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Modulational Stability of Nonlinear Saturated Gravity Waves

MARK SCHLUTOW Institut fur€ Mathematik, Freie Universitat€ Berlin, Berlin, Germany

(Manuscript received 15 March 2019, in final form 13 July 2019)

ABSTRACT

Stationary gravity waves, such as mountain lee waves, are effectively described by Grimshaw’s dissipative modulation equations even in high altitudes where they become nonlinear due to their large amplitudes. In this theoretical study, a wave-Reynolds number is introduced to characterize general solutions to these modulation equations. This nondimensional number relates the vertical linear with wave- number, pressure scale height, and kinematic molecular/ viscosity. It is demonstrated by analytic and numerical methods that Lindzen-type waves in the saturation region, that is, where the wave-Reynolds number is of order unity, destabilize by transient perturbations. It is proposed that this mechanism may be a generator for secondary waves due to direct wave–mean-flow interaction. By assumption, the primary waves are exactly such that altitudinal amplitude growth and viscous damping are balanced and by that the am- plitude is maximized. Implications of these results on the relation between mean-flow acceleration and wave breaking heights are discussed.

1. Introduction dynamic instabilities that act on the small scale comparable to the . For instance, Klostermeyer (1991) Atmospheric gravity waves generated in the lee of showed that all inviscid nonlinear Boussinesq waves are mountains extend over scales across which the back- prone to parametric instabilities. The waves do not im- ground may change significantly. The wave field can mediately disappear by the small-scale instabilities, rather persist throughout the layers from the to the perturbations grow comparably slowly such that the the deep atmosphere, the and beyond waves persist in their overall structure over several more (Fritts et al. 2016, 2018). On this range background . However, turbulence is produced. Lindzen temperature and therefore stratification and back- (1970) modeled the effect of turbulence on the wave by ground density may undergo several orders of magni- harmonic damping with a constant kinematic eddy vis- tude in variation. Also, dynamic viscosity and thermal cosity. The eddy viscosity is exactly such that it saturates conductivity cannot be considered constant on such a the wave, meaning that viscous damping and anelastic domain (Pitteway and Hines 1963; Zhou 1995). amplification are balanced (Lindzen 1981; Fritts 1984; Two predominant regimes for the waves can be identi- Dunkerton 1989; Becker 2012). Pitteway and Hines (1963) fied: the and the heterosphere (Nicolet 1954). referred to this instance as amplitude-balanced wave. These two are separated by the turbopause, usually Above the turbopause molecular viscosity takes over. somewhere in the region. Below the turbo- It is usually modeled by a constant dynamic viscosity. In pause molecular viscosity is negligible. Hence, if diffusion combination with the thinning background density, the occurs, it is caused by turbulence. Due to the missing kinematic viscosity increases exponentially with height damping and the thinning background air, mountain waves resulting in a rapid decrease in amplitude. amplify exponentially when extending upward. This phe- In the process of becoming saturated the amplitude nomenon is also called anelastic amplification. becomes considerably large such that the waves cannot At certain heights the amplitudes cannot grow any be considered linear. Pioneering work on the mathe- further due to limiting processes. Those may be static or matical description of nonlinear gravity waves was ac- complished by Grimshaw (1972, 1974). This paper aims Corresponding author: Mark Schlutow, mark.schlutow@fu- to extend Lindzen’s linear wave saturation theory with berlin.de the aid of Grimshaw’s nonlinear wave description.

DOI: 10.1175/JAS-D-19-0065.1 Ó 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 09/27/21 06:58 AM UTC 3328 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 76 y 2. How the modulation equations solve the pffiffiffiffiffiffiffiffir [ Fr 5 O(«1/2), (3b) compressible Navier–Stokes equations—A brief gLr review r y L r r r [ 5 «21 The nonlinear governing equations for our in- m Re O( ), (3c) r vestigations are the two-dimensional dissipative Grimshaw’s c m modulation equations being first introduced by Grimshaw p r [ 5 k Pr O(1), (3d) (1974). Before presenting them, we want to give a brief re- r view in this section how they solve the dimensionless r k m compressible, Reynolds-averaged Navier–Stokes equa- where r, pr, r, and r represent the reference density, tions (NSE) asymptotically. Detailed derivations can be pressure, thermal conductivity, and dynamic viscosity, also found in Achatz et al. (2010) and Schlutow et al. respectively. The constant k 5 2/7 is the ratio of the ideal (2017). Length and time are nondimensionalized via gas constant R to the specific heat capacity at constant pressure cp for diatomic gases. Equations (3) provide a 1 1 y distinguished limit for multiple-scale asymptotic analy- (x^, z^, ^t) 5 x*, z*, r t* , (1) Lr Lr Lr sis. Introducing compressed coordinates for the hori- zontal, vertical, and time axis, where Lr ’ 1...10 km denotes the reference wavelength ^ ^ ^ (hence the subscript r) and yr is the reference velocity. (x, z, t) 5 («x, «z, «t), (4) Note that variables labeled with an asterisk denote di- mensional quantities throughout the paper. To separate separates the slowly varying background from the fast the hydrostatic background from the flow field associ- oscillating wave fields. Under the assumption of stable ated with the wave the first ingredient necessary is a stratification to the leading order, the hydrostatic small scale separation parameter background is determined by the vertical temperature profile T(z). Two dimensionless background variables, « 5 Lr/Hu 1, (2) which vary on the large scale, will appear in the final modulation equations, the Brunt–Väisälä frequency or where Hu ’ 10...100 km is the potential temperature stratification measure N and the background density r. scale height. This choice for the scale separation pa- They have to be calculated from the temperature by rameter was introduced by Achatz et al. (2010). solving The authors considered inviscid flows. To take viscous damping into account, we need to compare inviscid and 1 dT N2 5 1 1 , (5) viscous terms. The (eddy) viscosity is not a constant T dz throughout the atmosphere and among others depends on temperature. Midgley and Liemohn (1966, their 1 dr 1 dT 1 52 1 , (6) Fig. 9) gave a realistic vertical profile of the effective r dz T dz k kinematic viscosity combining eddy and molecular ef- fects. Hodges (1969) computed the eddy diffusion by which originate from the hydrostatic assumption in combination with the ideal gas equation of state. gravity waves near the mesopause to be Kr(eddy) ’ 2 106...7 cm2 s 1, which compares well with Midgley and With the given scale separation of background and Liemohn (1966) and Lindzen (1981). In the scaling re- wave field, we can formulate the scaled two-dimensional gime of Achatz et al. (2010) as well as Schlutow et al. compressible NSE, (2017) these values correspond to Reynolds num- bers of Re ’ 10...100 when the Mach number is Ma ’ Dv 1 =^ 2 52« =^ 2 2 1 «L=^2 1 ... ^ p Nbez (Nb p N pez) v , 0.1...0.01. These numbers are also supported by a review Dt of Fritts (1984). (7a) Taking these arguments into account, a realistic flow Db 1 dN ^ regime in terms of Mach, Froude, Reynolds, and Prandtl 1 Nw 52« N2 1 wb 1 «L=2b 1 ... , (7b) D^t N dz number most suitable for internal gravity waves in the middle/upper atmosphere region is then found by ^ 1 dr = v 52« N2 1 w 1 ... , (7c) assuming r dz y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir [ Ma 5 O(«), (3a) in terms of the wave-field variables, velocity v, buoyancy 2 k r ^ T (1 )pr/ r Nb, and kinematic pressure p, where = 5 (›/›x^, ›/›z^)

Unauthenticated | Downloaded 09/27/21 06:58 AM UTC NOVEMBER 2019 S C H L U T O W 3329 and ez the unit vector pointing in the vertical direction; The modulation equations govern the evolution of ^ D/Dt denotes the material derivative, and L is the total vertical kz, wave action density ra,and kinematic viscosity. The dots represent higher-order mean-flow horizontal wind u.Equations(9a)–(9c) are terms that we do not give explicitly but must not be closed by neglected. The interested reader finds these terms in j j2 5 2 1 2 Schlutow et al. [2017, their (2.19)]. k kx kz , (9d) Wentzel–Kramers–Brillouin (WKB) theory provides Nk us with a multiple-scale method to solve the scaled NSE v 5 x 1 k u, (9e) asymptotically by the spectral ansatz jkj x

f « ^ ^ ^ « 5 1 i (x,z,t)/ 1 0 Nkxkz U(x, z, t; ) U0,0(x, z, t) [U0,1(x, z, t)e c.c. ] v 52 , (9f) jkj3 1 h.h. 1 O(«), (8) where k, v, and v0 represent the wavenumber vector, where c.c. stands for the complex conjugate and h.h. for extrinsic frequency, and vertical linear group velocity, higher harmonics. The vector U 5 (v, b, p)T contains the respectively. Note that primes denote derivative with prognostic variables and f is the wave’s phase. respect to the vertical wavenumber throughout this pa- We want to give a remark at this point. By construc- per. Extrinsic frequency is defined by the sum of intrinsic tion, the WKB ansatz is a nonlinear approach. In the frequency and Doppler shift. It is linked to the wave- limit « / 0 the amplitudes are finite. In fact, the ansatz number vector by the relation for non- converges to the nonlinear plane wave of Boussinesq hydrostatic gravity waves. It was shown in Schlutow theory, which is known to be an analytical solution. et al. (2017) that the modulation equations equally hold Mean-flow interaction and Doppler shift are leading- for hydrostatic waves, where the horizontal wavelength order effects, which is in contrast to weakly nonlinear is much larger than the vertical, if the theory where they appear as higher-order corrections. In is replaced by v 5 Nkx/jkzj 1 kxu. The prognostic var- the « limit, the weakly nonlinear approach converges to iables determine the asymptotic solution as described the linear plane wave. in the previous section and explained in Schlutow The nonlinear WKB ansatz is inserted into the com- et al. (2017). pressible NSE and terms are ordered with respect to The first equation, (9a), essentially describes the powers of « and harmonics. evolution of the phase in (8) as by definition =f [ k and 2›f/›t [ v. Cross differentiating these expressions, they add up to zero. The second equation, (9b), governs 3. Governing equations: Grimshaw’s dissipative the conservation of wave action density being the ratio modulation equations of wave energy density and the intrinsic frequency. In To leading order of the nonlinear WKB analysis of terms of the polarization relation the leading-order first the NSE with the turbulence model of Lindzen one harmonics are computable via obtains Grimshaw’s dissipative modulation equations ! T supported by a mean-flow horizontal kinematic pres- k v 2 k u v 2 k u k (v 2 k u)2 5 B 2 z x x 2 z x sure gradient [Grimshaw 1974, their (4.1)–(4.6)]. Note U0,1 i , i ,1, i 2 , k N N kx N that the leading-order WKB analysis of the dissipative x pseudoincompressible equations (Durran 1989) lead to the (10) same modulation equations. We present them in slightly pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with B 5 (v 2 k u)a/2 the wave’s buoyancy ampli- different notation as Grimshaw and with the prerequisite x tude. The third equation, (9c), accounts for the accel- that the wave field is horizontally periodic (0 , k 5 const) x eration of the mean flow. With a slight abuse of notation and only modulated in the z direction, (we drop the indices) we obtain the zero harmonics, ›k ›v z 1 5 0, (9a) U 5 (u,0,0,p)T. (11) ›t ›z 0,0 ›a ›v0ra The variable p corresponding to the mean-flow hori- r 1 52Ljkj2ra, (9b) ›t ›z zontal kinematic pressure is unknown and needs addi- tional investigation to close the system (9). 0 ›u ›v k ra ›p The governing equations, (9), can be reformulated in r 1 x 52 . (9c) ›t ›z ›x vector form:

Unauthenticated | Downloaded 09/27/21 06:58 AM UTC 3330 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 76 ð ! ›y ›F(y) F zLj j2 1 5 ( ), (12) 5 2 K ~ › › G y A 0 exp 0 dz . (19) t z rV 0 V with a flux F and an inhomogeneity G where y 5 (k , a, u)T z with F being the vertical wave action flux at z 5 0. is the prognostic vector.

5. Lindzen-type mountain lee wave and the wave- 4. General stationary solutions of the modulation Reynolds number equations So far, we considered all background variables of In this section we will explore general stationary our governing PDE as functions of z.Inthissection solutions before we focus on particular solutions for we will prescribe these functions in a piecewise fash- which we will present stability analysis in the up- ion in order to construct a typical mountain lee wave coming sections. that gets saturated by some small-scale instability In the inviscid limit (i.e., L / 0), the modulation process comparable to Lindzen (1981).Thecom- equations assume stationary solutions where ›p/›x 5 0, plete solution is divided into an unsaturated as well which can be computed analytically by a formula of as a saturated middle-atmospheric part and a deep- Schlutow et al. [2017, their (5.20)]. When we multiply atmosphere part. (9b) by kx and subtract (9c), we obtain a. The unsaturated middle-atmospheric solution › ›p r (k a 2 u) 5 2 k Ljkj2ra. (13) ›t x ›x x First, we assume that the background atmosphere is piecewise isothermal. From (5), this assumption implies Thus, to be consistent with the inviscid limit, that N 5 const, some typical value for the middle the dissipative modulation equations assume sta- atmosphere. Then (6) gives tionary solutions only if the right-hand side of (13) vanishes, which provides eventually a closure r 5 r 2z/H (z) 0e , (20) for the mean-flow horizontal kinematic pressure 2 gradient, where H 5 kN 2 5 const denotes the dimensionless (local) pressure scale height. Second, we assume that the › p 5 Lj j2r mean-flow horizontal wind is piecewise constant as well, kx k a. (14) ›x so U 5 const. These assumptions can be weakened, This result implicates that the mean-flow horizontal ki- which we will discuss in the concluding section 7.It nematic pressure gradient balances the viscous forces follows immediately by (18) and horizontal periodicity 5 acting on the horizontal mean-flow wind. Such a flow that Kz const making it a plane wave. configuration is referred to as antitriptic flow in the lit- In the middle atmosphere viscosity is negligible. erature (Jeffreys 1922). Therefore, below the breaking height zbreak the integral T in (19) vanishes and the amplitude of the wave grows Then a general stationary solution Y(z) 5 (Kz, A, U) depicting a mountain lee wave fulfills exponentially with the inverse density,

v 5 A0 (Kz, U) 0, (15) A(z) 5 in [0, z ). (21) r(z) break ›V0r A 2 52LjKj rA, (16) Because of the vanishing dissipation, the mean-flow ›z horizontal kinematic pressure gradient also vanishes U 5 U(z), (17) according to (14).

j j2 5 2 1 2 V0 5 v0 b. The saturated middle-atmospheric solution with K Kx Kz and (Kz). Note that we label the stationary solution by capital letters. Given any well- Above zbreak, the wave saturates by some small-scale behaved, slowly varying mean-flow horizontal wind instability process producing turbulence, which balances U(z), the remaining two variables of the general sta- the anelastic amplification. The exact kinematic eddy tionary solution are given explicitly by viscosity that keeps the amplitude leveled in (19), such sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that A 5 const, is then given by 2 N 2 K 52 2 K , (18) 0 2 z 2 x L5V 21j j 2 5 U H K const in [zbreak, zturbo). (22)

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In this region, the mean-flow horizontal kinematic amplitude is at its maximum and one can expect most likely pressure gradient depends on z only by the background nonlinear behavior. We assess stability by analyzing the density and can be computed inserting (22) into (14), evolution of small perturbations. Due to their smallness, we can linearize the governing equations in vector form (12) ›P 0 5 K H21V Ar in [z , z ). (23) around the stationary solution Y. Applying the ansatz ›x x break turbo y(z, t) 5 y^(z)elt (28) In particular, it is positive while the mean-flow horizontal wind is negative, which can be seen from (9e) and (15). for the perturbation, results in an eigenvalue problem c. The deep-atmosphere solution (EVP): › › › The saturated middle-atmospheric solution is valid l 1 F 5 G y › › y › y, (29) below the turbopause at zturbo as in the heterosphere the z y Y y Y molecular viscosity dominates, which is modeled by a where we dropped the hat over y. The Jacobian matrices constant dynamic viscosity mmol implying of the flux and inhomogeneity evaluated at Y are given by m L(z) 5 mol in [z , 1‘). (24) 0 1 r turbo V0 (z) 0 Kx ›F B 00 0 C 5 @ V A V 0 A, (30) In the deep-atmospheric region the integral in (19) has ›y 00 0 Y K V AKV 0 also an analytic solution for A, which decays quickly for x x / 1‘ 0 1 z . Here, typical values N and U for the deep at- 000 › B 2 00 C mosphere are used. G 5 1V 2 L @ H A 2 KzA 00A. (31) ›y d. The wave-Reynolds number Y 21V00 2 L KxH A 2Kx KzA 00 In Fig. 1, such a prototypical mountain wave, which Note that the second and third row of each Jacobian saturates, is illustrated. By inspection of (22), its be- are linearly dependent. Hence, we can solve the third havior in the different regions may be characterized by a equation of (29) for ‘‘wave-Reynolds’’ number. It can be written as u 5 K a, (32) 0 2 x V H21jKj 2 Y1. (25) L which reduces the dimension of the system, so with a T slight abuse of notation, y 5 (kz, a) and When reintroducing the dimensional variables by reversing 0 2 the nondimensionalization of Schlutow et al. (2017),so ›F V K 5 x , (33) › V00 V0 r y Y A « r Lr H 5 H*, L5 n*, N 5 N*, K 5 L K*, (26) L p m y r ›G 00 r r r 5 › 21V00 2 L . (34) y Y H A 2 KzA 0 and using the definitions (3), the wave-Reynolds number reads If one assumes that the perturbation decays sufficiently fast toward the edges of the region where Re 5 O(1) C D wave Re [ gz . (27) (in Fig. 1 at 70 and 110 km), in other words, that the wave n * edges have no influence on the perturbation, then one The dimensional linear vertical group velocity is deno- can extend the domain to the infinities. Consequently, we consider the differential operator associated with the ted by Cgz and represents the velocity scale for this type 2 5 *21j j*22 EVP as closed and densely defined on L , the space of of Reynolds number. The term D Hp K defines its length scale. Estimates of the wave-Reynolds number vector valued square integrable functions on the real in the different regions are depicted in Fig. 1. line equipped with the norm sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1‘ k k 5 2 1 2 1 2 6. Stability of the saturated wave y L2 kz a u dz. (35) 2‘ In this section we will investigate the saturated mountain wave with respect to stability. In particular, we are in- In conclusion, we translated the problem of stability to terested in the region where Rewave 5 O(1) as here the finding the spectrum of a linear operator of an EVP.

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which determine the spectrum of the linear operator of the EVP as curves in the complex plane parameterized by the spatial eigenvalue m, which can also be interpreted as the vertical wavenumber of the perturbation. The real part of l represents the instability growth rate, if it is positive, and the imaginary part is a frequency. Thus, (39) provides also a dispersion relation linking the perturba- tion’s wavenumber with its frequency.

One can readily show that either l1 or l2 of (39) has positive real part for reasonable wave solutions implying that they are unconditionally unstable. We want to point out that when H / ‘ and L / 0, (39) reduces to the spectrum of the inviscid nonlinear Boussinesq plane waves (Schlutow et al. 2019) having the classical mod- ulational stability criterion V00 . 0(Grimshaw 1977). a. Transient (in)stability FIG. 1. Schematic of a saturated plane mountain lee wave (thin gray line) with amplitude profile (thick black line) and effective In this subsection we want to investigate the charac- kinematic viscosity (thick blue line) being characterized by the teristics of the instability that is presented in the pre- wave-Reynolds number. vious section. We put the ‘‘in’’ of the section title into parentheses because there is the possibility that the Broadly speaking, the spectrum is the set of all l 2 C for primary wave ‘‘survives’’ the instability. One particular which the EVP has admissible solutions. type of those harmless instabilities is called transient The EVP is solved by the Fourier transform instability. Despite the fact that the instability’s norm ð grows exponentially in time, the instability vanishes at 1‘ each given point in space if one waits long enough. y 5 y~eimzdm, (36) 2‘ To show such characteristics a prerequisite for math- ematical rigor must be fulfilled: the linear differen- which yields an algebraic equation tial operator of the EVP must be well-posed, which means its spectrum has a maximum real part or, in other › › l 1 F m 2 G ~ 5 words, the instability growth rate is bounded. The reader › i › y 0. (37) y Y y Y finds this tedious verification in the appendix.Wewant to give an interesting remark at this point. The ‘‘well- It has nontrivial solutions only if the coefficient matrix is 00 posedness’’ depends on a criterion, V . 0, which turns singular [det(M) 5 0], which happens if out to be equivalent to the modulational stability 0 2 00 00 criterion of plane waves in Boussinesq theory. (l 1Vim)2 2 K2A[(2LK 2 H 1V )im 2V m2] 5 0. x z The key idea for transient instabilities (Kapitula and (38) Promislow 2013) is to ask for solutions of the EVP, (29), 2 in a weighted space La with a 2 R an exponential This characteristic polynomial has two zeros, weight, such that pffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k 5 k azk 0 00 00 y 2 ye 2 . (40) l (m) 52Vim 6 K A (2LK 2 H21V )im 2V m2 , La L 1,2 x z Doing so, we get the same curves as in (39) but with (39) im / im 2 a,so

pffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi la m 52V0 m 2 a 6 L 2 21V00 m 2 a 1V00 m 2 a 2 1,2( ) (i ) Kx A (2 Kz H )(i ) (i ) . (41)

When we choose provided V00 . 0 the spectrum is stabilized, so la # m 2 R 00 Re( ) 0 for all . The negative exponential 2LK 2 H21V 1,2 a , 1 pffiffiffiffiffiffiz , weight penalizes solutions at 2‘. To converge in the 00 0 00 0, (42) 2 V V 1V weighted norm, the perturbation in L2 must therefore

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2 FIG. 2. (a) Unstable spectrum (blue lines) in L and (b) stable spectrum in the exponentially 2 weighted La for the saturated wave of Fig. 3. The exponential weight is the largest a as defined by (42). decay exponentially at 2‘ for all times. But simulta- perturbation is found to be 1.6. The difference be- neously, its L2 norm grows exponentially in time. This tween theoretical and observed rate occurs because seeming paradox resolves when the perturbation the perturbation is not optimal. propagates sufficiently fast toward 1‘ (i.e., upward). Perturbations of this behavior are called transient 7. Summary and discussion instabilities (Sandstede and Scheel 2000). For illus- trative purpose, the unweighted and weighted spec- In this paper we investigated nonlinear mountain trum of an example wave, which we will discuss waves, which are governed by Grimshaw’s dissipative in more detail in the following section, are plotted modulation equations being asymptotically consis- in Fig. 2. tent with the compressible Navier–Stokes equations and the dissipative pseudoincompressible equations b. Numerical investigation of the transient instabilities likewise. We use the finite-volume numerical solver presented We introduced a wave-Reynolds number charac- in Schlutow et al. (2019) for the governing equations, terizing the stationary solution. When this dimen- (9), to compute the evolution of a tiny Gaussian initial sionless quantity is of order unity, the wave amplitude perturbation of the saturated wave in the region where saturates by small-scale instabilities and assumes a

Rewave 5 O(1). The results are shown in Fig. 3. The maximum as anelastic amplification by the thinning simulation is set up by N 5 1 and Kx 5 1. We discretize background air is exactly balanced by the turbulent the equations on 2000 grid points in z and integrate in t damping. We analyzed this regime with respect to over 6000 time steps. As can be seen in the figure, the modulational stability as nonlinearities dominate perturbation amplifies exponentially and propagates to for large-amplitude waves. It turned out that transient the right (i.e., upward), as theory suggests. We can also instabilities emerge that propagate upward. We tested observe that the perturbation’s wavelength compares this analysis solving the modulation equations nu- with the scale height. Or in other words the instability merically and found excellent agreement with the varies on the large scale. The corresponding spectra theory. to this case as computed by (39) and (41) are plotted In the framework of saturated nonlinear wave theory, in Fig. 2. The amplitude and the vertical wavenumber which we presented in this paper, the saturated moun- undergo strong modulations due to the exponentially tain wave does initially not accelerate the mean-flow growing perturbation. Furthermore, the initially con- horizontal wind. Instead, a mean-flow horizontal kine- stant mean-flow horizontal wind experiences acceler- matic pressure gradient emerges that keeps the hori- ation. The analytic maximum instability growth rate zontal wind constant by balancing the viscous forces. according to (A6) is 2.2 for this particular case. In The wave persists structurally and loses energy directly terms of an approximated L2 norm that we compute to turbulence, which in turn damps altitudinal amplifi- numerically, the actual growth rate of the Gaussian cation. Eventually, the mean flow is accelerated by a

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FIG. 3. Numerical corroboration of the governing equations, (9). Evolution of an initial Gaussian perturbation of the saturated wave centered at 78 km (blue lines). transient instability in the saturation zone propagating In our derivations we assumed piecewise analytic so- upward while growing and transferring kinetic energy to lutions being matched. This assumption can be weakened the mean flow. to almost arbitrary background, fulfilling hydrostatics and Our investigations have two implications being of in- the ideal gas law, as well as almost any mean-flow hori- terest for parameterizations in numerical zontal wind. However, these functions of height must be weather prediction and climate modeling. First, the restricted to converge to constant values at the infinities. induced mean flow behaves wavelike. Its evolution The resulting spectrum describing the temporal evolution is governed by a dispersion relation for the linear pertur- of the perturbation would be much more complicated but bation. In conclusion, it may be interpreted as an still analytically assessable by Fredholm operator theory upward-traveling secondary wave of larger scale than (Schlutow et al. 2019). In conclusion, our results are valid the primary wave. Secondary waves with wavelengths in a much more realistic atmosphere. comparable to the primary modulation scale were in- vestigated by Vadas and Fritts (2002), Vadas et al. Acknowledgments. The author thanks Prof. Ulrich é (2003),andBecker and Vadas (2018). The authors Achatz, Prof. Rupert Klein, and Prof. Erik Wahl n for propose a generating mechanism based on body forces many helpful discussions during the preparation of this produced by the dissipating primary wave. In contrast to article. The research was supported by the German this model, our secondary wave is generated by direct Research Foundation (DFG) through Grants KL 611/25-2 wave–mean-flow interaction that compares to Wilhelm of the Research Unit FOR1898 and Research Fellow- et al. (2018). ship SCHL 2195/1-1. Second, this novel picture of saturated mountain waves may explain the bias between the onset of small-scale APPENDIX instability and the actually observed mean-flow accel- eration (Achatz 2007). We give an extension to the es- Well-Posedness: Are the Instability Growth Rates tablished picture where waves become unstable at the Finite? breaking height and deposit their momentum and en- ergy onto the mean flow at this level. As it turns out, only In this appendix we demonstrate that the linear dif- in combination with modulational instability does an ferential operator of the EVP, (29), is well-posed. This initially saturated nonlinear wave induce a mean flow. property is essential for existence, uniqueness and This modulational instability has its own transient ve- continuous dependency of the solution for the per- locity and growth rate, which we can compute explicitly. turbation on the initial data. In section 6 we showed Then the mean-flow acceleration depends on these two that the spectrum of the operator extents into the quantities. right half of the complex plane rendering the

Unauthenticated | Downloaded 09/27/21 06:58 AM UTC NOVEMBER 2019 S C H L U T O W 3335 8 ! 21V00 2 L saturated wave unconditionally unstable. But is the > H 2 K 00 >arctan z , V , 0 spectrum bounded from above on the real axis? Let <> V00m us denote the discriminant of the square root ap- arg(D (m)) 5 ! Y > l m > 21V00 2 L pearing in 1,2 of (39) by DY( ). The real part being > H 2 K 00 :> z 6 p V . the instability growth rate may be expressed by arctan V00m , 0.   pffiffiffiffi 1/2 1 (A3) Re(l (m)) 56K AjD (m)j cos arg(D (m)) , 1,2 x Y 2 Y The function Re(l (m)) has no poles. Thus, we are only (A1) 1,2 concerned with its behavior at the infinities. We have where j m j1/2 5 jmj m / 6‘ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DY ( ) O( )as , (A4) j m j1/2 5 jmj 4 L 2 21V00 2m22 1V002 DY ( ) (2 Kz H ) , (A2)

8 00   <> O(1), V , 0 1 cos arg(D (m)) 5 21V00 1 L as m / 6‘. (A5) Y > 1 H 2 K 2 00 2 : 6 zm21 1 O(jmj 3), V . 0 2 V00

Therefore,

8 00 <> 6‘, V , 0 Re(l (m)) / L 2 21V00 as m / 6‘. (A6) 1,2 > 1 2 K H 00 : 6 zpffiffiffiffiffiffi , V . 0 2 V00

In conclusion the operator is well-posed (Kapitula and ——, and Coauthors, 2016: The Deep Propagating Gravity Wave Promislow 2013)ifV00 . 0 and ill-posed otherwise. Experiment (DEEPWAVE): An airborne and ground-based exploration of gravity wave propagation and effects from their sources throughout the lower and middle atmosphere. Bull. REFERENCES Amer. Meteor. Soc., 97, 425–453, https://doi.org/10.1175/ BAMS-D-14-00269.1. Achatz, U., 2007: Gravity-wave breaking: Linear and primary ——, and Coauthors, 2018: Large-amplitude mountain waves in the nonlinear dynamics. Adv. Space Res., 40, 719–733, https:// mesosphere accompanying weak cross-mountain flow during doi.org/10.1016/j.asr.2007.03.078. DEEPWAVE research flight RF22. J. Geophys. Res. Atmos., ——, R. Klein, and F. Senf, 2010: Gravity waves, scale asymptotics 123, 9992–10 022, https://doi.org/10.1029/2017JD028250. and the pseudo-incompressible equations. J. Fluid Mech., 663, Grimshaw, R., 1972: Nonlinear internal gravity waves in a slowly 120–147, https://doi.org/10.1017/S0022112010003411. varying medium. J. Fluid Mech., 54, 193–207, https://doi.org/ Becker, E., 2012: Dynamical control of the middle atmosphere. 10.1017/S0022112072000631. Space Sci. Rev., 168, 283–314, https://doi.org/10.1007/s11214- ——, 1974: Internal gravity waves in a slowly varying, dissipative 011-9841-5. medium. Geophys. Fluid Dyn., 6, 131–148, https://doi.org/ ——, and S. L. Vadas, 2018: Secondary gravity waves in the winter 10.1080/03091927409365792. mesosphere: Results from a high-resolution global circulation ——, 1977: The modulation of an internal gravity-, and model. J. Geophys. Res. Atmos., 123, 2605–2627, https://doi.org/ the resonance with the mean motion. Stud. Appl. Math., 56, 10.1002/2017JD027460. 241–266, https://doi.org/10.1002/sapm1977563241. Dunkerton, T. J., 1989: Theory of internal gravity wave saturation. Hodges, R. R., 1969: Eddy diffusion coefficients due to instabilities Pure Appl. Geophys., 130, 373–397, https://doi.org/10.1007/ in internal gravity waves. J. Geophys. Res., 74, 4087–4090, BF00874465. https://doi.org/10.1029/JA074i016p04087. Durran, D. R., 1989: Improving the anelastic approximation. Jeffreys, H., 1922: On the dynamics of wind. Quart. J. Roy. Meteor. J. Atmos. Sci., 46, 1453–1461, https://doi.org/10.1175/1520- Soc., 48, 29–48, https://doi.org/10.1002/qj.49704820105. 0469(1989)046,1453:ITAA.2.0.CO;2. Kapitula, T., and K. Promislow, 2013: Spectral and Dynamical Sta- Fritts, D. C., 1984: Gravity wave saturation in the middle bility of Nonlinear Waves, Applied Mathematical Sciences, Vol. atmosphere: A review of theory and observations. Rev. 185, Springer, 361 pp., https://doi.org/10.1007/978-1-4614-6995-7. Geophys. Space Phys., 22, 275–308, https://doi.org/10.1029/ Klostermeyer, J., 1991: Two- and three-dimensional paramet- RG022i003p00275. ric instabilities in finite-amplitude internal gravity waves.

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Geophys. Astrophys. Fluid Dyn., 61, 1–25, https://doi.org/ Schlutow, M., R. Klein, and U. Achatz, 2017: Finite-amplitude 10.1080/03091929108229035. gravity waves in the atmosphere: Travelling wave solutions. Lindzen, R. S., 1970: Internal gravity waves in atmospheres with J. Fluid Mech., 826, 1034–1065, https://doi.org/10.1017/ realistic dissipation and temperature part I. Mathematical jfm.2017.459. development and propagation of waves into the thermo- ——, E. Wahlén, and P. Birken, 2019: Spectral stability of non- sphere. Geophys. Fluid Dyn., 1, 303–355, https://doi.org/10.1080/ linear gravity waves in the atmosphere. Math. Climate Wea. 03091927009365777. Forecasting, 5, 12–33, https://doi.org/10.1515/mcwf-2019-0002. ——, 1981: Turbulence and stress owing to gravity wave and tidal Vadas, S. L., and D. C. Fritts, 2002: The importance of spatial breakdown. J. Geophys. Res., 86, 9707–9714, https://doi.org/ variability in the generation of secondary gravity waves from 10.1029/JC086iC10p09707. local body forces. Geophys. Res. Lett., 29, 1984, https://doi.org/ Midgley, J. E., and H. B. Liemohn, 1966: Gravity waves in a re- 10.1029/2002GL015574. alistic atmosphere. J. Geophys. Res., 71, 3729–3748, https:// ——, ——, and M. J. Alexander, 2003: Mechanism for the gener- doi.org/10.1029/JZ071i015p03729. ation of secondary waves in wave breaking regions. J. Atmos. Nicolet, M., 1954: Dynamic effects of the high atmosphere. The Sci., 60, 194–214, https://doi.org/10.1175/1520-0469(2003) Earth as a Planet, G. P. Kuiper, Ed., University of Chicago 060,0194:MFTGOS.2.0.CO;2. Press, 644–712. Wilhelm, J., T. R. Akylas, G. Bölöni, J. Wei, B. Ribstein, R. Klein, Pitteway, M. L. V., and C. O. Hines, 1963: The viscous damping of and U. Achatz, 2018: Interactions between mesoscale and atmospheric gravity waves. Can. J. Phys., 41, 1935–1948, submesoscale gravity waves and their efficient representation https://doi.org/10.1139/p63-194. in mesoscale-resolving models. J. Atmos. Sci., 75, 2257–2280, Sandstede, B., and A. Scheel, 2000: Absolute and convective in- https://doi.org/10.1175/JAS-D-17-0289.1. stabilities of waves on unbounded and large bounded domains. Zhou, Q., 1995: A simplified approach to internal gravity wave Physica D, 145, 233–277, https://doi.org/10.1016/S0167- damping by viscous dissipation. J. Atmos. Terr. Phys., 57, 2789(00)00114-7. 1009–1014, https://doi.org/10.1016/0021-9169(94)00107-Y.

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