Enhancement of wave transmissions in multiple radiative and convective zones
UUnder consideration for publication in J. Fluid Mech. Enhancement of wave transmissions inmultiple radiative and convective zones
Tao Cai † , Cong Yu ‡ and Xing Wei State Key Laboratory of Lunar and Planetary Sciences, Macau University of Science andTechnology, Macau, People’s Republic of China School of Physics and Astronomy, Sun Yat-sen University, Zhuhai, 519082, People’s Republicof China Department of Astronomy, Beijing Normal University, Beijing, People’s Republic of China(Received xx; revised xx; accepted xx)
In this paper, we study wave transmission in a rotating fluid with multiple alternatingconvectively stable and unstable layers. We have discussed wave transmissions in twodifferent circumstances: cases where the wave is propagative in each layer and caseswhere wave tunneling occurs. We find that efficient wave transmission can be achievedby ‘resonant propagation’ or ‘resonant tunneling’, even when stable layers are stronglystratified, and we call this phenomenon ‘enhanced wave transmission’. Enhanced wavetransmission only occurs when the total number of layers is odd (embedding stablelayers are alternatingly embedded within clamping convective layers, or vise versa). Forwave propagation, the occurrence of enhanced wave transmission requires that clampinglayers have similar properties, the thickness of each clamping layer is close to a multipleof the half wavelength of the corresponding propagative wave, and the total thicknessof embedded layers is close to a multiple of the half wavelength of the correspondingpropagating wave (resonant propagation). For wave tunneling, we have considered twocases: tunneling of gravity waves and tunneling of inertial waves. In both cases, efficienttunneling requires that clamping layers have similar properties, the thickness of eachembedded layer is much smaller than the corresponding e-folding decay distance, andthe thickness of each clamping layer is close to a multiple-and-a-half of half wavelength(resonant tunneling).
Key words:
Rotation; Stratified flow; Waves
1. Introduction
Inertial and gravito-inertial waves are important phenomena in rotating stars andplanets. Wave propagation can transport momentum and energy, therefore it may havesignificant impact on stellar or planetary structures and evolutions. For example, internal-gravity waves (IGWs) play an important role in transporting angular momentum whenthey propagate in the radiative zones of stars (Belkacem et al. a , b ; Pin¸con et al. et al. et al. et al. † Email address for correspondence: [email protected] ‡ Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] F e b can also be generated in rotating planets (Ogilvie & Lin 2004; Wu 2005 a ; Goodman &Lackner 2009). It has been found that the resonantly excited inertial wave has importantimpact on the tidal dissipation in planets (Wu 2005 b ).It is quite common for a star or planet to have a multi-layer structure. For exam-ple, superadiabatic region embedded in radiative layers may appear in neutron star’satmosphere because of the ionization of F e (Miralles et al. et al. et al. et al. N -barrier(low N layer embedded in high N layers and horizontal mean density varies contin-uously) and mixed- N (low N layer embedded in high N layers but horizontal meandensity varies discontinuously) profiles, where N is the square of buoyancy frequency.They found that wave transmission can be efficient by resonant transfers. Sutherland(2016) investigated the transmission of internal waves in a multi-layer structure sep-arated by discontinuous density jumps. He deduced an analytical solution for wavetransmission when the steps are evenly spaced, and predicted that waves with longerhorizontal wavelength and larger frequencies are more likely to transmit in the densitystaircase profile. Sutherland (1996) considered wave propagation in a profiles of piecewiselinear stratified layers with weaker stratification at the top. He discovered that large-amplitude IGWs incident from the bottom can partially transmit energy into the toplayer by the generation of lower frequency wave packet. Resonant tunneling of electrontransmission in double barriers is familiar in quantum physics, and has been widelyused in designing semiconductor devices, such as tunnel diode, NPN (negative-positive-negative) and PNP (positive-negative-positive) triodes (Singh 2010). In comparison withtunneling of electron transmission, it is expected that resonant tunneling also occurs forwave transmission in multi-layer structures.Wave transmission in a three-layer structure with rotational effects has been consideredby Gerkema & Exarchou (2008). They compared wave transmissions with and withouttraditional approximations (the horizontal component of rotation is neglected whentraditional approximation is adopted on a f -plane). For a three-layer structure with aconvective layer embedded in strongly stratified layers, waves cannot survive in bothconvective and stratified layers under the traditional approximation, while it is possibleif non-traditional effects are taken into account. They also showed that near-inertialwaves are always transmitted efficiently for stratified layers of any stratification. Belyaev et al. (2015) investigated the free modes of a multi-layer structure wave propagation withrotation at the poles and equator. They found that g-modes with vertical wavelengthssmaller than the layer thickness are evanescent. Andr´e et al. (2017) studied the effects ofrotation on free modes and wave transmission in a multi-layer structure at a generallatitude. They showed that transmission can be efficient when the incident wave isresonant with waves in adjacent layers with half-wavelengths equal to the layer depth.They also discovered that perfect wave transmission can be obtained at the criticallatitude. Pontin et al. (2020) studied the wave propagation in semiconvective regions ofnon-rotating giant planets in the full sphere. They found that wave transmissions areefficient for very large wavelength waves.In previous works (Wei 2020a,b; Cai et al. f -planes. Fora step stratification near the interface, we have found that the transmission generallyis not efficient if the stable layer is strongly stratified. In this paper, we investigate thewave transmission in a multi-layer structure on f -planes. Specifically, we will considerwave transmissions in two different mechanisms: wave propagation and wave tunneling.We find that wave transmission in a multi-layer structure can be significantly differentfrom that in a two-layer structure.
2. The model and result
For the Boussinesq flow in a rotating f -plane, the hydrodynamic equations can besynthesized into a partial differential equation on vertical velocity w (Gerkema & Shrira2005): ∇ w tt + ( f · ∇ ) w + N ( z ) ∇ h w = 0 , (2.1)where f = (0 , ˜ f , f ) is the vector of coriolis parameters, N is the square of the buoyancyfrequency, ∇ is Laplacian operator and ∇ h is its horizontal component, and the subscript t represents taking derivative with respect to time. By letting w = W ( χ, z ) exp( − iσt ),Gerkema & Shrira (2005) found (2.1) can be transformed into the following equation: AW χχ + 2 BW χz + CW zz = 0 , (2.2)where A = ˜ f s − σ + N , B = f ˜ f s , C = f − σ , ˜ f s = ˜ f sin α , and χ is a variablesatisfying x = χ cos α and y = χ sin α . Here x, y, z are the west-east, south-north, andvertical directions, respectively. We define A = ˜ f s − σ , so that A = A + N . Takingadvantage of plane waves and assuming W ( χ, z ) = ψ ( z ) exp( iδz + ikχ ), Gerkema & Shrira(2005) have further simplified (2.2) into an equation of wave amplitude ψ : ψ zz + k B − ACC ψ = 0 . (2.3)where δ = − kB/C . A wave solution then requires B − AC >
0, and the squaredwavenumber is r = k ( B − AC ) /C . If B − AC >
0, then r is a real number and the flowpropagates along the vertical direction as a wave. On the other hand, if B − AC <
0, then r is a pure imaginary number, and wave amplitude increases or decreases exponentially.In previous works (Wei 2020a; Cai et al. f -plane (a convective layer with N = 0 and a convectively stablelayer with N > N should be smaller than zero. In real stars,however, convection is generally efficient on transporting energy, leading to a nearlyadiabatic thermal structure. Thus N in convective layers are only slightly smaller thanzero. For this reason, we choose N = 0 for convective layers in our model. In this paper,we extend previous works to study the wave transmission among multi-layer setting f -plane. At this beginning stage, we use an ideal model by assuming N is a constant ineach layer. In all convective layers, N are equal and set to be zeros. In stable layers, N can be different but remains a constant in each layer, and its minimum and maximumvalues are N min and N max , respectively. We also assume that the propagation of inertialwaves is not affected by convection. The validity of this assumption requires that thenonlinear and viscous effects are small. Detailed discussion on relations and differencesbetween convection and inertial waves could be found in Zhang & Liao (2017). Here weattempt to use a toy model to gain some insights on wave transmissions in a multi-layerstructure.Cai et al. (2020) have made a detailed discussion on the frequency main of wavesolutions. If a wave could survive in a convective layer, then the following condition mustbe satisfied: ∆ c = B − A C > . (2.4)Similarly, condition for wave propagation in a stable layer is ∆ s = B − A C − N s C > . (2.5)Let us define σ , = (cid:20) ( f + ˜ f s ) ∓ (cid:113) ( f + ˜ f s ) (cid:21) / , (2.6) σ , = (cid:20) ( f + ˜ f s + N min ) ∓ (cid:113) ( f + ˜ f s + N min ) − N min f (cid:21) / , (2.7) σ , = (cid:20) ( f + ˜ f s + N max ) ∓ (cid:113) ( f + ˜ f s + N max ) − N max f (cid:21) / . (2.8)where σ = 0 and σ = f + ˜ f s are roots of ∆ c = 0, and σ , and σ , are roots of ∆ s = 0 when N s = N min and N s = N max , respectively. It is not difficult to verify thatthe relation σ (cid:54) σ (cid:54) σ (cid:54) σ (cid:54) σ (cid:54) σ holds.Fig.1(a) shows the frequency ranges for different waves in convectively unstable andstable layers, respectively. In the green region ( σ < σ < σ ), waves can survive andpropagate in both convective and stable layers, and we term this phenomenon ‘wavepropagation’. In the blue region σ < σ < σ , inertial waves can survive in convectivelayers but gravity waves cannot survive in stable layers. Inertial waves can transmitthrough a tunneling process, and we term this phenomena ‘tunneling of inertial wave’.In the purple region σ < σ < σ , gravity waves can survive in stable layers but inertialwaves cannot survive in convective layers. Similarly, gravity waves can transmit througha tunneling process, and we term this phenomena ‘tunneling of gravity wave’.In Cai et al. (2020), we have also deduced that c p c g has the same sign as C , where c p is the modified vertical component of wave phase velocity and c g is the vertical componentof the wave group velocity. The vertical component of phase velocity should be computedby σ/ ( δ ± r ), but here the tilted effect is excluded in the modified one c p = σ/ ( ± r ).Since the wave direction of energy propagation is determined by c g , a proper choice ofwave direction depends on whether the wave is sub-inertial ( C >
0) or super-inertial(
C <
Figure 1: Sketch plot of wave propagations in multiple convectively stable andunstable layers. Wave frequency ranges are shown above and below the middlearrow for convective and stable layers, respectively. Wave propagation occurs inthe green region, tunneling of inertial wave occurs in the blue region, andtunneling of gravity wave occurs in the purple region.(a) (b)Figure 2: Sketch plots of wave propagations in multiple convectively stable andunstable layers. (a)The convective layer is embedded between two stable layers.(b)The stable layer is embedded between two convective layers. Structure of thesquare of buoyancy frequency N is shown by the side. Wave propagation
In this section, we discuss wave propagation in both convective and stable layers, whichrequires the frequency is in the range σ < σ < σ . We consider different configurationswith different combinations of layer structures and wave directions. The incident wave canpropagate from the convective layer or stable layer, and the wave propagating directioncan be upward or downward. Since up/down symmetry holds in Boussinesq flow, it issufficient to discuss the cases with waves incident from the bottom. The cases with wavesincident from the top can be inferred from the up/down symmetry. Fig. 2 shows sketchplots of the two configurations in a three-layer setting f -plane. The structures with morelayers are similar. At each interface, two boundary conditions have to be satisfied: thevertical velocity is continuous; and the first derivative of the vertical velocity is continuous(Wei 2020a; Cai et al. m + 1 / m +1 / m -th and ( m + 1)-th stable layers, respectively; and we set the locations of the lowerand upper interfaces at z m and z (cid:48) m , respectively. The thickness of the convective layer is ∆z (cid:48) m = z (cid:48) m − z m , and the thickness of the stable layer is ∆z m = z m − z (cid:48) m − . For the m -thstable layer, we define its wavenumber square as s m = k ( B − A C − N m C ) /C . For the( m +1 / q m = k ( B − A C ) /C .In this paper, we only consider a simple case, in which the square of buoyancy frequencyis a constant within each convective or stable layer. Under this assumption, we find wavesolutions in the m -th stable layer is ψ m ( z ) = a m e − is m z + b m e is m z , z ∈ ( z (cid:48) m − , z m ) , (2.9)and in the ( m + 1 / ψ m +1 / ( z ) = c m e − iq m z + d m e − iq m z , z ∈ ( z m , z (cid:48) m ) , (2.10)respectively. As mentioned earlier, c p c g has the same sign as C, from which we conclude Sgn ( c g ) = Sgn ( C ) Sgn ( c p ). The sign of the modified vertical phase velocity c p is deter-mined by the sign of q m or s m . If Sgn ( C ) Sgn ( c p ) >
0, then the vertical group velocity c g is positive (wave direction is outgoing); on the other hand, if Sgn ( C ) Sgn ( c p ) < c g is negative (wave direction is incoming). We choose q m > C >
0, and q m < C <
0. For either case, the waves with wavenumbers q m and s m are outgoing waves,and the waves with wavenumbers − q m and − s m are incoming waves. Note that all q m and s m always have the same sign. Matching the boundary conditions at the interfaces z m and z (cid:48) m , we have a m e − is m z m + b m e is m z m = c m e − iq m z m + d m e iq m z m , (2.11)( − a m e − is m z m + b m e is m z m ) s m = ( − c m e − iq m z m + d m e iq m z m ) q m , (2.12) c m e − iq m z (cid:48) m + d m e iq m z (cid:48) m = a m +1 e − is m +1 z (cid:48) m + b m +1 e is m +1 z (cid:48) m , (2.13)( − c m e − iq m z (cid:48) m + d m e iq m z (cid:48) m ) q m = ( − a m +1 e − is m +1 z (cid:48) m + b m +1 e is m +1 z (cid:48) m ) s m +1 , (2.14)Wave propagations in multi-layer structures have been investigated in Belyaev et al. (2015), Andr´e et al. (2017) and Pontin et al. (2020). A useful approach on modelingwave propagations in a multi-layer structure is to build on relations of wave amplitudesby transfer matrices. From the above boundary conditions, we can derive the followingtransfer relations (see Appendix A): (cid:20) a m b m (cid:21) = T m,m +1 (cid:20) a m +1 b m +1 (cid:21) = (cid:34) (cid:98) T (cid:98) T (cid:98) T (cid:98) T (cid:35) (cid:20) a m +1 b m +1 (cid:21) , (2.15)where (cid:98) T = 14 e i ( s m z m − s m +1 z (cid:48) m ) [(1 + q m s m + s m +1 s m + s m +1 q m ) e iq m ∆z (cid:48) m (2.16)+(1 − q m s m + s m +1 s m − s m +1 q m ) e − iq m ∆z (cid:48) m ] , (cid:98) T = 14 e i ( s m z m + s m +1 z (cid:48) m ) [(1 + q m s m − s m +1 s m − s m +1 q m ) e iq m ∆z (cid:48) m (2.17)+(1 − q m s m − s m +1 s m + s m +1 q m ) e − iq m ∆z (cid:48) m ] , with (cid:98) T = (cid:98) T ∗ and (cid:98) T = (cid:98) T ∗ . Here the asterisk symbol represents the conjugate of acomplex number.2.1.1. The three-layer case
If we consider the special three-layer case (outgoing transmitted wave requires a = 0),we can quickly obtain (cid:20) a b (cid:21) = (cid:20) T T T T (cid:21) (cid:20) b (cid:21) , (2.18)where (cid:20) T T T T (cid:21) = T , , (2.19)Thus we have a = T b , (2.20) a = T b . (2.21)In Cai et al. (2020), we found that the averaged energy flux of a wave is (cid:104) F (cid:105) = C (cid:61) ( ψ z /ψ )2 k σ | ψ | , (2.22)where (cid:61) denotes the imaginary part of a complex number. Let us define the overalltransmission ratio as η = |(cid:104) F (cid:105) t ||(cid:104) F (cid:105) i | (2.23)where (cid:104) F (cid:105) i is the averaged energy flux of the incident wave at the lowermost interface,and (cid:104) F (cid:105) t is the averaged energy flux of the transmitted wave at the uppermost interface.For this three-layer case, we have η = | s s | | b || a | = | s s | | T | . (2.24)From (2.16) and the relation between T and T , we obtain | T | = 14 [(1 + s s ) cos ( q ∆z (cid:48) ) + ( q s + s q ) sin ( q ∆z (cid:48) )]= 14 (1 + s s ) cos ( q ∆z (cid:48) ) + 14 ( q s + s q ) sin ( q ∆z (cid:48) ) . (2.25)Therefore, the overall transmission ratio is η = (cid:18)(cid:114) s s + (cid:114) s s (cid:19) cos ( q ∆z (cid:48) ) + 14 (cid:115) q s s + (cid:114) s s q sin ( q ∆z (cid:48) ) − . (2.26)Note that the terms inside square roots are always positive, no matter what directionof the propagating wave is. It is easy to prove ( (cid:112) s /s + (cid:112) s /s ) / (cid:62)
1, and so asthe term before sin ( q ∆z (cid:48) ). Thus we can show η (cid:54) [cos ( q ∆z (cid:48) ) + sin ( q ∆z (cid:48) )] − = 1,which means that η is always smaller than or equal to 1. Gerkema & Exarchou (2008) hasobtained similar formula in their study on internal-wave transmission in weakly stratifiedlayers. Their equation (38) is a special case of our formula (2.26) with s = s . Someinteresting conclusions can readily be drawn from (2.26). In the previous investigation of atwo-layer structure (Wei 2020a; Cai et al. N / (2 Ω ) (cid:29) | q | ∆z (cid:48) → (cid:96)π ( (cid:96) is a positive integer number)and s /s →
1, we see that η →
1. For this case, there is no reflection and all of theincident wave is transmitted. This result is independent of N / (2 Ω ) , and it holds forboth weakly ( N / (2 Ω ) (cid:28)
1) and strongly ( N / (2 Ω ) (cid:29)
1) stratified rotating fluids. Tobetter understand the behavior of η , we separate η into two parts: the first part is thesolution at sin ( q ∆z (cid:48) ) = 0 and the second part is the solution at cos ( q ∆z (cid:48) ) = 0. Thetransmission ratio η of the general case is a weighted harmonic mean of η and η .For the first part, the condition sin ( q ∆z (cid:48) ) = 0 is equivalent to ∆z (cid:48) = (cid:96)π/ | q | . Inother words, it requires that the thickness of the middle convective layer is a multiple ofthe half wavelength of the propagating wave. In such case, the overall transmission ratiois η = (cid:34) (cid:18)(cid:114) s s + (cid:114) s s (cid:19) (cid:35) − . (2.27)From this equation, we see that η only depends on the wavesnumber ratio s /s of thestable layers. η decreases with s /s when s /s >
1, and increases with s /s when s /s <
1. The maximum value η = 1 is achieved at s /s = 1. Thus the transmissionis efficient when | N − N | / (2 Ω ) is small, or when the wave is at a critical colatitude θ c = cos − ± σ/ (2 Ω ) (because C → θ → θ c ), or when both stable layers are weaklystratified (because N , C (cid:28) B − A C ). The latter two points have also been observedin the two-layer structure (Cai et al. et al. (2017). Efficientwave transmission at critical colatitude and weakly stratified flow can be explained by acommon reason: the inertial and gravity waves separated by an interface have almost thesame vertical wavelengths, and thus these waves are ‘resonant’ at the interface. Similarreason can be used to explain the enhanced transmission when the degree of stratificationsin both clamping layers (in configuration 1, the embedding convective layer is embeddedwithin two neighboring clamping stable layers) are similar: the incident wave is ‘resonant’with waves in adjacent layers with wavelengths equal to free modes of the multi-layerstructure (Andr´e et al. η at three different combinations of N and N . In the uppermost panel (figs. 3(a-c)), both stable layers are strongly stratified, but N / (2 Ω ) is equal to N / (2 Ω ) . It clearly shows that the transmission is enhanced. Themiddle panel (figs. 3(d-f)) shows the result of the cases with one weakly and one stronglystratified layers. Apparently, the transmission is not as efficient as the cases shown in theuppermost panel. The lowermost panel (figs. 3(g-i)) presents the result of the cases withtwo weakly stratified stable layers. The transmission is efficient because the rotationaleffect is important. In previous study on wave transmission in a two-layer structure (Cai et al. α has important effecton the frequency range. Frequency range increases with increasing sin α . When stablelayers (or any of them) are strongly stratified, wave can only survive in a very thin regionif sin α is small. In the extreme case sin α = 0, the surviving frequency range vanishes.For the second part, the condition cos ( q ∆z (cid:48) ) = 0 is equivalent to ∆z (cid:48) = ( (cid:96) +1 / π/ | q | , which requires that the thickness of the middle convective layer is a multiple-and-a-half of the half wavelength of the propagating wave. In such case, the overalltransmission ratio is η = (cid:115) q s s + (cid:114) s s q − . (2.28)Similarly, η only depends on the wavenumber ratio s s /q . It decreases with s s /q when s s /q > s s /q when s s /q <
1. Let us consider two N /(2 ) =10 & N /(2 ) =10 & sin =0.1 cos / ( ) (a) N /(2 ) =10 & N /(2 ) =10 & sin =0.5 cos / ( ) (b) N /(2 ) =10 & N /(2 ) =10 & sin =1.0 cos / ( ) (c) N /(2 ) =0.1 & N /(2 ) =10 & sin =0.1 cos / ( ) (d) N /(2 ) =0.1 & N /(2 ) =10 & sin =0.5 cos / ( ) (e) N /(2 ) =0.1 & N /(2 ) =10 & sin =1.0 cos / ( ) (f) N /(2 ) =0.1 & N /(2 ) =0.1 & sin =0.1 cos / ( ) (g) N /(2 ) =0.1 & N /(2 ) =0.1 & sin =0.5 cos / ( ) (h) N /(2 ) =0.1 & N /(2 ) =0.1 & sin =1.0 cos / ( ) (i)Figure 3: Transmission coefficient η (the first part of (2.26)) for different N , / (2 Ω ) and sin α in a three-layer structure (one middle convective layerand two upper and lower neighbouring stable layers). The horizontal axis iscos θ , and the vertical axis is σ / (2 Ω ) . From the left to the right panels,sin α increases from 0.1 to 1.0. (a)-(c) Both stable layers are strongly stratified.(d)-(f) One stable layers are strongly stratified and the other is weaklystratified. (g)-(i) Both stable layers are weakly stratified. Wave propagation canonly occur in colored regions. Regions are left white if wave propagation isprohibited. cases C >
C <
0. For the first case we have | s , | < | q | , while for the second casewe have | s , | > | q | . Thus for C >
0, we obtain s s /q < η always increaseswith s s /q . While for the other case C <
0, we obtain s s /q > η alwaysdecreases with s s /q . Efficient transmission can occur if s s /q →
1, which basicallyrequires both N C/ ( B − A C ) and N C/ ( B − A C ) to be small. It indicates thatthe transmission ratio will decrease if both N / (2 Ω ) and N / (2 Ω ) increase, no matterwhat the sign of C is. Fig. 4 gives an example on such case. Apparently, it can be seen thatthe transmission ratio decreases when the stable layers are varied from weakly stratified(the lowermost panel) to strongly stratified (the uppermost panel). Also apparent is thattransmission is efficient near the critical colatitudes, where N , C/ ( B − A C ) → m + 1 /
2, and0 N /(2 ) =10 & N /(2 ) =10 & sin =0.1 cos / ( ) (a) N /(2 ) =10 & N /(2 ) =10 & sin =0.5 cos / ( ) (b) N /(2 ) =10 & N /(2 ) =10 & sin =1.0 cos / ( ) (c) N /(2 ) =0.1 & N /(2 ) =10 & sin =0.1 cos / ( ) (d) N /(2 ) =0.1 & N /(2 ) =10 & sin =0.5 cos / ( ) (e) N /(2 ) =0.1 & N /(2 ) =10 & sin =1.0 cos / ( ) (f) N /(2 ) =0.1 & N /(2 ) =0.1 & sin =0.1 cos / ( ) (g) N /(2 ) =0.1 & N /(2 ) =0.1 & sin =0.5 cos / ( ) (h) N /(2 ) =0.1 & N /(2 ) =0.1 & sin =1.0 cos / ( ) (i)Figure 4: Transmission coefficient η (the second part of (2.26)) for different N , / (2 Ω ) and sin α in a three-layer structure (one middle convective layerand two upper and lower neighbouring stable layers). The horizontal axis iscos θ , and the vertical axis is σ / (2 Ω ) . From the left to the right panels,sin α increases from 0.1 to 1.0. (a)-(c) Both stable layers are stronglystratified. (d)-(f) One stable layers are strongly stratified and the other isweakly stratified. (g)-(i) Both stable layers are weakly stratified. Wavepropagation can only occur in colored regions. Regions are left white if wavepropagation is prohibited. the neighbouring convective layers as m and m + 1. The neighbouring upper and lowerinterfaces of the stable layer m + 1 / z m and z (cid:48) m . By this setting, the transmissionratio can be deduced by simply interchanging q m with s m in (2.26). Therefore we obtainthe overall transmission ratio in configuration 2 is η = (cid:18)(cid:114) q q + (cid:114) q q (cid:19) cos ( s ∆z (cid:48) + 14 (cid:115) s q q + (cid:114) q q s sin ( s ∆z (cid:48) ) − . (2.29)The wavenumbers in the convective layers are all the same, thus we have η = (cid:34) cos ( s ∆z (cid:48) ) + 14 (cid:18) | s q | + | q s | (cid:19) sin ( s ∆z (cid:48) ) (cid:35) − . (2.30)1Again, we can show that wave transmission is enhanced when the thickness of the middlestable layer is a multiple of the half wavelength of the propagating wave.2.1.2. The Multiple-layer case
Now we consider the structure with more alternating layers. Here we first discuss thecase of 2 M + 1 alternating layers ( M + 1 stable layers and M convective layers), andboth the lowermost and uppermost layers are stable. From the recursive relation (2.15),we have (cid:20) a b (cid:21) = M (cid:89) m =1 T m,m +1 (cid:20) a M +1 b M +1 (cid:21) = (cid:20) T T T T (cid:21) (cid:20) a M +1 b M +1 (cid:21) . (2.31)Let a M +1 = 0, we obtain that the wave amplitude of the transmitted wave in theuppermost layer is b M +1 = T − b , and the transmission ratio is η = | s M +1 s | | T | . (2.32)Now we discuss the case of 2 M alternating layers ( M stable layers and M convectivelayers), and the lowermost layer is stable and the uppermost layer is convective. Using therecursive relation (2.15) and combining it with the boundary conditions at the uppermostinterface, we have (cid:20) a b (cid:21) = (cid:32) M − (cid:89) m =1 T m,m +1 (cid:33) S − M Λ M,M Q M,M (cid:20) c M d M (cid:21) = (cid:20) T (cid:48) T (cid:48) T (cid:48) T (cid:48) (cid:21) (cid:20) c M d M (cid:21) , (2.33)where T (cid:48) ij , i, j ∈ { , } is defined by the matrix multiplications shown in the middle of(2.33). Let c M = 0, we obtain that the wave amplitude of the transmitted wave in theuppermost layer is d M +1 = T (cid:48) b , and the transmission ratio is η = | q M s | | T (cid:48) | . (2.34)Again, following similar procedures, we can deduce the transmission ratio of configuration2 by interchanging q m with s m .We have shown that the transmission can be enhanced in the three-layer structure ofconfiguration 1. Now we investigate whether the enhancement occurs in a structure withmore layers. For the sake of simplicity, we assume that all convective layers have the samethickness ∆z c and wavenumber q , and all stable layers have the same thickness ∆z s andwavenumber s .We first discuss the case when the number of layers (2 M + 1) is odd. When s m = s , q m = q , and ∆z (cid:48) m = ∆z c , the transfer matrix T m,m +1 = T is (cid:98) T = 14 e − is∆z c (cid:26) [2 + ( qs + sq )] e iq∆z c + [2 − ( qs + sq )] e − iq∆z c (cid:27) , (2.35) (cid:98) T = 14 e i ( sz m + sz (cid:48) m ) (cid:20) ( qs − sq ) e iq∆z c − ( qs − sq ) e − iq∆z c (cid:21) , (2.36)with (cid:98) T = (cid:98) T ∗ and (cid:98) T = (cid:98) T ∗ . The eigenvalue satisfies the following equation( (cid:98) T − λ )( (cid:98) T − λ ) = (cid:98) T (cid:98) T . (2.37)After some manipulations, the equation can be written as λ − (cid:60) ( (cid:98) T ) λ + 1 = 0 , (2.38)2or in an explicit form λ − (cid:20) cos( q∆z c ) cos( s∆z c ) + 12 ( qs + sq ) sin( q∆z c ) sin( s∆z c ) (cid:21) λ + 1 = 0 , (2.39)where (cid:60) denotes the real part of a complex number. Let λ , be the two roots of theequation. Obviously we have λ λ = 1. Let ∆ λ = (cid:20) cos( q∆z c ) cos( s∆z c ) + 12 ( qs + sq ) sin( q∆z c ) sin( s∆z c ) (cid:21) − (cid:26) cos[( q − s ) ∆z c ] + 12 ( qs + sq −
2) sin( q∆z c ) sin( s∆z c ) (cid:27) − . (2.41)Then the eigenvalues λ , are real when ∆ λ >
0, and are complex when ∆ λ <
0. If λ , are real and λ (cid:54) = λ , then the maximum of | λ , | must be greater than one. For a multi-layer structure, the transfer matrix T M ∝ max( | λ , | ) M , which yields η ∝ max( | λ , | ) − M .Since max( | λ , | ) is greater than one, the transmission ratio decays with the number oflayers.To ensure that the transmission does not decay, the solution λ , must be on the unitcircle of the complex plane. This condition can be achieved when λ = λ = ±
1, or λ , are a complex pair. Therefore a necessary condition (not sufficient) for efficient wavetransmission is ∆ λ (cid:54) ∆ λ (cid:54) | T | , which could possibly be much greaterthan one even though the eigenvalues λ , are on the unit circle. Here we take a furtherstep to discuss when the value | T | will be close to one, so as to ensure an efficient wavetransmission.Let us further define z = 0 at the lowest interface, and α = exp( − is∆z s ) and α =exp( − is∆z c ). With such definitions, we have z m + z (cid:48) m = (2 m − ∆z s + (2 m − ∆z c andexp[ − is ( z m + z (cid:48) m )] = α m − α m − . The transfer matrix can be rewritten as T m,m +1 = (cid:34) α (cid:101) T α ∗ m − α ∗ m − (cid:101) T α m − α m − (cid:101) T ∗ α ∗ (cid:101) T ∗ (cid:35) , (2.42)where (cid:101) T = 14 (cid:26) [2 + ( qs + sq )] e iq∆z c + [2 − ( qs + sq )] e − iq∆z c (cid:27) , (2.43) (cid:101) T = 14 (cid:20) ( qs − sq ) e iq∆z c − ( qs − sq ) e − iq∆z c (cid:21) . (2.44)When m = 1, we note that the transfer matrix T , can be formulated as T , = (cid:34) α (cid:101) T α ∗ (cid:101) T α (cid:101) T ∗ α ∗ (cid:101) T ∗ (cid:35) = A (cid:20) α α ∗ (cid:21) , (2.45)where A = (cid:34) (cid:101) T (cid:101) T (cid:101) T ∗ (cid:101) T ∗ (cid:35) . (2.46)Here we consider a special case with α = α ∗ = 1, which can be achieved by letting | s | ∆z s = (cid:96) (cid:48) π , where (cid:96) (cid:48) is a non-negative integer. For this special case, it can be proved3that M (cid:89) m =1 T m,m +1 = A M (cid:20) α M α ∗ M (cid:21) . (2.47)Now we try to derive the explicit form of A M . It is obvious that (cid:101) T is a pure imaginarynumber, thus we can write A as A = (cid:34) (cid:60) ( (cid:101) T ) + i (cid:61) ( (cid:101) T ) i (cid:61) ( (cid:101) T ) − i (cid:61) ( (cid:101) T ) (cid:60) ( (cid:101) T ) − i (cid:61) ( (cid:101) T ) (cid:35) = cos( q∆z c ) I + i U , (2.48)where U = (cid:34) (cid:61) ( (cid:101) T ) (cid:61) ( (cid:101) T ) −(cid:61) ( (cid:101) T ) −(cid:61) ( (cid:101) T ) (cid:35) , (2.49)and it is easy to verify U = sin ( q∆z c ) I . (2.50)If sin( q∆z c ) = 0, we can show A = cos( q∆z c ) I and A M = cos M ( q∆z c ) I . If sin( q∆z c ) (cid:54) =0, then we obtain A M = (cos( q∆z c ) I + i U ) M (2.51)= I (cid:88) k ∈ even C Mk cos M − k ( q∆z c ) i k sin k ( q∆z c )+ U [sin( q∆z c )] − (cid:88) k ∈ odd C Mk cos M − k ( q∆z c ) i k sin k ( q∆z c ) (2.52)= cos( M q∆z c ) I + i sin( M q∆z c )[sin( q∆z c )] − U , (2.53)where C Mk = M ! / (( M − k )! k !) is the combination function. From the above calculation,the analytical solution of transmission ratio can be obtained. If sin( q∆z c ) = 0, we have T = α ∗ M cos M ( q∆z c ) , (2.54)and the transmission ratio is η = 1 . (2.55)If sin( q∆z c ) (cid:54) = 0, we have T = α ∗ M (cid:20) cos( M q∆z c ) − i
12 ( qs + sq ) sin( M q∆z c ) (cid:21) , (2.56)and the transmission ratio is η = (cid:20) cos ( M q∆z c ) + 14 ( qs + sq ) sin ( M q∆z c ) (cid:21) − . (2.57)(2.55) can be synthesized into (2.57), since sin( q∆z c ) = 0 implies sin ( M q∆z c ) = 0and cos ( M q∆z c ) = 1. Therefore we conclude that, under the condition | s | ∆z s = (cid:96) (cid:48) π ,the wave transmission ratio can be described by (2.57). Comparing with the result ofthree-layer structure case, we see that (2.26) is just a special case of (2.57) when M = 1.The discussion on efficiency of wave transmission based on (2.57) is similar to that inthree-layer structure case, and here we will not repeat it. From the analytical solution,it is clear that wave will be totally transmitted when M | q | ∆z c = (cid:96)π and | s | ∆z s = (cid:96) (cid:48) π .Analytical solution on the general cases of | s | ∆z s (cid:54) = (cid:96) (cid:48) π is more difficult, but some4insights can be provided from the discussion of eigenvalues of the transmission matrix.It is worth mentioning for the special case when sin q∆z c → s∆z c →
0, theeigenvalues | λ , | →
1. In this limit, it can be shown that ∆ λ ∼ [ mod (( q − s ) ∆z c , π )] > , (2.58)where mod is the modulo function. The eigenvalues are real numbers, and one of | λ , | is slightly greater than 1. The transmission decays slowly as the wave crosses each layer.The wave transmission can be efficient when the number of layers is not too large. Theeigenvalues can be estimated as λ , = (cid:112) ∆ λ ± (cid:112) ∆ λ ∼ ± [ mod (( q − s ) ∆z c , π )] √ . (2.59)Thus the decay rate of transmission ratio is approximately λ − M ∼ { − − / M [ mod (( q − s ) ∆z c , π )] } − . The transmission can be efficient when M (cid:28) M c = [ mod (( q − s ) ∆z c , π )] − . When mod (( q − s ) ∆z c , π ) →
0, the critical value M c is very large. Asa result, the transmission can be approximately efficient in this case. From the abovediscussion, we infer that the transmission can be efficient when sin( q∆z c ) (cid:28) s∆z c ) (cid:28)
1. This conclusion is useful when embedded convective layers are very thin.When the total number of layers is even (2 M ), we can consider it as (2 M −
1) layersplus an addition layer. The best scenario on transmission for (2 M −
1) layers is that theincident wave is totally transmitted to the (2 M − M − M )-th layer, and it can be considered as a two-layerproblem. For a two-layer problem, wave transmission is generally not efficient when thestable layer is strongly stratified (Wei 2020a; Cai et al. N m / (2 Ω ) = 10,sin α = 1 and the lowermost layer is stable. Thus in all cases, the stable layers arestrongly stratified. It clearly shows that the transmission is enhanced in fig. 5(a) and(c), where the total number of layers is odd and ∆z c and ∆z s satisfy the conditions M | q | ∆z c = (cid:96)π and | s | ∆z s = (cid:96) (cid:48) π . The transmission in figs. 5(b) is not enhanced because M | q | ∆z c (cid:54) = (cid:96)π . For the case of 102-layer structure (figs. 5(d) and (f)), the transmissionis not enhanced even when the above conditions are satisfied. For these two cases, waveis almost totally transmitted from the 1st layer to 101st layer (see figs. 5(a) and (c)).Wave transmission from the 101st layer to 102nd layer can be viewed as a two-layerproblem, and it is generally not efficient when the stable layer is strongly stratified.Therefore, if the stable layer is strongly stratified, the enhancement of transmission onlytakes place when the number of alternating layers is odd. In other words, enhanced wavetransmission only occurs in a multi-layer structure with stable layers embedded withinconvective layers, or convective layers embedded within stable layers. The enhancementof transmission also depends on the thicknesses of embedded layers ∆z s . Figs. 5(g-i)present results of three cases with | s | ∆z s = 0 . π and different values of | q | ∆z c . It can beseen that the transmission of the case | q | ∆z c = π is enhanced, while the transmission ofthe case | q | ∆z c = 0 . π is only partially enhanced.The result obtained in the configuration 1 is also true in the configuration 2. Here wedo not repeat the discussion on configuration 2.Andr´e et al. (2017) have also observed that wave transmission can be enhanced ina multi-layer structure. They provided a physical explanation for the enhancement inwave transmission when the incident wave is resonant with waves in adjacent layers with5 cos / ( ) (a) cos / ( ) (b) cos / ( ) (c) cos / ( ) (d) cos / ( ) (e) cos / ( ) (f) cos / ( ) (g) cos / ( ) (h) (i)Figure 5: Transmission ratios in multi-layer structures. For all the cases, N m / (2 Ω ) = 10 and sin α = 1, and the lowermost layer is stable. ∆z s = π/ | s | for (a-f) but ∆z s = 0 . π/ | s | for (g-i). In each case, all the convective layers havethe same thickness ∆z c and wavenumber q , and all the stable layers have thesame thickness ∆z s and wave number s . (a-c)The transmission ratio for a101-layer structure with ∆z c = (1 , . , . π/ | q | . (d-f) The transmission ratiofor a 102-layer structure with ∆z c = (1 , . , . π/ | q | . (g-i) Similar to (a-c) butwith different ∆z s . Wave propagation can only occur in colored regions.Regions are left white if wave propagation is prohibited. half-wavelengths equal to the layer depth. Our analysis verifies this phenomenon from amathematical point of view. 2.2. Wave tunneling
In the previous section, we have considered wave transmissions in multiple convectiveand radiative (stable) layers when the condition B − AC > (a) (b)Figure 6: Two configurations of tunneling of gravity and inertial waves.(a)Tunneling of gravity wave. Both top and bottom layers are stable. Gravitywave can propagate in the stable layer, but wave cannot propagate in theconvective layer. (b)Tunneling of inertial wave. Both top and bottom layers areconvective. Gravity wave can propagate in the convective layer, but wavecannot propagate in the stable layer. structures in fig. 6, although wave cannot propagate in the whole domain, it still cantransmit through a tunneling process (Mihalas & Mihalas 2013; Sutherland & Yewchuk2004). In the followings, we will discuss the tunneling of gravity and inertial waves,respectively.2.2.1.
Tunneling of gravity waves
For configuration 3 in fig. 6(a), wave can propagate in the stable layer but cannotpropagate in the convective layer. Thus the wave frequency must be in the range σ <σ < σ . The width of the frequency domain can be written as σ − σ = 12 [( N min − f − ˜ f s ) + (cid:113) ( N min − f − ˜ f s ) + 4 N min ˜ f s ] . (2.60)For the sake of convenience, we only discuss waves at the northern atmosphere ( θ ∈ [0 , π/ σ − σ always increases with θ , sin α , and N min / (2 Ω ) (see Appendix B). Therefore, the frequency domain is wider at equatorial regions thanpolar regions. Also, it is wider when the meridional wavenumber dominates the zonalwavenumber, and it is wider when the degree of stratification is stronger.Now we discuss the wave tunneling in configuration 3. We use the same setting as thatdiscussed in wave propagation. The derivation of wave transmission of tunneling is similarto that of wave propagation. The only difference is that now wave cannot propagate inthe convective layer, and thus q m = i ˆ q m is pure imaginary number. Then the transferrelation from m -th layer to ( m + 1)-th layer can be easily obtained by replacing q m with i ˆ q m in (2.15).Replacing q m with i ˆ q m in (2.15), we can obtain the transfer matrix. In this config-uration, we always have C = f − σ < σ > f ). Hence for an outgoingtransmitted wave, we have b M +1 = 0 (modified phase velocity has an opposite sign asthe group velocity). Again, let us first discuss the transmission ratio for a three-layerstructure. In such case, we have a = T a and T = 14 e − i ( s z − s z (cid:48) ) [(1 + i ˆ q s + s s + s i ˆ q ) e − ˆ q ∆z (cid:48) + (1 − i ˆ q s + s s − s i ˆ q ) e ˆ q ∆z (cid:48) ] . (2.61)Thus the wave transmission ratio is η = | s s | | T | (2.62)7= 16 ( (cid:114) s s + (cid:114) s s ) ( e − ˆ q ∆z (cid:48) + e ˆ q ∆z (cid:48) ) + ( (cid:115) ˆ q s s − (cid:114) s s ˆ q ) ( e − ˆ q ∆z (cid:48) − e ˆ q ∆z (cid:48) ) − (2.63)= 16 ( (cid:114) s s + (cid:114) s s ) ( e − ˆ q ∆z (cid:48) + e ˆ q ∆z (cid:48) ) + ( (cid:115) ˆ q s s + (cid:114) s s ˆ q ) − (cid:104) ( e − ˆ q ∆z (cid:48) + e ˆ q ∆z (cid:48) ) − (cid:105) − (2.64)Obviously, η depends on the values of s /s , ˆ q / ( s s ), and | ˆ q | ∆z (cid:48) . It increases with s /s when s /s <
1, and decreases with s /s when s /s >
1. Similarly, we seethat it increases with ˆ q / ( s s ) when ˆ q / ( s s ) <
1, and decreases with ˆ q / ( s s ) whenˆ q / ( s s ) >
1. It is also noted that it decreases with | ˆ q | ∆z (cid:48) . Thus efficient transmissionratio can be achieved when s → s , ˆ q → s s , and | ˆ q | ∆z (cid:48) →
0. The conditionˆ q → s s can be relaxed if | ˆ q | ∆z (cid:48) →
0. Therefore, efficient transmission requires thatthe wavenumbers ( s and s ) in the stable layers are similar in magnitudes, and thethickness of the convective layer ( ∆z (cid:48) ) is much smaller than the e-folding decay distance(1 / | ˆ q | ). Fig. 7 shows the contour plots of the transmission ratios for different N min and ∆z (cid:48) . In all the cases, we set s = s and sin α = 1. First, the figure clearly showsthat the frequency domain increases with N min and θ . It is consistent with the previousanalysis on the width of the frequency domain. Second, we see that the transmissionratio is mainly affected by the thickness of the convective layer ∆z (cid:48) . The shallower theconvective layer is, the higher the transmission ratio is. We also note that the transmissionratio is insensitive to N min . The effect of degree of stratification on the transmission ratiois insignificant.Now we consider the wave transmission in configuration 3 with more layers. Again,we assume that all of the stable layers have the same degree of stratification ( N ) andthickness ( ∆z s ). Similarly, we assume that all of the convective layers have the samethickness ∆z c . By these settings, s m = s and q m = i ˆ q are constants in the stable andconvective layers, respectively. When s m = s , q m = i ˆ q , and ∆z (cid:48) m = ∆z c , the transfermatrix T m,m +1 in (2.15) can be written as (cid:98) T = 14 e − is∆z c (cid:26)(cid:20) i ( ˆ qs − s ˆ q ) (cid:21) e − ˆ q∆z c + (cid:20) − i ( ˆ qs − s ˆ q ) (cid:21) e ˆ q∆z c (cid:27) , (2.65) (cid:98) T = 14 e is ( z m + z (cid:48) m ) (cid:26)(cid:20) ( ˆ qs + s ˆ q ) (cid:21) e − ˆ q∆z c − (cid:20) ( ˆ qs + s ˆ q ) (cid:21) e ˆ q∆z c (cid:27) , (2.66)with (cid:98) T ∗ = (cid:98) T and (cid:98) T = (cid:98) T ∗ . The eigenvalues of T m,m +1 are the roots of the followingequation λ − (cid:60) ( (cid:98) T ) λ + 1 = 0 . (2.67)Similarly, we can obtain a sufficient condition for efficient wave tunneling if ∆ λ = (cid:20)
12 ( e − ˆ q∆z c + e ˆ q∆z c ) cos( s∆z c ) + 14 (cid:18) ˆ qs − s ˆ q (cid:19) ( e − ˆ q∆z c − e ˆ q∆z c ) sin( s∆z c ) (cid:21) − , (2.68)and ∆ λ (cid:54)
0. It should be noted that ∆ λ (cid:54) | ˆ q | ∆z c (cid:29)
0, then ∆ λ is likely to be greaterthan zero. Therefore the probability is higher at small | ˆ q | ∆z c for the efficient transmissionto take place.By using similar technique as mentioned in section 2.1.2, we can find analytical solutionof transmission ratio for the special case | s | ∆z s = ( (cid:96) (cid:48) + 1 / π (so that | s | ∆z s − π/ (cid:96) (cid:48) π and ˆ T / exp( π/
2) is pure imaginary number, and the problem is analogous to that8 cos / ( ) (a) cos / ( ) (b) cos / ( ) (c) cos / ( ) (d) cos / ( ) (e) cos / ( ) (f) cos / ( ) (g) cos / ( ) (h) cos / ( ) (i)Figure 7: Contour plots of transmission ratios for the tunneling of gravity wavein a three-layer structure. (a-c)Transmission ratios when the buoyancyfrequency N min = 10, and ∆z (cid:48) = (0 . , , / | ˆ q | . (d-f)Transmission ratios whenthe buoyancy frequency N min = 1, and ∆z (cid:48) = (0 . , , / | ˆ q | . (g-i)Transmissionratios when the buoyancy frequency N min = 0 .
1, and ∆z (cid:48) = (0 . , , / | ˆ q | .Tunneling of gravity waves can only occur in colored regions. Regions are leftwhite if tunneling of gravity waves is prohibited. in section 2.1.2). Here we only give the result without showing the details. Under thecondition | s | ∆z s = ( (cid:96) (cid:48) + 1 / π , the transmission ratio in the multi-layer structure fortunneling of gravity wave is η = cos M β + 14 (cid:20) ( ˆ qs − s ˆ q )( e − ˆ q∆z c + e ˆ q∆z c ) (cid:21) (cid:34)(cid:18) ˆ qs + s ˆ q (cid:19) − ( e − ˆ q∆z c + e ˆ q∆z c ) (cid:35) − sin M β − , (2.69)where β = arg ( e − ˆ q∆z c + e ˆ q∆z c ) + i (cid:34)(cid:18) ˆ qs + s ˆ q (cid:19) − ( e − ˆ q∆z c + e ˆ q∆z c ) (cid:35) / , (2.70)and arg is the argument function operating on complex numbers. Note that sin M β ∝ (ˆ q/s + s/ ˆ q ) − ( e − ˆ q∆z c + e ˆ q∆z c ) , and there is no singularity problem in (2.69). From(2.69), we see that efficient tunneling of gravity waves generally requires | ˆ q | ∆z c to be9 cos / ( ) (a) cos / ( ) (b) cos / ( ) (c) cos / ( ) (d) cos / ( ) (e) cos / ( ) (f)Figure 8: Contour plots of transmission ratios for the tunneling of gravity wavein a multi-layer structure. The buoyancy frequency and thicknesses of theconvective layers are all the same with the values of N min and ∆z c , and thethicknesses of the stable layers are all the same with the value of ∆z s .(a-c)Transmission ratios when the buoyancy frequency N min = 10 , , . ∆z c = 0 . / | ˆ q | , and ∆z s = 0 . π/ | ˆ s | . (d-f)Transmission ratios when the buoyancyfrequency N min = 10 , , . ∆z c = 0 . / | ˆ q | , and ∆z s = π/ | ˆ s | . Tunneling ofgravity waves can only occur in colored regions. Regions are left white iftunneling of gravity waves is prohibited.. small. In the limit | ˆ q | ∆z c →
0, we find η →
1. Thus enhanced transmission of wavetunneling can occur at | ˆ q | ∆z c → | s | ∆z s = ( (cid:96) (cid:48) + 1 / π . We call this phenomenon‘resonant tunneling’.Fig. 8 shows the contour plots of the transmission ratios for different N min in a 101-layer structure (M=50). In this calculation, all of the stable layers are assumed to have thesame degree of stratification ( N min ) and thickness ( ∆z s ), and the convective layers areassumed to have the same thickness ( ∆z c ). The analysis of the three-layer structure showsthat efficient transmission only occurs when ∆z c is small. The mathematical analysis onthe eigenvalues of transfer matrices also indicates that efficient transmission is more likelyto take place when ∆z c is small. For this reason, we set ∆z c = 0 . / | ˆ q | for all the computedcases. From the figure, we see that the transmission ratio is insensitive to the degree ofstratification. Instead, the thickness of ∆z s is more important. In our calculations, wefind that the transmission is efficient when ∆z s = ( (cid:96) + 0 . π/ | s | , and inefficient when z s = (cid:96)π/ | s | , where (cid:96) is an integer. Therefore, the tunneling of gravity wave is efficientwhen each convective layer ∆z c is much shallower than the e-folding decay distance1 / | ˆ q | and the thickness of each stable layer is close to a multiple-and-a-half of the halfwavelength.2.2.2. Tunneling of inertial waves
In this section, we discuss the tunneling of inertial waves. Similarly, we consider a(2 M + 1)-layer structure with alternating M + 1 convective layers and M stable layers.The sketch plot is shown as configuration 4 in fig. 6(b). For this configuration, inertialwave can propagate in convective layers but no wave could propagate in stable layers,0and thus the frequency range is σ < σ < σ . The width of the frequency domain is σ − σ = 12 [( f + ˜ f s + N min ) − (cid:113) ( f + ˜ f s + N min ) − N min f ] . (2.71)Analysis shows that σ − σ decreases with θ and sin α , and increases with N min / (2 Ω ) (see Appendix C). Therefore the frequency domain is wider at polar regions thanequatorial regions. Also it is wider when the zonal wavenumber dominates the meridionalwavenumber, and it is wider when the degree of stratification is stronger.Now we discuss the wave transmission of tunneling of inertial wave. Again, we firstconsider a three-layer structure. It is not difficult to obtain the transmission ratio η = 16 e − ˆ s ∆z (cid:48) + e ˆ s ∆z (cid:48) ) + (cid:32)(cid:115) ˆ s q + (cid:115) q ˆ s (cid:33) − (cid:104) ( e − ˆ s ∆z (cid:48) + e ˆ s ∆z (cid:48) ) − (cid:105) − . (2.72)Again, we can show that efficient transmission requires that the thickness of the stablelayer is much smaller than the e-folding decay distance 1 / | ˆ s | . Figs. 9(a-c) show thetransmission ratios in the three-layer structure when the stable layer is strongly stratified.It clearly shows that the transmission is only efficient when ˆ s ∆z (cid:48) is small.Now we discuss the tunneling of inertial wave in structure with more layers. Similarly,we can use the recursive relations to calculate the transmission ratio. Here we only givethe result without showing details. Under the condition | q | ∆z c = ( (cid:96) (cid:48) + 1 / π , we find thetransmission ratio of tunneling of inertial waves in a multi-layer structure is η = cos M β + 14 (cid:20) ( ˆ sq − q ˆ s )( e − ˆ s∆z s + e ˆ s∆z s ) (cid:21) (cid:34)(cid:18) ˆ sq + q ˆ s (cid:19) − ( e − ˆ s∆z s + e ˆ s∆z s ) (cid:35) − sin M β − , (2.73)where β = arg ( e − ˆ s∆z s + e ˆ s∆z s ) + i (cid:34)(cid:18) ˆ sq + q ˆ s (cid:19) − ( e − ˆ s∆z s + e ˆ s∆z s ) (cid:35) / , (2.74)The middle and lower panels of fig. 9 show the transmission ratios in a 101-layerstructure. Again, we see that the efficiency of transmission mainly depends on thethicknesses of the convective and stable layers. For the transmission to be efficient, itrequires that the thickness of stable layer ( ∆z s ) is much smaller than the e-folding decaydistance 1 / | ˆ s | , and the thickness of each convective layer is close to a multiple-and-a-halfof the half wavelength. This result is similar to that obtained in the tunneling of gravitywaves.
3. Non-traditional effects
Sutherland (2016) has discussed wave transmission in a multi-layer structure in tra-ditional approximation. It is necessary to investigate the non-traditional effects on wavetransmission in the multi-layer structure. Under the traditional approximation ( f s = 0),the critical frequencies can be written as σ = 0 , σ = f , (3.1) σ = min( f , N min ) , σ = max( f , N min ) , (3.2) σ = min( f , N max ) , σ = max( f , N max ) . (3.3)1 cos / ( ) (a) cos / ( ) (b) cos / ( ) (c) cos / ( ) (d) cos / ( ) (e) cos / ( ) (f) cos / ( ) (g) cos / ( ) (h) cos / ( ) (i)Figure 9: Contour plots of transmission ratios for the tunneling of inertial wave.The upper panel is for the three-layer structure; and the middle and the lowerpanels are for the multi-layer structures. (a-c)Transmission ratios for N min = 10and ∆z (cid:48) = (0 . , , / | ˆ s | in a three-layer structure. (d-f)Transmission ratios for N min = (10 , , . ∆z c = 0 . π/ | ˆ q | , and ∆z s = 0 . / | s | in a 101-layer structure.(g-i)Transmission ratios for N min = (10 , , . ∆z c = π/ | ˆ q | , and ∆z s = 0 . / | s | in a 101-layer structure. Tunneling of inertial waves can only occur in coloredregions. Regions are left white if tunneling of gravity waves is prohibited. If stable layers are strongly stratified with f < N min , then σ = σ = σ = f , σ = N min , and σ = N max . In such case, wave cannot propagate in both convective andstable layers, while tunneling of gravity waves occurs at f < σ < N min and tunnelingof inertial waves occurs at 0 < σ < f (see the upper panel of fig. 10).If stable layers are weakly stratified with f > N max , then σ = σ = σ = f , σ = N min , and σ = N max . In such case, tunneling of gravity waves cannot occur,while wave can propagate in both convective and stable layers at N max < σ < f , andtunneling of inertial waves occurs at 0 < σ < N min (see the lower panel of fig. 10).If traditional approximation is made, wave propagation only occurs in weakly stratifiedflow, tunneling of gravity waves only occurs in strongly stratified flow, and tunnelingof inertial waves can occur in both strongly and weakly stratified flows. When non-traditional effects are included, however, no similar restriction is obtained in wavepropagation or tunneling. The non-traditional effects on wave propagation can already beseen by comparing the left panel of fig. 3 with other panels. In traditional approximation,2 Figure 10: Sketch plots of wave propagations in multiple convectively stable andunstable layers. Wave frequency ranges are shown above and below the middlearrow for convective and stable layers, respectively. Wave propagation occurs inthe green region, tunneling of inertial wave occurs in the blue region, andtunneling of gravity wave occurs in the purple region. ˜ f s is set to be zero and this can be achieved by setting sin α = 0. The left panel of fig. 3shows transmission ratios with small ˜ f s , which are similar to situations with traditionalapproximation. If ˜ f s = 0, the colored regions in figs. 3(a) and (d) vanish because wavepropagation is prohibited when N max (cid:62) f ; and the colored region in fig. 3(c) will shrinkinto a triangle region below the diagonal line σ / (2 Ω ) = cos θ (see fig. 11(a)). Forweakly stratified flow ( N max < f ), only sub-inertial waves ( σ < f ) can propagatewith traditional approximation. However, super-inertial waves ( σ > f ) can propagateif non-traditional effects are taken into account.With traditional approximation, tunneling of gravity waves can only occur whenstable layers are strongly stratified( N min > f ), and the wave frequency is smaller thanbuoyancy frequency ( σ < N min ). When non-traditional effects present, we see from fig. 8that tunneling of gravity waves can occur when stable layers are weakly stratified. Also,tunneling of gravity waves is possible for super-buoyancy-frequency waves.For tunneling of inertial waves, comparing figs. 11(c-e) with figs. 9(d-f), we see thatfrequency ranges are overestimated in traditional approximation. It is especially truewhen stable layers are weakly stratified. From fig. 11(c) and fig. 9(d), we also seethat traditional approximation has moderate effect on transmission ratio in the smallfrequency range.
4. Summary
In this paper, we have investigated wave transmissions in rotating stars or planetswith multiple radiative and convective zones. Two situations have been considered:3 cos / ( ) (a) cos / ( ) (b) cos / ( ) (c) cos / ( ) (d) cos / ( ) (e)Figure 11: Wave transmissions with traditional approximation. (a)Transmissionfor wave propagation. The setting is identical to that of fig. 3(g) except that˜ f s = 0 in this figure. (b)Tunneling of gravity waves. The setting is the same asthat in fig. 8(a) except that ˜ f s = 0 in this figure. (c-e) Tunneling of inertialwaves. The settings are the same as those in fig. 9(d-f) except that ˜ f s = 0 inthese figures. Colored and white regions have similar meaning as thosementioned in previous contour plots. wave propagation and wave tunneling. For wave propagation, waves could propagatein both convective and stable layers. Previous studies on wave propagation in a two-layer structure with step function of stratification (Wei 2020a; Cai et al. f = 0 and ˜ f s = 0. In such case, we can easily obtain thatˆ q m = k and s m = ( N m /σ − k , where σ < N m . Then the conclusion (4) can bedirectly applied to this special case.Table 1 summarizes conditions on efficient transmissions of wave propagation andtunneling when all stable layers have similar buoyancy frequencies. For tunneling waves,clamping layers should have similar properties. Table 1 only lists the conditions in suchstructures. The first column of table 1 lists four types of wave transmissions: propagationwith convective layers embedded, propagation with stable layers embedded, tunneling ofgravity waves, and tunneling of inertial waves. The second column gives the frequencyranges. The third and fourth columns show the conditions for efficient transmissions.From the table, we see that the conditions on efficient wave transmissions are significantlydifferent among tunneling and propagative waves.Our findings have interesting implications in gaseous planets. In a multi-layer structure,Belyaev et al. (2015) found that the g-mode with vertical wavelengths smaller thanthe layer thickness are evanescent in gaseous planets. It is true for tunneling of g-mode waves. However, if considering wave propagation, g-mode waves can transmit5 σ ∆z c ∆z s Propagation with CLs embedded in SLs ( σ , σ ) M∆z c ∼ (cid:96)λ c / ∆z s ∼ (cid:96) (cid:48) λ s / σ , σ ) ∆z c ∼ (cid:96) (cid:48) λ c / M∆z s ∼ (cid:96)λ s / σ , σ ) ∆z c (cid:28) λ c ∆z s ∼ ( (cid:96) + 1 / λ s / σ , σ ) ∆z c ∼ ( (cid:96) + 1 / λ c / ∆z s (cid:28) λ s Table 1: Summary on efficient wave transmissions in a multi-layer structure.Note: λ c is the wavelength or decay distance in convective layer. λ s is the wavelength ordecay distance in the stable layer. ∆z c and ∆z s are the thicknesses of the convective andstable layers, respectively. M is the number of embedded layers. (cid:96) and (cid:96) (cid:48) is a non-negativeinteger. IW and GW denote inertial and gravity waves, respectively. CLs and SLs denoteconvective and stable layers, respectively. Here we only consider the situation that all stablelayers have similar buoyancy frequencies, and clamping layers have similar properties. efficiently even when the wavelength is smaller than the layer thickness. Andr´e et al. (2017) have made promising progress in the study of wave transmission in multi-layerstructures, which reveals that wave transmission can be enhanced when the incident waveis resonant with waves in adjacent layers with half wavelengths equal to the layer depth.Their result is consistent with our derivations. By deriving a group of exact solutionsof wave transmission coefficients in multi-layer structures, we provide a mathematicalexplanation for why the transmission can be enhanced in a multi-layer structure. Inaddition, our analysis shows that wave transmission can also be enhanced in tunnelingof gravity waves or inertial waves. Conditions on ‘resonant propagation’ and ‘resonanttunneling’ have been provided. Pontin et al. (2020) have conducted interesting researchon wave propagation in a multi-layer structure in a non-rotating sphere, and found thatwave transmissions are efficient for very large wavelength waves. It already shares somesimilarities with our analysis on tunneling of gravity waves in the f -plane. It has to beemphasized that our model is derived under the assumptions in local f -plane. Applicationof the results to global sphere should be very careful because of the following reasons.First, only short wavelengths are considered in the local model. Wave transmissionsof global scale waves have not been discussed. Second, the geometrical effect has notbeen taken into account. This is important for waves propagating in a global sphere.For example, super-inertial waves propagating from equator to poles may change to sub-inertial waves across critical colatitudes, where waves can also be transmitted or reflectedin the meridional direction (Gerkema & Shrira 2005b; Shrira & Townsend 2010; Rieutord& Valdettaro 1997; Rieutord et al. et al. (2016)revealed that the Arctic Ocean has a rich transmission behavior. With their data, wecan have a simple estimation on wave transmission in Arctic Ocean by using our model.For the structure of the Arctic Ocean presented in Ghaemsaidi et al. (2016), N instratified layers are about 1 . × − − . × − s − ; the inertial frequency f is about1 . × − s − ; the thicknesses of embedded convective layers are of the order O (1); Thetotal depth of the multi-layer structure is about 30 m , which are approximately separatedinto 14 stable layers. Since N (cid:29) f , gravity waves are expected to transport across themulti-layer structure by tunneling. From table 1, we see that waves with wavelength6 λ s ∼ . m are possible to transmit efficiently by resonant tunneling. Near-inertial waveswith wavelengths 10 m − m are also possible to transmit efficiently. This has beenverified in Ghaemsaidi et al. (2016). Double diffusion also occurs in stars. Our modelmay also provide some insights for wave transmission in stars. The interior structures aredifferent for different types of stars. For example, as studied in Cai (2014), late-type starshave a convectively stable-unstable-stable three-layer structure; A-F type stars generallyhave complicated internal structures with two separated convectively unstable layers (forexample, some of them have a unstable-stable-unstable three-layer structure, and someof them have a stable-unstable-stable-unstable-stable five-layer structure), and massivestars have a unstable-stable two-layer structure. For waves excited at the innermostlayer, resonant wave propagation from the innermost to the outermost layers probablycan be taken place in late-type and A-F stars since the top and bottom layers are bothconvectively stable or unstable. However, if waves are excited in the second innermostlayer, enhanced wave transmission is unlikely to occur from this layer to the outermostlayer, because the bottom and top layers of the interested region are different. For massivestars, enhanced wave transmission are unlikely to take place because the properties ofthe top and bottom layers are different. We have to mention that the stratified structurespecified in our model is ideal. In our Boussinesq model, density variation and viscouseffect has been ignored. In real stars, however, density variation and viscous effect maybe important. In addition, our model assumes that buoyancy frequency changes abruptlyacross interfaces between convective and stable layers. For real stars, the change is likelyto be smoother. Previous investigations of wave transmission in two-layer structures withsmoothly varying buoyancy frequencies have shown significant differences. Models withmore realistic settings are desirable in the future. Acknowledgements
T.C. has been supported by NSFC (No.11503097), the Guangdong Basic and AppliedBasic Research Foundation (No.2019A1515011625), the Science and Technology Programof Guangzhou (No.201707010006), the Science and Technology Development Fund,Macau SAR (Nos.0045/2018/AFJ, 0156/2019/A3), and the China Space Agency Project(No.D020303). C.Y. has been supported by the National Natural Science Foundationof China (grants 11373064, 11521303, 11733010, 11873103), Yun-nan National ScienceFoundation (grant 2014HB048), and Yunnan Province (2017HC018). X.W. has beensupported by National Natural Science Foundation of China (grant no.11872246) andBeijing Natural Science Foundation (grant no. 1202015). This work is partially supportedby Open Projects Funding of the State Key Laboratory of Lunar and Planetary Sciences.
Declaration of Interests.
The authors report no conflict of interest.
Appendix A.
Equations (2.11-2.14) can be written in a matrix form S m (cid:20) a m b m (cid:21) = Λ m Q m (cid:20) c m d m (cid:21) , (A 1) (cid:101) Λ m (cid:101) Q m (cid:20) c m d m (cid:21) = (cid:101) S m (cid:20) a m +1 b m +1 (cid:21) . (A 2)7where S m = e − is m z m e is m z m − e − is m z m e is m z m , Q m = e − iq m z m e iq m z m − e − iq m z m e iq m z m , Λ m = q m /s m , (A 3) (cid:101) S m = (cid:20) e − is m +1 z (cid:48) m e is m +1 z (cid:48) m − e − is m +1 z (cid:48) m e is m +1 z (cid:48) m (cid:21) , (cid:101) Q m = (cid:20) e − iq m z (cid:48) m e iq m z (cid:48) m − e − iq m z (cid:48) m e iq m z (cid:48) m (cid:21) , (cid:101) Λ m = (cid:20) q m /s m +1 (cid:21) . (A 4)Synthesizing these equations, we obtain the recursive relation (cid:20) a m b m (cid:21) = T m,m +1 (cid:20) a m +1 b m +1 (cid:21) , (A 5)where the transfer matrix T m,m +1 = S − m Λ m Q m (cid:101) Q − m (cid:101) Λ − m (cid:101) S m = (cid:34) (cid:98) T (cid:98) T (cid:98) T (cid:98) T (cid:35) , (A 6)and (cid:98) T = 14 e i ( s m z m − s m +1 z (cid:48) m ) [(1 + q m s m + s m +1 s m + s m +1 q m ) e iq m ∆z (cid:48) m (A 7)+(1 − q m s m + s m +1 s m − s m +1 q m ) e − iq m ∆z (cid:48) m ] , (cid:98) T = 14 e i ( s m z m + s m +1 z (cid:48) m ) [(1 + q m s m − s m +1 s m − s m +1 q m ) e iq m ∆z (cid:48) m (A 8)+(1 − q m s m − s m +1 s m + s m +1 q m ) e − iq m ∆z (cid:48) m ] , with (cid:98) T = (cid:98) T ∗ and (cid:98) T = (cid:98) T ∗ . Appendix B.
For the tunneling of gravity wave, the width of the frequency domain is σ − σ = 12 [( N min − f − ˜ f s ) + (cid:113) ( N min − f − ˜ f s ) + 4 N min ˜ f s ] . (B 1)The monotonicity of the frequency width is equivalent to that of the function G ( θ, µ , µ ) = G ( θ, µ , µ ) + (cid:113) G ( θ, µ , µ ) + 4 G ( θ, µ , µ ) , (B 2)where G ( θ, µ , µ ) = µ − cos θ − µ sin θ , G ( θ, µ , µ ) = µ µ sin θ , µ = sin α , and µ = N min / (2 Ω ) . To analyze the monotonicity of G ( θ, µ , µ ) on θ , we compute G θ = G θ + G G θ + 2 G θ (cid:112) G + 4 G . (B 3)Because G θ = sin 2 θ (1 − µ ) > G θ = µ µ sin 2 θ >
0, we have G θ = ( (cid:112) G + 4 G + G ) G θ + 2 G θ (cid:112) G + 4 G (cid:62) . (B 4)Therefore, the frequency width always increases with θ .8To analyze the monotonicity of G on µ , we compute G µ = G µ + G G µ + 2 G µ (cid:112) G + 4 G (B 5)= − sin θ + (cid:112) G + 4 G + 4 µ cos θ (cid:112) G + 4 G sin θ (cid:62) . (B 6)Therefore, the frequency width always increases with µ .To analyze the monotonicity of G on µ , we compute G µ = G µ + G G µ + 2 G µ (cid:112) G + 4 G (B 7)= (cid:112) G + 4 G + G + 2 µ sin θ (cid:112) G + 4 G > . (B 8)Therefore, the frequency width always increases with µ . Appendix C.
For the tunneling of inertial wave, the width of the frequency domain is σ − σ = 12 [( f + ˜ f s + N min ) − (cid:113) ( f + ˜ f s + N min ) − N min f ] . (C 1)The monotonicity of the frequency width is equivalent to that of the function H ( θ, µ , µ ) = H ( θ, µ , µ ) − (cid:113) H ( θ, µ , µ ) − H ( θ, µ , µ ) . (C 2)where H ( θ, µ , µ ) = µ + cos θ + µ sin θ , H ( θ, µ , µ ) = µ cos θ , µ = sin α , and µ = N min / (2 Ω ) . The derivative of H ( θ, µ , µ ) to θ is H θ = H θ − H H θ − H θ (cid:112) H − H . (C 3)Since H θ = ( µ −
1) sin 2 θ and H θ = − µ sin 2 θ , we have H θ = ( (cid:112) H − H − H ) H θ + 2 H θ (cid:112) H − H (cid:54) . (C 4)Therefore the width of the frequency domain decreases with θ . The derivative of H ( θ, µ , µ ) to µ is H µ = H µ − H H µ − H µ (cid:112) H − H . (C 5)Since H µ = sin θ and H µ = 0, we have H µ = ( (cid:112) H − H − H ) sin θ (cid:112) H − H (cid:54) . (C 6)Therefore the width of the frequency domain decreases with µ . The derivative of H ( θ, µ , µ ) to µ is H µ = H µ − H H µ − H µ (cid:112) H − H . (C 7)9Since H µ = 1 and H µ = cos θ , we have H µ = (cid:112) H − H − H + 2 cos θ (cid:112) H − H . (C 8)It is found that H − H − ( H − θ ) = µ sin θ (cid:62) , (C 9)then we have H µ (cid:62) . (C 10)Thus the width of the frequency domain increases with µ . REFERENCESAerts, Conny, Mathis, St´ephane & Rogers, Tamara M
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