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# Atmospheric Waves

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1. Atmospheric Waves AgatheUntch e-mail: Agathe.Untch@ecmwf.int (office 011) Based on lectures by M. Miller and J.S.A. Green Atmospheric Waves

2. Introduction Why study atmospheric waves in a course on numerical modelling? Successful numerical modelling requires a good understanding of the system under investigation and its solutions. Waves are important solutions of the atmospheric system. Waves can be numerically demanding! (E.g. acoustic waves have high frequencies and large phase speeds => short time step in numerical integrations with explicit time-stepping schemes.) Atmospheric Waves

3. Introduction (2) If a wave type is not of interest and a numerical nuisance then filter these waves out, i.e. modify the governing equations such that this wave type is suppressed. How? To know how, we need to have a good understanding of the wave solutions and of which terms in the equations are responsible for the generation of the individual wave types. Question: How do the modifications to the governing equations to eliminate unwanted wave types affect the wave types we want to retain and study? Also: How do other commonly made approximations (e.g. hydro- static approximation) affect the wave solutions? Atmospheric Waves

4. Introduction (3) How are we going to address these questions? Ideally, we have to study analytically the wave solutions of the exact set of governing equations for the atmosphere first. Then we introduce approximations and study their effect on the wave solutions by comparing the new solutions with the exact solutions. Problem with this approach: Governing equations of the atmosphere are non-linear (e.g. advection terms) and cannot be solved analytically in general! We linearize the equations and study here the linear wave solutions analytically. Atmospheric Waves

5. Introduction (4) Question Are these linear wave solutions representative of the non-linear solutions? Answer Yes, to some degree. Non-linearity can considerably modify the linear solutions but does not introduce new wave types! Therefore, the origin of the different wave types can be identified in the linearized system and useful methods of filtering individual wave types can be determined and then adapted for the non-linear system. Atmospheric Waves

6. Method: Find analytically the wave solutions of the linearized basic equations of the atmosphere, first without approximations. Introduce approximations later and compare new solutions with the exact solutions. Objectives of this course • Discuss the different wave types which can be present in the atmosphere and the origin of these wave types. • Derive filtering approximations to filter out or isolate specific wave types. • Examine the effect of these filtering approximations and other commonly made approximations on the different wave types present in the atmosphere. Atmospheric Waves

7. Wave numbers Wave lengths Wave vector Frequency PeriodT Definition of basic wave properties (1) Mathematical expression for a 2-dimensional wave (periodic function in space and time) AmplitudeA (i=imaginary unit) Dispersion relation Atmospheric Waves

8. Phase: where is perpendicular to the wave fronts. Lz => Lx Phase velocity = velocity of wave fronts. Definition of basic wave properties (2) Wave fronts or phase lines = lines of constant phase (that is, all ) z x Atmospheric Waves

9. Vertical phase velocity !!! !!! Group velocity Definition of basic wave properties (3) Horizontal phase velocity Dispersive waves are waves with a phase velocity that depends on the wave number. Wave packetis a superposition of individual waves. Energy is transmitted with the group velocity. Waves travel with the phase velocity. Atmospheric Waves

10. (1) (2) Momentum equations: (3) (4) Continuity equation: (5) Thermodynamic equation: Equation of state: Complemented by Basic Equations We use height (z) as vertical coordinate. Atmospheric Waves

11. Remarks: (1) All source/sink terms are omitted in eqs (1)-(5) (2) Total time derivative is defined as , where R is the ideal gas constant and cp the heat capacity at constant pressure. (3) All other notations are standard. gives familiar set of equations in hydrostatic approximation (4) Setting We don’t make the hydrostatic approximation at present! It will be discussed later in detail. Atmospheric Waves

12. We would like to find analytically the wave solutions for the basic equations (1)-(5). Can we do this? No! Basic equations are non-linear partial differential equations! We have to linearize the basic equations (1)-(5) by using the perturbation method and solve the linearized system analytically. Atmospheric Waves

13. Change of variable: Replace T by where (potential temperature) Simplification: Neglect variation in y Introduce first some simplifications: Coordinates are now only (x, z, t)! Question: Has this simplification serious consequences for the wave solutions? Answer: Yes! The Rossby wave solution has been suppressed!! Rossby waves can only form if the Coriolis parameter f changes with latitude. (Detailed discussion of Rossby waves will follow later in this course.) Dependent variables (unknowns) are now u, v, w, ρ, p, Θ and they are functions only of x, zandt. Atmospheric Waves

14. Perturbation Method All field variables are divided into 2 parts: 1) a basic state part 2) a perturbation part (= local deviation from the basic state) + Non-linear equations are reduced to linear differential equations in the perturbation variables in which the basic state variables are specified coefficients. => Now we linearizethe set of equations (1)-(5) by using the perturbation method. Basic assumptions of perturbation theory are: a.) The basic state variables must themselves satisfy the governing equations. b.) Perturbations must be small enough to neglect all products of perturbations. Atmospheric Waves

15. define the basic atmospheric state and satisfy Apply perturbation method to basic equations (1)-(5) Consider small perturbations on an initially motionless atmosphere, i.e. basic state winds (u0,v0,w0)=0 Inserting into (1)-(5) and neglecting products of perturbations gives the linearizedbasic equations. Atmospheric Waves

16. No advection terms left! Here: static stability density scale height Linearized basic equations Perturbations (δu, δv, etc.) are now the dependent variables! For this set of equations it is now possible to find the wave solutions analytically. Atmospheric Waves

17. Introduction of tracer parameters This is a technique to help us save work and make consistent approximations. Introduce tracer parameters n1, n2, n3 and n4 to “mark” individual terms in the equations who’s effect on the wave solutions we want to investigate. These tracers have the value 1, but may individually be set to 0 to eliminate the corresponding term. For example n4 = 0 => hydrostatic approx. to pressure field. Atmospheric Waves

18. Boundary conditions: For simplicity we assume the atmosphere to be unbounded in x and z. Wave solutions: Since the coefficients f, B, g & H0 of the system (17)-(21) are independent of x & t, the solutions can be written in the from F(z) exp{i(kx + σt)} . Each dependent variable (perturbation) is of this form: Find wave solutions for system of linearized equations (17)-(21): Remarks: a.) The full solution is the appropriate Fourier sum of terms of this form over all wave numbers k. We study here only individual waves. b.) If the frequency is complex we have amplifying or decaying waves in time. We study only “neutral” waves, so σis assumed to be real. Atmospheric Waves

19. Inserting etc. into eqs (17)-(21) gives the following set of ordinary differential equations in z (derivatives only in z!): Dependent variables are now No x and t dependencies left! Operators ∂/∂x and ∂/∂t have been replaced by ik and iσ, respectively. Atmospheric Waves

20. Derive from this set of equations one differential equation in only one of the dependent variables: . Step 1: Find solution of this equation, i.e. and the dispersion relation σ(k, m, parameters of the system). Insert this solution for back into (22)-(26) to obtain solutions for the remaining dependent variables. Solve system of equations (22)-(26) for Strategy: Step 2: Step 3: Atmospheric Waves

21. Deriving from (22)-(26) a differential equation only in 1 From (22) and (23) we obtain Inserting û from (27) into (25), using (26) and the relation , where is the Laplacian speed of sound, transforms (25) into Using (26) to eliminate from (24) gives Atmospheric Waves

22. Deriving from (22)-(26) a differential equation only in 2 For simplicity we consider only constant (mean) values of B, H0 and c which are related by . Computing from (29) and inserting into (31) leads to the following second order ordinary differential equation governing the height variation of (32) In general the coefficients B, H0 and c are (known) functions of z. Finished step 1 !!!! Atmospheric Waves

23. plus the gas law , the hydrostatic equation Exercise: Verify that the following relation between the constants of the system holds exactly in a hydrostatic pressure field. You need the following definitions and to know that the gas constant R is related to the specific heat constants as Atmospheric Waves

24. Solution 1: Inserting σ = 0 into (22)-(26) gives for the winds: , i.e. geostrophic motion. Solution 2: For we have to solve Further solutions: (32a) Solutions of equation (32) (32) Not a wave! Lamb wave This solution will be discussed later in detail. Atmospheric Waves

25. (32a) Setting leads to a simpler differential equation for with no first derivatives (32b) (32b) has the form of a wave equation and, since we consider the fluid to be unbounded in z, is solution if m fulfills (33) = Dispersion relation (Combines the frequency with the wave numbers k & m and the parameters of the system (g, B, f, c & H0.)) Finding wave solutions of equation (32a) Atmospheric Waves

26. From with and we finally obtain as solution for the perturbation δw: (34) Free travelling wave in x and z with an amplitude changing exponentially with height! The remaining dependent variables are obtained from eqs. (25)-(29) by inserting Step 3: Final form of the solution for the perturbation δw Finished step 2 !!! Atmospheric Waves

27. From (33) with (36) Exact Solutions of the Linearized Equations By setting the tracer parameters to 1 (n1 = n2 = n3 = n4 = 1) in the solution we have derived (i.e. in the dispersion relationship (33) and in the expression for δw (34)) we obtain directly the solution for the exact linearized equations: dispersion relationfor the exact linearizedequations: From (34) => (36a) Amplitude of exact solution grows exponentially with height. Atmospheric Waves

28. (38) (39) Solutions of the dispersion relation (36) Re-arranging (36) gives a 4th order polynomial in σ: 4 solutions: pair of inertial-gravity waves pair of acoustic waves Atmospheric Waves

29. By using the following inequalities, valid for typical values of the system parameters , H0 , g , B and c in the atmosphere of the Earth, we can simplify expressions (38) and (39). (40) (38) (39) Use Taylor expansion of to first order in X around X= 0: With (40) => X << 1. Closer examination of the solutions (38) and (39) 1 Atmospheric Waves

30. Replacing by in (38) gives (38) (39) (39a) Equation (39) simplifies to Closer examination of the solutions (38) and (39) 2 (38a) Atmospheric Waves

31. (38a) , i.e. in a system with zero static stability and no rotation these waves can’t form! inertial-buoyancy waves or, more commonly, inertial-gravity waves Closer examination of solution (38a) 1 => Restoring forces (responsible for bringing the displaced air parcels back to the equilibrium location) for this wave type are the buoyancy force and the Coriolis force (inertial force). => These waves are called Atmospheric Waves

32. Short wave limit: (38a) For short waves in the horizontal (i.e. for large k) expression (38a) reduces to (41) These waves form only in stable stratification (for B > 0). For neutral stratification (B=0) => , i.e. no waves! For unstable stratification (B < 0) => is imaginary, no waves! Closer examination of solution (38a) 2 No Coriolis parameter f in (41)! These waves are too short to be (noticeably) modified by rotation, i.e. pure (internal) gravity waves. Restoring force is the buoyancy force. Atmospheric Waves

33. Closer examination of solution (38a) 3 From (41) => is called buoyancy frequency or Brunt-Väisäläfrequency (Often denoted by N in text books). (41) Very short wave limit: Buoyancy frequency is the upper limit to the frequency of gravity waves! Atmospheric Waves

34. We neglect the term in (41) for the following discussion. This is equivalent to assuming that the basic state density does not change with z ( ), i.e. the basic state is incompressible. Wave vector is perpendicular to phase lines, so is parallel to the phase lines. Some properties of pure (internal) gravity waves 1 => dispersion relation of gravity waves in this type of fluid is (41a) Slope of phase lines (= angle αto the local vertical ) Atmospheric Waves

35. Group velocity is perpendicular to the phase velocity ! Since Transversal waves: Particle path is parallel to the wave fronts. Some properties of pure (internal) gravity waves 2 Dispersive waves: Horizontal and vertical phase speeds depend on the wave numbers. Idealized cross section for internal gravity wave showing phases of p, T & winds. From J.R.Holton: An Introduction to Dynamic Meteorology Example: Lee waves Atmospheric Waves

36. (38a) Waves with are pure inertial waves. (Not influenced by buoyancy force.) Closer examination of solution (38a) 4 Long inertial-gravity waves These waves are influenced by the rotation of the earth. Their frequency is given by (38a). Long wave limit (k << ): * Small frequency but large horizontal phase speeds! * Dispersive waves. * Are a numerical nuisancebecause of their large phase speeds! Atmospheric Waves

37. 10000 10s 20min 2min 3h 1000 <— T 30h σ—> Horizontal Phase Speed [m/s] 100 ck 10 10000 1000 100 10 Horizontal Wavelength [km] k Dispersion Diagram ck(k) From J. S. A. Green: Dynamics lecture notes Atmospheric Waves

38. Dispersion curves for inertial-gravity waves 10000 10s 20min 2min 3h 1000 f L= ∞ 30h Horizontal Phase Speed [m/s] 100 L=20km L=10km 10 L=5km ... f 10000 1000 100 10 Horizontal Wavelength [km] k Atmospheric Waves

39. Closer inspection of equation (39a): Acoustic Waves 1 Here is the adiabatic (39a) (Laplacian) speed of sound. * are non-dispersive, i.e. ck is the same for all k. * have group velocity = phase velocity (in the horizontal) * are longitudinal waves (particle path is perpendicular to wave fronts) * have vertical phase lines (since ), i.e. horizontal propagation. For very shortwaves in the horizontal => phase speed is the speed of sound These waves transmit pressure perturbations with the adiabatic speed of sound, i.e. type of waves with dispersion relation (39a) are acousticorsound waves. Very short acoustic waves Atmospheric Waves

40. (long in the horizontal, i.e. very small k): Long acoustic waves (39a) These long acoustic waves * are dispersive * have large horizontal phase speeds * have almost horizontal phase lines ( ), i.e. mainly vertical propagation Closer inspection of equation (39a): Acoustic Waves 2 Atmospheric Waves

41. 10000 10s 20min 2min . . . L=10km L=20km 3h L= ∞ 1000 f ~300 30h L= ∞ Horizontal Phase Speed [m/s] 100 L=20km L=10km 10 L=5km . . . f 10000 1000 100 10 Horizontal Wavelength [km] k Dispersion curves of acoustic waves Atmospheric Waves

42. Acoustic waves are a numerical problem because of their high frequency and large phase speed. It would be good if we could filter them out of the system. How? By modifying the basic equation such that they don’t support this wave type. Atmospheric Waves

43. Now we will make use of the tracer parameters (n1, n2, n3, n4) we introduced earlier when we derived the solution of the linearized basic equations. We will make approximations to the linearized basic equations (17)-(21) by setting individual tracers to 0 in the equations to eliminate the corresponding terms. By setting these tracers to 0 also in the derived solutions we immediately obtain the solutions of the modified equations. Filtering Approximations We will learn how to modify the linearized basic equations so that they don’t support acoustic waves and/or gravity waves anymore as solutions. The physical principles behind these approximations can then be extended to achieve the same for the non-linear equations. We will also investigate the impact the hydrostatic approximation to the pressure field has on the different wave types and determine conditions under which it is valid. Atmospheric Waves

44. The elimination of acoustic waves (1) Acoustic waves occur in any elastic medium. Elastic compressibility is represented by in the continuity equation. This term can be removed from (20) by setting n2= 0. Setting n2= 0 in ( 33) immediately gives the dispersion expression for the modified set of equations. But we have to be careful!!! Atmospheric Waves

45. The elimination of acoustic waves (2) => We have to set always n2 = n3! (32) In eq. (32) n2 and n3 occur in the combination (n2-n3) which vanishes in the exact equation (i.e. when n1= n2= n3= n4 =1). => A spurious term will arise in (32) if we set n2 to zero but not n3 or vice-versa! Anelastic approximation is n2=0 &n3=0! Atmospheric Waves

46. The elimination of acoustic waves (3) (46) <=> (47) Consequently: (33) Acoustic waves have been eliminated. Inertial-gravity waves are not distorted by the anelastic approximation. Setting n2= n3= 0 and n1 = 1 = n4 in (33) => (46) This dispersion relation has only 2 roots in σ not 4 as (33) => only one wave type left! This is identical to the dispersion relation for inertial-gravity waves (38a). No acoustic waves! Atmospheric Waves

47. The elimination of acoustic waves (4) Compare (46) with the exact dispersion relation (36): (46) (36) When can we neglect in (36)? Re-arrange (36): (39a) can be neglected in (36) if Under what conditions is it OK to make the anelastic approximation? Atmospheric Waves

48. The elimination of acoustic waves (5) Acoustic filtered equations can be used with confidence for a detailed study of inertial-gravity waves in the atmosphere (e.g. for modelling of mountain gravity waves). => It is OK to make the anelastic approximation if the frequencies of the remaining waves are much smaller than the acoustic frequency. => This condition is satisfied for inertial-gravity waves. Atmospheric Waves

49. The hydrostatic approximation 1 Hydrostatic approximation = neglect of the vertical acceleration Dw/Dt in vertical momentum equation (3). (3) In the linearized momentum eq. (19) the vertical acceleration is represented by (since we assumed the basic state to be at rest). => set n4=0! (19) Questions we are going to address: How does the hydrostatic approximation to the pressure field affect the inertial-gravity waves and the acoustic waves? When is it alright to make this approximation? Atmospheric Waves

50. The hydrostatic approximation 2 (33) we see that the term containing n4 can be neglected if Hydrostatic approximation is OK for waves with frequencies much smaller than the buoyancy frequency! => Hydrostatic approximation affectsacoustic waves andvery short gravity waves. Inertial waves and long gravity waves are unaffected. => Validity criterion: From the dispersion relation (33) This condition is satisfied for inertial waves (because their frequency f 2 << gB) not satisfied for very short gravity waves not satisfied for acoustic waves Atmospheric Waves