Dynamic Meteorology (lecture 13, 2016)

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1 Dynamic Meteorology (lecture 13, 2016) Topics Chapter 3, lecture notes: High frequency waves (no rota;on) Boussinesq approxima4on Normal mode analysis of the stability of hydrosta4c balance Buoyancy waves in the atmosphere (dispersion-proper4es) Linear set of equations: method of normal modes For simplicity, neglect derivatives with respect to y Substitute solution of the form w = W (z)exp i αx ωt [ ( )] amplitude incompressible wavenumber frequency Period=2π/ω 1

2 Linear set of equations: method of normal modes For simplicity, neglect derivatives with respect to y Substitute solution of the form u =U(z)exp[ i( αx ωt) ] w = W (z)exp[ i( αx ωt) ] θ'= Θ(z)exp[ i( αx ωt) ] Π'= P(z)exp[ i( αx ωt) ] amplitude Substitute solution into these equations Helmholtz equation for z-dependent amplitude This leads to a system of four equations for the four amplitudes. This system can be reduced to one equation, for example for W(z): 2

3 Helmholtz equation for z-dependent amplitude This leads to a system of four equations for the four amplitudes, which can be reduced to one equation, for example for W(z): Brunt-Väisälä frequency: Helmholtz equation for z-dependent amplitude This leads to a system of four equations for the four amplitudes, which can be reduced to one equation, for example for W(z): m 2 determines transmission characteristics of the atmosphere If m 2 >0 the solution is wave-like in the vertical m 2 >0 if ω 2 <N 2! 3

4 Helmholtz equation for z-dependent amplitude This leads to a system of four equations for the four amplitudes, which can be reduced to one equation, for example for W(z): If m 2 >0 the solution is wave-like in the vertical i.e. W exp[ imz], provided ω 2 = N 2 α 2 α 2 + m 2 Helmholtz equation for z-dependent amplitude If m 2 >0 the solution is wave-like in the vertical i.e., provided ω 2 = N 2 α 2 W exp[ imz] α 2 + m 2 Dispersion relation for 2-dimensional buoyancy waves Let us define: m γ In order to distinguish: m : transmission coefficient; γ : vertical wave number 4

5 Interpretation of the dispersion relation for 2D buoyancy waves line of constant phase α and γ are the horizontal and vertical wavenumber, respectively. line of constant phase: line of constant phase x Interpretation of the dispersion relation for 2D buoyancy waves line of constant phase α and γ are the horizontal and vertical wavenumber, respectively. line of constant phase: line of constant phase x Dispersion relation for two-dimensional buoyancy waves 5

6 Dispersion properties of buoyancy waves Frequency does not depend on wavelength, only on the angle, φ For which value of does ω have a maximum value? What happens when φ=0? Why? Movie of buoyancy waves: Interpretation of the dispersion relation for 2D buoyancy waves Box 3.1, page 218 z φ L phase lines L=? phase velocity x 6

7 Interpretation of the dispersion relation for 2D buoyancy waves movement of air parcels wave vector phase lines Schematic structure of buoyancy wave Watch a short movie showing buoyancy waves in the real atmosphere: 7

8 Observations of buoyancy waves in the upper atmosphere The sodium (Na) layer is part of the remnant of elements formed when meteors and meteorites enter the atmosphere and begin to burn up or ablate. These elements remain in the lower part of the ionosphere and form layers, but little is known about these layers. Extra problem: What is the vertical component of the phase speed? What is the horizontal wavelength of these waves if the Brunt-Väisälä frequency is 0.02 s-1? Momentum transport by buoyancy waves z x (u'w') < 0 (see section 1.8, lecture notes) Mountains in Lake district, UK Momentum transport due to waves is downward! 8

9 Section 1.8 Turbulence: momentum transport Acceleration, a: a = du dt = u t + dx u dt x + dy u dt y + dz dt = u t + u2 x + uv y + uw z u u x + v y + w. z ( flux-form ) Registration of the wind velocity at 10 m during the 1990-storm u z u t + u u x + v u y + w u z =0 u = u + u';v = v + v';w = w + v' t+τ u 1 u dt τ t In meteorology : τ =10min Section 1.8 Turbulence continued: Reynolds Stress The time-average acceleration: a = u t + u u x + u v y + u w z + u'u' x + u'v' y + u'w' z. The latter three terms on the r.h.s. of this eq. can be interpreted also as a force or stress, called the Reynold s stress. This stress is due to the divergence of a momentum flux. Reynold s stress= u'w' u'u' x u'v' y u'w' z is the vertical component of this flux of momentum by the fluctuations 9

10 Section 3.5 Observations of momentum transport by waves u'w' 0 u'w' < 0 Both figures from Lilly&Kennedy, J.Atm.Sci. One dyne is the force that accelerates a mass of one gram at the rate of one centimeter per second per second. Expressed in SI-units, the dyne equals 10-5 N. Section 3.5 Observations of momentum transport by waves u'w' 0 u'w' < 0 u'w' >0 z Divergence of Reynolds stress, implying a drag force on the atmosphere. Gravity wave drag One dyne is the force that accelerates a mass of one gram at the rate of one centimeter per second per second. Expressed in SI-units, the dyne equals 10-5 N. 10

11 page 216: PROBLEM 3.1 PROBLEM 3.1. The peculiar dispersion of buoyancy waves. The dipersion relation for buoyancy waves in the plane y=0 (in which case β=0) is Extra: Draw dispersion diagram ω(α) for fixed γ (a) Show that the vertical component of the group velocity of buoyancy waves is opposite to the vertical component of the phase velocity of buoyancy waves. (b) Show that group velocity is directed parallel to the direction of movement of the oscillating air parcels (or lines of constant phase). HINT: first show, with the equation of continuity, that the air parcels oscillate perpendicular to the wave vector. Project 3: hand-in answer individually before end Thursday 12 January 2017 page 223 lecture notes PROBLEM 3.2 PROBLEM 3.2. Buoyancy waves and acoustic waves. In this exercise we repeat the analyis of section 3.3 without making the rather drastic approximation of incompressibility. Assume that the atmosphere is homogeneous in the y-direction, i.e. all derivatives with respect to y are equal to zero (this implies that the motion is twodimensional). Assume also that the motion is adiabatic, frictionless (inviscid) and neglect the rotation of the Earth. (a) Write down the x- and z-components of the equation of motion with u, w, Π and θ as unknown variables and t, x and z as independent variables. Linearise this system of equations around the steady hydrostatic state of rest (i.e. Π 0 / z =-g/θ 0, u 0 =0 and w 0 =0) by assuming that Π=Π 0 (z)+π'(x,z,t) and θ=θ 0 (z)+θ'(x,z,t), u=u'(x,z,t) and w=w'(x,z,t), where Π'<<Π 0 and θ'<<θ 0, and by neglecting products of perturbation quantities. Assume that θ 0 θ m =constant in the pressure gradient term and the buoyancy term (the shallow Boussinesq approximation)...parts b-f: see page

12 Next week and next year Friday, 16 December 2015: No lecture Wednesday, 21 December 2016: Retake of part 1 of the exam: room 079 BBG, 13:15-16:15 Friday, 23 December 2016: No lecture Wednesday, 11 January 2017: Presentations of project 2<<<<<<<<<<<<<<<<<<<<WHO? Problem 3.1 Thursday, 12 January 2017, 16:30: Deadline: hand-in - project 3 New years reception, IMAU, Friday, 13 January 2017: Quasi-geostrophic theory of vertical motion and frontogenesis (chapter 9) Wednesday, 18 January 2017: Presentations of project 2<<<<<<<<<<<<<<<<<<<<WHO? Problems 9.3, 9.4 Friday, 20 January 2017: Baroclinic instability in the two level model Interpretation of baroclinic instability using the omega equation (chapter 9) Wednesday, 25 January 2017: Problems 9.6, 9.7 Friday, 27 January 2017: Tutorial: discussion of project 3, problems 9.3, 9.4, 9.6 and 9.7 and subject matter for Exam-2 Wednesday, 1 February 2017: Exam 2 12