Atmospheric waves By Martin Miller European Centre for Medium-Range Weather Forecasts 1. INTRODUCTION In order to understand the many and varied approximations and assumptions which are made in designing a nu- merical model of the earths atmosphere (whether of small-scale features such as individual clouds, or mesoscale, regional, global weather prediction or climate models), it is necessary to study the various wave motions which can be present. The identification and appreciation of the mechanisms of these waves will allow us to isolate or elim- inate certain wave types and to better understand the viability and effectiveness of commonly made approximations such as assuming hydrostatic balance. Here we will attempt to identify the basic atmospheric wave motions, to emphasise their most important properties and their physical characteristics. Basic textbooks such as 'An Introduction to Dynamic Meteorology' by Holton or 'Numerical Prediction and Dynamic Meteorology' by Haltiner and Williams provide a good introduction and complement this short course well. Whereas these textbooks treat each wave motion separately starting from dif- ferent simplified equation sets, we will separate waves from a more complete equation set by various approxima- tions made at later stages thus making it easier to relate the impact of a particular approximation on several wave types. Nevertheless, to retain all wave types in the initial equation set is unmanageable and we will introduce sub- sets of equations later. We will restrict ourselves to neutral (non-amplifying, non-decaying) waves in which energy exchanges are oscil- latory. While the exact equations have been considerably simplified into a useful form for studies of atmospheric dynam- ics, these simplified equations cannot be solved analytically except in certain special cases. The main difficulty aris- es through the non-linear advective terms (v ⋅ ∇ )v , v ⋅ ∇ etc. Although these terms can considerably modify the linear solutions and are also physically significant because they represent transfer and feedback between different scales of motion, the linearized equations (obtained by removing second-order advective terms) are useful for iden- tifying the origin of distinct types of wave (acoustic, gravity and cyclone) in the equations. It should be understood that although nonlinearity will modify the acoustic, gravity and long waves it will not introduce any additional wave types. Since the origin of waves can be identified in the linearized equations and these can be solved analyt- ically, useful methods for filtering individual modes can be determined. 2. BASIC EQUATIONS The following mathematical analysis requires a choice of vertical coordinate. While it is common to use pressure or a pressure-based vertical coordinate in large-scale modelling, this is not a necessary theoretical restriction and ¢ ¢£¢ ¡ σ ⁄ we can write the equations in (height), (pressure) or (= * ), as follows: ¤¦¥§¡©¨ ¡ -coordinates ( , , , ) Meteorological Training Course Lecture Series ECMWF, 2002 1 Atmospheric waves ∂¢ D 1 -- ----¤ -- ------¨ - – = – (1) D ρ ∂ ∂¢ D 1 -- ---¥--- ------¨ - + = – (2) D ρ ∂ ∂¢ D 1 -- ----¡ -- ------¨ -- + = – (3) D ρ ∂ ∂ ∂ ∂ D ( ρ) -----¨ - ¥ ¡ ----¤ -- + ------ + ------- = ln (4) ∂ ∂ ∂ D D ( ) κ D ( ¢ ) ¨ -----¨ - ln = ------ ln (5) D D ∂ ∂ ∂ ∂ D κ ¥ ¡ ---¤ -- ----- ----- ¨ where -----¨ - = ----- + + + and = ----- D ∂ ∂ ∂ ∂ All source/sink terms are omitted. ¤¥§¢¨ ¢ -coordinates ( , , , ) An exact transformation of (1)–(5) to ¢ -coordinates gives: ∂Φ D ( ε) ----¤ --- ------¨ - – = – 1 + (6) D ∂ ∂Φ D ( ε) ----¤ --- ------¨ - + = – 1 + (7) D ∂ ∂Φ ( ε) ---¢ ---- ---¢ ------ = – 1 + ∂ (8) ∂ ∂ ∂ω D { ( ε)} -----¨ - ¥ ¢ ----¤ -- + ------ + ------- = ln 1 + (9) ∂ ∂ ∂ D D κω ¢ ------¨ -- = ------------ (10) D where φ ε 1 D 1 D D ¨ ¨ = ---------¨ -- = ----------- ------- (11) 2 D D D ¢ ∂ ∂ ∂ ∂ ¡ D ω Φ ω ¥ ¢ ---¤ -- ----- ------ ¨ ¨ and = ------¨ -- , = , and ------ = ----- + + + D ∂ ∂ ∂ ∂ 2 Meteorological Training Course Lecture Series ECMWF, 2002 Atmospheric waves ¢£¢ ¨ σ ( ¤¥, , σ ⁄ , ) -coordinates = * Either by transforming (1)(5) or (6)–(11) we can obtain the exact σ -coordinate set: ∂Φ ∂ D ¢ ( ε) ( ) ¤ ----¤ --- ----- ------¨ - – = – 1 + – ln (12) D ∂ ∂ * ∂Φ ∂ D ( ε) ( ¢ ) ¥ ----¥ --- ----- ------¨ - + = – 1 + – ln (13) D ∂ ∂ * ∂Φ ( ε) ---σ------ = – 1 + -∂---σ--- (14) ∂ ∂ ∂σ ˙ D ( ¢ ) D { ( ε)} -----¨ - ¥ ¨ ----¤ -- + ------ + ------ = – ln + ------ ln 1 + (15) ∂ ∂ ∂σ D * D σ D ˙ D ¢ κ ( ) -----¨ - ------¨ -- = --- + ln (16) D σ D * σ D ∂ ∂ ∂ ∂ σ σ˙ ˙ ¥ ---¤ -- ----- ------ ¨ ¨ where = ------¨ -- , and ------ = ----- + + + . D ∂ ∂ ∂ ∂σ All partial derivatives respect their coordinate system. ⁄ ¨ ε Note that putting D D = 0 in (3), and = 0 elsewhere gives the familiar large-scale equation sets, but we will not make this approximation at present. In principle the following linearized analysis can be done in any co- ordinate; we will use height but much of this analysis has been done in pressure and sigma coordinates elsewhere (e.g. Kasahara, 1974; Miller, 1974; Miller and White, 1984). For simplicity we will suppose that the motion is independent of ¥ and neglect the variation of the Coriolis param- β ∂ ⁄ ∂¥ eter with latitude ( = = 0 ). Also we will consider small disturbances on an initially motionless atmos- phere. A non-zero basic flow and restoration of β and non-zero ∂ ⁄ ∂¥ will be considered later. , , , In order to trace the effect of individual terms we will use ‘tracer parameters' 1 2 3 4 which have the value 1 but can be set to zero to eliminate the relevant term. ¢ κ Θ ≡ θ θ 00 ¢ 5 We define ln ,where = --¢ ------ and 00 is a reference pressure (e.g. 10 Pa), and write δ δ = 0 + = δ δ = 0 + = δ δ = 0 + = ρ ρ ( ¡ ) δρ = 0 + ¢ ¡ ¢ ¢ ( ) δ = 0 + Θ Θ ( ¡ ) δΘ = 0 + ¡ ¢ ¡ ¡ δ δ δ δρ δ¢ δΘ ρ ( ), ( ), Θ ( ) where , , , , , denote small∂¢ perturbations on a mean state and 0 0 0 define the 0 ρ basic horizontally stratified atmosphere with --∂---¡ ---- = – 0 . Since we consider small perturbations such that prod- δ¢ δρ ρ δΘ Θ ucts of perturbations can be negelcted and that /p0 « 1 , / 0 « 1 , / 0 « 1 we can write (1)–(5) as: Meteorological Training Course Lecture Series ECMWF, 2002 3 Atmospheric waves ∂δ ∂ δ¢ δ ---¤ -- ------ ---∂---¨ --- – + ∂ ρ = 0 (17) 0 ∂δ δ ---∂---¨ --- + = 0 (18) ¢ ∂δ ∂ δ¢ δ δΘ ¡ -------¨ --- + ----- ------ – ------ – = 0 (19) 4 ∂ ∂ ρ 3 ρ 0 0 ∂ δρ δ ∂δ ∂δ 1 ----¨ - ------ ¡ 2∂ ρ + --∂---¤ ---- + ---∂------- – ------------- = 0 (20) 0 0 ∂ δΘ δ ----¨ - + = 0 (21) ∂ ∂ 1 ∂ ( θ ) ( ρ ) ¡ where = ---¡ -- ln and ------- = – ----- ln . ∂ 0 ∂ 0 0 EXERCISE: Derive Eq. (19) ¤ ¨ The coefficients 1 ⁄ in (17)–(21) are independent of and , so these equations are linear and, in an 0 ¤ ¨ ¡ unbounded region, admit solutions of the separable form ( )exp( !( + σ )) where can be a complex func- ¤ tion, and and σ are the -wavenumber and the frequency respectively. Complex values of σ would imply am- plifying/decaying waves which are not considered here. The full solution is the appropriate Fourier sum of terms of this form over all wavenumbers. Since we shall be looking at individual waves we choose to discuss individual wave components rather than the Fourier sum. ¡ ¤ ¨ ¡ ¤ ¨ ! ! Inserting δ = ˆ ( )exp( ( + σ )) , δ = ˆ ( )exp( ( + σ )) , and the corresponding expressions for δ , ¢ ¤ ¨ δ , δρ , and δΘ into (17)-(21), and noting that the operators ∂ ⁄ ∂ , ∂ ⁄ ∂ can be replaced by " and σ , re- ¢ ˆ , , , ⁄ ρ , ρ ⁄ ρ , Θ spectively, yields the following set of ordinary differential equations in the unknowns ˆ ˆ ˆ ˆ 0 ˆ 0 : ¢ σ ˆ # " ˆ – ˆ + ρ----- = 0 (22) 0 σ # ˆ + ˆ = 0 (23) ¢ ¢ d ˆ ˆ ˆ σ ----- ----- Θ ˆ + ¡ – – = 0 (24) 4 ρ 3 ρ d 0 0 ρ ˆ ˆ d 1 " # σ----- 2 ρ + ˆ + ¡ ˆ – ---------- = 0 (25) 0 d 0 σΘˆ + ˆ = 0 (26) Then (22) and (23) give: 4 Meteorological Training Course Lecture Series ECMWF, 2002 ¢ σ ˆ ˆ = –---------------------- (27) 2 2 ρ σ – 0 ¢ $ ˆ ˆ = –---------------------- (28) σ2 2 ρ – 0 Eliminating ˆ between (25) and (27), and using (26) and the relation: ¢ ¢ ρ ρ δΘˆ 1 ˆ ˆ 1 ˆ ˆ = -- ¢ ----- – ----- = --------- – ----- γ ρ 2 ρ ρ 0 0 0 0 γ where = 0 is the Laplacian speed of sound, gives: 2 ¢ d 1 2 ˆ σ ----- ¡ ˆ + – ------- ˆ + ----- – ----------------- = 0 (29) 2 d 2 σ2 2 ρ 0 – 0 ˆ Θ " Using = ˆ -σ--- (30) to eliminate Θˆ from (24) yields: ¢ σ d ˆ ( σ2) ----- ˆ ¡ – + – = 0 (31) 3 ρ 4 d 0 ¡ ¢ ⁄ ⁄ ρ In general , 1 , are functions of so ˆ and ˆ are obtained by simultaneous solution of the two first 0 0 Θˆ ρ ⁄ ρ order equations (29) and (31) and the ˆ , ˆ , , ˆ 0 fields obtained from (27), (28), (30) and (25), respectively. However, for our present purposes it is sufficient to consider constant (mean) values of , 1 ⁄ and which are 0 ⁄ 2 ⁄ related by + = 1 . Then the differential equation for the height variation of ˆ is: 0 2 2 d 1 d 2 2 σ ( ) ( σ ) + – – ------- ¡ + – ----------------- – ----- 2 3 4 ¡ 2 d 2 2 2 d 0 σ – (32) 1 – – ------- ˆ = 0 3 2 0 3. EXACT SOLUTIONS OF THE LINEARIZED EQUATIONS The exact linearized equation for ˆ , obtained by setting 1 = 2 = 3 = 4 = 1 in (32), is: 2 2 d 1 d 2 1 1 σ ( σ ) – ------- ¡ + – -----------------–---- – – ------- ˆ = 0 (33) % ¡ 2 d σ2 2 2 d 0 – 0 Meteorological Training Course Lecture Series ECMWF, 2002 5 Atmospheric waves One solution of (33) is σ = 0 , and the corresponding dynamical structure can be determined by setting σ = 0 in ¥ ( ⁄ )( ¢ ⁄ ρ ) " & (22)–(26)