ATMOSPHERIC OSCILLATIONS 1

Transcription

1 7 ATMOSPHERIC OSCILLATIONS Introdution One of the properties of matter is that is an support transfers of mehanial energy, whether the matter is solid, liquid, or gas, without any net movement of the moleules involved. Suh transfers are referred to as "wave motion", or a transitory displaement of atoms within the matter that passes on kineti energy. This hapter disusses the physis of mehanial wave motion. Light also involves wave motion, but not of a mehanial sort, and will not be disussed in this ourse. We are all familiar with wave motion. The sounds that we hear are waves passing through the atmosphere, with the air ompressing and deompressing in an osillating fashion, passing the sound energy outward at hundreds of metres per seond. We are all familiar with the waves that expand outward after we drop a big rok in a pond or the waves that we reate by whipping a rope tied to a fene up and down. There are two kinds of waves: "ompression" or "longitudinal" waves, and "transverse" waves (Figure 7.1). In a ompression wave, the moleules move bak and forth in the diretion of the wave motion. Sound is a ompression wave. In a transverse wave, the moleules move bak and forth at a right angle to the diretion of motion. A rope being shaken with an up-and-down motion generates a transverse wave, and the ripples we see spreading aross a pond are also transverse waves. Figure 7.1 Two fundamentally different modes of wave propagation. In the atmosphere wave motion takes a speial plae among the multitude of motions and many different waves an be distinguished. Wave motion an be desribed with the aid of a linear or linearized system of differential equations, whih is relatively easy to treat mathematially. This approah is valid beause wave motion is important and overs the entire spetrum of atmospheri motions. 1 This hapter is based on the exellent text Dynamik der Atmosphäre by Helmut Pihler (see referenes) and has been translated and adapted to the notation used in the book by Hakim and Holton. Version 015_11 1

2 If we onsider the large-sale flow in the free atmosphere then in the mid-latitudes we will regularly see a hange from zonal to meridional flow and vie versa. This has led to the notion of troughs and ridges as a part of planetary waves with a wave number of 4-6. As a rule these waves travel eastward with a speed of 6 degrees longitude per day. For the weather in the mid-latitudes they are of the utmost importane. On a slightly smaller sale the barolini waves are important. They usually have a larger wave number (a smaller wavelength) and are ontrolled by the vertial windshear and the stati stability. Beyond a ritial windshear the waves beome unstable leading to ylogenesis where potential and internal energy of the atmosphere is transformed into kineti energy. Waves also our on smaller sales. If stably stratified air flows over an obstale (e.g. a mountain ridge) then lee waves will develop downstream of the obstale. These waves have a wavelength of 5 to 10 km and larger. The displaement of the air then is in the vertial. The nature of these waves is determined by the vertial profiles of wind and temperature and they are an example of internal gravity waves. Lee waves an reah into the stratosphere. Together with a strong vertial windshear in the upper atmosphere these waves may beome unstable and generate what is alled Clear Air Turbulene (CAT). The simplest forms of gravity waves are those that our on the disontinuity between two homogeneous fluids of different density, where due to Earth s gravity the denser fluid lies beneath the lighter, less dense fluid. The largest amplitude will our on the surfae of the disontinuity. These waves are alled internal gravity waves. If the density of the upper fluid dereases to, almost, zero then the surfae of disontinuity will be the upper boundary of just one fluid and perturbations of this surfae are alled external gravity waves. Ripples on the water surfae of a pond (Figure 7.) are a perfet example of suh waves. Figure 7. An example of external gravity waves. Waves in the atmosphere transport physial quantities suh as momentum and energy. A study of atmospheri waves is therefore important beause it will give us a better understanding of the physis of the atmosphere. Together with wave propagation in the atmosphere we will also dediate some time on the propagation of sound waves. Sound waves do not play an important role in largesale weather but must be onsidered beause of the stability of numerial modelling Version 015_11

3 used for weather foreasting. In order to obtain numerial stability, waves whih propagate with the speed of sound must be filtered out of the system. Therefore we need to onsider sound waves as well. 7. Representation and properties of waves In this doument we will only onsider waves travelling in one diretion (along the x- axis). The extension to two or three dimensions is omitted for brevity. Consider a partile exhibiting a periodi movement in spae i.e. an osillating partile. The energy of the periodi movement is transferred to neighbouring partiles whih will start to osillate: the osillation is propagating in spae. If this propagation is in one diretion only we are dealing with a one-dimensional (1D) wave. The mathematial representation of suh a wave is given by θ ( x t) = θ os( kx ν t), 0. (7.1) This is a one-dimensional (1D) wave propagating in the x-diretion (although the osillations may be in another diretion), see Figure 7.3. Figure 7.3 Representation of a simple one-dimensional wave (After Pihler, 1997). From Figure 7.3 we have the following quantities: - θ 0 is the amplitude; it is half the differene of the perturbation between a rest and a trough, - λ is the wavelength (in m) it is the distane between two rests (or troughs) of a wave, - τ is the period (in s) it the amount of time needed for one omplete osillation and it λ must equal the time required for the wave to travel one wavelength τ =, - k is the wave number (in radians m -1 or usually m -1 ); it is the number of waves per π unit length, hene π k =, (7.) λ Version 015_11 3

4 π - ν = is the angular (or irular) frequeny: π times the number of rests passing τ 1 a point in unit time (in Hz = s -1 ). The normal frequeny is given by: f =. τ - kx ν t is the phase of the wave. It indiates for a given time and loation whether a trough or a rest of the wave is loated there. For propagating waves the phase is onstant for an observer moving at the phase speed λ ν = = (7.3) τ k where the phase speed is in m s -1. This may be verified by observing that if the phase is to remain onstant following the motion, then D Dt Dx Dt ( k x ν t) = k ν = 0 Hene = = for the phase to be onstant. As by definition >0 then: - if >0 we have >0 in that ase x must inrease with inreasing t for the phase to remain onstant. The wave then propagates in the positive x-diretion; - if <0 we have <0 in that ase x must derease with inreasing t for the phase to remain onstant. The wave then propagates in the negative x-diretion In the one-dimensional ase with propagation along the x-axis we an rewrite (7.1) as ( x t) = θ os( kx ν t) = os[ k( x t) ] θ (7.4) Where we have used, 0 θ0 ν = k. Often it is easier to represent waves with the aid of Euler s equation (see Appendix A): this leads to: φ e i + θ ( φ) ( φ) = os i sin, (7.5) i( kx ν t) ik( x t) ( x, t) θ e = θ e = 0 0 Only the real part of (7.6) represents the periodi behaviour.. (7.6) 7.3 The wave equation It is easy to verify by diret substitution that (7.4) and (7.6) are the solutions of the following differential equation: = (7.7) Version 015_11 4

5 if the amplitude and phase speed, or wave number, remain onstant. This equation is alled the wave equation. It is a linear partial differential equation and therefore a ombination of two partiular solutions is also a solution of this equation. We will onsider two waves propagating in the same diretion but with a slight differene in frequeny ( ν) and onsequently also a slight differene in wave number ( k). A superposition of these two waves gives: = + = + (7.8) and leads to the formation of wave groups. From (7.8) we will have = + = + (7.9) whih, with the help of the relation ϕ iϕ iϕ os = e + e, leads to = + = Δ Δ (7.10) The superposition θ = θ 1 + θ an be regarded as a wave of whih the amplitude also shows a periodi hange (AM: amplitude modulation). This is depited in Figure 7.4. Figure 7.4 Representation of wave groups as a result of adding two slightly different waves traveling in the same diretion (From Pihler, 1997). The veloity of the wave groups, the group veloity ( g ) need not be the same as the phase speed of the two original waves. If we go from Δ and from Δ then we find for = = With the help of (7.11) we find for the phase of the amplitude modulation: (7.11) = (7.1) whih leads to the group veloity g : = (7.13) Version 015_11 5

6 Using (7.3) we an write for g g ν d = = k = + k = λ, (7.14) k k k dλ Equation (7.14) means that if the phase speed depends on the frequeny or the wavelength then the group veloity and the phase speed will be different. This phenomenon is alled dispersion. The following three possibilities an be diserned (for > 0): d > 0 dλ g < normal dispersion, d < 0 dλ g > anomalous dispersion, d = 0 dλ g = no dispersion. Sound waves are an example of non-dispersive waves: high and low tones travel with the same group/phase speed. Waves on a water surfae are an example of dispersive waves (normal dispersion, see Figure 7.5 and also e.g. Holton Figure 5.4). Figure 7.5 A superposition of two traveling waves illustrating the differene between the phase speed of the wave rests and the group veloity g of the envelope of the waves. In this ase g < indiating normal dispersion (From Gill, 198). In the ase of normal dispersion the wave with the largest wavelength has a larger phase speed than the wave with the smaller wavelength. In the ase of anomalous dispersion the opposite ours. Version 015_11 6

7 7.4 Unstable and damped waves Up to now we only onsidered waves where the amplitude remained onstant during time i.e. the amplitude did not grow or damp out. However, in nature both possibilities our. For ylogenesis the growth of the amplitude of a (barolini) wave is essential. With a growing or damping amplitude we need to have a omplex representation of the angular frequeny: ν = ν r + iν i. (7.15) where ν r = Reν ( ) is the real part and ν = Imν ( ) frequeny ν. i is the imaginary part of the omplex If you substitute (7.15) into the wave formula (7.7) then it follows diretly that, = (7.16) For ν > 0 the amplitude will grow with time, the wave is unstable. For ν < 0 the i amplitude will diminish with time approahing zero, the wave is stable or damped. For ν = 0 the amplitude will remain onstant in time, this is also a ase of a stable wave. In i general we will study wave propagation using ν = ν r ± iν. Using the priniple of superposition we will get the following wave representation i i, = + (7.17) One of the two amplitudes in (7.17) will grow with time, so that if ν 0 we will always have an unstable wave. Finally, we must realize that the amplitude fators θ 0, θ 01 and θ 0 themselves an be funtions of the spatial vetor. We will have to use this for the θ = z. propagation of ertain waves in the atmosphere e.g. ( ) 0 θ 0 i 7.5 Conept of perturbation and linearizing Waves will be introdued in the dynamis of the atmosphere with the aid of the so-alled perturbation theory. With this theory a nonlinear differential equation an be linearized so that an analytial solution an be found. Although suh a solution is only valid for relatively small perturbations, it is nonetheless very useful for the desription of waves in the atmosphere. The onept of perturbation theory is the following: we assume that the atmosphere is in a known basi state m ( m is either ρ, U, T or ρ or any ombination of these quantities). This basi state exatly satisfies the full nonlinear set of equations. Now a perturbation m is introdued, whih is a small deviation from this basi state : m << m. This perturbation then is superposed on the basi state, hene m = m+ m Version 015_11 7

8 and m is introdued into the full set of equations. Next, two essential steps are taken: 1. all terms ontaining m (and not m ) are subtrated from the set of equations. This is a valid proedure as m satisfies the original set of equations (this was our assumption from the outset);. from the remaining terms all terms ontaining two or more primed quantities will be removed. This is also a valid proedure as the produt of two small quantities is negligibly small. What remains after these two steps is a linearized equation for whih an analytial solution an be found. We will illustrate the omplete proedure using a simple nonlinear equation: y = ax + bx+ (7.18) As a matter of fat, an analytial solution for this quadrati equation is well known, but that is irrelevant in this ase. This equation is nonlinear beause of the term ontaining x. The basi state is defined by the known values x and y therefore we know then that the following holds exatly: y ax + bx+. (7.19) We define the perturbations y = y+ y with y << y, and x = x+ x with x << x and substitute these expressions in (7.18). This leads to ( x ) + bx+ bx y + y = ax + axx + a +. (7.0) Step 1: subtrat the basi state (7.19) from (7.0): ( x ) + bx y = axx + a Step : delete all terms ontaining produts of primed quantities: y = axx + bx = ( ax+ b)x. (7.1) What remains is a linear equation in the (primed) perturbation quantities y and x. It would now be easy to find a solution ontaining y and x of the linearized equation (7.1). Version 015_11 8

9 7.6 Linearization of the basi set of equations Using the perturbation onept we will introdue the following variables in the basi set of equations desribing atmospheri motions: U = U + U, p = p+ p, ρ = ρ+ ρ (7.) where the quantities with an overbar represent the variables of the basi state and the primed quantities represent the variables of the perturbation. At this point you may wonder why we also do not assume = + but it turns out that drops out of Eq. (7.34) altogether see Appendix B Equation of motion For the basi set of equations we will use Equation (.8) of Holton and neglet fritional fores: DU ρ + ρω U = p+ ρg. (7.3) Dt With the assumption (7.) this gives: ( ρ+ ρ ) ( U + U ) + ( U + U ) ( U + U ) + t ( ρ+ ρ ) Ω ( U + U ) = ( p+ p ) + ( ρ+ ρ )g (7.4) The basi state of the atmosphere satisfies U ρ + U U + ρω U = p+ ρg. (7.5) t Usually we assume that the basi state of the atmosphere is a zonal urrent, i.e. u = onstant and v = w= 0. Suh a flow will be in geostrophi and hydrostati equilibrium. With these assumptions, subtration of the basi state (7.5) from (7.4), and elimination of all terms with produts of the primed quantities, suh as U U and ρ U, leads to the following linear differential equation U ρ + U U + ρω U + ρ Ω U = p + ρ g (7.6) t James R. Holton and Gregory J. Hakim: An Introdution to Dynami Meteorology 5 th edition. Version 015_11 9

10 Using sale analysis (see Holton setion.4) the term ρ Ω U is very small ompared to the other Coriolis term and an be negleted. As a result we have the following linear equation of motion for a onstant zonal basi flow: U ρ + U U + ρω U = p + ρ g. (7.7) t If the basi flow has a horizontal or vertial wind shear then this an be inorporated into (7.7) Equation of ontinuity We an use the assumption (7.) and the same proedure to linearize the equation of ontinuity: Holton Equation (.31) This leads to Dρ + ρ U = 0. (7.8) Dt ρ + U ρ + U ρ+ ρ U = 0. (7.9) t If we furthermore assume that the density only depends on height (a barotropi ρ = ρ z, then the third term of (7.9) simplifies and we are left with atmosphere), i.e. ( ) ρ dρ + U ρ + ρ U + w = 0 (7.30) t dz Thermodynami energy equation The third equation we need to linearize is the thermodynami equation. Holton Equation (.41) states that DT Dα v + p = J. (7.31) Dt Dt For adiabati motions we have J = 0. Differentiating the equation of state p α = RT (7.3) and some algebrai manipulation (Appendix C) leads to Dp Dt p Dρ = RT. (7.33) Dt v Linearizing (7.33) is straightforward: p p p ρ dρ + U p + w = RT + U ρ + w, (7.34) t z v t dz Version 015_11 10

11 where R T = p ρ. With the system of equations (7.7), (7.30) and (7.34) we are now able to desribe the propagation of waves in the atmosphere. For brevity we will omit the prime of all primed quantities from now on beause we only have to deal with the perturbation quantities and the quantities of the basi state are indiated with an overbar. 7.7 Appliations Sound waves 3 Before we treat the general theory of the propagation of sound, or aousti, waves in the atmosphere, we onsider here the propagation of sound waves in an ideal gas that is at rest relative to the Earth s surfae. To this end we will use the following simplifiations in the basi set of equations (7.7), (7.30) and (7.34): - no Coriolis fore: all terms ontaining Ω are negleted - no gravity: all terms ontaining g are negleted, - no horizontal and vertial gradients of the thermodynami quantities of the basi state - an atmosphere at rest: U = 0 (this is not stritly neessary, see exerise 7.6 and Eq. 7.40) What remains then is the following set of equations, with the primes removed from the perturbation quantities: U ρ = p (7.35) t ρ = ρ U t (7.36) p t = p v ρ RT t (7.37) First we eliminate the density perturbation from (7.37) with the aid of (7.36). Next, we differentiate the results with respet to time and then eliminate the wind speed perturbation with the aid of (7.35). This leads to the following homogeneous 4 differential equation: p t = p v RT p= γ RT p. (7.38) After omparison with the wave equation (7.7) it is at one lear that (7.38) is a homogeneous wave equation. Hene, in a fore-free ideal gas perturbations propagate with a phase veloity of 3 This setion ompares with Holton and Hakim setion A linear differential equation is alled homogeneous if the following ondition is satisfied: If p(x.y,t) is a solution, so is p(x.y,t), where is an arbitrary (non-zero) onstant. Version 015_11 11

12 p = RT. (7.39) s ± v Aording to (7.31) the propagation of suh waves is by adiabati ompression and expansion of an ideal gas (Figure 7.6). Figure 7.6 Sound wave travelling in the positive x-diretion: pressure and density urves are in phase and in phase with the perturbation in horizontal veloity (arrows). Obviously we are dealing with sound waves, of whih the frequenies audible to the human ear are roughly between 16 Hz and 0 khz. Pressure flutuations audible to the human ear are in the range of 10 µpa (i.e times smaller than the mean sea level pressure!). Sound waves with frequenies higher than 0 khz are alled ultra sound waves, while those with frequenies lower than 16 Hz are alled infra sound waves. Sound waves osillate in the diretion of propagation, thus we are dealing with longitudinal waves, they an only propagate in a medium (air, water even solids). Furthermore, from (7.39) it is apparent that sound waves in an ideal gas have no dispersion: all wavelengths travel at the same veloity s. For a standard temperature of 73 K this veloity is approximately 331 ms -1. For a standard atmosphere the veloity of sound lies between 340 ms -1 near the Earth s surfae and 95 ms -1 near the tropopause (see Figure 7.7). Figure 7.7 Height distribution (US Standard Atmosphere) of temperature, sound veloity and potential temperature (After Gossard and Hooke, 1975). Version 015_11 1

13 The expression for the adiabati speed of sound (7.39) is slightly altered when the atmosphere is not at rest but when in the basi state the atmosphere has a onstant zonal veloity u, the result is then p = u± RT = u± s (7.40) v the phase speed depends on the wind speed. This priniple is used in the onstrution of soni anemometers where the small differene in travel times between upstream and downstream wave propagation is a measure for the wind speed in that diretion. The wind speed also hanges the wave number (ompared to the situation with no wind), and the two effets exatly anel, leaving the frequeny unhanged: ( u± ) ν ν ν = k = s = (7.41) u± s where double primed quantities refer to the situation when in the basi state the atmosphere has a onstant zonal veloity u. Note that this is ontrary to what is stated by Holton on p. 139! When the soure of sound, or the observer or both are moving, a differene in frequeny an be deteted by the observer leading to Doppler shifting. In this ase the frequeny as deteted by the observer (ν ) is then given by s ± uobserver ν " =, (7.4) s ± usoure ν and an observer will hear a higher/lower frequeny when the distane between soure and observer is dereasing/inreasing Shallow water waves 4 Shallow water waves our, by definition, in a fluid layer of onstant density in whih the horizontal sale of the flow is muh greater than the fluid layer depth. The fluid motion is fully determined by the momentum and ontinuity equations. However, for the momentum equation we annot use Eq. (7.7) diretly as we have to introdue the depth of the layer before linearizing the equation, hene we start with the nonlinearized momentum equation (7.3). We will use the following simplifiations: - no Coriolis fore: all terms ontaining Ω are negleted; - a single fluid layer with average depth H and density = ; - above the free surfae there is a fluid (air) of negligible density 4 This setion ompares with Holton and Hakim setion Version 015_11 13

14 Figure 7.8 A shallow water system over a flat solid surfae. The loal depth of the fluid olumn is indiated by h, its average value is indiated by H. With these assumptions the vertial omponent of the momentum equation (7.3) is redued to 0= (7.43) Beause density is assumed onstant we an integrate this expression from the bottom (z = 0 and p = p 0 ) to a ertain height z (Figure 7.8):,, =. (7.44) At the top of the fluid (z = h) the pressure is determined by the weight of the overlying air and this is assumed to be negligible. Thus p(z=h) = 0 whih yields,, = h,. (7.45) The onsequene is that the horizontal pressure gradient does not depend on z: = h. (7.46) where, as usual, + Next, we use (7.46) in the horizontal omponents (V = (u,v)) of the momentum equation together with the assumptions and obtain = = h (7.47) Linearizing (7.47) (reintroduing primes for the time being) using yields =0+ h, = +h, = h (7.48) Version 015_11 14

15 For the equation of ontinuity the above assumptions applied to Eq. (7.30) lead to and = + + = 0 = + = (7.49) (7.50) We integrate the above expression from z=0 to z=h: h 0 = h At the top of the fluid the vertial veloity flutuation is just the vertial motion of the fluid surfae: h h and at the bottom we have by definition (7.51a) 0 0 This results in h +h =0 (7.51b) (7.5) Using the linearizing assumptions and removing terms having produts of perturbation quantities yields h + =0 (7.53) The set of linear equations (7.48) and (7.53) an now be ombined by eliminating the veloity. To this end we differentiate (7.53) with respet to time and take the divergene of (7.48). This leads to h (7.54) h =0 whih may be reognized as a wave equation. Comparison with (7.7) yields the phase speed of shallow water waves: = (7.55) That is, the wave speed is proportional to the square root of the mean fluid depth and independent of wavenumber the waves do not show dispersion. This means that the shape of the wave is preserved as long as H does not hange. If, e.g. near oastlines H dereases then dereases as well and the wave builds up to form a tsunami. 7.8 Rossby waves In this setion we will onsider the large-sale waves having synopti or planetary length sales and whih propagate quasi-horizontally. These waves an be seen on daily Version 015_11 15

16 weather maps e.g. on the 50 kpa pressure surfae (geopotential) height analysis (Figure 7.9). In Figure 7.9 the Rossby wave of the 50 kpa surfae height shows two ridges and two troughs. The ridges are over Sandinavia / Western Europe and over the Atlanti Oean SW of Ieland. The troughs are over the Atlanti Oean just west of Ireland and over Russia and the Blak Sea. Rossby waves are very important for midlatitude weather development beause the ridges and troughs determine the large-sale irulation. The waves determine the loation and development of the polar front with its assoiated weather phenomena. Furthermore, Rossby waves are responsible for a part of the meridional transports of momentum, energy and water vapour and they are thus an integrating part of the global irulation. Figure 7.9 Weather maps. Left: the 50 kpa height (full) and temperature (dashed). Right: the orresponding surfae pressure (full). The driving fore behind the Rossby wave is the hange in Coriolis fore with latitude, the so-alled β-effet. In a barotropi atmosphere the Rossby wave is suh that the absolute vortiity (η), defined as η =ζ + f (7.56) is onserved. In (7.56) ζ is the (geostrophi) vortiity and f is the Coriolis parameter. The mehanism whih makes a partile osillate bak and forth between latitudes an be understood as follows (see Figure 7.10). A partile starts from point A with a given value of the horizontal wind speed and with ζ = 0. As it travels north f will inrease and, in order to onserve the absolute vortiity, ζ will have to derease. The partile will experiene an antiyloni urvature whih will deviate it from its original diretion. Inreasing antiyloni urvature will result in the partile reahing an northernmost position in point B. Beause of the existing urvature the partile will then travel southward and f will start to derease from here on. As f dereases ζ will have to inrease and the urvature will beome less and less antiyloni. When the partile reahes its starting latitude (point C) we again have ζ = 0 but its northward speed has reversed and it is heading south. As the partile rosses the latitude southward, f will derease so ζ will have to inrease: the partile will experiene a yloni urvature. The yloni vortiity and urvature will inrease until the partile reahes its southernmost latitude (point D) upon whih it will start moving northwards again, and so on. Version 015_11 16

17 Figure 7.10 Hypothetial path followed by a partile onserving absolute vortiity. From this example it is obvious that the hange of the Coriolis parameter with latitude: f β = (7.57) y generates a restoring fore. The restoring fore is horizontal and the osillation therefore is in the horizontal plane. The restoring fore is proportional to the distane from the original latitude, this means the partile is performing a harmoni osillation and a wave is generated. Another example of the generation of Rossby waves is given e.g. by Holton (p. 160). In a barolini atmosphere it is not the onservation of absolute vortiity, but onservation of potential vortiity whih is important in this respet. What we have desribed above is alled a free barotropi Rossby wave. These are only weakly exited in the atmosphere (Holton, 013). Of more importane are the fored stationary Rossby modes, whih are exited by longitudinally dependent diabati heating patterns (e.g. the oean-ontinent ontrasts in winter) or by flow over topography (e.g. flow over the Roky Mountains and the Himalaya) Rossby waves in a barotropi atmosphere In the derivation of the basi set of equations the Coriolis parameter (f 0 ) was assumed onstant and we need to reonsider the equations before we an ontinue. The simplest form of Rossby waves an be found in a barotropi, divergene free, atmosphere. In suh an atmosphere we start with the onservation of barotropi potential vortiity (Holton eq. 4.39): D ζ + Dt h f = 0. (7.58) Where the atmosphere is represented as a homogeneous inompressible fluid of variable depth h(x,y,z,t). Rewriting (7.58) yields: + + = (7.59) Version 015_11 17

18 where we have used f = f 0 + β y with β = onstant and also assumed that =0. We assume that the basi flow is (a) zonal, and (b) onstant i.e. u = onstant and v= 0 and hene ζ = 0 so: u = u+ u, v = v, ζ = ζ and h = H + h (with H = onstant). We linearize (7.59) with the perturbation method. This yields: ζ ζ f h h + u + v β = + u (7.60) t x H t x The perturbation veloities are now given on a planetary sale and assumed to be in geostrophi balane: v =+ g f h x and g β ζ = h + u' f f u' = g f h' y (7.61) So that (7.60) transforms into β f ' β f + + ' + h h + u h u h u h β = 0. (7.6) t g gh x g gh x For this differential equation we seek a solution of the form i( kx ν t) (7.63) h = Ae whih implies, using (7.61), that =0 (no y-dependene). Substituting expression (7.63) into (7.6) gives f f ( )( ν) ( ν) + u( k ) k u k + kβ = 0 k (7.64) gh gh This leads to the dispersion equation v uk k β = f k + gh (7.65) For short waves k >> f / gh, the dispersion equation then redues to: = Version 015_11 18

19 leading to = This is the form usually found in textbooks e.g. Holton equation (5.110), although Holton uses the two-dimensional equivalent expression where + and l is the meridional wavenumber. Using the definition (7.3) we find for the phase speed of the Rossby waves: = + / (7.66a) To make the analysis simpler we use the short-wave approximation so that the phase speed is = (7.66b) Equation (7.66) is one of the most fundamental equations of large-sale atmospheri dynamis. It gives the two-dimensional horizontal wave propagation on a planetary sale. The phase speed of these planetary waves, whih show anomalous dispersion, is ontrolled by the hange of the Coriolis parameter with latitude (β-effet). From (7.66b) we see that the phase speed is always negative relative to the basi state of the atmosphere beause β / k < 0 (Figure 7.11). For u = 0 we have < 0 and the Rossby waves propagate in the westward diretion, only if u is large enough will Rossby waves travel eastward, being dragged eastward as it were, by the basi zonal flow. From (7.66b) we an derive the wavelength for stationary waves: u λs = π (7.67) β For u = 10 m/s and latitude 50 N we find λ s = 5179 km (see Figure 7.11). Waves with wavelength λ < λ s have > 0 and are propagating from west to east. Waves with wavelength λ > λ s have < 0 and are propagating from east to west (retrograde motion). In the midlatitudes non-stationary Rossby waves usually travel eastward at a speed of 6 degrees longitude per day. Version 015_11 19

20 Figure 7.11 Phase speed of Rossby waves as a funtion of wavelength for φ = 50 N and a basi zonal flow of 10 m/s Rossby waves in two dimensions In reality Rossby waves do not travel in the East-West diretion only, but they will also move aross latitude irles. In that ase the derivation above needs to be extended to two dimensions. The one-dimensional wave funtion (7.6) then expands to θ i( kx+ ly ν t) ( x, y, t) θ e = 0 (7.68) where l is the wavenumber in the y-diretion. Without repeating the derivation in 7.8.1, we find the following dispersion formula: v uk k = k β f + l + gh (7.69) ompare with Eq. (7.65). This expression an be rewritten into β k ν = u k k + l (7.70) where the short wave approximation has been used again (see Holton eq ). For the group veloity in the two diretions we find (see Eq. (7.13) and Holton Eqs. (10.66) and (10.67)): ν k l gx = = u+ β k + ν kl = = β gy l + ( k l ) ( k l ) (7.71a) (7.71b) Version 015_11 0

21 Energy propagates at the group veloity, so from (7.71a and b) we find that energy always travels to the east relative to the basi flow and to either the south or the north depending on whether l is positive or negative. Referenes Gill, A.E., 198: Atmosphere-Oean Dynamis. International Geophysis series, Volume 30. Aademi Press, In. London, 66 pp. Gossard, E.E. and Hooke, W.H., 1975: Waves in the Atmosphere. Developments in Atmospheri Siene, vol.. Elsevier Sientifi Publishing Company, Amsterdam. Holton, J.R. and Hakim. G.J. 01: An introdution to Dynami Meteorology (5 th edition). Aademi Press, Amsterdam, 53 pp. Pihler, H., 1997: Dynamik der Atmosphäre, 3 aktualisierte Auflage. Spektrum Akademisher Verlag, Heidelberg, 57 pp. Vallis, G.K., 006: Atmospheri and oeani fluid dynamis, Cambridge University Press, 745 pp. Version 015_11 1

22 Appendix A. Making sense of Euler s equation Let us start 5 with the Taylor series, whih allows us to represent any funtion as an infinite sum of terms. If you want to know more about how a Taylor series is onstruted, then you will need to do some homework, but for our purposes the funtion e x an be represented as follows: =1+ 1! +! + 3! + 4! + 5! + Here x an represent any value, so we an substitute x with ix, where i = -1. Hene we get the following series: =1+ 1!! 3! + 4! + 5! + Next, we group the terms aording to whether or not they ontain i: = 1! + 4! + 1! 3! + 5! It is also possible to find a pair of Taylor series to represent the sine and osine funtions, whih leads to the following results: sin = 1! 3! + 5! 7! + os =1! + 4! 6! + Hene we an write e ix in terms of sin x and os x: =os + sin whih ompletes the derivation. For a graphial understanding of this, note that omplex numbers e.g. = + may also be represented in the omplex plane (Figure A.1). The omplex number A onsists of two parts: the real part (a in this ase) and the omplex part (b in this ase). In mathematial notation: = = 5 For this first part of the text the author greatly aknowledges Simon Singh from his book The Simpsons and their Mathematial Serets., Bloomsbury, London, 013. Version 015_11

23 All real numbers are loated on an infinite line strething from 0 to + to the right and from 0 to - to the left. This axis is alled the Real axis (Re in Figure A.1 left). The omplex part is loated similarly on the Imaginary axis (Im in Figure A.1 left). The omplex number A is now just a point in the omplex plane with oordinates a and b along, respetively, the Re-axis and the Im-axis. The distane (l) from the point A to the origin is then simply: = + Figure A.1 Definition sketh of the omplex number a+ib in the omplex plane (left) and the speial ase for all points at a distane 1 from the origin (right). For the angle (α) from the Re-axis, ounted positive ounter lokwise, we have: =tan =atan Now, for the omplex number e ix and Euler s equation we find that the oordinates along the Re-axis and the Im-axis are, respetively, os x and sin x (Figure A.1 right). For the distane to the origin we now have: = os + sin 1 This equation implies that for any value of x the point e ix is loated at a distane 1 from the origin i.e. it is loated on a irle with unit radius entered at the origin. For the angle (α) we find: =atan sin os =atantan = This means that the angle with the positive Re-axis is given by x (in radians). If x equals π we arrive at a very elegant equation ombining all five fundamental mathematial quantities: +1=0. Version 015_11 3

24 Appendix B. Inluding temperature perturbations In the ase that, next to perturbations in wind, pressure and density, also temperature perturbations are introdued, hene = + only the thermodynami energy equation (7.33) is involved and needs to be linearized again, leading possibly to an alternative expression of Eq. (7.34). We start by writing out (7.33): + = + Introdue basi state and perturbation: = ρ + + ρ + + ρ + + ρ If we now subtrat the basi state (i.e. the four terms indiated by b underneath) and all terms having two or more primes, and if we also apply the fat that whih implies:,,, = +,,, and,,, = +,,, =0 ; =0 ; =0 ; =0 the above equation is redued to: + +w = + ρ + + If we add the additional assumption that =0 The last term disappears and we are left with: + +w = + ρ + whih equals Eq. (7.34) and no longer ontains temperature perturbations. Version 015_11 4

25 Appendix C. Derivation of the thermodynami energy equation Start with eq. (7.31), with J = 0: + =0 = Next, differentiate the gas law = : + = = where the first equation has been used. Now, reorder terms giving: 1+ + =0 Use = + yielding + =0 The first term an be rearranged using resulting in or = = 1 = = = = whih is (7.33) and the equation of state (the gas law) has been applied in the last step. Version 015_11 5