responds with a southward heat flux that does not vary with P but is confined to the 1000-800 mb layer. The flux will vary with latitude. There are no other fluxes through the sides or top of this domain. a. Derive the general expression for the atmospheric heat transport across a latitude í where í lies between 30 and 52 S. This flux should have units of W. This flux will only be a function of latitude. You may prefer to write separate formulas for the two cases: í > 40 S and í < 40 S. b. Evaluate term C in eqn (4.31) from the heat flux information derived in part a. Hint: from information above, evaluation of term C should only give one term. That term should also have units of W. Assume that the efficiency factor is î = -0.04 to the south of 40 S and î = 0.04 north of 40 S. To check the magnitude, you may recall that the energy conversion found in an earlier problem set, for a steady rainfall was îQ/2 = 6 x 1012 W. 2. Complex arithmetic in theoretical dynamics. Let P be nondimensional perturbation pressure for an eddy defined as: P = Re{ è(z) exp(ikx) cos(ym) } where 0óxó2, -0.75óyó0.75, k=ã, m=2ã/3 and è(z) = cosh(àz) + i B sinh(àz) where à=2.0, B=1.6 and 0ózó1.0. Note that è is complex but P is not. a. Plot amplitude and phase angle of è at 6 levels: z = 0., .2, .4, .6, .8, 1.0. Note that the phase angle is calculated as þ = tan-1 { -Im(è) / Re(è) }. Hint: negative angle means location upstream from zero angle. b. write down the formulae for é (=þP/þz) and v (=þP/þx). c. derive the formula for the heat flux: vé at y=0, z=0. d. calculate the zonal mean [vé] at y=0. and z=0. Hint: northward flux is >0.