Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C5: PHYSICS OF ATMOSPHERES AND OCEANS TRINITY TERM 2016

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1 A11048W1 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C5: PHYSICS OF ATMOSPHERES AND OCEANS TRINITY TERM 2016 Tuesday, 14 June, 2.30 pm 5.45 pm 15 minutes reading time Answer four questions. Start the answer to each question in a fresh book. A list of physical constants and conversion factors accompanies this paper. The numbers in the margin indicate the weight that the Examiners anticipate assigning to each part of the question. Do NOT turn over until told that you may do so. With wavenumber ν measured in cm 1, the radiance from the surface of a black body of temperature T (in K) is given by B ν (T)dν = c 1ν 3 dν e c 2ν/T 1 Wm 2 sr 1, where c 1 = and c 2 = The following equation of motion for a fluid relative to coordinates stationary with respect to a rotating body may be assumed: DV Dt +2Ω V = 1 ρ p gk, where V is the velocity of the fluid, Ω is the angular velocity of the body, p is the pressure, ρ is the fluid density, g is the acceleration due to gravity and k is a unit vertical vector. Unless otherwise stated, x and y refer to the horizontal eastward and northward directions and z the upward direction with corresponding velocities u, v and w. 1000mb = 1000hPa = 10 5 Pa. 1

2 1. The albedo (A) of a cloud can be evaluated from the two-stream approximation as A (1 g)τ cld 2+(1 g)τ cld where g is the asymmetry factor and τ cld is the cloud optical depth. Show that da = A(1 A). dτ cld τ cld [4] Show that the cloud optical depth of a cloud of depth h formed of droplets of radius r and number density N can be approximated as τ cld = 2πNr 2 h, stating all assumptions. Using the definition of the liquid water content L = 4 3 πρnr3, where ρ is the density of liquid water, show that dτ cld dn = τ cld 3N ( 1+2 dlnl dlnn ) dlnh +3 dlnn and, stating any further assumptions, derive the expression for the albedo susceptibility to droplet number perturbations: da dn = A(1 A) 3N. [8] Qualitatively sketch da/dn for low and high droplet number concentrations and comment on the implications. [6] It has been proposed that global warming could be counteracted through artificial cloud brightening. Explain, with reference to the underlying physical concepts, how this could be tackled in practice. Based on the findings above and radiative considerations, what regions of the Earth might be most suitable for cloud brightening? [7] A11048W1 2

3 2. Starting from the first law of thermodynamics, derive the expression defining the potential temperature and explain its significance: Θ T ( p0 p ) κ. [6] Explain the concept of an air parcel in atmospheric physics. Show that the buoyancy of an air parcel can be expressed as B = g Θ p Θ e Θ e, where Θ p, Θ e are the potential temperatures of the air parcel and environment, respectively. Show how buoyancy is incorporated into the calculation of Convective Available Potential Energy (CAPE). [7] Use the expression for buoyancy to derive an expression for the vertical momentum equation of an air parcel: d 2 z dt 2 +N2 0 z = 0, where z is the vertical displacement from its equilibrium altitude. Discuss the range of solutions for this equation and explain its significance for atmospheric stability. [7] An unsaturated air parcel undergoes buoyancy oscillations in association with gravity waves excited by flow over a mountains with ridges at 10 km apart with a wind speed of 20 ms 1 and a temperature of T = 20 C. Show that the potential temperature is related to the environmental lapse rate Γ by 1 Θ Θ z = 1 T (Γ d Γ) where Γ d is the dry adiabatic lapse rate. Hence, or otherwise, determine the environmental lapse rate. [5] A11048W1 3 [Turn over]

4 3. Define what is meant by the phase function P(θ) and the asymmetry parameter g. [4] The intensity I of light of wavelength λ scattered by a particle of radius r λ from a beam of unpolarized light of irradiance E 0 is approximated by I=E 0 (1+cos 2 Θ) 2 ( 2π λ ) 4 ( n 2 ) 2 1 n 2 r 6, +2 where Θ is the scattering angle and n is the refractive index of the particle. Show that the phase function for the particle is P(Θ)= 3 4 (1+cos2 Θ). Sketch a polar plot of the phase function in the scattering plane and give a qualitative explanation for its shape. Evaluate the asymmetry parameter for this phase function. Describe how the phase function changes as the particle radius increases and becomes larger than the wavelength of the light. [10] Now consider the transmission of radiation respectively through a smoke plume whose particles are small compared to the wavelength of light, a water cloud (g = 0.85) and an ice cloud (g = 0.7) at a wavelength where neither the smoke particles, water drops nor ice particles absorb. If the optical depth is very large and is the same for each case, explain which cloud transmits the most light? [3] [8] A11048W1 4

5 4. The basic ozone chemistry in the stratosphere and mesosphere is described by the following reactions: O 2 +hν J 2 O+O (1) O+O 2 +M k 1 O 3 +M (2) O 3 +hν J 3 O 2 +O (3) O 3 +O k 3 O 2 +O 2 (4) where J 2 = s 1, J 3 = s 1, k 1 = m 6 s 1, k 3 = m 3 s 1. In the following it can beassumed that the atmospheric temperature (230 K) and the O 2 mole fraction (0.2) remain constant with altitude. Explain the role of the air molecule M in reaction (2) and, briefly, how these equations predict that the volume mixing ratio (VMR) of atomic oxygen (O) increases with height. [5] Show that there are two time constants associated with these equations, given by τ F = 1 and τ S = 1 ( )1 k1 n M 2 k 1 n 2 n M +J 3 2 J 2 J 3 k 3 where n 2,n M represent the molecular number densities of molecular oxygen (O 2 ) and air respectively, and it is assumed that τ F τ S, Determine the pressure at which the equilibrium concentrations of atomic oxygen and ozone are equal and give a physical interpretation of the asymptotic behaviour of τ F at higher altitudes. [10] Calculate the day-time VMRs of atomic oxygen and ozone at 0.01 hpa and show that the atomic oxygen concentration is approximately 150 times larger than for ozone. Taking these as the sunset values, estimate the subsequent VMR of ozone at sunrise. [10] A11048W1 5 [Turn over]

6 5. Define the term Noise Equivalent Power (NEP) and D as they relate to infrared detector performance. Why is the D generally a more useful figure of merit for a thermal infrared detector than the NEP? [5] A satellite radiometer is designed to measure sea surface temperature using the infrared atmospheric window at 833 cm 1. Themeasurement sequence consists of viewing a cold (273 K) blackbody calibration target, the scene (i.e., nadir-view of the earth), and a hot (300 K) blackbody calibration target, with 100 measurements of 1 s duration taken for each view. The figure below shows the recorded voltages during such a sequence. From this plot, estimate the Noise Equivalent Radiance (NER) of each 1 s measurement. If the area of the detector is 1 mm 2 and the f-number between the final focusing element and the detector is 2 (i.e., f/2) show that the étendue (AΩ) is approximately m 2 sr. Assumingaspectralbandwidthof1cm 1 withameantransmittance0.6, estimate the NEP and D of the detector. [11] Write downan equation that relates thesceneradiance B S to theobserved voltage V S in terms of thecalibration target radiances B H,B C andvoltages V H,V C. Hence show that the error in scene radiance δb S is related to the error in the measured voltages δv by: δb S = aδv S +bδv H (a+b)δv C where the subscripts S, H and C refer to the scene, hot and cold measurements respectively, and derive expressions for a and b. Hence derive the scene brightness temperature and its associated error based on average values for each 100 s view. [9] The Planck function B at 833 cm 1, and the temperature sensitivity dt/db at the scene temperature may be approximated as B 6.88(exp(1200/T) 1) 1 W/(m 2 sr cm 1 ), dt/db 630 (K m 2 sr cm 1 )/W A11048W1 6

7 6. The quasigeostrophic potential vorticity equation for a shallow fluid layer can be written: Q t +u g h Q = 0; Q = 2 h ψ +f ψ L 2 (1) d where the symbols have their usual definitions. Briefly describe the physical content of this equation and its historical significance for the development of numerical weather prediction models. Show that the coriolis parameter, f, close to the poles of a spherical planet of radius a and rotating at angular velocity Ω, can be written approximately as f 2Ω γr2 2, where r is the horizontal distance from the pole and γ = 2Ω/a 2. [7] The polar regions of a planetary atmosphere may be approximated as a cylindrical domain with radius r increasing in the direction of decreasing latitude θ and azimuth angle φ increasing with longitude λ. A streamfunction ψ for the geostrophic flow can be defined by: u r = 1 ψ r φ ; u φ = ψ r. Show that the quasi-geostrophic potential vorticity equation can be written as: Q t +J(ψ,Q) = 0; where J(ψ,Q) = 1 ( ψ Q r r φ ψ ) Q. φ r Considersolutions tothis (nonlinear)equation of theformψ = F(r)Re{expi(ωt nφ)}, where F(r) is a function of r only and n is an integer. If F(r) satisfies the eigenvalue equation 1 r d dr ( r df ) ( n 2 dr r ) L 2 F = α 2 F, d where α is a constant, show that the dispersion relation for such waves is given by ω = nγ α 2. Hence or otherwise, show that general solutions for F are of the form of Bessel functions of the first and second kind, J n (Kr) and Y n (Kr), with radial wavenumber K = α 2 1/L 2 d. [10] A planet exhibits a regular, hexagonal (n = 6) wave surrounding its north pole, propagating zonally without change of form or amplitude. It is centred at a latitude of 76 and with a latitudinal peak-to-peak amplitude of 5. The wave pattern is observed to drift in a retrograde direction with a speed of 4 m s 1 relative to the zonal flow at this latitude. Given the radius of the planet is km and it rotates about its axis in 10 h 33 m, estimate the value of the Rossby deformation radius L d at the latitude of the hexagonal wave, assuming it is a quasigeostrophic Rossby wave governed by Equation (1). [6] [ ] Bessel s equation is of the form y 2d2 W dr 2 +ydw dr +(y2 n 2 )W = 0. [2] A11048W1 7 [Turn over]

8 7. Consider an idealised circular vortex in gradient wind balance at 45 N. The forces (per unit mass) acting on a fluid parcel are given by V 2 /r, fv and (1/ρ) p/ r, where V > 0. Give adefinitionof theforces andeach of thequantities intheexpressions. Draw the force balances for a cyclonic and an anticyclonic vortex. What can you conclude about the pressure distribution around the vortices? [9] For an anticyclonic vortex, solve the system for V. How must the pressure field behave near the centre of the anticyclone? Let V g denote the geostrophic component of the wind. Express the ratio V g /V as a function of the Rossby number. Hence determine whether V g is greater or less than V for both cyclonic and anticyclonic vortices. [10] Sketch the pressure contours for cyclonic and anticyclonic vortices and indicate the geostrophic and ageostrophic winds. How would this sketch differ if vortices of the same wind speed were found at a lower latitude? Tornadoes are a regular hazard in the central US. Are these high- or low-pressure systems, and why? [6] A11048W1 8

9 8. Assume that the North Atlantic steady state ocean circulation is described by the following equations: fv= 1 ρ 0 p x, fu= 1 ρ 0 p y, ρg= p z, where u and v are the zonal and meridional velocity components, f is the Coriolis parameter, ρ 0 the constant reference ocean density, p and ρ are the in situ pressure and density respectively (which depend on x, y and z). Identify and name the balance of forces described by the above equations that maintains the time-mean ocean circulation. [3] Derive expressions for the vertical gradient of the zonal and meridional velocity as function density. Assuming that the vertical gradients are estimated over a depth z between a level of no motion (depth at which u = v = 0) and the surface, sketch how density variations are related to the surface geostrophic currents. Consider an oceanographic ship leaving the east coast of the US and heading at an angle θ with respect to the x (eastward) direction. The ship travels with a speed v s and takes measurements of density ρ along its track. Show that the expression for dρ/dt following the ship is given by dρ dt = fρ 0v s g z (u gsin θ v g cos θ), where u g and v g are eastward and northward surface geostrophic velocity components. [8] A second ship, leaving the same location simultaneously, travels due east (angle θ 0 = 0) at the same speed v s. Find an expression for the best estimate of u g and v g as a function of θ and dρ/dt on each ship. Numerically estimate the geostrophic velocity components u g and v g for the specific case with measured values dρ/dt = kg m 3 s 1 and kg m 3 s 1 from each ship respectively. Assume θ = 5, v s = 5.5 m s 1, f = 10 4 s 1, g = 10 m s 2, ρ 0 = 1028 kg m 3 and z = 1 km. If the ships measure ρ at 60 minute intervals and the standard error of the difference between successive observations is kg m 3, compute the error in the estimates of u g and v g due to the measurement errors. Are these errors correlated? If the first ship were to travel due north (θ = 90 ) rather than at θ = 5, explain how the errors in the estimates of u g and v g will be modified. [7] [7] Recall that the best estimate ˆx is given by ˆx = ( K T K ) 1 K T y, where y are the ob- servations and the K is the matrix representing the dynamics. The error covariance associated with the estimate ˆx is given by Sˆx = σn( 2 K T K ) 1, where σn is the standard error of the measurements. A11048W1 9 [LAST PAGE]