Dynamic Meteorology (lecture 9, 2014)

Transcription

1 Dynamic Meteorology (lecture 9, 2014) Topics Assessment criteria High frequency waves (no rota5on) Boussinesq approxima/on Normal model analysis of the stability of hydrosta/c balance Buoyancy waves and sound waves in the atmosphere Errors on page 355 (Box 1, sec7on 3) corrected!! This a8ernoon: problem 3.1. Homework (project 4): problem 3.2 (deadline: 12 December 2014)

2 Grade Exams (2x) on 7/11/2014 (week 45) (24%) and week 5 of 2015 (30/1/2015) (26%) (minimum average grade for 2 exams together should be 5/10) Project 1 (Problem 1.11) (written report 1 ) 12%) (26 September; deadline: Friday 14 November 2014) Project 2 (Problem 1.19) (oral* 2 ) (14%) (10 October; (hand in hypothesis on/before 7 November) Project 3 Problems and 1.26 (written report 1 ) (12%) (17 October; deadline: Friday 28 November 2014) Problem 3.2 (12%) (21 November; deadline: 12 December 2014) Projects 1, 2 and 3 can be performed in couples 1 <1000 words 2 15 minutes *in January 2015

3 Chapter 3 Hydrostatic balance ship waves Oscillations around hydrostatic balance (sections , of lecture notes) stationary propagating

4 Equations in terms of potential temperature and Exner-function dθ dt = J Π dv dt = θ Π gk ˆ 2Ω v dπ dt = RΠ v + RJ c v c v θ Eq Eq Eq Three differential equations with three unknowns! Neglect Coriolis effect

5 Any variable: Boussinesq approximation (1903) F(x, y,z,t) = F 0 (z)+ F'(x, y,z,t) F m = 1 H H F 0(z)dz 0 reference state Assume that Π=Π 0 (z)+π'(x,y,z,t) and θ=θ 0 (z)+θ'(x,y,z,t), Where Π'<<Π 0 and θ'<<θ 0, and Π 0 / z =-g/θ 0, i.e. the reference state is in hydrostatic balance

6 Any variable: Boussinesq approximation (1903) F(x, y,z,t) = F 0 (z)+ F'(x, y,z,t) F m = 1 H H F 0(z)dz 0 Assume that Π=Π 0 (z)+π'(x,y,z,t) and θ=θ 0 (z)+θ'(x,y,z,t), Where Π'<<Π 0 and θ'<<θ 0, and Π 0 / z =-g/θ 0, i.e. the reference state is in hydrostatic balance The three components of the { momentum equation become: Do the derivation of this equation du dt = θ 0 dv dt = θ 0 dw dt = θ 0 reference state Π' x Π' y Π' z + θ' θ 0 g

7 Buoyancy dw dt = θ 0 Π' z + θ' θ 0 g Buoyancy term (Archimedes principle) If potential temperature of an air parcel is higher than the ambient potential temperature at the same height, then the air parcel experiences an upward buoyancy force

8 Example Potential temperature profile dw dt = θ 0 Π' z + θ' θ 0 g Fig.3.1 Potential temperature, θ, as a function of height at Hemsby, UK (52.68 N, 1.68 W) on December 6, 1995, 00UTC Surface temperature North Sea: 10 C

9 Example Potential temperature profile dw Π' θ ' = θ 0 + g dt z θ 0-15 C at 850 hpa (1356 m) -42 C at 500 hpa (5140 m) stratosphere Buffalo & Lake Eerie Typical for the Arctic tropopause at about 5 km Γ=7.1 C/km=moist-adiabatic lapse-rate Potential temperature, θ, as a function of height at Buffalo, USA (42.93 N, W) on November 19, 2104, 00UTC troposphere Surface temperature of Lake Eerie: 12 C

10 Boussinesq approximation for shallow flow Ambient potential temperature varies little: θ 0 θ m = constant du dt = θ m dv dt = θ m dw dt = θ m Π' x Π' y Π' z + θ' θ m g Pressure gradient and buoyancy term is now linear!

11 Application of Boussinesq approximation to the pressure equation c v Π' RΠ 0 c v RΠ dπ dt = v + J θπ Applying the Boussinesq approximation: * Π' z << g 1 θ t + u Π' x + v Π' y + w Π' z c v g w + v = J RΠ 0 θ 0 θ 0 Π 0

12 Application of Boussinesq approximation to the pressure equation c v Π' RΠ 0 c v RΠ dπ dt = v + J θπ Applying the Boussinesq approximation: t + u Π' x + v Π' y + w Π' z c v g w + v = J RΠ 0 θ 0 θ 0 Π 0 With scale analysis (pressure gradients are of the order of 1 hpa/100 km and pressure tendencies are of the order of 1 hpa/hr) we can reduce this to* Where is pressure? v g c 0 2 w = J θ 0 Π 0 * Π' z << g 1 θ c v RΠ 0 = θ 0 c 0 2 speed of sound c 0 2 = γrt o

13 continued v g c 0 2 w = J θ 0 Π 0 If w is small and J=0 v = 0 atmosphere is incompressible?! Lamb: paradox

14 Stability of hydrostatic balance Linearization of equations The standard method to analyse the stability of a steady state to small perturbations is to linearise the equations around the steady state, thus deriving a set of linear equations for the evolution of small perturbations. If the steady state is the state of rest, and if perturbations are sufficiently small, the nonlinear advection terms (products of perturbations) can be neglected.

15 Stability of hydrostatic balance Linearization of equations The standard method to analyse the stability of a steady state to small perturbations is to linearise the equations around the steady state, thus deriving a set of linear equations for the evolution of small perturbations. If the steady state is the state of rest*, and if perturbations are sufficiently small, the nonlinear advection terms (products of perturbations) can be neglected. Drop primes: * u 0 = v 0 = w 0 = 0 u' u...

16 u t = θ m w t = θ m Linear set of equations: method of normal modes For simplicity, neglect derivatives with respect to y Π' x u x + w z = 0 θ' t = w θ 0 z Π' z + θ' θ m g Substitute solution of the form * amplitude incompressible u =U(z)exp[ i( αx ωt) ] wavenumber frequency Period=2π/ω Substitute solution* into this equation

17 u t = θ m w t = θ m Linear set of equations: method of normal modes For simplicity, neglect derivatives with respect to y Π' x u x + w z = 0 θ' t = w θ 0 z Π' z + θ' θ m g Substitute solution* into this equation Substitute solution of the form * u =U(z)exp[ i( αx ωt) ] wavenumber frequency amplitude Period=2π/ω incompressible This leads to a system of equations for the Amplitudes. This system can be reduced to one equation: d 2 dz W (z)+ 2 m2 W (z) = 0

18 Helmholtz equation for z-dependent amplitude d 2 dz 2 W (z)+ m2 W (z) = 0 m 2 = α 2 + gγα2 θ m ω 2 Γ dθ 0 dz m 2 determines transmission characteristics of the atmosphere If m 2 >0 the solution is wave-like in the vertical

19 Helmholtz equation for z-dependent amplitude d 2 dz 2 W (z)+ m2 W (z) = 0 m 2 = α 2 + gγα2 θ m ω 2 Γ dθ 0 dz m 2 determines transmission characteristics of the atmosphere If m 2 >0 the solution is wave-like in the vertical i.e. W exp[ iγz] provided ω 2 = N 2 α 2 N gγ θ m α 2 +γ 2 = N 2 sin 2 φ Brunt-Väisälä frequency Eq Draw dispersion diagram ω(α) Movie of buoyancy waves:

20 Interpretation of the dispersion relation for 2D buoyancy waves ω 2 = N 2 α 2 α 2 +γ 2 = N 2 sin 2 φ line of constant phase α and γ are the horizontal and vertical wavenumber, respectively. line of constant phase x

21 Interpretation of the dispersion relation for 2D buoyancy waves ω 2 = N 2 α 2 α 2 +γ 2 = N 2 sin 2 φ line of constant phase α and γ are the horizontal and vertical wavenumber, respectively. line of constant phase: αx +γz ωt = constant dz dx = α γ tanφ = α γ sinφ = line of constant phase α α 2 +γ 2 x

22 Interpretation of the dispersion relation for 2D buoyancy waves z L 2π /α 2π /γ φ L=? phase lines phase velocity x

23 Movie of buoyancy waves: 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 ω 2 = α2 N 2 α 2 +γ 2 (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.

24 Hand in the answer before end of 12 December 2014 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).

25 Project 2 Some of you have not yet formulated a hypothesis Please do so within two weeks! The NCEP website is probably the easiest source of data From this website you can select subsets of data more easily than from other websites The program PANOPLY ( is a nice program that allows you to view the data and convert data easily to normal text-files. Presentations in January

26 Next week The meteorological significance of buoyancy waves and sound waves The quasi-geostrophic approximation Omega equation, Q-vector and frontogenesis